Journal of Economic Dynmmcs and Control 13 (1989) 255-269. North-HoUand

LIQUIDITY-CONSTRAINED E M P L O Y M E N T CONTRACTS

John LEACH*

McMaster UmverstO,, Hamdton, Ont., Canada

Received January 1988, final vernon received August 1988

Kahn and Scheinkman (1985) have shown that optimal contracting between nsk-averse workers and risk-neutral but liqmdity-constramed firms results in underemployment if firms have better information than workers. This result is extended here by allowing the firm to divide its wealth between capital and liquid assets, so that fiquidity is a choice variable The firm generally chooses a portfolio under which it is opttmal not to choose the 'risk-neutral' (i.e., expected profit-maxi- mizing) employment contract. The underemployment result is thereby shown not to be strongly dependent on the choice of parameters.

1. Introduction

Kahn and Scheinkman (1985) have characterized the optimal asymmetric information contract between risk-neutral firms and risk-averse workers under the assumption that firms have limited wealth. The contract must be one in which the firm's liqttid assets exceed its losses in every state, since it would otherwise be unable to meet its contractual obfigations in some states. If this constraint is binding, the firm behaves as if it were risk-averse: it limits its losses in the bad state by transferring its wage obligations into the good states. The altered contract yields lower expected profits. 1 Kalm and Scheinkman show that the altered contract also entails underemployment (marginal prod- uct of labour above the marginal goods value of leisure) in the bad states. Of course, the constraint need not be binding; and if it is not, the contract does not imply the possibility of underemployment.

A similar approach is taken by Farmer (forthcoming), who assumes that firms must prove their solvency to workers by taking out a bank loan which at least equals the firm's losses in the worst possible state, thereby guaranteeing

*I should like to thank two anonymous referees for their comments. IEarlier papers by Grossman and Hart (1981,1983) and by Azariadis (1983) have shown that

optimal contracting generates underemployment if firms are risk-averse. T~e paper by Kahn and ~heinkman can be viewed as replacing the assumption of risk-averse firms with the more basic assumption of liquidity-constrained firms. This assumption has been rationalized by Greenwald, Stiglitz, and Weiss (1984) on the basis of informational asymmetries in the equity and credit markets.

0165-1889/89/$3.50© 1989, Elsevier Science Publishers B.V. (North-Holland)

256 J. Leach, Ltqutdtty-constramed employment contracts

the payment of wages. The bank is able to issue this loan because, unlike the worker (or at least more cheaply), it is able to monitor the firm's nonliquid assets. Farmer is then able to consider the effects of externally generated interest rate changes.

I have used elsewhere [Leach (1988)] a two-state, T-period framework to show that the firm with a fixed plant size might behave in a risk-averse fashion even if the liquidity constraint is not currently binding: it is sufficient that there be some sequence of realizations such that the liquidity constraint would become binding in the future. As in the Kahn-Scheinkman model, there is a level of asset holding sufficiently high that the firm will not exhibit risk-averse behaviour in any period. Thus, the incidence of such behaviour again depends on the (arbitrary) initial conditions of the model.

Suppose, however, that the firm chooses the division of its wealth between capital and liquid assets. The cost of easing the liquidity constraint is then a reduction in the scale of the firm's operations. Firms would not be willing to hold arbitrarily large quantities of liquid assets, making risk-averse behaviour more likely. In a dynamic model, the firm must consider the extent to which it will endanger its own future liquidity by risking large current losses; the endogenous shadow value of future liquidity plays much the same role as Farmer's exogenous interest rate.

Two T-period, two-state models of this type are considered below. The first assumes that investment is costlessly reversible with a one-period lag. The firm adopts the same policy every period, and this policy ensures that the liquidity constraint is binding in every period. These results conform with those obtained by Kahn and Scheinkman; but they do not require the assumption of a binding liquidity constraint. The binding constraint instead follows from optimal portfolio selection. The second model assumes that investment is irreversible. The firm chooses not to invest in the final periods of operations. Risk-neutral behaviour is unlikely over these periods because the firm accepts a relatively tight liquidity constraint when it invests for the last time. In addition, the firm always behaves in a risk-averse fashion during the initial periods of operation (when investment might still be profitable). These results indicate that liquidity constraints are best able co explain widespread under- emi.. \vment when they arise endogenously.

