Federal Reserve Bank of Minneapolis Research Department Staff Reports 223 and 277

February 1996 (Revised July 2001)

TECHNICAL APPENDIX: Can Sticky Price Models Generate Volatile and Persistent Real Exchange Rates?^{†}

V.V. Chari University of Minnesota and Federal Reserve Bank of Minneapolis

Patrick J. Kehoe Federal Reserve Bank of Minneapolis and National Bureau of Economic Research

Ellen R. McGrattan Federal Reserve Bank of Minneapolis and University of Minnesota

† The views expressed herein are those of the author and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.

Table of Contents

1. The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3. The Final Goods Producers . . . . . . . . . . . . . . . . . . 2 1.4. The Consumer’s Problem . . . . . . . . . . . . . . . . . . . 3 1.5. The Foreign Consumer’s Problem . . . . . . . . . . . . . . . . 4 1.6. The Intermediate Goods Producers . . . . . . . . . . . . . . . 5 1.7. The Government . . . . . . . . . . . . . . . . . . . . . . . 8 1.8. Additional Equilibrium Conditions . . . . . . . . . . . . . . . 9 1.9. Aggregates of Interest . . . . . . . . . . . . . . . . . . . . . 9 1.10. Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.10.1. Taylor Rule as Fed Policy . . . . . . . . . . . . . . . . 10 1.10.2. Incomplete Asset Markets . . . . . . . . . . . . . . . . 10 1.10.3. Additional Shocks . . . . . . . . . . . . . . . . . . . 12 1.10.4. Sticky Wages . . . . . . . . . . . . . . . . . . . . . 13

2. Computing an Equilibrium . . . . . . . . . . . . . . . . . . . . . 19

2.1. The Benchmark Economy . . . . . . . . . . . . . . . . . . . 19 2.1.1. Normalization in the Benchmark Economy . . . . . . . . . 19 2.1.2. Steady State in the Benchmark Economy . . . . . . . . . . 22 2.1.3. Linearized Equations in the Benchmark Economy . . . . . . 24 2.1.4. Solving the Linearized System in the Benchmark Economy . . 30

2.2. The Taylor-Rule Extension . . . . . . . . . . . . . . . . . . . 37 2.2.1. Normalization for the Taylor-Rule Extension . . . . . . . . 38 2.2.2. Steady State for the Taylor-Rule Extension . . . . . . . . . 38 2.2.3. Linearized Equations for the Taylor-Rule Extension . . . . . 38

2.2.4. Solving the Linearized System for the Taylor-Rule Extension . 39 2.3. The Incomplete-Markets Extension . . . . . . . . . . . . . . . 45

2.3.1. Normalization for the I-M Extension . . . . . . . . . . . . 46 2.3.2. Steady State for the I-M Extension . . . . . . . . . . . . 48 2.3.3. Linearized Equations for the I-M Extension . . . . . . . . . 48 2.3.4. Solving the Linearized System for the I-M Extension . . . . . 49

2.4. The Extension with Additional Shocks . . . . . . . . . . . . . . 52 2.4.1. New Code for the Benchmark Economy . . . . . . . . . . 53 2.4.2. New Code for the Taylor-Rule Extension . . . . . . . . . . 57 2.4.3. New Code for the I-M Extension . . . . . . . . . . . . . 60 2.4.4. Allowing for Accommodative Monetary Policy . . . . . . . 60

2.5. The Extension with Sticky Wages . . . . . . . . . . . . . . . . 65 2.5.1. Normalization in the Sticky-Wage Economy . . . . . . . . 66 2.5.2. Steady State in the Sticky-Wage Economy . . . . . . . . . 67 2.5.3. Linearized Equations in the Sticky-Wage Economy . . . . . 68 2.5.4. Solving the Linearized System in the Sticky-Wage Economy . 72

3. Formulas for Preferences . . . . . . . . . . . . . . . . . . . . . . 78 3.1. Separable Preferences . . . . . . . . . . . . . . . . . . . . . 79 3.2. Nonseparable Preferences I . . . . . . . . . . . . . . . . . . . 80 3.3. Nonseparable Preferences II . . . . . . . . . . . . . . . . . . 82

4. Are Real Exchange Rates Volatile and Persistent? . . . . . . . . . . . 83

5. Some Analytics for Special Cases . . . . . . . . . . . . . . . . . . . 87 5.1. Labor-Only Case with Nonseparable U, Exogenous Money, and N = 2 87 5.2. Labor-Only Case with Separable U, Exogenous Money, and N = 2 . 93

These notes contain derivations of expressions and results reported in the main text. We also provide intuition for some of the results and details on the computation of equilibria.

1. The Model

The model is a two-country business cycle model. Each country is populated by a large number of identical, infinitely-lived consumers. In both countries, intermediate goods are combined to form final goods which are country specific and cannot be shipped. All trade between the countries is in intermediate goods that are produced by monopolists who can charge different prices in the two countries. We assume that each intermediate goods producer has the exclusive right to sell his own good in the two countries. Thus, there is no possibility for arbitraging away price differences in intermediate goods.

1.1. Notation

Goods produced in the home country are subscripted with an H, while those produced in the foreign country are subscripted with an F. Allocations and prices in the foreign country are denoted with an asterisk. We use a caret over a variable to denote its logged deviation from the mean.

1.2. Uncertainty

In each period t, the economy experiences one of finitely many events s_{t}. We denote by s^{t} = (s_{0}, …, s_{t}) the history of events up through and including period t. The probability, as of period zero, of any particular history s^{t} is π(s^{t}). The initial realization s_{0} is given.

1.3. The Final Goods Producers

Final goods producers behave competitively and solve a static profit-maximization problem. In the home country in each period producers choose inputs y_{H}(i) for i ∈ [0, 1] and y_{F} (i) for i ∈ [0, 1] and output y to maximize profits given by

max P y − ∫_{0}^{1} P_{H}(i)y_{H}(i) di − ∫_{0}^{1} P_{F} (i)y_{F} (i) di  (1.1)

subject to

y = [ω_{1} (∫_{0}^{1} y_{H}(i)^{θ}di)^{ρ/θ} + ω_{2} (∫_{0}^{1} y_{F} (i)^{θ}di)^{ρ/θ}]^{\frac{1}{ρ}}  (1.2)

where y is the final good, P is the price of the final good, y_{H}(i) and y_{F} (i) are intermediate goods produced in the home and foreign countries, respectively, and P_{H}(i) and P_{F} (i) are their prices. These prices (P, P_{H}, P_{F}) are in units of the domestic currency.

The first-order conditions of the problem above with respect to y_{H}(i) and y_{F} (i) are

y_{H}(i) = [\frac{ω_{1}P y^{1−ρ} (∫ y_{H}(i)^{θ} di)^{\frac{ρ}{θ} −1}}{P_{H}(i)} ]^{\frac{1}{1−θ}}

y_{F} (i) = [\frac{ω_{2}P y^{1−ρ} (∫ y_{F} (i)^{θ} di)^{\frac{ρ}{θ} −1}}{P_{F} (i)} ]^{\frac{1}{1−θ}}

If we raise the first expression to the power θ, integrate across i, and solve for ∫ y_{H}(i)^{θ} di, we get

∫ y_{H}(i)^{θ} di = [ω_{1}P y^{1−ρ}]^{\frac{θ}{1−θ}} (∫ y_{H}(i)^{θ} di)^{\frac{ρ−θ}{1−θ}} ∫ P_{H}(i)^{\frac{θ}{θ−1}} di
= [ω_{1}P]^{\frac{θ}{1−ρ}} y^{θ} (∫ P_{H}(i)^{\frac{θ}{θ−1}} di)^{\frac{1−θ}{1−ρ}} .

Similarly, we can derive an expression for ∫ y_{F} (i)^{θ} di. Substituting these expressions into the first-order conditions, we get the input demand functions:

y_{H}(i) = \frac{[ω_{1}P]^{\frac{1}{1−ρ}} P̄H^{\frac{ρ−θ}{(1−ρ)(θ−1)}}}{P_{H}(i)^{\frac{1}{1−θ}}} y  (1.3)

y_{F} (i) = \frac{[ω_{2}P]^{\frac{1}{1−ρ}} P̄F^{\frac{ρ−θ}{(1−ρ)(θ−1)}}}{P_{F} (i)^{\frac{1}{1−θ}}} y  (1.4)

where P̄_{I} = (∫_{0}^{1} P_{I}(i)^{\frac{θ}{θ−1}} di)^{\frac{θ−1}{θ}} for I = H, F.

To get the price of the final good, we use the zero-profit condition which implies that

P = (ω1^{\frac{1}{1−ρ}} P̄H^{\frac{ρ}{ρ−1}} + ω2^{\frac{1}{1−ρ}} P̄F^{\frac{ρ}{ρ−1}} )^{\frac{ρ−1}{ρ}} .  (1.5)

If we solve the analogous problem for the foreigners, we get:

y_{F}^{∗} (i) = \frac{[ω_{1}P^{∗}]^{\frac{1}{1−ρ}} (P̄_{F}^{∗})^{\frac{ρ−θ}{(1−ρ)(θ−1)}}}{P_{F}^{∗}(i)^{\frac{1}{1−θ}}} y^{∗}  (1.6)
y_{H}^{∗}(i) = \frac{[ω_{2}P^{∗}]^{\frac{1}{1−ρ}} (P̄_{H}^{∗})^{\frac{ρ−θ}{(1−ρ)(θ−1)}}}{P_{H}^{∗}(i)^{\frac{1}{1−θ}}} y^{∗}  (1.7)
P^{∗} = (ω1^{\frac{1}{1−ρ}} (P̄_{F}^{∗})^{\frac{ρ}{ρ−1}} + ω2^{\frac{1}{1−ρ}} (P̄_{H}^{∗})^{\frac{ρ}{ρ−1}} )^{\frac{ρ−1}{ρ}} .  (1.8)

1.4. The Consumer’s Problem The consumer chooses consumption c, labor l, and real balances M/P to maximize:

∑_{t=0}^{∞} ∑_{s^{t}} β^{t}π(s^{t}) U (c(s^{t}), l(s^{t}), M(s^{t})/P (s^{t})) ,  (1.9)

subject to the sequence of budget constraints

P (s^{t})c(s^{t}) + M(s^{t}) + ∑_{s_{t+1}} Q(s^{t+1}|s^{t})B(s^{t+1})

≤ P (s^{t})w(s^{t})l(s^{t}) + M(s^{t−1}) + B(s^{t}) + Π(s^{t}) + T (s^{t})  (1.10)

and borrowing constraints B(s^{t+1}) ≥ −P (s^{t})\overline{b}. M and B are their holdings of money and contingent claims, Q is the price of the claims, w is the real wage, Π are profits, and T are government transfers.

The first-order conditions for the consumer are:

− \frac{U_{l}(s^{t})}{U_{c}(s^{t})} = w(s^{t})  (1.11)
\frac{U_{m}(s^{t})}{P (s^{t})} − \frac{U_{c}(s^{t})}{P (s^{t})} + β ∑_{s_{t+1}} π(s^{t+1}|s^{t})\frac{U_{c}(s^{t+1})}{P (s^{t+1})} = 0  (1.12)
Q(s^{t}|s^{t−1}) = βπ(s^{t}|s^{t−1})\frac{U_{c}(s^{t})}{U_{c}(s^{t−1})} \frac{P (s^{t−1})}{P (s^{t})}  (1.13)

where U(s^{t}) is shorthand notation for U(c(s^{t}), l(s^{t}), M(s^{t})/P (s^{t})).

Let R(s^{t}) and r(s^{t}) be the gross and net nominal interest rates, respectively; they are defined as follows:

\frac{1}{R(s^{t})} = \frac{1}{1 + r(s^{t})} = ∑_{s_{t+1}} Q(s^{t+1}|s^{t})
= ∑_{s_{t+1}} βπ(s^{t+1}|s^{t})\frac{U_{c}(s^{t+1})}{U_{c}(s^{t})} \frac{P (s^{t})}{P (s^{t+1})}  (1.14)

and r(s^{t}) = R(s^{t}) − 1. Notice that we can also write the money demand equation in (1.12) more simply,

\frac{U_{m}(s^{t})}{U_{c}(s^{t})} = \frac{r(s^{t})}{1 + r(s^{t})}  (1.15)

using the definition in (1.14).

1.5. The Foreign Consumer’s Problem

The foreign consumer solves a problem similar to that of the home consumer, namely to maximize:

∑_{t=0}^{∞} ∑_{s^{t}} β^{t}π(s^{t}) U (c^{∗}(s^{t}), l^{∗}(s^{t}), M^{∗}(s^{t})/P^{∗}(s^{t})) ,  (1.16)

subject to the sequence of budget constraints

P^{∗}(s^{t})c^{∗}(s^{t}) + M^{∗}(s^{t}) + ∑_{s_{t+1}} Q^{∗}(s^{t+1}|s^{t})B_{F}^{∗} (s^{t+1}) + ∑_{s_{t+1}} Q(s^{t+1}|s^{t})B_{H}^{∗}(s^{t+1})/e(s^{t})

≤ P^{∗}(s^{t})w^{∗}(s^{t})l^{∗}(s^{t}) + M^{∗}(s^{t−1}) + B_{F}^{∗} (s^{t}) + B_{H}^{∗}(s^{t})/e(s^{t}) + Π^{∗}(s^{t}) + T^{∗}(s^{t}) (1.17)

where B_{H}^{∗}(s^{t}) and B_{F}^{∗} (s^{t}) denotes the foreign consumer’s holdings of home and foreign country bonds in state s^{t}, respectively. The first-order conditions with respect to bonds are given by:

Q(s^{t}|s^{t−1}) = βπ(s^{t}|s^{t−1})\frac{U_{c}^{∗}(s^{t})}{U_{c}^{∗}(s^{t−1})} \frac{P^{∗}(s^{t−1})}{P^{∗}(s^{t})} \frac{e(s^{t−1})}{e(s^{t})}  (1.18)
Q^{∗}(s^{t}|s^{t−1}) = βπ(s^{t}|s^{t−1})\frac{U_{c}^{∗}(s^{t})}{U_{c}^{∗}(s^{t−1})} \frac{P^{∗}(s^{t−1})}{P^{∗}(s^{t})}  (1.19)

If we equate the bond prices derived here (in (1.18)) and above (in (1.13)), we get

\frac{U_{c}(s^{t})}{U_{c}(s^{t−1})} \frac{P (s^{t−1})}{P (s^{t})} = \frac{U_{c}^{∗}(s^{t})}{U_{c}^{∗}(s^{t−1})} \frac{P^{∗}(s^{t−1})}{P^{∗}(s^{t})} \frac{e(s^{t−1})}{e(s^{t})}.

We can iterate back to 0 and let q(s^{t}) = e(s^{t})P^{∗}(s^{t})/P (s^{t}) denote the real exchange rate. Then we find

q(s^{t}) = κ\frac{U_{c}^{∗}(s^{t})}{U_{c}(s^{t})}  (1.20)

where κ is the real exchange rate at 0 times the ratio of marginal utilities at 0. When computing an equilibrium, we normalize κ to 1.

1.6. The Intermediate Goods Producers

Intermediate goods producers are monopolistically competitive. They set prices for their goods, but they most hold them fixed for N periods. We assume that price-setting is done in a staggered fashion so that 1/N of the firms are setting in a particular period. We compute a symmetric equilibrium so we assume that all firms i ∈ [0, 1/N] behave the same way and all firms i ∈ [1/N, 2/N] behave the same way, and so on.

The problem solved by the home intermediate goods producers setting prices is to choose sequences of prices P_{H}, capital stocks k, investments x, and labor inputs l to maximize

∑_{τ=0}^{∞} ∑_{s^{τ}} Q̃(s^{τ})[P_{H}(i, s^{τ})y_{H}(i, s^{τ}) + e(s^{τ})P_{H}^{∗}(i, s^{τ})y_{H}^{∗}(i, s^{τ})
− P (s^{τ})w(s^{τ})l(i, s^{τ}) − P (s^{τ})x(i, s^{τ})] (1.21)

subject to the input demands (1.3) and (1.4), the production technology:

y_{H}(i, s^{t}) + y_{H}^{∗}(i, s^{t}) = F (k(i, s^{t−1}), l(i, s^{t})),  (1.22)

the law of motion for capital used in producing good i:

k(i, s^{t}) = (1 − δ)k(i, s^{t−1}) + x(i, s^{t}) − ϕ (\frac{x(i, s^{t})}{k(i, s^{t−1})}) k(i, s^{t−1})  (1.23)

and the following constraints on prices:

P_{H}(i, s^{t−1}) = P_{H}(i, s^{t}) = … P_{H}(i, s^{t+N−1})
P_{H}(i, s^{t+N}) = P_{H}(i, s^{t+N+1}) = … P_{H}(i, s^{t+2N−1})
⋮
P_{H}^{∗}(i, s^{t−1}) = P_{H}^{∗}(i, s^{t}) = … P_{H}^{∗}(i, s^{t+N−1})
P_{H}^{∗}(i, s^{t+N}) = P_{H}^{∗}(i, s^{t+N+1}) = … P_{H}^{∗}(i, s^{t+2N−1})
⋮  (1.24)

where Q̃(s^{τ}) is the τth period Arrow-Debreu price (that is, a product of the one-period Q(s^{t}|s^{t−1})’s).

The Lagrangian in this case is

L = … + Q̃(s^{t})\{[P_{H}(i, s^{t−1})^{\frac{θ}{θ−1}} Λ_{H}(s^{t}) + e(s^{t})P_{H}^{∗}(i, s^{t−1})^{\frac{θ}{θ−1}} Λ_{H}^{∗}(s^{t})
− P (s^{t})w(s^{t})l(i, s^{t}) − P (s^{t})x(i, s^{t})
+ ζ(s^{t})\{F (k(i, s^{t−1}), l(i, s^{t})) − Λ_{H}(s^{t})P_{H}(i, s^{t−1})^{\frac{1}{θ−1}}
− Λ_{H}^{∗}(s^{t})P_{H}^{∗}(i, s^{t−1})^{\frac{1}{θ−1}} \}
+ λ(s^{t}){(1 − δ)k(i, s^{t−1}) + x(i, s^{t})
− ϕ(x(i, s^{t})/k(i, s^{t−1}))k(i, s^{t−1}) − k(i, s^{t})}]

+ ∑_{s_{t+1}} Q(s^{t+1}|s^{t})[P_{H}(i, s^{t−1})^{\frac{θ}{θ−1}} Λ_{H}(s^{t+1}) + e(s^{t+1})P_{H}^{∗}(i, s^{t−1})^{\frac{θ}{θ−1}} Λ_{H}^{∗}(s^{t+1})

− P (s^{t+1})w(s^{t+1})l(i, s^{t+1}) − P (s^{t+1})x(i, s^{t+1})
+ ζ(s^{t+1})\{F (k(i, s^{t}), l(i, s^{t+1})) − Λ_{H}(s^{t+1})P_{H}(i, s^{t−1})^{\frac{1}{θ−1}}
− Λ_{H}^{∗}(s^{t+1})P_{H}^{∗}(i, s^{t−1})^{\frac{1}{θ−1}} \}
+ λ(s^{t+1}){(1 − δ)k(i, s^{t}) + x(i, s^{t+1})
− ϕ(x(i, s^{t+1})/k(i, s^{t}))k(i, s^{t}) − k(i, s^{t+1})}]
+ …}  (1.25)

where

Λ_{H}(s^{t}) = [ω_{1}P (s^{t})]^{\frac{1}{1−ρ}} P̄_{H}(s^{t−1})^{\frac{ρ−θ}{(1−ρ)(θ−1)}} y(s^{t})  (1.26)
Λ_{H}^{∗}(s^{t}) = [ω_{2}P^{∗}(s^{t})]^{\frac{1}{1−ρ}} (P̄_{H}^{∗}(s^{t−1}))^{\frac{ρ−θ}{(1−ρ)(θ−1)}} y^{∗}(s^{t}).  (1.27)

The variables ζ and λ are multipliers for constraints (1.22) and (1.23), respectively.

Taking the derivative of L in (1.25) with respect to the monopolist’s prices P_{H}(i, s^{t−1}) and P_{H}^{∗}(i, s^{t−1}), we find

∑_{τ} ∑_{s^{τ}} Q(s^{τ} |s^{t−1}) \{θP_{H}(i, s^{t−1})^{\frac{1}{θ−1}} Λ_{H}(s^{τ}) − ζ(s^{τ})P_{H}(i, s^{t−1})^{\frac{2−θ}{θ−1}} Λ_{H}(s^{τ})\}=0  (1.28)
∑_{τ} ∑_{s^{τ}} Q(s^{τ} |s^{t−1}) \{θe(s^{τ})P_{H}^{∗}(i, s^{t−1})^{\frac{1}{θ−1}} Λ_{H}^{∗}(s^{τ}) − ζ(s^{τ})P_{H}(i, s^{t−1})^{\frac{2−θ}{θ−1}} Λ_{H}^{∗}(s^{τ})\}=0. (1.29)

Taking the derivative of L with respect to l(i, s^{t}), we find:

−P (s^{t})w(s^{t}) + ζ(s^{t})F_{l}(i, s^{t}) = 0.  (1.30)

The derivative of L with respect to x(i, s^{t}) is:

−P (s^{t}) + λ(s^{t}) [1 − ϕ′ (\frac{x(i, s^{t})}{k(i, s^{t−1})})] = 0.  (1.31)

Finally, the derivative of L with respect to k(i, s^{t}) is:

−λ(s^{t}) + ∑_{s_{t+1}} Q(s^{t+1}|s^{t}){ζ(s^{t+1})F_{k}(i, s^{t+1}) + λ(s^{t+1})[1 − δ
− ϕ (\frac{x(i, s^{t+1})}{k(i, s^{t})}) + ϕ′ (\frac{x(i, s^{t+1})}{k(i, s^{t})}) \frac{x(i, s^{t+1})}{k(i, s^{t})}]\} = 0.  (1.32)

If we substitute expressions for the multipliers using (1.30) and (1.31) into (1.28), (1.29), and (1.32), we get

P_{H}(i, s^{t−1}) = \frac{∑_{τ} ∑_{s^{τ}} Q(s^{τ} |s^{t−1})P (s^{τ})mc(i, s^{τ})Λ_{H}(s^{τ})}{θ ∑_{τ} ∑_{s^{τ}} Q(s^{τ} |s^{t−1})Λ_{H}(s^{τ})}  (1.33)
P_{H}^{∗}(i, s^{t−1}) = \frac{∑_{τ} ∑_{s^{τ}} Q(s^{τ} |s^{t−1})P (s^{τ})mc(i, s^{τ})Λ_{H}^{∗}(s^{τ})}{θ ∑_{τ} ∑_{s^{τ}} Q(s^{τ} |s^{t−1})e(s^{τ})Λ_{H}^{∗}(s^{τ})}  (1.34)
\frac{U_{c}(s^{t})}{1 − ϕ′(i, s^{t})} = β ∑_{s_{t+1}} π(s^{t+1}|s^{t})U_{c}(s^{t+1})\{mc(i, s^{t+1})F_{k}(i, s^{t+1})
+ \frac{1}{1 − ϕ′(i, s^{t+1})} [1 − δ − ϕ(i, s^{t+1}) + ϕ′(i, s^{t+1})\frac{x(i, s^{t+1})}{k(i, s^{t})}]\}.(1.35)

Note that we have used the fact that marginal costs of producer i are given by:

mc(i, s^{t}) = w(s^{t})/F_{l}(i, s^{t}).  (1.36)

1.7. The Government

Monetary policy is modeled as an exogenous process for monetary growth rates, that is

M(s^{t}) = μ(s^{t})M(s^{t−1})  (1.37)

where μ is a stochastic process. The process that we use is

log μ(s^{t}) = ρ_{μ} log μ(s^{t−1}) + (1 − ρ_{μ}) log μ + ϵ_{μ,t}.  (1.38)

The home government budget constraint is given by:

T (s^{t}) = M(s^{t}) − M(s^{t−1}).  (1.39)

where T are transfers to consumers.

1.8. Additional Equilibrium Conditions We need some additional conditions before computing an equilibrium. The resource constraint in the home country is given by

y(s^{t}) = c(s^{t}) + ∫_{0}^{1} x(i, s^{t}) di.  (1.40)

The labor market clearing condition is given by

l(s^{t}) = ∫ l(i, s^{t}) di.  (1.41)

There are analogous conditions for the foreign country.

1.9. Aggregates of Interest Nominal and real net exports are defined as follows:

NX(s^{t}) = e(s^{t}) ∫ P_{H}^{∗}(i, s^{t−1})y_{H}^{∗}(i, s^{t}) di − ∫ P_{F} (i, s^{t−1})y_{F} (i, s^{t}) di  (1.42)
nx(s^{t}) = NX(s^{t})/P (s^{t}).  (1.43)

Nominal and real GDP are defined as follows:

GDP (s^{t}) = P (s^{t})y(s^{t}) + NX(s^{t})  (1.44)
gdp(s^{t}) = y(s^{t}) + ∫ y_{H}^{∗}(i, s^{t}) di − ∫ y_{F} (i, s^{t}) di.  (1.45)

1.10. Extensions

The economy just described is our benchmark economy. We now describe four extensions that we also consider.

1.10.1. Taylor Rule as Fed Policy

Above we assumed that there was an exogenous process for the monetary growth rates. We also consider cases where the Fed follows a Taylor-like interest rate-setting rule. In particular, we assume that the nominal interest rate is given by

r(s^{t}) = a′ [ \begin{matrix}r(s^{t−1}) \\ r(s^{t−2}) \\ r(s^{t−3}) \\ E_{t} log P (s^{t+1}) − log P (s^{t}) \\ log P (s^{t}) − log P (s^{t−1}) \\ log P (s^{t−1}) − log P (s^{t−2}) \\ log P (s^{t−2}) − log P (s^{t−3}) \\ log gdp(s^{t}) \\ log gdp(s^{t−1}) \\ log gdp(s^{t−2}) \end{matrix} ] + constant + ϵ_{r,t}  (1.46)

with the foreign rate defined analogously. In this case, we back out money from the money demand equation (1.15), U_{m}/U_{c} = r/(1 + r), once we know consumption, labor, and the interest rate.

1.10.2. Incomplete Asset Markets

When markets are complete, we can use the first-order conditions for state contingent bonds in the two countries (that is, (1.13) and (1.18)) to back out an expression for the real exchange rate, q(s^{t}) = U_{c}^{∗}(s^{t})/U_{c}(s^{t}). In the incomplete-markets extension, we assume that there is a bond market in the home country but no cross-country contingent claims. The budget constraints for the home and foreign country are then given by:

P (s^{t})c(s^{t}) + M(s^{t}) + ∑_{s_{t+1}} Q(s^{t+1}|s^{t})B(s^{t+1}) + V (s^{t})D(s^{t})

≤ P (s^{t})w(s^{t})l(s^{t}) + M(s^{t−1}) + B(s^{t}) + D(s^{t−1}) + Π(s^{t}) + T (s^{t}) (1.47)

P^{∗}(s^{t})c^{∗}(s^{t}) + M^{∗}(s^{t}) + ∑_{s_{t+1}} Q^{∗}(s^{t+1}|s^{t})B^{∗}(s^{t+1}) + V (s^{t})D^{∗}(s^{t})/e(s^{t})

≤ P^{∗}(s^{t})w^{∗}(s^{t})l^{∗}(s^{t}) + M^{∗}(s^{t−1}) + B^{∗}(s^{t}) + D^{∗}(s^{t−1})/e(s^{t})
+ Π^{∗}(s^{t}) + T^{∗}(s^{t}).  (1.48)

instead of (1.10) and (1.17). D and D^{∗} are one-period bonds in the home and foreign country respectively. Notice that in the foreign budget constraint, we no longer have the home and foreign contingent claims, B_{H}^{∗} and B_{F}^{∗}.

The first-order conditions corresponding to the choices of D and D^{∗} imply the following equations hold in equilibrium:

V (s^{t}) = ∑_{s_{t+1}} βπ(s^{t+1}|s^{t})\frac{U_{c}(s^{t+1})}{U_{c}(s^{t})} \frac{P (s^{t})}{P (s^{t+1})}  (1.49)
V (s^{t}) = ∑_{s_{t+1}} βπ(s^{t+1}|s^{t})\frac{U_{c}^{∗}(s^{t+1})}{U_{c}^{∗}(s^{t})} \frac{P^{∗}(s^{t})}{P^{∗}(s^{t+1})} \frac{e(s^{t})}{e(s^{t+1})}  (1.50)
D(s^{t}) + D^{∗}(s^{t}) = 0  (1.51)

Thus, if we eliminate the home budget constraint by Walras’ Law and eliminate D(s^{t}) as a variable using D + D^{∗} = 0, we are left with the foreign budget constraint and the two first-order conditions for bonds in the two countries. We keep these two as separate dynamic equations. Then we use the foreign budget constraint to get an expression for the nominal exchange rate:

e(s^{t}) = \frac{V (s^{t})D^{∗}(s^{t}) − D^{∗}(s^{t−1})}{P^{∗}(s^{t})[w^{∗}(s^{t})l^{∗}(s^{t}) − c^{∗}(s^{t})] + Π^{∗}(s^{t})}.  (1.52)

1.10.3. Additional Shocks

Above we assumed that the only shocks in the model were monetary shocks. We also consider extensions of the benchmark economy with changes in government spending and technology.

Because we want to consider alternative assumptions about the revelation of shocks, we will use the following notational convention. Let s^{t} = (s_{0}, …, s_{t}) be the history of events up through and including period t that is observed by consumers and final goods producers. Let z^{t} = (z_{0}, …, z_{t}) be the history of events up through and including period t that is observed by intermediate goods producers when making their pricing decisions. When making investment and hiring decisions, we assume that they have observed all of the information contained in s^{t}.

Our default assumption is that consumers and final goods producers know current (t) and past realizations (t − 1, t − 2, …) of the three shocks (i.e., money, government spending, and technology) when making their current t decisions. Thus, s_{t} are realizations of the three shocks. Intermediate goods producers, on the other hand, do not observe the current monetary shock and may or may not observe the current real shocks before making their current pricing decision. If they observe real shocks but don’t observe monetary shocks when making pricing decisions, then z_{t} contains only information on the government spending shock and the technology shock. If they see none of these events before deciding on prices, then z_{t} contains no information.

With spending included in the model, the resource constraint for the home country is given by

y(s^{t}) = c(s^{t}) + ∫_{0}^{1} x(i, s^{t}) di + g(s^{t}).

with an analogous constraint for the foreigners. With technology shocks, we need to modify the production function as follows:

y_{H}(i, s^{t}) + y_{H}^{∗}(i, s^{t}) = F (k(i, s^{t−1}), A(s^{t})l(i, s^{t}))

and therefore the expression for marginal costs,

mc(i, s^{t}) = \frac{w(s^{t})}{F_{l}(i, s^{t})A(s^{t})}.

Throughout, we will assume that g(s^{t}) and A(s^{t}) are AR(1) processes with means g and 1, respectively.

1.10.4. Sticky Wages

Finally, we allow for sticky wages in addition to sticky prices. One can think of the economy organized into a continuum of unions indexed by j. Each union j consists of all the consumers in the economy with labor of type j. This union realizes that it faces a downward sloping demand curve for its type of labor. It sets nominal wages for N periods at t, t + N, t + 2N, and so on. Thus, it faces constraints

W (j, s^{t−1}) = W (j, s^{t}) = … = W (j, s^{t+N −1})
W (j, s^{t+N}) = W (j, s^{t+N +1}) = … = W (j, s^{t+2N −1})

and so on in addition to the ones below.

