A Suite of Models for Dynare Description of Models

F. Collard, H. Dellas and B. Diba Department of Economics

Version 1.0 University of Bern

1

1

A REAL BUSINESS CYCLE MODEL

2

A real Business Cycle Model

The problem of the household is   ∞ X 1 max Et  β j ζ σ

c,t+j

j=0

1− 1

ct+jσ 1−

1 σ

 −

ψ −ν 1+ν  ζ h 1 + ν h,t+j t+j

ct + it = wt ht + zt kt − τt kt+1 = it + (1 − δ)kt The last two conditions can be combined to give kt+1 = wt ht + (zt + 1 − δ)kt − ct − τt Then the first order conditions are given by − σ1

1

σ ct ζc,t

−ν ν ψζh,t ht

= λt

(1)

= λ t wt

(2)

λt = βEt (λt+1 (zt+1 + 1 − δ))

(3)

The problem of the firm is max at ktα ht1−α − wt ht − zt kt which leads to the first order conditions wt = (1 − α)

yt yt and zt = α ht kt

The taxes, τt , finance an exogenous stream of government expenditures, gt , such that τt = gt . All shocks follow AR(1) processes of the type log(at+1 ) = ρa log(at ) + εa,t+1 log(gt+1 ) = ρg log(gt ) + (1 − ρg ) log(g) + εg,t+1 log(ζc,t+1 ) = ρc log(ζc,t ) + εc,t+1 log(ζh,t+1 ) = ρh log(ζh,t ) + εh,t+1

The general equilibrium is therefore represented by the following set of equations 1

− σ1

σ ζc,t ct

= λt

yt −ν ν ψζh,t ht = λt (1 − α) ht   yt+1 λt = βEt λt+1 (α + 1 − δ) kt+1 kt+1 = it + (1 − δ)kt yt = at ktα ht1−α yt = ct + it + gt and the definition of the shocks.

Department of Economics

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2

2

A NOMINAL MODEL WITH PRICE ADJUSTMENT COSTS

3

A Nominal Model with Price Adjustment Costs In all what follows, we will assume zero inflation in the steady state.

The problem of the household is    1− σ1 ∞ X 1 c ψ t+j −ν 1+ν  σ max Et  β j ζc,t+j 1 − 1 + ν ζh,t+j ht+j 1 − σ j=0 Bt + Pt ct + Pt it = Rt−1 Bt−1 Pt wt ht + Pt zt kt − Pt τt kt+1 = it + (1 − δ)kt The last two conditions can be combined to give Bt + Pt ct + Pt kt+1 = Rt−1 Bt−1 Pt wt ht + Pt (zt + 1 − δ)kt − Pt τt Then the first order conditions are given by 1

− σ1

σ ζc,t ct

−ν ν ht ψζh,t

= Λt Pt

(4)

= Λt Pt wt

(5)

Λt Pt = βEt (Λt+1 Pt+1 (zt+1 + 1 − δ)) Λt = βRt Et Λt+1

(6) (7)

The economy is comprised of many sectors. The first sector —the final good sector— combines intermediate goods to form a final good in quantity yt : Z

1

yt =

yt (i)

θ−1 θ

θ  θ−1 di

(8)

0

The problem of the final good firm is then Z max

{yt (i);i∈(0,1)}

Pt yt −

1

Pt (i)yt (i)di 0

subject to (8), which rewrites Z max

{yt (i);i∈(0,1)}

1

yt (i)

Pt

θ−1 θ

θ  θ−1 Z di −

0

1

Pt (i)yt (i)di

0

which gives rise to the demand function  yt (i) =

Pt (i) Pt

−θ yt

(9)

These intermediate goods are produced by intermediaries, each of which has a local monopoly power. Each intermediate firm i, i ∈ (0, 1), uses a constant returns to scale technology yt (i) = at kt (i)α ht (i)1−α Department of Economics

(10) University of Bern

2

A NOMINAL MODEL WITH PRICE ADJUSTMENT COSTS

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where kt (i) and ht (i) denote capital and labor. The firm minimizes its real cost subject to (10). Minimized real total costs are then given by st xt (i) where the real marginal cost, st , is given by st =

ztα wt1−α ςat

with ς = αα (1 − α)1−α . Intermediate goods producers are monopolistically competitive, and therefore set prices for the good they produce. However, it incurs a cost whenever it changes its price relatively to the earlier period. This cost is given by  2 ϕp Pt (i) − 1 yt 2 Pt−1 (i The problem of the firm is then to maximize the profit function  !  2 ∞ X ϕp Pt+j (i) Et  − 1 yt+j  Φt,t+j Pt+j (i)yt+j (i) − Pt+j st+j yt+j (i) − Pt+j 2 Pt+j−1 (i

(11)

j=0

where Φt,t+j is an appropriate discount factor derived from the household’s optimality conditions, and proportional to β j Λt+j Λt . The first order condition of the problem is given by  (1 − θ)

Pt (i) Pt

−θ

    Pt (i) −θ Pt (i) Pt Pt st ϕp − 1 yt yt − Pt (i) Pt Pt−1 (i) Pt−1 (i     Λt+1 Pt+1 Pt+1 (i) Pt+1 (i) ϕp − 1 yt+1 = 0 + βEt Λt Pt (i)2 Pt (i

yt + θ

Using Sheppard’s lemma we get the demand for each input wt = (1 − α)st zt = αst

yt (i) ht (i)

yt (i) kt (i)

