open economy macroeconomics literature. One goal of this research is to build a
model to replace or to update the workhorse small open economy model of ...
The New Open Economy Macroeconomics IS Curve: a Requiem and a Redux
Fernando de Holanda Barbosa*
Abstract
This paper shows that the infinitely-lived representative-agent model is not the proper theoretical framework to derive the IS curve for a small open economy with perfect capital mobility. In this framework, the natural rate of interest and the real international rate of interest would be equal only by chance. The paper also shows that this natural rate of interest inconsistency can be solved by using the infinitely-lived overlapping generations model.
JEL classification: E12; F41
Keywords: IS curve; Natural rate of interest; Intertemporal Optimizing Model
1. Introduction
Obstfeld and Rogoff (1995) initiated a research program that has become known as “The New Open Economy Macroeconomics” (NOEM). This research is based on dynamic intertemporal models featuring rational expectations, imperfect competition and nominal-price rigidity. Lane (2001) provides a comprehensive survey of the new open economy macroeconomics literature. One goal of this research is to build a model to replace or to update the workhorse small open economy model of Mundell-FlemingDornbusch. This model has resisted the micro-foundations revolution, in spite of the well-known Lucas critique. Recent work [Clarida, Galí and Gertler (2001), Galí and Monacelli (2005), McCallum and Nelson (2000), Svensson (2000)] present several models for a small open economy that does not influence foreign output, foreign price
*
Professor of Economics, Graduate School of Economics, Getulio Vargas Foundation.
1
level and foreign interest rate. These models have an IS curve and a Phillips curve. The new keynesian IS curve has the following specification: xt = xt +1 − α (ρ t − ρ ) where x is the output gap, ρ t the real rate of interest and ρ the natural rate of interest. The new open economy macroeconomics generalizes this closed economy specification by two routes: i) introducing mechanisms that change the coefficient α of the real rate of interest gap, and ii) adding new variables, such as the real exchange rate and world output. The NOEM models use the infinitely-lived intertemporal optimizing representative-agent model. It is common knowledge [see Barro and Sala-i-Martin (1995), Chapter 3] that this model, for a small open economy with perfect access to the world capital market and a fixed rate of time preference, yields awkward consequences that are not tenable either from a theoretical or from an empirical point of view. In a closed economy, the real rate of interest is equal to the rate of time preference in steady state. If they are different, the stock of capital adjusts to make up the difference. In an open economy, the real rate of interest is equal to the international rate of interest in steady state. In this case there is no mechanism to make the adjustment between the international real rate and the rate of time preference.1 The difference between the two rates, whether positive or negative, would imply consequences that are counterfactuals, unless an ad hoc endogenous risk premium is introduced in the model.2 The NOEM models that have derived an IS curve did not pay attention to the fact that in a small open economy, with perfect capital mobility, the natural rate of interest should be equal to the international real rate of interest. The goal of this paper is to show this theoretical inconsistency that invalidates the IS curve obtained from the infinitely-lived representative-agent model. The paper also shows that this inconsistency
1
To ensure the existence of stationary equilibrium either the consumer rate of time preference or the international rate of interest has to be made endogenous. The international rate of interest is endogenous by assuming that capital mobility is not perfect. Thus, the domestic interest rate is equal to the international rate of interest plus a risk premium, which is modeled with some ad hoc hypothesis. This is the solution method adopted by Kollmann (202). 2 Obstfeld and Rogoff [(1995); (1996), Chapter 10] set up a two-country model, where home and foreign individuals have identical preferences and the two countries have the same technology. This set up avoids the problem of natural rate inconsistency because the world is a closed economy. Thus, in such a case the real rate of interest is equal to the rate of time preference in steady state.
2
can be solved by using the infinitely-lived overlapping generations model [Weil (1989)]. This paper is organized as follows. Section 2 presents a framework that avoids the cumbersome algebra usually found in NOEM models, which allows a very simple and straightforward derivation of the IS curve, without loss of generality. We use deterministic variables because a stochastic environment is not necessary to show the natural rate inconsistency embedded in the NOEM IS curve. Section 3 derives the IS curve from the infinitely-lived overlapping generations model, in which the natural rate of interest is equal to the international real rate of interest. In this section we use continuous time since it simplifies the exposition. Section 4 concludes.
