Taxation and Aggregate Factor Supply: Preliminary Estimates Michael J. Bosk~n Lawrence J. Lau Comment Finis Welch Taxation and Aggregate Factor Supply: Preliminary Estimates M. J. Boskin L. J. Lau The life-cycle supply of the factors of production plays a crucial role in the analysis of many important economic problems. Among these are the optimal tax treatment of capital and labor incomes, the appropriate social rate of discount to be used for public proj­ ects, and the construction of indexes of economic welfare. Con­ sider, for example, the simple model (1) where U is a well-behaved utility function depending upon Cw, which is the annual flow of conS"Umption during working years, C,., is the annual flow of consumption during retirement, and L1n is the annual flow of leisure during working years. It then follows from recent analytical results 1 that the efficient (in the sense of minimizing the dead weight loss from raising a given tax revenue) taxation of Cw and C,. (Lw is assumed to be inherently too costly to attempt to tax) requires heavier taxation in the period in which consumption is a weaker substitute for leisure. The tax rates on Cu· and C,. will be equal only if utility is See, for example, Atkinson and Stiglitz, (1976). M. J. BOSKIN is with Harvard University and both he and L. J. LAU are with Stanford University and the National Bureau of Economic Research, Stanford, California. The authors would like to thank the U.S. Department of the Treasury, Office of Tax Analysis, for support of this research through a contract to NBER. The research assistance of Louis Garrison is also appreciated. (3) 4 COMPENDIUM OF TAX RESEARCH separable between goods and leisure; in general, the tax rates will be given by 2 (2) where N ii is the compensated cross elasticity of demand of i with respect to the price of j ( 1 =Cw, 2 = C1. , 3 = Lw). Hence, the optimal tax rate on capital income will be positive (tw<tr) or negative Ctw>t,. ) as N 13 is larger or smaller than N 2 ~ . Thus, the efficient taxation of capital income depends upon whether or not leisure and consumption during working years are complements, or at least weaker substitutes, than leisure and retirement consumption. Consider next the appropriate social rate of discount. Recently, Har berger ( 1968) and Sandmo and Dreze ( 1971) have derived the result that the appropriate social rate of discount is a weighted average of the gross of tax marginal product of private capital and the net of tax rate of return to saving, the weighU:; depending upon the interest elasticities of investment and saving, respective­ ly. Since business and personal income taxes cause the two rates to diverge sharply, a positive interest elasticity of saving would result in a much lower social rate of discount than the gross pri­ vate marginal product of capital. In the framework discussed above, the is,sue is the elasticity of Cw with respect to the forward price of Cr. 3 Also of interest is the almost total lack of attention given to future consumption in the calculation of indexes of economic wel­ fare. For example, saving is generally ignored in the calculation of constant utility index numbers, i.e., conceptually they often should be calculated as constant lifetime utility rather than as constant instantaneous utility. Fortunately, the empirical analysis of factor supply has re­ ceived renewed attention in recent years. The seminal work of Friedman ( 1957) , Modigliani and Brumberg and Ando and Modigliani ( 1954, 1963) on life-cycle consumption and of Becker ( 1964, 1965) on labor supply (broadly construed to include human investment and nonmarket work) rekindled interest among em- ~ See Harberger, (1964). If the two periods are of different lengths (e.g., a work life of 40 years and a retirement life of 20 years), the formula would be weighted to reflect this difference. 3 In the alternative approach which shadow prices public investment funds and discounts at the consumption rate of interest (e.g., Feldstein, 197 4b) , the shadow price depends on these elasticities. TAXATION AND AGGREGATE FACTOR SUPPLY 5 pirical economists on these issues. Curiously, however, empirical studies of the role of interest rates or forward prices on consump­ tion are few and far between.4 Most such studies conclude that interest rates have only a negligible effect on consumption or sav­ ing. Boskin (1976) has recently criticized much of this work, especially the structural interpretation of "Denison's Law," the alleged insensitivity of the saving rate to any economic variables (at full employment). He employed a variety of data sources and advances in econometric techniques to estimate an