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Unemployment insurance and the intertemporal substitution of consumption and labor supply
ARTHUR SNOW RONALD S. WARREN, JR. University of Georgia Athens, Georgia
Unemployment Insurance and the Intertemporal Substitution of Consumption and Labor Supply A central feature of modern business cycles is the procyclical pattern of consumption and labor supply. It has proved difficult, however, to construct a market-clearing model that is consistent with this fact and yet retains appealing assumptions and realistic implications. This paper demonstrates the possibility of positive comovements in contemporaneous consumption and labor supply in a model in which the unemployment insurance (UI) replacement ratio links future benefits to previous wage earnings. Specifically, in our model, an increase in the probability of future unemployment unambiguously increases present labor supply and may increase present consumption as well. One testable implication of our analysis is that the positive co-movement of consumption and labor supply is more pronounced in economies with an earnings-based UI system.
1. Introduction
One widely heralded feature of modern business cycles is the contemporaneous procyclical movement of consumption and labor supply. In contrast, the "new classical" market-clearing approach to macroeconomics predicts that consumption and labor supply move in opposite directions over the business cycle. This prediction rests on the intertemporal substitution of consumption and labor supply in response to random fluctuations in real interest rates or relative wage rates. Specifically, Lucas and Rapping (1969) argue that transitory increases in the present wage (interest) rate induce individuals to substitute present leisure (consumption) for future leisure (consumption), thereby increasing present labor supply (saving). This response arises because, in models with time-separable preferences, intertemporal substitution entails only an income effect on present spending so that, assuming the normality of goods and leisure, consumption (of goods) and labor supply move in opposite directions. Thus, as Diamond (1984, 51) observes, " . . . one has to work hard to develop a mechanism that explains why workers behave so differently in these two markets, consuming [and working] less" in response to the same shock. In any event, most empirical estimates Journal of Macroeconomics, Fall 1991, Vol. 13, No. 4, pp. 713-724 Copyright © 1991 by Louisiana State University Press 0164-0704/91/$1.50
713
Arthur Snow and Ronald S. Warren, Jr. of the i n t e r t e m p o r a l elasticity of substitution are too low to s u p p o r t the Lucas-Rapping explanation for the cyclical p a t t e r n o f c o n s u m p tion and labor supply. 1 Barro and King (1984), analyzing a time-separable m o d e l of intertemporal substitution, show that shocks to technology that cause a procyclical m o v e m e n t of the real wage can g e n e r a t e positive comovements in consumption and labor supply. However, as they note (p. 835), " . . . the real world significance of these types of shocks is u n c l e a r . " M o r e o v e r , the t r u e cyclical c h a r a c t e r o f the real wage remains an u n s e t t l e d issue. ~ Kydland and Prescott (1982) d e v e l o p a m o d e l with time-inseparable p r e f e r e n c e s in which c o n s u m p t i o n and labor supply m o v e in the same direction, p r o v i d e d that leisure c o n s u m e d in adjacent time periods is sufficiently substitutable. H o w e v e r , as Barro and King (1984, 820) remark, " . . . such d e p a r t u r e s from separability would m a t t e r significantly only for days or weeks, r a t h e r than for m o n t h s or years," and may, t h e r e f o r e , b e u n i m p o r t a n t for business cycle analysis. Moreover, Eichenbaum, Hansen, and Singleton (1988) cast d o u b t on the empirical usefulness of the K y d l a n d - P r e s c o t t specification of p r e f e r e n c e s for explaining the c o - m o v e m e n t s obs e r v e d in the data. W e retain the assumption of time-separable p r e f e r e n c e s and analyze a market-clearing m o d e l in which voluntary u n e m p l o y m e n t arises as a result of the c o m b i n e d effects of moral hazard and gove r n m e n t - p r o v i d e d u n e m p l o y m e n t insurance (UI). ~ Because private insurers and lenders c a n n o t effectively m o n i t o r work effort, the private sector provides workers with no opportunities to insure against the risk of low f u t u r e wages or to b o r r o w using risky f u t u r e wage income as collateral. H o w e v e r , if g o v e r n m e n t - p r o v i d e d UI benefits are high e n o u g h relative to the value o f the individual's best avail-
tPencavel (1986, 92) summarizes several studies by concluding that estimates of the intertemporal elasticity of substitution of male labor supply have a central tendency of 0.2. 2Blanchard and Fischer (1989, 19) conclude from a review of the relevant empirical research that the contemporaneous correlation between changes in the real wage and changes in real output " . . . is usually slightly positive but often statistically insignificant." 3Implicit contract theories, surveyed in Azariadis and Stiglitz (1983), efficiencywage theories, reviewed in Stiglitz (1987), and insider-outsider theory, developed in Lindbeck and Snower (1986), yield models in which labor markets may not clear, giving rise to involuntary unemployment. In contrast, models in the "'new classical" vein yield market-clearing equilibria wherein unemployment is voluntary. 714
Unemployment Insurance
able employment opportunity, then the individual chooses to be unemployed. W e show that, in this environment, consumption and labor supply may respond to business cycle shocks in a procyclical fashion when UI benefits vary directly with previous wage earnings. The model is set out in the next section. Our results are established in Section 3 and interpreted in Section 4. Section 5 offers concluding remarks.
