Contracts, Risk-Sharing, and Incentives

Transcription

1 Contracts, Risk-Sharing, and Incentives Bruno Van der Linden Université catholique de Louvain January 25, 2011 (ESL-UCL) 1 / 102

2 Introduction The labour relationship: Generally a long-term one The labour contract: Often a relationship of subordination specifying rights and duties; does not list the consequences of every possible eventuality. Remuneration: In some cases linked to performance, in others not. (ESL-UCL) 2 / 102

3 Introduction Aims Why firms and workers engage in long term relationships? How the tradeoff between insurance and incentives acts upon the remuneration rule for labor? Why firms make use of hierarchical promotions and internal markets? The links between seniority, experience and wages. The efficiency wage theory. (ESL-UCL) 3 / 102

4 Introduction Standard approach in economics Standard stylized (narrow) view about human motivation: Employees seek to earn as much money as possible with minimal effort on the job (or minimal hours worked). The work done has no value per se. No possibility of identification with the firm (hence, no effort of the firm to affect employee s identification). No norms of behaviour (Which effort is normal? Fairness?) Worker s preferences are the same whether or not there is an incentive scheme or a supervisor. Part 1: Standard approach in economics. Covered by Chap. 6 of Cahuc and Zylberberg (2004). Note: For this part, slides adapted from those of Bart Cockx. Part 2: Tries to enlarge (a bit) the standard view. (ESL-UCL) 4 / 102

5 Road-map Introduction 1 1. Stylized narrow view about motivation 1.1. The Labour Contract 1.2. Risk-sharing 1.3. Incentives if results verifiable 1.4. Incentives in the Absence of Verifiable Results 2 2. Extension 2.1. Intrinsic motivation 2.2. Envy 3 References (ESL-UCL) 5 / 102

6 1. Stylized narrow view about motivation 1.1. The Labour Contract 1. The Labour Contract Contract theory explains how such contracts can be understood as a rational response to: 1 uncertainty of the environment 2 private information of the employer or the employee Uncertainty To what extent do labour contracts ensure workers (employers?) against risks? Private information How do labour contracts provide incentives to reveal private information? (ESL-UCL) 6 / 102

7 1. Stylized narrow view about motivation 1.1. The Labour Contract Typology of contracts Key questions: Can the employee s (and the employer s) activity be observed and if so is it verifiable by a third party (a court)? Two types of contract can be distinguished: 1 Complete or explicit contracts All clauses of the contract can be verified by a third party, a court; At the moment of signing, all circumstances can be foreseen. 2 Incomplete or implicit contracts All the clauses of the contract cannot be verified by a court or circumstances are too numerous; The contract must then be self-enforcing: both parties have a mutual interest in continuing the relationship; Only feasible in long-term relationship. (ESL-UCL) 7 / 102

8 1. Stylized narrow view about motivation 1.1. The Labour Contract Agency model or principal-agent model We study contracts within a principal-agent framework: The principal (=employer) proposes a contract; The agent (=employee) can either accept or refuse. In this chapter, by assumption, no frictions, workers have no bargaining power, it make sense to talk about the output of the agent. Two textbook cases: 1 employee s effort is observed and verifiable but employee s output is random 2 employee s effort is not verifiable: employer faces a moral hazard problem (ESL-UCL) 8 / 102

9 1. Stylized narrow view about motivation 1.2. Risk-sharing 2. Risk-sharing Why could this be important? Assume a one-worker one-firm setting. A1 The agent s preferences are represented by a quasi-concave utility function U(C, L) = U(C, 1 h), where C denotes the agent s consumption, h the number of hours worked (or effort ) and, therefore, L = 1 h, the agent s leisure time. Hours observed and verifiable. A2 The production of the agent = y = f (h, ε), where f h > 0, f ε > 0, f hh 0 and f hε > 0. A very simple situation with two possible states: E ={ε 1, ε 2 }, ε 1 < ε 2. A3 The profit of the principal is defined by the equality Π = f (h, ε) W, W being real compensation (or earnings) for h hours of work (wage rate is then W /h). A4 The random variable ε is verifiable. (ESL-UCL) 9 / 102

10 1. Stylized narrow view about motivation 1.2. Risk-sharing Benchmark case Pure competition and spot markets Definition: spot markets = markets open when the realisation of ε is known (i.e. ex-post ). Interpretation: ε captures the business cycle. Under perfect competition, free entry implies Π = 0 = f (h, ε) W. In equilibrium, ε, the pair (h, W ) verifies: [ ] dw f (h, ε) = W and f h (h, ε) = = U L(W, 1 h) dh U W (W, 1 h) (see next figure) Note: unchanged if the principal was risk-averse and maximizing v(π), with v > 0 and v 0. U (ESL-UCL) 10 / 102

11 1. Stylized narrow view about motivation 1.2. Risk-sharing Benchmark case Pure competition and spot markets Figure: 6.1 Earnings and hours worked in a perfectly competitive spot market (ESL-UCL) 11 / 102

12 1. Stylized narrow view about motivation 1.2. Risk-sharing Benchmark case Pure competition and spot markets Two major implications: 1 The allocation of labour is efficient ex-post but, from an ex-ante perspective, 1 risks are not shared efficiently (a notion made precise below): In particular, the wage fluctuates (a lot) with the realisation of ε. 2 Variations in real earnings are greater than variations in hours worked if, as empirical analyses suggest, the elasticity of the labor supply is weak w.r.to wages. Although some care is needed (earnings wage rates and hours employment), empirical observations suggest that real wage rigidity and large fluctuations in employment are standard. 1 I.e. before the realisation of the state ε is observed (ESL-UCL) 12 / 102

13 1. Stylized narrow view about motivation 1.2. Risk-sharing Motivation of this literature (started in the 1970s) 1 Can real-wage rigidity be explained by the desire of workers to be insured against the risk of layoff (when they are not (fully) insured by a public unemployment insurance scheme)? 2 Can this real-wage rigidity explain an inefficient allocation of labour and/or employment fluctuations? (ESL-UCL) 13 / 102