The taxes, τt , finance an exogenous stream of government expenditures, gt , such that τt = gt In order to close the model, we add a Taylor rule that determines the nominal interest rate
log(Rt ) = ρr log(Rt−1 ) + (1 − ρ) log(R) + γπ (log(πt ) − log(π)) + γy (log(yt ) − log(y)) where πt = Pt /Pt−1 denotes aggregate inflation. All shocks follow AR(1) processes of the type log(at+1 ) = ρa log(at ) + εa,t+1 log(gt+1 ) = ρg log(gt ) + (1 − ρg ) log(g) + εg,t+1 log(ζc,t+1 ) = ρc log(ζc,t ) + εc,t+1 log(ζh,t+1 ) = ρh log(ζh,t ) + εh,t+1

Department of Economics

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3

A NOMINAL MODEL WITH STAGGERED PRICE CONTRACTS

5

The symmetric general equilibrium is therefore represented by the following set of equations 1

− σ1

σ ct ζc,t

= λt

yt −ν ν ψζh,t ht = λt (1 − α)st h t   yt+1 +1−δ λt = βEt λt+1 αst+1 kt+1   λt+1 λt = βRt Et πt+1  0 = (1 − θ)yt + θst yt − πt ϕp (πt − 1) yt + βEt

λt+1 πt+1 ϕp (πt+1 − 1) yt+1 λt



kt+1 = it + (1 − δ)kt yt = at ktα h1−α t ϕp (πt − 1)2 yt 2
log(Rt ) = ρr log(Rt−1 ) + (1 − ρ) log(R) + γπ (log(πt ) − log(π)) + γy (log(yt ) − log(y)) yt = ct + it + gt +

and the definition of the shocks.

3

A Nominal Model with Staggered Price Contracts

The main difference between this model and the previous one lies in the specification of the nominal rigidities. Intermediate goods producers are monopolistically competitive, and therefore set prices for the good they produce. We follow Calvo [1983] in assuming that firms set their prices for a stochastic number of periods. In each and every period, a firm either gets the chance to adjust its price (an event occurring with probability 1 − ξ) or it does not. This is illustrated in the following figure. ξ2

ξ

... Pt+2 = Pt?

Pt+1 = Pt? Pt = Pt?

ξ

ξ 1−ξ ? Pt+2 = Pt+2

1−ξ ? Pt+1 = Pt+1

t

t+1

t+2

time

When the firm does not reset its price, it just applies the price it charged in the last period such that Pt (i) = Pt−1 (i). When it gets a chance to do it, firm i resets its price, Pt? (i), in period t in order to maximize its expected discounted profit flow this new price will generate. In period t, the profit is given by Π(Pt? (i)). In period t + 1, either the firm resets its price, such that it will get Department of Economics

University of Bern

3

A NOMINAL MODEL WITH STAGGERED PRICE CONTRACTS

6

? (i)) with probability q, or it does not and its t + 1 profit will be Π(P ? (i)) with probability Π(Pt+1 t ξ. Likewise in t + 2. The expected profit flow generated by setting Pt? (i) in period t writes

max Et ?

Pt (i)

∞ X

Φt,t+j ξ j Π(Pt? (i))

j=0

subject to the total demand it faces:  yt (i) =

Pt (i) Pt

−θ yt

and where Π(Pt? (i)) = (Pt? (i) − Pt+j st+j ) yt+j (i). Φt,t+j is an appropriate discount factor related to the way the household value future as opposed to current consumption, such that Φt,t+j ∝ β j

Λt+j Λt

This leads to the price setting equation  !  ? −θ  ? −θ ∞ X Λt+j Pt+j Pt (i) Pt (i) Et  (βξ)j (1 − θ) yt+j + θ st+j yt+j  = 0 Λt Pt+j Pt+j (i) Pt+j j=0

from which it shall be clear that all firms that reset their price in period t set it at the same level (Pt? (i) = Pt? , for all i ∈ (0, 1)). This implies that Pt? = where

 ∞ X n  Pt = Et (βξ)j Λt+j j=0

and

Ptn Ptd

(12)

 θ P 1+θ st+j yt+j  θ − 1 t+j

  ∞ X θ Ptd = Et  (βξ)j Λt+j Pt+j yt+j  j=0

Fortunately, both Ptn and Ptd admit a recursive representation, such that θ n Λt Pt1+θ st yt + βξEt [Pt+1 ] θ−1 d = Λt Ptθ yt + βξEt [Pt+1 ]

Ptn =

(13)

Ptd

(14)

Recall now that the price index is given by Z Pt =

1

Pt (i)

1−θ

1  1−θ di

0

In fact it is composed of surviving contracts and newly set prices. Given that in each an every period a price contract has a probability 1 − ξ of ending, the probability that a contract signed in

Department of Economics

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3

A NOMINAL MODEL WITH STAGGERED PRICE CONTRACTS

7

period t − j survives until period t and ends at the end of period t is given by (1 − ξ)ξ j . Therefore, the aggregate price level may be expressed as the average of all surviving contracts   1 1−θ ∞ X j ? 1−θ   Pt = (1 − ξ)ξ Pt−j j=0

which can be expressed recursively as   1 1−θ 1−θ Pt = (1 − ξ)Pt? 1−θ + ξPt−1

(15)

Note that since the wage rate is common to all firms, the capital labor ratio is the same for any firm: kt (i) kt (j) kt = = ht (i) ht (j) ht we therefore have

 yt (i) = at

integrating across firms, we obtain Z 1

 yt (i)di = at

0

denoting ht =

R1 0

kt ht

α

kt ht

α Z

ht (i)