The problem now solved by a consumer of type j is

max ∑_{t=0}^{∞} ∑_{s^{t}} β^{t}π(s^{t}) U (c(j, s^{t}), L^{s}(j, s^{t}), M^{d}(j, s^{t})/P (s^{t})) ,

subject to the sequence of budget constraints, the definition of labor supply, and the labor demands of the firms.

P (s^{t})c(j, s^{t}) + M^{d}(j, s^{t}) + ∑_{s_{t+1}} Q(s^{t+1}|s^{t})B(j, s^{t+1})

≤ W (j, s^{t−1})L^{s}(j, s^{t}) + M^{d}(j, s^{t−1}) + B(j, s^{t}) + Π(s^{t}) + T (s^{t}) (1.53)

L^{s}(j, s^{t}) = ∫ l(i, j, s^{t}) di

l(i, j, s^{t}) = (\frac{W̄ (s^{t})}{W (j, s^{t−1})})^{\frac{1}{1−v}} L^{d}(i, s^{t}), for all i.

In this case, the consumer chooses the wage but agrees to supply whatever is demanded at that wage.

The Lagrangian in this case is

L = … β^{t} ∑_{s^{t}} π(s^{t})\{U (c(j, s^{t}), W̄ (s^{t})^{\frac{1}{1−v}} W (j, s^{t−1})^{\frac{1}{v−1}} L^{d}(s^{t}), \frac{M^{d}(j, s^{t})}{P (s^{t})})
+ ζ(s^{t})\{W̄ (s^{t})^{\frac{1}{1−v}} W (j, s^{t−1})^{\frac{v}{v−1}} L^{d}(s^{t}) + M^{d}(j, s^{t−1})

+ B(j, s^{t}) + Π(s^{t}) + T (s^{t})

− P (s^{t})c(j, s^{t}) − M^{d}(j, s^{t}) − ∑_{s_{t+1}} Q(s^{t+1}|s^{t})B(j, s^{t+1})}

+ β ∑_{s_{t+1}} π(s^{t+1}|s^{t})[U (c(j, s^{t+1}), W̄ (s^{t+1})^{\frac{1}{1−v}} W (j, s^{t−1})^{\frac{1}{v−1}} L^{d}(s^{t+1}), \frac{M^{d}(j, s^{t+1})}{P (s^{t+1})})

+ ζ(s^{t})\{W̄ (s^{t+1})^{\frac{1}{1−v}} W (j, s^{t−1})^{\frac{v}{v−1}} L^{d}(s^{t+1}) + M^{d}(j, s^{t})

+ B(j, s^{t+1}) + Π(s^{t+1}) + T (s^{t+1})

− P (s^{t+1})c(j, s^{t+1}) − M^{d}(j, s^{t+1}) − ∑_{s_{t+2}} Q(s^{t+2}|s^{t+1})B(j, s^{t+2})} …

where L^{d}(s^{t}) = ∫ L^{d}(i, s^{t}) di.

Taking the derivative of L with respect to W (j, s^{t−1}) we have

0 = ∑_{s^{t}} π(s^{t})\{\frac{1}{v − 1}W̄ (s^{t})^{\frac{1}{1−v}} W (j, s^{t−1})^{\frac{2−v}{v−1}} L^{d}(s^{t})U_{l}(j, s^{t})

+ \frac{v}{v − 1}ζ(s^{t})W̄ (s^{t})^{\frac{1}{1−v}} W (j, s^{t−1})^{\frac{1}{v−1}} L^{d}(s^{t})

+ \frac{1}{v − 1} ∑_{s_{t+1}} βπ(s^{t+1}|s^{t})W̄ (s^{t+1})^{\frac{1}{1−v}} W (j, s^{t−1})^{\frac{2−v}{v−1}} L^{d}(s^{t+1})U_{l}(j, s^{t+1})

+ \frac{v}{v − 1} ∑_{s_{t+1}} βπ(s^{t+1}|s^{t})ζ(s^{t+1})W̄ (s^{t+1})^{\frac{1}{1−v}} W (j, s^{t−1})^{\frac{1}{v−1}} L^{d}(s^{t+1}) + …

Using the equilibrium bond price and the fact that ζ(s^{t}) = U_{c}(j, s^{t})/P (s^{t}), we can simplify this first order conditions to get:

W (j, s^{t−1}) = −\frac{∑_{τ=t}^{t+N −1} ∑_{s^{τ}} Q(s^{τ} |s^{t−1})P (s^{τ})W̄ (s^{τ})^{\frac{1}{1−v}} L^{d}(s^{τ})U_{l}(j, s^{τ})/U_{c}(j, s^{τ})}{v ∑_{τ=t}^{t+N −1} ∑_{s^{τ}} Q(s^{τ} |s^{t−1})W̄ (s^{τ})^{\frac{1}{1−v}} L^{d}(s^{τ})}. (1.54)

The other first order conditions for the consumer are the same as in the benchmark economy, except for the fact that we need to record the type, e.g.,

\frac{U_{m}(j, s^{t})}{P (s^{t})} − \frac{U_{c}(j, s^{t})}{P (s^{t})} + β ∑_{s_{t+1}} π(s^{t+1}|s^{t})\frac{U_{c}(j, s^{t+1})}{P (s^{t+1})} = 0  (1.55)

Q(s^{t}|s^{t−1}) = βπ(s^{t}|s^{t−1})\frac{U_{c}(j, s^{t})}{U_{c}(j, s^{t−1})} \frac{P (s^{t−1})}{P (s^{t})}.  (1.56)

Notice that these conditions imply that

\frac{U_{c}(j, s^{τ})}{U_{c}(j, s^{t})} = \frac{U_{c}(k, s^{τ})}{U_{c}(k, s^{t})}

for all j and k. So, marginal utilities are equated up to a constant, namely the date 0 Lagrange multiplier on their budget constraint. Here, we assume that initial debts and transfers among the N types in either country are such that the multipliers are equalized. In that case, we can summarize these first-order conditions as

U_{c}(j, s^{t}) = U_{c}(k, s^{t})  (1.57)
U_{m}(j, s^{t}) = U_{m}(k, s^{t})  (1.58)

for all j and k. The equating of U_{m}’s from the money demand equations.

The problem solved by the intermediate goods producers setting prices in the home country is to choose sequences of prices P_{H}(i), capital stocks k(i), investments x(i), and labor inputs l(i, j), j = 1, …, N to maximize

∑_{τ=0}^{∞} ∑_{s^{τ}} Q̃(s^{τ})[P_{H}(i, s^{τ})y_{H}(i, s^{τ}) + e(s^{τ})P_{H}^{∗}(i, s^{τ})y_{H}^{∗}(i, s^{τ})
− ∫ W (j, s^{τ−1})l(i, j, s^{τ}) dj − P (s^{τ})x(i, s^{τ})]  (1.59)

subject to the input demands (1.3) and (1.4), the production technology:

y_{H}(i, s^{t}) + y_{H}^{∗}(i, s^{t}) = F (k(i, s^{t−1}), L^{d}(i, s^{t}))  (1.60)

the constraint on labor

L^{d}(i, s^{t}) ≤ [∫ l(i, j, s^{t})^{v} dj]^{\frac{1}{v}} ,  (1.61)

the law of motion for capital used in producing good i in (1.23), and the constraints on prices in (1.24). The firms take the wages as given.

L = … + Q̃(s^{t})\{[P_{H}(i, s^{t−1})^{\frac{θ}{θ−1}} Λ_{H}(s^{t}) + e(s^{t})P_{H}^{∗}(i, s^{t−1})^{\frac{θ}{θ−1}} Λ_{H}^{∗}(s^{t})

− ∫ W (j, s^{t−1})l(i, j, s^{t}) dj − P (s^{t})x(i, s^{t})

+ ζ(s^{t})\{F (k(i, s^{t−1}), L^{d}(i, s^{t})) − Λ_{H}(s^{t})P_{H}(i, s^{t−1})^{\frac{1}{θ−1}}

− Λ_{H}^{∗}(s^{t})P_{H}^{∗}(i, s^{t−1})^{\frac{1}{θ−1}} \}

+ λ(s^{t}){(1 − δ)k(i, s^{t−1}) + x(i, s^{t})

− ϕ(x(i, s^{t})/k(i, s^{t−1}))k(i, s^{t−1}) − k(i, s^{t})}]

+ κ(s^{t})\{[∫ l(i, j, s^{t})^{v} dj]^{\frac{1}{v}} − L^{d}(i, s^{t})\}

+ ∑_{s_{t+1}} Q(s^{t+1}|s^{t})[P_{H}(i, s^{t−1})^{\frac{θ}{θ−1}} Λ_{H}(s^{t+1}) + e(s^{t+1})P_{H}^{∗}(i, s^{t−1})^{\frac{θ}{θ−1}} Λ_{H}^{∗}(s^{t+1})

− ∫ W (j, s^{t})l(i, j, s^{t+1}) dj − P (s^{t+1})x(i, s^{t+1})

+ ζ(s^{t+1})\{F (k(i, s^{t}), L^{d}(i, s^{t+1})) − Λ_{H}(s^{t+1})P_{H}(i, s^{t−1})^{\frac{1}{θ−1}}

− Λ_{H}^{∗}(s^{t+1})P_{H}^{∗}(i, s^{t−1})^{\frac{1}{θ−1}} \}

+ λ(s^{t+1}){(1 − δ)k(i, s^{t}) + x(i, s^{t+1})

− ϕ(x(i, s^{t+1})/k(i, s^{t}))k(i, s^{t}) − k(i, s^{t+1})}]

+ κ(s^{t+1})\{[∫ l(i, j, s^{t+1})^{v} dj]^{\frac{1}{v}} − L^{d}(i, s^{t+1})\}

+ …}  (1.62)

where

Λ_{H}(s^{t}) = [ω_{1}P (s^{t})]^{\frac{1}{1−ρ}} P̄_{H}(s^{t−1})^{\frac{ρ−θ}{(1−ρ)(θ−1)}} y(s^{t})  (1.63)
Λ_{H}^{∗}(s^{t}) = [ω_{2}P^{∗}(s^{t})]^{\frac{1}{1−ρ}} (P̄_{H}^{∗}(s^{t−1}))^{\frac{ρ−θ}{(1−ρ)(θ−1)}} y^{∗}(s^{t}).  (1.64)

The variables ζ, λ, and κ are multipliers for constraints (1.60), (1.23), and (1.61), respectively.

Taking the derivative of L in (1.25) with respect to the monopolist’s prices P_{H}(i, s^{t−1}) and P_{H}^{∗}(i, s^{t−1}), the investment x(i, s^{t}), and the capital stock k(i, s^{t}), we find the same results as in the benchmark economy, namely (1.28),(1.29), (1.31), and (1.32).

The difference between the sticky wage economy and the benchmark economy is the labor market. Taking the derivative of L with respect to L^{d}(i, s^{t}), we find:

ζ(s^{t})F_{l}(i, s^{t}) − κ(s^{t}) = 0.

Taking the derivative with respect to l(i, j, s^{t}), we find:

−W (j, s^{t−1}) + κ(s^{t})l(i, j, s^{t})^{v−1} [∫ l(i, j, s^{t})^{v} dj]^{\frac{1}{v} −1} = 0

or,

W (j, s^{t−1}) = κ(s^{t})l(i, j, s^{t})^{v−1}L^{d}(i, s^{t})^{1−v}.  (1.65)

If we integrate both sides of (1.65) we get

κ(s^{t}) = [∫ W (j, s^{t−1})^{\frac{v}{v−1}} dj]^{\frac{v−1}{v}}
≡ W̄ (s^{t})  (1.66)

which implies that the multipler is equal to the aggregate wage. Substituting that back into (1.65), we have

l(i, j, s^{t}) = (\frac{W̄ (s^{t})}{W (j, s^{t−1})})^{\frac{1}{1−v}} L^{d}(i, s^{t}).

The solution to the decision problem of the intermediate goods producer can also be found in two steps. First, define an intermediate real wage w(s^{t}) as follows:

w(s^{t})L^{d}(i, s^{t}) ≡ min_{l(i,j,s^{t}),∀j} ∫ \frac{W (j, s^{t−1})}{P (s^{t})}l(i, j, s^{t}) dj
subject to L^{d}(i, s^{t}) ≤ [∫ l(i, j, s^{t})^{v} dj]^{\frac{1}{v}} .

Solving this problem yields:

w(s^{t}) = W̄ (s^{t})/P (s^{t})  (1.67)

where W̄ (s^{t}) is defined by (1.66). The second step is to return to (1.21) and everywhere we see l(i, s^{t}), we replace it with L^{d}(i, s^{t}). Thus, the solution for the intermediate goods producer’s problem in the benchmark and sticky-wage economies are the same, except that now we have a composite labor input and a composite wage. In other words, the first order conditions are given by (1.33)-(1.36) where F is evaluated at L^{d}(i, s^{t}), w is given in (1.67), and W̄ (s^{t}) is a function of the distribution of wages (which are part of the new state vector).

The last equations to be changed in the case of sticky wages are the resource constraints and the equating of money supply and money demand. We replace (1.40) with

y(s^{t}) = ∫_{0}^{1} c(j, s^{t}) dj + ∫_{0}^{1} x(i, s^{t}) di + g(s^{t}),  (1.68)

and we add

M^{s}(s^{t}) = ∫_{0}^{1} M^{d}(j, s^{t}) dj.

Analogous equations hold for the foreign country.

2. Computing an Equilibrium

We now describe how to compute an equilibrium for the four economies: the benchmark economy, the economy with the Fed using a Taylor rule, the incomplete-markets economy, and the sticky-wage economy.^{1} In each case, there are four steps taken. First, we normalize variables to make the problem stationary. Second, we derive equations for the steady states of the stationary variables. Third, we linearize the first-order conditions around the steady state. Fourth, we describe in detail the codes used for computing a solution to the linearized system of equations.

2.1. The Benchmark Economy

In this section, we describe the computation for the benchmark economy. Most of the derivations are done in this section since there are many common equations for the three economies that we consider.

To simplify things, we assume from here on (unless noted otherwise) that the ith group of monopolists (i ∈ {1, … N}) is the one who set prices i periods ago. Thus, in period t, monopolist 1 is assumed to have set prices conditional on seeing s^{t−1}, monopolist 2 set prices conditional on seeing s^{t−2}, and so on.

2.1.1. Normalization in the Benchmark Economy

Since we allow for positive money growth and inflation, we need to normalize prices in

1 For now we assume that there are just monetary shocks. Below, we extend the analysis to include other shocks. order to make the system of equations stationary. Let

\begin{matrix}p(s^{t}) \\ p_{I}(i, s^{t−1}) \\ p̄_{I}(s^{t−1}) \\ p^{∗}(s^{t}) \\ p_{I}^{∗}(i, s^{t−1}) \\ p̄_{I}^{∗}(s^{t−1}) \end{matrix} \begin{matrix}= P (s^{t})/M(s^{t−1}) \\ = P_{I}(i, s^{t−1})/M(s^{t−i}) \\ = P̄_{I}(s^{t−1})/M(s^{t−1}) \\ = P^{∗}(s^{t})/M^{∗}(s^{t−1}) \\ = P_{I}^{∗}(i, s^{t−1})/M^{∗}(s^{t−i}) \\ = P̄_{I}^{∗}(s^{t−1})/M^{∗}(s^{t−1}) \end{matrix}  (2.1)

where I = H or F. Notice that we normalize the intermediate goods prices by M(s^{t−i}) or M^{∗}(s^{t−i}). Since we assume that the ith group set prices i periods ago, we are effectively assuming

\frac{P_{I}(i, s^{t−1})}{M(s^{t−i})} = p_{I}(i, s^{t−1}) = p_{I}(s^{t−i}) = \frac{P_{I}(s^{t−i})}{M(s^{t−i})}.

A number of first-order conditions must be changed because they involve nonstationary variables. First, consider the equations derived from the final goods producer’s problem. The input demands depend on the prices. Using the normalization above, we can rewrite the input demand equation for the home intermediate goods as follows:

y_{H}(i, s^{t}) = \frac{[ω_{1}P (s^{t})]^{\frac{1}{1−ρ}} P̄_{H}(s^{t−1})^{\frac{ρ−θ}{(1−ρ)(θ−1)}}}{P_{H}(i, s^{t−1})^{\frac{1}{1−θ}}} y(s^{t})

= \frac{[ω_{1}p(s^{t})]^{\frac{1}{1−ρ}} p̄_{H}(s^{t−1})^{\frac{ρ−θ}{(1−ρ)(θ−1)}} M(s^{t−1})^{\frac{1}{1−θ}}}{p_{H}(i, s^{t−1})^{\frac{1}{1−θ}} M(s^{t−i})^{\frac{1}{1−θ}}} y(s^{t})

= \frac{[ω_{1}p(s^{t})]^{\frac{1}{1−ρ}} p̄_{H}(s^{t−1})^{\frac{ρ−θ}{(1−ρ)(θ−1)}} [μ(s^{t−i+1}) ⋯ μ(s^{t−1})]^{\frac{1}{1−θ}}}{p_{H}(i, s^{t−1})^{\frac{1}{1−θ}}} y(s^{t}). (2.2)

Note that the first relation is from (1.3) when the state is s^{t}. We can similarly rewrite y_{F}, y_{H}^{∗}, and y_{F}^{∗}.

The aggregate price in (1.5) is normalized as follows:

p(s^{t}) = (ω1^{\frac{1}{1−ρ}} p̄_{H}(s^{t−1})^{\frac{ρ}{ρ−1}} + ω2^{\frac{1}{1−ρ}} p̄_{F} (s^{t−1})^{\frac{ρ}{ρ−1}} )^{\frac{ρ−1}{ρ}}  (2.3)

and similarly for the foreign price. These equations replace (1.5) and (1.8).

Normalized first-order conditions for the consumer include

U_{m}(s^{t}) − U_{c}(s^{t}) + β ∑_{s_{t+1}} π(s^{t+1}|s^{t})U_{c}(s^{t+1})\frac{p(s^{t})}{p(s^{t+1})μ(s^{t})} = 0  (2.4)
Q(s^{t}|s^{t−1}) = βπ(s^{t}|s^{t−1})\frac{U_{c}(s^{t})}{U_{c}(s^{t−1})} \frac{p(s^{t−1})}{p(s^{t})μ(s^{t−1})}  (2.5)

and therefore

1/R(s^{t}) = ∑_{s_{t+1}} Q(s^{t+1}|s^{t})
= ∑_{s_{t+1}} βπ(s^{t+1}|s^{t})\frac{U_{c}(s^{t+1})}{U_{c}(s^{t})} \frac{p(s^{t})}{p(s^{t+1})μ(s^{t})}.  (2.6)

The normalized price of the intermediate goods producer is given by

p_{H}(i, s^{t−1}) = \frac{1}{M(s^{t−1})} \frac{∑_{τ} ∑_{s^{τ}} Q(s^{τ} |s^{t−1})p(s^{τ})mc(i, s^{τ})λ_{H}(s^{τ})M(s^{τ−1})^{\frac{2−θ}{1−θ}}}{θ ∑_{τ} ∑_{s^{τ}} Q(s^{τ} |s^{t−1})λ_{H}(s^{τ})M(s^{τ−1})^{\frac{1}{1−θ}}}

= \frac{∑_{τ} ∑_{s^{τ}} β^{τ−1}π(s^{τ} |s^{t−1})U_{c}(s^{τ})mc(i, s^{τ})λ_{H}(s^{τ}) (μ(s^{t}) ⋯ μ(s^{τ−1}))^{\frac{1}{1−θ}}}{θ ∑_{τ} ∑_{s^{τ}} β^{τ−1}π(s^{τ} |s^{t−1})U_{c}(s^{τ})/p(s^{τ})λ_{H}(s^{τ}) (μ(s^{t}) ⋯ μ(s^{τ−1}))^{\frac{θ}{1−θ}}}(2.7)

where

λ_{H}(s^{t}) = [ω_{1}p(s^{t})]^{\frac{1}{1−ρ}} p̄_{H}(s^{t−1})^{\frac{ρ−θ}{(1−ρ)(θ−1)}} y(s^{t}).  (2.8)

Expressions for p_{F}, p_{H}^{∗}, and p_{F}^{∗} can be derived similarly.

When we aggregate these prices, we get the following intermediate goods price index for the home country:

p̄_{H}(s^{t−1}) = [\frac{1}{N} ∑_{j=1}^{N} (\frac{P_{H}(s^{t−j})}{M(s^{t−1})})^{\frac{θ}{θ−1}} ]^{\frac{θ−1}{θ}}

= [\frac{1}{N} p_{H}(s^{t−1})^{\frac{θ}{θ−1}} + \frac{1}{N} (\frac{p_{H}(s^{t−2})}{μ(s^{t−1})})^{\frac{θ}{θ−1}} + ⋯

+ \frac{1}{N} (\frac{p_{H}(s^{t−N})}{μ(s^{t−1}) ⋯ μ(s^{t−N+1})})^{\frac{θ}{θ−1}} ]^{\frac{θ−1}{θ}} .  (2.9)

2.1.2. Steady State in the Benchmark Economy

When we linearize the first order conditions, we will do so around the steady state values derived below. We will assume that preferences, technologies, and processes for money are the same in the two countries. Therefore, steady state values of home and foreign allocations and prices will be equated.

We take the normalized first-order conditions, drop s^{t} arguments and solve for a fixed point. Consider doing this iteratively. Start with a guess for k(i), i = 1, … N, y, and P c/M. With the k(i)′s, we can back out the investments from the law of motion for capital

k(i) = (1 − δ)k(i − 1) + x(i) − ϕ (\frac{x(i)}{k(i − 1)}) k(i − 1), i = 1, …, N.

With y, we can get the steady state input demands and total product,

y_{H}(i) = y (\frac{ω_{1}p}{p̄_{H}})^{\frac{1}{1−ρ}} (\frac{p̄_{H}}{p_{H}} )^{\frac{1}{1−θ}} μ^{\frac{i−1}{1−θ}} = y_{F}^{∗} (i), i = 1, …, N
y_{H}^{∗}(i) = y (\frac{ω_{2}p}{p̄_{H}})^{\frac{1}{1−ρ}} (\frac{p̄_{H}}{p_{H}} )^{\frac{1}{1−θ}} μ^{\frac{i−1}{1−θ}} = y_{F} (i), i = 1, …, N
F (i) = y (\frac{p}{p_{H}} ) (\frac{p̄_{H}}{p_{H}} )^{\frac{θ}{1−θ}} μ^{\frac{i−1}{1−θ}} , i = 1, …, N

using (1.22), (2.2), and the fact that the price ratios p/p̄_{H} and p̄_{H}/p_{H} can be written as explicit functions of parameters,

p = (ω1^{\frac{1}{1−ρ}} p̄H^{\frac{ρ}{ρ−1}} + ω2^{\frac{1}{1−ρ}} p̄F^{\frac{ρ}{ρ−1}} )^{\frac{ρ−1}{ρ}} = p̄_{H} (ω1^{\frac{1}{1−ρ}} + ω2^{\frac{1}{1−ρ}} )^{\frac{ρ−1}{ρ}}
p̄_{H} = p_{H} [\frac{1}{N} (1 + μ^{\frac{θ}{1−θ}} + ⋯ + μ^{\frac{(N−1)θ}{1−θ}} )]^{\frac{θ−1}{θ}} .

These latter expressions follow directly from (2.3) and (2.9).

With the F (i)’s, k(i)’s, and x(i)’s we back out labor inputs via the production technology F (i) = F (k(i − 1), l(i)) and marginal costs via the Euler equations for capital:

\frac{1}{1 − ϕ′(i)} = β(mc(i + 1)F_{k}(i + 1)

+ \frac{1}{1 − ϕ′(i + 1)} [1 − δ − ϕ(i + 1) + ϕ′(i + 1)x(i + 1)/k(i)]), i = 1, … N

where ϕ(i) = ϕ(x(i)/k(i − 1)).

We can sum up the l(i)’s to get aggregate labor,

l = \frac{1}{N} ∑_{i} l(i),

and we can sum up the x(i)’s to get aggregate investment

x = \frac{1}{N} ∑_{i} x(i).

With x and y, we have consumption,

c = y − x.

With P c/M, we have P/M and therefore

p = μP/M = μ[P c/M]/c

given our definition of p in (2.1).

Now, we can use the following equations to check that we have a fixed point:

mc(i) = −U_{l}/(U_{c}F_{l}(i))
p_{H} = \frac{p}{θ} (\frac{mc(1) + mc(2)βμ^{\frac{1}{1−θ}} + mc(3)β^{2}μ^{\frac{2}{1−θ}} + … + mc(N)β^{N−1}μ^{\frac{N−1}{1−θ}}}{1 + βμ^{\frac{θ}{1−θ}} + β^{2}μ^{\frac{2θ}{1−θ}} + … + β^{N−1}μ^{\frac{(N−1)θ}{1−θ}}} )
U_{m} = U_{c}(1 − β/μ)

which are the equations for marginal costs ((1.36) with (1.11) substituted in), the intermediate goods price (2.7), and money demand (1.12).

Finally, our assumption about common preferences implies that q = 1 in a steady state.

2.1.3. Linearized Equations in the Benchmark Economy

We now use the steady state values in order to linearize the first order conditions. Here again we assume that the cohort 1 are the monopolists setting prices this period (in s^{t−1}), cohort 2 set last period and so on. First-order conditions for the final goods producers can be linearized to yield the following equations:

ŷ_{H,i,t} = ŷ_{t} + \frac{1}{1 − ρ}p̂_{t} + \frac{ρ − θ}{(1 − ρ)(θ − 1)}ˆp̄_{H,t−1} − \frac{1}{1 − θ} [p̂_{H,t−i} − μ̂_{t−i+1} … − μ̂_{t−1}] (2.10)
ŷ_{F,i,t} = ŷ_{t} + \frac{1}{1 − ρ}p̂_{t} + \frac{ρ − θ}{(1 − ρ)(θ − 1)}ˆp̄_{F,t−1} − \frac{1}{1 − θ} [p̂_{F,t−i} − μ̂_{t−i+1} … − μ̂_{t−1}] (2.11)
ŷ_{F,i,t}^{∗} = ŷ_{t}^{∗} + \frac{1}{1 − ρ}p̂_{t}^{∗} + \frac{ρ − θ}{(1 − ρ)(θ − 1)}ˆp̄_{F,t−1}^{∗} − \frac{1}{1 − θ} [p̂_{F,t−i}^{∗} − μ̂_{t−i+1}^{∗} … − μ̂_{t−1}^{∗}](2.12)
ŷ_{H,i,t}^{∗} = ŷ_{t}^{∗} + \frac{1}{1 − ρ}p̂_{t}^{∗} + \frac{ρ − θ}{(1 − ρ)(θ − 1)}ˆp̄_{H,t−1}^{∗} − \frac{1}{1 − θ} [p̂_{H,t−i}^{∗} − μ̂_{t−i+1}^{∗} … − μ̂_{t−1}^{∗}](2.13)
p̂_{t} = [\frac{p̄_{I}}{p} ]^{\frac{ρ}{ρ−1}} [ω1^{\frac{1}{1−ρ}} ˆ p̄_{H,t−1} + ω2^{\frac{1}{1−ρ}} ˆ p̄_{F,t−1}]  (2.14)
p̂_{t}^{∗} = [\frac{p̄_{I}}{p} ]^{\frac{ρ}{ρ−1}} [ω1^{\frac{1}{1−ρ}} ˆ p̄_{F,t−1}^{∗} + ω2^{\frac{1}{1−ρ}} ˆ p̄_{H,t−1}^{∗}]  (2.15)
ˆp̄_{H,t−1} = [1 + μ^{\frac{θ}{1−θ}} + … μ^{\frac{(N−1)θ}{1−θ}} ]^{−1} [p̂_{H,t−1} + μ^{\frac{θ}{1−θ}} (p̂_{H,t−2} − μ̂_{t−1}) + …
+ μ^{\frac{(N−1)θ}{1−θ}} (p̂_{H,t−N} − μ̂_{t−1} − … − μ̂_{t−N+1})]  (2.16)
ˆp̄_{F,t−1} = [1 + μ^{\frac{θ}{1−θ}} + … μ^{\frac{(N−1)θ}{1−θ}} ]^{−1} [p̂_{F,t−1} + μ^{\frac{θ}{1−θ}} (p̂_{F,t−2} − μ̂_{t−1}) + …
+ μ^{\frac{(N−1)θ}{1−θ}} (p̂_{F,t−N} − μ̂_{t−1} − … − μ̂_{t−N+1})]  (2.17)
ˆp̄_{F,t−1}^{∗} = [1 + μ^{∗}^{\frac{θ}{1−θ}} + … μ^{∗}^{\frac{(N−1)θ}{1−θ}} ]^{−1} [p̂_{F,t−1}^{∗} + μ^{∗}^{\frac{θ}{1−θ}} (p̂_{F,t−2}^{∗} − μ̂^{∗}_{t−1}) + …
+ μ^{∗}^{\frac{(N−1)θ}{1−θ}} (p̂_{F,t−N}^{∗} − μ̂_{t−1}^{∗} − … − μ̂_{t−N+1}^{∗})] (2.18)
ˆp̄_{H,t−1}^{∗} = [1 + μ^{∗}^{\frac{θ}{1−θ}} + … μ^{∗}^{\frac{(N−1)θ}{1−θ}} ]^{−1} [p̂_{H,t−1} + μ^{∗}^{\frac{θ}{1−θ}} (p̂_{H,t−2} − μ̂_{t−1}^{∗}) + …
+ μ^{∗}^{\frac{(N−1)θ}{1−θ}} (p̂_{H,t−N} − μ̂_{t−1}^{∗} − … − μ̂_{t−N+1}^{∗})] (2.19)

where p_{I} is the steady state value for the intermediate goods (i.e., for p_{H,t}, p_{F,t}, p_{H,t}^{∗}, and p_{F,t}^{∗}) and p is the steady state value of the final goods (i.e., for p_{t} and p_{t}^{∗}).

First-order conditions for the consumer can be linearized to yield the following equations:

ŵ_{t} = (\frac{U_{cl}}{U_{l}} − \frac{U_{cc}}{U_{c}} ) c ĉ_{t} + (\frac{U_{ll}}{U_{l}} − \frac{U_{cl}}{U_{c}} ) l l̂_{t} + (\frac{U_{lm}}{U_{l}} − \frac{U_{cm}}{U_{c}} ) M/P (μ̂_{t} − p̂_{t}) (2.20)

\frac{U_{m}}{U_{c}} \{(\frac{U_{cc}}{U_{c}} − \frac{U_{cm}}{U_{m}} ) c ĉ_{t} + (\frac{U_{cl}}{U_{c}} − \frac{U_{lm}}{U_{m}} ) l l̂_{t} + (\frac{U_{cm}}{U_{c}} − \frac{U_{mm}}{U_{m}} ) M/P (μ̂_{t} − p̂_{t})\}

= βE_{t}(\frac{U_{cc}c}{U_{c}}(ĉ_{t+1} − ĉ_{t}) + \frac{U_{cl}l}{U_{c}} (l̂_{t+1} − l̂_{t})

+ \frac{U_{cm}M/P}{U_{c}} (μ̂_{t+1} − p̂_{t+1} − μ̂_{t} + p̂_{t}) + p̂_{t} − p̂_{t+1} − μ̂_{t}).  (2.21)

Equation (2.20) is the linearization of wages from (1.11) and equation (2.21) is the money demand equation from (1.12).

