COLLEGIUM OF ECONOMIC ANALYSIS WORKING PAPER SERIES. Repeated moral hazard with costly self-control. Łukasz Woźny

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1 COLLEGIUM OF ECONOMIC ANALYSIS WORKING PAPER SERIES Repeated moral hazard with costly self-control Łukasz Woźny SGH KAE Working Papers Series Number: 06/007 October 06

2 Repeated moral hazard with costly self control Lukasz Woźny June 05 Abstract We consider a repeated principal-agent model, where a single agent exhibits problems of self control modelled using Gul and Pesendorfer (00) type temptation preferences. In such a setting, for a parameterized strength of self-control, we characterize the optimal contract setting using standard Rogerson (985) or Spear and Srivastava (987) techniques. Our analysis identifies a new channel of principal and agents interactions that can be used to incentivize an agent, this being the reduction of its self control costs. Presence of this new channel challenges typical results obtained in models with no-temptation on the agent s side. For example, incentive compatibility constraints may be non binding at the optimal solution. Moreover the standard martingale property equating marginal cost of high (low) action today and future expected marginal payments does not hold in our case as bonus (punishment) deterrence increases the cost of self control today. The higher the parameter measuring costly self control the larger the departure from the martingale property. We also consider an impact of self control on such issues as savings or renegotiation. Few other generalization follow. keywords: repeated moral hazard, self-control costs, temptation, principal-agent, optimal contract JEL codes: D86 Introduction and related literature Since the seminal work of Strotz (956) or more recently Laibson (997), there is now an extensive literature stressing the importance of temptation and self-control problems in explaining individual behavior in economic models. When studying dynamic models with such dynamically inconsistent preferences, economist have attempted to develop various solutions methods to explain the behavioral observations that have been found in the empirical literature. For example, Strotz (956) and Caplin and Leahy (006) used the language of recursive decision theory to compute time consistent solutions. Alternatively, following the contributions of Phelps and Pollak (968) or Peleg and Yaari (973) economists have reverted to studying game theoretic Warsaw School of Economics, Warsaw, Poland. Address: al. Niepodleg lości 6, Warszawa, Poland. lukasz.wozny@sgh.waw.pl. Phone: Fax:

3 constructions such as Markov perfect equilibria. Since the seminal contribution of Gul and Pesendorfer (00) (GP, henceforth), however, time consistent (or rational) representations of the models with temptation preferences, or preferences for commitment, have been presented. Their representation allows to solve many technical predicaments usually present, when using models with dynamically inconsistent preferences and, by doing so, allows to extend the analysis of temptation and self-control motives in the otherwise standard models. Temptation and commitment problems are also present in contractual framework, e.g., within a company between managers and employees, where employees face various temptations for not exerting a desired effort level or delaying a task or a proect. This is especially apparent in the multiperiod or repeated interactions. To identify these problems we construct a dynamic principal-agent model, but allow agent to exhibit GP preferences. Our analysis is parsimonious, i.e. we consider single-principal, single-agent model with two actions and two outputs, over two periods. From this perspective the contribution of our paper is threefold: first we characterize the optimal dynamic contract, when the agent finds it costly to control himself from not following the temptation in an otherwise standard dynamic principal-agent model. Second, we conduct the comparative statics exercise and answer how the strength of temptation changes the optimal wage scheme. Third, we verify how the presence of costly self control challenges typical results obtained in models with no-temptation on the agent s side. Finally, we consider some typical extensions including many periods, more general temptation preferences, the role of savings or renegotiation a.o. Our analysis identifies a new channel of principal and agents interactions that can be used to incentivize an agent, this being the reduction of its self control costs. Presence of this new channel challenges typical results obtained in models with no-temptation on the agent s side. For example, incentive compatibility constraints may be non binding at the optimal solution. Moreover the standard martingale property equating marginal cost of high (low) action today and future expected marginal payments does not hold in our case as bonus (punishment) deterrence increases the cost of self control today. The higher the parameter measuring costly self control the larger the departure from the martingale property e.g. This paper extends the analysis of Woźny (05), who considered the optimal contract of a static principal agent model with temptations on the agents side modelled using GP method, and is related to Yilmaz (03), who analyze the optimal contract in a dynamic principal agent, where agent s preferences are β δ. As compared to Woźny (05), here we show which results of the static principal-agent model are still valid in the dynamic model and what new channels of managing agents self control costs are available to the principal during repeated interaction. There are also few differences, when comparing our result versus Yilmaz (03). First, our paper offers a framework for considering contracts for a tempted agent that finds it costly but possible to refrain from temptations, rather then a time-inconsistent one that always follows temptation (so called Strotz case). Our analysis embeds Strotz model as a limiting case. Second, our model allows to extend the analysis easily to many or infinite number of periods following Gul