1

ht (i)di 0

ht (i)di, and making use of the demand for yt (i), we have 1

Z 0

Pt (i) Pt

−θ

diyt = at ktα ht1−α

Denote Z

1

∆t = =

0 ∞ X

Pt (i) Pt

−θ

(1 − ξ)ξ

di j

j=0



? Pt−j Pt

−θ

∞ X

? −θ Pt−j = (1 − ξ) + (1 − ξ)ξ Pt j=1  ? −θ X  −θ ∞ ? Pt `+1 Pt−`−1 = (1 − ξ) + (1 − ξ)ξ Pt Pt `=0  ? −θ  ?    ∞ Pt−`−1 −θ Pt Pt−1 −θ X = (1 − ξ) +ξ (1 − ξ)ξ ` Pt Pt Pt−1 `=0  ? −θ   Pt Pt−1 −θ = (1 − ξ) +ξ ∆t−1 Pt Pt  ? −θ Pt ∆t = (1 − ξ) + ξπtθ ∆t−1 Pt



Pt? Pt

−θ

j



Hence aggregate output is given by ∆t yt = at ktα ht1−α Department of Economics

University of Bern

3

A NOMINAL MODEL WITH STAGGERED PRICE CONTRACTS

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Hence the set of equations defining the general equilibrium is given by, where pnt = Ptn /Ptθ , pdt = Ptd /Ptθ−1 , λt = Λt Pt and πt = Pt /Pt−1 . 1

− σ1

σ ζc,t ct

= λt

yt −ν ν ψζh,t ht = λt (1 − α)st h  t   yt+1 +1−δ λt = βEt λt+1 αst+1 kt+1   λt+1 λt = βRt Et πt+1 θ θ pnt = λt st yt + βξEt [πt+1 pnt+1 ] θ−1 θ−1 d pdt = λt yt + βξEt [πt+1 pt+1 ]  n 1−θ pt + ξπtθ−1 1 = (1 − ξ) pdt  n −θ pt ∆t = (1 − ξ) + ξπ θ ∆t−1 pdt kt+1 = it + (1 − δ)kt ∆t yt = at ktα h1−α t yt = ct + it + gt
log(Rt ) = ρr log(Rt−1 ) + (1 − ρ) log(R) + γπ (log(πt ) − log(π)) + γy (log(yt ) − log(y)) and the definition of the shocks.

Department of Economics

University of Bern

4

4

A SMALL OPEN ECONOMY MODEL WITH STAGGERED PRICES

9

A Small Open Economy Model with Staggered Prices

The problem of the domestic household is    1− σ1 ∞ X 1 c ψ −ν 1+ν  t+j σ max Et  β j ζc,t+j 1 − 1 + ν ζh,t+j ht+j 1− σ j=0 χ f ? Bth + et Btf + Pt ct + Pt it = Rt−1 Bt−1 + et Rt−1 Bt−1 + Pt wt ht + Pt zt kt − Pt τt − Pt (et Btf )2 2  2 ! ϕk it kt+1 = it 1 − −δ + (1 − δ)kt 2 kt Then the first order conditions are given by 1

− σ1

σ ζc,t ct

= Λt Pt

(16)

−ν ν ψζh,t ht

= Λt Pt wt    it −δ Λt Pt = Qt 1 − ϕk kt Λt = βRt Et Λt+1 et+1 Λt+1 Λt (1 + χet Btf ) = βRt? Et et "

(17) (18) (19) (20)

Qt = βEt Λt+1 Pt+1 zt+1 + Qt+1

ϕk 1−δ+ 2



it+1 kt+1

!!#

2 −δ

2

(21)

where Λt and Qt denote respectively the Lagrange multiplier of the first and second constraint. The retailer firm combines foreign and domestic goods to produce a non–tradable final good. It determines its optimal production plans by maximizing its profit ? f max Pt Yt − Px,t xdt − et Px,t xt

{xdt ,xft }

? denote the price of the domestic and foreign good respectively, denominated in where Px,t and Pxt terms of the currency of the seller. The final good production function is described by the following CES function  1 1

1

ρ

yt = ω 1−ρ xdt + (1 − ω) 1−ρ xft

ρ

ρ

(22)

where ω ∈ (0, 1) and ρ ∈ (−∞, 1). Optimal behavior of the retailer gives rise to the demand for the domestic and foreign goods xdt

 =

Px,t Pt



1 ρ−1

ωyt and

xft

 =

? et Px,t Pt



1 ρ−1

(1 − ω)yt

(23)

xd and xf are themselves combinations of the domestic and foreign intermediate goods according to θ θ Z 1  θ−1 Z 1  θ−1 θ−1 θ−1 f f d d xt = xt (i) θ di and xt = xt (i) θ di (24) 0

0

where θ ∈ (−∞, 1). Department of Economics

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A SMALL OPEN ECONOMY MODEL WITH STAGGERED PRICES

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Profit maximization yields demand functions of the form: xdt (i)

 =

Pxt (i) Pxt

−θ

xft (i)

xdt ,

 =

? (i) Pxt ? Pxt

−θ

xft

At this stage, we need to take a stand on the behavior of the foreign firms in order to determine the demand for the domestic good by foreign agents. We assume that their behavior is symmetrical to the one observed in the domestic economy, such 

xd? t (i)

=

Pxt (i) ? et Pxt

−θ

xd? t

Plugging these demand functions in profits and using free entry in the final good sector, we get the following general price indexes 1  1−θ

1

Z

1−θ

Pxt =

Pxt (i)

di

(25)

0

 Pt =

ρ ρ−1

ωPxt

+ (1 −

ρ

? ρ−1 ω)(et Pxt )

 ρ−1 ρ

(26) (27)

The intermediate goods are produced by intermediaries, each of which has a local monopoly power. Each intermediate firm i, i ∈ (0, 1), uses a constant returns to scale technology xt (i) = at kt (i)α ht (i)1−α