2. The Model Our model is similar to the market-clearing framework employed by Barro and King (1984) but explicitly incorporates a moral hazard imperfection in the market for human capital. There are two periods, the present (period 1) and the future (period 2). The consumer is risk averse and has time-separable preferences for certain consumption represented by the von Neumann-Morgenstern utility function,
u(c , el) + au(c ,
(1)
where 8 E (0,1) is a fixed time discount factor and cj and e~ are, respectively, consumption and leisure in period j = 1, 2. W e assume that cj and ej are normal goods. The consumer is endowed with T > 0 units of leisure in both periods and with the certain, autonomous incomes al in period 1 and a2 in period 2. In period 1, the consumer is employed at the real wage w~ and can either save or borrow at the interest rate, r. Borrowing, however, requires non-human capital as collateral and thus cannot exceed a J ( 1 + r). Hence, the first-period budget constraint is ci + Wlel + s = al + w i T
and el~ T,
(2)
where s denotes first-period saving. In period 2 the consumer's endowment of income, denoted I, is the autonomous income a~ plus the income from saving, I = a~ + (1 + r)s >- O.
(3) 715
Arthur
S n o w a n d R o n a l d S. W a r r e n , Jr.
The consumer's second-period possible future states occurs. probability (1 - ~r), entails a chooses to be employed and straint
real wage depends on which of two State a, which occurs with known "high" wage, w a, and the consumer face the second-period budget con-
c2 + w°e2 = I + w a T
and e2<-T.
(4)
State b occurs with probability ~r, entails a "'low" wage, w a k > 0, and the consumer chooses to be unemployed and face the budget constraint -
-
cz=I+z
and
e2 = r ,
(5)
where z is the UI cash transfer, z = ~ + 13wI(T - e x ) ,
(6)
consisting of a lump sum ot >_ 0 and an amount proportional to firstperiod wage earnings that depends on the UI replacement ratio 13 >-0. We solve the consumer's problem by using backward induction. In the second period, once the state is revealed, the consumer chooses consumption and labor supply given the endowment of income, I. Thus, the optimal second-period choices depend on which state occurs and are functions of the consumption and labor supply chosen in the first period. Let v ( w , m) = m a x {u(c, e) l c + w e = m}
(7)
denote the indirect utSlity function which depends on the wage rate, w, and full income, m, and let e(w, m) represent the associated demand function for leisure. The budget constraint faced by the 716
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Insurance
consumer in the low-wage, u n e m p l o y e d state given in (5) can be equivalently expressed as c2 + Wbe2 = 1 + z + w b T - -
ms ,
(5')
where m b denotes full income in the u n e m p l o y e d state and w s is the shadow wage when the c o n s u m e r is unemployed. This shadow wage is a function of I + z and is implicitly defined by e ( w b, I + z + wbT) -- T .
(8)
Since e m p l o y m e n t is chosen in state a and u n e m p l o y m e n t in state b, it follows that v ( w a, m ~) > v ( w s, m s) > v [ w ~ - k, I + ( w ~ - k)T] ,
(9)
w h e r e m ~ - I + w a T denotes full income w h e n the c o n s u m e r is • employed. The c o n s u m e r chooses c~ and 21 to maximize J = u(cl, e~) + 8 E v ( w ' , m i) = u ( c l , el) + 8[*¢v(w b, m s) + (1 - ~r)v(w', ma)]
(10)
subject to the first-period b u d g e t constraint (2) and the relations (3), (4), (5'), (6), and (8). The first-order conditions for a maximum are
0 = Jc = uc - 8(1 + r)Ev~,(w', m~),
(11)
0 = J t = u t - 8w1(1 + r)Evm(W i, m') - 8"(r~WlVm(W b, m S ) ,
(12)
w h e r e the subscripts c, 2, and m denote partial differentiation with respect to ct, e~, and second-period income, respectively. 4
3. Results To analyze the effect on present consumption and labor supply caused by an increase in the probability of future u n e m p l o y m e n t , 4In deriving these conditions, we have substituted the constraints into the functions v(w ~, m ~) and used Roy's identity, v~ = -e(w b, mb)v~ = --Tvb=, so that, for example, avb/as = v~(1 + r) + (Tvb=+ v~)(awb/as) = v~(1 + r). 717
A r t h u r Snow and Ronald S. Warren, Jr. we use the first-order conditions, given in Appendix A, to derive the comparative statics equations
Hdcl/d~r = (-Uee + wluce)J~ - U~e~13WlVbm a
b
-- ~r132W~(1 + r)VmVmm +
~Z13w2x(1+
2
b
b
b
i
r ) ['ITVmm(V m -- V a ) -- V m E V m m ]
(13)
and
Hdhl/d~r
= -(-WlUcc
-Jr Uce)Jcl r -
Ucc~13Wl v b
b b b i + 6213w1(1 + r) 2 [T[Vmm(V m - V a) - vmEv~m]
(14)
