Underemployment with liquidity-constrained multi-period firms

JOURNAL OF ECONOMIC THEORY 4, 81-98 (1988)

Underemployment wit Liquidity-Constrained Multi-

JOHN LEACH

irms*

McMaster University, Hamiltoq Canada 58s 4M4

Received December 3, 1985; revised March 13, 1987

S. Grossman and 0. D. Hart [Amer. Econ. Rev. 71 (1981) 301-3071 have shown that underemployment can occur if firms are risk averse and have better information than workers. C. Kahn and J. Scheinkman [J. Econ. Theory 35 (19851, 343-3657 replace the assumption of risk aversion with the more basic assumption of limited liability, but they obtain risk-averse behaviour only when the liquidity constraint is currently binding. A T-period model is developed here to show that firms will also display risk aversion if the liquidity constraint might become binding in the future. Journal of Economic Literature Classification Numbers: 023. 026. 0 1988 Academic Press. Inc.

I. INTRODUCTIQN

Grossman and Hart [4] have shown that underemployment can occur if risk-averse firms contract with workers who are imperfectly informed about the current productivity of labour.’ That is, workers can be une when productivity is low even though their marginal product ex value of leisure. The contract requires low employment in these states to eliminate the firm’s incentive to behave as if productivity were low when it is in fact high, thereby significantly reducing the wage bill while incurring a comparatively small reduction in output. Kahn and Scheinkman [5f replace the assumption of risk aversion (on the part of firms) with the more basic assumption of limited liability. Their model is purely static: workers and firms contract for production in a single period prior to the revelation of the state of nature. A primary determinant of the structure of the contract is that workers must ensure that the contract does not offer, in any state of nature, more than the firm can deliver. This constraint causes a risk-neutral firm to behave as if it were risk averse. A limitation of this argument is that it demonstrates only that firms facing the threat of

* I thank Richard Amott, Peter McCabe, Martin Osborne, and an anonymous referee foor their comments. 1 am solely responsible for any remaining errors.

i See also Azariadis [ 11, Chari [2], and Green and Kahn 131.

@922-0531188 $3.00 Copyrigh: 0 1988 by Academic Press, Inc.

Ail rights of reproduckm in any form reserved.

82 JOHN LEACH

imminent bankruptcy will behave in a risk-averse fashion. Firms with current wealth so large that they will remain solvent under any state of nature will continue to be risk neutral. As the latter group is likely to form by far the greater part of the economy, the amount of underemployment in low productivity states could be quite small and would be confined to a readily identifiable group of firms.

A T-period model is developed below. It shows that firms will be risk averse, not only when the liquidity constraint is currently binding, but also when there is some possibility of it being binding in the future. Under- employment can be widespread even if no firms are currently liquidity constrained.

This result is developed in two stages. The lirst describes the behaviour of a risk-neutral, liquidity-constrained firm confronted with a convex set of feasible state-contingent earnings streams. The second shows that this set of earnings streams can be generated from the set of feasible employment contracts. Firms that are currently liquidity constrained, or that might be constrained in the future, will sometimes enter into contracts which call for underemployment in the adverse state.

II. INTERTEMPORAL OPTIMIZATION WITH LIQUIDITY CONSTRAINTS

Consider a firm which maximizes expected profits over the T periods of its operation. In each period the firm can be in either of two equally likely states. The firm incurs a loss y in state 0 and earns a profit c in state 1. The pair (y, c) constitutes the lirm’s earnings stream, which it chooses in each period prior to the realization of the state. The feasible earnings streams form a convex set with the boundary

c = thYI> (1)

where c and y are real numbers, and $ is an increasing and concave function of class C’. It will be shown, in Sections III-V, that each point on this curve corresponds to a single-period employment contract. These contracts maximize c-ky for some k, subject to an expected utility constraint and (in the case of asymmetric information) a pair of truth-telling conditions. Here, k is the firm’s marginal rate of substitution between earnings in the good and bad states. It will be shown that full information contracts display both complete risk-sharing and efficient production. These properties also characterize asymmetric information contracts when k is sufficiently near 1, but they are lost under both large and small values of k. For large k (and hence small y), the contract will generate underemployment in the bad state. The argument made in this section is that firms

LIQUIDITY-CONSTRAINED FIRMS 83

with low levels of liquidity choose contracts under which Y is ““small” SO that underemployment does occur in the bad state.

It is assumed that the function I,!I is fixed throughout the T periods of t

firm’s horizon. This would occur if the firm employed a new labour force i every period and thus could offer only single-period contracts. Nevertheless, this is a significant restriction: it rules out all of the potential benefits from long-term contracting.

