The relationship between income and consumption in life cycle models.

Ahmad, Eatzaz

I. INTRODUCTION

The traditional life-cycle theory of consumption has been extended in various directions. A major contribution to this subject is due to Heckman (1974). He showed that under an endogenous work-leisure decision, labour income unambiguously increases in response to anticipated increase in the wage rate over the life cycle. Further, if consumption and leisure are substitutes for each other, consumption also increases over the life cycle. This explains a positive relationship between consumption and current income in a life-cycle model. Browning, Deaton and Irish (1985) and MaCurdy (1981, 1983, 1985) further elaborated this theory and tested its empirical validity.

In this paper we provide an alternative interpretation to Heckman's 0974) result. Using the class of utility functions defined over consumption and leisure characterized by constant elasticities of intertemporal and intratemporal substitution, we show that the necessary and sufficient condition to have a positive relationship between consumption and current labour income is that the elasticity of substitution between consumption and leisure at a given age is greater than the elasticity of substitution between consumption (or leisure) at different ages. In addition, it is shown that the planned savings and anticipated labour income are positively correlated over the life cycle under quite general conditions.

II. THE MODEL

Consider an individual who is to live through a certain life of length T At each age t he has to allocate a fixed endowment of time E on work H(t) and leisure E-H(t). We employ the class of utility functions used by Auerbach, Kotlikoff and Skinner (1983):

U(C(t), E-H(t)) = 1/1 - 1/[sigma][[{ C(t)}.sup.1-1/[theta]] + [alpha] [{E-H(t)}.sup.1 - 1/[sigma]].sup.1 - 1/[sigma]/1 - 1/[theta]] ... ... ... (1)

where, [sigma], [theta], [alpha] > 0. The parameter [theta] is the (intratemporal) elasticity of substitution between consumption and leisure at a given age, [sigma] is the (intertemporal) substitution elasticity of consumption (or leisure) across two different ages and [alpha] measures the intensity of preference for leisure over consumption.

For simplicity and to avoid repetition we assume that the rate of interest and the subjective discount rate are zero. The individual does not receive or pay any transfers. The entireage path of the wage rate, W(t) is known with certainty. The individual's life-cycle problem is to maximize the lifetime utility function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... ... ... (2)

with respect to consumption, C(t) and leisure, E-H(t) subject to the lifetime budget constraint:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... ... ... (3)

and the boundary conditions: C(t) [greater than or equal to] 0, 0 [less than or equal to] H(t) [less than or equal to] E.

III. THE OPTIMAL SOLUTION

We solve our problem in two stages. We define the rate of expenditure at age t in the units of consumption good, M(t)as

M(t) = C(t) + W(t) {E -H(t)} ... ... ... (4)

The lower-stage problem is to maximize utility at each age t by choosing the rates of consumption and leisure assuming a given rate of expenditure M(t). The resulting consumption and leisure demand functions are substituted into the age-related utility function to obtain the indirect utility function. In turn, the indirect utility function is used at the upper stage to determine the optimal rates of expenditure for different ages.

At the lower stage we have to choose C(t) and H(t) which maximize the following Lagrange function such that [lambda](t) [greater than or equal to] 0.

1/1 - 1/[sigma]] [[{ C(t)}.sup.1 - 1/[theta]] + [alpha] [{ E-H(t)}.sup.1 - 1/[theta]]].sup.1 - 1/[sigma]/1 - 1/[theta]] + [lambda](t) [ M(t) - C(t) - W(t) { E-H(t)}] ... ... (5)

The parametrization of the utility function guarantees that the constraints C(t) [greater than or equal to] 0, H(t) [less than or equal to] E and (t) [greater than or equal to] 0 are not binding. For simplicity we assume that H(t)

[greater than or equal to] 0 is also non-binding, that is, the individual never retires from work. In appendix A it is shown that the conditional consumption and leisure demand functions are

C(t) = M(t)/1 + {[W(t)}.sup.1-[theta]] [[alpha].sup.[theta]] ... ... ... ... (6)

E-H(t) = {[W(t)}.sup.-[theta]] [[alpha].sup.[theta]] M(t)/1 + {[W(t)}.sup.1-[theta]] [[alpha].sup.[theta]] ... ... ... (7)

We can find the age-related indirect utility function by substituting these demand functions into the utility function. The result (derived in Appendix B) is

V(W(t), M(t)) = [ 1 + [{ W(t)}.sup.1 - [theta]] [[alpha].sup.[theta]]].sup.1 - [sigma]/[sigma](1 - [theta])] [{ M(t)}.sup.1 - 1/[sigma]/1 - 1/[sigma] ... (8)

Notice here that the parameter o can also be interpreted as the intertemporal substitution elasticity for expenditure.

