Simple Small Open Economy

Simple Small Open Economy

Lawrence Christiano

Department of Economics, Northwestern University

Outline

Starting point: Simple Closed Economy Model

Extension to Open Economy: building a fairly standard SOE for policy analysis in central banks

I Riksbank Ramses I and II:Adolfson, Laseen, Linde, and Villani, 2008, extended inChristiano, Trabandt and Walentin, 2011

I Copaciu-Nalban-Buleteof Romanian Central Bank

I Also,Castillo, Lama and Medina 2019,Lama and Medina 2020and references they cite.

I Technical appendix for ongoing work by me with: Santiago Camara and Husnu Dalgic.

Objective here:

I Extend the closed economy model to obtain a simple model of an open economy.

I Technical issue: scaling the variables, to accommodate (balanced) growth and inflation.

I Analyze the dynamic behavior of the model in response to shocks.

F Model has a strong Mundell-Fleming flavor.

Household

Simple Closed Economy Model

Results from closed economy model

I Household preferences:

E₀

∑∞ t=0

β^t{ u(C_t) −exp(τt)^N

1+ϕ t

1+ϕ

! , u(Ct) ≡log Ct

I Aggregate resources and household intertemporal optimization:

Y_t =p^∗_tA_tN_t, u_c,t =βEtu_c,t+1 R_t πt+1

I Law of motion of price distortion (seethis for details):

p_t^∗=



(1−θ) ¹−θ(πt)^ε−1 1−θ

!_ε−^ε₁ + ^θπ

εt

p^∗_t−1





−1

(4)

Simple Closed Economy Model

Equilibrium conditions associated with price setting:

PtYt

P_t^cCt

+Etπ^ε_t⁻₊¹₁βθFt+1=Ft (2)

Kt = ^ε

ε−1(1−ν)

Wt/(AtPt)

z }| {

Pc,t

PtAt

×

=^Wt_Pc

t by household optimization

z }| {

exp(τt)N_t^ϕ uc,t

× ^PtYt P_t^cCt

+E_tβθπ^ε_t+1K_t+1 (1)

I In simple closed economy model, Yt =Ct, not so here.

I Homogeneous good, Yt, and consumption good, Ct, different and have different prices, P_t and P_t^c, respectively (more below).

Cross-price restrictions Kt

Ft

=^{ 1}−θπ^ε_t⁻¹ 1−θ

₁₋¹_ε (3)

Extensions to Small Open Economy: 18 variables

rate of depreciation, exports, real (scaled) net foreign assets, terms of trade, real exchange rate, trend

z }| {

st, xt, d_t^∗, p^x_t, qt, zt

relative price of domestic consumption(c is composed of domestically produced goods & imports)

z }| {

p_t^c≡P_t^c/Pt relative price of imports

z}|{p^m_t

detrended nominal exchange rate

z}|{S˜_t

consumption price inflation

z}|{

π_t^c

closed economy variables

z }| {

Rt, πt, yt, Nt, ct, Kt, Ft, p_t^∗

Modifications to Simple Model to Create Open Economy

Unchanged:

I production of (domestic) homogeneous good, Y_t (=A_tp_t^∗N_t)

I Calvo pricing equations (with two adjustments listed above) Changes:

I household budget constraint includes opportunity to acquire foreign assets/liabilities.

I net foreign assets introduced into household utility for reasons explained below.

I Y_t =C_t no longer true.

I introduce exports, imports, balance of payments.

I exchange rate,

St =domestic currency price of one unit of foreign currency S_t = domestic money

foreign money

Monetary Policy Rule

Taylor rule log R_t

R



= ρRlog R_t−1 R



+ (1−ρR)Et[r_πlog

π^c_t

¯ π^c



+rylog y_t y



+rSlog

 Set



] +εR,t (17) where:

π_t^c consumer price inflation, and target,π¯^c εR,t iid, mean zero monetary policy shock yt =Yt/At, output scaled by technology Rt nominal rate of interest

εR,t mean zero monetary policy shock Se_t =S_t/ ψ^tS¯

˜ nominal exchange rate, S_t, relative to target, ψ^tS , ψ¯ >0.