14 1. Stylized narrow view about motivation 1.2. Risk-sharing 2.1 Symmetric or verifiable information An individual insurance contract model Assumptions (slightly different from above) A1 Preferences: Unchanged. Note: effort (understood as hours worked) h is observable and verifiable. A2 Same technology but ε is a continuous random variable defined over the interval E = [ε, ε + ] the density of which is denoted by g(ε). A3 The profit of the principal: Unchanged. A4 Verifiability: Unchanged. A5 The principal may be risk-averse: v(π), with v > 0 and v 0. (ESL-UCL) 14 / 102

15 1. Stylized narrow view about motivation 1.2. Risk-sharing An individual insurance contract model A contingent insurance contract A contingent insurance contract specifies ex-ante a level of earnings W and a number of hours worked for each possible realisation of ε (hence, the expression contingent contract ) Formally, contract A = {W (ε), h(ε)} solves: max Ev[Π(ε)] s. to EU (W (ε), 1 h(ε)) U max EU (W (ε), 1 h(ε)) s. to Ev[Π(ε)] Π where U (resp. Π) measures expected external opportunities or (ESL-UCL) 15 / 102

16 1. Stylized narrow view about motivation 1.2. Risk-sharing An individual insurance contract model Optimality conditions The optimality conditions lead to two strong properties. 1. Whatever the state ε, labour is allocated efficiently (!): U L [W (ε), 1 h(ε)] U C [W (ε), 1 h(ε)] = f h [h(ε), ε], ε E (3) 2. The so-called Arrow-Borch condition for optimal risk-sharing: U C [W (ε), 1 h(ε)] U C [W (θ), 1 h(θ)] = v [Π(ε)] v, (ε, θ) E2 [Π(θ)] Risk-sharing efficiency requires that the marginal rate of substitution between any pair of state contingent payoffs be equal for all economic agents (ESL-UCL) 16 / 102

17 1. Stylized narrow view about motivation 1.2. Risk-sharing An individual insurance contract model Real wage rigidity? Then, Employers typically own assets while for most workers their human capital is their only (or major) asset. Hence, employers can diversify risks in financial markets. So, if ε only captures diversifiable risks (hence, not macroeconomic nor systemic risks), taking employers as risk neutral is a sensible assumption. 1 the Arrow-Borch condition implies that the marginal utility of consumption has to be equal in all states ε; 2 this does not in general imply full real wage rigidity. (ESL-UCL) 17 / 102

18 1. Stylized narrow view about motivation 1.2. Risk-sharing An individual insurance contract model Properties of the contract if employers are risk neutral Differentiating the FOCs wrt to ε, we obtain ( ) U 2 CL U CC U LL dh f hh U C U CC dε = f hε and dw dε = U CL dh U CC dε (4) 1 If we assume that U CC < 0 and (U CL ) 2 U CC U LL < 0 (sufficient conditions for quasi-concavity 2 ), hours worked are an increasing function of ε, since f h increases in ε. No ambiguity substitution vs income effects. 2 Earnings W is insensitive to the cycle if U CL = 0. Reflects risk-sharing. 2 i.e. U CC (U L /U C ) 2 2U CL (U L /U C ) + U LL < 0 (C, L) (ESL-UCL) 18 / 102

19 1. Stylized narrow view about motivation 1.2. Risk-sharing An individual insurance contract model Properties of the contract if employers are risk neutral About dw dε = U CL U CC dh dε : 3 Earnings W are pro-cyclical only if U CL < 0. Usually, U CL > 0 is assumed. For leisure to be a normal good, it can be shown that the following condition should be fulfilled : U CC U L }{{} U CL U C < 0 A sufficient condition is therefore U CL when leisure is a normal good, in a bad state, there is more leisure and hence the marginal utility of consumption increases. Then, to equate this marginal utility across states, higher earnings are needed in states with more leisure. (ESL-UCL) 19 / 102

20 1. Stylized narrow view about motivation 1.2. Risk-sharing An individual insurance contract model Properties of the contract if employers are risk neutral 4 If leisure is a normal good, the real wage rate w evolves counter-cyclicly: Deriving W (ε) w(ε) h(ε) wrt ε and using (4), we find that: dw dε = (U CL wu CC ) dh dε < 0 hu CC 5 Other implications if leisure is a normal good: 1 the agent s utility is counter-cyclical: du/dε = U C dw /dε U L dh/dε = [U C (U CL /U CC ) U L ]dh/dε < 0. 2 the firm has an interest to announce to the worker that the state ε is good, since then the worker accepts a contract in which he works more and receives a lower compensation W : profits then increase (see below) (ESL-UCL) 20 / 102

21 1. Stylized narrow view about motivation 1.2. Risk-sharing An individual insurance contract model Exercise Consider the utility function U[W + φ(1 h)] with U > 0, U 0, φ > 0 and φ 0. What are the signs of U CL and U CC U L U CL U C? Knowing this revisit Properties 1 to 5 above. Properties 3, 4, 5-1 are odd. Extension: Insurance and ex-post labor mobility. Prop. 5-2 what if ε non verifiable? Extension: Asymmetric or unverifiable information. (ESL-UCL) 21 / 102

22 1. Stylized narrow view about motivation 1.2. Risk-sharing Insurance and ex-post labor mobility p What if ex-post, there is a spot market where the worker can get the competitive spot wage W (ε)? Assume a stationary framework: A1 The number of hours worked is fixed: utility of an employee is a function of wage only U(W ); infinite life and discount factor δ ]0, 1[. A2 f (h, ε) = ε A3 Expected lifetime profits = E[ε W (ε)]/(1 δ), A4 Verifiability: Unchanged. A5 The principal is risk neutral A6 The worker can quit at each moment and earn the outside wage W (ε), where W (ε) > 0 Without mobility, insurance contract constant W (and w). (ESL-UCL) 22 / 102