where hi = T - el denotes first-period labor supply and H > 0 is the Jacobian determinant of the first-order conditions. We now state and prove our main result. THEOaEM. I f the replacement ratio is zero (~ = 0), then dhl/drr > 0 and dcl/dCr < O. I f the replacement ratio is positive ([3 > 0), then dhl/dcr > O, but dcl/dlr ~ O. b G Proof. We demonstrate in Appendix B that v,~ > Vm. Since a v~ > v~, the expression for J ~ given in Appendix A shows that J~, < 0. The normality of c and of e imply that the first terms in parentheses in the comparative statics equations (13) and (14) are positive. When 13 = 0, the remaining terms in these equations vanish and, therefore, dhl/d~r > 0 and dcl/d~r < O. When 13 > 0, the effect of an increase in unemployment risk on labor supply remains positive. However, the response of consumption is theoretically ambiguous but could be positive as well. To establish that present labor supply increases when the risk of unemployment rises, observe that the last term in brackets in (14) can be written b
17
a
--'ITVmmV m __
(1
a __ ~T)VmmVm
which is positive given risk aversion. Since ucc is negative with risk aversion, dht/dlr > O. The sign of dcl/d~r could be positive or negative when 13
718
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Insurance
> 0 since, as established above, the first term on the right-hand side of (13) is negative, b u t the last t e r m is positive. To complete the proof, we show that a class of utility functions exists for which an increase in u n e m p l o y m e n t risk leads to a rise or decline in consumption d e p e n d i n g on the magnitude of the Arrow-Pratt index of absolute risk aversion. The right-hand side of (13) can be rearranged to yield -J,~
~ml)m Uee + 5~3w~ b
l)rn -- 1)a
137rAb - (1 + r)[cr(A ~ - A h) - A ~]
-
uce~fSw~v~,
where A i ~--- - - l ) mif n / l ) im > 0 is the index of absolute risk aversion. If the utility function is intratemporally separable, so that Uce = 0, and exhibits constant absolute risk aversion, in the sense that A b = A a = A, t h e n dcl/d~r > (<) 0 if and only if b a
l)ml)rn u . > (<) - ~ w ~b - a (1 + r + 13~)A . Vm -- 1)rn
Ceteris paribus, the greater (less) than inequality holds, and therefore consumption and labor supply move in the same (opposite) direction, w h e n the degree of absolute risk aversion is relatively high (low). Q . E . D .
4. Interpretation The model implies that consumption and labor supply move in opposite directions w h e n u n e m p l o y m e n t insurance transfer payments are not tied to previous wage earnings (13 = 0). In this case, the first-order conditions (11) and (12) imply Ue/U c =Wl,
which, along with the first-period b u d g e t (2), d e t e r m i n e s p r e s e n t consumption and labor supply, given saving (s). As a result, intertemporal substitution entails only an income effect on p r e s e n t behavior and, thus, consumption and labor supply move in opposite directions in response to changes in the probability of unemployment.
719
Arthur Snow and Ronald S. Warren, Jr. When the replacement ratio is positive ([3 > 0), however, we find that
ut/uc
=
wz[1 + ~rf3vb,,/(1 + r)Ev•].
Thus, the present-period labor-leisure trade-off is distorted because the replacement ratio causes the opportunity cost of leisure to rise above the market real wage (w~), thereby inducing the intratemporal substitution of consumption for leisure. The effect of this distortion carries over to the comparative statics of increased unemployment risk. An increase in the probability of unemployment induces intertemporal substitution of consumption and labor supply and also increases the present-period labor-leisure distortion. The latter effect reinforces the tendency to substitute present labor for future labor but dampens, and may even reverse, the tendency to substitute future consumption for present consumption. As a result, when the probability of future unemployment increases, present labor supply increases and present consumption may increase as well. It is interesting to contrast these predictions with those obtained when the consumer is unable to save or borrow. In that event, all terms except the second on the right-hand sides of the comparative statics equations (13) and (14) vanish. Thus, when UI provides only a lump sum payment (13 = 0), neither consumption nor labor supply is affected by an increase in unemployment risk. Although labor supply increases with greater unemployment risk when the replacement ratio is positive, the effect on consumption depends solely on the sign of uct. By contrast, we have shown that, when the consumer is able to save or borrow, consumption and labor supply move in opposite directions when unemployment risk increases and UI provides only a lump sum transfer. Moreover, when the replacement ratio is positive, the sign of dcl/d~r depends also on the degree of absolute risk aversion. Consequently, if UI benefit formulae incorporate a positive replacement ratio, then consumption and labor supply can move in the same direction when the individual is able to save or borrow, while they would move in opposite directions were saving and borrowing opportunities unavailable.