(28)

where kt (i) and ht (i) denote capital and labor. The firm minimizes its real cost subject to (28). Minimized real total costs are then given by st xt (i) where the real marginal cost, st , is given by st =

ztα wt1−α ςat

with ς = αα (1 − α)1−α . The price setting behavior is essentially the same as in a closed economy. The expected profit flow generated by setting Pet (i) in period t writes max Et Pex,t (i)

∞ X

Φt,t+j ξ j Π(Pex,t (i))

j=0

subject to the total demand it faces:  Pt (i) −θ xt (i) = xt with xt = xdt + xd? t Pt   and where Π(Pex,t (i)) = Pex,t (i) − Pt+j st+j xt+j (i). Φt,t+j is an appropriate discount factor related to the way the household value future as opposed to current consumption, such that 

Φt,t+j ∝ β j

Department of Economics

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4

A SMALL OPEN ECONOMY MODEL WITH STAGGERED PRICES

11

This leads to the price setting equation   !−θ ∞ X ex,t (i) P Pt+j Λ t+j j (1 − θ) Et  (βξ) yt+j + θ Λt Px,t+j Px,t+j (i) j=0

Pex,t (i) Px,t+j

!−θ

 st+j xt+j  = 0

from which it shall be clear that all firms that reset their price in period t set it at the same level (Pet (i) = Pet , for all i ∈ (0, 1)). This implies that Pex,t = where n Px,t

 ∞ X  = Et (βξ)j Λt+j j=0

and d Px,t

n Px,t d Px,t

(29)

 θ θ Pt+j Px,t+j st+j xt+j  θ−1

  ∞ X θ = Et  (βξ)j Λt+j Px,t+j xt+j  j=0

Fortunately, both

n Px,t

and

d Px,t

admit a recursive representation, such that θ θ n Λt Pt Px,t st xt + βξEt [Px,t+1 ] θ−1 θ d = Λt Px,t xt + βξEt [Px,t+1 ]

n Px,t =

(30)

d Px,t

(31)

Recall now that the price index is given by Z Px,t =

1

Px,t (i)

1−θ

1  1−θ di

0

In fact it is composed of surviving contracts and newly set prices. Given that in each an every period a price contract has a probability 1 − ξ of ending, the probability that a contract signed in period t − j survives until period t and ends at the end of period t is given by (1 − ξ)ξ j . Therefore, the aggregate price level may be expressed as the average of all surviving contracts  1 1−θ ∞ X j e 1−θ   = (1 − ξ)ξ Px,t−j 

Px,t

j=0

which can be expressed recursively as   1 1−θ 1−θ 1−θ Px,t = (1 − ξ)Pex,t + ξPx,t−1

(32)

Note that since the wage rate is common to all firms, the capital labor ratio is the same for any firm: kt (i) kt (j) kt = = ht (i) ht (j) ht we therefore have

 xt (i) = at

Department of Economics

kt ht

α ht (i) University of Bern

4

A SMALL OPEN ECONOMY MODEL WITH STAGGERED PRICES

integrating across firms, we obtain Z 1

 xt (i)di = at

0

denoting ht =

R1 0

kt ht

12

α Z

1

ht (i)di 0

ht (i)di, and making use of the demand for yt (i), we have Z

1

0

Px,t (i) Px,t

−θ

diyt = at ktα ht1−α

Denote Z

1

∆t = 0

Px,t (i) Px,t

−θ di

!−θ Pex,t−j Px,t !−θ !−θ ∞ X Pex,t−j Pex,t j + (1 − ξ)ξ Px,t Px,t j=1 !−θ !−θ ∞ X ex,t−`−1 P Pex,t (1 − ξ)ξ `+1 + Px,t Px,t `=0 !−θ !−θ   ∞ Pex,t−`−1 Px,t−1 −θ X Pex,t ` (1 − ξ)ξ +ξ Px,t Px,t Px,t−1 `=0 !−θ   Pex,t Px,t−1 −θ +ξ ∆t−1 Px,t Px,t !−θ Pex,t θ + ξπx,t ∆t−1 Px,t

∞ X = (1 − ξ)ξ j j=0

= (1 − ξ)

= (1 − ξ)

= (1 − ξ)

= (1 − ξ)

∆t = (1 − ξ)

Hence aggregate output is given by ∆t xt = at ktα ht1−α The behavior of the foreign economy is assumed to be very similar to the one observed domestically. We however assumethat foreign output and prices are exogenously given and modelled as AR(1) ? = P ? . The foreign household’s saving behavior is the same processes. We further assume that Px,t t as in the domestic economy (absent preference shocks), such that the foreign nomina interest rate is given by ? 1 ? − σ1 Pt yt? − σ = βRt? Et yt+1 ? Pt+1 We assume that foreign households do not buy domestic bonds, which implies that in equilibrium Btd = 0.