Next, the lifetime utility function can be expressed as a function of expenditure rates by substituting the indirect utility function into Equation (2). Using Equation (4), we can also express the budget constraint Equation (3) as a function of expenditure rates. The upper stage problem is to maximize the following Lagrange function with respect to M(t) for all t such that [lambda] [greater than or equal to] 0.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... ... (9)

Where, V(W(t), M(t)) is given by Equation (8). As before, the boundary conditions: M(t) [greater than or equal to] 0, [lambda] [greater than or equal to] 0 are not binding. The maximization conditions other than the budget constraint are

[[1 + {[W(t)}.sup.1-[theta]] [[alpha].sup.[theta]]].sup.1-[sigma]/[sigma](1-[theta])] {[M(t)}.sup.-1/[sigma]] = [lambda] ... ... (10)

We can find a solution for M(t) conditional upon [lambda], the marginal utility of wealth, from Equation (10) as follows:

M(t) = Z [1 + [{W(t)}.sup.1-[theta]] [[alpha.sup.[theta]]],.sup.1-[sigma]/1-[theta]] ... ... (11)

where Z = [[lambda].sup.-[sigma]] > 0. The anticipated changes in the wage rate over the life cycle are already incorporated into the maximization problem and do not induce any reallocation of expenditure. As a result [lambda], the marginal utility of wealth remains constant [See Ahmad (1988) and MaCurdy (1981, 1983)]. Thus the "[lambda]-constant" demand function for expenditure given by Equation (11) contains sufficient information to study the anticipated changes in wage rate. (1)

Finally, substituting the upper stage solution given by Equation (11) into the lower stage solution given by Equations (6) and (7) gives the [lambda]-constant demand functions for consumption and leisure:

C(t) = Z [1 + [{W(t)}.sup.1-[theta]][[alpha].sup.[theta]]].sup.[theta]-[sigma]/1-[theta]] ... ... ... (12)

E-H(t) = Z [{W(t)}.sup.-[theta][[alpha].sup.[theta]] [1 + [{W(t)}.sup.1-[theta]][[alpha].sup.[theta]].sup.[theta]- [sigma]/1-[theta] ... (13)

These demand functions are used to find the relationship between the anticipated changes in income and consumption in the next section.

IV. CHANGES IN INCOME AND CONSUMPTION OVER TIME

According to Equation (13) work hours vary over the life cycle only due to anticipated changes in the wage rate. Therefore, if the age path of the wage rate is fully anticipated, the age path of income Y(t) = W(t) H(t) is also known at the beginning of the horizon. The anticipated changes in income over the life cycle are given by the following age derivative, derived in Appendix C.

[??](t) = [H(t) + Z [{W(t)}.sup.-[theta]][[alpha].sup.[theta]] {[theta] + [sigma] [{W(t)}.sup.1- [theta]][[alpha].sup.[theta]]} B(t)] W(t) ... ... ... ... (14)

where,

B(t) = [! + [{W(t)}.sup.1-[theta]][[alpha].sup.[theta]]].sup.[theta]-[sigma]/1-[theta] - 1] > 0 ... ... (15)

Equation (14) shows that income varies over the life cycle in the same direction as the anticipated wage rate.

Next, the planned changes in consumption can be found by taking the age derivative of Equation (12). The result is obvious:

[??](t) = ([theta]-[sigma]) Z [{ W(t)}.sup.-[theta]] [[alpha].sup.[theta]] B(t) W(t) ... ... (16)

To find the marginal consumption rate out of anticipated income, we divide the rate of change in planned consumption by the rate of change in anticipated income. The result is:

dC/dt / dY/dt = ([theta] - [sigma]) Z [{ W(t)}.sup.-[theta]][[alpha].sup.[theta]] B(t)/H(t) + Z [{ W(t)}.sup.- [theta]][[alpha].sup.[theta]] [[theta] + [sigma] [{ W(t)}.sup.1-[theta]][[alpha].sup.[theta]] B(t) ... (17)

From this Equation we can derive two important results.