5. Concluding Remarks The observed procyclical pattern of consumption and labor supply has proved difficult to reconcile with the predictions of "new
720
Unemployment Insurance classical" macroeconomic models which embody market-clearing and intertemporal substitution. Previous attempts at reconciliation have relied on shocks to technology, time-inseparability, or other, often ad hoc, driving forces whose real-world significance is at best an unsettled issue. In this paper, we have shown that consumption and labor supply can respond in a procyclical fashion if UI benefits vary directly with previous wage earnings. Thus, our model is both institutionally realistic and predictively consistent with an important stylized fact of the business cycle. One testable implication of the model is that the positive co-movement of consumption and labor supply is more pronounced in economies with an earnings-based UI system. Receive& January 1990 Final version: November 1990
References
Azariadis, Costas, and Joseph E. Stiglitz. "'Implicit Contracts and Fixed Price Equilibria.'" Quarterly Journal of Economics 98 (Supplement 1983): 1-22. Barro, Robert J., and Robert G. King. "Time-Separable Preferences and Intertemporal-Substitution Models of Business Cycles." Quarterly Journal of Economics 99 (November 1984): 817-39. Blanchard, Olivier J., and Stanley Fischer. Lectures on Macroeconomics. Cambridge: The MIT Press, 1989. Diamond, Peter A. A Search-Equilibrium Approach to the Micro Foundations of Macroeconomics. Cambridge: The MIT Press, 1984. Eichenbaum, Martin, Lars Hansen, and Kenneth Singleton. "A Time Series Analysis of Representative Agent Models of Consumption and Leisure Choice under Uncertainty." Quarterly Journal of Economics 103 (February 1988): 51-78. Kydland, Finn, and Edward Prescott. "Time to Build and Aggregate Fluctuations." Econometrica 50 (November 1982): 1345-70. Lindbeck, Assar, and. Dennis J. Snower. "Wage Setting, Unemployment, and Insider-Outsider Relations." American Economic Review 76 (May 1986): 235-39. Lucas, Robert E., and Leonard Rapping. "Real Wages, Employment and Inflation." Journal of Political Economy 77 (September-October 1969): 721-54. Pencavel, John. "Labor Supply of Men: A Survey." In Handbook
721
A r t h u r S n o w and RonaId S. W a r r e n , Jr. o f L a b o r Economics, edited by Orley Ashenfelter and Richard
Layard, 3-102. Amsterdam: North-Holland, 1986. Stiglitz, Joseph E. "The Causes and Consequences of the Dependence of Quality on Price." Journal o f Economic Literature 25 (March 1987): 1-48.
Appendix A We derive the comparative statics equations (13) and (14) by differentiating the first-order conditions (11) and (12) and using Cramer's rule to solve for Hdcl/d~r = - J e e J ~ + J~eJe~
and Hdhl/d'rr = JccJe= - JceJ~ ,
where hi --- T - /?~ is first-period labor supply and H > 0 is the Jacobian determinant of the system. Substituting the relations Jcc = u~c + 8(1 + r)ZEvimm, Jet = nee + ~w12(1 + r)2Evimm + 2~'n'[3w~(1 + r)vbm + ~ Jce= U~e + 8w1(1 + r)2Evimm + ~'tr~Wl(1 +
2Wltgmm 2 b ,
r)vbm,
J ~ = -~(1 + r)(v~ - v~), Je~ = wlJ~, - ~f3WlVbm,
and rearranging terms, we arrive at (13) and (14).
Appendix B We demonstrate that the marginal utility of income is higher in the low-wage state than in the high-wage state. Although this conclusion is intuitive in the case of a risk averter, a demonstration is in order since the price of leisure as well as the level of income
722
Unemployment Insurance
differ in the two states. Let {w(t) m(t)} be a differentiable path from (wb, m b) at t = tb to (w~, m ~) at t = t,. We wish to show that a __ b ~I ta 0 > tgra t~rn =
Lb
OVm/Ot d t
=£° ~
(VmmOm/Ot + vm~Ow/Ot)dt a
= -
[VmtmOw/Ot + v m J e a w / o t - Om/Ot)]dt.
(B1)
The last step uses V~w = --tVm~ -- t~V,,, which is obtained by differentiating Roy's identity, t = - v d v m , with respect to m. The utility difference between employment and unemployment is, using
(9), 0 < v(w", m") - v ( w b, m b) =
(VmOm/Ot + vmOw/Ot)dt
= -
Vm(eOw/ot-
Om/Ot)dt. (B2)
Similarly, using (8), the difference in leisure demand is 0 > e(w", m") - e(w ~, m ~) =
(imam~Or + ewOw/ot)dt
~
a
=
-
[~wOwlOt
e~(eaw/at
-
am/at)]dt,
(B3)
where ~ is the compensated demand function for leisure associated with u. The integral differences in (B1)-(B3) are path-independent and we can select a path such that the integrands in (B2) and (B3) are uniformly signed; that is, t O w ~ a t - Om/Ot < 0
(B4)
723
Arthur Snow and Ronald S. Warren, Jr. and
ewdwlOt < em(edw/at
-
-
am/Ot).
(B5)
Using (B4) and risk aversion (Vmm < 0), the second term of the last integral in (B1) is positive. Since e , > 0 and ew < 0, (B5) implies that the first term is also positive. Hence, the inequality in (B1) holds.