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A SMALL OPEN ECONOMY MODEL WITH STAGGERED PRICES

13

The general equilibrium is then given by 1

− σ1

σ ζc,t ct

−ν ν ψζh,t ht

λt yt ∆t xt

= λt

(33)

px,t xt = λt (1 − α)st h   t  it = qt 1 − ϕk −δ kt χ = ct + it + gt + bft 2 2 = at ktα h1−α t

(34) (35) (36) (37)

1 ρ−1

xdt = px,t ωyt   1 px,t ρ−1 d? xt = (1 − ω)yt? rert

(38) (39)

1

xft = (rert p?x,t ) ρ−1 (1 − ω)yt xt =

xdt

+

(40)

xd? t

(41)

ρ ρ−1

ρ ρ−1

1 = ωpx,t + (1 − ω)(rert p?t ) λt+1 λt = βRt Et πt+1 ∆e λt (1 + χbft ) = βRt? Et t+1 λt+1 π " t+1 px,t+1 xt+1 + qt+1 qt = βEt λt+1 αst+1 kt+1

(42) (43) (44) ϕk 1−δ+ 2



it+1 kt+1

!!#

2 −δ

2

(45)

∆et ? f R b + px,t xt − yt πt t−1 t−1  2 ! ϕk it = it 1 − −δ + (1 − δ)kt 2 kt

bft = kt+1

(46) (47)

∆et p?t rert−1 πt p?t−1 πx,t px,t = px,t−1 πt
log(Rt ) = ρr log(Rt−1 ) + (1 − ρ) log(R) + γπ (log(πt ) − log(π)) + γy (log(yt ) − log(y)) θ θ pnx,t = λt pθx,t st xt + βξEt [pnx,t+1 πt+1 ] θ−1 θ−1 pdx,t = λt pθx,t xt + βξEt [pdx,t+1 πt+1 ]  1  !1−θ  1−θ 1−θ pnx,t px,t−1  px,t = (1 − ξ) +ξ d πt px,t !−θ pnx,t θ ∆t = (1 − ξ) + ξ∆t−1 πx,t px,t pdx,t rert =

1

1

? −σ yt? − σ = βR? Et yt+1

p?t p?t+1

(48) (49) (50) (51) (52) (53)

(54) (55)

where λt = Λt Pt , rert = et Pt? /Pt , px,t = Px,t /Pt , pnt = Ptn /Ptθ , pdt = Ptd /Ptθ−1 , πt = Pt /Pt−1 , Department of Economics

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6

A REAL SMALL OPEN ECONOMY MODEL

14

πx,t = Px,t /Px,t−1 , ∆et = et /et−1 . All shocks follow AR(1) processes of the type log(at+1 ) = ρa log(at ) + εa,t+1 log(gt+1 ) = ρg log(gt ) + (1 − ρg ) log(g) + εg,t+1 log(ζc,t+1 ) = ρc log(ζc,t ) + εc,t+1 log(ζh,t+1 ) = ρh log(ζh,t ) + εh,t+1 ? log(yt+1 ) = ρy log(yt? ) + (1 − ρy ) log(y) + ε?y,t+1

log(p?t+1 ) = ρp log(p?t ) + (1 − ρp ) log(p) + ε?p,t+1

5

A Nominal Small Open Economy Model with Price Adjustment Costs

When price contracts are replaced with price adjustment costs, equations (36)–(37) become yt = ct + it + gt +

χ f 2 ϕp b + (πx,t − 1)2 yt 2 t 2

xt = at ktα h1−α t and equations (51)–(54) are replaced with  (1 − θ)px,t xt + θst xt − ϕp πx,t (πx,t − 1)yt + βEt

6

 λt+1 πx,t+1 ϕp (πx,t+1 − 1)yt+1 = 0 λt

A Real Small Open Economy Model

All nominal aspects disappear, such that the general equilibrium becomes 1

− σ1

σ ct ζc,t

−ν ν ψζh,t ht

λt yt xt

= λt

(56)

px,t xt = λt (1 − α) h  t  it = qt 1 − ϕk −δ kt χ = ct + it + gt + bft 2 2 = at ktα ht1−α

(57) (58) (59) (60)

1 ρ−1

xdt = px,t ωyt  1  px,t ρ−1 d? (1 − ω)yt? xt = rert

(61) (62)

1

xft = (rert p?x,t ) ρ−1 (1 − ω)yt xt =

xdt

+

(63)

xd? t

ρ ρ−1

(64) ρ

1 = ωpx,t + (1 − ω)(rert p?t ) ρ−1

Department of Economics

(65)

University of Bern

6

A REAL SMALL OPEN ECONOMY MODEL

15

λt = βRt Et λt+1 λt (1 + χbft ) = βRt? Et λt+1 " qt = βEt λt+1 α

(66) (67) px,t+1 xt+1 + qt+1 kt+1

1−δ+

? bft = Rt−1 bft−1 + px,t xt − yt  2 ! ϕk it kt+1 = it 1 − −δ + (1 − δ)kt 2 kt 1

1

? −σ yt? − σ = βRt? Et yt+1

ϕk 2



it+1 kt+1

2

!!# − δ2

(68) (69) (70) (71)

together with the shocks.

Department of Economics

University of Bern

7

7

A NOMINAL 2–COUNTRY MODEL WITH STAGGERED PRICES

16

A Nominal 2–Country Model with Staggered Prices

The problem of the domestic household is    1− σ1 ∞ X 1 ct+j ψ −ν 1+ν  σ max Et  β j ζc,t+j 1 − 1 + ν ζh,t+j ht+j 1− σ j=0 f ? Bth + et Btf + Pt ct + Pt it = Rt−1 Bt−1 + et Rt−1 Bt−1 + Pt wt ht + Pt zt kt − Pt τt !  2 ϕk it −δ + (1 − δ)kt kt+1 = it 1 − 2 kt

Then the first order conditions are given by 1

− σ1

σ ζc,t ct

= Λt Pt

(72)

−ν ν ψζh,t ht

= Λt Pt wt    it Λt Pt = Qt 1 − ϕk −δ kt Λt = βRt Et Λt+1 et+1 Λt+1 Λt = βRt? Et et "

(73) (74) (75) (76)

Qt = βEt Λt+1 Pt+1 zt+1 + Qt+1

ϕk 1−δ+ 2



it+1 kt+1

2

!!# − δ2

(77)

where Λt and Qt denote respectively the Lagrange multiplier of the first and second constraint. The behavior of the foreign household, hereafter denoted by a ?, is totally symmetrical. We further have the following risk sharing condition Λt =