The firm brings wealth z, into period t (the periods being numbered from the end of the program). The firm’s initial wealth zT is given, and 2, (t < 9) IS z,+ I adjusted for the firm’s earnings in period t + 1. In each period the firm faces the liquidity constraint

YCdZl. (21

That is, the firm cannot initiate an earnings stream which includes a state 0 lsss larger than can be covered by its wealth. It is assumed that II/ satisfies

The first restriction implies that the firm will produce in every period: expected profits are positive even for a firm with no assets to draw upon. Under (4), however, higher expected profits are available to firms that can risk incurring positive losses,

The firm chooses T rules which determine the one-period earnings streams on the basis of the information available at the beginning of each period. The rules maximize the firm’s expected profits over the T-period horizon, given that the firm is constrained by (I ) and (2) in every period, Formally, the firm chooses the rules *J,[z.:] (t = 1, . . . . T) to solve t problem

max Y,,(z,; y

St. yt < zt, t= 1, . . . . T

z,=z,

{

7- Y zt-1= ;;+&,,

s, = 0,

s,= 1, t = 2, . ..) T,

where n(y) = $(y) - y and s is the state. The expected profits earned from period t onward are found by evaluating V using these rules:

84 JOHN LEACH

The optimal rules are found recursively. In the last period, expected profits are

A,(z,)=maxn(y,) s.t. yi dz,. Yl

(7)

The optimal rule is

(8)

where y* G arg maxy n(y). A, is a concave function which takes on the value rc(y*) for z, >y *. In any period other than the last, expected profits are

s.t. yc < z,, (9)

where

BYzrt Y,) = #IA,- I@, - YJ + At- lb, + $(Y,))l. (10)

Assume provisionally that A,-, is concave. Then B’ is also concave, and the solution to (9) is

7, = mWyF*, ~~1, (11)

where y:* satisfies

eJ,) + B;(z,, Y1) = 0. (12)

Also,

A;(z,) = K(zt, y”,), y”,=yt** n’(z,) + B,(Zt, 7,) + B*(zn y”th Y”*=zt. (13)

Differentiating (13) shows that A, is concave if A,- i is concave. Since A, is concave, the concavity of all A, and of all B’ follows by induction.

Now consider the solution functions:

PROPOSITION 1. The functions pr[z,] (t= 1, . . . . T) have the form

(a) Y”r=y2for z,3ty* (b) jj,<y*forz,<ty*.

ProoJ: The maximum value of the function Ar--l(~,-l) is (t- 1) z(y*) and is attained by choosing y * in each of the last (t - 1) periods. Assume for the moment that this policy is adopted when z, _ 1 3 (t - 1) y*. Combin- ing (9) and (10) shows that the optimal policy in period t is also y* if


A,+,(z, -y*)= (t- I) z(y*), that is, if z[>Q*. Part (a) now fol’lows by induction, since

(i) yI =y* for zr >y*, and

(ii) jYI=y* for z,BtY* if yteI=Y* for z,-.,>,(r-I)“j*~

To show (b), it need only be shown that Y:*[z,] <y* for z,< 0”. Since $‘(y*) = I, B,(z, p) = ~[A’(z + $(y*)) - A’(Z - y*)] < 0. Then (12) can be satisfied only for y < y *.

A firm which is risk averse but not liquidity constrained would forego some expected profits in order to reduce the magnitude on its state 0 loss. The risk-neutral, liquidity-constrained firm sometimes makes the same choice, and hence might appear (to an outside observer) to be dis~~ayi~~ risk-averse behaviour. This possibility was raise Scheinkman [S]. The last period results here are an Kahn and Scheinkman: the firm chooses a state 0 loss below y* when the liquidity constraint is binding. The liquidity constraint can also be binding outside of period 1, but in these periods there is an additional effect: the firm cuts its state 0 losses below y * when there is some sequence of realizations that would make the liquidity constraint binding in she future. Longer strings of adverse states are possible when t is higher, so that higher initial wealth is needed to protect the firm against the possibih liquidity constraint becoming binding at some future time. P,[zrj < y* over a wider range of wealth as the time to the en program rises. Risk-averse behaviour is not confined to very low wealk levels when the firm exists for a number of periods.

The solution functions need not be monotonic. Et can be shown that y,+ l(z) is non-decreasing if R,(z) z A:(z)/A:(z) is non-increasing. Restric- tions on the form of n can ensure that R, has the required R: fO (together with the restrictions on rr) does not im eon-mo~otouicit~es cannot be ruled out under even quite strong restrictions.