724
Unemployment Insurance and the Intertemporal Substitution of Consumption and Labor Supply A central feature of modern business cycles is the procyclical pattern of consumption and labor supply. It has proved difficult, however, to construct a market-clearing model that is consistent with this fact and yet retains appealing assumptions and realistic implications. This paper demonstrates the possibility of positive comovements in contemporaneous consumption and labor supply in a model in which the unemployment insurance (UI) replacement ratio links future benefits to previous wage earnings. Specifically, in our model, an increase in the probability of future unemployment unambiguously increases present labor supply and may increase present consumption as well. One testable implication of our analysis is that the positive co-movement of consumption and labor supply is more pronounced in economies with an earnings-based UI system.
1. Introduction
One widely heralded feature of modern business cycles is the contemporaneous procyclical movement of consumption and labor supply. In contrast, the "new classical" market-clearing approach to macroeconomics predicts that consumption and labor supply move in opposite directions over the business cycle. This prediction rests on the intertemporal substitution of consumption and labor supply in response to random fluctuations in real interest rates or relative wage rates. Specifically, Lucas and Rapping (1969) argue that transitory increases in the present wage (interest) rate induce individuals to substitute present leisure (consumption) for future leisure (consumption), thereby increasing present labor supply (saving). This response arises because, in models with time-separable preferences, intertemporal substitution entails only an income effect on present spending so that, assuming the normality of goods and leisure, consumption (of goods) and labor supply move in opposite directions. Thus, as Diamond (1984, 51) observes, " . . . one has to work hard to develop a mechanism that explains why workers behave so differently in these two markets, consuming [and working] less" in response to the same shock. In any event, most empirical estimates Journal of Macroeconomics, Fall 1991, Vol. 13, No. 4, pp. 713-724 Copyright © 1991 by Louisiana State University Press 0164-0704/91/$1.50
713
Arthur Snow and Ronald S. Warren, Jr. of the i n t e r t e m p o r a l elasticity of substitution are too low to s u p p o r t the Lucas-Rapping explanation for the cyclical p a t t e r n o f c o n s u m p tion and labor supply. 1 Barro and King (1984), analyzing a time-separable m o d e l of intertemporal substitution, show that shocks to technology that cause a procyclical m o v e m e n t of the real wage can g e n e r a t e positive comovements in consumption and labor supply. However, as they note (p. 835), " . . . the real world significance of these types of shocks is u n c l e a r . " M o r e o v e r , the t r u e cyclical c h a r a c t e r o f the real wage remains an u n s e t t l e d issue. ~ Kydland and Prescott (1982) d e v e l o p a m o d e l with time-inseparable p r e f e r e n c e s in which c o n s u m p t i o n and labor supply m o v e in the same direction, p r o v i d e d that leisure c o n s u m e d in adjacent time periods is sufficiently substitutable. H o w e v e r , as Barro and King (1984, 820) remark, " . . . such d e p a r t u r e s from separability would m a t t e r significantly only for days or weeks, r a t h e r than for m o n t h s or years," and may, t h e r e f o r e , b e u n i m p o r t a n t for business cycle analysis. Moreover, Eichenbaum, Hansen, and Singleton (1988) cast d o u b t on the empirical usefulness of the K y d l a n d - P r e s c o t t specification of p r e f e r e n c e s for explaining the c o - m o v e m e n t s obs e r v e d in the data. W e retain the assumption of time-separable p r e f e r e n c e s and analyze a market-clearing m o d e l in which voluntary u n e m p l o y m e n t arises as a result of the c o m b i n e d effects of moral hazard and gove r n m e n t - p r o v i d e d u n e m p l o y m e n t insurance (UI). ~ Because private insurers and lenders c a n n o t effectively m o n i t o r work effort, the private sector provides workers with no opportunities to insure against the risk of low f u t u r e wages or to b o r r o w using risky f u t u r e wage income as collateral. H o w e v e r , if g o v e r n m e n t - p r o v i d e d UI benefits are high e n o u g h relative to the value o f the individual's best avail-
tPencavel (1986, 92) summarizes several studies by concluding that estimates of the intertemporal elasticity of substitution of male labor supply have a central tendency of 0.2. 2Blanchard and Fischer (1989, 19) conclude from a review of the relevant empirical research that the contemporaneous correlation between changes in the real wage and changes in real output " . . . is usually slightly positive but often statistically insignificant." 3Implicit contract theories, surveyed in Azariadis and Stiglitz (1983), efficiencywage theories, reviewed in Stiglitz (1987), and insider-outsider theory, developed in Lindbeck and Snower (1986), yield models in which labor markets may not clear, giving rise to involuntary unemployment. In contrast, models in the "'new classical" vein yield market-clearing equilibria wherein unemployment is voluntary. 714
Unemployment Insurance
able employment opportunity, then the individual chooses to be unemployed. W e show that, in this environment, consumption and labor supply may respond to business cycle shocks in a procyclical fashion when UI benefits vary directly with previous wage earnings. The model is set out in the next section. Our results are established in Section 3 and interpreted in Section 4. Section 5 offers concluding remarks.