Households

Household preferences:

E0

∑∞ t=0

β^t{ u(Ct) −exp(τt) ^l

1+ϕ t

1+ϕ+ht

 S_tD_t^∗ P_t^c

! , u(C_t) ≡log C_t, where P_t^c denotes the price of the domestic consumption good.

Note that the real value of net foreign assets are included in the utility function.

Household Budget Constraint

’Uses of funds less than or equal to sources of funds’

StD_t^∗+P_t^cCt+Dt

≤Dt−1Rd ,t−1+StR_t^f₋₁D_t^∗₋₁+Wtlt+transfers and profitst

Domestic bonds

D_t end-of-period t stock of domestic loans R_{d ,t} rate of return on Dt−1

Foreign assets

D_t^∗ net, end-of-period t stock of foreign assets,

Household Intertemporal Conditions: Domestic Assets

First order condition:

1 P_t^cCt

= βEt

R_{d ,t} P_t^c₊₁Ct+1

Scaling (c_t ≡C_t/At, π_t^c₊₁ ≡P_t^c/P_t^c₋₁):

1

c_t =βEt

R_{d ,t}

π^c_t+1c_t+1exp(_∆at+1)^.(5) Technology:

at ≡log(At), ∆at =at−at−1.

Non-Pecuniary Impact of Foreign Assets

Define (more on this later):

h_t S_tD_t^∗ P_t^c



= −¹

2γ S_tD_t^∗ Z_tP_t^c −_Υt

2

, where

I Zt is a trend term that (in long run) grows at the same rate as At

(technology), details on this later...

I Υt is target for detrended (by Z_t) net foreign assets, measured in units of domestic consumption goods.

Marginal impact on utility of change in D_t^∗: dh_t_S

tD_t^∗ P_t^c



dD_t^∗ = −γ S_tD_t^∗ Z_tP_t^c −_Υt

 S_t Z_tP_t^c

If foreign assets above target (e.g., object in parentheses positive), then increase in D_t^∗ reduces utility

If foreign assets below target, then increase in D_t^∗ raises utility.

Household Intertemporal Conditions: Foreign Assets

Optimality of foreign asset choice (verify this by solving Lagrangian representation of household problem)

utility cost of 1 unit of foreign assets=St units of domestic currency or St/P_t^c units of Ct

z }| { u_c,tS_t

P_t^c

=

marginal utility benefit of extra net foreign assets

z }| {

dh_t_S

tD_t^∗ P_t^c



dD_t^∗ +βEt

conversion into utility units

z }| { u_c,t+1

×

quantity of domestic cons. goods purchased from the payoff of 1 unit of foreign currency

z }| {

St+1

foreign currency payoff next period from one unit of foreign currency today

z}|{R_{d ,t}^∗ P_t^c₊₁

Household Intertemporal Conditions: Foreign Assets

First order condition (p^c_t ≡P_t^c/Pt) for D_t^∗: S_t

P_t^cCt

= dht

StD_t^∗ P_t^c

 dD_t^∗ +βEt

S_t+1R_t^f P_t^c₊₁Ct+1

= −γ StD_t^∗ Z_tP_t^c −_Υt

 St

Z_tP_t^c +βEt

St+1R_t^f P_t^c₊₁C_t+1

Multiply by P_t^cAt/St: 1

c_t = −γ

 d_t^∗ z_tp_t^c −_Υt

 1 z_t +βEt

st+1R_t^f

π^c_t+1c_t+1exp(_∆at+1)

zt ≡ ^Z_A^t

t, ct ≡ ^C_A^t

t, d_t^∗ ≡ ^StD

t∗

PtAt

, st ≡ ^St St−1

= ψ S˜t

S˜_t−1(14)

Why Put Net Foreign Assets in the Utility Function?

One motivation is technical (see, e.g., Schmitt-Grohe and Uribe.) Because the domestic economy is assumed to be small, it has no impact on R_t^f, the return on foreign assets.

I From the point of view of domestic residents, the foreign asset represents a constant returns investment technology.

I Consequence: there does not exist a steady state level of net foreign assets that is independent of the initial net foreign asset position.

I Why? The answer resembles why there is no steady state capital stock independent of initial capital in the so-called Ak model:

F That is, if k starts low, there is no incentive to raise investment sharply to raise k to some steady state level. That’s because the marginal product of capital, A, is not higher when k is low.