4 and Pesendorfer (004) approach, rather than consider a dynamic model with time-inconsistent preferences that causes technical problems even with the solutions to the agent maximization problem (see example of Caplin and Leahy (006) or a discussion in Balbus, Reffett, and Woźny (05)). Third, many contributions following GP framework allow to consider and parameterize various aspects of self-control, including Dekel, Lipman, and Rustichini (009) random temptations, Olszewski (0) choice dependent temptations or dynamic temptations (see Noor (007)) bringing some new insight to the behavioral contracts literature. Such generalizations and analysis is possible in our framework. Finally, we mention Gilpatric (008) contribution, that shows, how the principal can use time-inconsistency of naive agents decisions to reduce the optimal cost of implementation. Our paper is related to a number of other papers studying temptation / time consistency in the contractual framework. See e.g. Eliaz and Spiegler (006), who characterize the optimal contract to screen naive agents with dynamically inconsistent preferences. Additionally, Heidhues and Koszegi (00) analyze credit markets, when borrowers have a taste for immediate gratification. See also Della Vigna and Malmendier (004), who characterize the optimal contract design for (partially) naive agents with β δ preferences. Finally, we refer the reader to the interesting work of Esteban and Miyagawa (e.g., 006), who characterize the optimal menu pricing, when consumers face temptation. The rest of the paper is organized as follows. Section introduces the setting, including agents preferences. Section 3 states the main result concerning characterization of the optimal contract and comparative statics exercise. Next, section 4 offers extensions to our basic model. Model Consider a moral hazard problem repeated over two periods. Specifically, there is a single principal and a single agent both leaving two periods. After accepting a contract, each period t the agent exerts a costly action a t {a h t, a l t} that parameterize the probability distribution over each period outputs qt h, qt. l The probability of output i = h, l, when action a t is chosen, is denoted by π i (a t ) (0, ) and is independent and the same for both periods. A long term contract specifies w = {w i, w i } i,=h,l, i.e. wage w = (w h, w l ) in the first period dependent on observed output i = h, l and wage w = (w h,h, w h,l, w l,h, w l,l ) in the second period dependent on the output observed in the first period (denoted by i = h, l) and that observed in the second period (denoted by = h, l). In the basic model we assume that agent cannot save nor renegotiate the contract in the second period. The principal s preferences are standard, while the agent s preferences allow for temptation. Gul and Pesendorfer (00, 004) show that such preferences can be represented using two utilities u, v, where u is a commitment utility function, while v is a temptation utility function. In particular, Gul and Pesendorfer show that the self-control preferences defined over the set of 3

