Contract Theory at Work

This section proposes several classical settings where the basic model of this chap- ter is useful. Introducing adverse selection in each of these contexts has proved to be a significative improvement of standard microeconomic analysis.

1. Regulation 

In the Baron and Myerson (1982) regulation model, the principal is a regulator who maximizes a weighted average of the consumers’ surplus S(q) − t and of a  regulated  monopoly’s  profit  U  = tθq,  with  a  weight  α <  1  for  the  firm’s profit. The principal’s objective function writes now as V  = S(q)θq − (1 − α)U . Because α < 1 it is socially costly to give up a rent to the firm. Maximizing expected social welfare under incentive and participation constraints leads to qSB = q for the efficient type and a downward distortion for the inefficient type, q¯SB  < q¯*, which is given by

Note that a higher value of α reduces the output distortion, because the regulator is less concerned by the distribution of rents within society as α increases. If α = 1, the firm’s rent is no longer costly and the regulator behaves as a pure efficiency maximizer implementing the first-best output in all states of nature.

The regulation literature of the last fifteen years has greatly improved our understanding of government intervention under asymmetric information. We refer to Laffont and Tirole (1993) for a comprehensive view of this theory and its various implications for the design of real world regula-tory institutions.

2. Nonlinear Pricing by a Monopoly 

In Maskin and Riley (1984), the principal is the seller of a private good with production cost cq who faces a continuum of buyers. The principal has thus a utility function V = t cq. The tastes of a buyer for the private good are such that  his  utility  function  is  U  = θu(q) − t,  where  q  is  the  quantity  consumed  and t his payment to the principal. Suppose that the parameter l of each buyer is drawn  independently  from  the  same  distribution  on with  respective probabilities 1 − v and v.

We are now in a setting with a continuum of agents. However, it is mathe- matically equivalent to the framework of section 2.5 with a single agent. Now the distribution of θ to be considered is the actual distribution of types, i.e., r is the frequency  of  type  θ¯ by  the  Law  of  Large  Numbers.  It  is  important  to  stress  this interpretation because it considerably enlarges the relevance of the principal-agent model analyzed before.

Incentive and participation constraints can as usual be written directly in terms  of  the  information  rents  as

The principal’s program now takes the following form:

The analysis is the mirror image of that of section 2.5, where now the efficient type is the one with the highest valuation for the good θ¯. Hence, (2.99) and (2.100) are the two binding constraints. As a result, there is no output distortion with respect  to  the  first-best  outcome  for  the  high  valuation  type  and ,  where .

However, there exists a downward distortion of the low valuation agent’s out- put with respect to the first-best outcome. We have qSB < q, where

So  the  unit  price  is  not  the  same  if  the  buyers  demand  q¯∗  or  qSB,  hence  the expression of nonlinear prices.

The literature on nonlinear pricing is huge. The interested reader will find in Tirole (1988), Varian (1988), and Wilson (1993) excel-lent reviews of this topic. In chapter 9, we discuss the link between direct revelation mechanisms and nonlinear prices, and in particular how and when the optimal direct mechanism can be implemented with a menu of simple linear prices in the case of a continuum of types that has been the focus of most of the nonlinear pricing literature.

3. Quality and Price Discrimination 

Mussa and Rosen (1978) studied a very similar problem to the one in section 2.15.2, where agents buy one unit of a commodity with quality q but are vertically differentiated with respect to their preferences for the good. The marginal cost (and  average  cost)  of  producing  one  unit  of  quality  q  is C(q) and  the  principal has  the  utility  function  V  = tC(q).  The  utility  function  of  an  agent  is  now U  = θqt  with  θ in ,  with  respective  probabilities  1 − v and  v.

Incentive and participation constraints can still be written directly in terms of the  information  rents as

The principal solves now:

Following procedures similar to what we have done so far, only (2.104) and (2.105) are binding constraints. Finally, we find that the high valuation agent receives  the  first-best  quality .  However,  quality  is  now reduced below the first-best for the low valuation agent. We have qSB < q, where

Interestingly, the spectrum of qualities (defined as the difference of qualities between what is obtained respectively by the high valuation and by the low valuation agent) is larger under asymmetric information than under complete information. This incentive of the seller to put a low quality good on the market is a well-documented phenomenon in the industrial organization literature. Some authors have even argued that damaging its own goods may be part of the firm’s optimal selling strategy when screening the consumers’ willingness to pay for quality is an important issue.