Section VI discusses the effects of this behaviour on ern~~oyme~t in states. Section VI can be read now, ahhough the nature of the resuhs depends upon a parameter k which has not yet been discussed. If k 6 1, a contract with y -C y* is an underemployment contract (that is, a contract under which underemployment occurs when the bad state is realized). L > 1, contracts with y < yC (where ‘J= < y*) are underem~~uyment contracts.

III. CONTRACTS AND EARNINGS STREAMS

Let ho and h, be the firm’s employment of labour in each state, and let W, and W, be its total wage payments in each state. The firm’s contract is

86 JOHN LEACH

represented by Q=(h,, h,, wo, ol), where oi= Wi-rhi (i=O, 1). There are a new contract and a new labour force every period. The contract implies a loss y for the firm in state 0 and a profit c in state 1, where

Y = w. - eOfcho) + rho (14)

c=8,f(h,)-rh,-~l. (15)

The production function is of(h), where 8 is a random variable which takes on the values 0, and 8, (0, > 0,) in states 1 and 0, respectively. It is an increasing and concave C2 function. It is assumed that lim,,,f’(h) = 00 and lim, -t co f’(h) =O. The shock to the production function is the only difference between states.

There are potentially two types of constraints on the firm’s choice of a contract. The first is that the firm must offer its employees an expected utility as high as they would get if they worked elsewhere. Each worker inelastically supplies one unit of labour, so there are h1 active workers in the good state and Iz, active workers in the bad state. Suppose that hl 2 ho. In state 1 the h, workers share equally in the wage payment W, so that the consumption of each worker is W,/h,. In state 0 the firm employs only ho workers but makes some payment to the 12, -ho workers that are laid off. Grossman and Hart [4] show that the firm will distribute W, so that the marginal utilities of the active and inactive workers are equal. Letting r be the pecuniary value of leisure, the required distribution is a payment of [ W,--rh,]/h, to each inactive worker and a payment of [W, + r(hl - h,)]/h, to each worker. The consumption of an active worker is therefore (ol /h, ) + r and (w,/h, ) + r in states 1 and 0, respectively. A similar argument show that these consumption levels would be (w,/h,) + r and (coo/ho) + r under a contract in which more workers were employed in the bad state than in the good state.2 The expected utility constraint is therefore

(16)

where h,,, is the larger of ho and h,, and v is expected utility under the best alternative contract. The worker’s utility u(.) is an increasing and concave C2 function of his consumption, with lim,,, u’(x) = 00 and

‘The possibility that hI <h, cannot be ruled out a priori. The contracts examined below maximize c-ky, where k is a positive constant. I f k is very large, a possible strategy for the firm is to have all workers active in the bad state (when earnings are very important to the firm) and to allow some of them to be inactive in the good state (when the cost of unemployment benelits is not very important). The existence of this option (which is never exer- cised) implies that expected utility is not everywhere a differentiable function of the contract.


lim, + m U’(X) = 0. When this constraint is binding, it can be expressed as (see the Appendix)

01 = ulhl,,, 00; 01, ?1>Q,‘Ir<O,%=>O. (17)

The function y maps ordered triplets (h,,,, L?)~, u) (where h,,, BO, “002 -en,,, v 3 u(r)) into co1 3 --~/h~~~.~ The first derivatives of this function have simple economic interpretations: q1 + r is the marginal Cost (in state 1 goods) of offering a contract to an additional worker and qz is the rate at which the firm can substitute wage payments in state 1 for wage payments in state 0.4

If both the firm and the workers can observe 8, the expected utility constraint is the only restriction on the structure of the contract. However, if the workers cannot observe 8, they will also require a contract under which the firm has no incentive to misrepresent the state of nature. Since there are only two states (and therefore only two ways to lie about the state), this requirement places two additional constraints on the structure of the contract:

B,f(h,)-vh,-o13B1f(lz,f-rho-os, (18)

B,f(h,)-rho-w,3B,f(h,)-rfz,-w,. fIas1

The left-hand side of each condition is the firm’s profits if it tells the truth about the state of nature, and the right-hand side is its profits if it does not tell the truth. These constraints simply state that it is ~~~ro~table to lie.

IV. FULL INFORMATION CONTRACTS

Assume that the current value of 8 is observed by both parties to the contract, and let c = $;(y) be the boundary of the set of feasible earnings

3 Note that some negative values of oi are admissible. This assumption invites questions about the enforceability of contracts, for if oi < 0 and hence W, < rh,,,, the workers can make themselves better off in state i by renouncing the contract. However, the negative o>, are introduced only to allow (20) and (33) to be solved fcr all possible (positive) k, or equivalently, for all possible (positive and negative) y. In equilibrium, the firm adopts only contracts for which 0 <y f y*. In equilibrium, o,>O (i=O, 1) over this interval so no enforcement problems occurs.