5 menus in two periods with a typical element A t have a representation: V (A ) = max u (a ) + v (a ) + δv (A ) max v (a ), a A a A where V (A ) = max a A u (a ) + v (a ) max a A v (a ), δ [0, ] is a discount factor, and A is determined by a pair (a, A ). In our case, as there is no borrowing nor savings, the choice set in both period is the same, say A, and gives utility: V (A) = max a A u (a ) + v (a ) max a A v (a ) + δ[max a A u (a ) + v (a ) max a A v (a )]. To incorporate Gul and Pesendorfer (00) type preferences into the principal-agent model with no borrowing/savings, first note that in our setting the menu of (possibly tempting) alternatives is the set of actions available to the agent. That is, even if the agent s utility depends on actions and wages, he is tempted by items in the action set only. Next, we model the agent s choice in two steps. First, the agent chooses whether to accept a contract or not. If contract is not accepted, he gets a reservation utility ũ. If, on the other hand, contract is accepted the agent gets utility V of the contract w and his action set is given by A = {a h, a l } each period. Then, in the second stage, after accepting a contract, the agent each period makes a choice from A and evaluates the action plan a, a in the maximization problem: max {u (w, a )+v (w, a )+δ[max u (w, a, a )+v (w, a, a ) max v (w, a, a )]} max v (w, a ). a A a A a A a A We assume that agents within period utility u is separable in reward and cost of action and given by u (w, a ) = i π i(a )u(w i ) c a, while v (w, a ) = α[ i π i(a )u(w i) c a ], similarly u (w, a, a ) = i π i(a ) π (a )u(w i ) c a, while v (w, a, a ) = α[ i π i(a ) π (a )u(w i) c a ]. Interpreting, the costs c a and c a denote the commitment and temptation costs of exerting action a A respectively, while the utility u is standard (i.e., is a differentiable, strictly increasing and strictly concave Bernoulli utility function satisfying Inada condition over random rewards). 3 The parameter α R + measures the strength of temptation. We assume that q h > q l, as well as that action a h is more costly that action a l, i.e., c a h > c a l and c a h > c a l. Also, we assume that the difference in costs satisfies the following: c a h c a l > c a h c a l, which reflects the fact that it is harder to motivate the tempted agent 4. We finally assume that probabilities satisfy the following: > π h (a h ) > π h (a l ) > 0, which implies that high effort shifts probability upward in the sense of the first order stochastic dominance. Also, for the two Following the representation of Gul and Pesendorfer (00) preferences (see section 4). That can be set ũ = V ({a l }), with u, u, v, v parameterized by the fixed wage w. 3 Note, here we assume that commitment and temptation utility differ only in the costs terms, while utility from wage is the same in both. Clearly this is done for simplicity. 4 Note, this assumption is critical for many results in our paper. 4

6 π output levels, this implies that monotone likelihood ratio property is satisfied: h (a l ) π h (a h ) < π l(a l ) π l (a h ). To sum up, when denoting agents utility over w = (w, w ) and a = (a, a ) by U(w, a) we obtain: ( + α) i π i (a )u(w i ) c a α c a α max a A( i π i (a )u(w i ) c a ) + + δπ h (a )[( + α) π (a )u(w h ) c a α c a α max a A( π (a )u(w h ) c a )] + + δ( π h (a ))[( + α) π i (a )u(w l ) c a α c a α max a A( π (a )u(w l ) c a )] where a t denotes maximal temptation in period t. This is a two period version of Woźny (05) static model, with time-separable preferences and identical, independent draws of q t each period. The principal s obective is given by a risk neutral utility over expected outputs and payments: π h (a )[q h w h + δ(π h (a )(q h w hh ) + ( π h (a ))(q l w hl ))] + () ( π h (a ))[q l w l + δ(π h (a )(q h w lh ) + ( π h (a ))(q l w ll ))]. () 3 The optimal contract and comparative statics The principal s problem is to maximize a risk neutral utility given in over actions and expected payments subect to constraints. To simplify we split the problem into two. First we solve for the costs of implementing each action profile: (a h, a h ), (a h, a l ) (a l, a h ), (a l, a l ). Second we the determine the optimal action profile. We start with the first case. Suppose the principal wants to implement a = a h and a = a h. The ex-ante participation constraint is: U(w, a h, a h ) ũ. and the second period incentive compatibility constraint: ( + α) ( + α) u(w i ) c a h α c a h α max a A( π (a l )u(w i ) c a l α c a l α max a A( π (a )u(w i ) c a ) π (a )u(w i ) c a ). for both i = l, h. Note the temptation term α max a A( π (a )u(w i) c a )] is the same in both sides of the inequalities hence: ( + α) u(w i ) c a h α c a h ( + α) π (a l )u(w i ) c a l α c a l. 5

