price level uncertainty, saving, and labor supply

PRICE LEVEL UNCERTAINTY, SAVING, AND LABOR SUPPLY

ARTHUR SNOW and RONALD S. WARREN, JR.’

This paper derives the implications, for individual saving and labor supply, of increased uncertainty about the future price level. The framework for the analysis is a two-period model in which saving and labor supply are alternative sources of both present disutility and future income. The individual is assumed to make simultaneously his saving and labor supply decisions prior to the resolution of the uncertainty about the future price level. W e find that, under theoretically plausible and empirically relevant assumptions about attitudes toward risk, an increase in future price level uncertainty increases individual saving and labor supply. These results imply that, for the economy as a whole, increased uncertainty about the future price level increases output and employment, while decreasing the real rate of interest, the present price level, and economic welfare.

I. INTRODUCTION

Several recent papers have examined the macroeconomic implications of increased uncertainty‘ about the future price level. For example, Lucas (1981, p. 141) argued that “. . . the higher the variance in average prices, the less ‘favorable’ will be the observed tradeoff” between inflation and unem- ployment. Klein (1975) suggested that an increase in long-term price uncertainty relative to short-term price uncertainty has implications for the term structure of nominal interest rates and new corporate debt.2

Unfortunately, the microeconomic foundations for a macroeconomic analysis of price level uncertainty are not well-established. In particular, it is commonplace to analyze the effects of increased price level uncertainty on a consumer’s saving and labor supply decisions in such a way that no distinction can be made between real and nominal price uncertainty. For example,

*Georgetown Liniversity and University of Georgia, respectiveIy. We thank Richard J. Sweeney, an anonymous referee, and participants in the Microeconomics Workshop at the University of Virginia for helpful comments on an earlier version of this paper.

1. We use the words “variability”, “uncertainty”, and ‘‘risk” interchangeably to r.?fer to Knight’s (1921, p. 233) concept of risk, in which individuals are assumed to know the probability distribution(s) generating values of the random variable(s). While much of the macroeconomic literature to which we refer deals with price level instability or variability-and these are not, of course, synonymous with uncertainty or unpredictability-the clear intent of this literature is to analyze the effects of the rondom (and, thus, uncertain) component of this variability. Klein (1975, p. 464, fn. 3) notes that variability is a good measure of unpredictability if the probability structure of the random variable is one of constant mean plus a disturbance.

2. For a descriptive examination of recent price level variability in the United States, see Gale (1981).

97 Economic Inquiry \’d. XXIV, January 1986

98 ECONOMIC INQUIRY

Eaton and Rosen (1980) motivated their analysis of individual labor supply under price level uncertainty by arguing that ‘‘. . . workers who contract to work at fixed money wages do not know the effect that changes in consumer prices will have on their real wages during the contract period (p. 366). However, as Killingsworth (1983, p. 241) has pointed out, price level uncertainty entails uncertainty about both real wages and the real return to saving. Since Eaton and Rosen ignore the saving decision, real and nominal wage uncertainty are equivalent in their model. Similarly, Sandmo (1970) and Hanson and Menezes (1978) analyzed the effect of interest rate uncertainty on individual saving with a model that ignores the labor supply decision. As a result, no distinction can be made in that model between price level uncertainty and uncertainty about the nominal interest rate.

In this paper we present a model that allows us to derive the implications of increased uncertainty about the future price level for an individual’s present labor supply and saving. In the next section, we introduce a model in which both saving and labor supply are sources of present disutility and future income. With wage and interest rates fixed in nominal terms, the individual is assumed to make his saving and labor supply decisions simultaneously, prior to the resolution of uncertainty about the future price level. After contrasting our model with the traditional approach, we proceed with the comparative statics analysis of an increase in price level uncertainty and provide an interpretation of the results. We find that, under theoretically plausible and empirically relevant assumptions about attitudes toward risk, an increase in price level uncertainty increases individual saving and labor supply. Applying these microeconomic results to a standard macroeconomic model of aggregate demand and supply leads to the conclusion that increased uncertainty about the future price level increases output and employment and decreases the real interest rate, the present price level, and economic welfare.