We can also linearize the interest rate r using the definition in (1.14) to get

−\frac{β}{μ}(r_{t} − r) = E_{t}(\frac{U_{cc}c}{U_{c}}(ĉ_{t+1} − ĉ_{t}) + \frac{U_{cl}l}{U_{c}} (l̂_{t+1} − l̂_{t})
+ \frac{U_{cm}M/P}{U_{c}} (μ̂_{t+1} − p̂_{t+1} − μ̂_{t} + p̂_{t}) + p̂_{t} − p̂_{t+1} − μ̂_{t}), (2.22)

where r_{t} is the level of the interest rate and all other variables are logged. Note also that the linearized money demand equation can be written

\frac{β(r_{t} − r)}{μr} = (\frac{U_{cm}}{U_{m}} − \frac{U_{cc}}{U_{c}} ) c ĉ_{t} + (\frac{U_{lm}}{U_{m}} − \frac{U_{cl}}{U_{c}} ) l l̂_{t}
+ (\frac{U_{mm}}{U_{m}} − \frac{U_{cm}}{U_{c}} ) M/P (μ̂_{t} − p̂_{t})  (2.23)

if we use the dynamic money demand equation (2.21) and the definition of r in (2.22).

From the foreign consumer’s problem we can linearize the expression for the real exchange rates:

q̂_{t} = −\frac{U_{cc}c}{U_{c}}(ĉ_{t} − ĉ_{t}^{∗}) − \frac{U_{cl}l}{U_{c}} (l̂_{t} − l̂_{t}^{∗}) − \frac{U_{cm}M/P}{U_{c}} (μ̂_{t} − p̂_{t} − μ̂_{t}^{∗} + p̂_{t}^{∗})  (2.24)

We next want to log-linearize the pricing equations such as (2.7). If there is positive inflation, the formulas are a bit messy so we do this in several steps. Consider log-linearizing the deterministic analogue of (2.7):

p_{H}(i, s^{t−1}) = \frac{∑_{τ} β^{τ−1}U_{c}(s^{τ})mc(i, s^{τ})λ_{H}(s^{τ}) (μ(s^{t}) ⋯ μ(s^{τ−1}))^{\frac{1}{1−θ}}}{θ ∑_{τ} β^{τ−1}U_{c}(s^{τ})/p(s^{τ})λ_{H}(s^{τ}) (μ(s^{t}) ⋯ μ(s^{τ−1}))^{\frac{θ}{1−θ}}}  (2.25)

where

λ_{H}(s^{t}) = [ω_{1}p(s^{t})]^{\frac{1}{1−ρ}} p̄_{H}(s^{t−1})^{\frac{ρ−θ}{(1−ρ)(θ−1)}} y(s^{t}).

We will add an expectations operator when we are done since we assume that E_{t−1}f(x(s^{t})) ≈ E_{t−1}f′(x)xx̂_{t}. But, for now, we ignore the expectations operator.

First, rewrite (2.25) as:

θp_{H}(i, s^{t−1}) [… + β^{j}U_{c}(s^{t+j})/p(s^{t+j})λ_{H}(s^{t+j}) (μ(s^{t}) ⋯ μ(s^{t+j−1}))^{\frac{θ}{1−θ}} + …]
= … + β^{j}U_{c}(s^{t+j})mc(i, s^{t+j})λ_{H}(s^{t+j}) (μ(s^{t}) ⋯ μ(s^{t+j−1}))^{\frac{1}{1−θ}} (2.26)

and then do the linearization of (2.26) in pieces:

U_{c}(s^{t+j})/p(s^{t+j})λ_{H}(s^{t+j})(μ(s^{t}) ⋯ μ(s^{t+j−1}))^{\frac{θ}{1−θ}}

≈ U_{c}/pλ_{H}μ^{\frac{θj}{1−θ}} [\frac{U_{cc}c}{U_{c}} ĉ_{t+j} + \frac{U_{cl}l}{U_{c}} l̂_{t+j} + \frac{U_{cm}M/P}{U_{c}} (μ̂_{t+j} − p̂_{t+j})

− p̂_{t+j} + λ̂_{H,t+j} + \frac{θ}{1 − θ} (μ̂_{t} + … μ̂_{t+j−1})]

U_{c}(s^{t+j})mc(i, s^{t+j})λ_{H}(s^{t+j})(μ(s^{t}) ⋯ μ(s^{t+j−1}))^{\frac{1}{1−θ}}

≈ U_{c}mc(i, j)λ_{H}μ^{\frac{j}{1−θ}} [\frac{U_{cc}c}{U_{c}} ĉ_{t+j} + \frac{U_{cl}l}{U_{c}} l̂_{t+j} + \frac{U_{cm}M/P}{U_{c}} (μ̂_{t+j} − p̂_{t+j})

+ m̂c_{i,t+j} + λ̂_{H,t+j} + \frac{1}{1 − θ} (μ̂_{t} + … μ̂_{t+j−1})]

λ̂_{H,t} = \frac{1}{1 − ρ}p̂_{t} + \frac{ρ − θ}{(1 − ρ)(θ − 1)}p̂_{H,t−1} + ŷ_{t}.  (2.27)

Therefore, the full equation is: θp_{H}U_{c}λ_{H}/p(1 + βμ^{\frac{θ}{1−θ}} + …)p̂_{H,i,t−1}

+ θp_{H}U_{c}λ_{H}/p\{… + (βμ^{\frac{θ}{1−θ}})^{j}[\frac{U_{cc}c}{U_{c}} ĉ_{t+j} + \frac{U_{cl}l}{U_{c}} l̂_{t+j} + \frac{U_{cm}M/P}{U_{c}} (μ̂_{t+j} − p̂_{t+j})

− p̂_{t+j} + λ̂_{H,t+j} + \frac{θ}{1 − θ} (μ̂_{t} + … μ̂_{t+j−1})] + …\}

= U_{c}λ_{H}\{… + (βμ^{\frac{1}{1−θ}})^{j−1}mc(i, j)[\frac{U_{cc}c}{U_{c}} ĉ_{t+j} + \frac{U_{cl}l}{U_{c}} l̂_{t+j} + \frac{U_{cm}M/P}{U_{c}} (μ̂_{t+j} − p̂_{t+j})

+ m̂c_{i,t+j} + λ̂_{H,t+j} + \frac{1}{1 − θ} (μ̂_{t} + … μ̂_{t+j−1})] + …\}.  (2.28)

Crossing out common coefficients in (2.28) and dividing by the coefficient on p_{H}, we get p̂_{H,i,t−1} =

\frac{p}{θp_{H}(1 + βμ^{\frac{θ}{1−θ}} + …)}\{… + (βμ^{\frac{1}{1−θ}})^{j}mc(i, j)[\frac{U_{cc}c}{U_{c}} ĉ_{t+j} + \frac{U_{cl}l}{U_{c}} l̂_{t+j}

+ \frac{U_{cm}M/P}{U_{c}} (μ̂_{t+j} − p̂_{t+j}) + m̂c_{i,t+j} + λ̂_{H,t+j} + \frac{1}{1 − θ} (μ̂_{t} + … μ̂_{t+j−1})] + …\}

− \frac{1}{(1 + βμ^{\frac{θ}{1−θ}} + …)}\{… + (βμ^{\frac{θ}{1−θ}})^{j}[\frac{U_{cc}c}{U_{c}} ĉ_{t+j} + \frac{U_{cl}l}{U_{c}} l̂_{t+j}

+ \frac{U_{cm}M/P}{U_{c}} (μ̂_{t+j} − p̂_{t+j}) − p̂_{t+j} + λ̂_{H,t+j} + \frac{θ}{1 − θ} (μ̂_{t} + … μ̂_{t+j−1})] + …\}. (2.29)

Finally, we use the steady state equation for p in (2.29) to get

p̂_{H,i,t−1} =

\{… + \frac{(βμ^{\frac{1}{1−θ}})^{j}mc(i, j)}{∑(βμ^{\frac{1}{1−θ}})^{j}mc(i, j)}[\frac{U_{cc}c}{U_{c}} ĉ_{t+j} + \frac{U_{cl}l}{U_{c}} l̂_{t+j} + \frac{U_{cm}M/P}{U_{c}} (μ̂_{t+j} − p̂_{t+j})

+ m̂c_{i,t+j} + λ̂_{H,t+j} + \frac{1}{1 − θ} (μ̂_{t} + … μ̂_{t+j−1})] + …\}

− \{… + \frac{(βμ^{\frac{θ}{1−θ}})^{j}}{∑(βμ^{\frac{θ}{1−θ}})^{j}} [\frac{U_{cc}c}{U_{c}} ĉ_{t+j} + \frac{U_{cl}l}{U_{c}} l̂_{t+j} + \frac{U_{cm}M/P}{U_{c}} (μ̂_{t+j} − p̂_{t+j})

− p̂_{t+j} + λ̂_{H,t+j} + \frac{θ}{1 − θ} (μ̂_{t} + … μ̂_{t+j−1})] + …\}.

The derivations are similar for the other three prices.

Putting expectations back in and doing the same exercise for the remaining prices yields the following linearized price equations:

p̂_{H,t−1} = E_{t−1} ∑_{j=0}^{N−1} ω_{1,j} (\frac{Û_{c,t+j}}{U_{c}} + λ̂_{H,t+j} + m̂c_{j+1,t+j} + \frac{1}{1 − θ} (μ̂_{t} + … μ̂_{t+j−1}))
− E_{t−1} ∑_{j=0}^{N−1} ω_{2,j} (\frac{Û_{c,t+j}}{U_{c}} + λ̂_{H,t+j} − p̂_{t+j} + \frac{θ}{1 − θ} (μ̂_{t} + … μ̂_{t+j−1}))  (2.30)

p̂_{F,t−1} = E_{t−1} ∑_{j=0}^{N−1} ω_{1,j} (\frac{Û_{c,t+j}^{∗}}{U_{c}} + λ̂_{F,t+j} + m̂c_{j+1,t+j}^{∗} + \frac{1}{1 − θ} (μ̂_{t} + … μ̂_{t+j−1}))
− E_{t−1} ∑_{j=0}^{N−1} ω_{2,j} (\frac{Û_{c,t+j}^{∗}}{U_{c}} + λ̂_{F,t+j} − p̂_{t+j} − q̂_{t+j} + \frac{θ}{1 − θ} (μ̂_{t} + … μ̂_{t+j−1}))(2.31)

p̂_{F,t−1}^{∗} = E_{t−1} ∑_{j=0}^{N−1} ω_{1,j} (\frac{Û_{c,t+j}^{∗}}{U_{c}} + λ̂_{F,t+j}^{∗} + m̂c_{j+1,t+j}^{∗} + \frac{1}{1 − θ} (μ̂_{t}^{∗} + … μ̂_{t+j−1}^{∗}))
− E_{t−1} ∑_{j=0}^{N−1} ω_{2,j} (\frac{Û_{c,t+j}^{∗}}{U_{c}} + λ̂_{F,t+j}^{∗} − p̂_{t+j}^{∗} + \frac{θ}{1 − θ} (μ̂_{t}^{∗} + … μ̂_{t+j−1}^{∗}))  (2.32)

p̂_{H,t−1}^{∗} = E_{t−1} ∑_{j=0}^{N−1} ω_{1,j} (\frac{Û_{c,t+j}}{U_{c}} + λ̂_{H,t+j}^{∗} + m̂c_{j+1,t+j} + \frac{1}{1 − θ} (μ̂_{t}^{∗} + … μ̂_{t+j−1}^{∗}))

− E_{t−1} ∑_{j=0}^{N−1} ω_{2,j} (\frac{Û_{c,t+j}}{U_{c}} + λ̂_{H,t+j}^{∗} − p̂_{t+j}^{∗} + q̂_{t+j} + \frac{θ}{1 − θ} (μ̂_{t}^{∗} + … μ̂_{t+j−1}^{∗}))(2.33)

where ω_{1,j} = (βμ^{\frac{1}{1−θ}})^{j}mc(j + 1)/ ∑(βμ^{\frac{1}{1−θ}})^{j}mc(j + 1) and ω_{2,j} = (βμ^{\frac{θ}{1−θ}})^{j}/ ∑(βμ^{\frac{1}{1−θ}})^{j} and Û_{c,t} is shorthand for the log-linearized marginal utility. Note that in the case with zero-inflation, the linearized pricing equations simplify to:

p̂_{H,t−1} = E_{t−1} ∑_{j=0}^{N−1} β^{j} (p̂_{t+j} + m̂c_{j+1,t+j} + μ̂_{t} + … μ̂_{t+j−1}) / ∑_{j=0}^{N−1} β^{j}
p̂_{F,t−1} = E_{t−1} ∑_{j=0}^{N−1} β^{j} (p̂_{t+j} + m̂c_{j+1,t+j}^{∗} + q̂_{t+j} + μ̂_{t} + … μ̂_{t+j−1}) / ∑_{j=0}^{N−1} β^{j}

p̂_{F,t−1}^{∗} = E_{t−1} ∑_{j=0}^{N−1} β^{j} (p̂_{t+j}^{∗} + m̂c_{j+1,t+j}^{∗} + μ̂_{t}^{∗} + … μ̂_{t+j−1}^{∗}) / ∑_{j=0}^{N−1} β^{j}

p̂_{H,t−1}^{∗} = E_{t−1} ∑_{j=0}^{N−1} β^{j} (p̂_{t+j}^{∗} + m̂c_{j+1,t+j} − q̂_{t+j} + μ̂_{t}^{∗} + … μ̂_{t+j−1}^{∗}) / ∑_{j=0}^{N−1} β^{j}.

The remaining equations for the monopolists are as follows:

y_{H}(i) ŷ_{H,i,t} + y_{H}^{∗}(i) ŷ_{H,i,t}^{∗} = F_{k}(i)k(i − 1) k̂_{i−1,t−1} + F_{l}(i)l(i) l̂_{i,t}  (2.34)
k̂_{i,t} = k̂_{i−1,t−1} + (ϕ′(i) − 1)x(i)/k(i) [k̂_{i−1,t−1} − x̂_{i,t}] .  (2.35)

0 = E_{t}\{\frac{U_{cc}c}{U_{c}}(ĉ_{t+1} − ĉ_{t}) + \frac{U_{cl}l}{U_{c}} (l̂_{t+1} − l̂_{t})

+ \frac{U_{cm}M/P}{U_{c}} (μ̂_{t+1} − p̂_{t+1} − μ̂_{t} + p̂_{t})

+ β(1 − ϕ′(i))mc(i + 1)F_{k}(i + 1)[ŵ_{t+1} + (\frac{F_{kk}(i + 1)}{F_{k}(i + 1)} − \frac{F_{kl}(i + 1)}{F_{l}(i + 1)}) k(i) k̂_{i,t}

+ (\frac{F_{kl}(i + 1)}{F_{k}(i + 1)} − \frac{F_{ll}(i + 1)}{F_{l}(i + 1)}) l(i + 1) l̂_{i+1,t+1}

− \frac{ϕ′′(i)}{1 − ϕ′(i)} \frac{x(i)}{k(i − 1)}(x̂_{i,t} − k̂_{i−1,t−1})]

+ β (\frac{1 − ϕ′(i)}{1 − ϕ′(i + 1)}) (1 − δ − ϕ(i + 1) + ϕ′(i + 1)\frac{x(i + 1)}{k(i)})

[\frac{−ϕ′′(i)}{1 − ϕ′(i)} \frac{x(i)}{k(i − 1)}(x̂_{i,t} − k̂_{i−1,t−1})

+ \frac{ϕ′′(i + 1)}{1 − ϕ′(i + 1)} \frac{x(i + 1)}{k(i)}(x̂_{i+1,t+1} − k̂_{i,t})]

+ β \frac{1 − ϕ′(i)}{1 − ϕ′(i + 1)}ϕ′′(i + 1) (\frac{x(i + 1)}{k(i)})^{2} (x̂_{i+1,t+1} − k̂_{i,t})\}

= E_{t}\{\frac{U_{cc}c}{U_{c}}(ĉ_{t+1} − ĉ_{t}) + \frac{U_{cl}l}{U_{c}} (l̂_{t+1} − l̂_{t})

+ \frac{U_{cm}M/P}{U_{c}} (μ̂_{t+1} − p̂_{t+1} − μ̂_{t} + p̂_{t})

+ β(1 − ϕ′(i))mc(i + 1)F_{k}(i + 1)[ŵ_{t+1} + (\frac{F_{kk}(i + 1)}{F_{k}(i + 1)} − \frac{F_{kl}(i + 1)}{F_{l}(i + 1)}) k(i) k̂_{i,t}

+ (\frac{F_{kl}(i + 1)}{F_{k}(i + 1)} − \frac{F_{ll}(i + 1)}{F_{l}(i + 1)}) l(i + 1) l̂_{i+1,t+1}]

− \frac{ϕ′′(i)}{1 − ϕ′(i)} \frac{x(i)}{k(i − 1)}(x̂_{i,t} − k̂_{i−1,t−1})

+ (1 − β(1 − ϕ′(i))mc(i + 1)F_{k}(i + 1))

[\frac{ϕ′′(i + 1)}{1 − ϕ′(i + 1)} \frac{x(i + 1)}{k(i)}(x̂_{i+1,t+1} − k̂_{i,t})]

+ β \frac{1 − ϕ′(i)}{1 − ϕ′(i + 1)}ϕ′′(i + 1) (\frac{x(i + 1)}{k(i)})^{2} (x̂_{i+1,t+1} − k̂_{i,t})\}  (2.36)
m̂c_{i,t} = ŵ_{t} − F_{kl}(i)k(i − 1)/F_{l}(i)k̂_{i−1,t−1} − F_{ll}(i)l(i)/F_{l}(i)l̂_{i,t}  (2.37)

These are linearizations of (1.22), (1.23), (1.35), and (1.36). Note that in deriving (2.36) we use the steady state Euler equation for capital to simplify terms.

Finally we need the labor market clearing condition and the resource constraint:

l̂_{t} = (l(1) l̂_{1,t} + l(2) l̂_{2,t} + … l(N) l̂_{N,t})/ ∑_{i} l(i)  (2.38)
ĉ_{t} = (y ŷ_{t} − [x(1) x̂_{1,t} + … x(N) x̂_{N,t}]/N)/c  (2.39)

which are linearizations of (1.41) and (1.40).

2.1.4. Solving the Linearized System in the Benchmark Economy The system of equations that we solve has 2N + 6 dynamic equations:

• 4 pricing equations, (2.30)-(2.33);

• 2N Euler equations for capital ((2.36) for home and similar for foreign);

• 2 money demand equations ((2.21) for home and similar for foreign);

• and static equations and definitions that determine:

∘ ŷ_{H,i}, y_{F,i}, y_{F,i}^{∗}, y_{H,i}^{∗} from (2.10) and analogues; ∘ p̂, p̂^{∗} from (2.14)-(2.15); ∘ ˆ p̄_{H}, ˆ p̄_{F}, ˆ p̄_{F}^{∗}, ˆ p̄_{H}^{∗} from (2.16)-(2.19) ∘ ŵ, ŵ^{∗} from (2.20) and foreign analogue; ∘ μ̂ − p̂, μ̂^{∗} − p̂^{∗} for money demands; ∘ q̂ from (2.24); ∘ λ̂_{H}, λ̂_{F}, λ̂_{F}^{∗}, λ̂_{H}^{∗} from (2.27) and analogues. ∘ l̂_{i}, l̂_{i}^{∗} from (2.34) and foreign analogue; ∘ x̂_{i}, x̂_{i}^{∗} from (2.35) and foreign analogue; ∘ m̂c_{i}, m̂c_{i}^{∗} from (2.37) and foreign analogue; ∘ l̂, l̂^{∗} from (2.38) and foreign analogue; ∘ ĉ, ĉ^{∗} from (2.39) and foreign analogue;

We can write the system of equations in terms of a subset of our variables and back out all variables via the static conditions listed above. We turn to this next.

We will use the following vectors in our computation:

z_{t} = [p̂_{H,t−1}, p̂_{F,t−1}, p̂_{F,t−1}^{∗}, p̂_{H,t−1}^{∗}, k̂_{1,t}, … k̂_{N,t}, k̂_{1,t}^{∗}, … k̂_{N,t}^{∗}, ŷ_{t}, ŷ_{t}^{∗}]′  (n_{z} × 1)
X_{t} = [p̂_{H,t−2}, …, p̂_{H,t−N}, p̂_{F,t−2}, …, p̂_{F,t−N}, p̂_{F,t−2}^{∗}, …, p̂_{F,t−N}^{∗}, p̂_{H,t−2}^{∗}, …, p̂_{H,t−N}^{∗},
k̂_{1,t−1}, …, k̂_{N,t−1}, k̂_{1,t−1}^{∗}, …, k̂_{Nt−1}^{∗}] (n_{X} × 1)
Z_{t} = [z_{t+N−1}, z_{t+N−2}, ...z_{t}, X_{t}, μ̂_{t+N−1}, …, μ̂_{t−N+1}, μ̂_{t+N−1}^{∗}, …, μ̂_{t−N+1}^{∗}]
Z_{t} = [z_{t}, z_{t−1}, …, z_{t−N+2}]′  (n_{Z} × 1)
S_{t} = [μ̂_{t}, …, μ̂_{t−N+1}, μ̂_{t}^{∗}, …, μ̂_{t−N+1}^{∗}]′  (n_{S} × 1)

The vector z_{t} contains the choice variables at time t. It has n_{z} = 2N + 6 elements. The vector X_{t} are the state variables at time t. There are n_{X} = 6N − 4 state variables. The vector Z_{t} contains all variables that appear in the residual equations. The vectors Z_{t} and S_{t} are used when we characterize the solution since it will take the form

Z_{t} = AZ_{t−1} + BS_{t}  (2.40)

where Z has n_{Z} = (N − 1)n_{z} elements and S has n_{S} = 2N elements.

The residual equations can be written succinctly as follows:

E [A_{1} [ \begin{matrix}X_{t+1} \\ Z_{t+N−1} \end{matrix} ] + A_{2} [ \begin{matrix}X_{t} \\ Z_{t+N−2} \end{matrix} ] + shock terms|Ω_{t}] = 0

where E implies that expectations are taken – but we will assume that different information sets for the different residual equations. For our example, the residuals are denoted R(Z) and the matrix A_{1} is given by

A_{1} = [ \begin{matrix}I_{n_{X},n_{X}} & 0_{n_{X},n_{z}} & 0_{n_{X},n_{z}} & … & 0_{n_{X},n_{z}} \\ 0_{n_{z},n_{X}} & \frac{dR}{dZ} (:, 1 : n_{z}) & \frac{dR}{dZ} (:, n_{z} +1 : 2n_{z}) & … & \frac{dR}{dZ} (:, (N −2)n_{z} +1 : (N −1)n_{z}) \\ 0_{n_{z},n_{X}} & 0_{n_{z},n_{z}} & I_{n_{z},n_{z}} & … & 0_{n_{z},n_{z}} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0_{n_{z},n_{X}} & 0_{n_{z},n_{z}} & 0_{n_{z},n_{z}} & … & I_{n_{z},n_{z}} \end{matrix} ]

(2.41)

and matrix A_{2} is given by:

A_{2} = [ \begin{matrix}−I_{1} & 0_{n_{X},n_{z}} & … & 0_{n_{X},n_{z}} & −I_{2} \\ \frac{dR}{dZ} (:, Nn_{z} +1 : Nn_{z} +n_{X}) & 0_{nz,nz} & … & 0_{n_{z},n_{z}} & \frac{dR}{dZ} (:, (N −1)n_{z} +1 : Nn_{z}) \\ 0_{n_{z},n_{X}} & I_{n_{z},n_{z}} & 0_{n_{z},n_{z}} & … & 0_{n_{z},n_{z}} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0_{n_{z},n_{X}} & 0_{n_{z},n_{z}} & … & I_{n_{z},n_{z}} & 0_{n_{z},n_{z}} \end{matrix} ] .

(2.42)

The matrices I_{1} and I_{2} in A_{2} are given by

I_{1} = [ \begin{matrix}I_{4,4} ⊗ [ \begin{matrix}0 & … & 0 & 0 \\ 1 & … & 0 & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ 0 & … & 1 & 0 \end{matrix} ] , & 0_{4N−4,2N} \\ 0_{2N,4N−4} & 0_{2N,2N} \end{matrix} ]

I_{2} = [ \begin{matrix}I_{4,4} ⊗ [ \begin{matrix}1 \\ 0_{N−2,1} \end{matrix} ] , & 0_{4N−4,2N+2} \\ 0_{2N,4} & [I_{2N,2N}, 0_{2N,2}] \end{matrix} ]

Using the method laid out in Blanchard and Kahn (1980), we construct eigenvalues of −A_{1}^{−1}A_{2} if A_{1} is invertible and generalized eigenvalues otherwise. Then, ignoring shock terms, we have

[ \begin{matrix}X_{t+1} \\ Z_{t+N−1} \end{matrix} ] = V ΛV^{−1} [ \begin{matrix}X_{t} \\ Z_{t+N−2} \end{matrix} ] .

We can sort eigenvalues inside and outside the unit circle. If there are n_{X} stable eigenvalues (which is the number of state variables in X), then we have a locally determinate system. Suppose that the eigenvectors in V and eigenvalues in Λ are sorted so that the upper left partition of Λ contains the stable eigenvalues. Then,

X_{t+1} = V_{11}Λ_{1}V_{11}^{−1} X_{t}
Z_{t+N−2} = V_{21}V_{11}^{−1} X_{t}.

The last n_{z} elements of Z_{t+N−2} are those of z_{t}. Therefore, we have a relationship between our decision variables z and the state variables X. If we want to write the system as (2.40), then we can use this relationship between z and X to fill in the elements of A. In particular, we set

A(1 : n_{z}, 1 : n_{z} − 2) = A_{zX}(:, [1, N, 2N − 1, 3N − 2, 4N − 3 : 6N − 4])
A(1 : n_{z}, n_{z} + 1 : n_{z} : n_{Z}) = A_{zX}(:, 2 : N − 1)
A(1 : n_{z}, n_{z} + 2 : n_{z} : n_{Z}) = A_{zX}(:, N + 1 : 2N − 2)
A(1 : n_{z}, n_{z} + 3 : n_{z} : n_{Z}) = A_{zX}(:, 2N : 3N − 3)
A(1 : n_{z}, n_{z} + 4 : n_{z} : n_{Z}) = A_{zX}(:, 3N − 1 : 4N − 4)
A(n_{z} + 1 : n_{Z}, 1 : n_{Z} − n_{z}) = I_{n_{Z}−n_{z},n_{Z}−n_{z}}

where A_{zX} comes from z_{t} = A_{zX}X_{t}.

The next step is to compute B. In this case, we use undetermined coefficients. This method boils down to solving a system of equations in the elements of B_{1} where B_{1} is the first partition of B which has dimension n_{z} × n_{S}, i.e.,

B = [ B_{1} ; 0_{n_{z},n_{S}} ; ⋮ ; 0_{n_{z},n_{S}} ] = [ I_{n_{z},n_{z}} ; 0_{n_{z},n_{z}} ; ⋮ ; 0_{n_{z},n_{z}} ] B_{1} ≡ SB_{1}.

We will use S below in order to reduce the problem of computing B to one of computing B_{1}.

The system of equations for B_{1} turns out to be linear so it is a simple computational problem. The only tricky part of the problem is getting the specification of the equations correct since expectations of intermediate goods producers and the other agents in the economy depend on different information sets.

To derive expressions for the elements of B, we first note that the residuals can be written as follows:

E[a_{0}Z_{t+N−1} + a_{1}Z_{t+N−2} + … + a_{N−1}Z_{t} + a_{N}Z_{t−1}
+ b_{0}S_{t+N−1} + b_{1}S_{t+N−2}.... + b_{N−1}S_{t}|Ω_{t}] = 0

We didn’t originally write them this way because we wanted to avoid lots of redundancies when computing the eigenvalues described above. However, here it is convenient to write it this way to show how we derive B. Using the definitions of Z and Z, we can write:

a_{0} = dR/dZ(:, 1 : (N − 1)n_{z})
a_{N−1}(:, 1 : n_{z}) = dR/dZ(:, (N − 1)n_{z} + 1 : Nn_{z})
b_{0}(:, 1 : N) = dR/dZ(:, Nn_{z} + n_{X} + 1 : Nn_{z} + n_{X} + N)
b_{0}(:, N + 1 : 2N) = dR/dZ(:, Nn_{z} + n_{X} + 2N : Nn_{z} + n_{X} + 3N − 1)
b_{N−1}(:, 2 : N) = dR/dZ(:, Nn_{z} + n_{X} + N + 1 : Nn_{z} + n_{X} + 2N − 1)
b_{N−1}(:, N + 2 : 2N) = dR/dZ(:, Nn_{z} + n_{X} + 3N : Nn_{z} + n_{X} + 4N − 2) (2.43)

with all other coefficients but a_{N} set equal to 0. The matrix a_{N} is nonzero but it is not used in computing B.

Using the solution in (2.40) we get:

E[a_{0} (A^{N}Z_{t−1} + BS_{t+N−1} + ABS_{t+N−2} + … + A^{N−1}BS_{t})

+ a_{1} (A^{N−1}Z_{t−1} + BS_{t+N−2} + ABS_{t+N−3} + … + A^{N−2}BS_{t}) + …
+ a_{N−1} (AZ_{t−1} + BS_{t}) + a_{N}Z_{t−1}
+ b_{0}S_{t+N−1} + b_{1}S_{t+N−2}.... + b_{N−1}S_{t}|Ω_{t}] = 0  (2.44)

We next derive expressions for E[MS_{t+j}|Ω_{t}] as a function of S_{t}, where M is assumed to be one of the coefficients in (2.44). First, using the fact that S_{t+1} = PS_{t} + ϵ_{t+1} we have

E[MS_{t+j}|Ω_{t}] = MP^{j}E[S_{t}|Ω_{t}].

For example, in our model,

P = [ \begin{matrix}[ \begin{matrix}ρ_{μ} & 0 & … & 0 & 0 \\ 1 & 0 & … & 0 & 0 \\ 0 & 1 & … & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & 1 & 0 \end{matrix} ] & 0_{N,N} \\ 0_{N,N} & [ \begin{matrix}ρ_{μ} & 0 & … & 0 & 0 \\ 1 & 0 & … & 0 & 0 \\ 0 & 1 & … & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & 1 & 0 \end{matrix} ] \end{matrix} ] .  (2.45)

Next, define M̂ as follows:

E[MS_{t}|Ω_{t}] = M̂S_{t}.

