Effects of the price of time on household saving: a life-cycle consistent model and evidence from micro-data.

• Author: Yan Wang

1. Introduction

   The United States has experienced a period of declining rates of personal saving since the mid-1970s. Meanwhile, average wage rates and females' labor force participation rate increased markedly, which may have affected households' consumption and saving behavior. Traditional models of saving often separated work behavior from saving behavior. Income was considered exogenous, and the cross price effect of leisure on saving was ignored. Alternative models proposed by Ghez and Becker |8~ and others incorporate the allocation of time and consumption over the life cycle, and consumption and leisure are found to be substitutes.(1) However, few recent studies have provided evidence as to what happens to saving when time becomes more expensive.(2)

   This study investigates the effects of the price of time (reservation wage rates) on saving of mature and stable households, focussing on couples staying married between 1983 and 1986. It develops a life-cycle consistent model of saving in which work/leisure and consumption/saving are simultaneously determined. The hypotheses of this paper concern mainly the cross price effect and wealth effect of time on household saving. Surveys of Consumer Finances (1983 and 1986) |3~ and an instrumental variable procedure are used in the empirical analysis. Section II presents the model. Section III provides an overview of data and the methodology. Section IV examines the results in the reservation wage estimation and the lifetime wealth projection. Section V discusses the findings in the estimation of the saving equations and the hypothesis testing. Conclusions can be found in the final section.

2. A Life-Cycle Consistent Model of Saving

   The relevant literature for this paper includes the well-documented life-cycle and permanent income hypotheses, Becker's theory of allocation of time and goods over the life cycle, and recent studies on life-cycle consistent models of consumption and labor supply. See King |10~, Gersovitz |7~ and Wang |16~ for surveys of the literature. The model used here is built upon the work of Blundell and Walker |4~. Assuming that household's intertemporal utility is additively separable over time, the lifetime utility function of a representative household in planning period t,|V.sub.t~, can be written as:(3)

   |V.sub.t~ = |summation of~ ||Delta~.sup.i-t~ |U.sub.i~(|X.sup.i~;|Z.sup.i~,|e.sub.i~) where i = t to L (1)

   where |X.sup.i~ is a choice vector in period i, containing leisure of the male and female spouses of the household, (|l.sub.mi~,|l.sub.fi~), and a composite consumption goods, |q.sub.i~. |Z.sup.i~ is a vector of exogenous taste shifters, and |e.sub.i~ a random error representing the unobserved factors influencing taste. i = (t,t + 1,t + 2,...L) is the time index for the remainder of the household's life span. |Delta~ is a time discount factor and |Delta~ = 1/(1 + |Sigma~), where |Sigma~ is the rate of time preference and 0 |is less than~ |Sigma~ |is less than~ 1.

   Let |P.sup.i~ be a price vector containing reservation wage rates for male and female spouses of the household, |W.sub.mi~ and |W.sub.fi~, and the price of consumption goods, |P.sub.qi~. The within-period budget constraint of the household is:

   |P.sup.i~|X.sup.i~ + |S*.sub.i~ = F|R.sub.i~

   where F|R.sub.i~ is the full resource (assets and time) available in period i, and |S*.sub.i~ is the planned saving defined as the desired change in non-human wealth (net worth) of the household from the beginning of period i,N|W.sub.i~, to the beginning of i + 1, N|W*.sub.i+1~:

   |S*.sub.i~ |is equivalent to~ N|W*.sub.i+1~ - N|W.sub.i~. (3)

   Let a "hat," |Mathematical Expression Omitted~, indicate discounted prices or values. The lifetime full wealth constraint of the household in current planning period t is:

   |Mathematical Expression Omitted~.

   (4) implies that the expected lifetime full expenditure must be equal to the lifetime full wealth expected at t,||Omega~.sub.t~, which consists of cumulated non-human wealth at t, |A.sub.t~, and expected labor and leisure wealth over the remainder of the household's life span. That is:

   |Mathematical Expression Omitted~

   where |l.sub.mi~(|l.sub.fi~) and |h.sub.mi~(|h.sub.fi~) are the hours of leisure and hours of work for male (female) spouse of the household.

   Two stage budgeting is conducted. In the first stage, household maximizes (1) subject to (4), with the optimal allocation of full resources given by:

   |Mathematical Expression Omitted~.

   To make this general model empirically manageable, I assume, further, that the utility function is homothetic with respect to F|R.sub.t~. This assumption, based on Friedman |6~, means that the intertemporal utility can be expressed as a monotonically increasing function of F|R.sub.t~. Solving for F|R.sub.t~ in the first stage budgeting, we got equation (7) (equivalent to Friedman's consumption function: |C.sub.p~ = k|Y.sub.p~), which implies that F|R.sub.t~ is proportional to the lifetime full wealth. This assumption not only simplifies (6) but also provides a device for demographic translation: Define |k.sub.t~ as the marginal propensity to consume (both goods and leisure) from ||Omega~.sub.t~ and express it as a function of expected rate of change in future prices, |Delta~p, demographic characteristics, |Z.sup.t~, and fractions of different kinds of wealth in ||Omega~.sub.t~, |f.sup.t~. The latter also follows Friedman |6~ in that the composition of full wealth (physical, financial and human wealth) is assumed to affect one's ability of transferring wealth between periods, and hence. have an impact on |k.sub.t~. One example is that human capital is not always considered a perfect collateral for bank lending. Thus, equation (6) becomes:

   F|R*.sub.t~ = |k.sub.t~ (|Delta~P,|Z.sup.t~,|f.sup.t~)||Omega~.sub.t~. (7)

   The second stage decision-making then is to maximize within-period utility subject to within-period budget constraint, taking F|R*.sub.t~ as given. The resulting demand function can be expressed in vector form as:

   |X.sup.t*~ = |F.sup.t~(|P.sup.t~,F|R*.sub.t~). (8)

   Substituting the desired demand for |l*.sub.mt~, and |l*.sub.ft~, and |q*.sub.t~ into the budget constraint, replacing F|R*.sub.t~ by (7), and solving for saving, one obtains the demand function for the planned saving:

   |S*.sub.t~ = |

   .t~)||Omega~.sub.t~). (9)

   In this replanning model, expectations on future prices are based on past information, and correction for errors is allowed at the beginning of each period. The observed saving, therefore, consists of the planned saving, |S*.sub.t~, and an error term...