Minimum Wages and Excessive E ort Supply

Minimum Wages and Excessive E ort Supply Matthias Kräkel y Anja Schöttner z Abstract It is well-known that, in static models, minimum wages generate positive worker rents and, consequently, ine ciently low e ort. We show that this result does not necessarily extend to a dynamic context. The reason is that, in repeated employment relationships, rms may exploit workers future rents to induce excessively high e ort. Key Words: bonuses; limited liability; minimum wages JEL Classi cation: D82; D86; J33. We would like to thank an anonymous referee for valuable advice. Financial support by the Deutsche Forschungsgemeinschaft (DFG), grant SFB/TR 15, is gratefully acknowledged. y University of Bonn, Adenauerallee 24-42, D-53113 Bonn, Germany, tel: +49 228 733914, fax: +49 228 739210, e-mail: m.kraekel@uni-bonn.de. z University of Bonn, Adenauerallee 24-42, D-53113 Bonn, Germany, tel: +49 228 739217, fax: +49 228 739210, e-mail: anja.schoettner@uni-bonn.de. 1

1 Introduction As is well-known in economics, minimum wages are ine cient for at least two reasons: On an aggregate level, they may prevent labor market clearing, and on a disaggregate level, they can imply ine ciently low e ort. The latter problem has been highlighted by contract-theoretic models analyzing moralhazard problems under limited liability (see, among many others, La ont and Martimort, 2002, chapter 4; Schmitz 2005): When agents are protected by minimum wages or limited liability, they usually earn positive rents under an incentive contract. These rents raise the principal s costs of eliciting e ort. Consequently, he optimally induces less than rst-best e ort. We show that this conclusion may no longer hold in a dynamic setting. To do so, we also consider a moral-hazard problem under minimum wages, which leads to positive rents and ine ciently low e ort in a static model. However, in a two-period model, the principal optimally uses second-period rents to generate extra incentives for the agent in the rst period. This is achieved by combining a bonus contract with an extension clause that allows the agent to sign a second-period contract only if he was successful in the rst period. If either the worker s reservation value or the expected secondperiod rent is large, the principal uses the extra incentives to induce more than rst-best e ort. In practice, this "reversed" ine ciency problem of minimum wages (i.e., excessively large e orts) should typically apply to low-skilled blue-collar workers. The introduction of minimum wages that are enforced by law (or collective agreements) usually compels rms to increase wages for unskilled labor, whereas wages for high-skilled employees are una ected because they already earn more than the minimum wage. Translating this to our model means that, for low-skilled workers, the minimum wage constraint is binding. Hence, blue-collar workers are likely to earn rents in a one-shot game. In a dynamic environment, rms then optimally respond by exploiting these rents, which may result in ine ciently high e ort. The paper is related to the contract-theoretic literature on moral hazard 2

and limited liability, which usually considers a static relationship. 1 Two exceptions are Ohlendorf and Schmitz (2008) and Kräkel and Schöttner (2009), who also analyze a two-period principal-agent relationship and show that the principal employs future rents to generate extra incentives. However, in those papers, the agent s e ort remains below rst-best. We obtain a contrary result by considering a situation where the rm can replace the worker after the rst period. 2 The Model A rm needs one worker to carry out a task in each of two periods. In each period, the rm can randomly hire a worker from a pool of homogeneous agents available on the labor market. Alternatively, in period 2, the rm may again employ the worker hired in period 1. We assume that the rm can only o er short-term employment contracts, where the period-1 contract is supplementable by an extension clause. That is, we consider an incomplete contracting framework. All players are risk neutral. The monetary output of the worker hired in period t (t = 1; 2) is V t 2 fv l t; v h t g, where 0 < v l t < v h t and Pr[V t = v h t je t ] = p(e t ). The variable e t denotes the worker s e ort in period t, and p(e t ) is a probability function with p(e t ) 2 [0; 1), p 0 (e t ) > 0, and p 00 (e t ) 0. For simplicity, we assume that the minimum attainable output is identical in both periods, v1 l = v2 l = v, and de ne v t := vt h v. Thus, the expected monetary output in period t is E[V t je t ] = v + p(e t )v t. The minimum output v is known to the rm. Furthermore, at the beginning of t = 1, the rm also knows v 1. However, due to uncertainty about the future, v 2 is still unknown and considered to be the realization of a non-negative random variable with commonly known cdf F (). The rm learns v 2 at the beginning of t = 2. E ort is not observable, but output V t is veri able. Hence, at the beginning of period t, the rm o ers a worker a bonus contract (b Lt ; b Ht ) contingent on output V t, where the low bonus b Lt is paid to the worker if V t = v, and 1 Note, however, that there is a rich literature on repeated moral hazard with risk averse agents. Contrary to our paper, the focus of these models is on consumption smoothing and the renegotiation of long-term contracts. 3