If Ω_{t} = {μ̂_{s}, μ̂_{s}^{∗}}_{s=0}^{t−1} (as is the case for the pricing equations in our model) and P is given by (2.45) then

E[M_{1}μ̂_{t} + … + M_{N} μ̂_{t−N+1} + M_{N+1}μ̂_{t}^{∗} + … + M_{2N}μ̂_{t−N+1}^{∗}|Ω_{t}]

= [0, M_{1}ρ_{μ} + M_{2}, M_{3}, … M_{N}, 0, M_{N+1}ρ_{μ} + M_{N+2}, M_{N+3}, … M_{2N}] S_{t}. If Ω_{t} = {μ̂_{s}, μ̂_{s}^{∗}}_{s=0}^{t} (as is the case for the capital Euler equations and the money demand equations), then M̂ = M. For our example, we have

E[((a_{0}B + b_{0})P^{N−1} + (a_{0}AB + a_{1}B + b_{1})P^{N−2} + ....+

(a_{0}A^{N−1}B + a_{1}A^{N−2}B + … a_{N−1}B + b_{N−1})P^{0})S_{t}|Ω_{t}] = E[MS_{t}|Ω_{t}] = M̂S_{t} where M and M̂ both have dimension n_{z} × n_{S}. Applying the method of undetermined coefficients, we want to find B_{1} such that every element of M̂ is equal to 0. Because of the timing of the pricing decisions, this will imply n_{z} × n_{S} − 8 equations in n_{z} × n_{S} − 8 unknowns. In other words, the coefficients on μ̂_{t} and μ̂_{t}^{∗} in the first four rows of B_{1} will be set equal to 0 because prices cannot respond immediately to the monetary shocks.

The following steps are taken to set up the system of equations. First, we stack the nonzero elements of M̂ in a vector. Second, we construct a matrix D that relates this vector to vec(M′). In our case, this relation is:

[ M_{1,1}ρ_{μ} + M_{1,2} ; M_{1,3} ; ⋮ ; M_{1,N} ; M_{1,N+1}ρ_{μ} + M_{1,N+2} ; M_{1,N+3} ; ⋮ ; M_{1,n_{S}} ; ⋮ ; M_{4,1}ρ_{μ} + M_{4,2} ; M_{4,3} ; ⋮ ; M_{4,N} ; M_{4,N+1}ρ_{μ} + M_{4,N+2} ; M_{4,N+3} ; ⋮ ; M_{4,n_{S}} ; M_{5,1} ; M_{5,2} ; ⋮ ; M_{n_{z},n_{S}} ] = \underbrace{[ \begin{matrix}I_{8,8} ⊗ Ψ & 0_{8N−8,n_{z}n_{S}−8N} \\ 0_{n_{z}n_{S}−8N,8N} & I_{n_{z}n_{S}−8N} \end{matrix} ]}_{D} \underbrace{[ M_{1,1} ; M_{1,2} ; M_{1,3} ; ⋮ ; M_{1,n_{S}} ; M_{2,1} ; M_{2,2} ; ⋮ ; M_{n_{Z},n_{S}} ]}_{vec(M′)}  (2.46)

where

Ψ = [ \begin{matrix}ρ_{μ} & 1 & 0 & 0 & … & 0 \\ 0 & 0 & 1 & 0 & … & 0 \\ 0 & 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & 0 & … & 1 \end{matrix} ] (N − 1 × N).

Third, we set Dvec(M′) equal to zero (which ensures that M̂ = 0),

Dvec(M′) = Dvec([a_{0}SB_{1}P^{N−1}]′ + [a_{0}ASB_{1}P^{N−2} + a_{1}SB_{1}P^{N−2}]′ + … +
[a_{0}A^{N−1}SB_{1}P^{0} + a_{1}A^{N−2}SB_{1}P^{0} + … + a_{N−1}SB_{1}P^{0}]′)
+ Dvec ([b_{0}P^{N−1} + b_{1}P^{N−2} + … + b_{N−1}P^{0}]′)
≡ Qvec(B′_{1}) + R.

To construct Q we need to use the fact that vec(ABC) is equal to [C′ ⊗ A]vec(B). At this point, we can write the equation explicitly in terms of B_{1} – or more precisely, the nonzero elements of B_{1}:

vec(B1′)(nonzero elements) = − [Q(:, nonzero elements)]^{−1} R.

For our example, the zero elements of B_{1} are: (i,1) and (i,N+1) for i =1, 2, 3, and 4. These are the coefficients on contemporaneous shocks in the pricing decision rules. All other elements are assumed to be nonzero.

2.2. The Taylor-Rule Extension

We now consider our first extension to the benchmark economy: the Fed follows (1.46) and money is determined residually from the money demand equation (1.15). The main differences in the computation from the benchmark economy are these:

• we add outputs and interest rates to the state vector;

• we add the Taylor Rules as residuals;

• we add r and r^{∗} to our choice variables;

• we define M/P via the money demand equation rather than as μ_{t}/p_{t};

• μ is constant.

2.2.1. Normalization for the Taylor-Rule Extension In the Taylor-Rule case, we assume that prices grow at the rate μ. To keep everything consistent with the equations derived above, we use the following normalization for prices:

\begin{matrix}p(s^{t}) \\ p_{I}(i, s^{t−1}) \\ p̄_{I}(s^{t−1}) \\ p^{∗}(s^{t}) \\ p_{I}^{∗}(i, s^{t−1}) \\ p̄_{I}^{∗}(s^{t−1}) \end{matrix} \begin{matrix}= P (s^{t})/μ^{t−1} \\ = P_{I}(i, s^{t−1})/μ^{t−i} \\ = P̄_{I}(s^{t−1})/μ^{t−1} \\ = P^{∗}(s^{t})/μ^{t−1} \\ = P_{I}^{∗}(i, s^{t−1})/μ^{t−i} \\ = P̄_{I}^{∗}(s^{t−1})/μ^{t−1} \end{matrix}

where I = H or F. In the equations of the benchmark economy, we replace all state dependent μ’s with a constant μ.

2.2.2. Steady State for the Taylor-Rule Extension The steady state is the same as in the benchmark economy.

2.2.3. Linearized Equations for the Taylor-Rule Extension We add the following equation to the set of linearized equations derived above:

r_{t} = a′ [ \begin{matrix}r_{t−1} \\ r_{t−2} \\ r_{t−3} \\ E_{t}p̂_{t+1} − p̂_{t} \\ p̂_{t} − p̂_{t−1} \\ p̂_{t−1} − p̂_{t−2} \\ p̂_{t−2} − p̂_{t−3} \\ ĝdp & t \\ ĝdp & t−1 \\ ĝdp & t−2 \end{matrix} ] + ϵ_{r,t}  (2.47)

and its foreign analogue, where

ĝdp_{t} = ŷ_{t} + ∑_{i} y_{H}^{∗}(i) (ŷ_{H,i,t}^{∗} − ŷ_{F,i,t})/(Ny).

Again, note that the interest rate is in levels while all other variables are logged. We back out money (also normalized by μ^{t}) from the money demand equation (1.15):

m̂_{t} − p̂_{t} = [(\frac{U_{cm}}{U_{m}} − \frac{U_{cc}}{U_{c}} ) c ĉ_{t} + (\frac{U_{lm}}{U_{m}} − \frac{U_{cl}}{U_{c}} ) l l̂_{t}
− \frac{β}{r} (r_{t} − r)]/ [(\frac{U_{mm}}{U_{m}} − \frac{U_{cm}}{U_{c}} ) M/P ]  (2.48)

where m̂_{t} = log(M(s^{t})/μ^{t}).

2.2.4. Solving the Linearized System for the Taylor-Rule Extension The system of equations that we solve has 2N + 8 dynamic equations:

• 2 Taylor rules ((2.47) for home and similar for foreign)

• 4 pricing equations, (2.30)-(2.33)

• 2N Euler equations for capital ((2.36) for home and similar for foreign)

• 2 equations for interest rates ((2.22) for home and similar for foreign)

• and static equations that are the same as the benchmark economy, except for real

balances which are now

∘ m̂ − p̂, m̂^{∗} − p̂^{∗} from (2.48) and foreign analogue; As before, we can write the system of equations in terms of a subset of our variables and back out all variables via the static first-order conditions.

When computing the Taylor-rule extension, we use the following vectors:

z_{t} = [p̂_{H,t−1}, p̂_{F,t−1}, p̂_{F,t−1}^{∗}, p̂_{H,t−1}^{∗}, k̂_{1,t}, … k̂_{N,t}, k̂_{1,t}^{∗}, … k̂_{N,t}^{∗}, ŷ_{t}, ŷ_{t}^{∗}, r_{t}, r_{t}^{∗}]′ (n_{z} × 1)
X_{t} = [p̂_{H,t−2}, …, p̂_{H,t−(N+3)}, p̂_{F,t−2}, …, p̂_{F,t−(N+3)}, p̂_{F,t−2}^{∗}, …, p̂_{F,t−(N+3)}^{∗},
p̂_{H,t−2}^{∗}, …, p̂_{H,t−(N+3)}^{∗}, k̂_{1,t−1}, …, k̂_{N,t−1}, k̂_{1,t−1}^{∗}, …, k̂_{Nt−1}^{∗},
ŷ_{t−1}, ŷ_{t−2}, ŷ_{t−1}^{∗}, ŷ_{t−2}^{∗}, r_{t−1}, r_{t−2}, r_{t−3}, r_{t−1}^{∗}, r_{t−2}^{∗}, r_{t−3}^{∗}]′ (n_{X} × 1)
Z_{t} = [z_{t+N−1}, z_{t+N−2}, ...z_{t}, X_{t}, ϵ_{r,t}, ϵ_{r,t}^{∗}]′

Z_{t} = [z_{t}, z_{t−1}, …, z_{t−N−1}]′ (n_{Z} × 1)
S_{t} = [ϵ_{r,t}, ϵ_{r,t−1}, ϵ_{r,t}^{∗}, ϵ_{r,t−1}^{∗}]′ (n_{S} × 1)

The vector z_{t} contains the choice variables at time t. It has n_{z} = 2N + 8 elements. The vector X_{t} are the state variables at time t. There are n_{X} = 6N + 18 state variables. The vector Z_{t} contains all variables that appear in the residual equations. The vectors Z_{t} and S_{t} are used when we characterize the solution, Z_{t} = AZ_{t−1} + BS_{t} where Z has n_{Z} = (N + 2)n_{z} elements and S has n_{S} = 4 elements.

The residual equations can be written as follows:

E [A_{1} [ X_{t+1} ; z_{t+N−1} ; ⋮ ; z_{t+1} ] + A_{2} [ X_{t} ; z_{t+N−2} ; ⋮ ; z_{t} ] + shock terms|Ω_{t}] = 0.

For our example, the residuals are denoted R(Z) and the matrix A_{1} is given by

A_{1} = [ \begin{matrix}I_{n_{X},n_{X}} & 0_{n_{X},n_{z}} & 0_{n_{X},n_{z}} & … & 0_{n_{X},n_{z}} \\ 0_{n_{z},n_{X}} & \frac{dR}{dZ} (:, 1 : n_{z}) & \frac{dR}{dZ} (:, n_{z} +1 : 2n_{z}) & … & \frac{dR}{dZ} (:, (N −2)n_{z} +1 : (N −1)n_{z}) \\ 0_{n_{z},n_{X}} & 0_{n_{z},n_{z}} & I_{n_{z},n_{z}} & … & 0_{n_{z},n_{z}} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0_{n_{z},n_{X}} & 0_{n_{z},n_{z}} & 0_{n_{z},n_{z}} & … & I_{n_{z},n_{z}} \end{matrix} ]

and matrix A_{2} is given by:

A_{2} = [ \begin{matrix}−I_{1} & 0_{n_{X},n_{z}} & … & 0_{n_{X},n_{z}} & −I_{2} \\ \frac{dR}{dZ} (:, Nn_{z} +1 : Nn_{z} +n_{X}) & 0_{nz,nz} & … & 0_{n_{z},n_{z}} & \frac{dR}{dZ} (:, (N −1)n_{z} +1 : Nn_{z}) \\ 0_{n_{z},n_{X}} & I_{n_{z},n_{z}} & 0_{n_{z},n_{z}} & … & 0_{n_{z},n_{z}} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0_{n_{z},n_{X}} & 0_{n_{z},n_{z}} & … & I_{n_{z},n_{z}} & 0_{n_{z},n_{z}} \end{matrix} ] .

The matrices I_{1} and I_{2} in A_{2} are given by

I_{1} = [ \begin{matrix}I_{4,4} ⊗ [ \begin{matrix}0 & … & 0 & 0 \\ 1 & … & 0 & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ 0 & … & 1 & 0 \end{matrix} ] , & 0_{4N+8,2N} & 0_{4N+8,4} & 0_{4N+8,6} \\ 0_{2N,4N+8} & 0_{2N,2N} & 0_{2N,4} & 0_{2N,6} \\ 0_{4,4N+8} & 0_{4,2N} & I_{2,2} ⊗ [ \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ] & 0_{4,6} \\ 0_{6,4N+8} & 0_{6,2N} & 0_{6,4} & I_{2,2} ⊗ [ \begin{matrix}0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ] \end{matrix} ]

I_{2} = [ \begin{matrix}I_{4,4} ⊗ [ \begin{matrix}1 \\ 0_{N+1,1} \end{matrix} ] , & 0_{4N+8,2N} & 0_{4N+8,2} & 0_{4N+8,2} \\ 0_{2N,4} & I_{2N,2N} & 0_{2N,2} & 0_{2N,2} \\ 0_{4,4} & 0_{4,2N} & I_{2,2} ⊗ [ \begin{matrix}1 \\ 0 \end{matrix} ] & 0_{4,2} \\ 0_{6,4} & 0_{6,2N} & 0_{6,2} & I_{2,2} ⊗ [ \begin{matrix}1 \\ 0 \\ 0 \end{matrix} ] \end{matrix} ]

Again, using the method laid out in Blanchard and Kahn (1980), we construct eigenvalues of −A_{1}^{−1}A_{2} if A_{1} is invertible and generalized eigenvalues otherwise. Then, ignoring shock terms, we have

[ X_{t+1} ; z_{t+N−1} ; ⋮ ; z_{t+1} ] = V ΛV^{−1} [ X_{t} ; z_{t+N−2} ; ⋮ ; z_{t} ] .

We can sort eigenvalues inside and outside the unit circle. If there are n_{X} stable eigenvalues (which is the number of state variables in X), then we have a locally determinate system. Suppose that the eigenvectors in V and eigenvalues in Λ are sorted so that the upper left partition of Λ contains the stable eigenvalues. Then,

X_{t+1} = V_{11}Λ_{1}V_{11}^{−1} X_{t}
[ z_{t+N−2} ; ⋮ ; z_{t} ] = V_{21}V_{11}^{−1} X_{t}.

Now we have a relationship between our decision variables z and the state variables X. If we want to write the system as (2.40), then we can use this relationship between z and X to fill in the elements of A. In particular, we set

A(1 : n_{z}, 1 : n_{z}) = A_{zX}(:, [1, N + 3, 2N + 5, 3N + 7, 4N + 9 : 6N + 9, ...

6N + 11, 6N + 13, 6N + 16])

A(1 : n_{z}, n_{z} + 1 : n_{z} : n_{Z}) = A_{zX}(:, 2 : N + 2) A(1 : n_{z}, n_{z} + 2 : n_{z} : n_{Z}) = A_{zX}(:, N + 4 : 2N + 4) A(1 : n_{z}, n_{z} + 3 : n_{z} : n_{Z}) = A_{zX}(:, 2N + 6 : 3N + 6) A(1 : n_{z}, n_{z} + 4 : n_{z} : n_{Z}) = A_{zX}(:, 3N + 8 : 4N + 8) A(1 : n_{z}, 2n_{z} − 3) = A_{zX}(:, 6N + 10) A(1 : n_{z}, 2n_{z} − 2) = A_{zX}(:, 6N + 12) A(1 : n_{z}, 2n_{z} − 1) = A_{zX}(:, 6N + 14) A(1 : n_{z}, 2n_{z}) = A_{zX}(:, 6N + 17) A(1 : n_{z}, 3n_{z} − 1) = A_{zX}(:, 6N + 15) A(1 : n_{z}, 3n_{z}) = A_{zX}(:, 6N + 18) A(n_{z} + 1 : n_{Z}, 1 : n_{Z} − n_{z}) = I_{n_{Z}−n_{z},n_{Z}−n_{z}}

where A_{zX} comes from z_{t} = A_{zX}X_{t}.

The next step is to compute B. In this case, we use undetermined coefficients. This method boils down to solving a system of equations in the elements of B_{1} where B_{1} is the first partition of B which has dimension n_{z} × n_{S}, i.e.,

B = [ B_{1} ; 0_{n_{z},n_{S}} ; ⋮ ; 0_{n_{z},n_{S}} ] = [ I_{n_{z},n_{z}} ; 0_{n_{z},n_{z}} ; ⋮ ; 0_{n_{z},n_{z}} ] B_{1} ≡ SB_{1}.

We will use S below in order to reduce the problem of computing B to one of computing B_{1}.

To derive expressions for the elements of B, we first note that the residuals can be written as follows:

E[a_{0}Z_{t+N−1} + a_{1}Z_{t+N−2} + … + a_{N−1}Z_{t} + a_{N}Z_{t−1}
+ b_{0}S_{t+N−1} + b_{1}S_{t+N−2}.... + b_{N−1}S_{t}|Ω_{t}] = 0.

Using the definitions of Z and Z, we can write:

a_{0}(:, 1 : Nn_{z}) = dR/dZ(:, 1 : Nn_{z})
b_{N−1}(:, 1) = dR/dZ(:, Nn_{z} + n_{X} + 1)
b_{N−1}(:, 3) = dR/dZ(:, Nn_{z} + n_{X} + 2)

with all other coefficients but a_{N} set equal to 0. The matrix a_{N} is nonzero but it is not used in computing B.

Again, using the solution in (2.40) we get:

E[a_{0} (A^{N}Z_{t−1} + BS_{t+N−1} + ABS_{t+N−2} + … + A^{N−1}BS_{t})

+ a_{1} (A^{N−1}Z_{t−1} + BS_{t+N−2} + ABS_{t+N−3} + … + A^{N−2}BS_{t}) + …
+ a_{N−1} (AZ_{t−1} + BS_{t}) + a_{N}Z_{t−1}
+ b_{0}S_{t+N−1} + b_{1}S_{t+N−2}.... + b_{N−1}S_{t}|Ω_{t}] = 0

We next derive expressions for E[MS_{t+j}|Ω_{t}] as a function of S_{t}, where M is assumed to be one of the coefficients in (2.44). First, using the fact that S_{t+1} = PS_{t} + ϵ_{t+1} we have

E[MS_{t+j}|Ω_{t}] = MP^{j}E[S_{t}|Ω_{t}].

For example, if the policy shocks ϵ_{r,t} and ϵ_{r,t}^{∗} are both serially correlated, then

P = [ \begin{matrix}ρ_{ϵ} & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & ρ_{ϵ} & 0 \\ 0 & 0 & 1 & 0 \end{matrix} ]  (2.49)

Next, define M̂ as follows:

E[MS_{t}|Ω_{t}] = M̂S_{t}.

If Ω_{t} = {ϵ_{r,s}, ϵ_{r,s}^{∗}}_{s=0}^{t−1} (as is the case for the pricing equations in our model) and P is given by (2.49) then

E[M_{1}ϵ_{r,t} + M_{2}ϵ_{r,t−1} + M_{3}ϵ_{r,t}^{∗} + M_{4}ϵ_{r,t−1}^{∗}|Ω_{t}]
= [0, M_{1}ρ_{ϵ} + M_{2}, 0, M_{3}ρ_{ϵ} + M_{4}] S_{t}.

If Ω_{t} = {ϵ_{r,s}, ϵ_{r,s}^{∗}}_{s=0}^{t} (as is the case for the capital Euler equations and the money demand equations), then M̂ = M. For our example, we have

E[((a_{0}B + b_{0})P^{N−1} + (a_{0}AB + a_{1}B + b_{1})P^{N−2} + ....+

(a_{0}A^{N−1}B + a_{1}A^{N−2}B + … a_{N−1}B + b_{N−1})P^{0})S_{t}|Ω_{t}] = E[MS_{t}|Ω_{t}] = M̂S_{t}

where M and M̂ both have dimension n_{z} × n_{S}. Applying the method of undetermined coefficients, we want to find B_{1} such that every element of M̂ is equal to 0. Because of the timing of the pricing decisions, this will imply n_{z} × n_{S} − 8 equations in n_{z} × n_{S} − 8 unknowns. In other words, the coefficients on μ_{t} and μ_{t}^{∗} in the first four rows of B_{1} will be set equal to 0 because prices cannot respond immediately to the monetary shocks.

The following steps are taken to set up the system of equations. First, we stack the nonzero elements of M̂ in a vector. Second, we construct a matrix D that relates this vector to vec(M′). In our case, this relation is:

[ M_{1,1}ρ_{ϵ} + M_{1,2} ; M_{1,3}ρ_{ϵ} + M_{1,4} ; M_{2,1}ρ_{ϵ} + M_{2,2} ; M_{2,3}ρ_{ϵ} + M_{2,4} ; ⋮ ; M_{4,1}ρ_{ϵ} + M_{4,2} ; M_{4,3}ρ_{ϵ} + M_{4,4} ; M_{5,1} ; M_{5,2} ; M_{5,3} ; M_{5,4} ; ⋮ ; M_{n_{z},4} ] = \underbrace{[ \begin{matrix}I_{8,8} ⊗ [ρ_{ϵ}, 1] & 0_{8,8N+16} \\ 0_{8N+16,16} & I_{8N+16,8N+16} \end{matrix} ]}_{D} \underbrace{[ M_{1,1} ; M_{1,2} ; M_{1,3} ; M_{1,4} ; ⋮ ; M_{n_{z},4} ]}_{vec(M′)} .

Dvec(M′) = Dvec([a_{0}SB_{1}P^{N−1}]′ + [a_{0}ASB_{1}P^{N−2} + a_{1}SB_{1}P^{N−2}]′ + … +
[a_{0}A^{N−1}SB_{1}P^{0} + a_{1}A^{N−2}SB_{1}P^{0} + … + a_{N−1}SB_{1}P^{0}]′)
+ Dvec ([b_{0}P^{N−1} + b_{1}P^{N−2} + … + b_{N−1}P^{0}]′)
≡ Qvec(B′_{1}) + R.

To construct Q we need to use the fact that vec(ABC) is equal to [C′ ⊗ A]vec(B). At this point, we can write the equation explicitly in terms of B_{1} – or more precisely, the nonzero elements of B_{1}:

vec(B1′)(nonzero elements) = − [Q(:, nonzero elements)]^{−1} R.

For our example, the zero elements of B_{1} are: (i,1) and (i,3) for i =1, 2, 3, and 4. These are the coefficients on contemporaneous shocks in the pricing decision rules. All other elements are assumed to be nonzero.

2.3. The Incomplete-Markets Extension

We turn next to the second extension: asset markets are incomplete. The main differences in the computation from the benchmark economy are these:

• we add foreign bond holdings (D_{−1}^{∗}) as a state variable;

• we add equation relating right hand sides of (1.49) and (1.50) to our residuals;

• we add D^{∗} to our choice variables;

• we use q̂ derived below rather than (2.24).

2.3.1. Normalization for the I-M Extension In the foreign budget constraint (1.48) impose T^{∗}(s^{t}) = M^{∗}(s^{t})−M^{∗}(s^{t−1}) and B(s^{t}) = 0. If we normalize variables as follows:

p^{∗}(s^{t}) = \frac{P^{∗}(s^{t})}{M^{∗}(s^{t−1})}, π^{∗}(s^{t}) = \frac{Π^{∗}(s^{t})}{M^{∗}(s^{t−1})}, d^{∗}(s^{t}) = \frac{D^{∗}(s^{t})}{M(s^{t})}, ε(s^{t}) = \frac{e(s^{t})M^{∗}(s^{t−1})}{M(s^{t−1})}

then the foreign budget constraint is given by:

p^{∗}(s^{t})c^{∗}(s^{t}) + V (s^{t})d^{∗}(s^{t})\frac{μ(s^{t})}{ε(s^{t})} = p^{∗}(s^{t})w^{∗}(s^{t})l^{∗}(s^{t}) + \frac{d^{∗}(s^{t−1})}{ε(s^{t})} + π^{∗}(s^{t}).

Solving for normalized nominal exchange rate ε(s^{t}), we get

ε(s^{t}) = \frac{V (s^{t})d^{∗}(s^{t})μ(s^{t}) − d^{∗}(s^{t−1})}{p^{∗}(s^{t})[w^{∗}(s^{t})l^{∗}(s^{t}) − c^{∗}(s^{t})] + π^{∗}(s^{t})}.  (2.50)

Now substitute out for profits in (2.50). Profits for the ith producer are

Π^{∗}(i, s^{t}) = \frac{P_{F} (i, s^{t−1})}{e(s^{t})}y_{F} (i, s^{t}) + P_{F}^{∗}(i, s^{t−1})y_{F}^{∗} (i, s^{t}) − P^{∗}(s^{t})[w^{∗}(s^{t})l^{∗}(i, s^{t}) + x^{∗}(i, s^{t})].

If we normalize these profits by the money supply we get

\frac{Π^{∗}(i, s^{t})}{M^{∗}(s^{t−1})} = \frac{P_{F} (i, s^{t−1})}{M^{∗}(s^{t−1})} \frac{M^{∗}(s^{t−1})}{ε(s^{t})M(s^{t−1})}y_{F} (i, s^{t}) + \frac{P_{F}^{∗}(i, s^{t−1})}{M^{∗}(s^{t−1})}y_{F}^{∗} (i, s^{t})

− \frac{P^{∗}(s^{t})}{M^{∗}(s^{t−1})}[w^{∗}(s^{t})l^{∗}(i, s^{t}) + x^{∗}(i, s^{t})]

or, using the definitions above,

π^{∗}(i, s^{t}) = \frac{P_{F} (i, s^{t−1})}{M(s^{t−1})} \frac{M(s^{t−i})}{ε(s^{t})M(s^{t−1})}y_{F} (i, s^{t}) + \frac{P_{F}^{∗}(i, s^{t−i})}{M^{∗}(s^{t−1})} \frac{M^{∗}(s^{t−i})}{M^{∗}(s^{t−i})}y_{F}^{∗} (i, s^{t})
− p^{∗}(s^{t})[w^{∗}(s^{t})l^{∗}(i, s^{t}) + x^{∗}(i, s^{t})]

= p_{F} (i, s^{t−1})y_{F} (i, s^{t})\frac{1}{ε(s^{t})} \frac{M(s^{t−i})}{M(s^{t−1})} + p_{F}^{∗} (i, s^{t−1})y_{F}^{∗} (i, s^{t})\frac{M^{∗}(s^{t−i})}{M^{∗}(s^{t−1})}

− p^{∗}(s^{t})[w^{∗}(s^{t})l^{∗}(i, s^{t}) + x^{∗}(i, s^{t})]

= \frac{p_{F} (i, s^{t−1})y_{F} (i, s^{t})}{ε(s^{t})μ(s^{t−1}) ⋯ μ(s^{t−i+1})} + \frac{p_{F}^{∗} (i, s^{t−1})y_{F}^{∗} (i, s^{t})}{μ^{∗}(s^{t−1}) ⋯ μ^{∗}(s^{t−i+1})}

− p^{∗}(s^{t})[w^{∗}(s^{t})l^{∗}(i, s^{t}) + x^{∗}(i, s^{t})] The last equation follows from the fact that M(s^{t−1}) = μ(s^{t−1}) ⋯ μ(s^{t−i+1})M(s^{t−i}). If we integrate across firms, we get

π^{∗}(s^{t}) = \frac{1}{ε(s^{t})} \frac{1}{N} ∑_{i=1}^{N} \frac{p_{F} (i, s^{t−1})y_{F} (i, s^{t})}{μ(s^{t−1}) ⋯ μ(s^{t−i+1})} + \frac{1}{N} ∑_{i=1}^{N} \frac{p_{F}^{∗} (i, s^{t−1})y_{F}^{∗} (i, s^{t})}{μ^{∗}(s^{t−1}) ⋯ μ^{∗}(s^{t−i+1})}
− p^{∗}(s^{t})[w^{∗}(s^{t})l^{∗}(s^{t}) + x^{∗}(s^{t})]  (2.51)

since π^{∗}(s^{t}) = ∑_{i=1}^{N} π^{∗}(i, s^{t})/N, l^{∗} = ∑_{i=1}^{N} l^{∗}(i)/N, and x^{∗} = ∑_{i=1}^{N} x^{∗}(i)/N.

Substitute for (2.51) in (2.50) to get our final equation for the normalized nominal exchange rate:

ε(s^{t}) = \frac{V (s^{t})d^{∗}(s^{t})μ(s^{t}) − d^{∗}(s^{t−1}) − \frac{1}{N} ∑ \frac{p_{F} (i,s^{t−1})y_{F} (i,s^{t})}{μ(s^{t−1})···μ(s^{t−i+1})}}{\frac{1}{N} ∑ \frac{p_{F}^{∗} (i,s^{t−1})y_{F}^{∗} (i,s^{t})}{μ^{∗}(s^{t−1})···μ^{∗}(s^{t−i+1})} − p^{∗}(s^{t})y^{∗}(s^{t})} .  (2.52)

The real exchange rate is then given by

q(s^{t}) = \frac{e(s^{t})P^{∗}(s^{t})}{P (s^{t})} = \frac{ε(s^{t})p^{∗}(s^{t})}{p(s^{t})}.  (2.53)

Finally, we normalize the equation for bond prices (1.50):

V (s^{t}) = ∑_{s_{t+1}} βπ(s^{t+1}|s^{t})\frac{U_{c}^{∗}(s^{t+1})}{U_{c}^{∗}(s^{t})} \frac{P^{∗}(s^{t})}{P^{∗}(s^{t+1})} \frac{e(s^{t})}{e(s^{t+1})}

= ∑_{s_{t+1}} βπ(s^{t+1}|s^{t})\frac{U_{c}^{∗}(s^{t+1})}{U_{c}^{∗}(s^{t})} \frac{P (s^{t})}{P (s^{t+1})} \frac{q(s^{t})}{q(s^{t+1})}

= ∑_{s_{t+1}} βπ(s^{t+1}|s^{t})\frac{U_{c}^{∗}(s^{t+1})}{U_{c}^{∗}(s^{t})} \frac{p(s^{t})}{p(s^{t+1})μ(s^{t})} \frac{q(s^{t})}{q(s^{t+1})}.  (2.54)

2.3.2. Steady State for the I-M Extension The steady state is the same as in the benchmark economy. The additional variables are equal to

ε = 1
d^{∗} = 0
V = β/μ

in the steady state.

2.3.3. Linearized Equations for the I-M Extension In addition to the equations for the benchmark economy, we have to linearize the nominal exchange rate (2.52) and the bond price (2.54).