4 Note that nz > -1 implies that the worker has higher consumption in the bad state than in the good state. This possibility has an analogue in the Green and Kahn model :3j, in which workers are worse off in good states than in bad ones because their higher consumption in good states is offset by higher work effort. In the current model, employment rises because the firm hires more workers, rather than increasing the hours of a single worker. All changes in utility are therefore the result of changing consumption levels.

88 JOHN LEACH

streams under this informational assumption. An earnings stream on the boundary is sustained by a contract of the form

di,(k) = arg m;x c - ky sat. (14)-( 16), (20)

where k is an exogenous parameter taking positive values. This section characterizes the contracts di,(k) and the function It/r.

The solutions to (20) have two features which simplify the maximization problem. First, (16) is binding at the maximum: if it were not binding, reducing the wage payments would increase c - ky without violating (16) so the initial contract could not have maximized c - ky. This result implies that (17) can replace (16) in the constraints of (20). Second, solutions to (20) have the property that h, B ho. This property is demonstrated by assuming that h, C/Z,,. If 0,S’(h,) > r, so that ~9,f’(/zi) > Y, an increase in h, raises c without violating (16). If 8J”(h,) < Y, reducing h, lowers y without violating (16). In either case the initial contract does not maximize c - ky. Thus, the inequaiity hi > ho can be added to the constraints in (20) without loss of generality. Under this restriction, h,,, is h,. The maximization problem in (20) now becomes

max L=@,f(h,)-vh,-~(hl,w,)+k[Bof(ho)--rho-w,] (21) fTl3 hi. 00

subject to

hl> ho. v-2)

Since L is concave and the constraint is linear, the necessary and sufficient conditions for a maximum are

B,f’(h,) = r + 2

Blf’(hl)=r+r/,-i

-qz=k

I(h, -ho) = 0; h, Bh,,

(23)

(24)

(25)

(26)

where 1 is a non-negative Lagrange multiplier. Equation (25) describes optimal risk-sharing: k is the rate at which the firm is willing to substitute income in the good state for income in the bad state, and -Y/~ is the same rate of substitution (holding utility constant) for the worker. Risk-sharing equates the two rates of substitution. Equation (24) determines h, by equating the marginal product of labour with its marginal cost. Equation (23) is the condition for efficient production. In the bad state the


firm will employ workers until the marginal product of labour is driven down to the goods value of leisure, unless this requires more workers than have contracts with the firm.

Proposition 2 characterizes the solution to (20). Proposition 3 examines the transformation of these contracts into earnings streams

~RQPQSITION 2. There is some k such that A = 0 for 0 < k < k and ho = h 1 for k G k < a. d,(k) is a C’ function ouer the domain 0 < k d I; such that ho is independent of k while h, and o0 are decreasing functions of k. d,(k) is a C’ function over the domain k < k < cc such that A is increasing in k and h and o0 are decreasing in k. bf(k) is Co over the domain 0 <k < CC.

Proof: Let /z = 0 and ho = h,. Then (23)-(25) can be solved sequentially to obtain values for h (the common value of k. and h,), coo, and k. This value of k is k. The constraint (22) is binding for larger k and not binding for smaller k. For 0 < k 6 R, Eqs. (23)-(25) can be sofved (with ;t = 0) to obtain ho, h, , and oO. The restrictions onSand u imply that the solution is in the mterior, so the implicit function theorem can be applied. The theorem implies that ho, h, , and o0 are C1 functions of k; and since w , is a C2 function of o0 and h,, it is also a C’ function of k. The properties of these functions are readily obtained by differentiating the first-order conditions. Similarly, (23k(25) can be solved for h, q,, and ;1. over the domain k Q k < co. The same argument shows that b,(k) is a C’ function of k over this interval, and differentiation of the first-order conditions gives its properties. Since k is contained in each of the two domains, df(k) is continuous at k and hence continuous for 0 < k < cx).

PROPOSITION 3. The function c = $r(y) is of class C2 except at a single value of?, where it is C’. It is increasing and concave.

Proo;S. Let (c(Q), y(Q)) be the earnings stream corresponding to 92, and let E,(k) = c(dF(k)) and f,(k) =$(&,(k)). Since c(Q) and y(Q) are c” functions, the composite functions Cr and yr are of the same class as 8,.

oth are strictly decreasing functions.5 By definition,

tirbJ)= ef(f;‘(Y)) 627)

so that $r is continuous. Differentiating (27) shows that

5 The comparative statics results can be used to show that f; 4 0 when the constraint is not binding and that c; 10 when the constraint is binding. Then both derivatives are negative everywhere since c^;/q; > 0.