the high bonus b Ht if V t = v h t. The rm s payment to the worker must be at least as high as the minimum wage, which is normalized to zero in both periods. Thus, b Ht ; b Lt 0. To supplement the period-1 bonus contract, the rm announces a probability q 2 [0; 1] of hiring the period-1 worker again in period 2, provided that the worker achieved a high output in period 1, i.e., if V 1 = v h 1 (extension clause). In each period, workers have a reservation value u 0. To guarantee that the rm is always interested in hiring a worker, we assume that the minimum output exceeds the worker s reservation value, v u. Exerting e ort e t entails cost c (e t ) with c (0) = c 0 (0) = c 00 (0) = 0 and c 0 (e t ) ; c 00 (e t ) > 0 for all e t > 0. Let e F t B denote rst-best e ort in period t, which is given by v t p 0 (e F t B ) = c 0 (e F t B ). Concavity of the rm s objective function in the second-best case is ensured by the technical assumption that the following function is convex: 2 G (e t ) := c0 (e t ) p 0 (e t ) p (e t) c (e t ) : (1) The timeline for each period t is the following: First, the rm observes v t. Then the rm o ers a bonus contract (b Lt ; b Ht ) supplemented by an extension probability q if t = 1. The worker accepts or rejects the contract. In case of acceptance, the worker chooses e ort e t. Finally, output and payo s are realized. 3 Solution to the Model We solve the problem by rst considering t = 2. If the worker has accepted the contract (b L2 ; b H2 ), his expected utility is EU 2 (e 2 ) = b L2 + (b H2 b L2 ) p (e 2 ) c (e 2 ) : (2) Hence, the worker optimally chooses e ort e 2 given by (b H2 b L2 ) = c 0 (e 2 ) =p 0 (e 2 ) ; (3) 2 For example, this assumption is satis ed for c(e t ) = e t and p(e t ) = e t with > 1, 2 (0; 1], and e t 2 [0; 1). In this case, we have G(e t ) = (= 1) e. 4