Λ?t et

The retailer firm combines foreign and domestic goods to produce a non–tradable final good. It determines its optimal production plans by maximizing its profit ? f max Pt Yt − Px,t xdt − et Px,t xt

{xdt ,xft }

? denote the price of the domestic and foreign good respectively, denominated in where Px,t and Pxt terms of the currency of the seller. The final good production function is described by the following CES function  1 1

1

ρ

yt = ω 1−ρ xdt + (1 − ω) 1−ρ xft

ρ

ρ

(78)

where ω ∈ (0, 1) and ρ ∈ (−∞, 1). Optimal behavior of the retailer gives rise to the demand for the domestic and foreign goods xdt

 =

Px,t Pt



1 ρ−1

ωyt and

xft

 =

? et Px,t Pt



1 ρ−1

(1 − ω)yt

Abroad, the behavior is symmetrical, such that  1 1 1 ρ ρ ρ yt? = ω 1−ρ xft ? + (1 − ω) 1−ρ xd? t Department of Economics

(79)

(80) University of Bern

7

A NOMINAL 2–COUNTRY MODEL WITH STAGGERED PRICES

and xd? t

 =

Px,t et Pt?



1 ρ−1

ω)yt?

(1 −

17

and

xft ?

 =

? Px,t Pt?



1 ρ−1

ωyt?

(81)

xd and xf are themselves combinations of the domestic and foreign intermediate goods according to θ θ  θ−1 Z 1  θ−1 Z 1 θ−1 θ−1 f f d d and xt = xt (i) θ di (82) xt = xt (i) θ di 0

0

where θ ∈ (−∞, 1). Likewise abroad xd? t

1

Z =

θ−1 θ di xd? t (i)

θ  θ−1

and

0

Z

xft ?

1

= 0

θ−1 xft ? (i) θ di

θ  θ−1

(83)

Profit maximization yields demand functions of the form: xdt (i)

 =

Pxt (i) Pxt

−θ



xdt ,

xft (i)

xd? t ,

xft ? (i)

=

? (i) Pxt ? Pxt

−θ

xft ,

similarly abroad xd? t (i)

 =

Pxt (i) Pxt

−θ

 =

? (i) Pxt ? Pxt

−θ

xft ?

Plugging these demand functions in profits and using free entry in the final good sector, we get the following general price indexes 1  1−θ

1

Z

1−θ

Pxt =

Pxt (i)

di

,

? Pxt

Z

Pt = Pt? =

(84)

0 ρ ρ−1

ωPxt  ω

1  1−θ

? Pxt (i)1−θ di

=

0



1

+ (1 −

Pxt et



ρ ρ−1

ρ

? ρ−1 ω)(et Pxt )

? + (1 − ω)Pxt

 ρ−1 ρ

(85) ρ ρ−1

! ρ−1 ρ

(86)

The intermediate goods are produced by intermediaries, each of which has a local monopoly power. Each intermediate firm i, i ∈ (0, 1), uses a constant returns to scale technology xt (i) = at kt (i)α ht (i)1−α

(87)

where kt (i) and ht (i) denote capital and labor. The firm minimizes its real cost subject to (87). Minimized real total costs are then given by st xt (i) where the real marginal cost, st , is given by st =

ztα wt1−α ςat

with ς = αα (1 − α)1−α . Similarly abroad x?t (i) = a?t kt? (i)α h?t (i)1−α Department of Economics

(88)

University of Bern

7

A NOMINAL 2–COUNTRY MODEL WITH STAGGERED PRICES

18

where kt? (i) and h?t (i) denote capital and labor. The firm minimizes its real cost subject to (87). Minimized real total costs are then given by s?t x?t (i) where the real marginal cost, s?t , is given by s?t =

zt? α wt? 1−α ςa?t

with ς = αα (1 − α)1−α . The price setting behavior is essentially the same as in a closed economy. The expected profit flow generated by setting Pet (i) in period t writes max Et Pex,t (i)

∞ X

Φt,t+j ξ j Π(Pex,t (i))

j=0

subject to the total demand it faces:  Pt (i) −θ xt (i) = xt with xt = xdt + xd? t Pt   and where Π(Pex,t (i)) = Pex,t (i) − Pt+j st+j xt+j (i). Φt,t+j is an appropriate discount factor related to the way the household value future as opposed to current consumption, such that 

Φt,t+j ∝ β j

Λt+j Λt

This leads to the price setting equation   !−θ ∞ X ex,t (i) Λ P Pt+j t+j j (1 − θ) Et  (βξ) yt+j + θ Λt Px,t+j Px,t+j (i) j=0

Pex,t (i) Px,t+j

!−θ

 st+j xt+j  = 0

from which it shall be clear that all firms that reset their price in period t set it at the same level (Pet (i) = Pet , for all i ∈ (0, 1)). This implies that Pex,t = where n Px,t

 ∞ X = Et  (βξ)j Λt+j j=0

and d Px,t

n Px,t d Px,t

(89)

 θ θ Pt+j Px,t+j st+j xt+j  θ−1

  ∞ X θ = Et  (βξ)j Λt+j Px,t+j xt+j  j=0

n and P d admit a recursive representation, such that Fortunately, both Px,t x,t

Department of Economics

θ θ n Λt Pt Px,t st xt + βξEt [Px,t+1 ] θ−1 θ d = Λt Px,t xt + βξEt [Px,t+1 ]

n Px,t =

(90)

d Px,t

(91)

