Endogenous Growth and External Balance in a Small Open Economy

Abstract

This paper puts forward an intertemporal model of a small open economy which allows for the simultaneous analysis of the determination of endogenous growth and external balance. The model assumes infinitely lived, overlapping generations that maximize lifetime utility, and competitive firms that maximize their net present value in the presence of adjustment costs for investment. Domestic securities are assumed perfect substitutes for foreign securities and the economy is assumed small in the sense of being a price taker in international goods and assets markets. It is shown that the endogenous growth rate is determined solely as a function of the determinants of domestic investment, such as the world real interest rate, the technology of domestic production and adjustment costs for investment and is independent of the preferences of domestic households and budgetary policies. The preferences of consumers and budgetary policies determine the savings rate. The current account and external balance are functions of the difference between the savings and the investment rates. The world real interest rate affects growth negatively but has a positive impact on external balance. The productivity of domestic capital affects growth positively but causes a deterioration in external balance. Population growth, government consumption and government debt affect the current account and external balance negatively, but do not affect the endogenous growth rate.

Appendix 1

1.1 The Derivation of the Aggregate Consumption Function

In this appendix we derive the aggregate consumption function (23) of the model, from the intertemporal optimization problem of households.

The household born at instant j chooses a consumption path in order to maximize,

$$ {U}_j={\displaystyle {\int}_{s=j}^{\infty }{e}^{-\rho s}} \ln {c}_{js} ds $$

(A.1)

subject to the instantaneous budget constraint,

$$ {\overset{\bullet }{a}}_{js}={r}_s{a}_{js}+{w}_{js}-{c}_{js} $$

(A.2)

and the household’s solvency (no-Ponzi game) condition,

$$ \underset{t\to \infty }{ \lim }{e}^{-{\displaystyle \underset{s=j}{\overset{t}{\int }}{r}_s ds}}{a}_{jt}=0 $$

(A.3)

where ρ is the pure rate of time preference, c _js is the consumption of household j at instant s, a _js is the non-human wealth of household j at instant s, w _js is non asset (labor) income of household j at instant s and r _s is the real interest rate. Instantaneous utility is assumed logarithmic implying that the elasticity of intertemporal substitution is equal to unity.

Maximization of (A.1) subject to (A.2) and (A.3) yields an Euler equation for consumption which takes the form,

$$ {\overset{\bullet }{c}}_{js}=\left({r}_s-\rho \right){c}_{js} $$

(A.4)

From (A.4) and the household’s present value budget constraint it follows that,

$$ {c}_{js}=\rho \left({a}_{js}+{\displaystyle \underset{\nu =s}{\overset{\infty }{\int }}{w}_{jv}{e}^{-{\displaystyle {\int}_{\psi =s}^{\nu }{r}_{\psi } d\psi}} d\nu}\right) $$

(A.5)

Household consumption is linear in total wealth because of the assumption of the unitary elasticity of intertemporal substitution. Furthermore, as we have assumed that the elasticity of intertemporal substitution is equal to 1, the propensity to consume out of wealth is independent of the real interest rate and only depends on the pure rate of time preference.

The size of the cohort born at time j equals nL _j , where L _j  = exp(nj) is the population size at time j. Population aggregates are defined as,

$$ {C}_t=n{\displaystyle \underset{j=-\infty }{\overset{t}{\int }}{c}_{jt}{e}^{nj} dj} $$

(A.6)

Aggregating over cohorts, assuming that newly born households do not inherit any wealth, and replacing the domestic real interest rate by the world real interest rate, yields,

$$ {\overset{\bullet }{C}}_t=\left(r\ast -\rho +n\right){C}_t- n\rho {A}_t $$

(A.7)

where C is aggregate private consumption and A the aggregate non-human wealth of domestic households.

Assuming that aggregate non human wealth consists of shares in domestic firms, government bonds and net foreign assets, and dividing through by domestic output, private consumption as a share of domestic output evolves according to,

$$ {\overset{\bullet }{c}}_t=\left({r}^{\ast }-\rho +n-g\right){c}_t- n\rho \left(q{k}_t+{b}_t+{f}_t\right) $$

(A.8)

which is Eq. (23) in the main text.