90 JOHN LEACH

so that $f is continuous except (possibly) at y = y”,(k), where the derivatives are undefined. This expression can be simplified. Total differentiation gives

44 - W,) dk

=<;-j$-ky^; (29)

while the envelope theorem gives

(30)

Combining (29) and (30) gives c”f/fi= k, so that (28), evaluated at Y = fdk), is

$f&(k)) = k > 0, k #k. (31)

Substituting k= y^;‘(y) into both sides of (31) and differentiating with respect to y gives

I/+; = l/y”;(k) < 0, k#R (32)

so that $” is continuous except at y = fr(k). Thus, II/ is an increasing function which is of class C* except at y = f,(k). It is at least a C’ function . . everywhere if llm,,E- $f(r(k))=lim,,iF+ t,b;(y(k)); and in light of (32), this condition also implies concavity. Equation (31) can be used to show that this requirement is satisfied. If lim,,k- $; > lim, +R+ 9; there are some earnings streams (f,(k), g,(k)) (specifically, those corresponding to values of k lying between the two limits) not contained in $r, which is ruled out by the definition of +r. If lim,, R- $; < lim, -,L+ $f there are some k which correspond to two earnings streams; but this cannot occur because d,(k) is unique. Hence, lim,,fi_ $; = lim, +k+ ~~ = k.

V. ASYMMETRIC INFORMATION CONTRACTS

Now assume that the current value of 0 is observed only by the firm, and let c = Ii/,(y) be the boundary of the set of feasible earnings streams. An earnings stream on the boundary is sustained by a contract of the form

d’,(k) G argJ2max c - ky s.t. (14)-(16), (IF?), (19). (33)

These contracts are characterized below, as is the boundary of the set of feasible earnings streams.


As under full information, (16) is binding at every aa( Also, combining (18) and (19) gives

>,~,-o,>,s,CSfhl)-S(h*)3-r(h1-il,) (34)

so the constraint h, >, h, is implied by the truth-tehing conditions. The maximization problem in (33) is therefore (21) with the constraints (18) and (19). The solution to (33) is obtained in steps. It is first shown (Lem- mas 1 and 2) that there is an interval k”, <k 6 l?, over which the full and asymmetric information contracts coincide. It is then shown (Lemma 3) that, for k f z, and g1 <k, one of the truth-telling conditions is binding. The properties of these contracts are described. Proposition 4 combines these results to obtain the complete solution to (33), and Proposition 5 describes the boundary of the set of feasible earnings streams.

There is some overlap between full and asymmetric information contracts:

Proof The maximization problems associated with these two contracts have the same objective function but different constraints. The constraints are tighter in (33) than they are in (20), so 6,(l) is &?,(I f if a,(l) satisfies the tighter constraints. Since (25) implies that w0 = w, when k = B, (48) becomes

6, f; f’(h) dh 2 r(h, -h,,).