2. The NOEM IS Curve: Micro-foundations
McCallum and Nelson (2000) model assumes that all imports and labor are used as inputs in the production of domestic goods. Thus, output in this small open economy can be written as:
y t = ω1 ct + ω 2 g t + ω 3 ex
where c, y, g, ex are logarithms of consumption, real output, government consumption and exports, while ω i is the steady-state ratio of the corresponding variable. Export demand is given as3 ext = y t* + η qt + κ
where y* is world output, η is the elasticity of substitution between imported materials and labor in production, κ is a constant and q is the real exchange rate defined by
3
The export demand equation, Ex
=γ (
P −η * ) Y =γ S P*
Qη Y * , is obtained solving a cost
minimization problem where the constraint is a CES production function. Taking logs we get the equation in the text.
3
q t = st + pt* − pt
Lower case letters denote the logs of the respective variables, S is the foreign exchange, P is the home-country money price of goods, P* is the foreign price index. The representative-agent can save to consume later or use her resources to consume now. In equilibrium we obtain the Euler equation:
u ' (Ct ) u ' (C t +1 ) = β (1 + rt ) Pt Pt +1 where 0 < β < 1 is the discount factor, r is the domestic nominal rate of interest, u is the instantaneous utility function and u ′ is the marginal utility of consumption. The representative-agent can save her resources abroad buying a foreign security that yields a nominal rate of interest r*. In order to buy such a security she buys foreign money and pays the exchange rate S. Next period she brings back her resources thru the foreign exchange market and buys consumption goods. In equilibrium
(
)
u ' (C t ) 1 + rt* u ' (Ct +1 ) =β S t +1 Pt St Pt +1
By combining the two Euler equations we obtain the uncovered interest parity condition:
(
1 + rt = 1 + rt*
) SS
t +1 t
The uncovered interest parity (UIP) for the nominal interest rate can be transformed into the uncovered interest parity for the real interest rate by simple algebra:
(
)
1 + rt 1 + rt* = Pt +1 / Pt Pt*+1 / Pt*
4
S t +1 Pt*+1 Pt +1 S t Pt* Pt
The Greek letter ρ denotes the domestic real interest rate, ρ * the foreign real rate of interest and Q the real exchange rate. Thus, the UIP condition for the real rate of interest can be written as
(
1 + ρ t = 1 + ρ t*
) QQ
t +1 t
Taking logs of this equation and using the approximation log(1 + x) ≅ 1 + x , the UIP condition becomes:
ρ t = ρ t* + qt +1 − q t We use the constant elasticity of substitution ( σ ) utility function
u (C ) =
C
1−
1
σ
1−
−1 1
σ
The Euler equation is given by:
−
Ct
1
σ
1 + ρ t −σ = C t +1 1+ δ
1
where 1 + δ = 1 / β , and δ is the rate of time preference. Taking logs of this expression and the approximation log(1 + x) ≅ 1 + x we get: log C t = −σ (ρ t − δ ) + log C t +1
We use the lower case letter notation to write this Euler equation as ct = −σ (ρ t − δ ) + ct +1
Inserting this equation into the output equation we obtain:
5
y t = −ω1 σ (ρ t − δ ) + ω1 ct +1 + ω 2 g t + ω 3 ext
The output equation can be written for the time period t+1 as
ω1 ct +1 = y t +1 − ω 2 g t +1 − ω 3 ext +1
The two last output equations imply: y t = y t +1 − ω1 σ (ρ t − δ ) + ω 2 ( g t − g t +1 ) + ω 3 ( ext − ext +1 )
Using the export demand equation we obtain: y t = y t +1 − ω1 σ (ρ t − δ ) + ω 2 ( g t − g t +1 ) + ω 3 ( y t* − y t*+1 ) + ω 3 η ( q t − qt +1 ) Output gap is the difference between real output and potential output: x = y − y and
x * = y * − y * . Thus, the NOEM IS curve is given by:
xt = xt +1 − ω1 σ (ρ t − ρ ) + ω 2 ( g t − g t +1 ) + ω 3 ( xt* − xt*+1 ) + ω 3 η ( qt − q t +1 )
where the equilibrium real rate of interest, the natural rate, is:
ρ =δ +
1
ω1 σ
∆y−
ω3 ∆ y* ω1 σ
The natural rate of interest depends upon the rate of time preference, the rate of growth of domestic potential output, the rate of growth of foreign potential output, the intertemporal elasticity of substitution, and the steady-state ratios of consumption and exports. In steady state, the real exchange rate is constant. Therefore, the international real rate of interest is the small open economy’s natural rate of interest:
ρ = ρ*
6
Only by chance the two natural rates would be equal. Thus, the infinitely-lived representative-agent model for a small open economy with free access to the world capital market yields an inconsistency: two natural rates of interest that would be equal just by chance.