2. The Model Our model is similar to the market-clearing framework employed by Barro and King (1984) but explicitly incorporates a moral hazard imperfection in the market for human capital. There are two periods, the present (period 1) and the future (period 2). The consumer is risk averse and has time-separable preferences for certain consumption represented by the von Neumann-Morgenstern utility function,
u(c , el) + au(c ,
(1)
where 8 E (0,1) is a fixed time discount factor and cj and e~ are, respectively, consumption and leisure in period j = 1, 2. W e assume that cj and ej are normal goods. The consumer is endowed with T > 0 units of leisure in both periods and with the certain, autonomous incomes al in period 1 and a2 in period 2. In period 1, the consumer is employed at the real wage w~ and can either save or borrow at the interest rate, r. Borrowing, however, requires non-human capital as collateral and thus cannot exceed a J ( 1 + r). Hence, the first-period budget constraint is ci + Wlel + s = al + w i T
and el~ T,
(2)
where s denotes first-period saving. In period 2 the consumer's endowment of income, denoted I, is the autonomous income a~ plus the income from saving, I = a~ + (1 + r)s >- O.
(3) 715
Arthur
S n o w a n d R o n a l d S. W a r r e n , Jr.
The consumer's second-period possible future states occurs. probability (1 - ~r), entails a chooses to be employed and straint
real wage depends on which of two State a, which occurs with known "high" wage, w a, and the consumer face the second-period budget con-
c2 + w°e2 = I + w a T
and e2<-T.
(4)
State b occurs with probability ~r, entails a "'low" wage, w a k > 0, and the consumer chooses to be unemployed and face the budget constraint -
-
cz=I+z
and
e2 = r ,
(5)
where z is the UI cash transfer, z = ~ + 13wI(T - e x ) ,
(6)
consisting of a lump sum ot >_ 0 and an amount proportional to firstperiod wage earnings that depends on the UI replacement ratio 13 >-0. We solve the consumer's problem by using backward induction. In the second period, once the state is revealed, the consumer chooses consumption and labor supply given the endowment of income, I. Thus, the optimal second-period choices depend on which state occurs and are functions of the consumption and labor supply chosen in the first period. Let v ( w , m) = m a x {u(c, e) l c + w e = m}
(7)
denote the indirect utSlity function which depends on the wage rate, w, and full income, m, and let e(w, m) represent the associated demand function for leisure. The budget constraint faced by the 716
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Insurance
consumer in the low-wage, u n e m p l o y e d state given in (5) can be equivalently expressed as c2 + Wbe2 = 1 + z + w b T - -
ms ,
(5')
where m b denotes full income in the u n e m p l o y e d state and w s is the shadow wage when the c o n s u m e r is unemployed. This shadow wage is a function of I + z and is implicitly defined by e ( w b, I + z + wbT) -- T .
(8)
Since e m p l o y m e n t is chosen in state a and u n e m p l o y m e n t in state b, it follows that v ( w a, m ~) > v ( w s, m s) > v [ w ~ - k, I + ( w ~ - k)T] ,
(9)
w h e r e m ~ - I + w a T denotes full income w h e n the c o n s u m e r is • employed. The c o n s u m e r chooses c~ and 21 to maximize J = u(cl, e~) + 8 E v ( w ' , m i) = u ( c l , el) + 8[*¢v(w b, m s) + (1 - ~r)v(w', ma)]
(10)
subject to the first-period b u d g e t constraint (2) and the relations (3), (4), (5'), (6), and (8). The first-order conditions for a maximum are
0 = Jc = uc - 8(1 + r)Ev~,(w', m~),
(11)
0 = J t = u t - 8w1(1 + r)Evm(W i, m') - 8"(r~WlVm(W b, m S ) ,
(12)
w h e r e the subscripts c, 2, and m denote partial differentiation with respect to ct, e~, and second-period income, respectively. 4
3. Results To analyze the effect on present consumption and labor supply caused by an increase in the probability of future u n e m p l o y m e n t , 4In deriving these conditions, we have substituted the constraints into the functions v(w ~, m ~) and used Roy's identity, v~ = -e(w b, mb)v~ = --Tvb=, so that, for example, avb/as = v~(1 + r) + (Tvb=+ v~)(awb/as) = v~(1 + r). 717
A r t h u r Snow and Ronald S. Warren, Jr. we use the first-order conditions, given in Appendix A, to derive the comparative statics equations
Hdcl/d~r = (-Uee + wluce)J~ - U~e~13WlVbm a
b
-- ~r132W~(1 + r)VmVmm +
~Z13w2x(1+
2
b
b
b
i
r ) ['ITVmm(V m -- V a ) -- V m E V m m ]
(13)
and
Hdhl/d~r
= -(-WlUcc
-Jr Uce)Jcl r -
Ucc~13Wl v b
b b b i + 6213w1(1 + r) 2 [T[Vmm(V m - V a) - vmEv~m]
(14)
