I. Theory

In the recent research on consumption (reviewed in Deaton (1986) and King (1985)), a number of important works (Hall (1978). Grossman and Shiller (1981). and Hansen and Singleton (1982)) have opened up the possibility of a direct estimation of the parameters of the intertemporal utility function that characterize the behavior of an (infinitely lived) representative consuming unit without requiring explicit solutions to the dynamic optimization problem.⁴

It can thus be posited, if this line of reasoning is followed, that aggregate consumption can be modeled as the outcome of optimizing decisions of a representative consumption unit (household). The household faces an economic environment in which future opportunities are uncertain, and its stationary utility function is given by

In equation (1), V_t is expected utility at t, E_t is the expectations operator conditional on information available at t, ρ is a constant discount factor, and F_t is actual family size. The parameter γ in equation (I) controls intertemporal substitution: large and negative values of γ characterize consumers who are willing to smooth consumption over time and who respond only to substantial changes in incentives. Finally, the function U_τ(.) takes, alternatively, one of the two following forms:

or

In equation (2), U_τ denotes household consumption in terms of equivalent adults, since H_τ is “effective” household size defined as

where d_τ is the proportion of household members in the young age bracket (for example, 0-14 years) or, in other words, the household’s dependency rate.⁵ In addition, γ is a proportionality factor converting children to equivalent adults for consumption purposes.⁶

Equation (2) has been used by a number of authors to describe the effect of changing family size on private consumption (Modigliani and Ando (1957), among others, and. more recently. Mariger (1986)). It implies that the within-period utility function is concave and increasing in private consumption per equivalent household member and it corresponds to intertemporal utility defined in terms of scaled quantities (C_τ/E_τ). Notice that the γ parameter (also called child costs or general equivalence scale) is independent of time.

In equation (3). demographic variables enter household preferences through the committed quantity or overhead cost ${\overline{c}}_{\tau },$ which is assumed to be linearly dependent on the number of young household members (B_τ):

Clearly, equation (5) implies an intertemporal utility function expressed in terms of uncommitted quantities $(C_{\tau }-{\overline{c}}_{\tau },\forall \tau )\mathrm{.}$

The household expects, for simplicity, the real rate of return to remain constant over time, and maximizes equations (1), (2), and (4) (or (1), (3), and (5)), subject to the usual intertemporal budget constraints, equating the discounted present value of consumption to the discounted present value of assets and noninterest income or, in period-to-period form:

where W_τ defines real assets at the end of period τ R is the constant real rate of return, and Y_τ is real net nonproperty income in period τ. It is assumed that some asset exists that is either held in positive amounts or for which borrowing is possible. As long as the optimum path lies in the interior of the budget set, simple perturbation arguments can be used to establish certain characteristics of the optimal path. At any point along an optimal path, the representative consumption unit cannot make itself better off by forgoing one unit of consumption at time t and using the proceeds to purchase any other good at any other point in time. Formally, at time t the marginal condition will be given by

which, apart from implicitly defining G_t+1, is satisfied for any free-traded risky asset and holds for consumers who expect with certainty to be there in the next period.

Under rational expectations and market clearing, the first-order condition (7) holds ex post except for an error term uncorrected with information available to the consumption unit at time t. In other words:

where ɛ_t+1 is the mean zero and constant variance (σ²) forecast error.

In the present case of time-separable, constant relative risk-averse preferences, and with lower-case letters other than “d” denoting natural logarithms and with Δ as the difference operator, both specifications of the within-period utility function imply⁷

in terms of consumption per (actual) household member.⁸ and where α = α(ρ,σ²,γ,r), and Φ = (λ—1) if equations (2) and (4) hold; whereas Φ = γω/(l - γ) if equations (3) and (5) hold, with ω = (β_tF_t)/C_t.

Therefore, in the equivalent adult model, -1 < ϕ < 0 and, in the light of the available evidence (Deaton and Muellbauer (1986)), Φ could well be around -.6. In the case of the overhead cost model, for ω = 1,—Φ would be equal to the coefficient of the expected real interest rate minus 1 if the latter were not assumed to be constant. In other words, for a given value of w, 6 provides direct information on consumers’ willingness to smooth their consumption path and, therefore, other things being equal, on the real interest rate elasticity of consumption. Since ω=[(β_τ)F_τ/C_τ], it can be regarded as the share of committed consumption (young member of the household) over total consumption per capita. In what follows, w is assumed to be independent of τ. Notice that Φ is greater, equal, or less than zero, depending on whether γ is negative, zero, or positive, respectively. In addition, if the household strongly prefers an even consumption path (that is. γ→-∞). the extent of the relationship between consumption growth and the expected dependency rate is given by the relative amount of overhead costs (that is, Φ→ω), and the rate of growth of “net” consumption is a constant. However, if the household tends to be extremely responsive to changes in incentives (that is, γ→1), consumption is likely to be shifted backward toward the present period, if effective expected family size is expected to increase (that is, Φ→—∞). This will imply, other things being equal, lower savings in the current period t. Therefore, the magnitude of the coefficient Φ turns out to possess useful information on consumer willingness to smooth consumption over time.