2. Contracts and earnings streams

A firm with a unit capital stock produces goods under the increasing and concave production function Of(h), where h is labour services and 0 is a random variable representing a shock to technology. The random variable takes on the value 0 0 in the bad state and 0~ in the good state. Each state occurs with equal probability. Prior to the revelation of the state, the firm contracts for the provision of labour with a group of risk-averse workers, each

J. Leach, Ltqutdtty-constramed employmer.t co,ttracts 257

of whom offer inelastically one unit of labour. There is a new labour force and a new contract every period. The contract f l - (h o, h t, W o, Wt) specifies the firm's employment in each state (h o and h~) and its total wage payments in each state (Wo and Wx). Corresponding to any contract is an earnings stream (~, c) which specifies the firm's loss ('t) in the bad state and its profit (c) in the good state. Profit and loss are

c ( ~ ) =O~f (h~) - W~, (1)

= W o - Oo/(ho). (2)

The firm's objective is to maximize c - s't, where s is a positive and finite parameter representing the firm's marginal rate of substitution Oetween earn- ings in the two states. A risk-neutral firm which did not have to satisfy a liquidity con~,~traint would always choose the contract that maximizes c - ~,, but one which mu~ satisfy a liquidity constraint will sometimes be forced to choose the contract that maximizes c - sl, with s ~ 1.

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. Since each worker inelastically supplies one unit of labour, there are h 1 active workers in the good state and h o active workers in the bad state. It can be shown that hi > h o under any optimal contract, so this restriction is imposed ex ante. In state 1 the h~ workers share equally in the wage payment W~ so that the consumption of each worker is W~/hl. In state 0 the firm employs only h o workers but makes some payment to the h~ - h 0 workers that are laid off. The firm will distribute W 0 so that the marginal utilities of the active and inactive workers are equaE Assume that the worker's utility in each state is u(c - oh), where c is consumption, h is work time, and the parameter # is the pecuniary value of leisure. The function u is increasing and concave. The required distribution of W o is then a payment of [W o - Oho]/hl to each inactive worker and a payment of [W o + o(h t - ho)]/h t to each active worker. The expected utility constraint is therefore

u(W1/ht) + u( [W o + p (h t - ho)]/hl) >_ 2v, (3)

where v is the expected utility under the best alternative contract. If both the firm and the workers can observe 0, the expected utility

constraint is the only restriction on the structure of the contract. However, if the workers cannot observe 0, 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 require-

258 J. Leach, l.zquMtty-constramed employment contracts

ment places two additional constraints on the structure of the contract:

Ozf(h~) - W~ > Ol f (ho) - W o, (4)

Oof(ho) - Wo> Oof(hl ) - W1. (@

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 unprofitable to lie.

I hav~ discussed the relationship between contracts and earnings streams elsewhere [Leach (1988, pp. 85-96)]. The following results, presented wlthont proof~ summarize the main elements of this relationship.

Result 1. Let f~*(s) maximize c - s7 subject to (1)-(5), and let $(s) and ~(s) be 7 and c evaluated ~t Q*(s). Then:

(a) "~(s) a"d ~(s) are (respectively) monotonically decreasing and monotoni- cally increasing C 1 functions of s for 0 < s < oo, and

(b) there is an increasing and concave C 1 function ~ such that ~(s) = ~(~/(s)) a n d ~ ' ( ~ ( s ) ) = s.

Result 2. Let 7 ° maximize expected profits I/ ,(7)- y. There exists some ~/< ~,o such that Oof'(ho) > p under all f~*(s) for which -~(s) < ~ and Oof'(ho) = p under all f~*(s) for which ~(s) > 7. Furthermore, for 7 < z/, the contract ~*(~/-1(7)) calls for the sm~\~= h0 as y falls.

