Solving and Closing a Small-Open-Economy. Model. Economics 826: International Finance. Nazim Belhocine. March 26th, 2004. 1 ...
Solving and Closing a Small-Open-Economy Model Economics 826: International Finance Nazim Belhocine March 26th, 2004
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The seminal work of Kydland and Prescott (1982) called naturally for an extension to a small-open-economy. However, a mere “opening” of a closed economy would feature an equilibrium dynamics that possess a random walk component. Many alternatives have been proposed to avoid this undesirable feature. This paper exposes these alternatives and uses one of them to solve a small-open-economy model. The organization of the paper is as follows. Section 1 explains why non-stationarity arises. Section 2 illustrates the different approaches adopted by the literature to induce stationarity or to “close” the model. Finally, section 3 illustrates one of these approaches by numerically solving a small-open-economy model.
1. The issue at hand A typical “textbook” small open economy model produces a smooth consumption path by assuming that the subjective discount rate ρ is equal to the interest rate r. As soon as a discrepancy between ρ and r is allowed, consumption behaves in a knife-edge fashion. This property can easily be seen from the Euler equation. Let β =
1 1+ρ
and U (.) denotes the utility function
with the usual regularity properties. The Euler equation in a deterministic representative agent model is given by: U 0 (ct ) = β(1 + r)U 0 (ct+1 ).
(1)
Regardless of the utility function consumption grows forever when β > 1 1+r
and shrinks forever when β < β>
1 1+r
since
1 ⇔ U 0 (ct+1 ) < U 0 (ct ) ⇔ ct+1 > ct . 1+r 2
(2)
Thus, the desired consumption path is tilted upward if the consumer is patient enough that β >
1 1+r
and downward in the opposite case. Therefore,
a constant steady-state consumption path is optimal only when β = a small-open-economy context, r is given exogenously so β >
1 1+r
1 . 1+r
In
leads to
an accumulation of financial assets over time. With ever-growing net foreign assets, the assumption that the small-economy always faces a fixed world interest rate becomes strained. In this framework, either (a) no stationary equilibrium exists when r and ρ are not preset to be equal, or (b) if the two are equal, the economy is always at a steady-state equilibrium that is consistent with any initial level of foreign asset holdings. The analysis so far was conducted in a deterministic setting to illustrate the issue at hand but it carries over naturally to a stochastic environment. However, computing business-cycle dynamics becomes problematic because with ρ different than r, the model generates nonstationary variables. This then precludes a well-defined stochastic steady-state equilibrium. In other words, certain variables of the model do not return to their initial steadystate values when they are subjected to a temporary shock. This introduces serious computational difficulties because all available techniques are valid locally around a given stationary path.
2. Closing a small-open-economy To resolve this problem, researchers resort to a number of modifications to the standard model that have no other purpose than induce stationarity of the equilibrium dynamics. Each alternative aims therefore to “close” the model. These stationary-inducing techniques are illustrated in the context 3
of a baseline model exposed in the following section.
2.1 The baseline model Consider a small-open-economy populated by a large number of identical households with preferences described by the following expected utility function: E0
∞ X
βt U (ct , ht )
(3)
t=0
where ct denotes consumption, ht denotes hours worked and βt denotes the discount factor. The discount factor is written in this general form to allow for an endogenous specification in section 2.2. Let us assume for now that βt = β t , ∀t. The evolution of financial wealth, bt , is given by bt+1 = (1 + rt )bt + tbt
(4)
where rt denotes the interest rate at which domestic residents can borrow in international markets in period t, and tbt denotes the trade balance. In turn, the trade balance is given by tbt = yt − ct − it − Φ(kt+1 − kt )
(5)
where yt denotes domestic output, it denotes gross investment, and Φ(·) denotes an adjustment cost function satisfying Φ(0) = Φ0 (0) = 0. Smallopen-economy models typically include capital adjustment costs to avoid excessive investment volatility in response to variations in the domestic-foreign interest rate differential. The restrictions imposed on Φ ensure that in the
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non-stochastic steady-state, adjustment costs are zero and the domestic interest rate equals the marginal product of capital net of depreciation. Output is produced by means of a linearly homogeneous production function: yt = At F (kt , ht )
(6)
where At is an exogenous stochastic productivity shock. Its law of motion is given by: ln At+1 = ρ ln At + t+1 ; t+1 ∼ N IID(0, σ2 ); t ≥ 0.
(7)
Finally, the stock of capital evolves according to kt+1 = it + (1 − δ)kt
(8)
where δ ∈ (0, 1) denotes the rate of depreciation of physical capital. Households choose processes {ct , ht , yt , it , kt+1 , bt+1 }∞ t=0 so as to maximize the utility function (3) subject to Equations (4)-(8) and a no-Ponzi constraint of the form dt+j ≤ 0. s=1 (1 + rs )
lim Et Qj
j→∞
(9)
The model can be solved after specifying the functional form of preferences and technologies. However, we need first to choose an appropriate alternative to close it.