In equation (9), the error term u,_t+1 reflects the impact of “news” about current levels of income and household size and composition. It is therefore orthogonal to all past information.⁹

As it stands, equation (9) refers to the case of a consuming unit executing intertemporal optimization through trading in perfectly competitive asset markets. The available empirical evidence (Rossi (1988)) strongly suggests, however, that contrary to the well-known life-cycle theory, a significant fraction of the population in developing countries is not able to shift consumption at will from a later date to an earlier date, for a number of reasons, including capital market imperfections. Although the importance of capital market imperfections continues to be a matter of debate even in countries with apparently sophisticated financial institutions and well-developed capital markets, such imperfections are likely to be exacerbated in developing countries. Following Muellbauer (1986) and the empirical analysis in Rossi (1988), the model therefore makes allowance for a departure by the representative consuming unit from optimal behavioral rules described by the theory, by simply adding a term [ζE_t(z_t+1-c_t)] to equation (9), where Z_t is real net nonproperty income. Consumers who are liquidity-constrained at t may not expect to be constrained at t +1. and may therefore be forced to let their consumption path follow more closely their income path. The coefficient ζ is expected to be positive if liquidity constraints are present (see Muellbauer (1986) for additional details).

II. Data and Estimates

A thorough empirical analysis of private savings behavior in developing countries raises several difficult statistical problems, mostly stemming from inadequacies in the data and their lack of comparability. A reasonable number of observations on aggregate time-series data are available on a consistent basis for only a few developing countries. In most of the cases, less than 20 annual observations are available. In such a situation, pooling cross-section and time-series data for a number of countries seems to be the most sensible procedure, provided allowance is made for obvious institutional and cultural differences among countries.

Following this line of research, the empirical analysis here is based on six sets of pooled time-series, cross-section data, each one referring to a geographical region that is assumed to be homogeneous. The first set covers 12 countries in sub-Saharan Africa, the second set includes 5 countries in the Middle East and North Africa, and the third one covers 9 countries in South and East Asia and the Pacific. The fourth and fifth sets cover eight and nine countries in Central America (including the Caribbean) and South America, respectively. Finally, the sixth includes six southern European countries. Since the sample as a whoie contains 11 low-income and 38 middle-income countries, low-income countries are somewhat underrepresented.¹⁰

The appendix in Rossi (1988) provides a description of the data set which has been constructed by the assembly of information from all available international, as well as national, sources as needed.

For estimation purposes, the model derived in Section I can be rewritten as follows:

where the suffix i identifies the i th country in each of the geographical areas referred to in the previous section. In other words, the constant term in equation (9), being a function of the variance of the forecast error, is allowed to differ among countries, because, for example, countries with a higher share of the product originating from agriculture are likely to face higher uncertainty. In addition, the original error term in equation (9) (that is, $u_{t+1}^i$) is now linearly decomposed in two random components with mean zero, although not necessarily homoscedastic, because the variances of different country forecast errors are likely to differ. The first component is country-specific and is uncorrected across countries $\left(v_{t+1}^i\right)$, whereas the second one is an area-wide component that equally affects all countries in a particular geographical area $\left({\overline{V}}_{t+1}\right)\mathrm{.}$. ¹¹ The obvious example of the latter component is given by the recent drought in sub-Saharan Africa.

If the expected nature of the variables on the right-hand side of equation (10) is disregarded for the time being, the appropriate estimator for the setting described by equation (10) is given by the between-within groups fixed-effects estimator, if each country is regarded as a group. As Mundlak (1978) shows, using this estimator is equivalent to the application of ordinary least squares to equation (10) expressed in terms of “transformed” variables, where the transformation takes the form

for a generic variable xⁱ_t, if T and N denote the number of time periods and the number of countries, respectively; that is, the transformed variable is the original variable minus the country and time means plus the total mean. Notice that the transformation eliminates the constant term and the area-wide term. In general, the transformation would eliminate all variables not simultaneously indexed on i and _t.