Let’s start with the exchange rate. Rewrite the equation as

ε(s^{t}) \frac{1}{N} ∑ \frac{p_{F}^{∗} (i, s^{t−1})y_{F}^{∗} (i, s^{t})}{μ^{∗}(s^{t−1}) ⋯ μ^{∗}(s^{t−i+1})} − ε(s^{t})p^{∗}(s^{t})y^{∗}(s^{t})

= V (s^{t})d^{∗}(s^{t})μ(s^{t}) − d^{∗}(s^{t−1}) − \frac{1}{N} ∑ \frac{p_{F} (i, s^{t−1})y_{F} (i, s^{t})}{μ(s^{t−1}) ⋯ μ(s^{t−i+1})}

Linearizing this yields

(\frac{p_{I}}{N} ∑_{i} \frac{y_{F}^{∗} (i)}{μ^{i−1}} − py) ε̂_{t} − py(p̂_{t}^{∗} + ŷ_{t}^{∗})

+ \frac{p_{I}}{N} (y_{F}^{∗} (1)[p̂_{F,t−1}^{∗} + ŷ_{F,1,t}^{∗}] + \frac{y_{F}^{∗} (2)}{μ}[p̂_{F,t−2}^{∗} + ŷ_{F,2,t}^{∗} − μ̂_{t−1}^{∗}] + …)

= βd_{t}^{∗} − d_{t−1}^{∗} − \frac{p_{I}}{N} (y_{F} (1)[p̂_{F,t−1} + ŷ_{F,1,t}] + \frac{y_{F} (2)}{μ}[p̂_{F,t−2} + ŷ_{F,2,t} − μ̂_{t−1}] + …)

or

ε̂_{t} = \frac{1}{Δ}[βd_{t}^{∗} − d_{t−1}^{∗} + py(p̂_{t}^{∗} + ŷ_{t}^{∗})

− \frac{p_{I}}{N} (∑_{i=1}^{N} y_{F} (i)[p̂_{F,t−i} + ŷ_{F,i,t} − μ̂_{t−1} − … μ̂_{t−i+1}])

− \frac{p_{I}}{N} (∑_{i=1}^{N} y_{F}^{∗} (i)[p̂_{F,t−i}^{∗} + ŷ_{F,i,t}^{∗} − μ̂_{t−1}^{∗} − … μ̂_{t−i+1}^{∗}])]

where Δ = p_{I} (\frac{1}{N} ∑_{i} \frac{y_{F}^{∗} (i)}{μ^{i−1}}) − py, and p_{I} is the steady value for all of the intermediate goods prices. Note that we have used the fact that the countries have the same steady state values (e.g., p = p^{∗}) when writing these expressions. Given ε̂_{t}, we can compute the real exchange rate,

q̂_{t} = ε̂_{t} + p̂_{t}^{∗} − p̂_{t}.  (2.55)

Next, we linearize (2.54) to get

−\frac{β}{μ}(r_{t} − r) = E_{t}(\frac{U_{cc}c}{U_{c}}(ĉ_{t+1}^{∗} − ĉ_{t}^{∗}) + \frac{U_{cl}l}{U_{c}} (l̂_{t+1}^{∗} − l̂_{t}^{∗})

+ \frac{U_{cm}M/P}{U_{c}} (μ̂_{t+1}^{∗} − p̂_{t+1}^{∗} − μ̂_{t}^{∗} + p̂_{t}^{∗})

+ p̂_{t} + q̂_{t} − p̂_{t+1} − q̂_{t+1} − μ̂_{t}).  (2.56)

where we have used the fact that V (s^{t}) = 1/(1 + r(s^{t})).

Given that V is the inverse of the gross real interest rate, we can also define it as V (s^{t}) = 1 − U_{m}(s^{t})/U_{c}(s^{t}) and therefore,

V (V_{t}−V) = \frac{U_{m}}{U_{c}} \{(\frac{U_{cc}}{U_{c}} − \frac{U_{cm}}{U_{m}} ) c ĉ_{t}+(\frac{U_{cl}}{U_{c}} − \frac{U_{lm}}{U_{m}} ) l l̂_{t}+(\frac{U_{cm}}{U_{c}} − \frac{U_{mm}}{U_{m}} ) M/P (μ̂_{t}−p̂_{t})\}.

(2.57)

2.3.4. Solving the Linearized System for the I-M Extension The system of equations that we solve has 2N + 7 dynamic equations:

• 4 pricing equations, (2.30)-(2.33)

• 2N Euler equations for capital ((2.36) for home and similar for foreign)

• 2 money demand equations ((2.21) for home and similar for foreign)

• 1 equation relating bond prices (set (2.22) equal to (2.56), eliminating r)

• and static equations and definitions as in the benchmark economy except ∘ q̂ from (2.55);

∘ V from (2.57).

We turn next to the computation. We will use the following vectors:

z_{t} = [p̂_{H,t−1}, p̂_{F,t−1}, p̂_{F,t−1}^{∗}, p̂_{H,t−1}^{∗}, k̂_{1,t}, … k̂_{N,t}, k̂_{1,t}^{∗}, … k̂_{N,t}^{∗}, d_{t}^{∗}, ŷ_{t}, ŷ_{t}^{∗}]′  (n_{z} × 1)
X_{t} = [p̂_{H,t−2}, …, p̂_{H,t−N}, p̂_{F,t−2}, …, p̂_{F,t−N}, p̂_{F,t−2}^{∗}, …, p̂_{F,t−N}^{∗}, p̂_{H,t−2}^{∗}, …, p̂_{H,t−N}^{∗},
k̂_{1,t−1}, …, k̂_{N,t−1}, k̂_{1,t−1}^{∗}, …, k̂_{Nt−1}^{∗}, d_{t−1}^{∗}] (n_{X} × 1)
Z_{t} = [z_{t+N−1}, z_{t+N−2}, ...z_{t}, X_{t}, μ̂_{t+N−1}, …, μ̂_{t−N+1}, μ̂_{t+N−1}^{∗}, …, μ̂_{t−N+1}^{∗}]
Z_{t} = [z_{t}, z_{t−1}, …, z_{t−N+2}]′  (n_{Z} × 1)
S_{t} = [μ̂_{t}, …, μ̂_{t−N+1}, μ̂_{t}^{∗}, …, μ̂_{t−N+1}^{∗}]′  (n_{S} × 1)

The vector z_{t} contains the choice variables at time t. It has n_{z} = 2N + 7 elements. The vector X_{t} are the state variables at time t. There are n_{X} = 6N − 3 state variables. The vector Z_{t} contains all variables that appear in the residual equations. As before, the vectors Z_{t} and S_{t} are used when we characterize the solution. (See (2.40).)

As in the benchmark economy, the residual equations can be written succinctly as follows:

E [A_{1} [ \begin{matrix}X_{t+1} \\ Z_{t+N−1} \end{matrix} ] + A_{2} [ \begin{matrix}X_{t} \\ Z_{t+N−2} \end{matrix} ] + shock terms|Ω_{t}] = 0.

The A_{1} and A_{2} matrices can again be written as in (2.41) and (2.42), respectively. However, I_{1} and I_{2} in the incomplete-markets example are given by

I_{1} = [ \begin{matrix}I_{4,4} ⊗ [ \begin{matrix}0 & … & 0 & 0 \\ 1 & … & 0 & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ 0 & … & 1 & 0 \end{matrix} ] , & 0_{4N−4,2N+1} \\ 0_{2N+1,4N−4} & 0_{2N+1,2N+1} \end{matrix} ]

I_{2} = [ \begin{matrix}I_{4,4} ⊗ [ \begin{matrix}1 \\ 0_{N−2,1} \end{matrix} ] , & 0_{4N−4,2N+3} \\ 0_{2N+1,4} & [I_{2N+1,2N+1}, 0_{2N+1,2}] \end{matrix} ]

We now construct eigenvalues of −A_{1}^{−1}A_{2} if A_{1} is invertible and generalized eigenvalues otherwise. Then, ignoring shock terms, we have

[ \begin{matrix}X_{t+1} \\ Z_{t+N−1} \end{matrix} ] = V ΛV^{−1} [ \begin{matrix}X_{t} \\ Z_{t+N−2} \end{matrix} ] .

We can sort eigenvalues inside and outside the unit circle. If there are n_{X} stable eigenvalues (which is the number of state variables in X), then we have a locally determinate system. Suppose that the eigenvectors in V and eigenvalues in Λ are sorted so that the upper left partition of Λ contains the stable eigenvalues. Then,

X_{t+1} = V_{11}Λ_{1}V_{11}^{−1} X_{t}
Z_{t+N−2} = V_{21}V_{11}^{−1} X_{t}.

The last n_{z} elements of Z_{t+N−2} are those of z_{t}. Therefore, we have a relationship between our decision variables z and the state variables X. If we want to write the system as (2.40), then we can use this relationship between z and X to fill in the elements of A. In particular, we set

A(1 : n_{z}, 1 : n_{z} − 2) = A_{zX}(:, [1, N, 2N − 1, 3N − 2, 4N − 3 : 6N − 3])
A(1 : n_{z}, n_{z} + 1 : n_{z} : n_{Z}) = A_{zX}(:, 2 : N − 1)
A(1 : n_{z}, n_{z} + 2 : n_{z} : n_{Z}) = A_{zX}(:, N + 1 : 2N − 2)
A(1 : n_{z}, n_{z} + 3 : n_{z} : n_{Z}) = A_{zX}(:, 2N : 3N − 3)
A(1 : n_{z}, n_{z} + 4 : n_{z} : n_{Z}) = A_{zX}(:, 3N − 1 : 4N − 4)
A(n_{z} + 1 : n_{Z}, 1 : n_{Z} − n_{z}) = I_{n_{Z}−n_{z},n_{Z}−n_{z}}

where A_{zX} comes from z_{t} = A_{zX}X_{t}.

The next step is to compute B. The computation is exactly the same as in the benchmark case except that the values for the a_{0}’s and b_{0}’s in (2.43) are different. In the incomplete-markets case, they are:

a_{0} = dR/dZ(:, 1 : (N − 1)n_{z})
a_{N−1}(:, 1 : n_{z}) = dR/dZ(:, (N − 1)n_{z} + 1 : Nn_{z})
b_{0}(:, 1 : N) = dR/dZ(:, Nn_{z} + n_{X} + 1 : Nn_{z} + n_{X} + N)

b_{0}(:, N + 1 : 2N) = dR/dZ(:, Nn_{z} + n_{X} + 2N : Nn_{z} + n_{X} + 3N − 1)
b_{N−1}(:, 2 : N) = dR/dZ(:, Nn_{z} + n_{X} + N + 1 : Nn_{z} + n_{X} + 2N − 1)
b_{N−1}(:, N + 2 : 2N) = dR/dZ(:, Nn_{z} + n_{X} + 3N : Nn_{z} + n_{X} + 4N − 2)

with all other coefficients but a_{N} set equal to 0. The matrix a_{N} is nonzero but it is not used in computing B. From here on, the steps of the computation of B are the same as in the benchmark case and therefore, the codes look the same.

2.4. The Extension with Additional Shocks

It is easy to include additional shocks to the three models described above. We need to make the following changes:

• Replace c = y − x by c = y − x − g in the steady state calculation.

• Replace (2.34) by

y_{H}(i) ŷ_{H,i,t} + y_{H}^{∗}(i) ŷ_{H,i,t}^{∗} = F_{k}(i)k(i − 1) k̂_{i−1,t−1} + F_{l}(i)l(i) (l̂_{i,t} + Â_{t}).

• Replace (2.37) by

m̂c_{i,t} = ŵ_{t} − F_{kl}(i)k(i − 1)/F_{l}(i)k̂_{i−1,t−1} − F_{ll}(i)l(i)/F_{l}(i)(l̂_{i,t} + Â_{t}) − Â_{t}.

• Replace (2.39) by

ĉ_{t} = (y ŷ_{t} − [x(1) x̂_{1,t} + … x(N) x̂_{N,t}]/N − g ĝ_{t})/c.

Notice that we did not adjust the Euler equations for capital since the A term in mc cancels with the A term in F_{k} leaving the linearization unchanged.

2.4.1. New Code for the Benchmark Economy

When we rewrite the code with multiple shocks, we need to adjust the Z and S vectors as follows:

Z_{t} = [z_{t+N−1}, z_{t+N−2}, ...z_{t}, X_{t}, μ̂_{t+N−1}, …, μ̂_{t−N+1}, μ̂_{t+N−1}^{∗}, …, μ̂_{t−N+1}^{∗},
ĝ_{t+N−1}, …, ĝ_{t}, ĝ_{t+N−1}^{∗}, …, ĝ_{t}^{∗}, â_{t+N−1}, …, â_{t}, â_{t+N−1}^{∗}, …, â_{t}^{∗}]′
S_{t} = [μ̂_{t}, …, μ̂_{t−N+1}, μ̂_{t}^{∗}, …, μ̂_{t−N+1}^{∗},
ĝ_{t}, ĝ_{t−1}, ĝ_{t}^{∗}, ĝ_{t−1}^{∗}, â_{t}, â_{t−1}, â_{t}^{∗}, â_{t−1}^{∗}]′.

The dimension of S is now 2N + 8 × 1.

Although the elements of A in (2.40) will be different when we turn on the other shocks, we will not have to change the code used to calculate it. The calculation of B on the other hand will change. We need to use new code for the b_{i}’s as follows:

b_{0}(:, 1 : N) = dR/dZ(:, Nn_{z} + n_{X} + 1 : Nn_{z} + n_{X} + N)

b_{0}(:, N + 1 : 2N) = dR/dZ(:, Nn_{z} + n_{X} + 2N : Nn_{z} + n_{X} + 3N − 1)

b_{0}(:, 2N + 1) = dR/dZ(:, Nn_{z} + n_{X} + 4N − 1)

b_{0}(:, 2N + 3) = dR/dZ(:, Nn_{z} + n_{X} + 5N − 1)

b_{0}(:, 2N + 5) = dR/dZ(:, Nn_{z} + n_{X} + 6N − 1)

b_{0}(:, 2N + 7) = dR/dZ(:, Nn_{z} + n_{X} + 7N − 1)

b_{1}(:, 2N + 1) = dR/dZ(:, Nn_{z} + n_{X} + 4N)

b_{1}(:, 2N + 3) = dR/dZ(:, Nn_{z} + n_{X} + 5N)

b_{1}(:, 2N + 5) = dR/dZ(:, Nn_{z} + n_{X} + 6N)

b_{1}(:, 2N + 7) = dR/dZ(:, Nn_{z} + n_{X} + 7N)

⋮ b_{N−2}(:, 2N + 1) = dR/dZ(:, Nn_{z} + n_{X} + 5N − 3)

b_{N−2}(:, 2N + 3) = dR/dZ(:, Nn_{z} + n_{X} + 6N − 3)

b_{N−2}(:, 2N + 5) = dR/dZ(:, Nn_{z} + n_{X} + 7N − 3)

b_{N−2}(:, 2N + 7) = dR/dZ(:, Nn_{z} + n_{X} + 8N − 3)

b_{N−1}(:, 2 : N) = dR/dZ(:, Nn_{z} + n_{X} + N + 1 : Nn_{z} + n_{X} + 2N − 1)

b_{N−1}(:, N + 2 : 2N) = dR/dZ(:, Nn_{z} + n_{X} + 3N : Nn_{z} + n_{X} + 4N − 2)

b_{N−1}(:, 2N + 1) = dR/dZ(:, Nn_{z} + n_{X} + 5N − 2)

b_{N−1}(:, 2N + 3) = dR/dZ(:, Nn_{z} + n_{X} + 6N − 2)

b_{N−1}(:, 2N + 5) = dR/dZ(:, Nn_{z} + n_{X} + 7N − 2)

b_{N−1}(:, 2N + 7) = dR/dZ(:, Nn_{z} + n_{X} + 8N − 2)

The matrix P in S_{t+1} = PS_{t} + ϵ_{t+1} will also change. With multiple shocks we have

P = [ \begin{matrix}I_{2,2} ⊗ [ \begin{matrix}ρ_{μ} & 0 & … & 0 & 0 \\ 1 & 0 & … & 0 & 0 \\ 0 & 1 & … & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & 1 & 0 \end{matrix} ] & 0_{2N,8} \\ 0_{8,2N} & [ \begin{matrix}ρ_{g} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & ρ_{g} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & ρ_{a} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & ρ_{a} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{matrix} ] \end{matrix} ]  (2.58)

To set up the matrix D needed to compute B in (2.40), we need to make assumptions about the timing of decisions.^{2} Suppose that Ω_{t} is the information set of the monopolists when setting prices. Let’s start with the case: Ω_{t} = {ω_{s}}_{s=0}^{t−1}, where ω_{t} =

2 See (2.46) for the benchmark case.

[μ̂_{t}, μ̂_{t}^{∗}, ĝ_{t}, ĝ_{t}^{∗}, â_{t}, â_{t}^{∗}]′. The interpretation in this case is that monopolists see none of the period t shocks before choosing their period t prices. In this case,

E[M_{1}μ̂_{t} + … + M_{N} μ̂_{t−N+1} + M_{N+1}μ̂_{t}^{∗} + … + M_{2N}μ̂_{t−N+1}^{∗}
+ M_{2N+1}ĝ_{t} + M_{2N+2}ĝ_{t−1} + M_{2N+3}ĝ_{t}^{∗} + M_{2N+4}ĝ_{t−1}^{∗}
+ M_{2N+5}â_{t} + M_{2N+6}â_{t−1} + M_{2N+7}â_{t}^{∗} + M_{2N+8}â_{t−1}^{∗}|Ω_{t}]
= [0, M_{1}ρ_{μ} + M_{2}, M_{3}, … M_{N}, 0, M_{N+1}ρ_{μ} + M_{N+2}, M_{N+3}, … M_{2N},
0, M_{2N+1}ρ_{g} + M_{2N+2}, 0, M_{2N+3}ρ_{g} + M_{2N+4},
0, M_{2N+5}ρ_{a} + M_{2N+6}, 0, M_{2N+7}ρ_{a} + M_{2N+8}] S_{t}.

In this case, D would be given by

D = [ \begin{matrix}I_{4,4} ⊗ Φ & 0_{4(n_{S}−6),n_{z}n_{S}−4n_{S}} \\ 0_{n_{z}n_{S}−4n_{S},4n_{S}} & I_{n_{z}n_{S}−4n_{S}} \end{matrix} ]

where

Φ = [ \begin{matrix}[ \begin{matrix}Ψ & 0_{N−1,N} \\ 0_{N−1,N} & Ψ \end{matrix} ] & 0_{2N−2,8} \\ 0_{4,2N} & [ \begin{matrix}ρ_{g} & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & ρ_{g} & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & ρ_{a} & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & ρ_{a} & 1 \end{matrix} ] \end{matrix} ]

Ψ = [ \begin{matrix}ρ_{μ} & 1 & 0 & 0 & … & 0 \\ 0 & 0 & 1 & 0 & … & 0 \\ 0 & 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & 0 & … & 1 \end{matrix} ] .

The matrix Φ has dimensions 2N + 2 × 2N + 8 while Ψ has N − 1 × N.

Now consider the case with Ω_{t} = {{ω_{s}}_{s=0}^{t−1}, a_{t}, a_{t}^{∗}} so that monopolists observe current technology shocks when choosing prices. In this case,

E[M_{1}μ̂_{t} + … + M_{N} μ̂_{t−N+1} + M_{N+1}μ̂_{t}^{∗} + … + M_{2N}μ̂_{t−N+1}^{∗}
+ M_{2N+1}ĝ_{t} + M_{2N+2}ĝ_{t−1} + M_{2N+3}ĝ_{t}^{∗} + M_{2N+4}ĝ_{t−1}^{∗}

+ M_{2N+5}â_{t} + M_{2N+6}â_{t−1} + M_{2N+7}â_{t}^{∗} + M_{2N+8}â_{t−1}^{∗}|Ω_{t}]

= [0, M_{1}ρ_{μ} + M_{2}, M_{3}, … M_{N}, 0, M_{N+1}ρ_{μ} + M_{N+2}, M_{N+3}, … M_{2N},

0, M_{2N+1}ρ_{g} + M_{2N+2}, 0, M_{2N+3}ρ_{g} + M_{2N+4},
M_{2N+5}, M_{2N+6}, M_{2N+7}, M_{2N+8}] S_{t}.

In this case, D would be given by

D = [ \begin{matrix}I_{4,4} ⊗ Φ & 0_{4(n_{S}−4),n_{z}n_{S}−4n_{S}} \\ 0_{n_{z}n_{S}−4n_{S},4n_{S}} & I_{n_{z}n_{S}−4n_{S}} \end{matrix} ]

where

Φ = [ \begin{matrix}[ \begin{matrix}Ψ & 0_{N−1,N} \\ 0_{N−1,N} & Ψ \end{matrix} ] & 0_{2N−2,8} \\ 0_{6,2N} & [ \begin{matrix}ρ_{g} & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & ρ_{g} & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] \end{matrix} ]

and Ψ is the same as above. The matrix Φ has dimensions 2N + 4 × 2N + 8.

Finally, suppose that Ω_{t} = {{ω_{s}}_{s=0}^{t−1}, ĝ_{t}, ĝ_{t}^{∗}, â_{t}, â_{t}^{∗}} so that monopolists observe current

real shocks when choosing prices. In this case,

E[M_{1}μ̂_{t} + … + M_{N} μ̂_{t−N+1} + M_{N+1}μ̂_{t}^{∗} + … + M_{2N}μ̂_{t−N+1}^{∗}
+ M_{2N+1}ĝ_{t} + M_{2N+2}ĝ_{t−1} + M_{2N+3}ĝ_{t}^{∗} + M_{2N+4}ĝ_{t−1}^{∗}
+ M_{2N+5}â_{t} + M_{2N+6}â_{t−1} + M_{2N+7}â_{t}^{∗} + M_{2N+8}â_{t−1}^{∗}|Ω_{t}]

= [0, M_{1}ρ_{μ} + M_{2}, M_{3}, … M_{N}, 0, M_{N+1}ρ_{μ} + M_{N+2}, M_{N+3}, … M_{2N},

M_{2N+1}, M_{2N+2}, M_{2N+3}, M_{2N+4},
M_{2N+5}, M_{2N+6}, M_{2N+7}, M_{2N+8}] S_{t}.

In this case, D would be given by

D = [ \begin{matrix}I_{4,4} ⊗ Φ & 0_{4(n_{S}−2),n_{z}n_{S}−4n_{S}} \\ 0_{n_{z}n_{S}−4n_{S},4n_{S}} & I_{n_{z}n_{S}−4n_{S}} \end{matrix} ]

where

Φ = [ \begin{matrix}[ \begin{matrix}Ψ & 0_{N−1,N} \\ 0_{N−1,N} & Ψ \end{matrix} ] & 0_{2N−2,8} \\ 0_{8,2N} & I_{8,8} \end{matrix} ]

The matrix Φ has dimensions 2N + 6 × 2N + 8. 2.4.2. New Code for the Taylor-Rule Extension The new Z and S vectors for the Taylor-Rule case are as follows:

Z_{t} = [z_{t+N−1}, z_{t+N−2}, ...z_{t}, X_{t}, ϵ_{r,t}, ϵ_{r,t}^{∗}, ĝ_{t+N−1}, …, ĝ_{t}, ĝ_{t+N−1}^{∗}, …, ĝ_{t}^{∗},

â_{t+N−1}, …, â_{t}, â_{t+N−1}^{∗}, …, â_{t}^{∗}]′

S_{t} = [ϵ_{r,t}, ϵ_{r,t−1}, ϵ_{r,t}^{∗}, ϵ_{r,t−1}^{∗}, ĝ_{t}, ĝ_{t−1}, ĝ_{t}^{∗}, ĝ_{t−1}^{∗}, â_{t}, â_{t−1}, â_{t}^{∗}, â_{t−1}^{∗}]′

The dimension of S is now 12×1.

To calculate the B matrix in (2.40) we need to update the b_{i}’s to take into account the real shocks. These are now:

b_{0}(:, 5) = dR/dZ(:, n_{Z} + n_{X} + 3) b_{0}(:, 7) = dR/dZ(:, n_{Z} + n_{X} + N + 3) b_{0}(:, 9) = dR/dZ(:, n_{Z} + n_{X} + 2N + 3) b_{0}(:, 11) = dR/dZ(:, n_{Z} + n_{X} + 3N + 3) b_{1}(:, 5) = dR/dZ(:, n_{Z} + n_{X} + 4) b_{1}(:, 7) = dR/dZ(:, n_{Z} + n_{X} + N + 4) b_{1}(:, 9) = dR/dZ(:, n_{Z} + n_{X} + 2N + 4) b_{1}(:, 11) = dR/dZ(:, n_{Z} + n_{X} + 3N + 4) ⋮ b_{N−1}(:, 1) = dR/dZ(:, n_{Z} + n_{X} + 1) b_{N−1}(:, 3) = dR/dZ(:, n_{Z} + n_{X} + 2) b_{N−1}(:, 5) = dR/dZ(:, n_{Z} + n_{X} + N + 2) b_{N−1}(:, 7) = dR/dZ(:, n_{Z} + n_{X} + 2N + 2) b_{N−1}(:, 9) = dR/dZ(:, n_{Z} + n_{X} + 3N + 2) b_{N−1}(:, 11) = dR/dZ(:, n_{Z} + n_{X} + 4N + 2)

with all of the other b coefficients equal to 0. Next, we adjust the P matrix. For this case it is given by

P = [ \begin{matrix}[ \begin{matrix}ρ_{ϵ} & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & ρ_{ϵ} & 0 \\ 0 & 0 & 1 & 0 \end{matrix} ] & 0_{4,8} \\ 0_{8,4} & [ \begin{matrix}ρ_{g} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & ρ_{g} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & ρ_{a} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & ρ_{a} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{matrix} ] \end{matrix} ]  (2.59)

We next set up the matrix D. If Ω_{t} = {ϵ_{r,s}, ϵ_{r,s}^{∗}, g_{s}, g_{s}^{∗}, a_{s}, a_{s}^{∗}}_{s=0}^{t−1}, then

E[M_{1}ϵ_{r,t} + M_{2}ϵ_{r,t−1} + M_{3}ϵ_{r,t}^{∗} + M_{4}ϵ_{r,t−1}^{∗} + M_{5}ĝ_{t} + M_{6}ĝ_{t−1} + M_{7}ĝ_{t}^{∗} + M_{8}ĝ_{t−1}^{∗} + M_{9}â_{t} + M_{10}â_{t−1} + M_{11}ĝ_{t}^{∗} + M_{12}â_{t−1}^{∗}|Ω_{t}]

= [0, M_{1}ρ_{ϵ} + M_{2}, 0, M_{3}ρ_{ϵ} + M_{4}, 0, M_{5}ρ_{g} + M_{6}, 0, M_{7}ρ_{g} + M_{8}, 0, M_{9}ρ_{a} + M_{10}, 0, M_{11}ρ_{a} + M_{12}] S_{t}.

For this case,

D = [ \begin{matrix}I_{4,4} ⊗ Φ & 0_{4(n_{S}−6),n_{z}n_{S}−4n_{S}} \\ 0_{n_{z}n_{S}−4n_{S},4n_{S}} & I_{n_{z}n_{S}−4n_{S}} \end{matrix} ]

Φ = [ \begin{matrix}ρ_{ϵ} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & ρ_{ϵ} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & ρ_{g} & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & ρ_{g} & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ρ_{a} & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & ρ_{a} & 1 \end{matrix} ]

Now consider the case with Ω_{t} = {{ω_{s}}_{s=0}^{t−1}, a_{t}, a_{t}^{∗}} with ω_{s} = [ϵ_{r,s}, ϵ_{r,s}^{∗}, g_{s}, g_{s}^{∗}, a_{s}, a_{s}^{∗}]′. Here, the monopolists observe the current technology shocks before changing prices. For this case,

E[M_{1}ϵ_{r,t} + M_{2}ϵ_{r,t−1} + M_{3}ϵ_{r,t}^{∗} + M_{4}ϵ_{r,t−1}^{∗}
+ M_{5}ĝ_{t} + M_{6}ĝ_{t−1} + M_{7}ĝ_{t}^{∗} + M_{8}ĝ_{t−1}^{∗}
+ M_{9}â_{t} + M_{10}â_{t−1} + M_{11}ĝ_{t}^{∗} + M_{12}â_{t−1}^{∗}|Ω_{t}]
= [0, M_{1}ρ_{ϵ} + M_{2}, 0, M_{3}ρ_{ϵ} + M_{4}, 0, M_{5}ρ_{g} + M_{6}, 0, M_{7}ρ_{g} + M_{8},
M_{9}, M_{10}, M_{11}, M_{12}] S_{t}

and

D = [ \begin{matrix}I_{4,4} ⊗ Φ & 0_{4(n_{S}−4),n_{z}n_{S}−4n_{S}} \\ 0_{n_{z}n_{S}−4n_{S},4n_{S}} & I_{n_{z}n_{S}−4n_{S}} \end{matrix} ]

where

Φ = [ \begin{matrix}ρ_{ϵ} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & ρ_{ϵ} & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & ρ_{g} & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & ρ_{g} & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ]

Finally, consider the case with Ω_{t} = {{ω_{s}}_{s=0}^{t−1}, â_{t}, â_{t}^{∗}, ĝ_{t}, ĝ_{t}^{∗}}. Here, the monopolists observe the current real shocks before changing prices. For this case,

E[M_{1}ϵ_{r,t} + M_{2}ϵ_{r,t−1} + M_{3}ϵ_{r,t}^{∗} + M_{4}ϵ_{r,t−1}^{∗}

+ M_{5}ĝ_{t} + M_{6}ĝ_{t−1} + M_{7}ĝ_{t}^{∗} + M_{8}ĝ_{t−1}^{∗}
+ M_{9}â_{t} + M_{10}â_{t−1} + M_{11}ĝ_{t}^{∗} + M_{12}â_{t−1}^{∗}|Ω_{t}]
= [0, M_{1}ρ_{ϵ} + M_{2}, 0, M_{3}ρ_{ϵ} + M_{4}, M_{5}, M_{6}, M_{7}, M_{8},
M_{9}, M_{10}, M_{11}, M_{12}] S_{t}

and

D = [ \begin{matrix}I_{4,4} ⊗ Φ & 0_{4(n_{S}−2),n_{z}n_{S}−4n_{S}} \\ 0_{n_{z}n_{S}−4n_{S},4n_{S}} & I_{n_{z}n_{S}−4n_{S}} \end{matrix} ]

where

Φ = [ \begin{matrix}[ \begin{matrix}ρ_{ϵ} & 1 & 0 & 0 \\ 0 & 0 & ρ_{ϵ} & 1 \end{matrix} ] & 0_{2,8} \\ 0_{8,4} & I_{8,8} \end{matrix} ]

2.4.3. New Code for the I-M Extension

The Z and S vectors and the b, and D matrices are the same in the benchmark economy and the incomplete-markets economy. Therefore, the extension of the incomplete-markets economy allowing for real shocks follows exactly as in the benchmark economy. (See above.)

2.4.4. Allowing for Accommodative Monetary Policy

Consider the following money growth rate processes:

μ̂_{t} = ρ_{μ}μ̂_{t−1} + γÂ_{t} + σ_{μ}ϵ_{μ,t} + σ_{μμ^{∗}}ϵ_{μ^{∗},t}  (2.60)
μ̂_{t}^{∗} = ρ_{μ}μ̂_{t−1}^{∗} + γÂ_{t}^{∗} + σ_{μ^{∗}}ϵ_{μ^{∗},t}.  (2.61)

With the technology shocks given by

[ Â_{t} ; Â_{t}^{∗} ] = [ \begin{matrix}ρ_{a} & 0 \\ 0 & ρ_{a} \end{matrix} ] [ Â_{t−1} ; Â_{t−1}^{∗} ] + [ \begin{matrix}σ_{a} & σ_{a,a^{∗}} \\ 0 & σ_{a^{∗}} \end{matrix} ] [ \begin{matrix}ϵ_{a,t} \\ ϵ_{a^{∗},t} \end{matrix} ]

we can rewrite (2.60) as follows:

μ̂_{t} = ρ_{μ}μ̂_{t−1} + γρ_{a}Â_{t−1} + σ_{μ}ϵ_{μ,t} + σ_{μμ^{∗}}ϵ_{μ^{∗},t} + γσ_{a}ϵ_{a,t} + γσ_{aa^{∗}}ϵ_{a^{∗},t}

and the foreign growth rate as

μ̂_{t}^{∗} = ρ_{μ}μ̂_{t−1}^{∗} + γρ_{a}Â_{t−1}^{∗} + σ_{μ^{∗}}ϵ_{μ^{∗},t} + γσ_{a^{∗}}ϵ_{a^{∗},t}

The complete system (ignoring g) looks like

[ μ̂_{t} ; μ̂_{t}^{∗} ; Â_{t} ; Â_{t}^{∗} ] = [ \begin{matrix}ρ_{μ} & 0 & γρ_{a} & 0 \\ 0 & ρ_{μ} & 0 & γρ_{a} \\ 0 & 0 & ρ_{a} & 0 \\ 0 & 0 & 0 & ρ_{a} \end{matrix} ] [ μ̂_{t−1} ; μ̂_{t−1}^{∗} ; Â_{t−1} ; Â_{t−1}^{∗} ] + [ \begin{matrix}σ_{μ} & σ_{μμ^{∗}} & γσ_{a} & γσ_{aa^{∗}} \\ 0 & σ_{μ^{∗}} & 0 & γσ_{a^{∗}} \\ 0 & 0 & σ_{a} & σ_{aa^{∗}} \\ 0 & 0 & 0 & σ_{a^{∗}} \end{matrix} ] [ ϵ̂_{μ,t} ; ϵ̂_{μ^{∗},t} ; ϵ̂_{a,t} ; ϵ̂_{a^{∗},t} ] (2.62)

or x_{t} = Ax_{t−1} + Cϵ_{t} where

A = [ \begin{matrix}ρ_{μ}I_{2} & γρ_{a}I_{2} \\ 0_{2} & ρ_{a}I_{2} \end{matrix} ] , C = [ \begin{matrix}Σ_{μ} & γΣ_{a} \\ 0_{2} & Σ_{a} \end{matrix} ] .