2.2 Allowing for an endogenous discount factor The most commonly used approach, introduced first by Obstfeld (1981), endogenizes the discount factor. Suppose that instead of being equal to β t ,
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the discount rate is given by the following recursive relation: β0 = 1, βt+1 = θ(ct )βt .
(10) (11)
This type of preferences were introduced by Uzawa (1968) and are discussed thoroughly in Obstfeld (1990). Recent papers using this type of preferences include Mendoza (1991, 1995), Uribe (1997) and Cook and Devereux (2000). It is assumed that θ0 (ct ) < 0 i.e, agents become more impatient the more they consume. The reason why this modification makes the steady-state independent of initial conditions becomes clear from inspection of the Euler equation U 0 (ct ) = θ(ct )(1 + rt )Et U 0 (ct+1 ). In the steady-state, this equation reduces to θ(c)(1 + r) = 1, which pins down the steady-state level of consumption solely as a function of r and the parameters defining the function θ(.).
2.3 Allowing for a debt-elastic interest rate A second stationary inducing technique assumes a debt-elastic interest rate premium. It has been used in recent papers by Mendoza and Uribe (2000) and Schmitt-Groh´e and Uribe (2001). Suppose that the interest rate faced by domestic agents, rt , is increasing in the country level of foreign debt, dt . Specifically, rt is given by rt = r + p(dt )
(12)
where r denotes the world interest rate and p(.) is a country-specific interest rate premium. To see why this device induces stationarity, rewrite the Euler 6
equation as U 0 (ct ) = β[1 + r + p(dt )Et ]U 0 (ct+1 ).
(13)
In the steady-state, equation (13) implies that β[1 + r + p(d)] = 1. This expression determines the steady-state net foreign asset position as a function of r and the parameters that define the premium function p(.) only.
2.4 Allowing for convex portfolio adjustment costs Suppose now that agents face convex costs of holding assets in quantities ¯ Equation (4) describing the law of different from some long-run level, d. motion of financial wealth can then be rewritten as dt+1 = (1 + rt )dt − tbt +
ψ ¯2 (dt − d) 2
(14)
where dt stands for foreign debt and ψ and d¯ are constant parameters defining the portfolio adjustment cost function. This equation states that the cost of increasing asset holdings by one unit is greater than one because it includes a marginal cost of adjusting the size of the portfolio. The Euler equation thus becomes: ¯ = β(1 + rt )Et U 0 (ct+1 ) U 0 (ct )[1 − ψ(dt − d)]
(15)
where it can be seen that current consumption is increased by one unit minus ¯ In the steady-state, this expression the marginal adjustment cost ψ(dt − d). ¯ = β(1 + r) which implies a steady-state level of simplifies to 1 − ψ(d − d) foreign debt that depends only on parameters of the model. This way of ensuring stationarity has recently been used by Neumeyer and Perri (2001).
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2.5 Discussion A natural question at this stage is whether these alternatives produce similar economic dynamics. Schmitt-Groh´e and Uribe (2003) show that regardless of how stationarity is induced, the models’ predictions regarding second moments and impulse response functions are virtually identical. They conclude that in solving small-open-economy models, a researcher should chose the variant of the model she finds easiest to approximate numerically.
2.6 Digression: more alternatives Two other methods are available to close small-open-economy models. Although they do not fit in our representative agent, small-open-economy model with incomplete asset markets, they are mentioned for comprehensiveness. The first method naturally arises in the context of complete asset markets. In this setup, the marginal utility of consumption is proportional across countries for all dates and under all contingencies. Therefore, one of the equi0
librium conditions will state that U 0 (ct ) = ζU ∗ (c∗t ), where ζ is a constant parameter determining differences in wealth across countries. Stars denote foreign variables. Because the domestic economy is small, c∗t must be an exogenous variable. Since we are interested in the effect of domestic productivity shocks, c∗t is taken to be constant. This stationarity of c∗t will imply stationarity of ct . The second method was developed by Cardia (1991) and uses an overlapping generation model with perpetually young agents as in Blanchard (1985).
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The finite probability of death drives a wedge between the world interest rate and the subjective discount rate that becomes a function of financial wealth. If the individual’s discount rate modified by the probability of death is higher than
1 , 1+r
the intertemporal consumption profile shifts towards the present.
However, when individuals die, their estate is transferred to insurance companies which cover their outstanding debts. This guarantees that the economy in aggregate cannot indefinitely accumulate assets or debts vis − a ` − vis the rest of the world. Therefore, there is a well-defined steady-state even if the modified discount rate does not equal the world interest rate.
