Wage structure when wage offers are private

LABOUR ECONOMICS

ELSEVIER Labour Economics 2 (1995) 19-32

Wage structure when wage offers are private

Kit-Chun Lama, Pak-Wai Liub, *, Yue-Chim Wang” “Department of Economics, Hong Kong Baptist College, Waterloo Road, Kowloon, Hong Kong

‘Department of Economics, Chinese Unioersity of Hong Kong, Shatin, N.T., Hong Kong ‘School of Economics and Finance, Unioersity of Hong Kong, Pakfulam, Hong Kong

Received June 1992; final version received October 1994

Abstract

In this paper we analyse the structure of wages of workers in contract firms for a two-period economy in which there is interfirm mobility. A contract firm provides specific training for a worker during the first period, which increases his productivity if he stays in the second period, but the worker may quit to join an alternative firm after a successful search. When the worker cannot borrow in capital markets, the motive for consumption smoothing dominates and the contract firm acts as a banker and sets wage above marginal product in the first period and below it in the second. When the worker can borrow, insurance is the dominant concern and the contract firm acts instead as an insurer bjr setting the first-period wage below marginal product and the second-period wage above it. This dichotomy will fade away if the contract includes an exit fee as a quit penalty.

Keywords: Wage structure; Private wage offers; Consumption smoothing; Insurance

JEL classification: D82, D83, 531

1. Introduction

This paper examines the structure of intertemporal wage profiles in relation to the productivity profiles in a two-period economy. Specifically, it analyses the wage structure in an economy in which there are contracting, specific training, interfirm mobility and privately observed wage offers. The backbone of our

*Corresponding author. The authors wish to thank participants of a seminar at University of Sydney and two anonymous referees for their helpful comments on an earlier version of this paper. Any errors that remain arc our responsibility.

0927-5371/95/%09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0927-5371(94)00018-2

20 K.-C. Lam et al. /Labour Economics 2 (1995) 19-32

model is laid out as follows. Identical workers enter into ex ante contracts with contract firms which stipulate the level of specific training that contract firms provide in the first period and intertemporal wages. Specific training enhances workers’ second-period productivity in contract firms. Contract firms’ investments in training, however, are affected by workers’ mobility between contract firms and alternative firms. Specifically, workers search in the first period for higher paying second-period jobs and will quit if they receive attractive outside wage offers. Outside wage offers are private to workers. This paper analyzes the effects of alternative stochastic wage offers on properties of the optimal contract. Specifically it derives results on intertemporal consumption smoothing and insurance in wage contracts under different assumptions on the workers’ borrowing constraints in capital markets.

This paper follows the same general approach of studies on wage insurance which emerge as a branch of the implicit contract literature. These studies include Freeman (1977), Harris and Holmstrom (1982), Weiss (1984) and Hal- tiwanger and Waldman (1986). Their main concern is the provision of insurance through the contract against the uncertainty workers face concerning their own future productivity under different capital market assumptions. One of their main results is that the wage profile is non-decreasing. For instance, Harris and Holmstrom (1982) show that the threat of quitting will force the wage to be bid up. The rising earnings profile can be generated purely by an insurance effect.

This paper addresses two deficiencies in these studies. First, with the exception of a brief discussion in Weiss (1984) there has been no treatment of consumption smoothing in these studies. The reason for this is because most previous studies are mainly concerned with insurance against different outcomes of the workers’ productivity over time but not with the intertemporal structure of wage. However, in an intertemporal contract consumption smoothing should be a major concern as it will have to be reconciled with the rising wage profile. Besides, the intertemporal wage structure may be affected by tension between consumption smoothing and insurance. Our paper will deal with these issues. In this connection it should be noted that in another different but somewhat similar class of intertemporal contracts, namely repeated principal-agent contracts, consumption smoothing has been considered by Allen (1985), Fudenberg et al. (1987), and Malcomson and Spinnewyn (1988). In particular, Malcomson and Spinnewyn’s model which analyses short-term and long-term principal-agent contracts bears some similarity to our model with regard to consumption smoothing under different liquidity assumptions.

Second, previous studies typically assume that workers can quit ex ante but ex post there is no mobility.’ For reasons of feasibility, these models typically constrain the wage to prevent all workers from being bid away by other firms so

1 An exception is Bernhardt and Timmis (1990).

K.-C. Lam et al. /Labour Economics 2 (1995) 19-32 21

that there is no turnover ex post. The lack of interfirm mobility, besides being counterfactual, characterises an inefficient labor market which is uninteresting. In this paper we explicitly allow for interfirm mobility. This enables us to analyse the intertemporal wage structure in the presence of tension between productive efficiency on the one hand, and consumption smoothing or insurance on the other.

