A Multitask Model without Any Externalities

Transcription

1 A Multitask Model without Any Externalities Kazuya Kamiya and Meg Sato Crawford School Research aper No 6 Electronic copy available at:

2 A Multitask Model without Any Externalities Kazuya Kamiya y and Meg Sato z First Version: July 27, 2011 Revised Version: August 22, 2011 Abstract This paper shows that o ering a xed wage maximizes the principal s welfare when the agent needs to engage in multitask and that the e ort needed to achieve one task can be induced by suppressing the e ort needed for the other task, in the absence of externalities. In the existing literature, it is argued that these results are obtained because externalities exist between the costs of tasks or production of tasks. The former is typically represented by a perfect substitute in the cost function. In this paper, we demonstrate that if the agent is engaged in multitask in which one task produces veri able output and the other task produces unveri able output, the same results are obtained without externalities. Keywords: Multitask; Unveri able Outputs; Unveri able Investments JEL Codes: D86; J41; J31 We would like to thank Bruce Chapman, Cleo Fleming, Hideshi Itoh, and Judith abian for helpful comments. y Faculty of Economics, University of Tokyo, Bunkyo-ku, Tokyo , Japan. kkamiyaate.utokyo.ac.jp z Crawford School of Economics & Government, The Australian National University, 132 Lennox Crossing, Canberra, ACT 0200 Australia. meg.satoatanu.edu.au 1 Electronic copy available at:

3 1 Introduction Conventional study on typical multitask problems argues that o ering a xed wage maximizes the principal s welfare only if some externalities exist (Holmstr½om and Milgrom 1991). 1 2 Typically, the problems of ownership pattern 3 and allocation of e ort among the tasks 4 have been examined in this framework. (For example, Holmstr½om and Milgrom 1991, Itoh 1992, Hemmer 1995, Dewatripont and Tirole 1999, MacDonald and Marx 2001, and Akai, Mizuno, and Osano 2010.) multitask analysis. In short, without any externalities, there is no In this paper, we aim to make two contributions to the literature on multitask problems, by focusing on two tasks in which cost functions are additive separable and outputs are stochastically independent. Speci cally, we show that even without introducing any externalities, we can obtain similar results with two multitask problems; we rst show that o ering a xed wage becomes optimal; and then we further show that the e ort of one task can be induced by suppressing the e ort of the other task. In other words, we identify the environment in which a xed wage becomes optimal and the e ort allocation problem between two tasks can be explained without externalities. Both contributions are obtained when one task produces veri able output and the other produces unveri able output and only the wage for the veri able output can be bargained. The bargaining power is obtained by acquiring the rm-speci c skill to produce veri able output or unveri able output or both. These ndings are explained in a simple two-period wage contract model, as follows. We start with de ning a short-term contract and a long-term contract. Under a two-period 1 Suppose Task B is relatively important than task A. When the agent s cost functions for producing the outputs for Task A and B are perfect substitute, Task A will be sacri ced in order to enhance Task B, if the principal wishes to cut down on costs. 2 Another assumption in Holmstrom and Milgrom (1991) which is crucial in obtaining their result is that they assume disutility function which satis es the following properties: f 0 (x) < 0 for 0 < x < x; f 0 (x) > 0 for x < x; and f 00 (x) > 0 for all x: This implies that they assume that making some e orts enhances the agent s utility. In this paper, we assume f 0 (x) > 0 and f 00 (x) > 0 for all x, overriding the ad hoc assumption. 3 For example, Holmstrom & Milgrom (1991) explain why employment is sometimes superior to independent contracting, by using the externality in the cost function. 4 Suppose Task A is relatively important than Task B, and externality in the cost is large between these two tasks. Then, the incentive to engage in Task A can be increased by reducing the incentive in engaging in Task B. 2

