New Result in Theory of Consumption:

Changes in Savings and Income Growth

Cheng K. Wu

May 1994

Copyright 1993 Cheng K. Wu

New Result in Theory of Consumption:

Changes in Savings and Income Growth

Introduction

In extending the Life Cycle-Permanent Income hypothesis, where an agent maximizes its utility based upon its assets, present and expected future labor income, Hall (1978) theorized that future consumption could be written only as a function of its current consumption; in effect, weakening the link between changes in income and consumption. One consequence of his result is that consumption theory has being slow to systematically incorporate research on factors affecting income, such as productivity and foreign trade. Our article examines this discontinuity. To help explain changes in savings, we show the Modigliani and Brumberg's proposition (1952, published 1979) that changes in savings are also a function of income growth.

Following the works of Hall (1978) and Sargent (1978), Flavin (1981) made a "convincing" derivation showing that, under the permanent income hypothesis (PIH) and rational expectations, changes in permanent income, or consumption should depend only upon revisions in expectations of future income (when these terms are assumed to be zero, Hall's martingale holds). However, from the start, this theoretical result was questioned by Flavin (1981) - and Hayashi (1982) - who found evidence that consumption is "excess-sensitive" to current income. Similarly, Deaton (1987) found evidence that changes in consumption are "excess-smooth" compared with changes in labor income. The causes of this discrepancy have been the subject of an extensive literature. Some authors, such as Deaton (1987) on aggregate consumption behavior and indefinitely lived agent, Zeldes (1989) on liquidity constraints, Campbell and Mankiew (1987) and Koenig (1990) on nonseparabilities of the utility function of consumers and Deaton (1991) on rate of time preference and interest rates, have examined the microfoundations of the PIH's and rational expectations; while others, Hall and Mishkin (1982) on current consumption response to an innovation in income, Quah (1991) on agent's information of its permanent and transitory components of labor income and Deaton (1991) on income growth as the income process, have emphasized the nature of the income process.

Since many of the above mentioned authors have employed Flavin's derivation as the starting point, we also think appropriate to examine this increasing divergence between theory and evidence based upon her original approach. In section I, we show her three basic equations, and in section II, we replaced one of her equations. Under the same initial conditions, we show that the permanent income for the period t+1 should have the sum of incomes varying from period 1 to infinite instead of 0 to infinite, i.e. the loss of one labor income should be partially compensated for by an increase in assets (in the Appendix, we derive a generic permanent income equation and also discuss Flavin's possible mispecification). Graphically,

Flavin	Our Approach

Sum of Labor Income varying from 0 to infinite	varying from 0 (or 1) to infinite

at period t	|---------- ----->	|---------- ----->

at period t+1	|---------- ----->	|------ ----->

0	infinite	0 1	infinite

(loss of 1 labor income due to aging)

With this adjustment for the period t+1, we show that changes in consumption should also depend upon labor income and change in assets. In section III, we apply this result to savings and show that its change may depend upon interest rate and growth of labor income.

I The Three Basic Equations

Following Flavin (1981), an agent's permanent income can be expressed as

At is the agent's real non-human wealth at the beginning of the period t,

r is the real rate of return, assumed constant,

is (1 + r)-1

yt is labor income at the end of period t,

Et is the agent's expectation at period t.

Assets varying according to:

And, Flavin (1981) considered agents consuming exactly their permanent income,

where consumption, ct, is also paid at the end of period t.

(From the seminal works of Friedman (1957), Modigliani and Brumberg (1954), Muth (1961) and the already mentioned authors, these equations basically assume a quadratic utility with constant risk, infinitely lived and forward-looking agents, little or no liquidity constraints, and rate of time preference equal to interest rate. It also assumes rate of interest and income exogenous to consumption.)

II What, However, Is Permanent Income for Period t+1?

While Flavin (1981) wrote permanent income at t+1 as,

in the Appendix A, we show that, assuming interest rates were to remain constant, an agent's permanent incomes over time should be:

Specifically, under an "Euler equation approach," if there is a change in the reference system, from t to t+1, permanent income at t+1 should be written as

If, as in eq. (3), the change in consumption with stochastic, or transitory components of consumption were,

Substituting eq. (6) and (1) into (7), we find that,

and since,

eq. (8) can be written as

Eq. (9) shows that changes in consumption depend upon more than just changes caused by revisions in expectations on income. Note that, by working out the variances on eq. (9), it easy to see that the addition of the changes in assets less current income may help explain why changes in consumption are "excessively smooth" compared with the changes in income, i.e. the Deaton's paradox (see Flavin 1992).

Furthermore, if we simplify eq. (9) assuming that,

then changes in consumption are given by

III Was the Decline of Savings Caused by a Slower Growth of Income?

From the definition of total income or "measured" income,

eq. (11) can be written as

thus,

From eq. (13), we could conclude that, for constant rates, the recent decline of savings may, in theory, be partially attributed to a slower growth in labor income. This result was anticipated by Modigliani and Brumberg (1952, published 1979). The striking innovation of their model over earlier ones - and advantage over more recent ones - was to rationalize zero-negative saving in terms of the stationary-dynamic state of the economy.

For the past two decades, this decline in growth seems to be supported by a growing number of research. Among them, Levi (1989) found evidence of slower growth of family income and inequalities in income distribution; Bosworth, Burtless and Sabelhaus (1991), after an extensive discussion on demographics, income distribution and capital gains, also raised the possibility of savings decline being related to a slower growth of income. In practice, to derive its optimal consumption model, Deaton (1991) has also accepted the growth of income as the income process.

Appendix A

A.1 A consumer maximizes

Thus, under the "Euler equation approach," which is found in many textbooks, including Sargent (1987) - chapters IX and XII, optimal consumption for period t is given by

Obviously, repeating the same optimization technique applied for eq. (iii), one should find that consumption, ct+1, ct+2, . . . , ct+n, should be given by,

. . .

Assuming Rb = 1, the difference between (vi) and (iii) is,

and since,

writing equation (vii) in terms of their power series results in,

A.2 On the other hand, though Flavin provided no explanation on how she arrived at equation (4), it would be a mispecification to reach her conclusion based upon "shifting forward one period and writing out the polynomial in L explicitly." That is, dividing eq. (iii) by L results in

or,

which may be confused with Flavin's equation (4):

Eq. (xii) is clearly not a particular case of eq. (vi). The flaw is that, multiplying eq. (iii), which is a particular optimal consumption, either by a constant or variable, or worse, utilizing the lag operator definition to establish beforehand that there is a relationship between ct and ct+1 , will not necessarily result in another optimal solution.

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