The function c--~(~,) is the boundary of the set of earnings streams that are feasible under asymmetric information. An earnings stream (7, c) lies on the boundary if and only if it is supported by a contract which is optimal for some s. The s for which the supporting contract is optimal is given by the slope of the boundary (~') at that point. This function is illustrated in fig. 1. Sections 3 and 4 examine the manner in which the liquidity-constrained firm positions itself on this boundary. Since the main objective of this analysis is to discover the circumstances under which underemployment is an outcome of optimal contracting, a link between underemployment and earnings streams is required. This link is provided by Result 2: underemployment occurs if the chosen 7 is sufficiently small or, equivalently, if the firm'~ marginal rate of substitution is sufficiently high. This result also goes some way toward explain- ing how large s must be before underemployment occurs.

Earlier findings by Cooper (1983) and Green and Kahn (1983) have shown that, for utility ft:nctions of the sort considered here, there is no underemploy- ment under the optimal contract if the firm maximizes expected profits (so that s = 1). Risk aversion, in the sense that the firm maximizes the expected value

J. Leach, LzquMzly-comtramed employment contracts 259

..~o

0

1

I

Fig 1

of a concave function of its earnings, is necessary and sufficient for the ex ante optimal contract to generate ex post inefficiencies.

By c o n t r o l the firms considered here are not risk-averse since they maxi- mize a linear function of the returns in each state, but their choices give the appearance of risk aversion. A firm with a ~aarginal rate of substit:~:ion s greater than unity chooses an earnings strean, at which ~' is greater than unity. Relative to the earnings stream that ma~L~zes expected profts this earnings stream involves both smaller profits and smaller losses (see fig. 1), so that an outside observer might believe the firm to be .i~'oiding variation in its earnings across states. The observer might therefore infer risk aversion on the part of the firm.

Why does underemployment occur u~der the contr.~cts associated with sufficiently large marginal rates of substitution for the firm? Suppose that (4) is not initially binding. An increase in s causes the firm to offer a contract which lowers the firm's loss in the bad state at the cost of lower profits in the good state. The new contract will differ from the old one in two ways. First, the firm will move wage obligations from the bad state to the good state. Second, the firm will reduce its work force in the good state. This adjustment lowers production in the good state, but permits wage obligations to be reduced in both states since the state 1 workforce must be compensated in both states. Continued adjustments of this kind (as s increases) r~t~t eventually cause (4) to become binding. Further adjustments (again necessffated by increases in s) can only be made if employment in the bad st~¢~. ~ reduced to prevent violations of (4). Thus, as s rises above some ¢ ~ c a l value (which cannot be

260 J. Leach, LIqmdtty-constramed employment contracts

smaller than unity by the Cooper and Green-Kahn results), employment in the bad state falls off. Associated with this critical value of s is the state 0 loss ~'.

The size of ~ depends upon a number of parameters, but it is sufficient to examine its relationship to v (the expected utility under the best alternative contract). When v is high, ~ is equal to yo, which is the state 0 loss associated with the expected profit-maximizing contract. The nature of the optimal contract is then very simple. If the firm maximizes expected profits, there will be no underemployment in the bad state. If the firm's desire to avoid state 0 losses causes it to adopt any other contract, there will be underemployment in the bad state. When v is small, ~ lies below yo so that there are other possibilities. A finn which maximizes expected profits will again adopt a contract which does not generate underemployment. However, a firm which does not adopt this contract will not necessarily underemploy workers in the bad state. Specifically, there will be no underemployment if the firm chooses so that ~ _< y < yo; and there will be underemployment if the firm chooses y so that y < ~.

The boundary of the set of feasible earnings streams is assumed to satisfy the inequalities

> o, (6)

> 1. (7)

The first assumption ensures that the firm will choose to operate in every period. It is introduced to avoid the nonconvexities engendered by bankruptcy. The second provides the firm with an incentive (in the form of higher expected profits) to accept positive losses in state 0. If this inequality were reversed, the liquidity constraint would never be binding.