3. Infinitely-lived Overlapping Generations Model: IS Curve
An agent born at time s lives forever and maximizes at time t her utility function,
∫
∞ t
e −δ (v −t )u [C (s, v )] dv
subject to the flow budget constraint: A& (s, v ) = W (v ) − τ (v ) − C (s, v ) + ρ A (s, v ) where δ is the pure rate of time preference, u[C(s, v)] is the instantaneous utility function, C(s, v) stands for the consumption at time v by an agent born at time s, A(s, v) are real financial assets, W (v) is her wage rate, τ (v ) is a lump sum tax levied by the government, and ρ is the real rate of interest. Both, the wage and the tax rate are not cohort specific. Newly born agents are not linked altruistically to existing cohorts and they are born with only human wealth but no financial assets. Thus: A (v, v) =0. The current value Hamiltonian is
Η = u [C (s, v )] + λ [W (v ) − τ (v ) + C (s, v ) + ρ A (s, v )] where λ is a co-state variable. The first order conditions are: ∂Η = u ′ [C (s, v )] − λ = 0 ∂C
7
λ& = δλ −
∂Η = δλ − ρλ ∂A
∂Η & = A (s, v ) = W (v ) − τ (v ) − C (s, v ) + ρ A (s, v ) ∂λ
and the transversality condition is given by: lim v→∞ λ A e − ρv = 0
By combining the first and second equation we obtain the consumption Euler equation:
λ& u ′′ (C )C& = =δ −ρ u ′ (C ) λ
We assume a constant elasticity of substitution instantaneous utility function. Therefore, the consumption rate of growth is given by: C& (s, v ) = σ (ρ − δ ) C (s , v )
The transversality condition allows us to write the flow budget constraint as the following stock budget constraint: ∞
A (s, t ) + H (t ) = ∫ e − ρ (v −t )C (s, v ) dv t
The present value of the agent’s consumption is equal to the sum of financial and human wealth, where human wealth is the present value of lifetime after-tax wage income: ∞
H (t ) = ∫ e − ρ (v −t ) [W (v ) − t (v )] dv t
8
We use the rate of grow of consumption equation and the stock budget constraint to get: ∞
A (s, t ) + H (t ) = ∫ e − ρ (v −t )C (s, t ) e σ ( ρ − δ ) ( ν −t ) dv t
Thus, consumption is proportional to total wealth: C (s, t ) = θ [A (s, t ) + H (t )] where θ = ρ + σ (δ − ρ ) . We use the normalization that population at date t is equal to one: P (t) =1. The number of agents in the economy grows at rate η . At time s < t total population was: P ( s) = e − η ( t − s ) . Therefore the size of the cohort born at time s was η P (s) . Aggregating over all agents at time t for any variable x(s, t) the aggregate is defined as
X (t ) = ∫ η P (s ) x (s, t ) ds = ∫ η e t
t
−∞
−∞
n ( s −t )
x (s, t ) ds
By aggregating all agents in the economy we obtain the aggregate consumption at time t:
C (t ) = ∫ η e n ( s −t )C (s, t ) ds t
−∞
It follows from this equation that:
dC (t ) = η [C (t , t ) − C (t )] + σ (ρ − δ )C (t ) dt Since C (t ) = θ [A (t ) + H (t )] and C (t , t ) = θ [ A (t , t ) + H (t )] = θ H (t ) because the agent is born without financial assets, we have: C (t , t ) − C (t ) = −θ A (t ) . Thus: dC (t ) = σ (ρ − δ )C (t ) − η θ A (t ) dt
9
By the same token, aggregate financial asset is defined by:
A (t ) = ∫ η eη ( s −t ) A (s, t ) ds t
−∞
We differentiate this expression with respect to time to obtain: t t dA (t ) = η A (t , t ) − η ∫ η eη ( s −t ) A (s, t ) ds + ∫ η eη ( s −t ) A& (s, t ) ds − ∞ − ∞ dt
By taking into account the hypothesis that A (t, t) =0 and the flow budget constraint we have the following differential equation:
dA (t ) = (ρ − η ) A (t ) + W (t ) − τ (t ) − C (t ) dt
The system of differential equations for aggregate consumption and aggregate financial wealth is given by:4
C& = σ (ρ − δ ) C − η θ A & A = (ρ − η ) A + W − τ − C
This small open economy with perfect access to the world capital market can have a steady state for aggregate consumption and aggregate financial wealth even when the rate of time preference is different from the real interest rate, as shown in the phase diagrams of Figures 1 and 2. Figure 1 supposes that the real interest rate is greater than the rate of time preference (a creditor country) and Figure 2 is drawn under the hypothesis that the rate of time preference is greater than the real interest rate (a debtor country). The Jacobian matrix of this system is:
4 We assume that this system of first-order differential equations is autonomous, i.e., the variable time (t) does not appear explicitly in the system of equations. The wage rate and the lump sum tax do not change with time. The real rate of interest is equal to the international real rate of interest.