University of Bern

7

A NOMINAL 2–COUNTRY MODEL WITH STAGGERED PRICES

19

Likewise abroad, θ ? θ ? ? n? Λ?t Pt? Px,t st xt + βξEt [Px,t+1 ] θ−1 ? θ ? d? = Λ?t Px,t xt + βξEt [Px,t+1 ]

n? Px,t =

(92)

d? Px,t

(93)

Recall now that the price index is given by 1

Z Px,t =

Px,t (i)

1−θ

1  1−θ di

0

In fact it is composed of surviving contracts and newly set prices. Given that in each an every period a price contract has a probability 1 − ξ of ending, the probability that a contract signed in period t − j survives until period t and ends at the end of period t is given by (1 − ξ)ξ j . Therefore, the aggregate price level may be expressed as the average of all surviving contracts  1 1−θ ∞ X j e 1−θ   = (1 − ξ)ξ Px,t−j 

Px,t

j=0

which can be expressed recursively as  1  1−θ 1−θ 1−θ Px,t = (1 − ξ)Pex,t + ξPx,t−1

(94)

Note that since the wage rate is common to all firms, the capital labor ratio is the same for any firm: kt (i) kt (j) kt = = ht (i) ht (j) ht we therefore have

 xt (i) = at

integrating across firms, we obtain Z 1

 xt (i)di = at

0

denoting ht =

R1 0

kt ht

α

kt ht

α Z

ht (i)

1

ht (i)di 0

ht (i)di, and making use of the demand for yt (i), we have Z 0

Department of Economics

1

Px,t (i) Px,t

−θ

diyt = at ktα ht1−α

University of Bern

7

A NOMINAL 2–COUNTRY MODEL WITH STAGGERED PRICES

20

Denote Z

1

∆t = 0

Px,t (i) Px,t

−θ di

!−θ Pex,t−j Px,t !−θ !−θ ∞ X Pex,t Pex,t−j j + (1 − ξ)ξ Px,t Px,t j=1 !−θ !−θ ∞ X Pex,t−`−1 Pex,t `+1 + (1 − ξ)ξ Px,t Px,t `=0 !−θ !−θ   ∞ Pex,t−`−1 Pex,t Px,t−1 −θ X ` +ξ (1 − ξ)ξ Px,t Px,t Px,t−1 `=0 !−θ   Pex,t Px,t−1 −θ +ξ ∆t−1 Px,t Px,t

∞ X = (1 − ξ)ξ j j=0

= (1 − ξ)

= (1 − ξ)

= (1 − ξ)

= (1 − ξ) therefore ∆t = (1 − ξ)

Pex,t Px,t

!−θ θ + ξπx,t ∆t−1

Hence aggregate output is given by ∆t xt = at ktα ht1−α In a general equilibrium, we will have Btd + Btd? = 0 and Btf + Btf ? = 0. The general equilibrium is then given by 1

− σ1

σ ζc,t ct 1 σ

? − σ1

? ζc,t ct

−ν ν ψζh,t ht ? −ν ? ν ψζh,t ht

λt λ?t yt yt?

= λt

(95)

= λ?t

(96)

xt = λt (1 − α)px,t st ht x? = λ?t (1 − α)p?x,t s?t t? ht    it = qt 1 − ϕk −δ kt   ?  it ? = qt 1 − ϕ k −δ kt? = ct + it + gt =

∆t xt = ∆?t x?t

Department of Economics

=

c?t

+ i?t + gt? at ktα ht1−α a?t kt? α h?t 1−α

(97) (98) (99) (100) (101) (102) (103) (104)

University of Bern

7

A NOMINAL 2–COUNTRY MODEL WITH STAGGERED PRICES

21

1 ρ−1 ωyt xdt = px,t   1 px,t ρ−1 d? xt = (1 − ω)yt? rert

(105) (106)

1

xft = (rert p?x,t ) ρ−1 (1 − ω)yt xft ? = p?x,t

1 ρ−1

(107)

ωyt?

(108)

xt = xdt + xd? t x?t =

xft

+

(109)

xft ?

ρ ρ−1

(110) ρ ρ−1

1 = ωpx,t + (1 − ω)(rert p?x,t )  ?  ρ ρ px,t ρ−1 ? ρ−1 1 = ωpx,t + (1 − ω) rert ? λ λt = t rert λt+1 λt = βRt Et π " t+1 px,t+1 xt+1 + qt+1 1 − δ + qt = βEt λt+1 αst+1 kt+1 " p?x,t+1 x?t+1 ? + qt+1 1−δ+ qt? = βEt λ?t+1 αs?t+1 ? kt+1  2 ! ϕk i t + (1 − δ)kt −δ kt+1 = it 1 − 2 kt  2 ! ϕk i?t ? ? kt+1 = it 1 − −δ + (1 − δ)kt? 2 kt?

(111) (112) (113) (114) ϕk 2



ϕk 2



it+1 kt+1

2

i?t+1 ? kt+1

2

!!# − δ2

(115) !!#

− δ2

(116) (117) (118)

Rt = Rt? Et ∆et+1 ∆e π ? rert = t t rert−1 πt πx,t px,t−1 px,t = πt ? πx,t p?x,t = ? p?x,t−1 πt

(119) (120) (121) (122)


log(Rt ) = ρr log(Rt−1 ) + (1 − ρ) log(R) + γπ (log(πt ) − log(π)) + γy (log(yt ) − log(y))
? log(Rt? ) = ρr log(Rt−1 ) + (1 − ρ) log(R) + γπ (log(πt? ) − log(π)) + γy (log(yt? ) − log(y)) θ θ pnx,t = λt pθx,t st xt + βξEt [pnx,t+1 πt+1 ] θ−1 θ ?θ pn? λ?t p?x,t θ s?t x?t + βξEt [pn? x,t = x,t+1 πt+1 ] θ−1 θ−1 pdx,t = λt pθx,t xt + βξEt [pdx,t+1 πt+1 ] ?θ−1 ? ? θ ? d? pd? x,t = λt px,t xt + βξEt [px,t+1 πx,t+1 ]