23 1. Stylized narrow view about motivation 1.2. Risk-sharing Insurance and ex-post labor mobility Expected present value outside the contract: V (ε) = U[W (ε)] + δe V (θ) Expected present value in a firm offering a contract A = {W (ε)}: V (ε) = U [W (ε)] + δemax [ V (θ), V (θ) ] State-specific participation constraints: V (ε) V (ε), ε If U denotes EU[W (ε)], the participation constraints can be written as: U [W (ε)] + δ δ EU [W (θ)] U[W (ε)] + U, ε (6) 1 δ 1 δ (ESL-UCL) 23 / 102

24 1. Stylized narrow view about motivation 1.2. Risk-sharing Insurance and ex-post labor mobility Maximizing expected profits s. to the participation constraints (6): ε W (ε) max A={W (ε)} 1 δ [ +λ(ε) U [W (ε)] + δ δ EU [W (θ)] U[W (ε)] 1 δ 1 δ U where λ(ε) 0 is the Lagrangian multiplier associated to (6). For any ε, the F.O.C. w.r. to W (ε) can be written as: ] g(ε)dε [(1 δ)λ(ε) + δeλ(θ)] U [W (ε)] = 1 (7) (ESL-UCL) 24 / 102

25 1. Stylized narrow view about motivation 1.2. Risk-sharing Insurance and ex-post labor mobility Exploiting (7), i.e. [(1 δ)λ(ε) + δeλ(θ)] U [W (ε)] = 1: Let Λ + = {ε λ(ε) > 0} (participation constraints are binding). From (7), W (ε) is no more fully rigid for ε Λ +. Let ε 1 > ε 2, both in Λ +. Subtracting equalities (6), i.e. leads to U [W (ε)] + δ δ EU [W (θ)] = U[W (ε)] + 1 δ 1 δ U U [W (ε 1 )] U [W (ε 2 )] = U [ W (ε 1 ) ] U [ W (ε 2 ) ] > 0 sincew (ε) > 0 W (ε) > 0 over the set Λ +. Then, from (7), in Λ +, as U < 0, one has λ (ε) > 0. Λ + = {ε ε ε λ }, i.e. the set Λ + is made of all ε above a certain threshold value (ε λ ). (ESL-UCL) 25 / 102

26 1. Stylized narrow view about motivation 1.2. Risk-sharing Insurance and ex-post labor mobility Figure: 6.2 The wage contract with labor mobility If principal could break the contract if profits become too low, then wages should not be completely rigid downwards. (ESL-UCL) 26 / 102

27 1. Stylized narrow view about motivation 1.2. Risk-sharing Asymmetric or Unverifiable Information Static model Only the (risk-neutral) principal observes the true values of the productivity shock ε. The principal may have an incentive to hide this information to the agent. How can one design a contract such that the principle has an incentive to reveal the correct information? The Revelation Principle Consider any contingent contract A = {W (ε), h(ε)}. For any true state ε, the principal announces the state m(ε) that maximizes its profits: m(ε) = ArgMax {f [h(θ), ε] W (θ)} (8) θ (ESL-UCL) 27 / 102

28 1. Stylized narrow view about motivation 1.2. Risk-sharing The Revelation Principle Consider now a contingent contract  = {Ŵ (ε), ĥ(ε)} = {W (m(ε)), h(m(ε))} Contract  is incentive-compatible, i.e. the principal can never make more profit by lying then by revealing the truth. Assume that  is in force, the principal announces that the state is θ while it is actually ε. Then the principal makes a loss: ] f [ĥ(θ), ε Ŵ (θ) f {h[m(θ)], ε} W [m(θ)] Max {f [h(s), ε] W (s)} s f [h(m(ε)), ε] W (m(ε)) = f [ĥ(ε), ε] Ŵ (ε) Contracts A and  lead to the same allocations and compensations. (ESL-UCL) 28 / 102

29 1. Stylized narrow view about motivation 1.2. Risk-sharing Asymmetric or Unverifiable Information The Incentive-Compatible contract Since it is possible to associate any contract with an incentive-compatible contract that leads to the same allocation of resources, the search for the optimal contract can be confined to the set of incentive-compatible contracts. This second-best contract solves: max E Π(ε) s. to A EU (W (ε), 1 h(ε)) U f [h(ε), ε] W (ε) f [h(θ), ε] W (θ), (ε, θ) (I-C) In general, complex problem! (ESL-UCL) 29 / 102

30 1. Stylized narrow view about motivation 1.2. Risk-sharing Asymmetric or Unverifiable Information The first-best contract (i.e. the one when (I-C) contraints are ignored) is also optimal under asymmetric information if leisure is not affected by the income level. For then, from the optimality conditions, announcing that the state is better than the true one increases real output and the wage bill to the same extent (Formula (12) in CZ2004). If U CL 0 (hence, leisure normal good), we saw (see Prop 5.2 above) that the first-best contract is characterized by dh/dε > 0 > dw /dε. So, the firm has an interest to announce to the worker that the state is good (actually, ε + ) since then the worker accepts a contract in which he works more and receives a lower compensation W. If leisure is an inferior good, the firm announces ε. (ESL-UCL) 30 / 102

31 1. Stylized narrow view about motivation 1.2. Risk-sharing Asymmetric or Unverifiable Information An example with two equiprobable states of nature ε + > ε A3 f (h, ε) = εh ( f h (h, ε) = ε). The Principal s Problem Max (h i,w i ) i=+, Subject to constraints: [ 1 ( ε + h + W +) + 1 ( ε h W )] U(W +, 1 h + ) U(W, 1 h ) U (14) ε + h + W + ε + h W (15) ε h W ε h + W + (16) (ESL-UCL) 31 / 102