II. THE MODEL AND BASIC RESULTS

The setting for the analysis is a partial equilibrium, one consumer model consisting of two periods, the “present” and the “future”. The consumer is endowed with M units of the numeraire (money capital) and T units of time (human capital) and derives utility from present consumption of the numeraire (c) and leisure or nonmarket time ( E ) and from future real income (Y). We assume that c, 1, and Y are normal goods. Future real income consists of the nominal income from saving (s T - l ) , deflated by the future price level, p - ‘ . The numeraire, c, has a price of unity. The wage rate and (gross) interest rate (one plus the interest rate) are denoted w and r, respectively. The present value of nominal future income from saving and labor amounts to

M - c ) and market labor ( h

K = ( M - c ) + w / r ( T - 1 ) > 0 ,

SNOW AND WARREN: PRICE LEVEL UNCERTAINTY 99

so that future real income is Y = p r K . We assume that the consumer is risk averse and maximizes the expected value of a von Neumann-Morgenstern utility function U(c, Z, Y), which is thrice continuously differentiable and intertemporally separable so that U,, = 0 = U,y.

Uncertainty about the future price level is represented by a probability distribution function F ( p ; y), which is assumed to be continuously differentiable, non-negative on a compact interval containing the possible values of p , and dependent upon a risk parameter, y. An increase in y induces an increase in risk about p , in the sense of Rothschild and Stiglitz (1970), i.e., a mean-preserving spread in the probability density function for p . Thus, we assume

(2) p ( T ; y)& 2 0

for all p , with the equality in (2) holding for the extreme values of p and the inequality holding for some non-extreme values. (Note that we omit the limit of integration when it is one of the extreme values of p.)

The consumer’s problem is to choose c and 1 to maximize

where we have substituted for Y in U and used the definitions for K , s, and h. We denote the consumer’s choice of a present consumption bundle by ( co , 1’) so that optimal saving and labor supply are, respectively, so = M - c* and h” = T - 1‘.

Before proceeding further, we emphasize that the consumer chooses both saving and labor supply prior to the resolution of uncertainty about the future price level, and that the returns to labor and saving accrue contemporaneously. Thus, saving and labor supply are alternative sources of future income, while the present income endowment can be viewed as the cumu- lative result of past decisions. This framework combines the labor supply problem analyzed by Block and Heineke (1973), Tressler and Menezes (1980), and Eaton and Rosen (1980) and the model of saving behavior inves- tigated by Sandmo (1970) and Hanson and Menezes (1978). In these models, there is only one source of future income (labor supply or saving) so it is impossible to distinguish between price level uncertainty and uncertainty

3. By assuming K > 0, we confine our attention to interior solutions. Alternatively, we could place restrictions on preferences which ensure interior solutions.

4. The consumer’s problem can be interpreted as a rolling plan model with a one-period horizon. When the future period becomes the present, Y becomes the endowment 111. Further- more, if the intertemporal separability assumption is relaxed, third-order, cross-partial deriva- tives of the utility function would have to be incorporated into the sufficient conditions on preferences required, in order to determine the comparative statics effects of price level uncertainty. In particular, assumptions that “own” effects (U,.,.,.) dominate “cross” effects (e.g.. U,.,.,) would be required, in a condition analogous to (17) below, in order to extend our results for the separable case to the nonseparable case.

100 ECONOMIC INQUIRY

about the nominal return to the choice variable. In our approach, in contrast, labor supply and saving are distinct sources of present disutility and alternative sources of future income. As a result, it is possible to investigate the implications of price level uncertainty when the nominal returns to labor and saving are fixed. Previous studies that do allow for the consumer's choice of both labor supply and saving (Block and Heineke (1975) and Kill- ingsworth (1983, sec. 5.3)) have not considered the separate issue of price level risk.

To analyze the effect of increased uncertainty about the future price level, we formulate the consumer's problem as if it were solved in two stages. In the first stage, demand functions for the present consumption goods c and 1 are determined under the assumption that the level of present spending is fixed. In the second stage, these demand functions are used to arrive at an intertemporal allocation of spending between the present and the future. The optimal level of present spending determined in this second stage implies optimal levels of c and 1 from the first stage demand functions and these, in turn, determine optimal levels of saving and labor supply.