The boundary of the set of feasible earnings streams, as described above, summarizes the options of a firm operating a single unit of capital. The assumption that the capital stock is fixed will now be relaxed. Interpret h as the labour-capital ratio, and let k be the capital stock. Let OF(k h, k) be a linearly homogeneous production function such that F(h,1)ffif(h). Then expanding the capital stock is equivalent to replicating plants, each of which operates with a unit of capital. The firm's state 0 losses are ~,k and its state 1 profits are ~,(3,)k.

Sections 3 and 4 describe the behaviour of a firm that operates for T periods. In each period, before the current state is revealed, the firm chooses an earnings stream subject to the constraint that its liquid assets cannot be smaller than its state 0 losses. This constraint is needed because the firm could otherwise enter into a contract which it would be unable to fulfill if the adverse state occurred. The presence of the constraint can prevent the firm

J'. Leach, laquadtty-constramed employment contracts 261

from choosing the (expected) profit-maximizing earnings stream (.to ,k(`io)). However, the firm itself determines the tightness of this constraint since it chooses the division of its wealth between liquid assets and capital. Section 3 assumes that capital can be freely converted into liquid assets with a one-period lag, while section 4 assumes that installed capital cannot be converted into liquid assets until the end of the program (and then only at a positive cost). The proofs of the propositions in section 4 are contained in the appendix.

3. Reversible investment

Assume that the f tm operates for T periods, and number these periods so that its operations begin in period T and end in period 1. Let Yt be the firm's wealth (capital plus liquid assets) at the end of period t. The firm maximizes its expected terminal wealth, Eyt, by making a sequence of investment and production decisions. Its investment decision in period t is the selection of stocks of capital (kt) and liquid assets (z,'y subject to the adding-up constraint

Y,÷I ffi k, + z,. (8)

Its production decision is the choice of "it, where `i indexes the earnings streams. Once these decisions have been made, the state is revealed and production occurs. The firm's wealth Yt is Yt+l adjusted by the profit or loss in period t.

It is convenient to treat the investment decision as if it preceded the production decision. The state of the firm prior to the period t investment decision is represented by the pair (kt+:, Yt+l). Alternatively, the state can be represented by (kt+ 1, qt+l), where qt+l -Yt+l/kt+l • The latter representation is more convenient, since the central issue is the amount of liquidity backing each unit of physical capital. The expected value of the firm's terminal wealth, conditioned on the state existing prior to the period t investment decision, is

Vt(kt+l, qt+,)- E(y~lkt+~,q,+~). (9)

Similarly, the state existing after the period t investment decision is repre- sented by (k o rt), where r t -y t+I /kr The expected value of the firm's te~-nal wealth, conditioned on this state, is

W , ( k , , r , ) - E(y, lk,,r,). (10)

The firm's optimal policies are found re, cursively, working from the future into the present. The first step is to consider the period 1 production decision. Since the firm is risk-neutral, it will choose `il so as to maximize expected profits subject to the constraint that the firm's total losses in the bad state

262 J. Leach, Liquidity-constrained employment contracts

(3'k~) do not exceed its liquid assets (z~). Expected profits are therefore kt~(rx), where

, r ( r t ) -=ma x{ [~ (3 ' ) - 3 ' ] s.t. 3 ' :~r1-1 . (11) Y

The constraint here is the liquidity constraint 3'k ~ y - k = (r - 1)k. Since 3'0 maximizes ~(3') - 3', the optimal production policy 3't*(rl) is min[3' °, r 1 - 1].

The next decision is the investment decision in period 1. Now Y2 is known; and the firm chooses k t and rx so as to maximize, subject to the budget constraint, expected terminal profits:

= k , [ r ! + (12)

The first-order condition for a maximum is

~r(rl) -- rl~r' (rl). (13)

The opthnal investment policy rl* is r**, where r** satisfies (13). Since ~'(r**) > 0 while ~,(3'o + 1) -- 0, r** < 3'0 + 1 and the constraint in (11) will always be bindhlg. 2 The firm, even though risk-neutral, adopts a policy which ensures that the constraint in (11) will be binding in the last period. It must therefore adopt a contract which avoids large losses in the bad state. The firm would appear, to an outside observer, to be risk-averse.