32 1. Stylized narrow view about motivation 1.2. Risk-sharing Asymmetric or Unverifiable Information An example with two equiprobable states of nature ε + > ε The optimal contract when leisure is a normal good The principal only has an incentive to lie if the bad state realises; The incentive compatible contract avoids this, since it requires the principal to pay a high wages if it announces a good state. wages (and hours) and economic conditions unambiguously move in the same direction: h + > h, W + > W. The FOC also results in over-employment in good states, but not in underemployment in bad states: MRS CL (ε + ) = U L(W +, 1 h + ) U C (W +, 1 h + ) > ε+ = MPL(ε + ) (22) MRS CL (ε ) = U L(W, 1 h ) U C (W, 1 h ) = ε = MPL(ε ) (23) (ESL-UCL) 32 / 102

33 1. Stylized narrow view about motivation 1.2. Risk-sharing Risk-sharing Conclusion It is standard to conclude that taking risk-sharing between employers and employees into account can help understanding the low variability of real wages, can help understanding the procyclicity of hours and compensation, does not yield insight into the reasons for underemployment in bad states. The last conclusion is however questioned by e.g. Drèze (1997) who stresses that coordination failures sustained by wage rigidities cause pervasive underutilisation of resources. For him, risk-sharing is a natural reason for these wage rigidities. (ESL-UCL) 33 / 102

34 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable 3. Incentives if results are verifiable 3.1 The principal-agent model with hidden action The problem 1 Actions followed by the realisation of some random process lead to results of an agent s activity. 2 Actions of the agent (effort) are not verifiable by a third party, 3 but results of actions are. 3 Consequence: a trade-off between providing incentives (to induce effort) and insurance. The solution: An explicit performance contract: 1 Define the action of the agent as a reaction to a wage contract, before a random shock, affecting the result of this action, realises. 2 Determine the choice of the principal, anticipating the action of the agent 3 In the words of some authors: effort is not contractible. (ESL-UCL) 34 / 102

35 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable The principal-agent model with hidden action Assumptions and Notations: Timing of decisions Decisions unfold in the following sequence: 1 the principal offers a contract; 2 the agent accepts the contract, or turns it down; 3 if the agent turns it down, the protagonists go their separate ways, but if the agent accepts it, he or she then supplies an effort; let e denote the optimally chosen effort level. 4 a random event ε which affects the result of the agent s effort e occurs; 5 the principal and the agent observe the result; 6 the principal remunerates the agent according to the terms of the contract Note: This requires that the notion of the result of an agent makes sense. OK for a salesman. Also in the case of teams of workers??? (ESL-UCL) 35 / 102

36 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable The principal-agent model with hidden action Assumptions and Notations A1 Preferences of the agent of the CARA type: U [W C(e)] = exp { a [W C(e)]} (24) W is wage income, e the agent s unverifiable effort, C(e) is the cost of effort, where C > 0 4 and C > 0, and a > 0 is the index of absolute risk aversion, equal to U /U. A2 Effort results in a certain level of verifiable but random production: y = e + ε, where ε is a normal random variable with zero mean and standard error σ. A3 The explicit performance wage contract is linear in y: W = w + by, where w is a fixed wage and b is a piece-rate on production. 4 With a more general specification U(W, e) one could let U/ e > 0 over some range. Yet, the optimum cannot be in this range since the worker would unilaterally increase e and hence profits and his own utility. (ESL-UCL) 36 / 102

37 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable The principal-agent model with hidden action The agent s effort level If the agent accepts the contract, he takes the wage contract as given and chooses his effort as to maximise expected utility: EU = exp { a [w + be C(e)]} E [exp( abε)] { = exp a [w + be C(e) ab2 σ 2 ]} 2 since the exponential of a Normal random [ variable ] X N (µ, σ 2 ) is log-normally distributed with mean exp µ + σ2 2. The chosen effort e trades-off the benefits and costs of marginally increasing effort: C (e ) = b defines e = e (b) with de db = 1 C (e ) (ESL-UCL) 37 / 102

38 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable The principal-agent model with hidden action The Principal s Behaviour The expected profit of the principal is E(y W ) = E(1 b)(e + ε) w = (1 b)e w. Max [(1 {w, b} b)e w] s. to C (e ) = b, EU U exp{ a x} (25) The binding participation constraint can be rewritten as: w = x be + C(e ) + ab2 σ 2 The optimisation problem can therefore be rewritten as: [ ] Max e (b) C[e (b)] a {C [e (b)]} 2 σ 2 x {b} 2 2 (26) (ESL-UCL) 38 / 102

39 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable The principal-agent model with hidden action The Optimal Remuneration Rule The piece rate (or variable part of total remuneration) is b = ac (e )σ 2 < 1 b a b < 0, < 0, (27) σ2 and w is given by (26) evaluated at b. (27) captures the trade-off between providing incentives and insurance: The magnitude of the piece rate decreases with absolute risk-aversion the variance of the noise ε the concavity of the cost of effort (i.e. how the marginal cost of effort varies) Note: risk-aversion is taken into account by the principal because of the participation constraint EU U. (ESL-UCL) 39 / 102

40 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable The principal-agent model with hidden action The Optimal Remuneration Rule: An example Let C(e) = c e 2 /2, c > 0. Then as C (e) = c i.e. is constant: b = 1/(1 + a σ 2 c), where aσ 2 is called the risk factor, and e = b /c. Contrary to Formula (29) in the book, I find that w = x 1 a σ 2 c 2c (1 + a σ 2 c) 2 = x (b ) 2 1 a σ2 c 2c A rise in σ or in a induces an increase in the fixed wage component w if a σ 2 c < 3. (ESL-UCL) 40 / 102

41 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable The principal-agent model with hidden action Two limit cases for comparison Comparison with the first-best outcome: e verifiable is part of the contract chosen by the firm. The latter solves: Max [e (w + be)] subject to w + be C(e) ab2 σ 2 {w, b,e} 2 Indexing the first-best with superscript o: b o = 0, C (e o ) = 1, w o = x + C(e o ) i.e. full insurance; e o > e since C (e ) = b < 1. If risk-neutral workers (a = 0), b = 1. x (ESL-UCL) 41 / 102

42 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable In practice, how does the employer know the employee s best response? The employer can arrive at an estimate of the best effort response C (e ) = b by varying b and observing the effects on output since on average Ey = e It is here assumed that the estimate e (b) is accurate. (ESL-UCL) 42 / 102