implying that EU 2 (e 2 ) (1) = b L2 +G (e 2 ). The strictly increasing function G (e 2 ) denotes the worker s expected gain from exerting e ort e 2, i.e., the resulting expected wage increase net of e ort cost. The rm maximizes v + v 2 p (e 2 ) b L2 (b H2 b L2 ) p (e 2 ), taking into account the worker s incentive constraint (3), the participation constraint (PC) EU 2 (e 2 ) u, and the minimum-wage condition (MWC) b L2 0. Thus, using (1), the rm s Lagrangian reads as follows: 3 L 2 (b L2 ; b H2 ) = v +v 2 p (e 2 ) b L2 G(e 2 ) c(e 2 )+ 1 [b L2 + G(e 2 ) u]+ 2 b L2 : Maximization leads to our rst result: 4 Proposition 1 (i) If u < G (e 2) with e 2 being implicitly de ned by v 2 p 0 (e 2) c 0 (e 2) G 0 (e 2) = 0, only the MWC will be binding and the rm induces e 2. (ii) If u 2 G (e 2) ; G e F 2 B, both MWC and PC will be binding and the rm implements e 2 with G (e 2 ) = u. (iii) If u > G e F 2 B, then only the PC will be binding and the rm implements e F 2 B. (iv) We have e 2 < e 2 < e F 2 B. Proposition 1 shows that an increasing reservation value u relaxes the MWC, thus leading to higher implemented e ort. The worker earns a positive rent if and only if case (i) applies. Because e 2 and, consequently, G (e 2) is increasing in v 2, case (i) occurs if v 2 is su ciently large. More precisely, v 2 needs to exceed the threshold ^v implicitly de ned by G (e 2(^v)) = u. Since we are interested in situations where workers may earn rents, we assume that v 2 > ^v occurs with positive probability. Thus, before uncertainty about v 2 is resolved, the expected rent of the period-2 worker is R := (1 F (^v)) (E [EU 2 (e 2) jv > ^v] u) > 0. We now turn to the optimal contract for t = 1. The period-1 worker earns R in the second period if V 1 = v h 1 and he is hired again in period 2 (which happens with probability q). Thus, a worker s expected utility from 3 Note that b L2 0 together with (b H2 b L2 ) = c 0 (e 2 ) =p 0 (e 2 ) ensures that b H2 0. Recall that e 2 is a function of (b L2 ; b H2 ). 4 All proofs are in the appendix. 5

accepting a contract in t = 1 is given by EU 1 (e 1 ) = b L1 + u + b H1 b L1 + Rq p (e 1 ) c (e 1 ). Hence, the worker chooses e ort according to b H1 b L1 + Rq = =p 0 (e 1 ). For t = 1, the PC is EU 1 (e 1 ) 2u and the MWC is b L1 ; b H1 0. We obtain the following result: Proposition 2 The rm optimally sets q = 1. If only the participation constraint is binding under the optimal contract (i.e., u is su ciently large), the rm will always implement e 1 > e F B 1. By contrast, if the participation constraint is not binding under the optimal contract, there exists a threshold ^R such that the rm implements e 1 > e F B 1 if and only if R > ^R. 5 According to Proposition 2, the rm optimally combines a bonus contract with an extension clause that guarantees a period-1 worker another contract in t = 2 in case of success (i.e., V 1 = v h 1 ). Thereby, the rm can use the entire expected second-period rent R to generate extra incentives in t = 1. The rent R reduces both absolute and marginal implementation costs for e 1. Concerning absolute implementation costs, the rm substitutes bonus payments by implicit incentives via R. Due to reduced marginal implementation costs, the rm has an incentive to implement higher e ort levels. If only the PC is binding in t = 1 (i.e., the worker earns no rent), the e ect of lower marginal implementation costs always leads to higher optimal e ort compared to a situation where q = 0 or, equivalently, a static employment setting. Since the rm would already implement rst-best e ort if q = 0, more than rst best e ort is implemented if q = 1. If the PC is non-binding, the rm faces a trade-o : On the one hand, the worker earns a positive rent G (e 1 ) which increases in the implemented e ort level; this e ect makes high e ort levels unattractive to the rm. On the other hand, reduced marginal implementation costs favor higher e ort levels. If the rent R is su ciently large, 5 For example, consider the case c(e t ) = e 2 t and p(e t ) = e t with e t 2 [0; 1). Then, G(e t ) = e 2 t and e F 1 B = v 1 =2. According to the proof of Proposition 2, the rm induces more than rst-best e ort if u > G(e F 1 B ) = v1=4 2 or R > ^R = maxfu; 2 p u; v 1 g. (Assuming v 1 + R < 2 ensures that the induced e ort and thus the success probability remains smaller than 1.) 6 u,