3. Application: Mendoza (1991) Let us now close our baseline model using the endogenous discount factor approach. Assume that the utility function has the following form: U (ct , ht ) =
[ct −
hω t 1−γ ] ω
−1
1−γ
(16)
where ω > 1 and γ > 1. Suppose also that βt = β(ct , ht ) = [1 + ct −
hωt −ψ ] . ω
(17)
The production function is given by F (kt , ht ) = ktα ht1−α
(18)
where α ∈ (0, 1) is the share in national income of capital expenditure. Finally, the cost of adjustment function has the form: Φ(kt+1 − kt ) = 9
φ (kt+1 − kt )2 2
(19)
where φ > 0. These specifications along with the calibrated parameters in Table 1 follow Mendoza (1991). The calibrated parameters were computed for the Canadian economy. Table 1: Calibrated parameters values γ
ω
ψ
α
φ
r
δ
ρ
σ
2
1.455
.11
.32
.028
.04
.1
.42
.0129
3.1 Approximate solution Contrary to Mendoza (1991) who solves the model by iteration, we approximate the solution by log-linearizing the equilibrium conditions around the steady-state. The appendix shows how we proceed.
3.2 Predictions of the model Table 2 displays some unconditional second moments of interest implied by the model. Because the parameters ρ, σ and φ were picked to match volatility of output and investment, and the serial correlation of output, it comes as no surprise that the model does very well along these dimensions. The model also correctly implies a volatility ranking featuring investment above output and output above consumption. Also, as in the data, the trade balance is negatively correlated with output. The model overestimates the correlation of hours and consumption with output. In particular, the implied correlation between hours and output is perfect. This prediction is due to the assumed functional form for the period utility. To see this, note
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that the optimality condition equating the marginal product of labor to the marginal rate of substitution between consumption and leisure can be written as hω−1 = At Fh (kt , ht ) ⇔ hωt = (1 − α)yt . This last equality implies that ht t and yt are perfectly correlated. Table 2: Business cycle properties: Data and Model Variable
Canadian Data
Model
σ xt
ρxt ,xt−1
ρxt ,GDPt
σ xt
ρxt ,xt−1
ρxt ,GDPt
y
2.8
0.61
1
3.1
0.61
1
c
2.5
0.7
0.59
2.3
0.7
0.94
i
9.8
0.31
0.64
9.1
0.07
0.66
h
2
0.54
0.8
2.1
0.61
1
tb y
1.9
0.66
-0.13
1.5
0.33
-0.012
1.5
0.3
0.026
ca y
Figure 1 displays impulse response functions to a technology shock of size 1 in period 0. The model predicts an expansion in output, consumption, investment, and hours worked. The increase in consumption is larger than the increase in output, which results in a deterioration of the trade balance. This latter implication is in line with its countercyclical behavior observed in the data. The current account features a similar behavior.
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Figure 1: Impulse Responses to a Unit Technology Shock Consumption
Output
2
1.5
1.5
1
1 0.5
0.5 0
0
2
4 6 Investment
8
0
10
1.5
5
1
0
0.5
0
2
4 6 8 Trade Balance / GDP
0
10
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
0
2
4
6
2
4
6
8
10
4 6 8 Current Account / GDP
10
Hours
10
−5
0
8
−1
10
0
2
0
2
4
6
8
10
References [1] Blanchard, O., (1985) “Debt, deficits, and finite horizons.” Journal of Political Economy 93, 223-247. [2] Cardia, E., (1991) “The dynamics of a small open economy in response to monetary, fiscal, and productivity shocks.” Journal of Monetary Economics 28, 411-434. [3] Cook, D., Devereux, M. (2000) “The macroeconomic effects on international financial panics.” mimeo, University of British Columbia.