where hi = T - el denotes first-period labor supply and H > 0 is the Jacobian determinant of the first-order conditions. We now state and prove our main result. THEOaEM. I f the replacement ratio is zero (~ = 0), then dhl/drr > 0 and dcl/dCr < O. I f the replacement ratio is positive ([3 > 0), then dhl/dcr > O, but dcl/dlr ~ O. b G Proof. We demonstrate in Appendix B that v,~ > Vm. Since a v~ > v~, the expression for J ~ given in Appendix A shows that J~, < 0. The normality of c and of e imply that the first terms in parentheses in the comparative statics equations (13) and (14) are positive. When 13 = 0, the remaining terms in these equations vanish and, therefore, dhl/d~r > 0 and dcl/d~r < O. When 13 > 0, the effect of an increase in unemployment risk on labor supply remains positive. However, the response of consumption is theoretically ambiguous but could be positive as well. To establish that present labor supply increases when the risk of unemployment rises, observe that the last term in brackets in (14) can be written b
17
a
--'ITVmmV m __
(1
a __ ~T)VmmVm
which is positive given risk aversion. Since ucc is negative with risk aversion, dht/dlr > O. The sign of dcl/d~r could be positive or negative when 13
718
Unemployment
Insurance
> 0 since, as established above, the first term on the right-hand side of (13) is negative, b u t the last t e r m is positive. To complete the proof, we show that a class of utility functions exists for which an increase in u n e m p l o y m e n t risk leads to a rise or decline in consumption d e p e n d i n g on the magnitude of the Arrow-Pratt index of absolute risk aversion. The right-hand side of (13) can be rearranged to yield -J,~
~ml)m Uee + 5~3w~ b
l)rn -- 1)a
137rAb - (1 + r)[cr(A ~ - A h) - A ~]
-
uce~fSw~v~,
where A i ~--- - - l ) mif n / l ) im > 0 is the index of absolute risk aversion. If the utility function is intratemporally separable, so that Uce = 0, and exhibits constant absolute risk aversion, in the sense that A b = A a = A, t h e n dcl/d~r > (<) 0 if and only if b a
l)ml)rn u . > (<) - ~ w ~b - a (1 + r + 13~)A . Vm -- 1)rn
Ceteris paribus, the greater (less) than inequality holds, and therefore consumption and labor supply move in the same (opposite) direction, w h e n the degree of absolute risk aversion is relatively high (low). Q . E . D .
4. Interpretation The model implies that consumption and labor supply move in opposite directions w h e n u n e m p l o y m e n t insurance transfer payments are not tied to previous wage earnings (13 = 0). In this case, the first-order conditions (11) and (12) imply Ue/U c =Wl,
which, along with the first-period b u d g e t (2), d e t e r m i n e s p r e s e n t consumption and labor supply, given saving (s). As a result, intertemporal substitution entails only an income effect on p r e s e n t behavior and, thus, consumption and labor supply move in opposite directions in response to changes in the probability of unemployment.
719
Arthur Snow and Ronald S. Warren, Jr. When the replacement ratio is positive ([3 > 0), however, we find that
ut/uc
=
wz[1 + ~rf3vb,,/(1 + r)Ev•].
Thus, the present-period labor-leisure trade-off is distorted because the replacement ratio causes the opportunity cost of leisure to rise above the market real wage (w~), thereby inducing the intratemporal substitution of consumption for leisure. The effect of this distortion carries over to the comparative statics of increased unemployment risk. An increase in the probability of unemployment induces intertemporal substitution of consumption and labor supply and also increases the present-period labor-leisure distortion. The latter effect reinforces the tendency to substitute present labor for future labor but dampens, and may even reverse, the tendency to substitute future consumption for present consumption. As a result, when the probability of future unemployment increases, present labor supply increases and present consumption may increase as well. It is interesting to contrast these predictions with those obtained when the consumer is unable to save or borrow. In that event, all terms except the second on the right-hand sides of the comparative statics equations (13) and (14) vanish. Thus, when UI provides only a lump sum payment (13 = 0), neither consumption nor labor supply is affected by an increase in unemployment risk. Although labor supply increases with greater unemployment risk when the replacement ratio is positive, the effect on consumption depends solely on the sign of uct. By contrast, we have shown that, when the consumer is able to save or borrow, consumption and labor supply move in opposite directions when unemployment risk increases and UI provides only a lump sum transfer. Moreover, when the replacement ratio is positive, the sign of dcl/d~r depends also on the degree of absolute risk aversion. Consequently, if UI benefit formulae incorporate a positive replacement ratio, then consumption and labor supply can move in the same direction when the individual is able to save or borrow, while they would move in opposite directions were saving and borrowing opportunities unavailable.