4 model, there are two possibilities of wage contract. One is a short-term contract in which the wage of only one period is contracted at the beginning of the relevant period. The other is a long-term contract in which the wages of all n periods are signed at the beginning of the rst period. Under both contracts, when and how to contract the second period wage becomes an issue. 5 Furthermore, whether to o er a short- or a long-term contract depends on how important one task is to the other, and/or on the e ciency of the e ort between two tasks. 6 We assume that the agent must engage in two tasks. Task A produces veri able output x and Task B produces unveri able output y. We consider a situation in which there exist no externalities between the tasks. This situation corresponds to, for example, a baseball player who is expected to make a lot of hits (veri able output) and to exercise leadership within the team (unveri able output). 7 As in practice, the wage for the output x is paid as incentive pay as it is observable, whereas the wage for the output y is xed. The agent needs to make an e ort to obtain the skills to produce x and y: We assume skills to produce x and y are both rm-speci c to some extent, in that once the agent obtains the skill, he gains bargaining power to negotiate the second period wage under the short-term contract. In this simple setting, we show that when the principal values the output x to the output y; the principal o ers a long-term contract with an incentive pay contract on the second period wage. If the principal values the output y to the output x, she o ers a short-term contract with a xed wage contract for the second period wage. 8 behind these results are as follows. The logic If the principal o ers a wage contract before the agent makes an e ort (this corresponds to the long-term contract in which the second 5 Note that the rst period wage is not an issue as it is determined before the agent makes an e ort under both contracts. 6 The e ort needed in order to produce one task is more e cient than the e ort needed in order to produce the other task if both tasks yield the same amount of output but the output from the former task is produced with less e orts. 7 In this case, there may be some externalities in the production function. That is, a baseball player s popularity may increase if he makes a lot of hits. However, unlike other existing literatures which require a large amount of externalities, our results hold in both the case in which the externality is small and the case in which there is no externality. 8 We show in section two that if the agent is risk neutral, it can be either xed wage or incentive pay. We further show in section three if the agent is risk averse, the principal o ers xed wage contract. 3

5 period wage is signed at the beginning of the rst period), the agent will be deprived of an incentive to engage in producing the output y, as his wage would only depend on the output x in the second period. However, the agent will be given an incentive to engage in producing the output x. In fact, the principal can write a contract that can induce the agent to achieve the rst-best level e ort in producing x. On the other hand, if the principal is going to o er the agent a wage contract after the agent has made an e ort (this corresponds to the short-term contract in which the second period wage must be contracted at the beginning of the second period), the agent is given an incentive to make an e ort needed in the production of x and y in the second period during the rst period, as he will wish to obtain bargaining power to negotiate second period wage. However, as the principal will also obtain bargaining power, the agent will not be granted the entire wage he wishes and hence the hold-up problem arises. Therefore, if the principal values the output x produced in Task A relatively more than the output y produced in Task B, the principal o ers the contract before the agent makes his e ort, and thereby induces a stronger incentive to engage in Task A by suppressing the agent s incentive to engage in Task B. If the principal values Task B more than Task A, the opposite holds true. One may wonder if complex and sophisticated contracts (such as Edlin and Reichelstein 1996, Maskin and Tirole 1999a, 1999b, and Moore and Repullo 1988) are introduced in an environment which we discuss in this paper, we may attain the rst-best outcome. However, due to various reasons, the rst-best would not be attained even if complex contracts were to be introduced. For example, Maskin and Tirole (1999b) and Noldeke and Schmidt (1995) introduce option contracts, and the former demonstrates that an option to sell contracts can induce the right incentive to invest under some environments. If we introduce an option contract into our environment, the story would be as follows: the principal has a right to sell her property to the agent and the agent buys the entire property (project). Then the principal will exercise the option if the investment is not e cient, but will not exercise the option if the investment is e cient. However, under the limited liability constraint, the agent cannot purchase the project and hence this is not a 4