The main results of this paper are as follows:

(i) Tension between productive efficiency on the one hand, and consumption smoothing (insurance) on the other, will accentuate (attenuate) the slope of the wage profile but in all cases the profile remains upward-sloping.

(ii) When workers cannot borrow in capital markets, consumption smoothing dominates and the equilibrium contract will prescribe a rising wage profile which is flatter than the productivity profile.

(iii) When workers can borrow in capital markets, wage insurance dominates and the wage profile will be steeper than the productivity profile.

(iv) The dichotomy in (ii) and (iii) will fade away if the contract includes an exit fee as a quit penalty.

The rest of the paper is organized as follows. The model is contained in Section 2. In Section 3 contracts when workers have no access to capital markets are analysed with a characterisation of the wage structure. The analysis is repeated in Section 4 when workers are assumed to have access to capital markets. Contracts with severance payments are analyzed in Section 5. The paper concludes in Section 6.

2. The model

The economy produces a single numeraire commodity with labor as the sole factor using constant returns to scale technology. The price of the commodity is stationary and is normalised to one. There are no productivity shocks. Firms are competitive, risk neutral and maximise expected profit.

Labor is supplied inelastically by workers with their unit of working time normalised to one. There is no disutility of effort. Workers are also identical except with respect to their match qualities with different firms. They are risk averse with intertemporal utility function U(.), which is additively separable. The von Neumann-Morgenstern utility function u( *), defined over consumption, has the usual properties of being continuous and twice differentiable. Also u’ > 0, u” c 0 and u’(O) = cc. The last property u’(O) = co effectively precludes zero consumption in any state. For simplicity we assume both the subjective discount rate and the interest rate are zero.

Workers live for two periods, indexed 1 and 2. At the beginning of the first period, a worker enters into a wage contract with a contract firm which pays


him wages w1 and w2 in the two periods. In the first period, time is spent working and investing in specific skills in the firm. All training costs are time costs. Specific training is given by the time-equivalent unit x with 0 G x < 1. Hence x is the fraction of time in period 1 spent investing which is observable to both parties. Let m be the workers’ first-period productivity per unit time in the contract firm. Specific training in period 1 will augment his productivity in the same firm in period 2, written as h(x), where h is the specific skill production function with a strictly concave technology. It has the properties of h’ > 0, h” < 0 and h(O) = m. The workers’ productivity profile over time in the contract firm is, therefore, rising due to the presence of specific investment (provided that it is non-zero).

The exchange relation between the worker and the firm is characterised by a contract agreed upon ex ante at the beginning of period 1. It is assumed that the contract does not bind the worker irrevocably to the firm; involuntary servitude is prohibited. By virtue of the assumption of stationary output price, uncertainty in the product market is removed and the contract firm will not lay off workers.2 At the end of the first period, the worker will search for alter- natives.3 Search is assumed to be costless. The outside wage offers he may receive are stochastic and depend on match qualities with the alternative firms.4 The distribution of maximum wage offers which the worker receives in the second period is denoted by F(G2) where f2 is the stochastic maximum wage offer. It is continuously differentiable with density functionf(G2). Depending on the outcome of his search, the worker either stays with the contract firm in the second period and receives wage w2 or quits to join an alternative firm. Outside wage offers are private to the worker. 5 In the second and last period of the model, all time is spent working; there is no further investment or search.

‘Many studies which focus on wage insurance but not unemployment simply assume that the contract binds the firm from discharging the worker. See, for example, Arnott (1982) Harris and Holmstrom (1982), Weiss (1984), Amott and Stightz (1985), Haltiwanger and Waldman (1986) and Berkovitch (1986). For a detailed argument why a model with voluntary quits is important for studying in the implicit contract literature as opposed to a model with involuntary unemployment, see Ito (1988).

3 Renegotiation of the contract at the end of the first period will not take place because the firm’s individual rationality constraint which we will formulate in Section 3 is in expected terms and the distribution function of outside job offers is time-invariant.