elasticity of private saving with respect to the real net rate of return of approximately 0.4. The purpose of the present paper is threefold: first, to see if this sensitivity of saving with respect to the real net rate of return continues to obtain when explicit account is taken of leisure de­ mand and also of the potential influence of social security; second, to estimate the parameters of the demand functions expressly derived from maximizing a utility function of the form of equation ( 1) ; and, third, to begin to account expressly for such life-cycle phenomena as changes in the age distribution of the population. We hope to be able at a later date to use the estimates to draw some provisional conclusions about the issues discussed earlier. Toward this end, we present in the next section our basic model, its properties and estimating equations. Following that is a discussion of the data used in this study, i.e., the national income and wealth accounts developed by Chris­ tensen and Jorgenson (1972) and estimates of forward ·prices, etc., developed by the authors. A fourth section reports our preliminary empirical results, which are indeed striking. While many refinements are still to be made, our preliminary results strongly reject the structural inter­ pretation of "Denison's Law." Our estimates suggest that the consumption saving choice is strongly influenced by relative prices, including the forward price of future consumption. Our estimates, which appear to be measured quite precisely, suggest that leisure and future consumption are much stronger substitutes than leisure and current consumption. We conclude with some provisos and a long list of extensions, refinements, and improvements we hope to make relative to these preliminary results. ~ Thus, Break (1974) notes "Unfortunately, empirical evidence on the inter­ est elasticity of the saving rate is rare. 6 COMPENDIUM OF TAX RESEARCH The General Model The Direct Utility Function We assume that each consumer unit (an individual or a house­ hold) has a utility function of the form U(Li, ClJ L2, C2, B; t, A, D, T (A, t) -A, R(t) -A, s), where L1 =quantity of leisure in period 1, L2 =quantity of leisure in period 2, C1 =quantity of consumption in period 1, C2 =quantity of consumption in period 2, B =quantity of bequest, t = calendar time, A=age, D =size of the household in equivalent adult consumption units, T (A, t) =life expectancy of an individual (or head of household) at age A in year t, R ( t) = expected retirement age at time t, and s = sex of individual or head of household. As a first pass, we make three simplifying assumptions: 1. R ( t) is given exogenously, that is, it is not a choice variable of the consumer unit. 2. L2 is assumed to be fixed and equal to L, that is, the consumer unit supplies no labor in the second (postretirement) period. 3. Band C2 will be treated as a composite commodity. Utility Maximization The consumer unit i's assumed to maximize utility with respect to L1, CH and C2, subject to a wealth constraint: peC +peC = (l-µ)W1(L-L1) +WP+ (1-A.) W~, 1 1 2 2 • where W1 = wage rate, µ=effective tax rate on labor income, p~ =price of consumption in period 1, p~ =forward price of consumption in period 2, WP= private wealth, W8 = social security wealth, and A.= "discount" factor associated with social security wealth. TAXATION AND AGGREGATE FACTOR SUPPLY 7 As usual, one can transform the budget constraint into the canoni­ cal form where w· and pc 2 ~·- . :! (1-µ)w,L+ W 11 + (l-,\) W.. The Indirect Utility Function Under mild regularity conditions an indirect utility function of the form s) V(w*,1 pc;1•, pc2•, t, A, D, T(A,t) -A, R(t) -A, exist;.'). If one specifies an indirect utility function, then the de­ mand functions are given through Roy's Identity as oV ow; Li= V V V ' _.E_w• + y__pc• + y__pc• aw: 1 oP~· 1 oP~· 2 and oV (}pc• 1 Ci , i= 1, 2. oV w•+ oV pc•+ oV pc• aw· 1 ()pc• 1 ()pc• 2 1 1 2 If we assume that the indirect utility function has the homoge­ neous translog form, we obtain the linear logarithmic expenditure system of Lau and Mitchell (1971), which gives the expenditure . hares as 3 · P~X.=ai+t t ~ f3i7 ln p~I -t:.13itt+{3;AA+f3;rfT(A, t) -A] i 1 +f3iR(R(t) -A) +f3i.~s, i=l, ... , 3, (3) where the subscripts 1, 2, and 3 stand for Li, C11 and C2, and 8 COMPENDIUM OF TAX RESEARCH 3 ~ ai=l; I (3ij=O, V j ; ~ f3it=O; ~ /3iA=O; i=l i=l i=l i=l 3 3 3 ~ f3;T=O; ~ f3rn=O; ~ f3i .~= 0 and /3ij=f3ii' ¥i, j. i=l i= 1 i=l The homogeneous translog indirect utility function is a homo­ thetic utility function that implies that as total full wealth in­ creases, holding prices constant, all expenditures will increase proportionally.