6 feasible contract. 9 The rest of the paper is organized as follows. Section two analyzes the case of a risk neutral agent. We also discuss the case with limited liability constraint. Section three discusses the case in which the agent is risk averse. The nal section concludes. 2 The Model: The Case of Risk Neutral Agent There is a principal and an agent. We assume that both of them are risk neutral. There are two types of outputs: an observable and contractible output x 0 and an observable but non-contractible output y 0. There are two contractible output levels, x H and x L, where x H > x L > 0. The probabilities of x H and x L are denoted by H 2 [0; 1] and L = 1 H. There are two non-contractible output levels, y H and y L, where y H > y L > 0. Note that 0 is a parameter introduced for later use. The probabilities of y H and y L are denoted by Q H 2 [0; 1] and Q L = 1 Q H. In the rst period, the agent makes two types of investments, I c 0 and I n 0. We assume that both I c and I n are observable but non-contractible, and that H and Q H in the second period are functions of I c and I n, denoted by H (I c ) and Q H (I n ), respectively. As will be formally stated in Assumptions 1 and 2, we assume that the random variables x and y are stochastically independent and that H = Q H = 0 in the rst period. That is, we assume that the investments in the human capital made in the rst period will lead to an increase in skills from the second period onwards. The agent incurs disutilities in making investments, denoted by D c (I c ) and D n (I n ): Let 2 (0; 1) be the discount factor. The wages for each period are paid at the end of each period, or after the realization of outputs in each period. Since only x is contractible, the wage depends only on the realization of x: the wages in the cases of x H and x L are denoted by w H and w L, respectively. w i, i = H; L, in period t is denoted by wt; i t = 1; 2. Note that because of the risk-neutrality, w2 i need not depend on the realization of an output in the rst period. We rst investigate the model without the limited liability constraint, and later show that similar results can be obtained with 9 If there is no limited liability constraint, the rst-best is attained when the agent is risk neutral. However, when the agent is risk averse, the rst-best is not attained. 5

7 the constraint. Note that there is no externality between I c and I n, since x and y are stochastically independent and the total cost of the investments are additive separable, i.e., D c (I c ) + D n (I n ). Throughout this section we make the following three assumptions. The assumptions on D c ; D n ; H, and Q H are standard. Assumption 1 1. dd i di i > 0, d2 D i di 2 i > 0, D i (0) = 0, and d2 D i (0) di 2 i = 0, i = c; n. 2. d H di c > 0 and d2 H di 2 c < dq H di c > 0 and d2 Q H di 2 c < The random variables x and y are stochastically independent. For simplicity, we make the following assumption. Assumption 2 In the rst period, the probabilities of x H and y H are zero. By the above assumption, the principal needs not determine w H 1 in the rst period. We suppose that the market of workers without rm-speci c skills are competitive. We also assume that the agent obtains some rm-speci c skills in the rst period and hence he has bargaining power to negotiate his wage at the beginning of the second period. 10 Therefore, when the principal hires an agent without rm-speci c skills, she posts a takeit-or-leave-it wage o er. After the agent has obtained the skills, the principal and the agent bargain over the wage at the beginning of the second period. For simplicity, we adopt Nash bargaining with the threat point set at (0; 0). That is, we assume that their bargaining powers are the same and that if they lose a partner, they cannot nd a new one, i.e., they can access the labor market just once and their reservation utilities are zero. It is worth noting that we can obtain similar results even if they have di erent bargaining 10 Alternatively, we assume that the agent who made investments in the rst period, that is the agent with I c > 0 or I n > 0; has bargaining power. We could assume that bargaining power is only given to the agent with I c > 0 or I n > 0; but we can obtain the same result even when we give bargaining power to the agent who did not make any investments. 6

8 power or their reservation utilities are non-zero in the second period. Note that in the discussion of renegotiation-proofness in the following theorems we consider the bargaining, where the status quo is the wage contract signed in the previous periods. Assumption 3 When a contract is signed at the beginning of the rst period, the principal posts a take-it-or-leave-it wage o er. When a contract is signed at the beginning of the second period, they Nash bargain over wages with the threat point held at (0; 0). We consider two types of wage contracts: a short-term contract and a long-term contract. In the short-term contract, the wages are determined at the beginning of each period and paid at the end of each period. In the long-term contract, the wages for both periods are determined at the beginning of the rst period but paid at the end of each period. As will be shown later, the equilibrium contracts are renegotiation-proof. We also discuss limited liability constraint at the end of this section, and demonstrate that almost the same results can be obtained when the constraint is imposed. 2.1 Short-Term Contract Under the short-term contract, the principal and the agent sign the contract on the rst period wage at the beginning of the rst period, and they bargain over the second period wage at the beginning of the second period. The agent can make investments in human capital during the rst period. By Assumption 3, the contracting problem of the shortterm contract in the rst period is a take-it-or-leave-it o er on the rst period wage, subject to the individual rationality constraint and the incentive compatibility constraint on investments: max x L w1 L;Ic;In w L 1 + y L + V p 2 (I c ; I n ) (1) s.t. w L 1 D c (I c ) D n (I n ) + V a 2 (I c ; I n ) u; (2) w L 1 D c (I c ) D n (I n ) + V a 2 (I c ; I n ) (3) w L 1 D c (I 0 c) D n (I 0 n) + V a 2 (I 0 c; I 0 n); 8 I 0 c; I 0 n; 7