F Similarly, when k is high, there is no reason to let the stock of capital fall. That is, when k is high, A is not lower.

Why Put Net Foreign Assets in the Utility Function?

Standard solution methods assume that variables have a steady state that is independent of initial conditions.

Small open economy models require a small adjustment.

Consider the first order condition for foreign assets:

1 c_t =

non-pecuniary part

z }| {

−^γ z_t

 d_t^∗ z_tp^c_t −_Υt

 +

pecuniary part

z }| {

βEt

st+1R_t^f

π^c_t+1c_t+1exp(_∆at+1)(7) The payoff on the foreign asset corresponds to the two parts of the term on the right of the equality:

I Pecuniary part: money (pecuniary) part of the return on the foreign assets.

I Non-pecuniary part: makes overall return higher when households’ net foreign asset position is below target,Υt, giving households incentive to accumulate more assets. Similarly, the overall return is lower when the net foreign asset position is above target, giving households an incentive to accumulate less.

So, model has a unique steady state, for γ>0.

Why Put Net Foreign Assets in the Utility Function?

We have described a purely technical reason for including foreign debt in the utility function:

I want a steady state value of d_t^∗

d^∗=p^cΥ.

I this can be accomplished with a tiny value of γ>0, without affecting model dynamics.

Later, we will provide an empirical reason for γ>0

I help to account for observation that uncovered interest parity (UIP) does not hold in the data.

Household

Final Domestic Consumption Goods

Produced by representative, competitive firm using:

C_t =

"

(1−ωc)^{ηc1 }C_t^d^ηc−1

ηc +ω

1

cηc (C_t^m)^ηc

−1 ηc

# ^ηc

ηc−1

where

C_t^d domestic homogeneous output good, price P_t C_t^m imported good, price P_t^m ≡StP_t^f

C_t final consumption good, P_t^c

ηc elasticity of substitution, domestic and imported goods.

The firm takes the prices, Pt, P_t^m, P_t^c, as given and beyond its control.

Final Domestic Consumption Goods

Profit maximization by representative firm:

max

Ct,C_t^m,C_t^d

P_t^cC_t−P_t^mC_t^m−P_tC_t^d, subject to production function.

First order conditions associated with maximization:

C_t^m : P_t^c

=

 ωc Ct

C mt

¹

ηc

z }| { dC_t

dC_t^m =P_t^m, C_t^d : P_t^c

=



(1−ωc)^Ct

C dt

¹

ηc

z }| { dC_t

dC_t^d =P_t so that the demand functions are:

C_t^m =ωc

 P_t^c P_t^m

ηc

C_t, C_t^d = (1−ωc)^{ P}

tc

Pt

ηc

C_t.

Final Good Prices

Substituting demand functions back into the production function:

C_t = [(1−ωc)^ηc1

 C_t P_t^c

Pt

ηc

(1−ωc)

^ηc

−1 ηc

+ω

1

cηc

 ωc

 P_t^c P_t^m

ηc

C_t

^ηc

−1 ηc

]^ηcηc−1, to obtain,

1=^{ P}

tc

P_t

ηc



(1−ωc)^ηc1 ((1−ωc))^ηc

−1

ηc +ω

1

cηc

 ωc

 P_t P_t^m

ηc^ηc

−1 ηc





ηc ηc−1

,

or (p_t^c ≡P_t^c/Pt, p^m_t ≡P_t^m/Pt):

p^c_t =

marginal cost, in units of the homogeneous good

z }| {

h(1−ωc) +ωc(p^m_t )^1−ηci

1 1−ηc

(8)

Pass-Through

Multiplying (8) by Pt ’price = marginal cost’:

P_t^c = ^h(1−ω_c) (P_t)^1−ηc +ω_c(P_t^m)^1−ηci

1 1−ηc

, or, using P_t^m =S_tP_t^f :

P_t^c =



(1−ω_c) (P_t)^1−ηc +ω_c



S_tP_t^f1−ηc₁₋¹

ηc

.

Note that if the exchange rate depreciates, i.e., S_t rises, then marginal cost rises so that the depreciation is ’passed through’

marginal cost and into the final good price, P_t^c. This pass-through occurs, no matter how sticky the prices underlying Pt are.