Department of Economics

(123) (124) (125) (126) (127) (128)

University of Bern

8

A NOMINAL 2–COUNTRY MODEL WITH PRICE ADJUSTMENT COSTS

 px,t = (1 − ξ)  p?x,t = (1 − ξ)

∆t = (1 − ξ) ∆?t

= (1 − ξ)

pnx,t

!1−θ

 +ξ

pdx,t pn? x,t pd? x,t

!1−θ

pnx,t

!−θ

+ξ

px,t pdx,t pn? x,t



22

px,t−1 πt

1−θ



1 1−θ

(129)

 

1

1−θ  p?x,t−1 1−θ  πt?

(130)

θ + ξ∆t−1 πx,t

(131)

? θ + ξ∆?t−1 πx,t

(132)

!−θ

p?x,t pd? x,t

? /P ? , pn = P n /P θ , where λt = Λt Pt , λ?t = Λ?t Pt? , rert = et Pt? /Pt , px,t = Px,t /Pt , p?x,t = Px,t t t t t n? /P ?θ , pd? = P d? /P ?θ−1 , π = P /P ? = P ? /P ? , π pdt = Ptd /Ptθ−1 , pn? = P , π = P /P , t t t−1 x,t x,t x,t−1 t t t t t t t t t−1 ? = P ? /P ? e = e /e πx,t , ∆ . t t−1 x,t t x,t−1

All shocks follow AR(1) processes of the type log(at+1 ) = ρa log(at ) + ρ?a log(a?t ) + εa,t+1 log(a?t+1 ) = ρ?a log(at ) + ρa log(a?t ) + ε?a,t+1 log(gt+1 ) = ρg log(gt ) + (1 − ρg ) log(g) + εg,t+1 ? log(gt+1 ) = ρg log(gt? ) + (1 − ρg ) log(g) + ε?g,t+1

log(ζc,t+1 ) = ρc log(ζc,t ) + εc,t+1 ? ? log(ζc,t+1 ) = ρc log(ζc,t ) + ε?c,t+1

log(ζh,t+1 ) = ρh log(ζh,t ) + εh,t+1 ? ? log(ζh,t+1 ) = ρh log(ζh,t ) + ε?h,t+1

8

A Nominal 2–Country Model with Price Adjustment Costs

When price contracts are replaced with price adjustment costs, equations (101)–(104) become ϕp (πx,t − 1)2 yt 2 ϕp ? yt? = c?t + i?t + gt? + (π − 1)2 yt? 2 x,t xt = at ktα ht1−α yt = ct + it + gt +

x?t = a?t kt? α h?t 1−α and equations (125)–(132) are replaced with 

 λt+1 πx,t+1 ϕp (πx,t+1 − 1)yt+1 0 = (1 − θ)px,t xt + θst xt − ϕp πx,t (πx,t − 1)yt + βEt λt  ?  λt+1 ? ? ? ? ? ? ? ? ? ? 0 = (1 − θ)px,t xt + θst xt − ϕp πx,t (πx,t − 1)yt + βEt π ϕp (πx,t+1 − 1)yt+1 λ?t x,t+1

Department of Economics

University of Bern

9

9

A REAL 2–COUNTRY MODEL

23

A Real 2–Country Model

All nominal aspects disappear, such that the general equilibrium becomes 1

− σ1

σ ζc,t ct 1 σ

? − σ1

? ζc,t ct

= λt

(133)

= λ?t

(134)

xt ht x? ? −ν ? ν ψζh,t ht = λ?t (1 − α)p?x,t t? h   t  it −δ λt = qt 1 − ϕk kt   ?  it ? ? λ t = qt 1 − ϕ k −δ kt? yt = ct + it + gt −ν ν ψζh,t ht = λt (1 − α)px,t

yt?

=

c?t

+

i?t

+

(135) (136) (137) (138) (139)

gt?

(140)

1 ρ−1

xdt = px,t ωyt   1 px,t ρ−1 d? xt = (1 − ω)yt? rert

(141) (142)

1

xft = (rert p?x,t ) ρ−1 (1 − ω)yt xft ? = p?x,t

1 ρ−1

(143)

ωyt?

(144)

xt = xdt + xd? t

(145)

xft

(146)

x?t = xt = x?t =

+ xft ? at ktα h1−α t ? ? α ? 1−α at kt ht ρ ρ−1

(147) (148) ρ ρ−1

1 = ωpx,t + (1 − ω)(rert p?x,t )  ?  ρ ρ px,t ρ−1 ? ρ−1 1 = ωpx,t + (1 − ω) rert ? λt = λt " px,t+1 xt+1 qt = βEt λt+1 α + qt+1 1 − δ + kt+1 " p?x,t+1 x?t+1 ? ? qt = βEt λ?t+1 α + qt+1 1−δ+ ? kt+1  2 ! ϕk it kt+1 = it 1 − −δ + (1 − δ)kt 2 kt  2 ! ϕk i?t ? ? kt+1 = it 1 − −δ + (1 − δ)kt? 2 kt?

Department of Economics

(149) (150) (151) ϕk 2



ϕk 2



it+1 kt+1

2

i?t+1 ? kt+1

2

!!# − δ2

(152) !!#

−δ

2

(153) (154) (155)

University of Bern