43 Empirical findings p Stylized narrow view about motivation 1.3. Incentives if results verifiable 1 Empirical findings confirm theory: move to incentive contract raises productivity; increases variance of performance; reduces (increases) quit rate of most (least) productive workers. attracts more productive applicants. 2 However, financial incentives can have counterproductive effects: experimental and field evidence indicates that extrinsic motivation (contingent rewards) can sometimes conflict with intrinsic motivation (the individual s desire to perform the task for its own sake) (Bénabou and Tirole, 2003, p. 490) : In what cases should financial incentives be used with caution? A growing literature that requires a less narrow view about human motivation... see Part 2. (ESL-UCL) 43 / 102

44 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable 3.2 Should Remuneration Always Be Individualised? Initial idea: Why a contrat influenced exclusively by individual production? If there are other verifiable variables, their utilization could lead to more efficient contracts. The Agency Model with Two Signals A5 The principal observes a verifiable signal ε N(0, σ 2 ) that is possibly correlated with the component of individual production y that is unrelated to effort, i.e. with ε N(0, σ 2 ): cov(ε, ε) = ρσ 2. Main application: If the results of the agent s effort are correlated to the results of colleagues (because, say, of common shocks). A3 The linear wage contract takes the following form: W = w + by b ε (ESL-UCL) 44 / 102

45 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable The Agency Model with Two Signals Now, the expected utility is: [ [ EU = exp { a [w + be C(e)]} E exp( a bε b ε ] ] ) { = exp a [w + be C(e) aσ2 ( b b ) ]} 2 2ρb b The chosen effort still verifies: C (e ) = b which defines e = e (b) As E ε = 0, the problem of the principal is: Max {w,b, b} [(1 b)e w] s.to C (e ) = b, EU exp{ a x} (ESL-UCL) 45 / 102

46 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable The Optimal Compensation Rule The binding participation constraint can be rewritten as: w = x be + C(e ) + aσ2 2 ( b 2 + b ) 2 2ρb b (30) The optimisation problem can therefore be rewritten as: Max [e (b) C[e (b)] aσ2 ( b 2 + {b, b} 2 b ) ] 2 2ρb b x (ESL-UCL) 46 / 102

47 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable The Optimal Compensation Rule From the f.o.c. s: b = a[1 ρ 2 ]C (e )σ 2 < 1 b b > 0; ρ a, b σ 2 < 0, b = ρb d b dρ = b + ρ b ρ W = w + b y b ε, w from (30) > 0 if ρ > 0, 1 ρ = 0 b = 0: The second signal just adds in noise ignore it 2 1 ρ > 0 b b > 0. An increase in the signal ε reduces the compensation W since the positive correlation induces that (statistically) y is higher because ε is bigger, too. 3 1 ρ < 0 b < 0 < b. (ESL-UCL) 47 / 102

48 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable 3.3 Some reasons why performance pay may be inefficient Multitasking A1 The preferences of the agent are as in A1: CARA utility function but with two efforts; effort (time) devoted to two distinct tasks is perfectly substitutable: C(e 1, e 2 ) = C(e 1 + e 2 ). A2 y 1 = e 1 + ε 1, y 2 = e 2 + ε 2 where ε i N(0, σ 2 i ), i = 1, 2. cov(ε 1, ε 2 ) = σ 12 A3 W = w + b 1 y 1 + b 2 y 2. Hence, W = W (ε 1, ε 2 ). The agent s behaviour: Max EU {e 1,e 2 } { [ = Max exp a w + b 1 e 1 + b 2 e 2 C(e 1 + e 2 ) {e 1,e 2 } a ]} 2 Var[W (ε 1, ε 2 )] (ESL-UCL) 48 / 102

49 Multitasking 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable The FOC s are: b i = C (e1 + e 2 ), for i = 1, 2. This implies that b1 = b 2. The principal must remunerate the effort devoted to each activity equally. Otherwise the agent will devote all effort to the activity that is best remunerated.it can be checked that: b 1 = b 2 = ac (e 1 + e 2 )(σ σ 12 + σ 2 2 ) Consequence: Suppose that the performance of one of the agent s tasks cannot be measured: e.g. σ 1. Then b1 = b 2 = 0. So, whenever it s difficult to measure the performance of one task, performance pay becomes inefficient if e 1, e 2 perfect substitutes. (ESL-UCL) 49 / 102

50 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable Supervision and rent-seeking Often the principal does not observe agent s output but well supervisors who are themselves agents: 1 To avoid friction with collaborators, supervisors tend to write favourable performance reports problem of measurement of performance. 2 Or, agents try to influence performance reports by undertaking actions that attempt to "impress" supervisors: rent-seeking or lobbying The book focuses on case 2. (ESL-UCL) 50 / 102

51 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable A Model with Rent-Seeking A1 Preferences and cost of effort as in A1. In addition, there is a cost for rent-seeking activity α: K (α), where K > 0 and K > 0. A3 Output is y but the supervisor reports to the principal that it is y + α W = w + b(y + α) Observe that α has no productive value: Assumption A2 is maintained: y = e + ε The behaviour of the agent C (e ) = K (α ) = b The Optimal Pay Scheme [ Max b e (b) C[e (b)] K [α (b)] a {C [e (b)]} 2 σ 2 2 x ] (ESL-UCL) 51 / 102

52 1. Stylized narrow view about motivation 1.3. Incentives if results verifiable A Model with Rent-Seeking The Optimal Pay Scheme b = C (e ) K (α ) + ac (e )σ 2 The less costly rent-seeking, the smaller K (α ), the smaller is the performance pay. With risk-neutral agents (a = 0) the first-best solution, b = 1, can no longer be attained. Conclusion: If performance is not verifiable due to complex tasks, components of which are not measurable or due to rent seeking activities, then other contractual should be designed to provide incentives: Promotions on the basis of relative performance; Seniority rules in long-term contracts. (ESL-UCL) 52 / 102