Formally, in the first stage the consumer chooses c and I to maximize

(4)

subject to the constraint

EU(c, I , p [TM + WT - TI ] )

(5) c + ( W h ) 1 = I ,

where I is the (fixed) present value of spending on present consumption, (c, 1). The optimum bundle (co, l o ) satisfies the first-order condition

(6) EU,/EU, = W / T ,

along with the budget constraint (5 ) . Now, let D'(w/T, I , y ) and D'(w/T, I, y ) denote, respectively, the resulting

demand functions for c and 1. In the second stage, the consumer chooses I to maximize

(7) The necessary condition for intertemporal (expected) utility maximization is

EU(D", D', p r [ M + (w/T)T - I ] ) .

where

(9) R ( I , y ) =E(U,.D," + U,D,')/EprUy

is the marginal rate of substitution between present spending ( I ) and present spending foregone ( K ) . Let lo( y ) denote the optimal level of present spending. Then, the consumer's optimal saving and labor supply are, respectively,

SNOW A N D WARREN: PRICE LEVEL LINCERTAINTY 101

(10) s" = M - D'(w/r , Z"(y) , y )

(11)

saving and labor supply are given by

h" = T - D ' ( W / T , I " ( - / ) , y ) .

The comparative statics effects of an increase in price level uncertainty on

d s " / d y = -D: - D f d Z " / d y

d h " / d y = -0: - D:dZ"/dr .

To explore the signs of these effects, note first that, since preferences are assumed to be intertemporally separable, the marginal rate of substitution between c and 1 does not depend on the level of future income, Y. As a result, D' and D' are not affected by any uncertainty about future income, including that arising from future price level risk, so that D l = D J = 0. Moreover, since c and 1 are normal goods, Df and D: are positive. Thus, saving and labor supply both increase (decrease) if and only if I" decreases (increases) with the increase in uncertainty.

The effect of an increase in price level uncertainty on I" can be analyzed by differentiating the marginal condition (8) for intertemporal optimality to obtain

(14) dZ" /dy = - R , / R , .

The second-order condition for the utility-maximizing choice of I requires R , < 0. Hence, the direction of change in I" has the sign of R,. These remarks establish the following theorem:

Theorem. Saving and labor supply both increase (decrease) if and only if R , < (>,) 0.

To evaluate R,, we differentiate (9) and use (8) to arrive at

(15) R , = - [EprU,] - ' J rpU, f , dp .

Applying integration by parts twice, we obtain

If R , has the same sign for any initial risk, then (ZU,, + YU,,,) is mono- tonic in Y. It follows from (2) and the theorem that saving and labor supply increase (decrease) if and only if

(17) ZU,, + YUy,, > (<) 0 .

Menezes et. al., (1980) have introduced a definition of increasing downside risk and shown that the intuitively appealing assumption of "aversion to downside risk" is equivalent to the condition U,,, > 0. However, simple risk aversion implies U,, < 0. Thus, the sign of the expression in (17) is indeter-


minate given only risk aversion and aversion to downside risk. A more strin- gent characterization of risk preferences is necessary to obtain an unam- biguous relationship between increased price level uncertainty, saving, and labor up ply.^

II I . INTERPRETATION

The intertemporal equilibrium condition (8) states that the marginal rate of substitution of present real spending (I) for present spending foregone ( K ) is equal to one. That is, the consumer’s discounted marginal rate of time preference ( R ) is unity at the optimum. The way in which R responds to an increase in price level uncertainty governs the direction of the inequality in (17) and, hence, determines whether saving and labor supply increase or decrease. That is, if and only if an increase in price level variability causes a ceteris paribus ( c and 1 constant) decrease (increase) in the consumer’s discounted marginal rate of time preference, the response is an increase (decrease) in both saving and labor supply. The crucial issue, then, is the direction of the response of the discounted rate of time preference to an increase in uncertainty about the future price level.