Note that the matrices Σ_{μ} and Σ_{a} are the coefficient matrices on the disturbances in (2.62). Let V be the variance-covariance matrix for this system. This 4×4 matrix solves

V = AV A′ + CC′.

Doing the algebra, we have

V_{11} = ρ_{μ}^{2}V_{11} + 2γρ_{μ}ρ_{a}V_{12} + γ^{2}ρ_{a}^{2}V_{22} + Σ_{μ}Σ′_{μ} + γ^{2}Σ_{a}Σ′_{a}
V_{12} = ρ_{μ}ρ_{a}V_{12} + γρ_{a}^{2}V_{22} + γΣ_{a}Σ′_{a}
V_{22} = ρ_{a}^{2}V_{22} + Σ_{a}Σ′_{a}

We want to set parameters so that we can govern the correlation between the growth rates of money and the technology shocks across countries. To do this, we first introduce parameters ϱ_{a} and ϱ_{μ} which satisfy:

σ_{aa^{∗}} = ϱ_{a}σ_{a}/\sqrt{1 − ϱ_{a}^{2}}
σ_{a^{∗}} = σ_{a}/\sqrt{1 − ϱ_{a}^{2}}
σ_{μμ^{∗}} = ϱ_{μ}σ_{μ}/\sqrt{1 − ϱ_{μ}^{2}}
σ_{μ^{∗}} = σ_{μ}/\sqrt{1 − ϱ_{μ}^{2}}

and are used to define the elements of Σ_{μ} and Σ_{a}.

It turns out that ϱ_{a} is simply the correlation between Â_{t} and Â_{t}^{∗}. To see this, solve for V_{22} above,

V_{22} = \frac{σ_{a}^{2}}{(1 − ρ_{a}^{2})(1 − ϱ_{a}^{2})} [ \begin{matrix}1 & ϱ_{a} \\ ϱ_{a} & 1 \end{matrix} ]

and note that the correlation between Â_{t} and Â_{t}^{∗} is equal to the (1,2) element of V_{22} divided by the square roots of the (1,1) and (2,2) elements.

Unfortunately, constructing the correlation of μ̂ and μ̂^{∗} is not so simple when there is feedback from technology shocks. So, we will calculate this correlation in steps. First, note that

V_{11} = \frac{1}{1 − ρ_{μ}^{2}} [\frac{γ^{2}(1 + ρ_{μ}ρ_{a})}{(1 − ρ_{μ}ρ_{a})(1 − ρ_{a}^{2})} \frac{σ_{a}^{2}}{(1 − ϱ_{a}^{2})} [ \begin{matrix}1 & ϱ_{a} \\ ϱ_{a} & 1 \end{matrix} ] + \frac{σ_{μ}^{2}}{(1 − ϱ_{μ}^{2})} [ \begin{matrix}1 & ϱ_{μ} \\ ϱ_{μ} & 1 \end{matrix} ]]

= κ_{1} [ \begin{matrix}1 & ϱ_{a} \\ ϱ_{a} & 1 \end{matrix} ] + \frac{κ_{2}}{(1 − ϱ_{μ}^{2})} [ \begin{matrix}1 & ϱ_{μ} \\ ϱ_{μ} & 1 \end{matrix} ]

after substituting in for V_{22} and then for V_{12}. We want to set ϱ_{μ} so that the correlation between μ̂ and μ̂^{∗} can be set (i.e., the (1,2) element of V_{11} divided by either the (1,1) element or the (2,2) element),

cor(μ̂, μ̂^{∗}) = \frac{κ_{1}ϱ_{a} + κ_{2}ϱ_{μ}/(1 − ϱ_{μ}^{2})}{κ_{1} + κ_{2}/(1 − ϱ_{μ}^{2})}.

Rewriting this, we have a quadratic equation in ϱ_{μ}:

ϱ_{μ}^{2} + [\frac{κ_{2}}{κ_{1} (cor(μ̂, μ̂^{∗}) − ϱ_{a})}] ϱ_{μ} − [1 + cor(μ̂, μ̂^{∗})\frac{κ_{2}}{κ_{1} (cor(μ̂, μ̂^{∗}) − ϱ_{a})}] = 0

If we choose cor(μ̂, μ̂^{∗}), we can back out the value for ϱ_{μ} that ensures this (assuming there is only one root inside the unit circle).

Next, we modify our definitions for P and D. Recall that S_{t} is given by

S_{t} = [μ̂_{t}, …, μ̂_{t−N+1}, μ̂_{t}^{∗}, …, μ̂_{t−N+1}^{∗},
ĝ_{t}, ĝ_{t−1}, ĝ_{t}^{∗}, ĝ_{t−1}^{∗}, â_{t}, â_{t−1}, â_{t}^{∗}, â_{t−1}^{∗}]′

and P is the coefficient matrix in S_{t+1} = PS_{t} + ϵ_{t+1}. With accommodative money, we have

P = [ \begin{matrix}I_{2,2} ⊗ [ \begin{matrix}ρ_{μ} & 0 & … & 0 & 0 \\ 1 & 0 & … & 0 & 0 \\ 0 & 1 & … & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & 1 & 0 \end{matrix} ] & [ \begin{matrix}0 & 0 & 0 & 0 & γρ_{a} & 0 & 0 & 0 \\ ⋮ & & ⋮ & & ⋮ & & ⋮ \\ 0 & 0 & 0 & 0 & 0 & 0 & γρ_{a} & 0 \\ ⋮ & & ⋮ & & ⋮ & & ⋮ \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} ] \\ 0_{8,2N} & [ \begin{matrix}ρ_{g} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & ρ_{g} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & ρ_{a} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & ρ_{a} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \end{matrix} ] \end{matrix} ]

Now consider setting up the matrix D. Let’s start with the assumption that: Ω_{t} = {ω_{s}}_{s=0}^{t−1}, where ω_{t} = [μ̂_{t}, μ̂_{t}^{∗}, ĝ_{t}, ĝ_{t}^{∗}, â_{t}, â_{t}^{∗}]′. That is, we assume that monopolists see none of the period t shocks before choosing their period t prices. In this case,

E[M_{1}μ̂_{t} + … + M_{N} μ̂_{t−N+1} + M_{N+1}μ̂_{t}^{∗} + … + M_{2N}μ̂_{t−N+1}^{∗} + M_{2N+1}ĝ_{t} + M_{2N+2}ĝ_{t−1} + M_{2N+3}ĝ_{t}^{∗} + M_{2N+4}ĝ_{t−1}^{∗}

+ M_{2N+5}â_{t} + M_{2N+6}â_{t−1} + M_{2N+7}â_{t}^{∗} + M_{2N+8}â_{t−1}^{∗}|Ω_{t}]
= [0, M_{1}ρ_{μ} + M_{2}, M_{3}, … M_{N}, 0, M_{N+1}ρ_{μ} + M_{N+2}, M_{N+3}, … M_{2N},
0, M_{2N+1}ρ_{g} + M_{2N+2}, 0, M_{2N+3}ρ_{g} + M_{2N+4},

0, M_{1}γρ_{a} + M_{2N+5}ρ_{a} + M_{2N+6}, 0, M_{N+1}γρ_{a} + M_{2N+7}ρ_{a} + M_{2N+8}] S_{t}.

In this case, D would be given by

D = [ \begin{matrix}I_{4,4} ⊗ Φ & 0_{4(n_{S}−6),n_{z}n_{S}−4n_{S}} \\ 0_{n_{z}n_{S}−4n_{S},4n_{S}} & I_{n_{z}n_{S}−4n_{S}} \end{matrix} ]

where

Φ = [ \begin{matrix}[ \begin{matrix}Ψ & 0_{N−1,N} \\ 0_{N−1,N} & Ψ \end{matrix} ] & 0_{2N−2,8} \\ [ [ \begin{matrix}0 \\ 0 \\ γρ_{a} \\ 0 \end{matrix} ] 0_{4,N−1} [ \begin{matrix}0 \\ 0 \\ 0 \\ γρ_{a} \end{matrix} ] 0_{4,N−1} ] & [ \begin{matrix}ρ_{g} & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & ρ_{g} & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & ρ_{a} & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & ρ_{a} & 1 \end{matrix} ] \end{matrix} ]

Ψ = [ \begin{matrix}ρ_{μ} & 1 & 0 & 0 & … & 0 \\ 0 & 0 & 1 & 0 & … & 0 \\ 0 & 0 & 0 & 1 & … & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & 0 & … & 1 \end{matrix} ] .

Now consider the case: Ω_{t} = {{ω_{s}}_{s=0}^{t−1}, a_{t}, a_{t}^{∗}} so that monopolists observe current technology shocks when choosing prices. In this case,

E[M_{1}μ̂_{t} + … + M_{N} μ̂_{t−N+1} + M_{N+1}μ̂_{t}^{∗} + … + M_{2N}μ̂_{t−N+1}^{∗}
+ M_{2N+1}ĝ_{t} + M_{2N+2}ĝ_{t−1} + M_{2N+3}ĝ_{t}^{∗} + M_{2N+4}ĝ_{t−1}^{∗}

= [0, M_{1}ρ_{μ} + M_{2}, M_{3}, … M_{N}, 0, M_{N+1}ρ_{μ} + M_{N+2}, M_{N+3}, … M_{2N},

0, M_{2N+1}ρ_{g} + M_{2N+2}, 0, M_{2N+3}ρ_{g} + M_{2N+4},

M_{1}γ + M_{2N+5}, M_{2N+6}, M_{N+1}γ + M_{2N+7}, M_{2N+8}] S_{t}.

In this case, D would be given by

D = [ \begin{matrix}I_{4,4} ⊗ Φ & 0_{4(n_{S}−4),n_{z}n_{S}−4n_{S}} \\ 0_{n_{z}n_{S}−4n_{S},4n_{S}} & I_{n_{z}n_{S}−4n_{S}} \end{matrix} ]

where

Φ = [ \begin{matrix}[ \begin{matrix}Ψ & 0_{N−1,N} \\ 0_{N−1,N} & Ψ \end{matrix} ] & 0_{2N−2,8} \\ [ [ \begin{matrix}0 \\ 0 \\ γ \\ 0 \\ 0 \\ 0 \end{matrix} ] 0_{6,N−1} [ \begin{matrix}0 \\ 0 \\ 0 \\ 0 \\ γ \\ 0 \end{matrix} ] 0_{6,N−1} ] & [ \begin{matrix}ρ_{g} & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & ρ_{g} & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ] \end{matrix} ]

and Ψ is the same as above.

Finally, suppose that Ω_{t} = {{ω_{s}}_{s=0}^{t−1}, â_{t}, â_{t}^{∗}, ĝ_{t}, ĝ_{t}^{∗}} so that monopolists observe current

real shocks when choosing prices. In this case,

E[M_{1}μ̂_{t} + … + M_{N} μ̂_{t−N+1} + M_{N+1}μ̂_{t}^{∗} + … + M_{2N}μ̂_{t−N+1}^{∗}

+ M_{2N+1}ĝ_{t} + M_{2N+2}ĝ_{t−1} + M_{2N+3}ĝ_{t}^{∗} + M_{2N+4}ĝ_{t−1}^{∗}

+ M_{2N+5}â_{t} + M_{2N+6}â_{t−1} + M_{2N+7}â_{t}^{∗} + M_{2N+8}â_{t−1}^{∗}|Ω_{t}]

= [0, M_{1}ρ_{μ} + M_{2}, M_{3}, … M_{N}, 0, M_{N+1}ρ_{μ} + M_{N+2}, M_{N+3}, … M_{2N},

M_{2N+1}, M_{2N+2}, M_{2N+3}, M_{2N+4},

M_{1}γ + M_{2N+5}, M_{2N+6}, M_{N+1}γ + M_{2N+7}, M_{2N+8}] S_{t}.

In this case, D would be given by

D = [ \begin{matrix}I_{4,4} ⊗ Φ & 0_{4(n_{S}−2),n_{z}n_{S}−4n_{S}} \\ 0_{n_{z}n_{S}−4n_{S},4n_{S}} & I_{n_{z}n_{S}−4n_{S}} \end{matrix} ]

Φ = [ \begin{matrix}[ \begin{matrix}Ψ & 0_{N−1,N} \\ 0_{N−1,N} & Ψ \end{matrix} ] & 0_{2N−2,8} \\ [ [ \begin{matrix}0 \\ 0 \\ 0 \\ 0 \\ γ \\ 0 \\ 0 \\ 0 \end{matrix} ] 0_{8,N−1} [ \begin{matrix}0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ γ \\ 0 \end{matrix} ] 0_{8,N−1} ] & I_{8,8} \end{matrix} ]

2.5. The Extension with Sticky Wages We now consider extending the benchmark economy to include sticky wages. For completeness, we will allow for both monetary and real shocks. The main differences in the computation from the benchmark economy with additional shocks are these:

• we add W (s^{t−2}), …, W (s^{t−N}) and foreign analogues as state variables;

• we add equation (1.54) and one for W^{∗} to our residuals;

• we add W (s^{t−1}) and W^{∗}(s^{t−1}) as choice variables;

• we drop w = −U_{l}/U_{c};

• we add N − 1 static equations U_{c}(j) = U_{c}(k) to determine c(j, s^{t})’s and do the same for c^{∗};

• we add N − 1 static equations U_{m}(j) = U_{m}(k) to determine M^{d}(j, s^{t})’s and do the same for M^{d}^{∗};

• we add N static equations L^{s}(j) = (W̄ /W (j))^{\frac{1}{1−v}} ∫ L^{d}(i) di to determine the L^{s}(j, s^{t})’s, and we do the same for the foreign country.

2.5.1. Normalization in the Sticky-Wage Economy

In addition to normalizing prices as we had in (2.1), we need to normalize the wages of the j consumer-types. Let

ω(j, s^{t−1}) = W (j, s^{t−1})/M^{s}(s^{t−j})
ω(s^{t−1}) = W (s^{t−1})/M^{s}(s^{t−1})
ω̄(s^{t}) = W̄ (s^{t})/M^{s}(s^{t−1})
ω^{∗}(j, s^{t−1}) = W^{∗}(j, s^{t−1})/M^{s}^{∗}(s^{t−j})
ω^{∗}(s^{t−1}) = W^{∗}(s^{t−1})/M^{s}^{∗}(s^{t−1})
ω̄^{∗}(s^{t}) = W̄^{∗}(s^{t})/M^{s}^{∗}(s^{t−1})
m^{d}(j, s^{t}) = M^{d}(j, s^{t})/P (s^{t}) = (\frac{M^{d}(j, s^{t})}{M^{s}(s^{t})}) (\frac{M^{s}(j, s^{t−1})μ(s^{t})}{P (s^{t})})
m^{d}^{∗}(j, s^{t}) = M^{d}^{∗}(j, s^{t})/P^{∗}(s^{t}) = (\frac{M^{d}^{∗}(j, s^{t})}{M^{s}^{∗}(s^{t})}) (\frac{M^{s}^{∗}(j, s^{t−1})μ^{∗}(s^{t})}{P^{∗}(s^{t})})

As in the case of prices, we assume that cohort 1 is the group changing wages. Therefore, when we normalize the wage equation, we have

ω(s^{t−1})M^{s}(s^{t−1})

= \frac{− ∑_{τ} ∑_{s^{τ}} β^{τ−1}π(s^{τ} |s^{t−1})ω̄(s^{τ})^{\frac{1}{1−v}} L^{d}(s^{τ})U_{l}(τ −t+1, s^{τ})M^{s}(s^{τ−1})^{\frac{1}{1−v}}}{v ∑_{τ} ∑_{s^{τ}} β^{τ−1}π(s^{τ} |s^{t−1})ω̄(s^{τ})^{\frac{1}{1−v}} L^{d}(s^{τ})U_{c}(τ −t+1, s^{τ})/p(s^{τ})M^{s}(s^{τ−1})^{\frac{v}{1−v}}}

or

ω(s^{t−1})  (2.63)

= \frac{− ∑_{τ} ∑_{s^{τ}} β^{τ−1}π(s^{τ} |s^{t−1})ω̄(s^{τ})^{\frac{1}{1−v}} L^{d}(s^{τ})U_{l}(τ −t+1, s^{τ}) (μ(s^{t}) … μ(s^{τ−1}))^{\frac{1}{1−v}}}{v ∑_{τ} ∑_{s^{τ}} β^{τ−1}π(s^{τ} |s^{t−1})ω̄(s^{τ})^{\frac{1}{1−v}} L^{d}(s^{τ})U_{c}(τ −t+1, s^{τ})/p(s^{τ}) (μ(s^{t}) … μ(s^{τ−1}))^{\frac{v}{1−v}}}

Notice that the indices for the marginal utilities are τ − t + 1 which is 1, 2, …, N when we write out the sums.

The relationship between the aggregate and individual wages is normalized as follows:

ω̄(s^{t}) = [\frac{1}{N} ∑_{j=1}^{N} (\frac{W (s^{t−j})}{M^{s}(s^{t−1})})^{\frac{v}{v−1}} ]^{\frac{v−1}{v}}

= [\frac{1}{N} ω(s^{t−1})^{\frac{v}{v−1}} + \frac{1}{N} (\frac{ω(s^{t−2})}{μ(s^{t−1})})^{\frac{v}{v−1}} + ⋯

+ \frac{1}{N} (\frac{ω(s^{t−N})}{μ(s^{t−1}) … μ(s^{t−N +1})})^{\frac{v}{v−1}} ]^{\frac{v−1}{v}}

2.5.2. Steady State in the Sticky-Wage Economy

Let’s follow the same strategy as in the benchmark and change what we need to. First, guess the capital stocks, k(i), i = 1, … N, output y, the consumption levels c(j), j = 2, … N, and the money demands m^{d}(j), j = 1, … N. Next get the x(i)’s, the y_{H}(i)’s, the y_{H}^{∗}(i)’s, and the F (i)’s as in the benchmark case. Using F (i)’s and k(i)’s, we can back out the labor demands for each firm i, i.e., L^{d}(i), using the production technology.

Having the L^{d}(i)’s we can determine the F_{k}(i)’s and then back out marginal costs via the usual capital Euler equations. The marginal cost of i is the real wage (w̄/p) divided by the marginal product F_{l}(i). We have everything needed to construct the marginal product but the real wage is a function of all of the U_{l}(j)’s and U_{c}(j)’s – which we don’t yet have. But to get these marginal utilities, we need the c(1), the L^{s}(j)’s and the M^{d}(j)’s.

The first consumption level is derived with the resource constraint

c(1) = N (y − x − g) − ∑_{j=2}^{N} c(j).

We can back out L^{s}(j)’s from the labor demand functions:

L^{s}(j) = (\frac{μ^{j−1}ω̄}{ω} )^{\frac{1}{1−v}} \frac{1}{N} ∑_{i=1}^{N} L^{d}(i) = (\frac{μ^{j−1}ω̄}{ω} )^{\frac{1}{1−v}} L^{d}

Note that L^{s}(j) is a function of the μ’s and the total labor demand which we know. We can back out the aggregate price from the definition of m^{d}(j) as follows:

p = μ/[\frac{1}{N} ∑_{j} m^{d}(j)].

With all of the consumptions, labor supplies, and money demands we can compute marginal utilities and, in turn, the steady state wage:

ω = −\frac{p}{v} (\frac{U_{l}(1) + U_{l}(2)βμ^{\frac{1}{1−v}} + U_{l}(3)β^{2}μ^{\frac{2}{1−v}} + … + U_{l}(N)β^{N −1}μ^{\frac{N −1}{1−v}}}{U_{c}(1) + U_{c}(2)βμ^{\frac{v}{1−v}} + U_{c}(3)β^{2}μ^{\frac{2v}{1−v}} + … + U_{c}(N)β^{N −1}μ^{\frac{(N −1)v}{1−v}}} )

With ω, we can derive the steady state aggregate wage as follows:

ω̄ = ω [\frac{1}{N} (1 + μ^{\frac{v}{1−v}} + … μ^{\frac{(N −1)v}{1−v}} )]^{\frac{v−1}{v}} .

We can use the following equations to check that we have a fixed point:

mc(i) = ω̄/(pF_{l}(i)), i = 1, …, N
p_{H} = \frac{p}{θ} (\frac{mc(1) + mc(2)βμ^{\frac{1}{1−θ}} + mc(3)β^{2}μ^{\frac{2}{1−θ}} + … + mc(N)β^{N−1}μ^{\frac{N−1}{1−θ}}}{1 + βμ^{\frac{θ}{1−θ}} + β^{2}μ^{\frac{2θ}{1−θ}} + … + β^{N−1}μ^{\frac{(N−1)θ}{1−θ}}} )
U_{m}(1) = U_{c}(1)(1 − β/μ)
U_{c}(j) = U_{c}(1), j = 2, …, N
U_{m}(j) = U_{m}(1), j = 2, …, N .

Finally, our assumption about common preferences implies that q = 1 in a steady state.

2.5.3. Linearized Equations in the Sticky-Wage Economy

Most of the first-order conditions look the same in the benchmark and sticky-wage economies. The utility functions must be indexed but the form of the linearized equations remain the same. In this section, we derive formulas for the additional equations needed in the stickywage economy.

Let’s start with the deterministic wage equation:

ω(s^{t−1}) = \frac{− ∑_{τ} β^{τ−1}ω̄(s^{τ})^{\frac{1}{1−v}} L^{d}(s^{τ})U_{l}(τ −t+1, s^{τ}) (μ(s^{t}) … μ(s^{τ−1}))^{\frac{1}{1−v}}}{v ∑_{τ} β^{τ−1}ω̄(s^{τ})^{\frac{1}{1−v}} L^{d}(s^{τ})U_{c}(τ −t+1, s^{τ})/p(s^{τ}) (μ(s^{t}) … μ(s^{τ−1}))^{\frac{v}{1−v}}}

First, rewrite this as:

vω(s^{t−1}) [… + β^{i}U_{c}(i + 1, s^{t+i})/p(s^{t+i})ω̄(s^{t+i})^{\frac{1}{1−v}} L^{d}(s^{t+i}) (μ(s^{t}) ⋯ μ(s^{t+i−1}))^{\frac{v}{1−v}} + …]
= … − β^{i}U_{l}(i + 1, s^{t+i})ω̄(s^{t+i})^{\frac{1}{1−v}} L^{d}(s^{t+i}) (μ(s^{t}) ⋯ μ(s^{t+i−1}))^{\frac{1}{1−v}} − … (2.64)

and then do the linearization of (2.64) in pieces:

U_{c}(i + 1, s^{t+i})/p(s^{t+i})ω̄(s^{t+i})^{\frac{1}{1−v}} L^{d}(s^{t+i}) (μ(s^{t}) ⋯ μ(s^{t+i−1}))^{\frac{v}{1−v}}

≈ U_{c}/pω̄^{\frac{1}{1−v}} L^{d}μ^{\frac{vi}{1−v}} [\frac{U_{cc}(i + 1)c(i + 1)}{U_{c}} ĉ_{i+1,t+i} + \frac{U_{cl}(i + 1)L^{s}(i + 1)}{U_{c}} L̂_{i+1,t+i}^{s}

+ \frac{U_{cm}(i + 1)M^{d}(i + 1)/P}{U_{c}(i + 1)} m̂_{i+1,t+i}^{d} − p̂_{t+i} + \frac{1}{1 − v} ˆ ω̄_{t+i} + L̂_{t+i}^{d}

+ \frac{v}{1 − v} (μ̂_{t} + … μ̂_{t+i−1})]

U_{l}(i + 1, s^{t+i})ω̄(s^{t+i})^{\frac{1}{1−v}} L^{d}(s^{t+i}) (μ(s^{t}) ⋯ μ(s^{t+i−1}))^{\frac{1}{1−v}}

≈ U_{l}(i + 1)ω̄^{\frac{1}{1−v}} L^{d}μ^{\frac{i}{1−v}} [\frac{U_{cl}(i + 1)c(i + 1)}{U_{l}(i + 1)}ĉ_{i+1,t+i} + \frac{U_{ll}(i + 1)L^{s}(i + 1)}{U_{l}(i + 1)}L̂_{i+1,t+i}^{s}

+ \frac{U_{lm}M^{d}(i + 1)/P}{U_{l}(i + 1)} m̂_{i+1,t+i}^{d} + \frac{1}{1 − v} ˆ ω̄_{t+i} + L̂_{t+i}^{d}

+ \frac{1}{1 − v} (μ̂_{t} + … μ̂_{t+i−1})]

Therefore, the full equation is:

vω/pω̄^{\frac{1}{1−v}} L^{d}U_{c}(1 + βμ^{\frac{v}{1−v}} + …)ω̂_{t−1}

+ vω/pω̄^{\frac{1}{1−v}} L^{d}\{… + (βμ^{\frac{v}{1−v}})^{i}U_{c}[\frac{U_{cc}(i + 1)c(i + 1)}{U_{c}} ĉ_{i+1,t+i}

+ \frac{U_{cl}(i + 1)L^{s}(i + 1)}{U_{c}} L̂_{i+1,t+i}^{s} + \frac{U_{cm}(i + 1)M^{d}(i + 1)/P}{U_{c}} m̂_{i+1,t+i}^{d}

− p̂_{t+i} + \frac{1}{1 − v} ˆ ω̄_{t+i} + L̂_{t+i}^{d} + \frac{v}{1 − v} (μ̂_{t} + … μ̂_{t+i−1})] + …\}

= ω̄^{\frac{1}{1−v}} L^{d}\{… − (βμ^{\frac{1}{1−v}})^{i}U_{l}(i + 1)[\frac{U_{cl}(i + 1)c(i + 1)}{U_{l}(i + 1)}ĉ_{i+1,t+i}

+ \frac{U_{ll}(i + 1)L^{s}(i + 1)}{U_{l}(i + 1)}L̂_{t+i}^{s} + \frac{U_{lm}(i + 1)M^{d}(i + 1)/P}{U_{l}(i + 1)} m̂_{i+1,t+i}^{d}

+ \frac{1}{1 − v} ˆ ω̄_{t+i} + L̂_{t+i}^{d} + \frac{1}{1 − v} (μ̂_{t} + … μ̂_{t+i−1})] − …\}  (2.65)

Crossing out common coefficients in (2.65) and dividing by the coefficient on ω, we get

ω̂_{t−1} =

\frac{p}{vωU_{c}(1 + βμ^{\frac{v}{1−v}} + …)}\{… − (βμ^{\frac{1}{1−v}})^{i}U_{l}(i + 1)[\frac{U_{cl}(i + 1)c(i + 1)}{U_{l}(i + 1)}ĉ_{i+1,t+i}

+ \frac{U_{ll}(i + 1)L^{s}(i + 1)}{U_{l}(i + 1)}L̂_{t+i}^{s} + \frac{U_{lm}(i + 1)M^{d}(i + 1)/P}{U_{l}(i + 1)} m̂_{i+1,t+i}^{d}

+ \frac{1}{1 − v} ˆ ω̄_{t+i} + L̂_{t+i}^{d} + \frac{1}{1 − v} (μ̂_{t} + … μ̂_{t+i−1})] − …\}

− \frac{1}{1 + βμ^{\frac{v}{1−v}} + …}\{… + (βμ^{\frac{v}{1−v}})^{i}[\frac{U_{cc}(i + 1)c(i + 1)}{U_{c}} ĉ_{i+1,t+i}

+ \frac{U_{cl}(i + 1)L^{s}(i + 1)}{U_{c}} L̂_{i+1,t+i}^{s} + \frac{U_{cm}(i + 1)M^{d}(i + 1)/P}{U_{c}} m̂_{i+1,t+i}^{d}

− p̂_{t+i} + \frac{1}{1 − v} ˆ ω̄_{t+i} + L̂_{t+i}^{d} + \frac{v}{1 − v} (μ̂_{t} + … μ̂_{t+i−1})] + …\}.  (2.66)

Finally, we use the steady state equation for p in (2.66) to get ω̂_{t−1} =

\{… + \frac{(βμ^{\frac{1}{1−v}})^{i}U_{l}(i + 1)}{∑(βμ^{\frac{1}{1−v}})^{i}U_{l}(i + 1)}[\frac{U_{cl}(i + 1)c(i + 1)}{U_{l}(i + 1)}ĉ_{i+1,t+i}

+ \frac{U_{ll}(i + 1)L^{s}(i + 1)}{U_{l}(i + 1)}L̂_{t+i}^{s} + \frac{U_{lm}(i + 1)M^{d}(i + 1)/P}{U_{l}(i + 1)} m̂_{i+1,t+i}^{d}

+ \frac{1}{1 − v} ˆ ω̄_{t+i} + L̂_{t+i}^{d} + \frac{1}{1 − v} (μ̂_{t} + … μ̂_{t+i−1})] − …\}

− \{… + \frac{(βμ^{\frac{v}{1−v}})^{i}}{∑(βμ^{\frac{v}{1−v}})^{i}} [\frac{U_{cc}(i + 1)c(i + 1)}{U_{c}} ĉ_{i+1,t+i}

+ \frac{U_{cl}(i + 1)L^{s}(i + 1)}{U_{c}} L̂_{i+1,t+i}^{s} + \frac{U_{cm}(i + 1)M^{d}(i + 1)/P}{U_{c}} m̂_{i+1,t+i}^{d}

− p̂_{t+i} + \frac{1}{1 − v} ˆ ω̄_{t+i} + L̂_{t+i}^{d} + \frac{v}{1 − v} (μ̂_{t} + … μ̂_{t+i−1})] + …\}.