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[4] Kydland, F.E., Prescott, E.C. (1982) “Time to build and aggregate fluctuations.” Econometrica 50, 1345-1370. [5] King, R.G., Plosser, C.I, Rebelo, S.T, (1988) “Production, growth and business cycles: 1. The basic neoclassical model.” Journal of Monetary Economics 21, 195-232. [6] Mendoza, E., (1991) “Real business cycles in a small open economy.” American Economic Review 81, 797-889. [7] Mendoza, E., (1995) “The terms of trade, the real exchange rate, and economic fluctuations.” International Economic Review 36, 101-137. [8] Mendoza, E., Uribe, M., (2000) “Devaluation risk and the businesscycle implications of exchange-rate management”. Carnegie-Rochester Conference Series on Public Policy 53, 239-296. [9] Neumeyer, P.A., Perri, F., (2001) “Business cycles in emerging economies: the role of interest rates.” Working Papers 01-12, New York University, Department of Economics. [10] Obstfeld, M., (1981) “Macroeconomic policy, exchange rate dynamics and optimal asset accumulation.” Journal of Political Economy 89, 1142-1161. [11] Obstfeld, M., (1990) “Intertemporal dependence, impatience, and dynamics.” Journal of Monetary Economics 26, 45-75. [12] Schmitt-Groh´e, S., Uribe, M., (2001) “Stabilization policy and the costs of dollarization.” Journal of Money, Credit and Banking 33, 482-509. [13] Schmitt-Groh´e, S., Uribe, M., (2003) “Closing small open economy models.” Journal International Economics 61, 163-185. [14] Uribe, M., (1997) “Exchange rate based inflation stabilization: the initial real effect of credible plans.” Journal of Monetary Economics 39, 197-221. 13
[15] Uzawa, H., (1968) “Time preference, the consumption function and optimum asset holdings.” In: Wolfe, J.N. (Ed.), Value, Capital and Growth: Papers in Honor of Sir John Hicks. The University of Edinburgh Press, Edinburgh, pp.485-504.
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Appendix: Solution Method We follow King, Plosser and Rebelo (1988) by solving the model using a first-order Taylor expansion around the non-stochastic steady-state The Lagrangian of the model of section 2.2 is given by E0
∞ X
βt {U (ct , ht ) + λt [(1 + r)bt + At F (kt , ht ) + (1 − δ)kt − ct
t=0
−kt+1 − Φ(kt+1 − kt ) − bt+1 ] + ηt [
βt+1 − θ(ct , ht )]} βt
The system of first-order conditions are: λt = θ(ct , ht )(1 + rt )Et λt+1 λt = Uc (ct , ht ) − ηt θc (ct , ht ) ηt = −Et U (ct+1 , ht+1 ) + Et ηt+1 θ(ct+1 , ht+1 ) −Uh (ct , ht ) + ηt θh (ct , ht ) = λt At Fh (kt , ht ) λt [1+Φ0 (kt+1 −kt )] = θ(ct , ht )Et λt+1 [At+1 Fk (kt+1 , ht+1 )+1−δ+Φ0 (kt+2 −kt+1 )] Let xˆt ≡ log( xxt ) denote the log-deviation of xt from its steady-state value, x. Taking a first-order log-linearization of the model we get: ˆ t = θc cˆt + θh h ˆ t + Et λ ˆ t+1 λ
ˆt = λ
(1 − θ)c θc ˆ t] − ˆ t] [cc cˆt + ch h [ˆ ηt + θcc cˆt + θch h (1 − θ)c − θθc (1 − θ)c − θθc
ˆ t+1 ] + θ[Et ηˆt+1 + θc cˆt + θh h ˆ t] ηˆt = (1 − θ)[c Et cˆt+1 + h Et h
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(1 − θ)h θh ˆ t] + ˆ t] [hc cˆt + hh h [ˆ ηt + θhc cˆt + θhh h (1 − θ)h + θθh (1 − θ)h − θθh ˆ t + Aˆt + αkˆt − αh ˆt =λ
ˆ t + φk kˆt+1 − φk kˆt = θc cˆt + θh h ˆ t + Et λ ˆ t+1 + θ(θ−1 + δ − 1)[Et Aˆt+1 λ ˆ t+1 − (1 − α)kˆt+1 + θφkEt kˆt+2 − θφk kˆt+1 +(1 − α)Et h stb stb ˆ ˆ t + sc cˆt + si kˆt+1 − (1 − δ)si kˆt (1 + r)ˆbt − Aˆt − αkˆt − (1 − α)h bt+1 = r r δ δ
Aˆt = ρAˆt−1 + t where θc ≡ ch ≡
hUch , Uc
cθc , θ
stb ≡
θh ≡ tb , y
hθh , θ
θcc ≡
cθcc , θc
θch ≡
cθch , θc
c ≡
cUc , U
cc ≡
cUcc , Uc
sc ≡ yc , si ≡ yi .
This procedure allowed us to rewrite the non-linear original system of the form Et f (xt+1 , xt ) = 0
(20)
where all the variables are elements of the vector xt , to a linear system of the form AEt xt+1 = Bxt
(21)
where A and B are 7x7 matrices whose elements are functions of all the structural parameters. The 7 equations that form the above linearized equilibrium model ˆ t, λ ˆ t , and contain 3 state variables, kˆt , ˆbt , and Aˆt and 4 control variables cˆt , h 16
ηˆt . Finally, the system has three initial conditions kˆ0 , ˆb0 , Aˆ0 and we naturally impose the boundary condition lim |Et xt+j | = 0.
j→∞
17
(22)