53 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results 4. Incentives when results are not verifiable What if both, effort and individual performance are unverifiable? Double moral hazard problem. Answers: 4.1 An internal market + a system of promotions 1 if relative performance (= ordinal) is easier to measure than absolute performance (= cardinal); 2 if relative performance is verifiable: One can publicly announce in advance the wage increase to which the promotion entitles and the number of promoted workers = verifiable clauses; One should limit rent-seeking behaviour: (i) if the firm systematically favours candidates without merit, this will harm the firm s reputation and one will no longer be willing to work in this firm. (ii) One may design procedures such that rent-seeking behaviour is hampered: e.g. assign outside members to the promotion committee. 4.2 Compensation rules based on seniority (ESL-UCL) 53 / 102

54 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results 4.1 Promotions and Tournaments A (single period) Tournament Model A1P (Homogeneous) agents are risk-neutral: U = W C(e), where C(0) = 0, C > 0 and C > 0. A2P Effort (e 0) results in a certain level of unverifiable and random production: y = e + ε, where ε is a normal random variable with zero mean and standard error σ. CDF Φ( ), φ( ) Φ ( ). All the workers specific ε are. A3P A large firm with N employees, among which L will be promoted and get a bonus b in addition to the base wage w 0. Wage bill = w 0 N + bl. Announcing {w 0, b, L} is formally equivalent to a contract {w 0, b, y}, where promotion takes place if y > ȳ > 0. Reason: if N sufficiently large, L N = [1 Φ(ȳ e )]. A4P The expected profit per capita of the firm is: Eπ = E(y W ) = e w 0 b [1 Φ(ȳ e )] (ESL-UCL) 54 / 102

55 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results A Tournament Model The Behaviour of the Agent MaxEU = Max {w 0 + b [1 Φ(ȳ e)] C(e)} (33) e e The FOC leads to the following incentive constraint: b φ(ȳ e ) = C (e ) e = e (b, ȳ) (34) The 2 nd order condition requires: bφ + C > 0. One can check that de /db 0 and de /dȳ 0. The Behaviour of the Principal The principal maximises expected per capita profits taking the incentive and participation constraints (EU > U) into account. The latter binds ex ante (not ex post for those promoted) and verifies the following expression: w 0 + b [1 Φ(ȳ e )] = U + C(e ) (36) (ESL-UCL) 55 / 102

56 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results A Tournament Model Inserting these in the expression of expected profits (per capita) results in the following optimisation problem: { Max e (b, ȳ) C[e (b, ȳ)] U } {b,ȳ} The FOC determines the optimal level of effort (if de /db 0 and de /dȳ 0): C (e ) = 1 (37) Note that: Given the assumed properties of C( ), e is unique. Condition (37) entails the first-best level of effort: a consequence of risk-neutrality of the agent. (ESL-UCL) 56 / 102

57 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results The contract is characterized by {w 0, b, y}. Given the first best value e, this vector solves the incentive constraint: and b φ(ȳ e ) = C (e ) = 1 (34) w 0 + b [1 Φ(ȳ e )] = U + C(e ) (36) where as before U is exogenously given. So there remains a degree of freedom (say, w 0 ). In the original paper (Malcomson, 1984), the model has 2 periods and the promotion takes place between the 1 st and the 2 nd one (in between the worker can leave and take a job in a period-2 spot market ). This complicates the model and imposes more constraints on the optimal contract. The number of promoted is L = N [1 Φ(ȳ e )] (assuming that the law of large numbers can be applied!). (ESL-UCL) 57 / 102

58 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results A Tournament Model Increasing risk i.e. σ The density of a normal distribution satisfies the following conditions: φ(0) = 1/σ 2Π and φ (0) = 0 Therefore, if e is sufficiently close to ȳ, the incentive and participation constraints (34 and 36) jointly with the FOC (37) can be approximated as follows: b σ 2Π and [1 Φ(ȳ e )] C(e ) + U w 0 σ 2Π Increasing risk (σ) increases the wage gap, b, and reduces the proportion of promotions, (1 Φ). (ESL-UCL) 58 / 102

59 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results A Tournament Model Increasing risk i.e. σ If the risk increases with the hierarchical grade (e.g. because jobs become increasingly complex), we should observe the wage gain to increase with this number; (Unmodelled) workers risk-aversion, on the other hand, should decrease the wage gain. The following figure illustrates the first prediction. It concerns a large US firm of the service sector. Eight levels are distinguished. The average wage corresponding to each grade increases at an increasing rate as we move up the hierarchy. This figure also shows a dispersion of wages within each grade ( in addition to the tournament, another incentive scheme is in place). (ESL-UCL) 59 / 102

60 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results A Tournament Model Empirical evidence Figure: Salary ranges by level of hierarchy (ESL-UCL) 60 / 102

61 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results Tournaments and Rent-seeking Are seniority rules so bad after all? A1P In addition to A1, the worker can engage in a rent-seeking activity, α to impress his supervisor. This entails cost, K (α), where K > 0 and K > 0. A3P If y + α = e + ε + α > r, then W = w 0 + b, b > 0; Otherwise, W = w 0. For any r and b, the agent maximises expected utility wrt e and α: Max {w 0 + b [1 Φ( r e α)] C(e) K (α)} {e,α} This results in the following FOC: bφ( r e α ) = C (e ) = K (α ) (40) (ESL-UCL) 61 / 102

62 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results Tournaments and Rent-seeking Are seniority rules so bad after all? Inserting this together with the participation constraint w 0 + b [1 Φ( r e α )] = U + C(e ) + K (α ) (41) in the maximisation problem of the principal, this yields: Max {b, r} {e (b, r) C[e (b, r)] K [α (b, r)]} This results in the following FOC: K (α ) = C (e ) = K (α ) C (e ) + K (α ) < 1 (42) The interest in staging promotions is lessened since effort falls. Other problems: sabotage by competing colleagues; issues of fairness. (ESL-UCL) 62 / 102