Both theoretical considerations and empirical evidence suggest that increased uncertainty about future income decreases the rate of time preference so that, according to the theorem, both saving and labor supply increase. Perhaps the earliest discussion of this response of the rate of time preference, and its effect on present saving, was presented by Fisher (1930, pp. 76-78, emphases in the original):

Future income is always subject to some uncertainty, and this uncertainty must naturally have an influence on the rate of time preference, or degree of impatience, of its possessor. . . . If, as is very common, the possessor of income regards his immediate future income as fairly wefl assured, but fears for the safety or certainty of his income in a more remote period, he may be aroused to a high appreciation of the needs of that remote future and hence may feel forced to save out of his present relatively certain abundance in order to supplement his relatively uncertain income later on. He is likely to have a low degree of impatience for a certain dollar added to a remoter uncertain income. . . . This tendency is expressed in the phrase “to lay up for a rainy day.” The greater the risk of rainy days in the future, the greater the impulse to provide for them at the expense of the present.

More recent literature provides further theoretical support for the assumption of decreasing marginal rate of time preference (DMRTP) with respect to an increase in real income uncertainty. Menezes and Auten (1978) demon-

5. Of course, risk aversion alone implies that the individual is worse off as a result of the increase in price level risk. The effect on expected utility is given by aEV/ar = r2K21Uy, [IPF,dr]dp, which is negative given Uy,. < Uand condition (2).


strated that DMRTP is equivalent to Leland's (1968) assumption of decreasing risk aversion to concentration (DRAC), defined as follows:6

Definition: DRAC: EU(D'(Z), Dr(Z), Y) > E U ( D ' ( I ' ) , Dr(Z') , Y') when U(D'(Z) , Dr(Z) , EY) = U(Dc(Z' ) , D'(Z'), EY') and EY > EY'.

Leland introduced DRAC as an intuitively appealing extension of decreasing absolute risk aversion, defined for a single argument (Y) utility function by Arrow (1965) and Pratt (1964) in the following manner:

Definition: DARA: a(-U,,/U,)/aY < 0.

DARA states that an individual becomes less averse to risk (more inclined to accept a fair lottery) as he becomes richer in the asset that is risky. DRAC states that this decrease in risk aversion occurs when other variables are adjusted so as to hold the utility of the fair lottery constant. Thus accepting the hypothesis of DRAC is equivalent to accepting Fisher's hypothesis DMRTP and implies that labor supply and saving increase in response to greater price level variability.

It can be shown that DMRTP is implied by DARA and a coefficient of relative risk aversion (RRA) no less than two.' Hence, the reasonableness of the DMRTP/DRAC assumption hinges on the empirical relevance of DARA and RRA 2 2 as characterizations of attitudes toward risk. With regard to DARA, this assumption is intuitively appealing and has been used in a wide variety of settings to model individual behavior toward risk.8 Moreover, Cohn, et. aZ., (1975) and Projector and Weiss (1966) have presented evidence that is consistent with DARA. Although the size of the coefficient of RRA is an unsettled matter, the preponderance of existing estimates implies that R R A > 2. For example, Weber [(1970) (1975)l reported estimates of the intertemporal elasticity of substitution in consumption which, under his maintained assumptions of homotheticity and separability, can be interpreted as finding that RRA > 2, uniformly across alternative ~pecifications.~ Additionally, Friend and Blume (1975, p. 920) reported

6. The equivalence derived by hlenezes and Auten (1978) treats the case of absolute (addi- tive) uncertainty about Y. Their demonstration applies as well to the case of proportional (multiplicative) risk, which is the type of risk associated with future price level uncertainty. Also, Leland (1968) defined DRAC for a two-good utility function. In our statement of DRAC, the two goods are I and Y.

7. By definitio?, DARA requires U , , , > U,?U, and RRA 2 2 requires -Y U,,./U, 2 2 or, equivalently, U y , - / U , 5 -2U, , /Y. Thus, DARA and RRA 2 2 imply U , . , , > -ZCJ,,/Y, which is the condition for DRAC, since preferences exhibit DRAC = DMRTP if and only if the greater than inequality obtains in (17).

8. In addition to its use in studies of portfolio and saving behavior, the DARA assumption has been invoked to model labor market behavior under uncertainty. See Danforth (1979) for an application to the problem of job search.