5. Concluding Remarks The observed procyclical pattern of consumption and labor supply has proved difficult to reconcile with the predictions of "new
720
Unemployment Insurance classical" macroeconomic models which embody market-clearing and intertemporal substitution. Previous attempts at reconciliation have relied on shocks to technology, time-inseparability, or other, often ad hoc, driving forces whose real-world significance is at best an unsettled issue. In this paper, we have shown that consumption and labor supply can respond in a procyclical fashion if UI benefits vary directly with previous wage earnings. Thus, our model is both institutionally realistic and predictively consistent with an important stylized fact of the business cycle. One testable implication of the model is that the positive co-movement of consumption and labor supply is more pronounced in economies with an earnings-based UI system. Receive& January 1990 Final version: November 1990
References
Azariadis, Costas, and Joseph E. Stiglitz. "'Implicit Contracts and Fixed Price Equilibria.'" Quarterly Journal of Economics 98 (Supplement 1983): 1-22. Barro, Robert J., and Robert G. King. "Time-Separable Preferences and Intertemporal-Substitution Models of Business Cycles." Quarterly Journal of Economics 99 (November 1984): 817-39. Blanchard, Olivier J., and Stanley Fischer. Lectures on Macroeconomics. Cambridge: The MIT Press, 1989. Diamond, Peter A. A Search-Equilibrium Approach to the Micro Foundations of Macroeconomics. Cambridge: The MIT Press, 1984. Eichenbaum, Martin, Lars Hansen, and Kenneth Singleton. "A Time Series Analysis of Representative Agent Models of Consumption and Leisure Choice under Uncertainty." Quarterly Journal of Economics 103 (February 1988): 51-78. Kydland, Finn, and Edward Prescott. "Time to Build and Aggregate Fluctuations." Econometrica 50 (November 1982): 1345-70. Lindbeck, Assar, and. Dennis J. Snower. "Wage Setting, Unemployment, and Insider-Outsider Relations." American Economic Review 76 (May 1986): 235-39. Lucas, Robert E., and Leonard Rapping. "Real Wages, Employment and Inflation." Journal of Political Economy 77 (September-October 1969): 721-54. Pencavel, John. "Labor Supply of Men: A Survey." In Handbook
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A r t h u r S n o w and RonaId S. W a r r e n , Jr. o f L a b o r Economics, edited by Orley Ashenfelter and Richard
Layard, 3-102. Amsterdam: North-Holland, 1986. Stiglitz, Joseph E. "The Causes and Consequences of the Dependence of Quality on Price." Journal o f Economic Literature 25 (March 1987): 1-48.
Appendix A We derive the comparative statics equations (13) and (14) by differentiating the first-order conditions (11) and (12) and using Cramer's rule to solve for Hdcl/d~r = - J e e J ~ + J~eJe~
and Hdhl/d'rr = JccJe= - JceJ~ ,
where hi --- T - /?~ is first-period labor supply and H > 0 is the Jacobian determinant of the system. Substituting the relations Jcc = u~c + 8(1 + r)ZEvimm, Jet = nee + ~w12(1 + r)2Evimm + 2~'n'[3w~(1 + r)vbm + ~ Jce= U~e + 8w1(1 + r)2Evimm + ~'tr~Wl(1 +
2Wltgmm 2 b ,
r)vbm,
J ~ = -~(1 + r)(v~ - v~), Je~ = wlJ~, - ~f3WlVbm,
and rearranging terms, we arrive at (13) and (14).
Appendix B We demonstrate that the marginal utility of income is higher in the low-wage state than in the high-wage state. Although this conclusion is intuitive in the case of a risk averter, a demonstration is in order since the price of leisure as well as the level of income
722
Unemployment Insurance
differ in the two states. Let {w(t) m(t)} be a differentiable path from (wb, m b) at t = tb to (w~, m ~) at t = t,. We wish to show that a __ b ~I ta 0 > tgra t~rn =
Lb
OVm/Ot d t
=£° ~
(VmmOm/Ot + vm~Ow/Ot)dt a
= -
[VmtmOw/Ot + v m J e a w / o t - Om/Ot)]dt.
(B1)
The last step uses V~w = --tVm~ -- t~V,,, which is obtained by differentiating Roy's identity, t = - v d v m , with respect to m. The utility difference between employment and unemployment is, using
(9), 0 < v(w", m") - v ( w b, m b) =
(VmOm/Ot + vmOw/Ot)dt
= -
Vm(eOw/ot-
Om/Ot)dt. (B2)
Similarly, using (8), the difference in leisure demand is 0 > e(w", m") - e(w ~, m ~) =
(imam~Or + ewOw/ot)dt
~
a
=
-
[~wOwlOt
e~(eaw/at
-
am/at)]dt,
(B3)
where ~ is the compensated demand function for leisure associated with u. The integral differences in (B1)-(B3) are path-independent and we can select a path such that the integrands in (B2) and (B3) are uniformly signed; that is, t O w ~ a t - Om/Ot < 0
(B4)
723
Arthur Snow and Ronald S. Warren, Jr. and
ewdwlOt < em(edw/at
-
-
am/Ot).
(B5)
Using (B4) and risk aversion (Vmm < 0), the second term of the last integral in (B1) is positive. Since e , > 0 and ew < 0, (B5) implies that the first term is also positive. Hence, the inequality in (B1) holds.
724