7 The first period incentive compatibility constraint requires: ( + α) i [π i (a h ) π i (a l )]u(w i ) + + δ[π h (a h ) π h (a l )][( + α) + δ(π l (a h ) π l (a l ))[( + α) c a h + α c a h c a l α c a l. π (a )u(w h ) c a α c a α max a A( π (a )u(w l ) c a α c a α max a A( π (a )u(w h ) c a )] + π (a )u(w l ) c a )] Using standard technique of replacing wages by inverse utilities u i to calculate cost of implementing high actions in both period (C(a h, a h )), we need to solve some strictly convex minimization problem. The presence of self control cost does not change the concavity of the problem (as our constraints are linear in utility values u i ). The minimization problem can, therefore, be solved using standard Kuhn-Tucker conditions. That is, by letting λ denote Lagrange multiplier of the (the first period) participation constraint, µ the Lagrange multiplier of (the first period) incentive compatibility constraint, µ i the Lagrange multiplier of (the second period) incentive compatibility constraint, the necessary and sufficient condition for the optimal w i is: ( + α)u (w i ) = λ[ α + α while the necessary and sufficient condition for the optimal w i is: π i (a ) π i (a h ) ] + µ[ π i(a l ) π i (a h ], (3) ) ( + α)u (w i ) = [λ + µ( π i(a l ) π i (a h ) )][π (a h ) α + α π (a )] + µ i δπ i (a h ) [π (a h ) π (a l )]. (4) Few observations follow: Lemma The participation constraint is binding, i.e. λ > 0. Proof: Summing equations (3) for i =, we obtain: λ = i= π i (a h ) u (w i ). Since u is strictly increasing and π i (a h ) > 0 we conclude that λ > 0. Lemma The most tempting item in both periods is a = al and a = al. Proof: We consider four cases. First, assume µ = µ = 0. Then equations (3) and (4) imply that a ah and a ah as otherwise wages would be flat and incentive compatibility conditions will not be satisfied. 6

8 Second, assume µ > 0 but µ = 0. Then a ah as otherwise equation (4) implies w il = w ih and the second period incentive compatibility could not be satisfied. To see that a ah observe that equation (4) implies w ih w il by assumed MLRP. Next, argue by contradiction that a h = a = arg max a A i π i(a )u(w i ) c a. Then, ( + α) i (π i(a h ) π i (a l ))u(w i ) ( + α)[ c a h c a l] > c a h c a l + α[ c a h c a l] by increasing differences in costs. But this conditions, together with w ih w il, contradicts that the first period incentive compatibility condition is satisfied with equality, as required by µ > 0. Third, assume µ = 0 but µ > 0. This implies that the second period incentive compatibility constraint is satisfied with equality requiring that a = al, by increasing differences in costs. To see that a = al observe that µ = 0 implies that expected payment for the second period is independent on first period result, and hence w h = w l. This implies that the first period incentive compatibility reduces to: ( + α)[π h (a h ) π h (a l )][u(w h ) u(w l )] c a h + α c a h c a l α c a l, for which to hold, it is necessary that w h > w l, and hence, by equation (3), we obtain a = al. Fourth, assume µ > 0 but µ > 0. As above, the second period incentive compatibility implies that a = al. To see that a = al observe, that otherwise ( + α) i (π i(a h ) π i (a l ))u(w i ) > c a h c a h + α[ c a h c a h] which means that for the first period incentive compatibility to hold with equality we must have: ( + α) π (a )u(w h ) c a α c a α max a A( π (a )u(w h ) c a ) < (5) ( + α) π (a )u(w l ) c a α c a α max a A( π (a )u(w l ) c a )], (6) implying π (a )u(w h ) < π (a )u(w l ). This means that the expected payment in the second period following low output in the first period, must be higher that this following the high output. But this contradicts that the first order condition (4) stating that: u (w h ) = [λ + µ( π h(a l ) π h (a h ) )] > [λ + µ( π l(a l ) π l (a h ) )] = u (w l ). This contradiction implies that a = al. The previous lemma implies that we can obtain a variable pay implementing high action even, if incentive compatibility condition is not binding. Put it differently, it means in fact that in our case, the incentive compatibility condition may be binding or not. We next analyse the monotonicity of the optimal contract: Proposition We have: w h > w l and w ih > w il for any i. 7