4.  Financial Contracts 

Asymmetric information significantly affects the financial markets. For instance, in Freixas and Laffont (1990) the principal is a lender who provides a loan of size k to a borrower. Capital costs Rk to the lender since it could be invested elsewhere in the economy to earn the risk-free interest rate R. The lender has thus a utility function V = t Rk. The borrower makes a profit U = θf(k) − t where θf(k) is the production with k units of capital and t is the borrower’s repayment to the lender. We assume that f > 0 and f < 0. The parameter θ is a productivity shock  drawn  from with  respective  probabilities  1 − v and  v.

Incentive and participation constraints can again be written directly in terms of  the  borrower’s  information  rents  as

The principal’s program takes now the following form:

We let the reader check that (2.109) and (2.110) are now the two binding con- straints. As a result, there is no capital distortion with respect to the first-best outcome  for  the  high  productivity  type  and .  In  this case, the return on capital is equal to the risk-free interest rate. However, there also exists a downward distortion in the size of the loan given to a low productivity borrower with respect to the first-best outcome. We have kSB < k, where

Screening borrowers according to the size of their loans amounts to some kind of rationing for the low productivity firms. This phe- nomenon is well documented in the finance literature, and we refer to Freixas and Rochet (1999, chap. 5) for a more complete analysis and further ref- erences on screening in financial contracts. We will see in section 3.6 that instead of using the loan size the lender may also rely on other screening devices, like auditing or the threat of termination, to get valuable information about the firm requesting financing.

5.  Labor Contracts 

Asymmetric information also undermines the relationship between a worker and the firm for which he works. In Green and Kahn (1983) and Hart (1983b) among others, the principal is a union (or a set of workers) providing its labor force l to a firm. To simplify the analysis, we assume that the union has full bargaining power in determining the labor contract with the firm and that the latter has a zero reservation utility.

The  firm  makes  a  profit  θf(l) − t,  where  f(l) is  the  return  on  labor  and  t is the worker’s payment. We assume that f > 0 and f < 0. The parameter θ is a  productivity  shock  drawn  from with  respective  probabilities  1 − v and v. In the labor contracting literature, the firm knows the realization of the shock and the union ignores its value. The firm’s objective is to maximize its profit U  = θf(l) − t. Workers have a utility function defined on consumption and labor. If their disutility of labor is counted in monetary terms and all revenues from the firm  are  consumed,  they  get  V  = v(tl) where  l  is  their  disutility  of  providing  l units  of  labor  and  v(·) is  increasing  and  concave  (v > 0p v < 0).

In this context, the firm’s boundaries are determined before the realization of the shock and contracting takes place ex ante. The ex ante participation constraint is written in a way similar to (2.71). It should be clear that the model is similar to that of section 2.11.2 with a risk-averse principal and a risk-neutral agent.

Using the results of that section, we know that the risk-averse union will propose a contract to the risk-neutral firm which provides full insurance and implements  the  first-best  levels  of  employments  l¯*  and  l,  defined  respectively  by

Let us now turn to the more difficult case where workers have a utility func- tion exhibiting an income effect, and let us assume, to simplify, that V = v(t) − l. The first-best optimal contract would still require efficient employment in both states of nature. Moreover, it would also call for equating the worker’s marginal utility of income across states:

and making the firm’s expected utility equal to zero:

Solving those latter two equations for the pair of transfers immediately yields ,  where  E(·)  denotes  the expectation  operator  with  respect  to  θ and  θ˜ is  the  random  effect.

Inserting this value of the transfer into the union’s objective function, the principal chooses the levels of employment that are obtained as solutions to

We immediately find the first-best levels of labor,


It  follows  that .  Using  the  fact  that  f < 0,  we  finally  obtain that  l¯*  > l.  The  firm  uses  more  labor  when  the  productivity  shock  is  larger.