41nvestments in specific skills will raise the worker’s productivity in the contract firm but not elsewhere, thus increasing his comparative advantage in staying in the contract firm. However, it is assumed that the distribution of match qualities among alternative firms are sufficiently disperse so that even though specific investments may reduce quit probability, it does not completely eliminate the possibility of a better match elsewhere.

’ The assumption of private wage offers is natural. For a justification, see Ito (1988). Other studies which make this assumption include Kahn (1985), Moore (1985), and Arnott et al. (1988).


3. Contracts with no access to capital markets

We first analyse the structure of the equilibrium contract under the assumption that the worker can save and lend but cannot borrow in capital markets. The lack of access to borrowing facilities can be justified on the ground that human capital cannot be used as collateral in borrowing. Under this assumption, it can be shown that the worker will not save and lend as the contract firm will tailor the wage profile to suit his consumption plan. The problem is identical to the case where he can neither lend nor borrow.(j Hence, we need only to analyse the contract when the worker has no access at all to capital markets.

The worker’s utility function is

U(w1, w2) = a(w1) + w72Mw2) + s

m Wz)f(+2)df2 = 4wA + Ww2),

w2

where the first term on the RHS of the first equation is his utility in period 1; the second term is his utility in period 2 if he stays weighted by his probability of stay; the third term is his expected utility if he quits in period 2. The expectation operator E with expectation taken over staying and quitting is introduced to simplify notations, which is defined as follows:

I

m Eu(w2) = F(wz)u(wz) + 4~2)_0~2) dG2.

w2

The following problem can be solved:

max u(wl) + Eu(w2), X, WI 1 w2

subject to

(1)

(1 - x)m - WI + F(w#l(x) - w2) = 0, (2)

Wl,W2 a 0, (3) where (2) is the individual rationality constraint of the contract firm which specifies that expected profit is zero under perfect competition and constraint (3) requires non-negative consumption.

By virtue of the assumption u’(0) = cc, constraint (3) cannot be satisfied as equalities. The first-order conditions of the problem with interior solutions for w1 and w2 can now be written in the following compact form (the arguments of functions are suppressed where they are obvious):

x:$( -m + Fh’) = 0, xE(O, l),

< 0, x = 0,

20, x=1, (4)

6 For this result see Harris and Holmstrom (1982). See also Lam et al. (1987).


w1: u’(w,) - Ic/ = 0, (5)

w2: FU’(W2) + II/[f(h - w2) - F] = 0, (6)

I): (1 - x)m - Wl + F(h - w2) = 0, (7)

where $ is the Lagrange multiplier associated with (2). There may be multiple local optima which solve the problem. We will proceed

to analyse the properties possessed by any of the optima (including the global ones). The equilibrium wage structure satisfies the following proposition.

Proposition 1. If the worker has no access to capital markets and specijic investments are non-zero, the contract will set wage above marginal product in the first period and below it in the second period.

Proof: From (5),

* = u’(w1).

This can be substituted into (6) to yield

h _ w _ [. “twd - u’(w2)

2-f U’(Wd . (8)

We can prove h - w2 > 0 by contradiction. Suppose not and h - w2 < 0. Then (8) implies u’(wI) < u’(w2), that is w1 > w2 by concavity of u. On the other hand, from the zero profit constraint, wr < (1 - x)m because h - w2 < 0. We now have the following relations:

w2 3 h(x) 2 h(0) = m > (1 - x)m > wl * w2 > wl.

We have a contradiction. Hence h - w2 > 0. From the zero profit constraint, it follows that w1 > (1 - x)m. Q.E.D.

Proposition 1 and its proof show that the wage profile is rising over time with w2 > w1 but is flatter than the productivity profile. This result is similar to that in Weiss (1984) which did not consider private wage offers.

The economic meaning of our results in Propositions 1 is clear. The worker’s marginal rate of substitution of w2 for w1 derived from (5) and (6) is

auPW2 wW2) < F -_=- ) 87J/8W, 4Wl)

which means that the worker is trading off between period 1 consumption and period 2 consumption if he stays without regard for the consumption if he quits.


That is, the worker desires a wage contract which evens out the consumption stream over time but not one which insures his consumption between staying and quitting. The motive for consumption smoothing is dominant and insurance is not a concern here. The firm responds by setting w1 and w2 in the contract without reference to the worker’s expected outcome if he quits. In order to even out consumption over time, the contract sets w2 so low as to be below the post-training marginal product.