9 Proof: This is implied by equations (3) and (4), assuming MLRP and recalling results of lemma. We next aim to verify is the optimal contract satisfy the martingale property. Rearranging equations (3) and (4) we obtain: or u (w i ) = µ( π i(a l ) π i (a h ) ) + λ = ( + α)u (w i ) + λ α + α π i (a ) π i (a h ), u (w i ) u (w i ) = α(λ[ π i(a ) π i (a h ) ] + µ[ π i(a l ) π i (a h ]). (7) ) This shows, how α and hence presence of self control costs scales the difference from the martingale property. This is summarized in the next result: Proposition The martingale property does not hold, unless α = 0, In fact for α > 0, we have: (i) (ii) u (w l ) u (w l ) > 0, u (w h ) u (w h ) < 0. Proof: Lemma assures that a = al. Then we obtain the required inequalities from equation (7). Next consider the other cases, i.e. when the principal wants to implement low action in both period, or high action in only one of the periods. The wage scheme to implement the low action in both period is flat and the same in both periods. To implement the high action in the first period only, the second period wages are flat and the dynamics of the contract is governed by the equation: u (wi ) u (wi ) = α(λ[ π i(a ) π i (a h ) ] + µ[ π i(a l ) π i (a h ) ]). If α = 0 then wage for a high (low) output today is equal to the wage tomorrow conditioned on high (low) output the first period. The higher the α the higher the difference between the first period high (low) wage and the second period wage conditioned on a high (low) output the first period. Finally to implement the high action the next period only, the first period wage is flat and the martingale property is satisfied indicating that: u (w l ) = u (w l ) = u (w h ) = u (w h ). We finish this section with the following comparative statics result. 8

10 Proposition 3 We have the following: (i) if µ = 0, then u (w h ) u (w l ) is decreasing in α, (ii) if µ i = 0, then u (w ih ) u (w il ) is decreasing in α, (iii) if µ i > 0, then u(w ih ) u(w il ) is increasing in α, (iv) if µ > 0, then u(w h ) u(w l ) or u(w hh ) u(w lh ) (u(w hl ) u(w ll )) or u(w hl ) u(w ll ) is increasing in α. Proof: (i) follows from equations (3). In fact we have: u (w h ) u (w l ) = α α π l (a ) +α π l (a h ) π h (a ) +α π h (a h ) = + α[ π l(a ) π l (a h ) ] + α[ π h(a ) π h (a h ) ], which is decreasing in α by MLRP. Similarly we obtain (ii), for a fixed period one output i from equations (4): u (w ih ) u (w ih ) π [λ + µ( i (a l ) π = i (a h ) )][ [λ + µ( π i(a l ) π i (a h ) )][ α π l (a ) +α π l (a h ) ] α π h (a ) +α π h (a h ) ] = + α[ π l(a ) π l (a h ) ] + α[ π h(a ) π h (a h ) ] which is decreasing in α by MLRP. To see (iii) observe that the second period incentive compatibility constraint must be satisfied with equality hence: [ π (a l )]u(w i ) = [π h (a h ) π h (a l )][u(w ih ) u(w il )] = + α [c a h c a l]+ α + α [ c a h c a l], which is increasing as c a h c a l > c a h c a l. To see (iv) observe the first period incentive compatibility constraint must be satisfied with equality hence: (π h (a h ) π h (a l ))[u(w h ) u(w l )] + δ(π h (a h ) π h (a l ))(π h (a h ) δ + α (π h(a h ) π h (a l ))[u(w hl ) u(w ll )] = + α (c a h c a l) + α + α ( c a h c a l). α + α π h(a ))[u(w hh ) u(w lh ) (u(w hl ) u(w ll ))] + As the right hand side is increasing in α at least one of the terms on the left hand side must be increasing in α. 9