9 where u > 0 is the reservation utility determined in the competitive market, and V p 2 (I c ; I n ) and V a 2 (I c ; I n ) are the principal s value and the agent s value when the investments are I c and I n. (2) and (3) are the the individual rationality constraint and the incentive compatibility constraint, respectively. Note that V p 2 (I c ; I n ) and V a 2 (I c ; I n ) are determined by the backward induction given below. The agent has bargaining power at the beginning of the second period. max w H 2 ;wl 2 Applying Assumption 3, the principal and the agent Nash bargain over wages: for a given (I c ; I n ), ( ) ( ) j (I c )(x j w2) j + g(i n ; ) j (I c )w j 2 ; where g(i n ; ) = i=h;l Qi (I n )y i. Since both players are risk neutral, their utilities are the same in the Nash bargaining solution and are equal to a half of the total utility, i.e., their utilities are which is equal to V p 2 (I c ; I n ) and V a 2 (I c ; I n ). 2.2 Long-Term Contract ( ) 1 j (I c )x j + g(i n ; ) ; 2 Under the long-term contract, the principal and the agent sign the contract on the rst and second periods wages at the beginning of the rst period. The agent can make investments during the rst period. By Assumption 3, the contracting problem is a take-it-or-leave-it o er on the rst and second period wages, subject to the individual rationality constraint and the incentive compatibility constraint on investments: max x L w1 L;Ic;In;wH 2 ;wl 2 w L 1 + y L + s.t. w L 1 D c (I c ) D n (I n ) + j (I c )(x j w j 2) + g(i n ; ) w L 1 D c (I c ) D n (I n ) + w L 1 D c (I 0 c) D n (I 0 n) + 8! (4) j (I c )w j 2 u; (5) j (I c )w j 2 (6) j (I 0 c)w j 2; 8 I 0 c; I 0 n:

10 The principal s utility is (4), and the agent s utility is the left-hand side of (5). (5) and (6) are expressions satisfying the individual rationality and the incentive compatibility of the agent. 2.3 Comparison of two types of contracts Below, we explain why the principal and agent sign a variety of contracts depending on the parameters, such as relative e ciency of investments. In the long-term contract, at the beginning of the rst period, the principal can write a xed amount of the second period wage depending on the output x the agent is going to produce in the second period. However, she cannot write the second period wage to re ect the amount of y the agent is going to produce in the second period, as y may be observable but unveri able. Then, it is clear that the long-term contract deprives the principal of investing on I n, an e ort to improve his skill to produce y: The bene t of the long-term contract is that the principal can motivate the agent to invest a lot on I c than it can under short-term contracts, since the long-term contract can induce the rst-best level of I c. As shown in the Appendix A, the agent s incentive to invest on I c under the short-term contract is smaller than the rst-best level. Note that in order to motivate the agent to invest on I c, w2 H must be larger than w2 L. In the short-term contract, the bargaining position of the agent at the beginning of the second period depends on his skill to produce y as well as on his skill to produce x: Therefore, the agent has an incentive to invest on I n; where this is also bene cial for the principal. However, the agent has less incentive in investing on I c under the short-term contract than under the long-term contract. This is because the principal obtains half of the bene t generated from the agent s investment on I c ; through Nash bargaining. Note that because the contract on the second period wage is signed after the investments, there is no need to motivate the agent to invest on I c ; that is, w2 H can be equal to w2 L. Thus, the principal chooses a type of contract depending on the relative e ciency of investments. That is, if the principal values x relatively more than y, and if the principal expects that the investment the agent makes for x is e cient, she prefers the long-term 9