Putting expectations back in, we get the following linearized wage equation:

ω̂_{t−1} = E_{t−1} ∑_{i=0}^{N −1} ω_{1,i}(\frac{Û_{l,i+1,t+i}}{U_{l}(i + 1)} + \frac{1}{1 − v} ˆ ω̄_{t+i} + L̂_{t+i}^{d} + \frac{1}{1 − v} (μ̂_{t} + … μ̂_{t+i−1}))

− E_{t−1} ∑_{i=0}^{N −1} ω_{2,i}(\frac{Û_{c,i+1,t+i}}{U_{c}} − p̂_{t+i} + \frac{1}{1 − v} ˆ ω̄_{t+i} + L̂_{t+i}^{d}

+ \frac{v}{1 − v} (μ̂_{t} + … μ̂_{t+i−1}))  (2.67)

where

ω_{1,i} = \frac{(βμ^{\frac{1}{1−v}})^{i}U_{l}(i + 1)}{∑(βμ^{\frac{1}{1−v}})^{i}U_{l}(i + 1)}, ω_{2,i} = \frac{(βμ^{\frac{v}{1−v}})^{i}}{∑(βμ^{\frac{v}{1−v}})^{i}}

and Û_{l,i,t}, U_{c,i,t} is shorthand for the log-linearized marginal utilities. Note that in the case with zero-inflation, the linearized pricing equation simplifies to:

ω̂_{t−1} = E_{t−1} ∑_{i=0}^{N −1} β^{i}(p̂_{t+i} + \frac{Û_{l,i+1,t+i}}{U_{l}} − \frac{Û_{c,i+1,t+i}}{U_{c}} + μ̂_{t} + … μ̂_{t+i−1})/ ∑_{i=0}^{N −1} β^{i}

From the first-order conditions of consumer j, we also get a relationship for labor supply which when linearized is

L̂_{j,t}^{s} = L_{t}^{d} + \frac{1}{1 − v} (ˆω̄_{t} − ω̂_{t−j} + μ̂_{t−j+1} + … + μ̂_{t−1}).  (2.68)

This equation depends on the individual wages, which will be in the state vector, and the aggregate given by

ˆω̄_{t} = [1 + μ^{\frac{v}{1−v}} + … μ^{\frac{(N −1)v}{1−v}} ]^{−1} [ω̂_{t−1} + μ^{\frac{v}{1−v}} (ω̂_{t−2} − μ̂_{t−1}) + …
+ μ^{\frac{(N −1)v}{1−v}} (ω̂_{t−N} − μ̂_{t−1} − … − μ̂_{t−N +1})]. (2.69)

Note that the labor demands are going to be derived by the same equation as in the benchmark, except that we use the notation L^{d} here:

y_{H}(i) ŷ_{H,i,t} + y_{H}^{∗}(i) ŷ_{H,i,t}^{∗} = F_{k}(i)k(i − 1) k̂_{i−1,t−1} + F_{l}(i)L^{d}(i) (L̂_{i,t}^{d} + Â_{t}).  (2.70)

Similarly, with marginal costs, we use L^{d}:

m̂c_{i,t} = ŵ_{t} − F_{kl}(i)k(i − 1)/F_{l}(i)k̂_{i−1,t−1} − F_{ll}(i)L^{d}(i)/F_{l}(i)(L̂_{i,t}^{d} + Â_{t}) − Â_{t} (2.71)

where ŵ_{t} is now:

ŵ_{t} = ˆ ω̄_{t} − p̂_{t}.  (2.72)

From the first-order conditions (1.57) and (1.58), we get

U_{cc}(j)c(j)ĉ_{j,t} + U_{cl}(j)L^{s}(j)L̂_{j,t}^{s} + U_{cm}(j)M^{d}(j)/P m̂_{j,t}^{d}
= U_{cc}(k)c(k)ĉ_{k,t} + U_{cl}(k)L^{s}(k)L̂_{k,t}^{s} + U_{cm}(k)M^{d}(k)/P m̂_{k,t}^{d}  (2.73)
U_{cm}(j)c(j)ĉ_{j,t} + U_{lm}(j)L^{s}(j)L̂_{j,t}^{s} + U_{mm}(j)M^{d}(j)/P m̂_{j,t}^{d}
= U_{cm}(k)c(k)ĉ_{k,t} + U_{lm}(k)L^{s}(k)L̂_{k,t}^{s} + U_{mm}(k)M^{d}(k)/P m̂_{k,t}^{d}  (2.74)

for all j, k ∈ {1, …, N }.

Finally, we need the labor market clearing condition, the money market clearing condition, and the resource constraint:

L̂_{t}^{d} = (L^{d}(1) L̂_{1,t}^{d} + L^{2}(2) L̂_{2,t}^{d} + … L^{d}(N) L̂_{N,t}^{d})/ ∑_{i} L^{d}(i)  (2.75)
μ̂_{t} − p̂_{t} = \frac{1}{N} (m̂_{1,t}^{d} + m̂_{2,t}^{d} + … + m̂_{N ,t}^{d})  (2.76)
ŷ_{t} = ([c(1) ĉ_{1,t} + … c(N) ĉ_{N ,t}]/N + [x(1) x̂_{1,t} + … x(N) x̂_{N,t}]/N + g ĝ_{t})/y(2.77)

2.5.4. Solving the Linearized System in the Sticky-Wage Economy The system of equations that we solve has 2N + 7 dynamic equations:

• 4 pricing equations, (2.30)-(2.33);

• 2N Euler equations for capital ((2.36) for home and similar for foreign);

• 2 money demand equations ((1.55) for home and similar for foreign);

• 2 wage-setting equations, ((2.67) for home and similar for foreign);

• and static equations and definitions that determine:

∘ ŷ_{H,i}, y_{F,i}, y_{F,i}^{∗}, y_{H,i}^{∗} from (2.10) and analogues; ∘ p̂, p̂^{∗} from (2.14)-(2.15); ∘ ˆ p̄_{H}, ˆ p̄_{F}, ˆ p̄_{F}^{∗}, ˆ p̄_{H}^{∗} from (2.16)-(2.19) ∘ q̂ from (2.24); ∘ λ̂_{H}, λ̂_{F}, λ̂_{F}^{∗}, λ̂_{H}^{∗} from (2.27) and analogues. ∘ L̂_{i}^{d}, L̂^{d}_{i}^{∗} from (2.70) and foreign analogue; ∘ x̂_{i}, x̂_{i}^{∗} from (2.35) and foreign analogue; ∘ m̂c_{i}, m̂c_{i}^{∗} from (2.71) and foreign analogue; ∘ L̂_{j}^{s}, L̂^{s}_{j}^{∗} from (2.68) and foreign analogue; ∘ ŵ, ŵ^{∗} from (2.72) and foreign analogue; ∘ ĉ_{j}, m̂_{j}^{d}, ĉ_{j}^{∗}, m̂^{d}_{j}^{∗} from (2.73), (2.74), (2.76), (2.77) and the foreign analogues.

We can write the system of equations in terms of a subset of our variables and back out all variables via the static conditions listed above. We turn to this next.

We introduce a new index ℵ = max(N, N) because we will need to record sufficient lags and leads of the variables. We will use the following vectors in our computation:

z_{t} = [p̂_{H,t−1}, p̂_{F,t−1}, p̂_{F,t−1}^{∗}, p̂_{H,t−1}^{∗}, k̂_{1,t}, … k̂_{N,t}, k̂_{1,t}^{∗}, … k̂_{N,t}^{∗}, ŷ_{t}, ŷ_{t}^{∗}, ω̂_{t−1}, ω̂_{t−1}^{∗}]′  (n_{z} × 1)
X_{t} = [p̂_{H,t−2}, …, p̂_{H,t−N}, p̂_{F,t−2}, …, p̂_{F,t−N}, p̂_{F,t−2}^{∗}, …, p̂_{F,t−N}^{∗}, p̂_{H,t−2}^{∗}, …, p̂_{H,t−N}^{∗},

k̂_{1,t−1}, …, k̂_{N,t−1}, k̂_{1,t−1}^{∗}, …, k̂_{Nt−1}^{∗}, ω̂_{t−2}, …, ω̂_{t−N} , ω̂_{t−2}^{∗}, …, ω̂_{t−N}^{∗}] (n_{X} × 1)

Z_{t} = [z_{t+ℵ−1}, z_{t+ℵ−2}, …, z_{t}, X_{t}, μ̂_{t+ℵ−1}, …, μ̂_{t−ℵ+1}, μ̂_{t+ℵ−1}^{∗}, …, μ̂_{t−ℵ+1}^{∗},
ĝ_{t+ℵ−1}, …, ĝ_{t}, ĝ_{t+ℵ−1}^{∗}, …, ĝ_{t}^{∗}, â_{t+ℵ−1}, …, â_{t}, â_{t+ℵ−1}^{∗}, …, â_{t}^{∗}]′
Z_{t} = [z_{t}, z_{t−1}, …, z_{t−ℵ+2}]′  (n_{Z} × 1)
S_{t} = [μ̂_{t}, …, μ̂_{t−ℵ+1}, μ̂_{t}^{∗}, …, μ̂_{t−ℵ+1}^{∗},
ĝ_{t}, ĝ_{t−1}, ĝ_{t}^{∗}, ĝ_{t−1}^{∗}, â_{t}, â_{t−1}, â_{t}^{∗}, â_{t−1}^{∗}]′ (n_{S} × 1)

The vector z_{t} contains the choice variables at time t. It has n_{z} = 2N + 8 elements. The vector X_{t} are the state variables at time t. There are n_{X} = 6N + 2N − 6 state variables.

The vector Z_{t} contains all variables that appear in the residual equations. The vectors Z_{t} and S_{t} are used when we characterize the solution, Z_{t} = AZ_{t−1} + BS_{t}, where Z has n_{Z} = (ℵ − 1)n_{z} elements and S has n_{S} = 2ℵ + 8 elements.

The residual equations can be written succinctly as follows:

E [A_{1} [ \begin{matrix}X_{t+1} \\ Z_{t+ℵ−1} \end{matrix} ] + A_{2} [ \begin{matrix}X_{t} \\ Z_{t+ℵ−2} \end{matrix} ] + shock terms|Ω_{t}] = 0

where E implies that expectations are taken – but we will assume that different information sets for the different residual equations. For our example, the residuals are denoted R(Z) and the matrix A_{1} is given by

A_{1} = [ \begin{matrix}I_{n_{X},n_{X}} & 0_{n_{X},n_{z}} & 0_{n_{X},n_{z}} & … & 0_{n_{X},n_{z}} \\ 0_{n_{z},n_{X}} & \frac{dR}{dZ} (:, 1 : n_{z}) & \frac{dR}{dZ} (:, n_{z} +1 : 2n_{z}) & … & \frac{dR}{dZ} (:, (ℵ−2)n_{z} +1 : (ℵ−1)n_{z}) \\ 0_{n_{z},n_{X}} & 0_{n_{z},n_{z}} & I_{n_{z},n_{z}} & … & 0_{n_{z},n_{z}} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0_{n_{z},n_{X}} & 0_{n_{z},n_{z}} & 0_{n_{z},n_{z}} & … & I_{n_{z},n_{z}} \end{matrix} ]

(2.78)

and matrix A_{2} is given by:

A_{2} = [ \begin{matrix}−I_{1} & 0_{n_{X},n_{z}} & … & 0_{n_{X},n_{z}} & −I_{2} \\ \frac{dR}{dZ} (:, ℵn_{z} +1 : ℵn_{z} +n_{X}) & 0_{nz,nz} & … & 0_{n_{z},n_{z}} & \frac{dR}{dZ} (:, (ℵ−1)n_{z} +1 : ℵn_{z}) \\ 0_{n_{z},n_{X}} & I_{n_{z},n_{z}} & 0_{n_{z},n_{z}} & … & 0_{n_{z},n_{z}} \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0_{n_{z},n_{X}} & 0_{n_{z},n_{z}} & … & I_{n_{z},n_{z}} & 0_{n_{z},n_{z}} \end{matrix} ] .

(2.79)

The matrices I_{1} and I_{2} in A_{2} are given by

I_{1} = [ \begin{matrix}I_{4,4} ⊗ [ \begin{matrix}0 & … & 0 & 0 \\ 1 & … & 0 & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ 0 & … & 1 & 0 \end{matrix} ]_{N−1} , & 0_{4N−4,2N} & 0_{4N−4,2N −2} \\ 0_{2N,4N−4} & 0_{2N,2N} & 0_{2N,2N −2} \\ 0_{2N −2,4N−4} & 0_{2N −2,2N} & I_{2,2} ⊗ [ \begin{matrix}0 & … & 0 & 0 \\ 1 & … & 0 & 0 \\ ⋮ & ... & ⋮ & ⋮ \\ 0 & … & 1 & 0 \end{matrix} ]_{N −1} \end{matrix} ]

I_{2} = [ \begin{matrix}I_{4,4} ⊗ [ \begin{matrix}1 \\ 0_{N−2,1} \end{matrix} ] , & 0_{4N−4,2N} & 0_{4N−4,2} & 0_{4N−4,2} \\ 0_{2N,4} & I_{2N,2N} & 0_{2N,2} & 0_{2N,2} \\ 0_{2N −2,4} & 0_{2N −2,2N} & 0_{2N −2,2} & I_{2,2} ⊗ [ \begin{matrix}1 \\ 0_{N −2,1} \end{matrix} ] \end{matrix} ]

Using the method laid out in Blanchard and Kahn (1980), we construct eigenvalues of −A_{1}^{−1}A_{2} if A_{1} is invertible and generalized eigenvalues otherwise. Then, ignoring shock terms, we have

[ \begin{matrix}X_{t+1} \\ Z_{t+ℵ−1} \end{matrix} ] = V ΛV^{−1} [ \begin{matrix}X_{t} \\ Z_{t+ℵ−2} \end{matrix} ] .

We can sort eigenvalues inside and outside the unit circle. If there are n_{X} stable eigenvalues (which is the number of state variables in X), then we have a locally determinate system. Suppose that the eigenvectors in V and eigenvalues in Λ are sorted so that the upper left partition of Λ contains the stable eigenvalues. Then,

X_{t+1} = V_{11}Λ_{1}V_{11}^{−1} X_{t}
Z_{t+ℵ−2} = V_{21}V_{11}^{−1} X_{t}.

The last n_{z} elements of Z_{t+ℵ−2} are those of z_{t}. Therefore, we have a relationship between our decision variables z and the state variables X. If we want to write the system as (2.40), then we can use this relationship between z and X to fill in the elements of A. In particular, we set

A(1 : n_{z}, 1 : n_{z} − 4) = A_{zX}(:, [1, N, 2N − 1, 3N − 2, 4N − 3 : 6N − 4]) A(1 : n_{z}, n_{z} + 1 : n_{z} : (N − 1)n_{z}) = A_{zX}(:, 2 : N − 1) A(1 : n_{z}, n_{z} + 2 : n_{z} : (N − 1)n_{z}) = A_{zX}(:, N + 1 : 2N − 2) A(1 : n_{z}, n_{z} + 3 : n_{z} : (N − 1)n_{z}) = A_{zX}(:, 2N : 3N − 3) A(1 : n_{z}, n_{z} + 4 : n_{z} : (N − 1)n_{z}) = A_{zX}(:, 3N − 1 : 4N − 4) A(1 : n_{z}, n_{z} − 1 : n_{z} : (N − 1)n_{z}) = A_{zX}(:, 6N − 3 : 6N + N − 5) A(1 : n_{z}, n_{z} : n_{z} : (N − 1)n_{z}) = A_{zX}(:, 6N + N − 4 : 6N + 2N − 6) A(n_{z} + 1 : n_{Z}, 1 : n_{Z} − n_{z}) = I_{n_{Z}−n_{z},n_{Z}−n_{z}} where A_{zX} comes from z_{t} = A_{zX}X_{t}.

The next step is to compute B:

B = [ B_{1} ; 0_{n_{z},n_{S}} ; ⋮ ; 0_{n_{z},n_{S}} ] = [ I_{n_{z},n_{z}} ; 0_{n_{z},n_{z}} ; ⋮ ; 0_{n_{z},n_{z}} ] B_{1} ≡ SB_{1}.

We will use S below in order to reduce the problem of computing B to one of computing B_{1}.

To derive expressions for the elements of B, we first note that the residuals can be written as follows:

E[a_{0}Z_{t+ℵ−1} + a_{1}Z_{t+ℵ−2} + … + a_{ℵ−1}Z_{t} + a_{ℵ}Z_{t−1}
+ b_{0}S_{t+ℵ−1} + b_{1}S_{t+ℵ−2} + … + b_{ℵ−1}S_{t}|Ω_{t}] = 0

Using the definitions of Z and Z, we can write:

a_{0} = dR/dZ(:, 1 : (ℵ − 1)n_{z})

a_{ℵ−1}(:, 1 : n_{z}) = dR/dZ(:, (ℵ − 1)n_{z} + 1 : ℵn_{z})

b_{0}(:, 1 : ℵ) = dR/dZ(:, ℵn_{z} + n_{X} + 1 : ℵn_{z} + n_{X} + ℵ)

b_{0}(:, ℵ + 1 : 2ℵ) = dR/dZ(:, ℵn_{z} + n_{X} + 2ℵ : ℵn_{z} + n_{X} + 3ℵ − 1)

b_{0}(:, 2ℵ + 1) = dR/dZ(:, ℵn_{z} + n_{X} + 4ℵ − 1)

b_{0}(:, 2ℵ + 3) = dR/dZ(:, ℵn_{z} + n_{X} + 5ℵ − 1)

b_{0}(:, 2ℵ + 5) = dR/dZ(:, ℵn_{z} + n_{X} + 6ℵ − 1)

b_{0}(:, 2ℵ + 7) = dR/dZ(:, ℵn_{z} + n_{X} + 7ℵ − 1)

b_{1}(:, 2ℵ + 1) = dR/dZ(:, ℵn_{z} + n_{X} + 4ℵ)

b_{1}(:, 2ℵ + 3) = dR/dZ(:, ℵn_{z} + n_{X} + 5ℵ) b_{1}(:, 2ℵ + 5) = dR/dZ(:, ℵn_{z} + n_{X} + 6ℵ)

b_{1}(:, 2ℵ + 7) = dR/dZ(:, ℵn_{z} + n_{X} + 7ℵ)

⋮

b_{ℵ−2}(:, 2ℵ + 1) = dR/dZ(:, ℵn_{z} + n_{X} + 5ℵ − 3)

b_{ℵ−2}(:, 2ℵ + 3) = dR/dZ(:, ℵn_{z} + n_{X} + 6ℵ − 3)

b_{ℵ−2}(:, 2ℵ + 5) = dR/dZ(:, ℵn_{z} + n_{X} + 7ℵ − 3)

b_{ℵ−2}(:, 2ℵ + 7) = dR/dZ(:, ℵn_{z} + n_{X} + 8ℵ − 3)

b_{ℵ−1}(:, 2 : ℵ) = dR/dZ(:, ℵn_{z} + n_{X} + ℵ + 1 : ℵn_{z} + n_{X} + 2ℵ − 1)

b_{ℵ−1}(:, ℵ + 2 : 2ℵ) = dR/dZ(:, ℵn_{z} + n_{X} + 3ℵ : ℵn_{z} + n_{X} + 4ℵ − 2)

b_{ℵ−1}(:, 2ℵ + 1) = dR/dZ(:, ℵn_{z} + n_{X} + 5ℵ − 2)

b_{ℵ−1}(:, 2ℵ + 3) = dR/dZ(:, ℵn_{z} + n_{X} + 6ℵ − 2)

b_{ℵ−1}(:, 2ℵ + 5) = dR/dZ(:, ℵn_{z} + n_{X} + 7ℵ − 2)

b_{ℵ−1}(:, 2ℵ + 7) = dR/dZ(:, ℵn_{z} + n_{X} + 8ℵ − 2)

with all other coefficients but a_{ℵ} set equal to 0. The matrix a_{ℵ} is nonzero but it is not used in computing B.

Using the solution in (2.40) we get:

E[a_{0} (A^{ℵ}Z_{t−1} + BS_{t+ℵ−1} + ABS_{t+ℵ−2} + … + A^{ℵ−1}BS_{t}) + a_{1} (A^{ℵ−1}Z_{t−1} + BS_{t+ℵ−2} + ABS_{t+ℵ−3} + … + A^{ℵ−2}BS_{t}) + …

+ a_{ℵ−1} (AZ_{t−1} + BS_{t}) + a_{ℵ}Z_{t−1}
+ b_{0}S_{t+ℵ−1} + b_{1}S_{t+ℵ−2} + … + b_{ℵ−1}S_{t}|Ω_{t}] = 0  (2.80)

The matrix P in S_{t+1} = PS_{t} + ϵ_{t+1} is the same as in the benchmark case except that we use ℵ in place of N. (See equation (2.58).)

If we assume that the timing of the household/union wages are the same as that for the monopolist, then we have to do a little more work to get D. Suppose first that households and firms see Ω_{t} = {μ̂_{s}, μ̂_{s}^{∗}, ĝ_{s}, ĝ_{s}^{∗}, â_{s}, â_{s}^{∗}}_{s=0}^{t−1} when making their pricing decisions. Taking expectations is the same as in the benchmark economy with multiple shocks except here we have 2 additional equations to restrict – those related to the wage rates. The matrix D in this case is

D = [ \begin{matrix}I_{4,4} ⊗ Φ & 0_{4(n_{S}−6),n_{z}n_{S}−6n_{S}} & 0_{4(n_{S}−6),2n_{S}} \\ 0_{n_{z}n_{S}−6n_{S},4n_{S}} & I_{n_{z}n_{S}−6n_{S}} & 0_{n_{z}n_{S}−6n_{S},2n_{S}} \\ 0_{2(n_{S}−6),4n_{S}} & 0_{2(n_{S}−6),n_{z}n_{S}−6n_{S}} & I_{2,2} ⊗ Φ \end{matrix} ]

If agents can see technology shocks, then

D = [ \begin{matrix}I_{4,4} ⊗ Φ & 0_{4(n_{S}−4),n_{z}n_{S}−6n_{S}} & 0_{4(n_{S}−4),2n_{S}} \\ 0_{n_{z}n_{S}−6n_{S},4n_{S}} & I_{n_{z}n_{S}−6n_{S}} & 0_{n_{z}n_{S}−6n_{S},2n_{S}} \\ 0_{2(n_{S}−4),4n_{S}} & 0_{2(n_{S}−4),n_{z}n_{S}−6n_{S}} & I_{2,2} ⊗ Φ \end{matrix} ]

Finally, if agents can see both technology and government spending shocks, then

D = [ \begin{matrix}I_{4,4} ⊗ Φ & 0_{4(n_{S}−2),n_{z}n_{S}−6n_{S}} & 0_{4(n_{S}−2),2n_{S}} \\ 0_{n_{z}n_{S}−6n_{S},4n_{S}} & I_{n_{z}n_{S}−6n_{S}} & 0_{n_{z}n_{S}−6n_{S},2n_{S}} \\ 0_{2(n_{S}−2),4n_{S}} & 0_{2(n_{S}−2),n_{z}n_{S}−6n_{S}} & I_{2,2} ⊗ Φ \end{matrix} ]

3. Formulas for Preferences

We will consider three alternative functional forms for the utility function. In these notes, I refer to them as ‘separable preferences,’ ‘nonseparable preferences I’ and ‘nonseparable preferences II’.

3.1. Separable Preferences

The separable preferences are given by

U(c, l, m) = \frac{1}{1 − σ} [(ω c^{\frac{η−1}{η}} + (1 − ω) m^{\frac{η−1}{η}} )^{\frac{η}{η−1}} ]^{1−σ} + ψ(1 − l)^{1−ξ}/(1 − ξ) (3.1)

≡ \frac{1}{1 − σ} Φ(c, m)^{1−σ} + ψ(1 − l)^{1−ξ}/(1 − ξ)

where Φ(c, m) = Ψ^{\frac{η}{η−1}} . The first and second partials of this function are as follows

U_{c} = Φ^{−σ}Ψ^{\frac{1}{η−1}} ωc^{− \frac{1}{η}}
U_{l} = −ψ(1 − l)^{−ξ}
U_{m} = Φ^{−σ}Ψ^{\frac{1}{η−1}} (1 − ω)m^{− \frac{1}{η}}
U_{cc} = −σΦ^{−σ−1}(Ψ^{\frac{1}{η−1}} ωc^{− \frac{1}{η}})^{2}
+ Φ^{−σ}Ψ^{\frac{1}{η−1} −1}ω^{2}/ηc^{− \frac{2}{η}}
− Φ^{−σ}Ψ^{\frac{1}{η−1}} ω/ηc^{− \frac{1}{η} −1}
U_{cl} = 0
U_{lm} = 0
U_{ll} = −ψξ(1 − l)^{−ξ−1}
U_{cm} = −σΦ^{−σ−1}(Ψ^{\frac{1}{η−1}} ωc^{− \frac{1}{η}})(Ψ^{\frac{1}{η−1}} (1 − ω)m^{− \frac{1}{η}})
+ Φ^{−σ}Ψ^{\frac{1}{η−1} −1}ω(1 − ω)/ηc^{− \frac{1}{η}} m^{− \frac{1}{η}}
U_{mm} = −σΦ^{−σ−1}(Ψ^{\frac{1}{η−1}} (1 − ω)m^{− \frac{1}{η}})^{2}
+ Φ^{−σ}Ψ^{\frac{1}{η−1} −1}ω^{2}/ηm^{− \frac{2}{η}}
− Φ^{−σ}Ψ^{\frac{1}{η−1}} (1 − ω)/ηm^{− \frac{1}{η} −1}.

Again, ignoring m, consider calculating the labor supply elasticity. In this case, the elasticity holding U_{c} fixed will be the same as that holding c fixed since U_{c} is not a function of l. With w = ψ(1 − l)^{−ξ}c^{σ}, we get

\frac{d ln l}{d ln w} = \frac{U_{l}}{U_{ll}l} = \frac{1 − l}{ξl} .

If l = 1/4 and ξ = 1.5 then d ln l/d ln w = 2.

3.2. Nonseparable Preferences I For the nonseparable preferences, type I, we assume

U(c, l, m) = \frac{1}{1 − σ} [(ω c^{\frac{η−1}{η}} + (1 − ω) m^{\frac{η−1}{η}} )^{\frac{η}{η−1}} (1 − l)^{ψ}]^{1−σ}  (3.2)

≡ \frac{1}{1 − σ} Φ(c, l, m)^{1−σ}

where

Φ(c, l, m) = Ψ(c, m)^{\frac{η}{η−1}} (1 − l)^{ψ}

where m denotes real balances (M/P). The first and second partial derivatives of this function are as follows

U_{c} = Φ^{−σ}Ψ^{\frac{1}{η−1}} ωc^{− \frac{1}{η}} (1 − l)^{ψ} = (1 − σ)ωUc^{− \frac{1}{η}} /Ψ
U_{l} = −Φ^{−σ}Ψ^{\frac{η}{η−1}} ψ(1 − l)^{ψ−1} = −(1 − σ)ψ \frac{U}{1 − l}
U_{m} = Φ^{−σ}Ψ^{\frac{1}{η−1}} (1 − ω)m^{− \frac{1}{η}} (1 − l)^{ψ}
U_{cc} = −σΦ^{−σ−1}(Ψ^{\frac{1}{η−1}} ωc^{− \frac{1}{η}} (1 − l)^{ψ})^{2}

+ Φ^{−σ}Ψ^{\frac{1}{η−1} −1}ω^{2}/ηc^{− \frac{2}{η}} (1 − l)^{ψ} − Φ^{−σ}Ψ^{\frac{1}{η−1}} ω/ηc^{− \frac{1}{η} −1}(1 − l)^{ψ}

= \frac{U_{c}U_{c}}{U} \frac{1 − ησ}{η(1 − σ)} − \frac{1}{η} \frac{U_{c}}{c}

U_{cl} = σΦ^{−σ−1}Ψ^{\frac{η}{η−1}} ψ(1 − l)^{ψ−1}Ψ^{\frac{1}{η−1}} ωc^{− \frac{1}{η}} (1 − l)^{ψ}

− Φ^{−σ}Ψ^{\frac{1}{η−1}} ωc^{− \frac{1}{η}} ψ(1 − l)^{ψ−1}

= \frac{U_{c}U_{l}}{U}

U_{lm} = σΦ^{−σ−1}Ψ^{\frac{η}{η−1}} ψ(1 − l)^{ψ−1}Ψ^{\frac{1}{η−1}} (1 − ω)m^{− \frac{1}{η}} (1 − l)^{ψ}

− Φ^{−σ}Ψ^{\frac{1}{η−1}} (1 − ω)m^{− \frac{1}{η}} ψ(1 − l)^{ψ−1}

U_{ll} = −σΦ^{−σ−1}(ψ(1 − l)^{ψ−1}Ψ^{\frac{η}{η−1}} )^{2} + Φ^{−σ}Ψ^{\frac{η}{η−1}} ψ(ψ − 1)(1 − l)^{ψ−2}

= \frac{U_{l}U_{l}}{U} + \frac{U_{l}}{1 − l}

U_{cm} = −σΦ^{−σ−1}(Ψ^{\frac{1}{η−1}} ωc^{− \frac{1}{η}} (1 − l)^{ψ})(Ψ^{\frac{1}{η−1}} (1 − ω)m^{− \frac{1}{η}} (1 − l)^{ψ})

+ Φ^{−σ}Ψ^{\frac{1}{η−1} −1}ω(1 − ω)/ηc^{− \frac{1}{η}} m^{− \frac{1}{η}} (1 − l)^{ψ}

= \frac{U_{c}U_{m}}{U} \frac{1 − ησ}{η(1 − σ)}

U_{mm} = −σΦ^{−σ−1}(Ψ^{\frac{1}{η−1}} (1 − ω)m^{− \frac{1}{η}} (1 − l)^{ψ})^{2}

+ Φ^{−σ}Ψ^{\frac{1}{η−1} −1}ω^{2}/ηm^{− \frac{2}{η}} (1 − l)^{ψ} − Φ^{−σ}Ψ^{\frac{1}{η−1}} (1 − ω)/ηm^{− \frac{1}{η} −1}(1 − l)^{ψ}

where the arguments of the utility function and its derivatives have been dropped for convenience.

Ignoring m, consider two calculations of the labor supply elasticity. First, hold U_{c} fixed. If λ = U_{c}, then

λ = c^{−σ}(1 − l)^{ψ(1−σ)}.

Inverting this we have

c = [\frac{(1 − l)^{ψ(1−σ)}}{λ} ]^{\frac{1}{σ}} .

The equilibrium equation is w = −U_{l}/U_{c} = ψc/(1 − l). With c substituted in, we get

w = \frac{ψ}{λ^{\frac{1}{σ}}} (1 − l)^{\frac{ψ(1−σ)}{σ} −1}

Totally differentiating, we get

dw = −\frac{ψ}{λ^{\frac{1}{σ}}} [\frac{ψ(1 − σ)}{σ} − 1] (1 − l)^{\frac{ψ(1−σ)}{σ} −2} dl

= [1 − \frac{ψ(1 − σ)}{σ}] \frac{w}{1 − l} dl.

Rearranging this equation we get the following elasticity:

\frac{d ln l}{d ln w} = [\frac{σ}{σ − ψ(1 − σ)}] \frac{1 − l}{l}

If l = 1/4 and σ = 1, the elasticity is 3. Suppose that we set ψ = 2. As σ increases from 1 to 10, the elasticity falls from 3 to roughly 1.

The labor elasticity holding c fixed is simpler. Again, ignoring m, we have w = ψc/(1 − l) and, therefore,

\frac{d ln l}{d ln w} |_{c fixed} = \frac{d(1 − ψc̄/w)}{dw} \frac{w}{l} = \frac{1 − l}{l}

If l = 1/4, the elasticity is 3.