The high degree of pass through in this model reflects its simplicity.

SeeCTW for a discussion of how this model can be modified to slow down the pass through of exchange rate changes into final good prices.

Consumer Price Inflation

Consumption good inflation and homogeneous good inflation:

π^c_t ≡ ^P

tc

P_t^c₋₁ = ^Ptp

tc

P_t−1p^c_t−1 =π_t

"

(1−ωc) +ωc(p^m_t )^1−ηc (1−ωc) +ωc p_t^m₋₁
1−ηc

#₁₋¹

ηc

(10)

Exports

Foreign demand for domestic goods:

X_t =^{ P}

tx

P_t^f

−ηf

Y_t^f = (p^x_t)^−ηf Y_t^f

Y_t^f foreign demand shifter

P_t^f foreign currency price of foreign good P_t^x foreign currency price of export good Foreign demand is exogenous to the domestic economy.

Exporters are competitive and simply sell the homogeneous good to foreigners.

I P_t^x=Pt/St (i.e., if St depreciates then P_t^x drops).

I Problem: evidence is that export prices are sticky in dollars (more on this in a later lecture).

Temporary Diversion on Balanced Growth

The source of growth in the model is A_t.

We require the growth to be balanced, so that when growing variables are scaled by At the ratios converge in steady state.

Mathematically, balanced growth requires that after all variables are scaled by At, At itself disappears from the system.

I This places certain restrictions on preferences and technology, restrictions which our model satisfies.

So, in the case of Y_t^f, balanced growth requires that Y_t^f grows at the same rate as A_t.

An obvious way to proceed is to assume Y_t^f =y_t^fAt,

where y_t^f is an exogenous shock to Y_t^f. But, this formulation implies that a shock to technology simultaneously expands the demand for exports.

I Seems implausible.

I Inconvenient when we compute impulse response functions. Here, we want to study the effects of a disturbance that originates in just one part of the system.

Back to Exports

Preceding slide suggests we want to express Y_t^f in the following form:

Y_t^f =y_t^fZt,

where Zt grows with At, yet Zt responds extremely slowly to At. We apply the approach in Christiano-Trabandt-Walentin(2011, section 2.3):

Y_t^f =y_t^fZ_t, Z_t =A¹_t^−δZ_t^δ₋₁, 0<δ<1, zt ≡ ^Zt

A_t =^{ At−1} A_t

Zt−1

A_t−1

δ

=exp(−δ∆at)z_t^δ₋₁(18) xt = (p_t^x)^−ηf y_t^fzt(11)

Note: with δ close, but less than, unity

I Z_t grows at the same as A_t in the sense that Z_t/At converges to a constant in steady state.

I Z_t hardly responds to a shock to a shock in A_t.

Household

Aggregate Conditions and Variables

We’ve discussed all the agents.

Next, turn to aggregate conditions:

I market clearing, balance of payments

Aggregate variables: real exchange rate and GDP.

Homogeneous Goods Market Clearing

Clearing in domestic homogeneous goods market:

output of domestic homogeneous good, Yt

=uses of domestic homogeneous goods or,

Y_t =

goods used in production of final consumption, Ct

z}|{

C_t^d +

exports

z}|{X_t +

govenment

z}|{G_t

= (1−ωc)(p_t^c)^ηcC_t+X_t+G_t.

Note: we assume that government spending is on homogeneous good, which is purely domestically produced.

Aggregate Employment and Uses of Homogeneous Goods

Substituting out in previous expression for Y_t:

A_tp^∗_tl_t= (1−ωc) (p^c_t)^ηcC_t+X_t+G_t, or,

p^∗_tl_t = (1−ωc) (p^c_t)^ηcc_t+x_t+g_tz_t,(6) c_t≡ ^Ct

At

, x_t ≡ ^Xt At

, G_t =g_tZ_t, z_t= ^Zt At

.

Also,

y_t = ^Yt

A_t =p_t^∗l_t(16)

For an extended discussion of (16), see thisincluding the footnotes.