The same policy is followed in every period of the firm'~ operations. There is a single optimal division of the firm's wealth between fiquid and liquid assets and, since portfolio adjustments are costless, the firm adopts this portfolio at the beginning of each period.

Proposition L i f r ( - r**, then r t* = r** for all t.

.Proof. Let R be the number [r** + ~(r**)]/r**. As~:Jme provisionally that t~ =yt+lR t. Then

Wt+x= max ½R'{ k,+t[rr+ t + ~(y) ] ¥

+kt+t[r t+l - 3']} s.t. 3' < r t+l - 1

'There is an interior solution (r**> 1) if and only if ~'(0)> ~(0). If there is an interior solution, r** is unique because (a) there must be a stationary point and (b) any stationary point is a maximum.

J. Leach, l.aquidi~-constrained employment contracts 263

Sin~ ~t+i"* maximiTes W~+ 1 subject to the budget constraint, rt* t = r** and Vt+ 1 ffi Yt+2 Rt+l. Since V 1 •y2R, the proposition is proved by induction.

These results are very like those of Kahn and Scheinkman (1985) in that the firm's choice of a contract is directly restricted by the liquidity constraint. However, the firm's tight liquidity is the result of its own optimizing decisions rather than being exogenously imposed on it. The results are therefore more robust in that they are independent of the initial conditions.

The above model assumes that, after a one-period lag, capital is as liquid as money, so that portfolio adjustments are costless. Since an important charac- teristic of physical investment is that it is hard to undo, this assumption should be relaxed. At the opposite extreme is a model in which capital is illiquid, and this model will be examined now.

4. Irreversible investment

Now suppose the capital, once installed, must remain in place until the finn ends its operations. The terminal value of each unit of capital is 1 - 8, where 8 > 0. The terminal value is the scrap value of the plant and equipment if 8 < L and it is the shut-down (or moth-bailing) cost if 8 > 1. It will be shown that there is a period J which is the last period in which the expected profits from new investment can be positive. There is never any investment in periods J - 1 through 1; and in periods T through J investment occurs only when the firm's liquidity is sufficiently high. This section examines the finn's choice of over each of the two parts of its decision horizon.

The firm's choice problem must first be considered from the perspective of the final period. Define the function

(14)

which represents expected terminal wealth per unit of capital: r~ - 8 is the wealth brought into the last period, net of shut-down costs, and to this is added the expected value of profits during the last period. If ~r°- = ~.(~,o) and r ~ - 7 ° + 1, A 1 ffi r 1 + ~r ° - 8 for r 1 > r ° That is, A x is linear once r 1 is high enough to allow the finn to adopt the profit-maximizing contract. Expected terminal wealth is

Vl(k2,q2)-- maxk2q2[Al(r)/r] s.t. q2> r. (15) r

The constraint states that investment is irreversible (the ratio of wealth to capital cannot be larger after the investment decision than it was before the investment decision).

264 .I. Leach, I~qui&ty-cor~strained employment contracts

The solution to the maximization problem in (15) is most easily character- ized by introducing a new concept. Let ~ be the optimal value of rl for a firm that is not subject to the irreversibility constraint; that is, Vl ffi arg max[A~(r)/r]. The are several possibilities. First, ~l is infinite if 8 > ~°. The shut-down costs are so large relative to the expected profits that can be earned in a single period, that expected terminal wealth rises as the firm's capital stock is reduced. The firm should choose not to hold any capital in period 1; it should not operate. Second, ~t is any value greater than or equal to r ° if 8 ffi ~r °. The expected profits from one period of operations just compensate the firm for the shut-down costs that it incurs, provided that the faro is able to adopt the risk-neutral contract. This requirement places a lower bound on ~1. Third, F1 < r° if ~r ° > 8. Since expected profits are large relative to shut-down costs, the firm expands its operations so much that the liquidity constraint prevents it from adopting the risk-neutral contract. In this case ~1 is unique, occurring at either a stationary point or the comer (rt ffi 1). The uniqueness of ~x follows from the concavity of A~.

