Unii,~rsity of I xhester, Rochester, NY 14617, USA. Received ... is als0 shown that optimal allocations involve lotteries over employment and eotl!Umption.
Journal of Monetary Economics 21 (1988) 3-16. North-Holla.Old
INDIVlSiiii..E LABOR, LOITERIES AND EQUIUBRIUM Richard ROGERSON* Unii,~rsity
of I xhester, Rochester, NY 14617, USA
Received January 1987, final version received June 19'l7 This paper considers an economy where labor is indivisible and agents are identical. Although the discontinuity in labor supply at the individual level disappears as a result of ag,gregation, it is shown that indivisible labor bas strong consequences for the aggregate ':>ehavior of •he economy. It is als0 shown that optimal allocations involve lotteries over employment and eotl!Umption.
l, Introduction During the last decade general equilibrium theory has become an increasingly common framework in which to study the aggregate properties cf economies [see, e.g., Lucas (1981)). In its original form [see Debreu (1959)] general equilibrium theory was developed in the context of convex economic enviror.ments, but has since been extended to non-convex environments [~ Aumann (1966) and Mas-Colell (1977)). The concern of these authors was I.he existence of equilibrium. They show that if there is a continuum of agents, then equilibrium exists even with non-convexities at the indivi::ntions considered by the firsi weifare theorem. Re.call that the standard nethod of proof for the first welfare theorem involves an argument that if individuals prefer an allocation to their individual allocation then it -.n•st cost too much relative to their budget or else they would have purchased ii.. Th.le; argument does not hold here because the allocation involving lotteries is not vi.,wed as a feasib!e one by consumers with consumption set X. l-lr)wever, the result is still troubling because it suggests tl1at not all gains to :rarie are being realized evGtl though apparently all m.arkets are in operation. In h; remainder of this paper it is shown how the consumption set X can be ""dified so that allocations like the one described above caZl be obtained as a ~mpetitive allocation.
4. F.quilibrimn with lotteries
In ttis section the consumption .sei i:; cxp.w.!cm, c,.P
subject to c=wct>+r,
c;:.::: 0, 0::;; c/>::;; 1.
Note that by the strict concavity of u( ·) this problem has a unique solution. Since all agents are identical it follows that c and c/> are independent of t. Hence, finding an equilibrium now reduces to finding a list ( c, c/>, K, N, r, w) such that (i) ( c, ct>) solves problem (P-2). (ii) (K, N) solves profit maximization problem. (iii) ct>=N,K=l,c=F(K,N). This is identical to the equilibrium one would obtain for an economv with technology /(K, N), with one agent whose utility is specified by u(c)- mn v.rith consumption set ·
X= {, C,
.p
subject to
c ::;;f(l, if>), This is the social planning problem for E which maximizes utility. Because this economy now appears identical to one without any non-convexities, the standard results on equivalence of competitive and optimal allocations can be applied [see, e.g., Negishi (1960)]. Proposition I. If {c*, c/>*, K*, N*, w*, r*) is a competitive equilibrium for then ( c*, c/>*) is the solution to problem ( P-3).
E,
R. Rogerson, lndii·!sible labor, lotteries and equili/J.rium
11
Proposition 2. If ( c*, 4>*) is the solution to problem ( P-3), then tr.ere exists K*,N*, w*, and r* such that (c*, -p•, K*, N*, w*, r*) is a competitive equilibrium for E. Since (P-2) is a strictly concave programming problem, the above two results imply the existence of a unique equilibrium. One of the reasons for adding lotteries to the consumption set was the potential gain in welfare. In essence, making labor indh-isfo1e creates a barrier to trade and the introduction of lotteries is one way to overcome prut of this barrier. It should be noted that adding lotteries of the type considerecf here to an environment similar to E but without the indivisibility in labor would have no effect on equilibrium. Finally, in Example 1 it was demonstrated that an allocation involving lotteries Pareto-dominated the equilibrium allocation for E. That this result is general should be clear, but a formal statement is:
Proposition 3. If (c*(t), n*(t), k*(t), K*, N*) is an equilibrium allocation/or E and (c1(t), k 1(t), c2 (t), k 2 (t), 4i(t), K, N) is an equilibriw1< allocation for E, then u( c(t )) - n(t )ms ~(t )[ u( c1(t)) - m]
+ (1- ~(t )}u( c2 (t )),
· - ·· ·, with strict inequality if n( t) is not constant for all t. Proof.
Define ct>=
[l n*(t)dt and
10
Define an allocation for
c= 110 c*(t)dt.
E as follows:
c1 (t)=c2(t)=c, allt, 4>(t) =;j;, all t, k1(t)=k2(t)=l, allt, K=K*,
N=N*.
This allocation is clearly feasible, and by definition of an equilibrium for E
u(c*(t))-n*(t)m= [1(u(c*(t))-r.*(t)m)dt
·o
s u(fc*(t)dt
)-m f n*(t)dt= u(c)- ~m,
12
R. Rogerson, !ndiuisib!c labor, lotteries and equilif:r;iJm
where the inequality follows from Jensen's ineqi.•ality and is strict if c(t) is not constant. Note that the right-hand side is simply the utility resulting from one particular feasible allocation in E. By Proposition 1, the equilibrium for E must result in utility at least thls large. II 5. Stochastic environments itDc1 computation
The previous analysis continues to hold for stochastic environments. Suppose that there is a random variable s taking values in a finite subset S of RN. Let s 1 index the realizations of s and P; be the probability that s = s;. In state i assume that preferences are given by
and te.chnology is given by
/(K, N, s;), where these functions are assumed to have the same properties as before for each value of s;. If lottc1ic:; are intrcduced ~!:en optimal allocations ac = - -In W +constant. 1 -a As cru1 b~ seen by comparison of these two expressions, the case where labor supply is indivisible pro(~u •. ,~s ? sl.opr: which is larger. In fact, in this example the heterogl:'neity has no impact on f,1e elasticity of labor supply. Note that the elasticity in the indivisible case is found by setting fJ equal to one in the corresponding expression for the d ivisfole labor case. Hence, the same result concernin~ linearity holds in this case. Care should be taken not to misinterpret this result. It is derived in the context of a static deterministic environment. Computing equilibrium allocations for dynamic stochastic environments (like that of Hansen) is ultimately the object of interest, but is too difficult a task to be undertaken here for the case of hetetogeneous cor.sumer:,. If there were no lotteries in !.he economy with indivisibie labor, tnen each agent simply decides whether or not to work. BecauSt the m;'s differ across agents, this decision will differ across agents. In particular, an agent supplies labor if W" - m; > 0 and doesn't supply labor if W"' - m; < 0. When equality holds the individual is indillerent. In this economy each individual has a different reservation wage which causes them to enter the market, given by w; = m}I"'. Note that in the economy with lotteries individuals always have the choice of '/>; = 1 or cf>1 = 0, so implicitly it follows that lotteries are improving welfare.
16
R. Rogerson, Indivisible luhor, lotteries and equilibrium
7. Conclusion This paper has analyzed the problem of indivisible labor in an economy with identical individuals. The main conclusion. of this paper is that nf n-convexities at the individual level may have important aggregate effects even if there are a large number of individuals. In the case studied here, the aggregate economy behaves as if there were no non-convexities but all individuals have preferences which are linear in leisure even though no individual in the economy has such preferences.
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