However, only incomplete consumption smoothing is provided.’ In setting wages consumption smoothing requires that w2 be set relatively low but when w2 is below marginal product, productive inefficiency appears as the worker will be quitting when his marginal product is higher inside the contract firm than outside. Therefore, to reduce inefficient quits, w2 must be set relatively high. The contract must trade off consumption smoothing against ‘stay incentives’ and attain the (constrained) efficient level of stays by stipulating an appropriate level of w2. The tension between productive efficiency and consumption smoothing accentuates the slope of the wage profile.

To summarise, when the worker cannot borrow, the motive for consumption smoothing via the wage contract is dominant. The tension between productive efficiency and consumption smoothing accentuates the slope of the wage profile. The wage profile is upward-sloping but flatter than the productivity profile.

4. Contracts with lending and borrowing

We now relax the assumption that the worker cannot borrow and assume that he has access to capital markets. To simplify matters we will only deal with capital markets where lenders will not bear default risk arising from uncertainty on debtors’ ability to repay. There is no collateral. Borrowers must borrow against assured future income and therefore there will be no bankruptcy risk and credit rationing. In our model the worker has an assured income equal to the sum of contract wages wi + w2 in the two periods and that is the maximum consumption that he can have in period 1.

When the worker can lend and borrow, his utility function is

s

m U(w1, w2, Cl, c2) = 4Cl) + W,)u(c,) + 4~2hf(~2) dG2

w2

= Ncl) + J3@2),

’ If there were complete smoothing, the expected marginal utility of consumption over time would be equated at the margin; i.e. the marginal rate of substitution of w2 for wi should be unity and given that there is uncertainty in the level of consumption in period 2, a declining net wage profile would entail with w1 > w2. If there is no uncertainty, a flat wage profile would be the result of complete consumption smoothing.


where ci is the worker’s consumption in the ith period and & his stochastic consumption in period 2 if he quits.

The problem can be characterised by

max u(cl) + Eu(c~), (9) X,WI .W!.

c,*cS? subject to

(1 - x)m - Wl + F(w,)(h(x) - WJ = 0, (10)

cl ~argmax u(w1,w2,c;,cz), (11) c; EC1

c2 = w1 + w2 - Cl, (12)

c”2 = WI + 62 - Cl, (13)

Cl,C2 a 0, (14)

Cl < w1 + w2. (15)

Given the contract the worker will choose cl from the consumption set Cr to maximise his utility, hence constraint (11). His consumption in period 2, c2, can be obtained as a residual from the budget constraint (12). The budget constraint for the worker if he quits is given by (13). Constraint (14) requires that consumption be non-negative. The restriction on borrowing only against assured future claims is given by constraint (15) but this is implied by (12) and (14) and therefore can be dropped from the list of constraints.

The wage structure can be characterised by

Proposition 2. If the worker can lend and borrow the contract will set wage below marginal product in the first period and above it in the second period.

Proof: Since U is strictly concave in cl, constraint (11) can be replaced by the first-order stationary condition

u’(cl) - Eu’(c2) = 0, (11’)

where

En’&) = F(wz)LJ’(c~) + s

m ~'(~dfbWd~2

w2

The first-order conditions of the problem with an interior solution for w1 and w2 can now be written as

x:1(-m+Fh’)=O, x~(O,l),

< 0, x = 0,

20, x=1, (16)


wl: Eu’(cz) - A - &Eu”(cJ = 0, (17)

w2: Fu’(cz) + A[f(h - w2) - F] - ~Eu”(c~) = 0, (18)

1: (1 - x)m - wr + F(h - WJ = 0, (19)

4: u’(cr) - Eu’(cz) = 0, (20)

cl: c$[u”(c~) + Eu”(cz)] = 0, (21)

where A and 4 are Lagrange multipliers associated with constraints (10) and (11’). From (21), C$ = 0, which has the interpretation that the firm is indifferent to the worker’s choice of cr.

From (17) Lagrange multiplier 1 = Eu’(cZ) > 0. This is substituted into (18). The equilibrium wage structure is now characterised by

h _ w = F Eu’(c2) - u’(c2)

2 f Eu’(c2) ’ (22)

But

s

to Eu’(c2) - u’(c2) = u’(E2)fdG2 - (1 - F)u’(c2)

w2

< (1 - F)u’(cz) - (1 - F)u’(cz) = 0.

The inequality is due to concavity of u. Hence h - w2 < 0. From (19), it follows that w1 -C (1 - x)m. Q.E.D.