the second e ect dominates and the rm prefers to implement more than rst-best e ort despite a positive worker rent. Appendix Proof of Proposition 1: We have @L 2 = v 2 p 0 (e 2 ) @e 2 1 [G 0 (e 2 )+c 0 (e 2 )] @e 2 + 1 1 + G 0 (e 2 ) @e 2 + 2 = 0 @b L2 @b L2 @b L2 @b L2 (4) and @L 2 @b H2 = v 2 p 0 (e 2 ) @e 2 @b H2 [G 0 (e 2 ) + c 0 (e 2 )] @e 2 @b H2 + 1 G 0 (e 2 ) @e 2 @b H2 = 0: (5) Adding both optimality conditions, using that, by (3), @e 2 @b H2 = p 0 (e 2 ) (b H2 b L2 ) p 00 (e 2 ) c 00 (e 2 ) = @e 2 @b L2 > 0; (6) yields 1 + 2 = 1. Hence, either (i) only the MWC is binding, or (ii) both the MWC and the PC are binding, or (iii) only the PC is binding. In case (iii) we have 1 = 1 and 2 = 0. Inserting into (5) shows that the rm implements rst-best e ort: v 2 p 0 e F 2 B = c 0 e F 2 B. From the binding PC and the non-binding MWC b L2 > 0 we obtain G e F 2 B < u. In case (i), 1 = 0, 2 = 1 and b L2 = 0. Inserting into (5) gives v 2 p 0 (e 2 ) c 0 (e 2 ) G 0 (e 2 ) = 0: (7) Obviously, the solution to (7), e 2, satis es e 2 < e F B 2. The non-binding PC yields G (e 2) > u. Finally, in case (ii), 1 ; 2 > 0 and b L2 = 0. From the binding PC optimal e ort in this scenario, e 2, is characterized by G (e 2 ) = u. Solving (5) for the multiplier 1 yields 1 = 1 v 2 p 0 (e 2 ) c 0 (e G 0 (e 2 ) 2 ) : (8) 1 < 1 implies that v 2 p 0 (e 2 ) c 0 (e 2 ) > 0 and, hence, e 2 < e F B 2. From (8) 7

and 1 > 0 we obtain v 2 p 0 (e 2 ) c 0 (e 2 ) G 0 (e 2 ) < 0: (9) Since G() is a convex function, v 2 p () c () G () is concave in e ort. Consequently, comparison of (7) and (9) gives e 2 > e 2. Proof of Proposition 2: In period 1, the rm s problem is: Inserting for b H1 simpli es to: max v + v 1p (e 1 ) b L1 (b H1 b L1 ) p (e 1 ) b H1 ;b L1 ;e 1 ;q max b L1 ;e 1 ;q2[0;1] s.t. b H1 b L1 = c0 (e 1 ) p 0 (e 1 ) Rq b L1 + b H1 b L1 + Rq p (e 1 ) c (e 1 ) u b L1 ; b H1 0 q 2 [0; 1]: b L1 and using the de nition for G () from (1), this problem v + v 1 p (e 1 ) s.t. b L1 = max Rq b L1 + p 0 (e 1 ) p 0 (e 1 ) ; u G (e 1) ; 0 Rq p(e 1 ) (10) Here, the term in square brackets corresponds to the rm s implementation costs for a given e 1. Restriction (10) follows from the MWC for b H1, the PC, and the MWC for b L1, respectively. We now show that, under the optimal contract, the rm sets q = 1. To do so, rst consider the optimal bonus b L1 for a given e ort level e 1 and a given q. According to restriction (10), there are three possible cases: (i) b L1 = Rq, (ii) b p 0 (e 1 ) L1 = u G (e 1 ), or (iii) b L1 = 0. First observe that, if R u, case (i) does not occur. This is because u G (e 1 ) Rq p 0 (e 1 ), u Rq [p (e1 ) 1] c0 (e 1 ) p 0 (e 1 ) c (e 1 ), where the last inequality holds since the left-hand side is non-negative for all 8