Once the variables are transformed as in equation (11). the coefficients of equation (10) can be estimated, using instrumental variables for Δd_t+1 and z_t+1. All variables known at time t are good candidates as instruments. ^{Table 1} reports the parameter estimates for the six subsamples. their White’s (1980) heteroscedasticity-consistent standard errors, the standard error of the regression (σ). as well as the Durbin-Watson statistics for fixed-effects model given in Bhargava. Franzini, and Narendra-nathan (1982), and the test of over-identifying restrictions (including the validity of instruments) due to Sargan (1964).¹² The set of instruments includes lagged consumption, lagged real net nonproperty income, the dependency rate lagged once and twice, total population lagged once and twice, and a time trend. In addition, if the overhead cost model is assumed to be correct, the table reports the values of γ that would be implied by the estimated Φs if ω were alternatively given the values 0.3 and 0.7. For each geographical region, two sets of estimates are presented. The first row refers to equation (10) above, but with ζ set to zero (no liquidity constraints). The second row allows for liquidity constraints.

Apart from confirming the importance of the proxy for liquidity constraints, which turns out to be irrelevant only in the case of Central America and the Caribbean,¹³ Table 1 provides a number of interesting indications. First of all, in all cases except Central America and the Caribbean.—∞ < Φ < ω, as predicted by the more general case given by equations (3) and (5)—that is, by the overhead cost model. In only one case is the point estimate of ω between 0 and -1, as the equivalent adult model would also predict.¹⁴ It remains true, however, that except for Southern Europe, the estimates of Φ are highly imprecise, and therefore no definite conclusions can be drawn.

Table 1.

Parameter Estimates and Test Statistics

Note: White’s (1980) standard errors are shown iti parentheses; σ denotes the standard error of the regression: DW denotes the Durbin-Watson statistic; χ²(.) denotes the test of overidentifying restrictions: and γ is intertemporal substitution for two values of ω For each region, the first row of estimates refers to equation (10). but with ζ (liquidity constraints) set to zero; the second row allows for liquidity constraints.

^aThe regressions also include a dummy variable that takes a value of 1 in 1973 and -1 in 1984 for Swaziland. It accounts for two large outliers but does not affect the remaining coefficients. Its coefficient takes a value of -0.51 (0.02) in the first row of regressions, and -0.41 (0.08) in the second row of regressions.

^bWith six degrees of freedom.

^cWith five degrees of freedom.

^dThe regressions also include two dummy variables taking a value of 1 in 1974 for both Cyprus and Portugal. They do not affect the remaining coefficients, and their coefficients take the following values: -0.20 (0.01) for the first dummy variable in the first row of regressions, and—0.10 (0.02) for the second row: and 0.09 (0.02) for the second dummy variable in the first row of regressions, and 0.08 (0.01) for the second row.

Table 1.

Parameter Estimates and Test Statistics

	Number of	Parameter Estimates					γ
Region	Observations	Φ	ζ	σ	DW	χ2(.)	(ω =0.3)	(ω =0.7)
Sub-Saharan Africaa	104	-2.41 (3.51)	—	0.070	1.75	20.9b	0.9	0.8
		-1.70 (2.97)	0.37 (0.10)	0.057	1.80	5.5c	0.0	0.7
Middle East and	44	-0.42 (4.66)	—	0.072	1.81	25.3b	0.3	-1.3
North Africa		-0.42 (4.68)	0.28 (0.09)	0.061	1.66	23.8c	0.3	-1.3
South and East Asia	84	-1.27 (2.36)	—	D.I 133	2.11	4.0b	0.8	0.6
and the Pacific		-1.63 (2.43)	0.21 (0.12)	0.026	2.14	3.5c	0.8	0.7
Southern Europed	56	-0.44 (3.56)	—	0.045	1.61	15.6	0.3	-1.3
		-4.92 (2.74)	0.44 (0.12)	0.029	1.59	9.4	0.9	0.12
Central America and	74	9.23 (5.66)	—	0.048	1.14	6.1b	—	—
the Carribbean		9.23 (5.73)	0.01 (0.29)	0.048	1.15	6.1c	—	—
South America	88	-0.89 (6.02)	—	0.053	1.56	40.2b	0.3	-1.3
		-2.42 (4.33)	0.75 (0.18)	0.038	1.18	12.8c	0.9	0.8

Note: White’s (1980) standard errors are shown iti parentheses; σ denotes the standard error of the regression: DW denotes the Durbin-Watson statistic; χ²(.) denotes the test of overidentifying restrictions: and γ is intertemporal substitution for two values of ω For each region, the first row of estimates refers to equation (10). but with ζ (liquidity constraints) set to zero; the second row allows for liquidity constraints.

^aThe regressions also include a dummy variable that takes a value of 1 in 1973 and -1 in 1984 for Swaziland. It accounts for two large outliers but does not affect the remaining coefficients. Its coefficient takes a value of -0.51 (0.02) in the first row of regressions, and -0.41 (0.08) in the second row of regressions.