The optimal policy r~* is the solution to the m 'a~dmiz~tion problem in (15), which includes the irreversibility constraint. Since Al(r)/r is nondecreasing over the interval [1, Pl], rl* ffi min[Pt, q2]. The firm chooses ~t if this ratio is not precluded by the h-reversibility constraint; otherwise it approaches ~t as nearly as it can.

The expectations of terminal wealth formed in earlier periods are found recursively. If t - 1 < J (so that there is no i~vestment in period t - 1),

where

Vrffi maxkt+lq,+1[At(r)/r ] s.t. qt+l>-r, (16) #.

A,(,,) = n ax ,4,_i(r, + + Y

s .t . rt_ I > ¥.

(17)

It is readily demonstrated that A t is a concave function if At_ 1 is concave. Since A 1 is concave, the concavity of all A t follows by induction.

Let ~ be the optimal r t for a firm that can freely choose its current capital stock k t, but must satisfy the irreversibility constraint in all future periods. That is, ~ - arg max[ At( r ) /r ]. The appendix shows (see the proof of Proposi- tion 2) that the three possibilities described for period 1 arise again, with the outcome now depending upon the relative sizes of 8 and t~r °. As before, r~* ffi mln!~, q,+l].

Investment is not optimal in any period t in which ~ .is infinite, for then rt*ffi~ qt+1, implying kt= kt+ I. In a period t in which ~ is finite, investment occurs if the firm is sufficiently wealthy. Since J is the latest period in

J. Leach, l.zquldtty-constrmned employment contracts 265

which investment might occur, it has the property that ~ is finite but (t -- J - 1, . . . , 1) are not. Then:

Proposition 2. J is the integer that satisfies the inequality ( J - 1)~ "° < 8 ~ j~.o.

There are two limiting cases. First, J = 1 implies that there is no period beyond which investment will not be profitable. In each period, investment will occur if the firm's liquid assets are sufficiently large. Second, J > T implies that investment is not even profitable in the first period of the firm's program. If the firm enters period T with no capital, it will choose not to operate. If the initial conditions specify that the firm enters period T with an 'endowment' of capital, it will operate that capital but it will not make further investments. More generally, the firm will go through two phases. Investment in periods T through J will depend on the liquidity of the firm at each step in the program. There will be no investment in periods J - 1 through 1.

The 'risk-neutral' policy 3, ° is unlikely to be adopted in period J:

Proposition 3. ¥.t* --- ¥ ° if and only i f ( a ) q: +1>1 +,/70 and ( b ) 8 = J~r °.

For a firm with an exogenously fixed capital stock and a J-period horizon, ,,,p = ~,o if and only if the firm is so liquid that even an unbroken sequence of bad states would not cause the liquidity constraint to become binding in the future [Leach (1988)]. If q:+l is the initial ratio of wealth to capital for this firm, the necessary liquidity is described by condition (a). When the capital stock is endogenous, there is a J-period horizon over which the firm chooses to fix the capital stock; but 7:* ffi ~,o is much less likely in this case than in the case where the capital stock is exogenous. This earnings stream is adopted only when both (a) and the knife-edge condition (b) are satisfied.

There are two reasons why the firm might not choose the risk-neutral contract. First, the firm might not be able to set r: equal to F: because too much of its wealth is in the form of capital. Second, ~: might itself correspond to a capital stock so large that concern for liquidity prevents the adoption of this contract. Conditions (a) and (b) respectively rule out these two possibili- ties. The stringency of (b) follows from the endogeneity of J: 8 < J~r ° implies values of F: which are too small (for ~,* ffi ~,o), while 8 > j~o is not possible by Proposition 2.