q while the right-hand side is non-positive for all e 1. Hence, from the rm s simpli ed problem we see that q = 1 minimizes the rm s implementation costs for each given e 1. Thus, q = 1 if R u. Now assume that R > u. Then, case (i) occurs for su ciently small e 1 and large q. To distinguish which of the cases (i) (iii) is the relevant one for a given e 1, we de ne two thresholds ^e and e. The threshold ^e is given by u ~e < ^e such that R c0 (~e) p 0 (~e) = u G (~e), then e = ~e. Note that, because G 0 (e) < d de one ~e. If such an ~e does not exist, then e is given by G (^e) = 0. If there is an h i c 0 (e) for all e, there is at most p 0 (e) R c0 (e) p 0 (e) = 0. Let b L1 (e 1 ), b H1 (e 1 ) and q(e 1 ) denote the bonuses and the extension probability, respectively, that minimize the rm s implementation costs for a given e 1. If e 1 e, we are either in case (ii) or in case (iii) and hence q(e 1 ) = 1. In the next step, we show that the rm does not want to induce an e ort level e 1 < e. The proof is by contradiction. Assume that the rm wishes to induce an e 1 with e 1 < e. First, consider a situation where e < ^e. Then, q(e 1 ) is given by Rq p 0 (e 1 ) = u G (e 1). Hence, if the rm induces e 1, the PC is binding. Furthermore, b H1 (e 1 ) = 0 and b L1 (e 1 ) > 0. For all e ort levels e 0 2 (e 1 ; e] we also have b H1 (e 0 ) = 0. However, b L1 (e 0 ) < b L1 (e 1 ). Consequently, the rm prefers to induce a higher e ort level than e 1 by increasing q and lowering b L1 appropriately. Now consider the case e ^e. Then, q(e 1 ) is given by Rq p 0 (e 1 ) = maxfu G (e 1) ; 0g. Again, the rm is always better o by inducing an e ort level e 0 > e 1. In particular, if e 1 ^e, the rm bene ts from keeping both bonuses at zero but 9

increasing q. Thus, if R > u the rm will induce an e ort level e 1 e and set q = 1: Now de ne z such that z = 0 if R u and z = e otherwise. Then, the rm s problem can be transformed to: max e 1 z ( v + v 1 + R p (e 1 ) [G (e 1 ) + c (e 1 )] if G (e 1 ) u v + v 1 + R p (e 1 ) [u + c (e 1 )] otherwise. (11) It is now easy to characterize situations where the rm induces more than rst-best e ort e F 1 B. For example, if z = 0 and G(e F 1 B ) < u, the rm inducese e 1 > e F 1 B since the marginal bene t of increasing e ort is v 1 + R > v 1. Moreover, if z > e F 1 B, we also obtain that e 1 > e F 1 B. We have z > e F 1 B if, e.g., R > maxfu; c 0 (^e)=p 0 (^e); v 1 g. Here, R > maxfu; c 0 (^e)=p 0 (^e)g implies that z = e and e is implicitly given by R c0 (e) = 0. Furthermore, if R > v p 0 (e) 1, then e > e F 1 B c because v 0 (e F B 1 p 0 (e F 1 B ) = 0 and c0 (e) is strictly increasing in e. p 0 (e) References 1 ) Kräkel, M. and A. Schöttner (2009): Internal Labor Markets and Worker Rents. Discussion Paper, University of Bonn. La ont, J.-J. and D. Martimort (2002): The Theory of Incentives. Princeton University Press. Ohlendorf, S. and P.W. Schmitz (2008): Repeated Moral Hazard, Limited Liability, and Renegotiation. CEPR Discussion paper No. 6725. Schmitz, P.W. (2005): Workplace Surveillance, Privacy Protection, and E ciency Wages, Labour Economics 12, 727 738. 10