11 contract to the short-term contract. Otherwise, she prefers the short-term contract. In other words, in the former case, in order to induce the agent to invest on I c, the principal chooses a contract where the agent has no incentive to invest on I n : In the latter case, in order to induce I n, the principal restricts the agent s incentive to invest on I c by choosing the short-term contract. Theorem 1 1. The investment for the contractible output x is larger under the longterm contract than under the short-term contract. 2. In the long-term contract, w2 H is strictly larger than w2 L, and in the short-term contract, the xed wage, i.e., w H 2 = wl 2, can be o ered. 3. There exists a > 0 such that the principal prefers the long-term contract to the short-term contract at the beginning of the rst period for 2 [0; ), and prefers the short-term contract to the long-term contract for 2 ( ; 1). Moreover, the equilibria are renegotiation-proof. roof: See the Appendix A. Below, we discuss limited liability constraints. We consider two types of constraints: (i) all wages are nonnegative and (ii) w L 1 + w i 2 0; i = H; L. In the case of short-term contract, we can set w H 2 = w L 2 = V a 2 (I c ; I n) 0. Then holds, where I c w L 1 + w i 2 = D c (I c ) + D n (I n) + u 0; i = H; L and I n are investments chosen under the short-term contract. (See the Appendix A.) Therefore, the limited liability constraint of type (ii) is always satis ed. Moreover, if D c (I c ) + D n (I n) V a 2 (I c ; I n) + u > 0; (7) holds, w L 1 can be nonnegative, i.e., (i) is satis ed. 10

12 In the case of long-term contract, we can set w H 2 = x H r and w L 2 = x L r, where r is the principal s utility in period two. (See the Appendix A.) Then holds, where I c r = w L 1 D c (Ic ) + j (Ic )x j u is the investment chosen under the long-term contract. Thus w L 1 + w H 2 > w L 1 + w L 2 = x L + D c (I c ) j (Ic )x j + u: The right-hand side is positive for su ciently large u; since I c does not depend on u. Thus the limited liability constraint of type (ii) is not binding for su ciently large u. Note that we can also nd a su ciently large u such that (i) is also satis ed. If we consider the case in which u is not su ciently large, the limited liability constraint is binding under the long-term contract, and hence the total utility is smaller than the case without the constraint. Even in this case, when = 0, the long-term contract is better than the short-term contract. Indeed, setting w L 2 = 1 2 xl > 0, w H = 1 2 xh > 0 and w L 1 = D c (I c ) + D n (I n) V a 2 (I c ; I n) + u; (8) it can be shown that the agent chooses Ic and the principal obtains the same utility as in the short-term contract. (See the Appendix A.) Moreover, the principal can choose the wage di erences (w2 H w2 L ) larger than 1 2 (xh x L ): The principal can also keep the expected wages constant. Therefore, she can obtain larger gain. Thus in the case of type (ii) limited liability constraint, Theorem 1.3 still holds with smaller. This is because if we consider the short-term contract, the principal obtains the same gain as in the case without limited liability constraints. If we consider the long-term contract, the principal s gain is smaller. Moreover, in the case of type (i) limited liability constraint, if (7) is satis ed, the same results can be obtained by using the same argument as in the case of type (ii). Theorem 2 In the case with limited liability constraint, the contracts have the following properties. 11

13 1. Under the long-term contract, w2 H is larger than w2 L. Under the short-term contract, the xed wage, i.e., w2 H = w2 L, can be o ered. 2. In the case of type (ii) limited liability constraint, there exists a > 0 such that the principal prefers the long-term contract to the short-term contract at the beginning of the rst period for 2 [0; ), and prefers the short-term contract to the long-term contract for 2 ( ; 1). Next, if the condition for limited liability constraint (i) for the short-term contract, expressed as : D c (I c ) + D n (I n) V a 2 (I c ; I n) + u > 0; is satis ed, the same results can be obtained. 3 The Case of Risk Averse Agent In this section, we adopt the same model as in the previous section, except that the agent s utility regarding his wage w; is expressed as U(w) = w 1 ; where 0 < 1, i.e., the case of constant relative risk aversion, and that the domain of w is the set of nonnegative real numbers, i.e., we adopt the limited liability constraint of type (i). We can show that same results as in the previous section hold for close to zero, since it can be shown that all equilibrium values are continuous functions of (; ). Theorem 3 Suppose D c (I c ) + D n (I n) V a 2 (I c ; I n) + u > 0 and H (I c ) 2 (0; 1) for all I c. properties hold for all 2 (0; ] : Then there exists a 2 (0; 1) such that the following 1. Under the long-term contract, w2 H is larger than w2 L, and under the short-term contract, the xed wage, i.e., w2 H = w2 L, is o ered. 2. There exists a > 0 such that the principal prefers the long-term contract to the short-term contract at the beginning of the rst period for 2 [0; ), and prefers the short-term contract to the long-term contract for 2 ( ; 1). 12