Result 1: The marginal consumption response to anticipated income changes is positive (negative) if and only if the intratemporal substitution elasticity [theta] is greater (less) than the intertemporal substitution elasticity [sigma]. If the two parameters are equal, the marginal consumption rate would be zero.

This result is explained as follows. An anticipated increase in the wage rate at age t affects consumption on two accounts. First, it makes consumption cheaper relative to leisure at age t. As a result the individual would consume more at age t. The magnitude of this effect depends on the size of the intratemporal elasticity of substitution between consumption and leisure, [theta]. Second, the anticipated increase in the wage rate at age t makes expenditure (on consumption and leisure) relatively more expensive at that age in comparison to other ages. Therefore, the individual plans to consume less at age t. The magnitude of this effect depends on the size of the intertemporal substitution elasticity of expenditure (or consumption) at different ages, [sigma]. Since income varies directly with the wage rate, the sign of the marginal consumption response to in anticipated income follows the pattern described above.

Result 2: The marginal consumption rate out of anticipated income is less than one irrespective of the size of the two substitution elasticities.

Thus, while the individual may increase consumption in response to an anticipated increase in the wage rate under certain conditions on the substitution elasticities, he will increase savings under more general conditions.

V. CONCLUSIONS

A life-cycle model of consumption and work hours is studied. Using the class of utility functions with constant intertemporal and intratemporal elasticities of substitution, changes in labour income and consumption over the life cycle are derived. It is shown that the marginal propensity to consume out of anticipated labour income is positive if and only if the (intratemporal) elasticity of substitution between consumption and leisure at a given age is greater than the (intertemporal) elasticity of substitution between expenditure on consumption and leisure across two different ages. In addition, it is shown that the marginal propensity to save out of anticipated labour income is positive under quite general conditions. This supports the idea that the Keynesian Absolute Income Hypothesis of the consumption function can be supported even in the life-cycle context.

Comments on "The relationship between income and consumption in life cycle models"

Let me start by saying that the paper by Dr Ahmad is an interesting and useful contribution to the debate on the relationship between income and consumption. The recent interest over this relationship is partly due to the paper by Lester Thurow in an 1969 issue of the American Economic Review in which he presented empirical evidence supporting a positive relationship between income and consumption. Thus, contradicting the life-cycle consumption theory which predicts no necessary relationship between consumption and income at any age. Lester Thurow explained this positive relationship in terms of credit market restrictions by arguing that credict market restrictions prevent consumers from borrowing as much against their future income as they desire at the going interest rate. Since income tends to increase with age and discounted future income cannot be fully transferred at the borrowing rate, this leads to an increase in the consumer's net worth which causes consumption to increase with age. Nagatani (1972) has explained the same phenomenon in terms of uncertainty of future income by arguing that a typical consumer buys less than he would in a riskless environment with the same expected income. However, consumption plans are successfully revised once expected income is realized. Subsequently, Heckman (1974) presented an alternative neo-classical model to explain Thurow's result. Rather than resorting to credit market restrictions or uncertainty, Heckman treats earnings as resulting from a life-cycle labour supply decision, where individuals are free to set their hour of work and wage rate change systematically over the life cycle. Heckman demonstrates that in such an environment the consumption path of market goods depend on the wage rate at each age. Heckman in his paper considered a very general form of the utility function to show the positive relation between income and consumption. Dr Ahmad in his paper has considered a very specific form, namely the constant elasticity of substitution, to derive the necessary and sufficient condition for a positive relation between income and consumption. I would imagine that it should be possible to obtain similar conditions by considering different forms of the utility function.

Let me now turn to some specific comments on the paper.

Although the type of model employed in the paper has been widely used in many other studies, the specific form adopted in this paper is highly simplistic. For instance, both the interest rate and the rate of time preference are assumed to be zero. Similarly, initial wealth has been excluded from the analysis. However, because of the model being simplistic the results holds under some very special circumstances. For instance, within the life-cycle framework, the positive relationship between income and consumption holds only if the interest rate equals the rate of time preference, i.e., as long as the consumers are at a steady state equilibria. It has been shown elsewhere that if the interest rate and the rate of time preference differ, the association between income and consumption is not precise. Since the author has used a specific form of the utility function it would be interesting to know whether in a more general case the positive relation between income and consumption is possible over certain ranges of values of the parameters.