11 4 Extensions 4. More than two actions So far we have assumed there are only two possible actions. This has been done to simplify. It can be shown that our main results extend to more extend easily to the more actions case. To see that consider the set of actions A = {a, a,..., a n } each period, with c i, c i strictly increasing in i. Denoting π (a i ) the probability of output if a i was chosen and assuming the principal wants to implement action a k in both periods, we have the first order conditions: ( + α)u (w i ) = λ[ α + α π (a k ) ( + α)u (w i ) = [λ + s k π i (a ) π i (a k ) ] + s k µ s [ π i(a s ) π i (a k ) ], µ s ( π i(a s ) π i (a k ) )][π (a k ) α + α π (a )] + s k µ is δπ i (a k ) [π (a k ) π (a s )]. Although we cannot argue that a = a in both periods we can show that, if a = a m, then m < k in both periods. This in turn implies that we still observe two channels of incentivising agent, direct (standard) one and indirect via reduced temptation. As a result, again the incentive compatibility constraints may be not binding, including the case of a local downward incentive constraint satisfied. Moreover the martingale property is still not satisfied: u (w i ) u (w i ) = α(λ[ π i(a ) π i (a k ) ] + µ s [ π i(a s ) π i (a k ) ]), s k unless α = 0. Corresponding result are obtained for a continuum of actions case. 4. More periods To analyse T period problem we use the recursive method. Here, for the references, we only derive the cost of implementation of high effort in every period. So consider period T problem. By v we denote the promised utility. Let V T (v) = max w h,w l subect to period T incentive compatibility condition: [q w ], ( + α) u(w ) c a h α c a h ( + α) π (a l )u(w ) c a l α c a l and the promised utility condition (with the associated Lagrange multiplier λ T := λ(v)): ( + α) u(w ) c a h α c a h α max a A( i π i (a )u(w ) c a ) v. 0

12 Denote the argument solving this problem by (w T (v, q )) =h,l. With the change of variables (w = u (u )) the problem can be replaced with the strictly concave program with linear constraints. Therefore V T is concave. By the envelope theorem we have V T (v) = λ T = u (w T (v, q )). Period T problem is to V T (v) = max [q w + δv T (v )], w h,w l,v h,v l subect to period T incentive compatibility constraint: [ π (a l )](( + α)u(w ) + δv ) c a h + α c a h c a l α c a l. and the promised utility condition: (( + α)u(w ) + δv ) c a h α c a h α max a A ( π (a )u(w ) c a ) v. Similarly we obtain the optimal solution (w T (v, q ), v T (v, q )) =h,l and that V T (v) = λ T = u (w T (v, q )). Proceeding this way we obtain V (ũ) and we track (w t (v t, q ), v t+ (v t, q )) =h,l ) T t= with v T +( ) = 0. Now the first order conditions for w t are: ( + α)u (w t (v, q )) = λ t[ α + α while the first order conditions for v are: which gives: π (a t) ] + µ t[ π (a l ) ] δv t+(v) = δµ[ π (a l )] + δλ t, u (w t (v, q )) = V t+(v ) = µ t ( π (a l ) ) + λ t = ( + α)u (w t (v, q )) + λ α π (a t) + α, or u (w t (v, q )) u (w t (v, q i )) = α(λ t [ π i(a t ) π i (a h ) ] + µ t [ π i(a l ) π i (a h ) ]),