14 roof: See the Appendix B. Even if we consider renegotiation under the long-term contract, the principal o ers the same wages and the agent invests the same amount of I c as the case without renegotiation. Further, the parties would agree to have xed wage in the renegotiation, since the agent is risk averse. That is, the agent accepts a xed wage larger than the certainty equivalent, and the parties share the gain (which is the di erence between the expected wage and the certainty equivalent) from the renegotiation. 4 Conclusion In traditional multitask problems, a xed wage is introduced when externalities between the costs of tasks are large. Similarly, the incentive to engage in one task has to be reduced in order to shift the incentive to engage in the other task, if externalities are large between the tasks. In this paper, we have shown that similar results for these two cases can be obtained in the absence of externalities. That is, when the agent is expected to engage in two tasks in which one task produces veri able output and the other task produces unveri able output, a xed wage can become optimal and one task is prioritized over the other task. This depends on whether the principal values the veri able output more than unveri able output or vice-versa, the investments on the agent s human capital is rmspeci c, and incentive pay contracts can be made only for a task that produces veri able outcome. 13

15 References Akai, N., K. Mizuno, and H. Osano 2010 Incentive Transfer Schemes with Marketable and Nonmarketable ublic Services. Journal of Institutional and Theoretical Economics, 166, Dewatripont, M., and J. Tirole 1999 Advocates. Journal of olitical Economy, 107, Edlin, A. S. & S. Reichelstein 1996 Holdups, Standard Breach Remedies, and Optimal Investment. The American Economic Review, 86(3), Hemmer, T On the Interrelation between roduction Technology, Job Design, and Incentives. Journal of Accounting and Economics, 19, Hildenbrand, W. 1974, Core and Equilibria of a Large Economy. rinceton University ress, rinceton, New Jersey. Holmstr½om, B. and. Milgrom 1991 Multitask rincipal-agent Analyses: Incentive Contracts, Asset Ownership, and Job Design. Journal of Law, Economics, and Organizations, 7(Supplement), Itoh, H Cooperation in Hierarchical Organizations: An Incentive erspective. Journal of Law, Economics, and Organizations, 8(2), MacDonald, G. and L. M. Marx 2001 Adverse Specialization. Journal of olitical Economy, 109(4), Maskin, E and J. Tirole 1999 Unforeseen Contingencies and Incomplete Contracts. The Review of Economic Studies, 66(1), Maskin, E and J. Tirole 1999b Two Remarks on the roperty-rights Literature. The Review of Economic Studies, 66(1), Moore, J. and R. Repullo 1990 Nash Implementation: A Full Characterization. Econometrica, 58(5),

16 Appendices A The roof of Theorem 1 The Short-term Contract In the rst period, the agent chooses I c and I n satisfying the incentive compatibility constraint: max w L 1 D c (I c ) D n (I n ) ( The rst-order condition yields and j (I c )x j + g(i n ; ) ) : (9) dd c (I c ) di c = 1 2 d H (I c ) di c (x H x L ); (10) dd n (I n ) di n = 1 n; n : Note that by Assumption 1 the second-order condition is satis ed. Let the solutions of the above equation be I c and I n. On the other hand, by the individual rationality constraint, the principal must set w L 1 = D c (I c ) + D n (I n) V a 2 (I c ; I n) + u: (11) Then the principal s utility is obtained as follows: x L w1 L + y L + V p 2 (Ic ; In) = x L + y L D c (Ic ) D n (In) + 2V p 2 (Ic ; In) u = x L + y( L D c (Ic ) D n (In) ) + j (Ic )x j + g(in; ) u: (12) Finally, we can see that the principal can choose a xed wage, i.e., ( ) w2 H = w2 L = V2 a (Ic ; In) = 1 j (Ic )x j + g(i 2 n; ) : 15