^bWith six degrees of freedom.

^cWith five degrees of freedom.

^dThe regressions also include two dummy variables taking a value of 1 in 1974 for both Cyprus and Portugal. They do not affect the remaining coefficients, and their coefficients take the following values: -0.20 (0.01) for the first dummy variable in the first row of regressions, and—0.10 (0.02) for the second row: and 0.09 (0.02) for the second dummy variable in the first row of regressions, and 0.08 (0.01) for the second row.

View Table

Table 1.

Parameter Estimates and Test Statistics

	Number of	Parameter Estimates					γ
Region	Observations	Φ	ζ	σ	DW	χ2(.)	(ω =0.3)	(ω =0.7)
Sub-Saharan Africaa	104	-2.41 (3.51)	—	0.070	1.75	20.9b	0.9	0.8
		-1.70 (2.97)	0.37 (0.10)	0.057	1.80	5.5c	0.0	0.7
Middle East and	44	-0.42 (4.66)	—	0.072	1.81	25.3b	0.3	-1.3
North Africa		-0.42 (4.68)	0.28 (0.09)	0.061	1.66	23.8c	0.3	-1.3
South and East Asia	84	-1.27 (2.36)	—	D.I 133	2.11	4.0b	0.8	0.6
and the Pacific		-1.63 (2.43)	0.21 (0.12)	0.026	2.14	3.5c	0.8	0.7
Southern Europed	56	-0.44 (3.56)	—	0.045	1.61	15.6	0.3	-1.3
		-4.92 (2.74)	0.44 (0.12)	0.029	1.59	9.4	0.9	0.12
Central America and	74	9.23 (5.66)	—	0.048	1.14	6.1b	—	—
the Carribbean		9.23 (5.73)	0.01 (0.29)	0.048	1.15	6.1c	—	—
South America	88	-0.89 (6.02)	—	0.053	1.56	40.2b	0.3	-1.3
		-2.42 (4.33)	0.75 (0.18)	0.038	1.18	12.8c	0.9	0.8

Note: White’s (1980) standard errors are shown iti parentheses; σ denotes the standard error of the regression: DW denotes the Durbin-Watson statistic; χ²(.) denotes the test of overidentifying restrictions: and γ is intertemporal substitution for two values of ω For each region, the first row of estimates refers to equation (10). but with ζ (liquidity constraints) set to zero; the second row allows for liquidity constraints.

^aThe regressions also include a dummy variable that takes a value of 1 in 1973 and -1 in 1984 for Swaziland. It accounts for two large outliers but does not affect the remaining coefficients. Its coefficient takes a value of -0.51 (0.02) in the first row of regressions, and -0.41 (0.08) in the second row of regressions.

^bWith six degrees of freedom.

^cWith five degrees of freedom.

^dThe regressions also include two dummy variables taking a value of 1 in 1974 for both Cyprus and Portugal. They do not affect the remaining coefficients, and their coefficients take the following values: -0.20 (0.01) for the first dummy variable in the first row of regressions, and—0.10 (0.02) for the second row: and 0.09 (0.02) for the second dummy variable in the first row of regressions, and 0.08 (0.01) for the second row.

It is interesting, however, that the implied estimates of γ suggest a very high degree of intertemporal substitution—far higher than the one estimated on the basis of the relationship between the rate of growth of consumption and the expected real rate of return on financial assets (see Rossi (1988)).

III. Concluding Remarks

This paper has presented two simple demographic extensions of the representative household’s stochastic dynamic optimization problem, for the purpose of clarifying the long-debated issue of the relationship between the savings rate and the household dependency rate in developing countries. The implications of this relationship cannot be overemphasized, given the present state of international capital markets. Lower rates of domestic savings can no longer be offset by an increased reliance on foreign capital, and. therefore, growth targets have to be revised downward.

As has been shown here, the magnitude of the above relationship can well depend on consumers’ willingness to smooth consumption over time and also, therefore, on the degree of intertemporal substitution in developing countries. Given the difficulties in estimating the latter, it is easier to understand why existing studies have generated such widespread controversy, as documented in the introduction to this paper.

The empirical evidence provided in the paper is only disappointing in that, with the exception of Southern Europe, the estimates of the coefficients linking the rate of growth of consumption to the expected change in the dependency rate are highly imprecise, making it impossible to draw any definite conclusions. However, as it turns out. Southern Europe is one of the few regions where independent evidence has suggested that the degree of intertemporal substitution is likely to be sizable and accurately estimated (Rossi (1988)).

Finally, the methodology outlined in the paper suggests that, in developing countries where rates of return on financial assets are unobserv-able or irrelevant, the expected path of demographic variables can provide information on the savings responsiveness to changes in the real interest rate.