13 which is the counterpart of the two period model analyzed so far. 5 Conclusions In this paper we have discussed costly self control implications on the shape of optimal dynamic contract in the repeated principal agent model with moral hazard. What remains to be done is further analysis of costly self control on renegotiation and agent s willingness to save, when having access to the credit market. Few other observations: how does α influences the cost of implementation. Its not clear. On the one hand increase in α increase costs (of self control) so we the principal needs to pay more. On the other, such increase change the marginal utility of a given reward. As a result, even for u(w) = ln w the direction is not clear in general. Hence, changes in α may cause nonlinear distortions in costs of implementation and hence the actions the principal wants to implement. If one separates these two channels: costs vs. marginal utility, i.e. set α = and analyze the comparative statics with with respect to value of = c ah c al only. Then the marginal utility channel is closed and the higher the the higher the cost of implementation. more general u + v v new argument for endogenous outside option. Wage dependent in this case. α does not change risk aversion. Right? Is renegotiation proof. Reneging the contract. One sided limited commitment on the agent side...additional constraint for the second period. α reduces incentives smoothing... so also willingness to break the contract the next period. It is clear for Strotz α =. But In fact if α is high so that IC in the first period is satisfied but not binding then v h = v l and so the optimal contract with and without commitment on the agent side is the same. Compare with time-inconsistent preferences where commitment can be beneficial for the planner. in the case when v h = v l then optimal dynamic contract is also spot implementable. Savigs/borrowing. Timing. Chose action. Observe w. Chose s. Case. Borrowing / Saving is not tempting. Then save after w h. for any monotone second period wage: u > E q=h π u > E q=h u > [(+α)π h (a h ) απ h (a t )]u(w hh )+[(+α)π l (a h ) απ l (a t )]u(w hl ). Observe that period temptation is still low, as the richer one gets the lower the action it chooses. Will such saving alter period action? NO necerrairly as the IC was not necessairly binding before. Case. Add temptation to consume the whole income each

14 period (say no borrowing allowed). Then agent may want to save less or not at all as marginal utility ( + α)u for high α IC not binding and hence optimal dynamic contract is spot implementable Chiappori, Macho, Rey, and Salanie (994). 6 Acknowledgments I would like to thank Hector Chade, Eddie Dekel, Kevin Reffett, Ed Schlee as well as all the participants of my talk at the seminar at UFAE, Universitat Autonoma de Barcelona, especially: Mikhail Drugov, Susanna Esteban and Ines Macho for helpful comments and discussions during the writing of this paper. All the usual caveats apply. The proect was financed by NCN grant DEC-03//D/HS4/0383. References Balbus, L., K. Reffett, and L. Woźny (05): Time consistent Markov policies in dynamic economies with quasi-hyperbolic consumers, International Journal of Game Theory, 44(), 83. Caplin, A., and J. Leahy (006): The recursive approach to time inconsistency, Journal of Economic Theory, 3(), Chiappori, P.-A., I. Macho, P. Rey, and B. Salanie (994): Repeated moral hazard: The role of memory, commitment, and the access to credit markets, European Economic Review, 38(8), Dekel, E., B. L. Lipman, and A. Rustichini (009): Temptation-driven preferences, Review of Economic Studies, 76(3), Della Vigna, S., and U. Malmendier (004): Contract design and delf-control: theory and evidence, Quarterly Journal of Economics, 9(), Eliaz, K., and R. Spiegler (006): Contracting with diversely naive agents, Review of Economic Studies, 73(3), Esteban, S., and E. Miyagawa (006): Temptation, self-control, and competitive nonlinear pricing, Economics Letters, 90(3), Gilpatric, S. M. (008): Present-biased preferences, self-awareness and shirking, Journal of Economic Behavior and Organization, 67(3-4), Gul, F., and W. Pesendorfer (00): Temptation and self-control, Econometrica, 69(6),

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