17 The Long-term Contract By (6), I n joint utility: = 0 is chosen. Since both the principal and the agent are risk neutral, the x L + y L D c (I c ) + is maximized with respect to I c. Indeed, setting w j 2 = x j j (I c )x j + g(0; )! ; (13) r; j = H; L, where r is the principal s utility in period two, (6) yields the following rst-order condition for maximizing (13): Let I c dd c (I c ) di c = d H (I c ) di c (x H x L ): (14) be the solution. Then by the individual rationality constraint, w L 1 = D c (I c ) j (Ic )w j 2 + u: (15) Then the principal s utility is expressed as follows:! x L w1 L + y L + j (Ic )(x j w2) j + g(0; )! = x L D c (Ic ) + y L + j (Ic )x j + g(0; ) u: (16) Finally, from w j 2 = x j r; j = H; L, w H 2 is larger than w L 2. The Comparison of Two Types of Contracts First, comparing (10) and (14), the agent makes more investment on I c under the long-term contract than under the short-term contract, i.e., I c < I c. When = 0, the principal prefers the long-term contract to the short-term contract, i.e., (16) is larger than (12). Indeed, when = 0, I n = 0 is chosen even in the short-term contract and thus (16) (12) = D c (Ic ) + j (Ic )x j D c (Ic ) +! j (Ic )x j > 0: 16

18 The last inequality follows from (14), i.e., Ic satis es the rst-order condition for maximizing D c (I c ) + j (I c )x j. In order to investigate the e ect of on the choice of contracts, we only need to investigate in (9), since I c does not depend on. and Let Then by envelope theorem D n (I n ) g(i n; ) () = max I n D n (I n ) g(i n; ) h() = arg max I n 0 () = 1 2 D n (I n ) g(i n; ) Therefore is a strictly increasing function of ; and goes to +1 as goes to = i=h;l Qi (h())y i y L > 0. This implies that the principal s utility under the short-term contract (12) also goes to +1 as goes to +1. Since when = 0 the principal strictly prefers the long-term contract to the short-term contract, then there exists a > 0 such that the principal prefers the long-term contract to the short-term contract for 2 [0; ), and prefers the short-term contract to the long-term contract for 2 ( ; 1). B The roof of Theorem 3 The Short-term Contract The contracting problem in period two is as follows: for a given (I c ; I n ), ( ) ( ) j (I c )(x j w2) j + g(i n ; ) j (I c )U(w2) j ; max w H 2 ;wl 2 where g(i n ; ) = i=h;l Qi (I n )y i. Note that w H 2 ; w L 2 0 is shown later. The rst-order 17

19 condition with respect to w H 2 and w L 2 are as follows: for i = H; L. This yields j (I c )U(w j 2) = U 0 (w i 2) w H 2 = w L 2 : j (I c )(x j w j 2) + g(i n ; ) That is, a xed wage is o ered. On the other hand, from U 0 (w) = (1 )w,! w 2 = w2 H = w2 L = 1 j (I c )x j + g(i n ; ) 0 2 holds. Thus the value of the agent in the second period, denoted by V2 a (I c ; I n ; ; ), is equal to w 1 2. Note that V2 a (I c ; I n ; 0; ) is equal the value of the risk neutral agent in the second period obtained in Section two. The value of the principal is obtained as follows:! V p 2 (I c ; I n ; ; ) = 1 j (I c )x j + g(i n ; ) : 2! In the rst period, the agent chooses I c and I n satisfying the incentive compatibility constraint: max w L 1 D c (I c ) D n (I n ) + The rst-order condition yields dd c (I c ) = 1 di c 2 and dd n (I n ) = 1 di n 2 d H (I c ) di c (x H x L )(1 ) " ( 1 1 j (I c )x j + g(i n ; ))# : (17) 2 " ( 1 j (I c )x j + g(i n ; ))# ; (18) 2 " n ; ) 1 (1 ) j (I c )x j + g(i n ; ))# n 2 Note that by Assumption 1 the second-order condition is satis ed and the solutions of the above equation, denoted by I c () and I n(; ), are continuous functions. On the other hand, by the individual rationality constraint, the principal must set w L 1 = D c (I c ()) + D n (I n(; )) V a 2 (I c (); I n(; )) + u: (19) 18