9. If the utility function is homothetic and (completely) separable-as assumed, for example, by U'eber (1975)-RRA equals the inverse of the intertemporal elasticity of substitution in consumption.

104 ECONOMIC INQLURY

that “. . . the coefficient of proportional risk aversion is more likely to be in excess of two.” More recently, Summers (1982) concluded, from a survey of previous research and his own estimates derived from alternative estimating equations, that RRA = 3.

If DARA and RRA > 2 (hence DMRTP) are considered reasonable assumptions regarding attitudes toward risk, then our analysis implies that an increase in price level variability increases both saving and labor supply. A comparison of our result with that obtained by Hanson and Menezes (1978) is complicated by the fact that they omit the labor supply decision so that uncertainty about the nominal interest rate ( r ) and the pricelevel ( p ) cannot be distinguished in their treatment of real capital income ( r p ) risk. However, if r is held constant so that r p uncertainty is interpreted as arising from price level variability, then Hanson and Menezes also obtain the result that DMRTP implies an increase in saving in response to increased price level uncertainty. Similarly, the model of real wage uncertainty and labor supply specified by Eaton and Rosen (1980) cannot be used to analyze separately the effects of nominal wage and price level uncertainty. With the nominal wage assumed to be known with certainty (which accords with our approach and their motivation for modelling real wage uncertainty), the condition’O derived by Eaton and Rosen (1980) for determining the effect of (price level) uncertainty on labor supply reduces, for the case of separable utility (and no taxes), to our equation (17). Thus, our result that DMRTP implies an increase in labor supply as a result of price level variability is analogous to the implication of the Eaton-Rosen analysis that DMRTP leads to increased labor supply, as a result of real wage uncertainty arising from price level uncertainty with a fixed nominal wage.

Finally, although our analysis is microeconomic in nature, we briefly in- dicate some of its macroeconomic implications. Consider a conventional IS-LM model of aggregate demand and assume a classical (perfectly inelas- tic) aggregate supply curve. Under reasonable assumptions about attitudes toward risk, an increase in future price level uncertainty increases both labor supply and saving. The increase in labor supply increases employment, thereby increasing output and lowering the present price level, so that the aggregate supply curve shifts to the right. The increase in saving shifts the IS curve down,” reducing the real rate of interest. This reduction in the real interest rate is reinforced by the outward shift in the LM curve occasioned by the lower price level. While current employment and output increase and the present price level falls as a result of an increase in uncertainty about

10. Their equation (3) on page 367. 11. U’e abstract here from the effect of price level uncertainty on the investment function

Dietrich and Heckerman (1980) have analyzed the effect of uncertain inflation in a partial equilibrium, maximizing model of firm behavior and found that greater uncertainty about inflation increases the firm’s demand for capital. This latter effect would shift the investment function and tend to dissipate the downward shift in the IS curve associated with the increased saving.


the future price level, it is important to remember that welfare is reduced since individuals are assumed to be risk averse. Thus, as Samuelson (1972) has emphasized, any attempt to manufacture price (level) instability is un- ambiguously harmful.

IV. CONCLUDING REMARKS

This paper has analyzed the effect of increased uncertainty about the future price level on an individual’s labor supply and saving decisions. To conduct this analysis, we introduced a model in which both labor and saving are sources of present disutility and sources of future income. With this specification, the returns to both factors accrue contemporaneously so that two margins of intertemporal choice are present. As a result, nominal and real factor returns can be meaningfully distinguished and the model can determine the implications of greater uncertainty about the absolute price level when relative prices are stable.

The model predicts that both labor supply and saving increase when uncertainty about the future price level increases if and only if an individual’s preferences exhibit the property of decreasing marginal rate of time preference or, equivalently, decreasing risk aversion to concentration. For this property to obtain, it is sufficient that preferences exhibit decreasing absolute risk aversion and a coefficient of relative risk aversion no less than two. Since available empirical evidence is consistent with these conditions on preferences, we conclude that labor supply and saving increase in response to greater uncertainty about the price level. Application of these results to a standard model of the macroeconomy implies that an increase in future price level risk increases present output and employment and decreases the real interest rate, the present price level, and economic welfare.

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price level uncertainty, saving, and labor supply

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