3.3. Nonseparable Preferences II

The nonseparable preferences, type II, are given by

U(c, l, m) = \frac{1}{1 − σ} [\frac{1}{1 − κ}c^{1−κ} + \frac{ω}{1 − 1/η} m^{1− \frac{1}{η}} − \frac{ψ}{1 + ξ} l^{1+ξ}]^{1−σ}  (3.3)

≡ \frac{1}{1 − σ} Φ(c, m, l)^{1−σ}.

The first and second partials of this function are as follows

U_{c} = Φ^{−σ}c^{−κ}
U_{l} = Φ^{−σ}l^{ξ}(−ψ)
U_{m} = Φ^{−σ}m^{−1/η}ω
U_{cc} = −U_{c}(σ/Φc^{−κ} + κ/c)
U_{cl} = U_{c}σ/Φl^{ξ}ψ
U_{lm} = U_{m}σ/Φl^{ξ}ψ
U_{ll} = U_{l}(σ/Φl^{ξ}ψ + ξ/l)
U_{cm} = −U_{c}σ/Φm^{−1/η}ω
U_{mm} = −U_{m}(σ/Φm^{−1/η}ω + 1/(ηm))

Again, ignoring m, consider calculating the labor supply elasticity. The elasticity holding U_{c} fixed is complicated when σ is greater than 0. For now, let’s do the simpler case of holding c fixed. Since w = ψl^{ξ}c^{κ},

\frac{d ln l}{d ln w} = \frac{1}{ξ}

If ξ = 1, then the elasticity is 1. If ξ = .5, the elasticity is 2. Note that this is the elasticity for the σ = 0 (i.e., completely separable preferences) case.

4. Are Real Exchange Rates Volatile and Persistent?

When we log-linearize the real exchange rate in the benchmark economy, we get

q̂_{t} = −\frac{U_{cc}c}{U_{c}}(ĉ_{t} − ĉ_{t}^{∗}) − \frac{U_{cl}l}{U_{c}} (l̂_{t} − l̂_{t}^{∗}) − \frac{U_{cm}M/P}{U_{c}} (m̂_{t} − m̂_{t}^{∗})  (4.1)

where m̂_{t} = μ̂_{t} − p̂_{t}. Suppose that we chose preferences in such a way as to guarantee that increases in money would not have a large effect on utility or marginal utility; that is, we assume that quantitatively the third term in (4.1) is small. In that case we could think of preferences defined over consumption c and labor l. In this case, the variance of q̂_{t} is given by

var q̂ = (−U_{cc}c/U_{c})^{2} σ_{Δc}^{2} + (−U_{cl}l/U_{c})^{2} σ_{Δl}^{2} + 2 (U_{cc}U_{cl}cl/U_{c}^{2}) σ_{Δc,Δl}^{2}  (4.2)

where σ_{Δc}^{2} = var(ĉ − ĉ^{∗}), σ_{Δl}^{2} = var(l̂− l̂^{∗}), and σ_{Δc,Δl}^{2} = cov(ĉ − ĉ^{∗}, l̂− l̂^{∗}).

If we construct variances of the real exchange rate, the difference in logged and detrended consumption, and the difference in logged and detrended employment for U.S. and European data, we find that the variance of q̂ is relatively large. Therefore, to get the model to generate sufficiently volatile real exchange rates, we require that the coefficient on ĉ−ĉ^{∗} or l̂− l̂^{∗} in (4.1) is large in absolute value with U_{cl} either positive or not too negative.

Consider the signs of partial derivatives with respect to c and l. We know that U_{c} > 0 and U_{cc} < 0 so that the first coefficient in (4.1) is positive. The cross-product U_{cl} can be negative or positive as long as U_{cc}U_{ll} > U_{cl}^{2} (for concavity). That implies that the second coefficient in (4.1) is either positive or negative. If it is too negative, then we cannot generate much volatility in exchange rates since the covariance term in (4.2) would be negative (with U_{cc} < 0 and U_{cl} > 0). So, for now, we will assume that −U_{cc}c/U_{c} is a large positive number and −U_{cl}l/U_{c} is either positive or not very negative. This ensures volatility.

What about persistence? As we show in our earlier work, it is crucial that prices don’t jump in response to changes in marginal costs. Otherwise, the effects of monetary shocks are short-lived. If the production function is Cobb-Douglas with a capital share equal to α, then the linearized expression for firm i’s marginal cost is:

m̂c_{i,t} = ŵ_{t} − αk̂_{i−1,t−1} + αl̂_{i,t}.  (4.3)

This expression depends on the wage ŵ_{t} which, when linearized, is given by

ŵ_{t} = (\frac{U_{cl}}{U_{l}} − \frac{U_{cc}}{U_{c}} ) c ĉ_{t} + (\frac{U_{ll}}{U_{l}} − \frac{U_{cl}}{U_{c}} ) l l̂_{t} + (\frac{U_{lm}}{U_{l}} − \frac{U_{cm}}{U_{c}} ) M/P m̂_{t}.  (4.4)

Again, let’s assume that the coefficients on money are quantitatively small so that we can ignore the last term in (4.4). What can we say about the other two given the restrictions above. Since preferences are concave in consumption and leisure (−l), we know that −U_{cc}c/U_{c} and U_{ll}l/U_{l} are both positive. And, above, we assume that −U_{cc}c/U_{c} was large. If U_{cl} < 0, then the first coefficient in (4.4) is large and positive and the second coefficient is positive. This follows from the fact that U_{c} > 0 and U_{l} < 0. With at least one large coefficient on c or l, the theory won’t generate persistent responses to positive monetary shocks because marginal costs will jump up immediately and firms will respond by immediately increasing their prices. What if U_{cl} > 0? If this cross-product is large enough, then we can get small coefficients in (4.4). However, having U_{cl} > 0 leads to less volatility in q̂_{t}. (Recall that above we wanted −U_{cl}l/U_{c} positive.)

The arguments make clear the tension: there is no obvious way to ensure both volatility and persistence of real exchange rates – unless we choose parameters to get volatility and then assume that prices are stuck for long periods.

Let’s consider the specific preferences of the last section to see this tension more clearly. For the separable utility function in (3.1), we have

q̂_{t} = [\frac{ωc^{\frac{η−1}{η}}}{Ψ} (σ − \frac{1}{η} ) + \frac{1}{η} ] (ĉ_{t} − ĉ_{t}^{∗})

+ [\frac{(1 − ω)(M/P)^{\frac{η−1}{η}}}{Ψ} (σ − \frac{1}{η} )] (m̂_{t} − m̂_{t}^{∗}).

In this case U_{cl} = 0. If we choose a value of ω near 1, then all that we have to do to guarantee a lot of volatility is to choose a large value for σ.

Now consider the wage rate for the separable-utility case:

ŵ_{t} = [\frac{ωc^{\frac{η−1}{η}}}{Ψ} (σ − \frac{1}{η} ) + \frac{1}{η} ] ĉ_{t} + \frac{ξl}{1 − l} l̂_{t} + [\frac{(1 − ω)(M/P)^{\frac{η−1}{η}}}{Ψ} (σ − \frac{1}{η} )] m̂_{t} (4.5)

with U defined in (3.1). With ω near 1 and σ large, we will find that wages, and in turn marginal costs, will jump when there is an increase in the growth rate of money. This is because the coefficient (what we called γ in our earlier) on output in the price equation will be large if the coefficients on ĉ and l̂ in (4.5) are large.

What about nonseparable preferences of type I in (3.2)? In this case, the real exchange rate is

q̂_{t} = [\frac{ωc^{\frac{η−1}{η}}}{Ψ} (σ − \frac{1}{η} ) + \frac{1}{η} ] (ĉ_{t} − ĉ_{t}^{∗}) + \frac{(1 − σ)ψl}{1 − l} (l̂_{t} − l̂_{t}^{∗})

+ [\frac{(1 − ω)(M/P)^{\frac{η−1}{η}}}{Ψ} (σ − \frac{1}{η} )] (m̂_{t} − m̂_{t}^{∗})

where l̂ appears because U_{cl} is not equal to 0. Notice though that a large σ implies a very negative value for U_{cl}. Thus, we can’t generate volatile real exchange rates by simply increasing σ. And, if money is not playing an important role, there is really no way to get volatile real exchange rates. (To see this, note that ωc^{(η−1)/η} ≈ Ψ so that the coefficient on ĉ_{t} − ĉ_{t}^{∗} is approximately equal to σ.)

In summary, we see that standard choices for preferences lead to a negative result: it is not easy to generate both volatile and persistent real exchange rates in the benchmark economy.

What happens when we assume that markets are incomplete? When markets are incomplete, we replace q_{t} = U_{ct}^{∗}/U_{ct} with the following condition

E_{t} \frac{U_{c}(s^{t+1})P (s^{t})}{U_{c}(s^{t})P (s^{t+1})} = E_{t} \frac{U_{c}^{∗}(s^{t+1})P (s^{t})q(s^{t})}{U_{c}(s^{t})P (s^{t+1})q(s^{t+1})}.

When we linearize this, we get

E_{t}q̂_{t+1} − q̂_{t} = E_{t} [−\frac{U_{cc}c}{U_{c}}(ĉ_{t+1} − ĉ_{t+1}^{∗}) − \frac{U_{cl}l}{U_{c}} (l̂_{t+1} − l̂_{t+1}^{∗}) − \frac{U_{cm}M/P}{U_{c}} (m̂_{t+1} − m̂_{t+1}^{∗})]

+ \frac{U_{cc}c}{U_{c}}(ĉ_{t} − ĉ_{t}^{∗}) + \frac{U_{cl}l}{U_{c}} (l̂_{t} − l̂_{t}^{∗}) + \frac{U_{cm}M/P}{U_{c}} (m̂_{t} − m̂_{t}^{∗})

This is only a slight modification of what we had before – it is simply an expectational difference of (4.1). As before, we see that the sign of U_{cl} will matter. If it is positive, then the model will generate persistence. If it is negative, then the model will generate volatility. But, again, it will be hard to generate both.

What happens when we allow for sticky wages? Consider the simplest case with μ = 1, β ≈ 1, and N = 2. In that case, the normalized nominal wage set by household type 1 is equal to

ω̂_{t−1} = \frac{1}{2}E_{t−1}[p̂_{t} + p̂_{t+1} + μ̂_{t} + (\frac{U_{cl}}{U_{l}} − \frac{U_{cc}}{U_{c}} ) c(ĉ_{1,t} + ĉ_{2,t+1})

+ (\frac{U_{ll}}{U_{l}} − \frac{U_{cl}}{U_{c}} ) l(l̂_{1,t} + l̂_{2,t+1}) + (\frac{U_{lm}}{U_{l}} − \frac{U_{cm}}{U_{c}} ) \frac{M}{P} (m̂_{1,t} + m̂_{2,t+1})].

Note that the μ_{t} term appears because of our choice of normalizing by M. The linearized real wage in the sticky-wage case is given by

ŵ_{t} = ˆ ω̄_{t} − p̂_{t} = \frac{1}{2}(ω̂_{t−1} + ω̂_{t−2} − μ̂_{t−1}) − p̂_{t}.

Thus, we have lags and leads of the terms in (4.4). But, to get persistence of real exchange rates, we still require that firms do not raise prices too quickly. The sticky wages will slow the response down somewhat – but not entirely. Unless U_{cl} is sufficiently negative (to offset −U_{cc}/U_{c} or U_{ll}/U_{l}), we will have the same problem as before: in response to an increase in μ, wages will rise quickly, marginal costs will rise quickly, and firms will increase prices. This in turn will imply no persistence in consumption and, therefore, no persistence in real exchange rates.

5. Some Analytics for Special Cases

5.1. Labor-Only Case with Nonseparable U, Exogenous Money, and N = 2

Let’s first consider the case that we studied in our earlier work. Assume that preferences are given by (3.2) and that F (k, l) = l. In the labor-only case, we have c(s^{t}) = y(s^{t}) and therefore ĉ_{t} = ŷ_{t}. Assume that prices and monies are not growing over time so that P and M are stationary variables. In this case μ = 1. Finally, assume that there are two cohorts of firms (N = 2). With N = 2, aggregate labor is

l(s^{t}) = (l(1, s^{t}) + l(2, s^{t}))/2

= (y_{H}(1, s^{t}) + y_{H}^{∗}(1, s^{t}) + y_{H}(2, s^{t}) + y_{H}^{∗}(2, s^{t}))/2

= \frac{1}{2} [ω_{1}P (s^{t})]^{\frac{1}{1−ρ}} P̄_{H}(s^{t−1})^{\frac{ρ−θ}{(1−ρ)(θ−1)}} y(s^{t}) (P_{H}(s^{t−1})^{\frac{1}{θ−1}} + P_{H}(s^{t−2})^{\frac{1}{θ−1}} ) +

\frac{1}{2} [ω_{1}P^{∗}(s^{t})]^{\frac{1}{1−ρ}} P̄_{H}^{∗}(s^{t−1})^{\frac{ρ−θ}{(1−ρ)(θ−1)}} y^{∗}(s^{t}) (P_{H}^{∗}(s^{t−1})^{\frac{1}{θ−1}} + P_{H}^{∗}(s^{t−2})^{\frac{1}{θ−1}} ) .

If we linearize this, we get

ll̂_{t} = y_{H} (\frac{1}{1 − ρ}P̂_{t} + \frac{ρ − θ}{(1 − ρ)(θ − 1)} ˆP̄_{H,t−1} + ŷ_{t} + \frac{1}{θ − 1}(P̂_{H,t−1} + P̂_{H,t−2})/2)

+ y_{H}^{∗} (\frac{1}{1 − ρ}P̂_{t}^{∗} + \frac{ρ − θ}{(1 − ρ)(θ − 1)} ˆP̄_{H,t−1}^{∗} + ŷ_{t}^{∗} + \frac{1}{θ − 1}(P̂_{H,t−1}^{∗} + P̂_{H,t−2}^{∗})/2)

= αl (\frac{1}{1 − ρ}(P̂_{t} − ˆ P̄_{H,t−1}) + ŷ_{t}) + α^{∗}l (\frac{1}{1 − ρ}(P̂_{t}^{∗} − ˆ P̄_{H,t−1}^{∗}) + ŷ_{t}^{∗})

= αl (\frac{α^{∗}}{1 − ρ} [ ˆP̄_{F,t−1} − ˆ P̄_{H,t−1}] + ŷ_{t}) + α^{∗}l (\frac{α}{1 − ρ} [ ˆP̄_{F,t−1}^{∗} − ˆ P̄_{H,t−1}^{∗}] + ŷ_{t}^{∗})

= \frac{αα^{∗}l}{1 − ρ} [ ˆP̄_{F,t−1} + ˆ P̄_{F,t−1}^{∗} − ˆ P̄_{H,t−1} − ˆ P̄_{H,t−1}^{∗}] + αl ŷ_{t} + α^{∗}l ŷ_{t}^{∗}

where the parameter α corresponds to the ratio y_{H}/(y_{H} + y_{H}^{∗}). (In the paper we chose this to be 0.984 because imports from Europe are roughly 1-.984 or 1.6% of GDP.) The parameter α^{∗} is 1 − α. We will need the above expression for the labor input when we write out the pricing equations.

We will also need the wage rate which is given by −U_{l}(s^{t})/U_{c}(s^{t}) and therefore depends on consumption, the labor input, and real balances. The linearized wage equation is given by

ŵ_{t} = [\frac{ωc^{\frac{η−1}{η}}}{Ψ} (\frac{η − 1}{η}) + \frac{1}{η} ] ĉ_{t} + \frac{l}{1 − l} l̂_{t} + [\frac{(1 − ω)(M/P)^{\frac{η−1}{η}}}{Ψ} (\frac{η − 1}{η})] (M̂_{t} − P̂_{t})

= [\frac{ωc^{\frac{η−1}{η}}}{Ψ} (\frac{η − 1}{η}) + \frac{1}{η} ] ŷ_{t}

+ (\frac{l}{1 − l} ) \{\frac{αα^{∗}}{1 − ρ} [ ˆP̄_{F,t−1} + ˆ P̄_{F,t−1}^{∗} − ˆ P̄_{H,t−1} − ˆ P̄_{H,t−1}^{∗}] + α ŷ_{t} + α^{∗} ŷ_{t}^{∗}\}

+ [\frac{(1 − ω)(M/P)^{\frac{η−1}{η}}}{Ψ} (\frac{η − 1}{η})] (M̂_{t} − P̂_{t}).

where we have used the formulas above for partial derivatives of the nonseparable (type I) utility function.

The nominal exchange rate appears in two of the pricing equations. We can write this in terms of the real exchange rate since e = qP/P^{∗}. The real exchange rate is a ratio of the marginal utilities. Using the nonseparable (type I) preferences, we get the following expression for the real exchange rate:

q̂_{t} = [\frac{ωc^{\frac{η−1}{η}}}{Ψ} (σ − \frac{1}{η} ) + \frac{1}{η} ] (ĉ_{t} − ĉ_{t}^{∗}) + \frac{(1 − σ)ψl}{1 − l} (l̂_{t} − l̂_{t}^{∗})

+ [\frac{(1 − ω)(M/P)^{\frac{η−1}{η}}}{Ψ} (σ − \frac{1}{η} )] (M̂_{t} − P̂_{t} − M̂_{t}^{∗} + P̂_{t}^{∗}).

We can substitute in c = y and the formula for the labor input derived above to get q in terms of outputs, prices, and money.

The money demand equation is given by

M̂_{t} − P̂_{t} = ŷ_{t} − ηβ(r_{t}/r − 1).

If η ≈ 0 then money demand is interest-inelastic and output and real balances are equal in equilibrium. Suppose this is true. Then wages are equal to:

ŵ_{t} = (1 + \frac{αl}{1 − l} ) (M̂_{t} − P̂_{t}) + (\frac{α^{∗}l}{1 − l} ) (M̂_{t}^{∗} − P̂_{t}^{∗})

+ (\frac{αα^{∗}l}{(1 − l)(1 − ρ)}) [ ˆP̄_{F,t−1} + ˆ P̄_{F,t−1}^{∗} − ˆ P̄_{H,t−1} − ˆ P̄_{H,t−1}^{∗}]

and the real exchange rate is equal to

q̂_{t} = (σ + (1 − σ)\frac{ψ(α − α^{∗})l}{1 − l} ) (M̂_{t} − P̂_{t} − M̂_{t}^{∗} + P̂_{t}^{∗})

+ (\frac{2ψ(1 − σ)αα^{∗}l}{(1 − l)(1 − ρ)} ) [ ˆP̄_{F,t−1} + ˆ P̄_{F,t−1}^{∗} − ˆ P̄_{H,t−1} − ˆ P̄_{H,t−1}^{∗}]

We will now derive the pricing equation for P_{H} assuming that money demand is interest-inelastic. The intermediate goods price is a function of nominal marginal costs. In this example, the linearized marginal nominal cost is given by

P̂_{t} + ŵ_{t} = (1 + \frac{αl}{1 − l} ) M̂_{t} + (\frac{α^{∗}l}{1 − l} ) M̂_{t}^{∗}

− (\frac{αl}{1 − l} ) [α ˆP̄_{H,t−1} + α^{∗} ˆP̄_{F,t−1}] − (\frac{α^{∗}l}{1 − l} ) [α ˆP̄_{F,t−1}^{∗} + α^{∗} ˆP̄_{H,t−1}^{∗}]

+ (\frac{αα^{∗}l}{(1 − l)(1 − ρ)}) [ ˆP̄_{F,t−1} + ˆ P̄_{F,t−1}^{∗} − ˆ P̄_{H,t−1} − ˆ P̄_{H,t−1}^{∗}]

If we substitute this expression into the pricing equation we get

P̂_{H,t−1} = \frac{1}{1 + β} E_{t−1}\{(1 + \frac{αl}{1 − l} ) M̂_{t} + (\frac{α^{∗}l}{1 − l} ) M̂_{t}^{∗}

− (\frac{αα^{∗}l}{1 − l} ) ((\frac{1}{1 − ρ} + \frac{α}{α^{∗}} ) (P̂_{H,t−1} + P̂_{H,t−2}) /2

+ (\frac{1}{1 − ρ} + \frac{α^{∗}}{α} ) ˆ P̄_{H,t−1}^{∗} − \frac{ρ}{1 − ρ} ( ˆP̄_{F,t−1} + ˆ P̄_{F,t−1}^{∗}))

+ β[(1 + \frac{αl}{1 − l} ) M̂_{t+1} + (\frac{α^{∗}l}{1 − l} ) M̂_{t+1}^{∗}

− (\frac{αα^{∗}l}{1 − l} ) ((\frac{1}{1 − ρ} + \frac{α}{α^{∗}} ) (P̂_{H,t} + P̂_{H,t−1}) /2

+ (\frac{1}{1 − ρ} + \frac{α^{∗}}{α} ) ˆ P̄_{H,t}^{∗} − \frac{ρ}{1 − ρ} ( ˆP̄_{F,t} + ˆ P̄_{F,t}^{∗}))]\}

Thus, we get finally,

P̂_{H,t−1} = \frac{1}{1 + β} [1 + \frac{1}{2} (\frac{αα^{∗}l}{1 − l} ) (\frac{1}{1 − ρ} + \frac{α}{α^{∗}} )]^{−1}

× E_{t−1}\{(1 + \frac{αl}{1 − l} ) (M̂_{t} + βM̂_{t+1}) + (\frac{α^{∗}l}{1 − l} ) (M̂_{t}^{∗} + βM̂_{t+1}^{∗})

− (\frac{αα^{∗}l}{1 − l} ) ((\frac{1}{1 − ρ} + \frac{α}{α^{∗}} ) (P̂_{H,t−2} + βP̂_{H,t})/2

+ (\frac{1}{1 − ρ} + \frac{α^{∗}}{α} ) ( ˆP̄_{H,t−1}^{∗} + β ˆP̄_{H,t}^{∗})

− \frac{ρ}{1 − ρ} ( ˆP̄_{F,t−1} + ˆ P̄_{F,t−1}^{∗} + β ˆP̄_{F,t} + β ˆP̄_{F,t}^{∗}))\}

If we drive α and β both to 1 and let γ = 1/(1 − l), then we get the following difference equation:

E_{t−1}P̂_{H,t} − 2\frac{1 + γ}{1 − γ} P̂_{H,t−1} + P̂_{H,t−2} = −\frac{2γ}{1 − γ} E_{t−1}(M̂_{t} + M̂_{t+1})

which is exactly what we found for our closed economy example. Recall that the solution to this is:

P̂_{H,t−1} = aP̂_{H,t−2} + (1 − a)M̂_{t−1}

where a = (1 − \sqrt{γ})/(1 + \sqrt{γ}).

To derive the equations for P_{H}^{∗} and P_{F} we need to linearize P w/e and P^{∗}w^{∗}e, respectively. When money demand is interest-insensitive, the linearization of P w/e is given by

P̂_{t} + ŵ_{t} − ê_{t} = P̂_{t}^{∗} + ŵ_{t} − q̂_{t}

= (1 − σ + (α − ψ(1 − σ)(α − α^{∗})) \frac{l}{1 − l} ) M̂_{t}

+ (σ + (α^{∗} + ψ(1 − σ)(α − α^{∗})) \frac{l}{1 − l} ) M̂_{t}^{∗}

− (1 − σ + (α − ψ(1 − σ)(α − α^{∗})) \frac{l}{1 − l} ) [α ˆP̄_{H,t−1} + α^{∗} ˆP̄_{F,t−1}]

+ (1 − σ − (α^{∗} + ψ(1 − σ)(α − α^{∗})) \frac{l}{1 − l} ) [α ˆP̄_{F,t−1}^{∗} + α^{∗} ˆP̄_{H,t−1}^{∗}]

+ (\frac{αα^{∗}l[1 − 2ψ(1 − σ)]}{(1 − l)(1 − ρ)}) [ ˆP̄_{F,t−1} + ˆ P̄_{F,t−1}^{∗} − ˆ P̄_{H,t−1} − ˆ P̄_{H,t−1}^{∗}] .

The linearization of P^{∗}w^{∗}e is given by

P̂_{t}^{∗} + ŵ_{t}^{∗} + ê_{t} = P̂_{t} + ŵ_{t}^{∗} + q̂_{t}

= (1 − σ + (α − ψ(1 − σ)(α − α^{∗})) \frac{l}{1 − l} ) M̂_{t}^{∗}

+ (σ + (α^{∗} + ψ(1 − σ)(α − α^{∗})) \frac{l}{1 − l} ) M̂_{t}

− (1 − σ + (α − ψ(1 − σ)(α − α^{∗})) \frac{l}{1 − l} ) [α ˆP̄_{F,t−1}^{∗} + α^{∗} ˆP̄_{H,t−1}^{∗}]

+ (1 − σ − (α^{∗} + ψ(1 − σ)(α − α^{∗})) \frac{l}{1 − l} ) [α ˆP̄_{H,t−1} + α^{∗} ˆP̄_{F,t−1}]

+ (\frac{αα^{∗}l[1 − 2ψ(1 − σ)]}{(1 − l)(1 − ρ)}) [ ˆP̄_{H,t−1}^{∗} + ˆ P̄_{H,t−1} − ˆ P̄_{F,t−1}^{∗} − ˆ P̄_{F,t−1}]

which looks the same as the first except that we swap domestic and foreign variables. Notice that these formulas simplify considerably in the case of σ = 1, for example:

P̂_{t} + ŵ_{t} − ê_{t} = \frac{αl}{1 − l} M̂_{t} + (1 + \frac{α^{∗}l}{1 − l} ) M̂_{t}^{∗}

− \frac{αl}{1 − l} [α ˆP̄_{H,t−1} + α^{∗} ˆP̄_{F,t−1}]

− \frac{α^{∗}l}{1 − l} [α ˆP̄_{F,t−1}^{∗} + α^{∗} ˆP̄_{H,t−1}^{∗}]

+ (\frac{αα^{∗}l}{(1 − l)(1 − ρ)}) [ ˆP̄_{F,t−1} + ˆ P̄_{F,t−1}^{∗} − ˆ P̄_{H,t−1} − ˆ P̄_{H,t−1}^{∗}]

because in this case ê = M̂ − M̂^{∗}. Therefore, when σ = 1 the expressions for P_{H} and P_{H}^{∗} differ only in the money terms.

What if we set η = 0, β = 1, α = α^{∗} = 1/2 and ρ = 1/3? Then the equation for P_{H} is given by

P̂_{H,t−1} = \frac{1}{2 + 5(γ − 1)/8}E_{t−1}\{\frac{1}{2}(1 + γ)(M̂_{t} + M̂_{t+1}) + \frac{1}{2}(γ − 1)(M̂_{t}^{∗} + M̂_{t+1}^{∗})

− \frac{1}{4}(γ − 1)[\frac{5}{2} (P̂_{H,t−2} + P̂_{H,t}) /2 + \frac{5}{2} ( ˆP̄_{H,t−1}^{∗} + ˆ P̄_{H,t}^{∗})

− \frac{1}{2} ( ˆP̄_{F,t−1} + ˆ P̄_{F,t−1}^{∗} + ˆ P̄_{F,t} + ˆ P̄_{F,t}^{∗})]\}.

Notice that this can be rewritten as

E_{t−1}P_{H,t} + ϕP_{H,t−1} + P_{H,t−2} = r.h.s. terms

where ϕ = [2 + 5(γ − 1)/8]/[5/16(γ − 1)] which is greater than 2. (This follows because γ = 1/(1 − l) > 1.) The root satisfying λ^{2} + ϕλ + 1 that is inside the unit circle lies somewhere between -1 and 0. Therefore, to generate persistence, we need that the other prices have a quantitatively important affect on P_{H}. The equation with the right hand side terms written out is

E_{t−1}P_{H,t} + 2 (1 + \frac{16}{5(γ − 1)}) P_{H,t−1} + P_{H,t−2}

= E_{t−1}\{\frac{8}{5}(1 + γ)(M̂_{t} + M̂_{t+1}) + \frac{8}{5}(γ − 1)(M̂_{t}^{∗} + M̂_{t+1}^{∗})

+ 2( ˆP̄_{H,t−1}^{∗} + ˆ P̄_{H,t}^{∗}) − \frac{2}{5} ( ˆP̄_{F,t−1} + ˆ P̄_{F,t−1}^{∗} + ˆ P̄_{F,t} + ˆ P̄_{F,t}^{∗})\}

5.2. Labor-Only Case with Separable U, Exogenous Money, and N = 2

We need to rederive the equation for the wage rate and the real exchange rate. The linearized wage equation with separable preferences is given by

ŵ_{t} = [\frac{ωc^{\frac{η−1}{η}}}{Ψ} (σ − \frac{1}{η} ) + \frac{1}{η} ] ĉ_{t} + \frac{ξl}{1 − l} l̂_{t} + [\frac{(1 − ω)(M/P)^{\frac{η−1}{η}}}{Ψ} (σ − \frac{1}{η} )] (M̂_{t} − P̂_{t})

= [\frac{ωc^{\frac{η−1}{η}}}{Ψ} (σ − \frac{1}{η} ) + \frac{1}{η} ] ŷ_{t}

+ (\frac{ξl}{1 − l} ) \{\frac{αα^{∗}}{1 − ρ} [ ˆP̄_{F,t−1} + ˆ P̄_{F,t−1}^{∗} − ˆ P̄_{H,t−1} − ˆ P̄_{H,t−1}^{∗}] + α ŷ_{t} + α^{∗} ŷ_{t}^{∗}\}

+ [\frac{(1 − ω)(M/P)^{\frac{η−1}{η}}}{Ψ} (σ − \frac{1}{η} )] (M̂_{t} − P̂_{t}).

The real exchange rate is now

q̂_{t} = [\frac{ωc^{\frac{η−1}{η}}}{Ψ} (σ − \frac{1}{η} ) + \frac{1}{η} ] (ĉ_{t} − ĉ_{t}^{∗})

+ [\frac{(1 − ω)(M/P)^{\frac{η−1}{η}}}{Ψ} (σ − \frac{1}{η} )] (M̂_{t} − P̂_{t} − M̂_{t}^{∗} + P̂_{t}^{∗})

which is the same as before except that now there is no term with the labor inputs.

If we assume that money demand is interest-inelastic, then the wage and real exchange rate simplify to:

ŵ_{t} = (σ + \frac{ξαl}{1 − l} ) (M̂_{t} − P̂_{t}) + (\frac{ξα^{∗}l}{1 − l} ) (M̂_{t}^{∗} − P̂_{t}^{∗})

+ (\frac{ξαα^{∗}l}{(1 − l)(1 − ρ)}) [ ˆP̄_{F,t−1} + ˆ P̄_{F,t−1}^{∗} − ˆ P̄_{H,t−1} − ˆ P̄_{H,t−1}^{∗}]

q̂_{t} = σ(M̂_{t} − P̂_{t} − M̂_{t}^{∗} + P̂_{t}^{∗}).

If we drive α and β both to 1 and let γ = 1/(1 − l), then we get the following difference equation for P_{H}:

E_{t−1}P̂_{H,t} − 2\frac{1 + γ̂}{1 − γ̂} P̂_{H,t−1} + P̂_{H,t−2} = −\frac{2γ̂}{1 − γ̂} E_{t−1}(M̂_{t} + M̂_{t+1})

where

γ̂ = σ + ξ(γ − 1).

Therefore, if σ = 1 and ξ = 1, we have the same equation as in the nonseparabe utility case. If σ and ξ are both small then we can get persistence – changes in outputs and other firms prices do not imply that a producer adjusts his price immediately. However, we lose volatility of exchange rates since σ must be large in order to amplify q.