This condition is always satisfied under the contract &,(I j since h, = Jz, if F<l and8,f’(h,)=r+~,>rifk,)l.Equation(19)becomes

~~~h,~~ also satisfied under 8,(l) since h, = h, for k < 1 and @,f’(h,) = r , .

The extent of the overlap between the fu11 and asymmetric information contracts depends upon the value of k. If R> 1, neither trut~~tel~i~g condition is binding under the contract fi,( 1). Since di,(k) is continuous, there are values of k on both sides of 1 for which bif(k) satisfies the tr~th~te~~~n~ conditions. That is, there is an interval g, <k d x1 over which a,(k) =fidk). One of the truth-telling conditions is binding at each

92 JOHN LEACH

endpoint. Lemma 2 describes the properties of the endpoints of this interval.

LEMMA 2. Assume that k > 1. Then there is a positive number &, < 1 such that (19) is binding under d,(k), and there is afinite number x, > 1 such that ( 18) is binding under fii,(k).

ProoJ: There are two parts to the proof. The first part shows that (19) is binding at k, and (18) is binding at E, if positive, finite endpoints exist, The second part shows that positive, finite endpoints exist.

(i) By definition, one truth-telling condition is binding at each endpoint. Equations (22) and (24) imply that (18) can be binding only under a contract fif(k) in which wi 3 oO. Then (25) implies that it can only be binding under a contract df(k) if k > 1, and hence it is not binding under fir&J. Similarly, (22) and (23) imply that (19) can be binding only under a contract d,(k) in which oi GO,. Then (25) implies that (19) is binding under df(k) only if k > 1, so it cannot be binding under 8,(x,).

(ii) Let x,(Q) and x,(Q) be the difference between the left- and right- hand sides of (18) and (19), respectively. Since xi(Q) (i = 0, 1) and fif(k) are continuous functions, xi(6,(k)) (i= 0, 1) is also continuous. By definition, x,(6&J) = 0 for i= 0, L6 It has already been shown that xi(b,(l))>O (i=O, l), so the mean value theorem implies that positive, finite numbers k, and E, exist if lim,,, x,(fi,(k)) ~0 and lim,,, x,(df(k)) ~0. Consider the former limit first. Under full information, h, lies between R, and Ai, where e,f(h,) =r. In light of (23), a lower bound on the value of the expression on the right-hand side of (34) is found by evaluating it at h, = h,, h, = h,. Call this negative, finite value m,. Since u1 - o0 < ?I(&,, fnO) - oO, lim,,, xO(Br(k)) < 0 if lim, _ 0 [~(h,, oO) - o,] < m, under the full information contract. Equation (25) can be written as k= -1/2>u’([oo/~o]+r)/u’([r(/;o, w,)/h,]+r), which implies lim, _ 0 w0 = co. Hence, lim,,, [r(hl, oO) - wO] = --co cm,, as required. Now consider the second limit. In light of (24), an upper bound on the expression on the left-hand side of (34) is found by evaluating it at ho = ho, h, = h, : call this value mr . The value of w1 - w0 is bounded below by y(h,, aJ-wo, so lim,,, xl@,(k)) < 0 if lim,, m [I~(~,, a,) - aJ> ml. Under (25), k = --a2 < u’( [Iwo/h,] + r)/u’( [v(A,, w,)/h,] + r) so lim,,,~(R,,~O)=oO. Hence, lim,,, [~(h,,o,)--o,]=~x,>m, as required.

6 It will become evident that the equations xj(6,-(ki)) = 0 are satisfied by unique ki. Since uniqueness has not yet been shown, &, and WI would be more formally defined as the largest k, and the smallest k,

LfR<l, (18) becomeso,<o,and (19) becomesw,>,o,under~,(k)for kad;; but (25) shows that di,(k) violates (19) when it < I and violates ( 18 ) when k > 1. The oniy value of k near 1 for which d,(k) = tif’,(k) is 1 its&. In this case, E, and R, are equal to unity.

Hn summary there is an interval (possibly degenerate) [E,, k,] over which fia(k)= df(k). There are also intervals over which one of the two truth-telling conditions is binding under the contract G,Jk). If the state I truth-telling condition is binding, the first-order conditions are

B,f’(hl)-r-y, =o (351

kC&J-‘W,) - ~1 - PIC~LP~%~ - VI= 0 (36)

(Y*+ki+Pl1(412-1)=~~ (37)

where p 1 3 0 is the Lagrange multiplier, A contract violates (18 ) if h 1 - h, is too small relative to W, - W,: the firm would forego the output tI,[f(h!)-f(&)] in order to reduce its wage liabilities by -w,. A comparison of (36) and (37) with (23) and (25) shows that the aracter of the contract has been altered to prevent these situations, Equation (37) states that there is incomplete risk-sharing; the firm is willing to give up

re state 1 goods to obtain a unit of state 0 goods than is the worker. ks would be more effectively shared if W0 were reduc

increased, but doing so would create too large a gap between Both bracketed terms in (36) must be positive, since k > pI by (37), so BoS’(h,) > Y when p1 > 0. The implied underemployment of labour is not eliminated because raising h, would make h, -h, too small.

If the state 0 truth-telling condition is binding, the first-order conditions are

where ~~30 is the Lagrange multiplier. The full information contract is incentive incompatible because the tirm would be willingly incur additional wage payments of WI - W,, in order to obtain S,[f(ki) -f(hO)] units of output, so that it would be profitable to announce that the state is 1 w it is actually 0. The payment to labour in state 1 must rise and the payrne~~ in state 0 must fall until it is no longer profitable to make a false announcement. Equation (39) shows that the change in the wage payments raises qI until B,f’(h,) < r + q I (that is, the marginal product of labour is below its marginal cost). The other result is (again) incomplete risk-sharing. There would be more effective risk-sharing if W. were higher and WI