The resultant wage profile is rising and steeper than the productivity profile. When the worker can lend and borrow, consumption over time can always be evened out regardless of the wage profile. Free of consideration for consumption smoothing, the motive for insuring second-period consumption now dominates. Since staying is the ‘bad’ state because of the lower wage and quitting the ‘good state because of the higher wage, the worker will seek to insure his consumption against staying. This can be readily observed by examining the worker’s marginal rate of substitution of w2 for wl:

a u/aw2 FUR ---=- XJ/aw, Eu’(c2)’

The worker is trading off between expected consumption for staying and expected consumption over staying and quitting in period 2.

Now insurance requires w2 be set relatively high so as to even out variation in consumption between the two states: staying and quitting. But when w2 is set above marginal product, productive inefficiency due to excessive inefficient stays sets in as the worker will be staying in the contract firm when his productivity is

28 K.-C. Lam et al. JLabour Economics 2 (199.5) 19-32

higher outside. The contract must therefore trade off insurance against quit incentives and stipulate an appropriate level of w2 to achieve the (constrained) efficient level of quits. The tension between productive efficiency and insurance therefore attenuates the slope of the wage profile but it is still rising and steeper than the productivity profile. It would not be flatter than the productivity profile for if it were, there would be loss in both insurance and productive efficiency as excessive inefficient quits would now be encouraged.

In summary, when the worker can lend and borrow, the motive for insurance dominates. The tension between productive efficiency and insurance attenuates the slope of the wage profile. The wage profile will be rising and steeper than the productivity profile.

5. Contracts with severance payments

Under the two different capital market environments we have a dichotomy in the wage profile. The motive for consumption smoothing dominates and the equilibrium contract will prescribe a rising wage profile which is flatter than the productivity profile when the worker cannot borrow in capital markets. When the worker can lend and borrow, however, the desire to insure second-period consumption dominates and the wage profile is steeper than the productivity profile.

This dichotomy will disappear if severance payment is introduced into the contract to mediate the tension between consumption smoothing/insurance and productive efficiency. Specifically the inclusion of an exit fee as a contract parameter introduces an extra degree of freedom into the contract. In that case the wage contract will be given more latitude in carrying out its smoothing function whereas the quit penalty assumes more of the role of providing appropriate ‘stay incentives’ and of evening out consumption between staying and quitting. The result is a better trade off (from an efficiency standpoint) between consumption smoothing/insurance and productive efficiency than could be possible when severance payment is disallowed. The analysis is as follows.

Suppose now the contract includes a severance payment Z, positive or negative, which will be paid by the contract firm to the worker if he quits in the second period. A negative severance payment refers to an exit fee which is a penalty for quitting whereas a positive severance payment is a reward for quitting.* When the worker has no access to capital markets, his utility function

s Here we assume that quits and layoffs are distinguishable and maintain our earlier assumption there are quits but no layoffs stated in Section 2. This is different from Carmichael (1983), and MacLeod and Malcomson (1989) who assume that quits and fires are indistinguishable.

K.-C. Lam et al. JLabour Economics 2 (1995) 19-32 29

can be written as

WI, w2,a = eh) + F(w2 - 04w2) + i

O” u(G + Z)f(G)d3, W2-Z

where the first term on the RHS is his utility in period 1, the second term is his utility in period 2 if he stays weighted by his probability of stay which depends on his reservation wage w2 - Z; the third term is his expected utility if he quits in period 2.

The second-best problem can be characterised as follows:

max u(w1,w2,Z), X,XIIW2.Z

subject to

(23)

(1 - x)m - wi + F(w2 - Z)@(x) - w2) - (1 - F(w2 - Z))Z = 0,

w1,wz > 0.

The following proposition can be proved.’

(24)

(25)

Proposition 3. If the worker has no access to capital markets, at the equilibrium, (a) the wage projle will be declining over time, i.e. w1 > w2; (b) the contract will set the reservation wage above the second-period productivity, i.e. w2 - Z > h; and (c) the severance payment in the contract will be a penalty for quits, i.e. Z < 0.

Proof See appendix.

The presence of the quit penalty allows the wage contract to be tailored for better consumption smoothing. It should be noted that the equilibrium does not entail setting the quit penalty at a level so that the reservation wage (w2 - Z) equal the second-period productivity (h), to eliminate all inefficient quits. In- stead, the reservation wage is set above h thus bringing in productive inefficiency from a different direction; the worker will be staying in the contract firm when alternative marginal products are higher outside. The reason for this result is that the worker also desires insurance against staying besides consumption smoothing.