63 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results Exercise 1 What is an explicit performance contract? To answer that question, you do not need to model such a contract. Explain in words the economic assumptions needed for such a contract and the main features of it. 2 Why do employers offer such a contract rather than a fixed wage contract? 3 Linear performance contracts, maximizing profits, usually index wages only partially and not completely to performance (single task, no rent-seeking). What are the key factors that determine the degree in which wages should be linked to performance? How do they affect the optimal performance contract and thereby profits? What s the intuition behind? 4 Why may it be sensible rather than directly relating wages to measured performance - to link wages to a system of promotions? (ESL-UCL) 63 / 102

64 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results 4.2 The role of seniority The shirking model Dynamic model where the agent s production is not verifiable. We search for a self-enforcing contract (the partners stay together if and only if the two parties have a mutual interest to do so). A1S The worker is risk neutral and exerts two levels of effort: 0 or e > 0 to which are associated costs of respectively 0 and C > 0. A2S Production at date t, y t > 0, is only realised if e > 0, but due to costs of supervision the firm inspects the production level (and therefore effort) only with an exogenous probability p (0, 1). A3S Since effort is unverifiable the wage cannot be linked to the level of effort. Instead the employer offers a contract {w t ; t = 0, 1,.., + } that is paid until the worker is caught shirking; if caught the worker is paid the wage until the end of the period and is fired. (ESL-UCL) 64 / 102

65 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results The model The Behaviour of the Principal In a discrete time framework, δ (0, 1] being the discount factor and q (0, 1) an exogenous probability of separation, the expected discounted profit solves: Π t = y t w t + δ [ (1 q)max(π t+1, Π s t+1 ) + q Π t+1 ] (43) where Π t+1 designates the profit if the contractual relationship ends and Π s t+1 the expected profit is the firm cheats (i.e. breaks the contract, wrongly claiming that the worker shirks ). The latter is given by Π s t = y t w t + δ Π t+1 (44) (ESL-UCL) 65 / 102

66 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results The shirking model The Behaviour of the Principal The employer s incentive constraint is Π t Π s t : Π t Π s t = δ(1 q) [ Max(Π t+1, Π s t+1 ) Π t+1 ] 0, t 0 Π t+1 Π t+1, t 0 Only future rents matter! If one also considers the participation constraint, Π 0 Π 0, we obtain Π t Π t, t 0 (ESL-UCL) 66 / 102

67 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results The shirking model The Behaviour of the Agent V t = w t C + δ [ (1 q)max(v t+1, V s t+1 ) + q V t+1 ] (45) where Vt+1 s is the expected lifetime utility of the worker who shirks and V t+1 is the outside utility. The former is equal to V s t = w t + (1 p)δ [ (1 q)max(v t+1, V s t+1 ) + q V t+1 ] + pδ Vt+1 (46) The worker s incentive constraint is then V t V s t = C + pδ(1 q) [ Max(V t+1, V s t+1 ) V t+1 ] 0, t 0 V t+1 V t+1 C > 0, t 0 (47) pδ(1 q) (ESL-UCL) 67 / 102

68 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results The shirking model Rent and Set of Feasible Contracts The set of feasible contracts, P, is defined by: { } (Π t, V t ) Π t Π t, V t+1 V C t+1 pδ(1 q), V 0 V 0, t 0 (48) Surplus and the Existence of Self-Enforcing Contracts The global surplus is the sum of rents procured by the contract: S t V t V t + Π t Π t, t 0 By adding up (43) and (45), the surplus is defined by: S t δ(1 q)s t+1 = y t C + δ( Π t+1 + V t+1 ) ( V t + Π t ), t 0 (49) the surplus is exogenous since the RHS is taken as exogenous by the partners. (ESL-UCL) 68 / 102

69 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results The shirking model Rent and Set of Feasible Contracts Using that Π t Π t = S t (V t V t ) for all t 0, the set P can be characterised in relation to the exogenous surplus. So, P can be rewritten as the set of {V t } t 0 such that: S t+1 V t+1 V t+1 C pδ(1 q), S 0 V 0 V 0 0, t 0 (50) A self-enforcing contract will therefore only exist if P S 0 0 and S t+1 C, t 0 (51) pδ(1 q) The non-verifiability of effort and performance leads to Pareto inefficiency: if the surplus is positive, but not large enough, mutually advantageous production is not realised: explains why workers with a low productivity are excluded from long-term contractual relationships. (ESL-UCL) 69 / 102

70 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results The shirking model Rent and the Optimal Contract The optimal contract is found by maximising the expected profits Π t = V t + ( V t + Π t + S t ) over P, t. As V t + Π t + S t is exogenous, this results in choosing the lowest value of V t over the set P: (i) if P, then V 0 = V 0, V t+1 = V t+1 + C/pδ(1 q), t 0 (ii) if P =, no self-enforcing contract exists. (52) The worker does not obtain any rent at time t = 0: V 0 = V 0 ; Incentives to provide effort result from a strictly positive rent in all subsequent periods (t > 0); The principal captures the entire surplus at t = 0: Π 0 Π 0 = S 0 0. (ESL-UCL) 70 / 102

71 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results Seniority, Experience and Wage The Reasons Why Wages Rise with Seniority 1 The accumulation of general human capital leads to an increasing relationship between experience (i.e. employment durations) and wages; 2 The accumulation of specific human capital leads to an increasing relationship between seniority (i.e. duration of employment in the same firm) and wages if workers have some bargaining power; 3 On-the-job search leads to an increasing relationship between experience and wages; 4 Revelation of information on the worker s abilities allows to assign to worker to tasks that match their abilities and may therefore lead to an increasing relationship between seniority and wages; 5 A deferred payment scheme provides incentives to work hard and explain why wages rise with seniority. (ESL-UCL) 71 / 102