94 JOHN LEACH

lower; but any such adjustment would make a false announcement of the state profitable.

These contracts satisfy (33) for extreme values of k:

LEMMA 3. Let type 1 contracts satisfy the equality form of (18) and (35)-(37) with p1 >O, and let type 0 contracts satisfy (19) and (38)-(40) with p0 > 0. Then

(i) The type 1 contracts are a C’ function of k over the domain El <k < co with the property that h,, hl, and o0 fall as k rises.

(ii) The type 0 contracts are a C’ function of k over the domain 0 < k < &, with the property that h, = R, while h 1 and o0 fall as k rises.

(iii) The type 1 and 0 contracts correspond to d,(k) over their respec- tive domains.

Proof: When one or the other of the truth-telling conditions is binding, the contract is in the interior and the second-order conditions are satisfied (so that the bordered Hessian is non-zero). In these circumstances, the implicit function theorem shows that the contract is C”. The comparative statics properties are obtained by differentiating the equation systems. Since &,(it,) is a type 1 contract with ,ui = 0 and since ,~i rises with k, type I contracts occur over the domain g1 < k < 00. Since .&(&) is a type 0 contract with p0 = 0 and since ,U falls as k rises, type 0 contracts occur over the domain 0 < k<EO. To show that these contracts satisfy (33), it need only be shown that (19) is satisfied under type 1 contracts and that (18) is satisfied under type 0 contracts. Equation (34) shows that these relationships hold if h, 3 h, under both types of contracts. The inequality holds for type 0 contracts: it holds for 6,(&J and the comparative statics of the type 0 contracts imply that it holds for smaller k if it holds at &,. It also holds for type 1 contracts. Since --y/? > 1 under 6,@,) and since -ylz rises with k under type 1 contracts7 all such contracts have the property o1 3 w,; (34) shows that co1 3 o,-, implies h, 2 h, under a binding state 1 truth-telling condition.

The complete solution to (33) is now evident:

’ Formally,

where doo/dh, has been obtained from (35).


FIGURE 1

PROPOSITION 4. G,(k) (0 < k -=z m ) is a C1 function except at no more than two distinguishable points (z, and k,) where it is only Co. The constraint (19) is binding for 0 <k < it,, no constraint is binding for k, < k <El, and (18) is binding for It, Sk < co.

ProoJ: d=(k) has already been described over the three domains (0, J&j, [&,,E,], and (x1, co). The continuity of fiSi,(k) follows from the fact that & and k, are both endpoints of two adjacent domains.

PROPOSITKON 5. The function c = +,(y) is of class C2 except at no more than two distinguishable points (&, and ,?1), at which it is C’. It is increasing and concave.

ProoJ Let t,(k) = c(b,(k)) and g,(k) = y(fi,(k)). By de~nitio~, cl/,(y)=Z,($;‘(y)). The proof of this proposition now parallels that of Proposition 3, with C$ and y^h not defined at &, and K,, where the constraints become binding.

Figures 1 and 2 summarize the main results of Sections III-V. The set of feasible earnings streams is bounded by an increasing and concave function

FIGURE 2

96 JOHN LEACH

under either informational assumption. The Le Chatalier principle implies that $a(y) d $Xy) for all y, since the constraints are tighter under asymmetric information than they are under full information ((34) implies (22) but (22) does not imply (34)). The figures show the relative positions of the two loci when E < 1 and when E > 1. In the former case the two curves touch only when y = y,(l) = f,(l), or equivalently, when y = y *. In the latter case the two loci coincide over an interval [&,, xi]. In either case there is underemployment in state 0 if and only n, > 0.

VI. DISCUSSION

This section combines previous results to show that asymmetric information contracts can generate underemployment in the bad state, The cases k 6 1 and k > 1 are considered separately.

If k G 1 there is state 0 underemployment under all contracts for which y <y,(l), and the amount of underemployment rises as y falls. Since y,(l) = y* (both of these values have the property that $H(y)= l), this inequality can be written as y < y *, Section II has already described the circumstances in which the contract chosen corresponds to a y smaller than y*: y”,(z,) < y* when z, < ty* and y”,(z,) = y* when z, > ty*.

Underemployment contracts (that is, contracts with y < y*) are more likely at the beginning of a firm’s program than at the end. If it is optimal for the firm to choose y = y* in one period, it will be optimal for the firm to choose y = y * in every succeeding .period: if z, 2 ty *, then z, _ 1 > z, - y * 3 (t - 1) y *. No string of state 0 realizations can induce the firm to adopt an underemployment contract. On the other hand, a firm which initially offers underemployment contracts can often, if it encounters enough favourable realizations, accumulate enough wealth to switch to the contract d,( 1). Define the function