20 Then the principal s value; x L w L 1 + y L + V p 2 (I c (); I n(; )); (20) is a continuous function of (; ), since Ic is a continuous function of and In is a continuous function of (; ). Note that w1 L is positive for su ciently close to 0, since Ic (0) and In(0; ) are the investments in the case of risk neutral agent and wages are positive. That is, the limited liability constraint is satis ed. The Long-term Contract The principal s problem is as follows: max x L w1 L0;Ic0;In0;wH 2 0;wL 2 0 w L 1 + y L + j (I c )(x j w j 2) + g(i n ; )! (21) s.t. (w L 1 ) 1 D c (I c ) D n (I n ) + Clearly, I n = 0 is chosen. (w L 1 ) 1 D c (I c ) D n (I n ) + (w L 1 ) 1 D c (I 0 c) D n (I 0 n) + j (I c )(w j 2) 1 u; (22) j (I c )(w j 2) 1 (23) j (I 0 c)(w j 2) 1 ; 8 I 0 c; I 0 n: Below, we show by Berge s maximum theorem (See, for example, Hildenbrand (1974)) that the value of the above problem is a continuous function of. Let B = D c (I c + 1) + u + 1, where I c is the rst-best investment obtained in the case of risk neutral agent. Then we can restrict the domain of investments and wages in the compact set = f(i c ; w L 1 ; w H 2 ; w L 2 ) j 0 I c I c + 1; 0 (w L 1 ) 1 ; (w H 2 ) 1 ; (w L 2 ) 1 Bg. Below, we show that the feasible set in the above problem is a continuous correspondence of. Then, since the objective function is continuous, the continuity of the maximum value in follows from Berge s maximum theorem. Let () be the feasible set of the principal s problem, i.e., the set of (I c ; w L 1 ; w H 2 ; w L 2 ) satisfying (22) and (23). Let () = () \. Below, we show that is a continuous correspondence of. The upper hemi-continuity 19

21 clearly follows from the continuity of the functions in the constraints. The lower hemicontinuity can be obtained as follows. First, note that by the strict concavity of H and strict convexity of D c, the optimal I c in (23) is a continuous function of (; w H 2 ; w L 2 ), denoted by I c (; w H 2 ; w L 2 ). For ^ 2 [0; 1), let (^I c ; ^w L 1 ; ^w H 2 ; ^w L 2 ) 2 (^) and k 2 [0; 1); k = 1; 2 : : : ; be a sequence converging to ^. Suppose ^w L 1 ; ^w H 2 ; and ^w L 2 are larger than 0 and smaller than B, it is easy to nd a sequence (w Lk 1 ; w Hk 2 ; w Lk 2 ); k = 1; 2 : : : ; satisfying (22) with = k and I c = I c ( k ; w Hk 2 ; w Lk 2 ), and converging to ( ^w L 1 ; ^w H 2 ; ^w L 2 ): Suppose some of ^w L 1 ; ^w H 2 ; and ^w L 2 are equal to 0 or to B. If all of such wages are equal to zero, then (22) is not satis ed because u > 0. Thus some of such wages must be positive. If at least one of them is less than B, it is easy to nd a sequence (w Lk 1 ; w Hk 2 ; w Lk 2 ); k = 1; 2 : : : ; satisfying (22) with = k and I c = I c ( k ; w Hk 2 ; w Lk 2 ), and converging to ( ^w L 1 ; ^w H 2 ; ^w L 2 ): If at least two of them are equal to B, then (22) is satis ed with strict inequality, and thus it is easy to nd (w Lk 1 ; w Hk 2 ; w Lk 2 ) satisfying (22) with = k and I c = I c ( k ; w Hk 2 ; w Lk 2 ), and converging to ( ^w L 1 ; ^w H 2 ; ^w L 2 ): Clearly, I c ( k ; w Hk 2 ; w Lk 2 ) converges to ^I c. Thus is a lower hemi-continuous correspondence. Then, together with the continuity of the objective function, the continuity of the maximum value in follows from Berge s maximum theorem. Moreover, since I n = 0 always holds and g(0; ) is a continuous function of, then the maximum value of the principal in the long-term contract is a continuous function of (; ). The Comparison of the Two Types of Contracts Thus values of the principal in the short-term contract and the long term contract are continuous functions of (; ) and they coincide with those in the case of risk neutral agent at = 0. Thus 9 2 (0; 1); 8 2 (0; ] satisfying the same property as in the case of risk neutral agent. 20