72 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results Seniority, Experience and Wage The Optimal Wage Profile in the Shirking Model Combining the expression for the expected utility of the worker (45) and the relation (52) defining an optimal contract and assuming that the outside opportunities procure an instantaneous utility w t C, the optimal wages are: w 0 = w 0 C p < w 0 (55) w t = w t + C [ ] 1 p δ(1 q) 1 > w t, t 1 (56) Proof of (56): By (45) in the set P, V t = w t C + δ(1 q) [ ] V t+1 V t+1 + δv t+1 w t = V t δv }{{ t+1 + C + C [ ] 1 } p δ(1 q) 1 (53) w t C (ESL-UCL) 72 / 102

73 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results Seniority, Experience and Wage The Optimal Wage Profile in the Shirking Model Figure: The profile of optimal wages with general human capital and deferred payments. (ESL-UCL) 73 / 102

74 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results Seniority, Experience and Wage A bonding mechanism The optimal wage profile described by relations (55) and (56) is interpretable as an incentive mechanism in which there is a bond the agent is obliged to post at the time of hiring which will be gradually paid back to him or her. Is such a bonding mechanism plausible? Arguably, not (see Section 4.4.). (ESL-UCL) 74 / 102

75 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results Seniority, Experience and Wage The Optimal Wage Profile in the Shirking Model The deferred payment takes an extreme form: Only the hiring wage w 0 is lower than the outside wage w 0 (e.g. the competitive wage). As of t = 1, dw t /dw t = 1. General human capital affects the outside wage and hence w t but specific human capital does not affect w t, hence nor w t (recall that workers bargaining power = 0!) More generally, a less extreme form of deferred payment if heterogeneity in the ability of workers; and time required for these abilities to be revealed to the employer. (ESL-UCL) 75 / 102

76 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results 4.3 Empirical Elements on Experience, Seniority, and Wages Estimating the return to seniority Let w ijt be the wage of individual i in firm j in year t. With "ed" for education, "ex" for experience and "te" for seniority and ignoring non linearities, the basic equation writes: Problem: ln w ijt = β + β t t + β ed ed ijt + β ex ex it + β te te ijt + ε ijt (57) 1 Unobserved heterogeneity bias: More efficient workers also have larger seniority and experience. 2 Endogeneity bias: participation and therefore experience is positively related to wages; quits are negatively related to wages and therefore seniority positively. (ESL-UCL) 76 / 102

77 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results Estimating the return to seniority To see this more clearly we decompose the error term as follows: ε ijt = η i + µ ij + ζ ijt (58) where η i is a fixed unobserved individual component, µ ij is an unobserved match specific component of worker i employed in firm j and ζ ijt is a random term. In general, η i and µ ij are correlated with experience and seniority and the OLS estimator of equation (57) is therefore inconsistent. (ESL-UCL) 77 / 102

78 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results Estimating the return to seniority Solutions 1. Abraham and Farber (1987) ln w ijt = β + β t t + β ed ed ijt + β ex ex ijt + β te te ijt + β du du ij + ε ijt where du ij is the total effective job duration of worker i in firm j. Problems: right censoring of these durations; does not account for correlation between unobserved individual effects and experience; It does not account for variation of the match specific effect over the job spell: µ ijt instead of µ ij. (ESL-UCL) 78 / 102

79 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results Estimating the return to seniority Solutions 2. Topel (1991) ln w ijt = b + β t t + b ed ed ijt + b ex ex 0 ij + b te te ijt + ε ijt where the term ex 0 ij designates the experience of individual i at the time he or she is hired for job j. 1 Estimate this equation in differences, retaining only workers who do not change jobs between 2 dates. This identifies a consistent estimator of b te ; 2 In a second stage Topel fits the following equation: ln w ijt ˆb te te ijt ˆβ t t = b + b ed ed ijt + b ex ex 0 ij + ε ijt (59) where ˆb te ˆb ex is the return to seniority. (ESL-UCL) 79 / 102

80 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results Estimating the return to seniority Topel (1991) Problems: 1 First step is consistent only if the wages grow at same rate independently of whether workers stay or not within the firm. Justifies this by the finding that all wages vary according to a random walk; 2 In the second step, b ed may be biased upwards, since experience can be positively correlated with the unobserved productivity components of workers ˆb te ˆb ex is a lower bound to the return of seniority. Conclusion Researchers tackling these problems find results close to Topel s: 10 years of seniority increases wages by 25% and 10 years of experience by 34%. One also finds that returns to seniority in an industry are more important than within a firm. (ESL-UCL) 80 / 102

81 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results On the Mechanism of Deferred Payment 1 See figure 6.5 in CZ: The time profile of the wage income of a salesperson follows closely the productivity profile (output of the salesperson is arguably verifiable); For managers the wages start off below productivity and ends above (output arguably not verifiable). [ 2 Formula (56), namely w t = w t + C 1 p δ(1 q) ], 1 predicts w t / p < 0. Firms (hospitals) pay higher wages the lower the supervisor/employee ratio (proxy for the probability p that effort is verified). (ESL-UCL) 81 / 102

82 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results 4.4. Efficiency wages and involuntary unemployment Since credit markets are imperfect, Shapiro and Stiglitz (1984) argue that the bonding mechanism is infeasible; Assume instead that employer must pay the same wage in every period This results in involuntary unemployment as an incentive device. Shapiro and Stiglitz (1984) consider an economy populated with homogeneous workers and firms. (ESL-UCL) 82 / 102

83 1. Stylized narrow view about motivation 1.4. Incentives in the Absence of Verifiable Results Efficiency wages A particular stationary version of the shirking model Shapiro and Stiglitz (1984) assume a stationary version where V t = V, t 0. Then, (53) simplifies: w t = w = V t δv }{{ t+1 + C + C [ ] 1 } p δ(1 q) 1 (1 δ)v (60) Moreover in the set P, t, one has V V = C/pδ(1 q) > 0, where V is here interpreted as the expected utility of an unemployed person: V = z + δ [ sv + (1 s)v ] C (1 δ)v = z + s p(1 q) in which z is the gains in unemployment and s is the probability of being hired. Note: shirking = a misconduct z unemployment benefits as these are normally only paid to the involuntary unemployed. (ESL-UCL) 83 / 102