Yt-l(Z,) - zt + W,(zJ) (41)

so that a firm’s wealth rises from z, to y,- ,(zt) if the good state occurs in the current period. It is evident that yt- ,(z,) >z,. A firm chooses an underemployment contract in period t and in every succeeding period if and only if

Yl(Y2(Y3’.. (zt)...))<Y** (42)

If this condition is met, z, d zy* (for z = t, t - 1, . . . . 1) even when the good state is realized for all r. If this condition is not met, an unbroken string of state 1 realizations causes the firm to abandon underemployment contracts


in or before the last period. Since z,+(t-l)ll/(0)<yl(...(zr).‘.), (42) is not satisfied when z, > y* - (t - 1) r/j(O). For suhiciently large t, even firms with no wealth in period t might eventually (with luck) become wealthy enough the abandon underemployment contracts.

If E > 1 there is a critical value y, = y^,(&) such that all contracts with y < yC are underemployment contracts and all contracts with y 3 yC are not underemployment contracts. Even though the firm is behaving in a risk- averse fashion (in the sense that the contract chosen is b,(k) with k > l), it need not be choosing an underemployment contract. Risk aversion is a necessary but not sufficient condition for underemployment. However, in each period there is a level of wealth sufliciently small that the firm will choose an underemployment contract.

The possible non-monotonicity of the solution functions jr(z,) implies that the extent of the underemployment need not be a nonincreasing function of wealth. Although underemployment rises as y falls, y might rise as wealth falls. If I; d 1, it is at least true that in every period there is a level of wealth such that underemployment occurs (does not occur) for lower (higher) levels of wealth. If a > 1, even this conclusion cannot be drawn. Underemployment occurs for all levels of wealth below some value, and underemployment does no occur at any ievel of ~ealtb greater than a

particular value; but these two critical values do not necessarily coincide.’ Overemployment exists when 67(/z) -=c r. No overemployment occurs in

state 0: Oof’(h,) 2 r under all contracts corresponding to points on the locus c = $,(y). (This result is not surprising in light of the form sf the utility function, for Green and Kahn [3] show that the state 0 truth+ condition can be binding only if leisure is a normal good.) Some of contracts would generate overemployment in state 1, since (39) irnp~i~~ 8, f’(h, ) < r for sufficiently large pLo. However, the firm chooses onIy contracts d,(k) with k > 1; and cl0 can be positive only for b,(k) with k < I.

ence, overemployment does not occur in this state either. Finally, consider the effects of a change in t e value of a~te~~at~ve

employment (v). Most of these effects are uncert n. An example of this ambiguity is that an increase in u can either increase of reduce y*, so that the level of wealth at which the firm becomes risk neutral (ty*) rise or fall. However, an increase in v always lowers k. Thus, un metric information, risk aversion on the part of firms is both a necessary and sufficient condition for underemployment when u is high; but it is only a necessary condition for underemployment when Y is low.

’ Since y^, is a continuous function p,(O) = 0 and y^,(ly*) = y*, there is an odd number of z, at which $Jz,) = yc. The lowest and highest of these are the first and second critical values. They coincide when there is only one solution.

98 JOHN LEACH

VII. CONCLUSIONs

It has been shown that the risk-neutral firm will display risk-averse behaviour if it is currently liquidity constrained or if it might become liquidity constrained in the future. The latter condition can be satisfied by firms with high levels of wealth if they expect to operate for a number of periods more. Under asymmetric information, such firms might adopt contracts that call for underemployment in the adverse state.

APPENDIX: PROPERTIES OF THE FUNCTION W1=q(hmax,~)

Differentiating (16) with respect to or and o,, gives

q2= -(43/u;)<@ (Al)

where u; is the marginal utility of consumption in state i. Differentiating (16) with respect to or and h,,, and substituting from (Al) gives

YIl = (01 i w%)/hmax ‘0. (AZ)

Similarly,

r22=~2Xlhmax>O (A3)

~12 = --OoGXhwxJ2 -co (A4)

rll= f~o)2~J/&xJ3~O~ (A5)

where x=(u6/ub)-1?2(U;/U~)<0.

It is easy to show that v11q22 - q12q12 = 0.

(A6)

REFERENCES

1. C. AZARIADIS, Employment with asymmetric information, Quart J. Econ. 98, Suppl. (1983), 157-172.

2. V. V. CHARI, Involuntary unemployment and implicit contracts, Quart. J. Econ. 98, Supplement (1983), 107-122.

3. J. GREEN AND C. KAHN, Wage-employment contracts, Quart. J. Econ. 98, Suppl. (1983), 133-188.

4. S. GROSSMAN AND 0. D. HART, Implicit contracts, moral hazard and unemployment, Amer. Econ.Rev. I1 (1981), 301-307.

5. C. KAHN AND J. SCHEINKMAN, Optimal employment contracts with bankruptcy constraints, J. Econ. Theory 35 (1985), 343-365.

Underemployment with liquidity-constrained multi-period firms

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