Insurance implies that variation in consumption between quitting and staying has to be evened out. This means that the contract will specify a high w2 to raise the income of staying and/or a high quit penalty (low Z) and, ipso facto, a high

‘This result is reported in Lam et al. (1990).


reservation wage w2 - 2, to reduce the net income after quitting. The higher the reservation wage, the larger is the insurance. But if w2 - 2 is set so high as to be above h, excessive inefficient stays in the contract firm will ensue. The result is a tradeoff of insurance against productive efficiency. At the equilibrium, w2 - 2 > h entails. The insurance is incomplete.”

Now consider the case in which the worker can lend and borrow. It will be easy to show that as in the case of no access to capital markets, the contract will specify an exit fee as a penalty for quits. The reservation wage will, as before, be set above the second-period productivity to provide insurance but as long as outside wage offers are stochastic, insurance will be incomplete. The major difference with the previous case is in consumption smoothing. Consumption smoothing will be complete since the worker can lend and borrow to even out consumption. The wage profile therefore plays no role in smoothing consumption. It does not have a unique shape.”

In our previous analysis we have shown that there is a dichotomy in the wage profile under alternative capital market environments. It arises because there is tension between consumption smoothing/insurance and productive efficiency and the rising wage profile plays a dominant role in mediating this tension. Once this role is mainly taken over by another parameter in the contract such as an exit fee, the wage profile is no longer necessarily rising, and the dichotomy fades away.

6. Conclusion

In this paper we incorporate elements of a human capital investment model, a principal-agent model, an interfirm mobility model and an intertemporal liquidity model into a labor contract to analyse the intertemporal wage structure. The main finding of the paper is that the motive for consumption smoothing dominates and the equilibrium contract always sets wage above marginal product in the first period and below it in the second period if the worker has no access to capital markets. The desire to insure second-period consumption dominates and the reverse is true for the wage profile if the worker can lend and borrow. This dichotomy will fade away if the contract includes an exit fee as a quit penalty.

lo This result is similar to that obtained by Kahn (1985). Kahn examined second-best contracts with severance penalties used as an incentive device to keep employees attached to firms and concluded that when firms cannot verify outside offers, only incomplete insurance will be provided.

‘i The proof is straightforward. It proceeds along the same line as the proof for Proposition 3. Among the three first-order conditions for w,, w2 and Z, only two are independent.


Appendix: Proof of Proposition 3

The relevant first-order conditions of the problem with interior solutions for wl, w2 and Z can be written in compact form:

WI: U’(W1) - p = 0, (A.1)

w2: Fu’(w2) + p[f(h - w2 + Z) -F-J = 0, (A.2)

s

Go z: u’( + Z)fdG + p[F -Ah - w2 + Z) - l] = 0, (A.3)

w2-z

where ,u is the Lagrange multiplier associated with (24).

(a) From (A.l) we have

p = u’(wJ.

Solving (A.3) and (A.4) together,

j

a, U’(Wl) = Fu’(w2) + ~‘(6 + Z)jdG < Fu’(w2) + u’(w2)

s * fd@

w2-z w2-z

= Fu’(w2) + u’(wJ(1 - F) = u’(w2). (A.4)

The inequality is due to concavity of u. Therefore, u’(wl) -C u’(w2) implying w1 > w2. The wage profile is declining.

(b) Solving (A.l) and (A.2) yields

h _ w 2

+ z = [. ml) - u’b2)

f 4wd . But u’(wl) < u’(w2) from (a), therefore h - w2 + Z < 0 or w2 - Z > h. (c) We can prove Z < 0 by contradiction. Suppose Z 2 0. Then the zero profit constraint gives

(1 - x)m - Wl = z - F(h - w2 + Z)

> 0,

since h - w2 + Z < 0 from (b) and Z 2 0 by assumption. We conclude (1 - x)m > wl. Now w2 > h + Z * w2 > h because Z 2 0 by assumption. Fur- thermore w1 > w2 from (a). We have the following string of relation:

w1 > w2 > h 2 h(0) = m 2 (1 - x)m.

In other words wl > (1 - x)m. We have a contradiction. Hence Z cannot be non-negative. Q.E.D.


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Wage structure when wage offers are private