Balance of Payments

Sources equal uses of funds.

acquisition of new net foreign assets, in domestic currency units

z }| { S_tD_t^∗ + expenses on imports_t

=receipts from exports_t +

receipts from existing stock of net foreign assets

z }| {

S_tR_{d ,t}^∗ ₋₁D_t^∗₋₁

Balance of Payments, the Pieces

Exports and imports:

expenses on imports_t =StP_t^f

=C_t^m

z }| {

ωc

 p_t^c p_t^m

ηc

Ct

receipts from exports_t =StP_t^xXt.

Balance of payments:

S_tD_t^∗+S_tP_t^fωc

 p_t^c p_t^m

ηc

C_t

=S_tP_t^xX_t+S_tR_{d ,t}^∗ ₋₁D_t^∗₋₁.

Balance of Payments, Scaling

Scaling by P_tA_t :

≡d_t^∗

z }| { StD_t^∗ P_tA_t +^StP

tf

P_t ωc

 p_t^c p_t^m

ηc

ct

=

=1

z }| { S_tP_t^x

Pt

xt+^StR

∗

d ,t−1D_t^∗₋₁ PtAt

, or,

d_t^∗+p^m_t ωc

 p_t^c p^m_t

ηc

ct =xt+ ^stR

∗

d ,t−1d_t^∗₋₁ πtexp(_∆at)^,(15)

where d_t^∗ is ’scaled, homogeneous goods value of net foreign assets’

Gross Domestic Product

GDP: ’C + I + G + Net Exports’.

I Problem: these are different goods, with different prices.

GDP in domestic consumption units - nominal divided by P_t^c:

GDPt ≡

nominal expenditures on consumption

z }| {

P_t^cAtct +

nominal government exp

z }| { PtgtZt

P_t^c

+

nominal exports

z }| { xtAtPt −

nominal imports

z }| {

StP_t^fωc

 p^c_t p^m_t

ηc

Ct

P_t^c

=At

≡gdpt (=GDPt/At)

z }| {

"

ct+^gtzt p_t^c + ^xt

p^c_t −^{ p}

tm

p_t^c

1−ηc

ωcct

#

So, GDP (in consumption units) growth is:

log(GDP_t) −log(GDP_t−1) =_∆at+log(gdp_t) −log(gdp_t−1).

Real Exchange Rate

Real Exchange Rate, q

p_t^m = ^P

tc

P_t^c

zero profits for importers, P_t^m=StP_t^f

z}|{P_t^m P_t

=p_t^c×

real exchange rate, qt ≡^S_P^tPc^tf t

z}|{q_t (9)

Scaling:

1

zero profit condition for exporters

z}|{= ^StP

tx

Pt

= ^P

tcS_tP_t^fP_t^x

PtP_t^cP_t^f =q_tp^x_tp_t^c(12) Also,

q_t qt−1

= ^p

tm

p^m_t−1 p_t^c₋₁

p_t^c =st

π^f_t

π^c_t,(13)_{, πt}^f ≡ ^P

tf

P_t^f₋₁,

terms of trade

z}|{p_t^x =P_t^x P_t^f

Household

Pulling the Equations Together

The 18 endogenous variables:

Kt, Ft, lt, yt, πt, ct, p_t^∗, Rd ,t, d_t^∗, p_t^m, p^c_t, qt, p_t^x, π^c_t, xt, st, eSt, zt

The 8 exogenous variables: Υt, τt,∆at, εR,t, gt, π^f_t, y_t^f, R_{d ,t}^∗ Equilibrium conditions resembling those in closed economy:

Kt = ^ε(1−ν)

ε−1 exp(τt)l_t^ϕyt+βθEtπ_t^ε₊₁Kt+1 (1) F_t = ^yt

p_t^cc_t +E_tπ^ε_t⁻₊¹₁βθFt+1 (2) K_t

F_t= 1−θπ_t^ε−1 1-θ

1−¹ε

(3)

p_t^∗ =



(1−θ) ¹−θ(πt)^ε−1 1−θ

!_ε−^ε₁ + ^θπ

εt

p_t^∗₋₁





−1

(4) 1

ct

=βEt

Rt

π^c_t+1ct+1exp(_∆at+1) ⁽⁵⁾ y_t = (1−ωc) (p^c_t)^ηcc_t+x_t+g_tz_t (6)