10
∂C& ∂C J = ∂A& ∂A
∂C& ∂A σ (ρ − δ ) − η θ = ρ −η ∂A& − 1 ∂A
The determinant of this matrix is negative if the following restriction is satisfied: J < 0 ⇒ ρ < δ +
n
σ
. Given this condition, the steady state is a saddle-point,
and the saddle paths SS are upward sloping as shown in Figures 1 and 2.
C& = 0
C
S
A& = 0
E
C S
A
A Figure 1: ρ − δ > 0
We define the financial asset consumption ratio as a=A/C. We denote the equilibrium ratio as a . It follows from the first equation of the system of differential equations that we can write:
c& =
C& = σ (ρ − δ ) − η θ a C
By adding and subtracting ηθ a to the right hand side of this equation we obtain:
11
c& = σ (ρ − ρ ) − η θ (a − a ) where the natural rate of interest rate ρ is equal to the international rate of interest:
ρ =δ +
ηθ a = ρ* σ
In this set up the adjustment between the rate of time preference and the international real rate of interest is thru the asset consumption ratio, which can be positive or negative depending on the degree of impatience of the small open economy compared to the world real rate of interest. C
C& = 0
S
A& = 0
C
S
A
A
Figure 2: ρ − δ < 0
The IS curve for a small open economy can be obtained from an infinitely-lived overlapping generations model by combining the aggregate consumption Euler equation with McCallum and Nelson (2000) approach. We differentiate y = ω1 c + ω 2 g + ω 3 ex with respect to time and we use the Euler equation and the export demand equation to write the IS curve as
12
y& = ω1 σ (ρ − ρ ) − ω1 η θ (a − a ) + ω 2 g& + ω 3 y& * + ω 3 ξ q&
The IS curve depends upon the real interest rate, the asset consumption ratio, and the rates of growth of government expenditures, world output and real exchange rate. In this specification the asset consumption ratio gap, besides the interest rate gap, affects the rate of growth of domestic output in the short run.
4. Conclusion
The value added of this paper is twofold. Firstly, it shows that the infinitelylived representative- agent model is not the proper theoretical framework to derive the IS curve for a small open economy with perfect access to the world capital market. Secondly, the paper shows that the IS curve for a small open economy can be derived from an infinitely-lived overlapping generations model. In the infinitely-lived representative-agent framework, the natural rate of interest and the international real rate of interest would be equal only by chance. If the two rates were different there would be opportunities for arbitrage and capital flows that lead to paradoxical conclusions, either the country owns the world’s wealth or it mortgages all of its capital and all of its wage income. These results are incompatible with a steady state. Thus, a requiem for the NOEM IS curve is warranted. In the infinitely-lived overlapping generations model the natural rate of interest is equal to the real international rate of interest, given that certain conditions are satisfied. Thus, in this overlapping generations framework the IS curve has no natural rate of interest inconsistency.
References
Barro, Robert J. and Xavier Sala-i-Martin (1995). Economic Growth, New York, McGraw-Hill.
13
Clarida, Richard H, Jordi Galí and Mark Gertler (2001). Optimal Monetary Policy in Closed Versus Open Economies: An Integrated Approach, American Economic Review, 91, 248-252.
Galí, Jordi and Tommaso Monacelli (2005). Monetary Policy and Exchange Rate Volatility in a Small Open Economy, Review of Economic Studies, 72, 707-734.
Kollmann, Robert (2002). Monetary Policy Rules in the Open Economy: Effects on Welfare and Business Cycles, Journal of Monetary Economics, 49, 989-1015.
Lane, Phillip (2001). The New Open Economy Macroeconomics: A Survey, Journal of International Economics, 54, 235-266.
McCallum, Bennett T. and Edward Nelson (2000). Monetary Policy for an Open Economy: An Alternative Framework with Optimizing Agents and Sticky Prices, Oxford Review of Economic Policy, 16, 74-91.
Obstfeld, Maurice and Kenneth Rogoff (1995). Exchange Rate Dynamics Redux, Journal of Political Economy, 103, 624-660.
Obstfeld, Maurice and Kenneth Rogoff (1996). Foundations of International Macroeconomics, Cambridge, MA, MIT Press.
Svensson, Lars E. O. (2000). Open-Economy Inflation Target, Journal of International Economics, 50, 155-183.
Weil, Philippe (1989). Overlapping Families of Infinitely-Lived Agents, Journal of Public Economics, 38, 183-198.
14