If the conditions of Proposition 3 are met, 3't* ffi3 '° for t = J, J - 1,...,1. Even an uninterrupted string of bsd states does not lower r fast enough to move the firm off the linear portion of A (which is the region over which the risk-neutral contract is optimal). If these conditions are not met, ~9" < yo, but the firm would eventually switch to the risk-neutral policy if it encountered enough good states to drive r, above 1 + t~/°. Rising liquidity allows the firm to adopt less risk-averse contracts; the firm will not use any of this liquidity to increase its capital stock.

266 J. Leach, l~quidtty-constramed employment contracts

Now consider the optimal policies in periods T through J + 1, which are also found recursively. For J < t < T, (16) continues to hold. However, invest- ment might now be optimal. The optimal investment policy rt*(qt+x) is again min[qt+ l, ~1 where ~=arg m a x [ A i r ] . There is investment in period t if r t*< qt+t (so that the irreversibility of past investments is not a binding constraint), and no investment if r,* = qt+x (so that it is a binding constraint). Substituting the optimal policy into (16) gives

Vt=kt+xBt(qt+l). (18)

Here

Bt( qt+ l) ~ max[ qt+ xAt( rt)/rt, At( qt+ I)] (19)

expresses expected terminal wealth as a function of qt+r This equation follows from the observation that no investment occurs (and r t equals q,+0 if qt+~ is smaller than ~, and that investment otherwise occurs until r t is driven down to ~. Then (for J < t < T - 1)

At+l(rt+l) --~ max½{ Bt(r,+ ~ + ~(Vt+x)) + Bt(rt+l-~t+l)} 7t+1

s.t. rt+~ - 1 : ' ~ t + l , (20)

and, by the definitions of qt+2 and rt+ 1,

k t+l = k,+2 qt+ z/rt+ x. (21)

Combining (20) and (21) gives the up.dated version of (16). Eqs. (19) and (20) show that ?t* t(rt+x) < ,/o if and only if rt+ l - Yt*l < ~. Alternatively, ~t*l = ~,o if and only if both rt+ 1 - ~,o and rt+ 1 + ~,(yo) are on the linear part of the function B t, so that the firm invests in period t even if the bad state occurs in period t + 1. This relationship can be used to show that the firm will not adopt the risk-neutral policy under any circumstances during the first part of its program:

Proposition 4. yt*(rt*) < y ° for t = T, T - 1, . . . . J + 1.

Since J is independent of 7", a lengthening of the firm's horizon increases the number of periods over which the firm would consider making an invest- ment without changing the number of periods over "which it would not consider making an investment. Since the firm does not adopt the risk-neutral contract during the interval when investment is still a viable option, an

J. LeacK l~¢mdzty-constramed employment contracts 267

increase in T increases the number of periods over which it is certain that the firm will not adopt the risk-neutral contract (this number being T - J ) .

The results of this section are summarized as follows. In periods T through J + 1, ~t*< .re. in period J the firm makes its last (nontrivial) investment decision; and it is almost always the case that y f < 7 °. If y f ffi yo, the earnings stream corresponding to ~o is adopted in every subsequent period. If 7 . < 7o the firm might subsequently become wealthy enough (by encounter- ing the good state sufficiently often) to adopt the earnings stream (7o ~(~o)) in all remaining periods. As in the case of reversible investment, the assump- tion that the firm chooses its capital stock makes 'risk-averse' behaviour on the part of a multi-period firm more likely. This finding reinforces the argument that unemployment can be explained by optimal contracting between a risk-neutral but liquidity-constrained firm and a group of risk-averse workers.

5. Summary

The portfolio behaviour of firms that must satisfy a liquidity constraint in every period has been examined. Wealth is divided between capital and liquid assets. Additional capital allows the firm to expand the scale of its operations, while additional liquid assets loosen the liquidity constraint. It is generally the case that the firm will choose an allocation under which it is optimal not to choose the employment contract that maximizes expected profits. The firm instead chooses a contract with smaller losses in the bad state; this behaviour is characterized as being risk-averse. Such risk aversion is a necessary condi- tion for underemployment under an optimal asymmetric information contract.