While the condition for positive relation between income and consumption, derived in this paper is important in its own right, it would be interesting to know under what sort of circumstances this is likely to hold and what policy implications are likely to emerge.

Finally, it has also been shown in the paper that under quite general circumstances, planned savings and anticipated labour income are positively correlated. With both interest rate and the rate of time preference assumed to be zero, it is not clear that in the model what motivates the consumers to save.

Nadeem A. Burney

Pakistan Institute of

Development Economics, Islamabad.

Appendix

A. THE DEMAND FUNCTIONS AT THE LOWER STAGE

The maximizing conditions for the Lagrange problem Equation (5) are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ... ... (A1)

We can eliminate k(t) by taking ratio on the two sides of the first two equations. The result after simplification is:

C(t) = (W(t)/[[alpha]}.sup.[theta]] {E-H(t)} ... ... ... (A2)

Solving Equations (A1) and (A2) simultaneously gives the lower stage solution in terms of the conditional demand functions for consumption and leisure given by Equations (6) and (7) respectively.

B. THE INDIRECT UTILITY FUNCTION

To find the indirect utility function we substitute the consumption and leisure demand functions ((6) and (7)) into the utility function (1) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Simplifying further, we obtain the indirect utility function (8).

C. THE AGE PATHS OF WORK HOURS AND INCOME

Differentiate Equation (13) with respect to t, we find the age path of work hours:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Collecting the common terms on the right hand side, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A4)

where, B(t) is given by Equation (15).

Next, the age derivative of income Y(t) = W(t)H(t)is [??](t) = H(t)[??](t)+W(t)[??](t) which can be inferred from (A4). The result is given by Equation (14).

REFERENCES

Ahmad, E. (1988) Consumption and Work Hours in Life Cycle Models under Uncertain Lifetimes: An Individual Analysis and an Equilibrium Analysis with Overlapping Generations. Unpublished Ph D. Dissertation, Department of Economics, McMaster University.

Auerbach, A. J., L. J. Kotlikoff and J. Skinner (1983) The Efficiency Gains from Dynamic Tax Reform. International Economic Review 24 : 1 81-100.

Browning, M. J., A. Deaton and M. Irish (1985) A Profitable Approach to Labour Supply and Commodity Demands over the Life-cycle. Econometrica 53 : 3 503-543.

Fisher, I. (1930) The Theory of Interest. New York: The Macmillan Company.

Friedman, M. (1957)A Theory of Consumption Function. Princeton: Princeton University Press.

Heckman, J. J. (1974) Life Cycle Consumption and Labour Supply: An Explanation of the Relationship between Income and Consumption over the Life Cycle. American Economic Review 64 : 1 188-194.

MaCurdy, T. E. (1981) An Empirical Model of Labour Supply in a Life-cycle Setting. Journal of Political Economy 89 : 6 1059-1085.

MaCurdy, T. E. (1983) A Simple Scheme For Estimating an Intertemporal Model of Labour Supply and Consumption in the Presence of Taxes and Uncertainty. International Economic Review 24 : 2 265-289,

MaCurdy, T. E. (1985) Interpreting Empirical Models of Labour Supply in an Intertemporal Framework with Uncertainty. In J. J. Heckman and B. Singer (eds) Longitudinal Analysis of Labour Market Data. Cambridge: Cambridge University Press.

Modigliani, F., and R. Brumberg (1954) Utility Analysis and the Consumption Function: An Interpretation of Cross-section Data. In K. K. Kurihara (ed) Post-Keynesian Economics, 1954. New Brunswick, New York: Rutgers University Press.

(1) The term "[lambda]-constant" is due to MaCurdy (1981). He derived the [lambda]-constant demand functions for consumption and leisure to study the response of consumption and work hours to anticipated changes in wage rate over the life-cycle.

EATZAZ AHMAD, The author is Assistant Professor at the Department of Economics, Quaid-i-Azam University, Ishmabad.