Appendix

L e m m a 1. A t ( r t ) --- r t - 8 + t~r ° f o r r t ~ 1 + t y °.

P r o o f . Suppose that A t _ z ( r t _ z ) ffi rt_ 1 - 8 + ( t - 1)~r ° (t - 1)7 °. If r t >__ 1 + ty °, (17) impfies Yt* ffi 7 ° and, therefore,

for rt_z > 1 +

A,= {{[r, + , ( : ) - 8 + ( t - 1):1 + + ( t - 1 ) : ] }

-- r t - 8 + t ~ °.

Since A~ = r l - 8 + ~o for r > 1 + ~,o, the Lemma follows by induction.

268 j . Leach, laquidtty-constramed employment contracts

Proof of Proposition 2. Differentiation of At/r t gives

0 [ A / r ]~Or l, + tyo = [ ~ - tfr ° ] / r 2.

If t~r ° < 8, A / r is monotonically increasing (since any stationary point has to be a maximum); ~ is infinite so t < J. If t~r ° = 8, ~ is not unique: A / r takes its maximum value at all r > 1 + ty °. If t¢ ° > 8, ~ (which is now unique) is either a stationary point or the comer solution r = 1. In either case, ~t is finite so that t > J. Hence, the condition ( t - 1)~r ° < 8 < t~r ° implies t - 1 < J and t >__ J , that is, t -- J.

Proof of Proposition 3. The Kuhn-Tucker condition for 3's* requires

+ >_0,

with the equality holding whenever rz - 1 > 7. Since ~,(3'o) = 1, this condition holds with 3' = 7 ° if and only if rj - 3'0 and rj + ~,(7 °) both lie on the linear section of A. Lemma 1 shows that this occurs when r s is no smaller than 1 + J 7 °. Under conditions (a) and (b), r~* = ~ > 1 + j3'o: (b) is necessary and sufficient for ~ > 1 + J7 ° (as shown in the proof of Proposition 2), while (a) states that qJ+l is large enough to permit r~* = ~.

Proof of Proposition 4. It can readily be shown that B is a concave function so that an interior solution to the maximization problem in (20) is character- ized by the usual first-order condition. Differentiating this condition shows that d 3 ' * / d r < 1 for interior solutions and, of course, d y * / d r = 1 for comer solutions. Then the inequality r - yt*l(r) < ~t holds for all r = rt* 1 if it holds for r = rt+ t (this being the largest possible value of rt* ~). Thus, the proposition is proved if ~+ 1 - 3't* l(~t+ 1) < ~t. There are two possibilities: either Yt* t(rt* 1) is a comer solution or it is an interior solution.

(i) If the inequality constraint in (20) is binding, ~+l satisfies

r B / ( r + ~ / ) ( l + ~ / ( r - 1 ) ) - B t ( r + ~ ) - B t ( r - 3 ' ) = O . (A.1)

Suppose that ~+1--¥t*l(~t+l) > ~t, SO that (a) yt*l(~+l) ffi 3 '0 and (b) both r + ~, and r - 3' lie on the linear section of B t. The linearity of B t reduces (A.1) to ( 't + 1)~'ffi 1 + ~ . Since 3't*l(~t+l) ffi 3' ° (at which ~ ' - -1 ) , this condition further reduces to the equation ~ ( ¥ o ) _ 3'0 •0 which is contradicted by the initial assumption that there is some earnings stream under which expected profits are positive. Hence, ~t+1-3't* 1(~+ 1) < ~t as required.

J. Leach, Ltqutdtty-constramed employmen.t contracts 269

(ii) If the inequality constraint in (20) is not binding, the envelope theorem implies that Pt. 1 satisfies

r[Bt(r+~)+ B t ' ( r - y ) ] =Bt(r+d/)+ Bt(r-y).

If r t+ l - Yt*l(~+l) > ~t, this condition reduces to ~k(y °) -yo__ 0 by (a) and (b) above. The contradiction implies ~t+l - Yt*l(~t+l) < ~t as required.

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