Let us now consider the case of asymmetric information. The firm’s incentive compatibility constraints in both states of nature are written as

in  the  good  state  θ¯,  and

in the bad state θ.

Summing (2.117) and (2.118), we immediately obtain the implementability condition

Note   that   the   first-best   menu    satisfies   (2.117)   but   vio-lates (2.118). Therefore, let us look for an optimal incentive feasible contract where the binding incentive constraint prevents the firm from claiming that a high shock  θ¯ has  realized  when,  in  fact,  a  low  shock  θ has  realized.  In  this  example, the  bad  type  θ wants  to  mimic  the  good  type  θ¯.  This  may  be  surprising  in  view of the previous analysis carried out in this chapter. To understand this result, it is useful to look at figure 2.10 where we have represented the first-best contracts A and  B of  the  θ¯-firm  and  the  θ-firm  respectively.

Figure 2.10: First-Best Labor Contracts

The indifference curves of the firm are concave in the (l, t) space with higher levels  of  utility  obtained  as  one  moves  towards  the  southeast.  The  θ¯-indifference curve  has  a  greater  slope  than  the  θ-one  so  that  the  single  crossing  property  still holds in our context.

At the first-best, the union requests a constant wage because it is averse to monetary risk. A and B are thus on the same horizontal line. If the union offers this pair of contracts under asymmetric information, the firm has an incentive to always claim that state θ¯ realized and obtain more labor. To avoid this, the union may propose the incentive compatible menu (A, C). The firm is then indifferent between  A and  C  in  state  θ and  strictly  prefers  A to  C  in  state  θ¯.  This  menu is good from an allocative point of view, because the levels of employment are still equal to their first-best values. However, this contract makes the union bear an excessive risk in monetary transfers. This risk can be reduced by decreasing the  gap  between  l¯ and  l.  Because  of  the  impact  of  asymmetric  information  on the marginal utility of income of the union members in both states of nature, it is hard to assess graphically how the second-best optimal contract should be chosen.

Let us move therefore to a formal derivation of this second-best contract. The union’s problem writes now as

where (2.120) is the firm’s ex ante participation constraint.

Let us denote the respective multipliers of these constraints with λ and µ.

Optimizing  with  respect  to  t¯ and  t  leads  to

Summing  (2.121)  and  (2.122),  we  immediately  obtain  that . As expected, the firm’s participation constraint (2.120) is bind- ing.  Using  (2.121)  and  (2.122),  we  also  get .  To satisfy (2.118) and (2.119), it must be that t¯SB  tSB. Moreover, the equality holds only if we have bunching and l¯ = l. For l¯ > l, the inequality is strict. In this case, λ > 0 and the incentive constraint (2.118) is also binding. This allows us to finally express the optimal second-best transfers as

Substituting the transfers into the principal’s objective function and optimiz- ing  with  respect  to  l¯ and  l,  we  finally  obtain

Or, to put it differently:



The system of four nonlinear equations (2.127) through (2.130) is quite hard to solve. One reason for this difficulty is that one cannot immediately deduce from (2.127) and (2.128) that l¯SB  is smaller and lSB is larger than the corresponding first-best values. Indeed, the marginal utility of income µ changes between both settings.

Nevertheless, one point is worth making. Assume that v”’ > 0; then, by Jensen’s inequality

Using (2.128), we observe that

Conditionally  on  the  level  of  labor  l¯SB   in  state  θ¯ (which,  keep  in  mind,  is no  longer  equal  to  the  first-best  level  l¯*)  the  inequality  (2.132)  shows  that  there is an incentive to expand output in state θ above what would be optimal under complete information. This can be interpreted as a source of overemployment in the model.

In   state  θ¯,   things   are   less   clear.   To   reduce   the   costly   incentive   con-straint (2.118), the union wants θ¯SB  to be chosen closer to lSB  than at the first-best. However, because lSB has a priori been shifted upward, this does not imply that l¯SB   is  below  its  first-best  value.  The  complete  comparison  with  the  first-best  levels of employment depends on the utility function.

This section has illustrated how income effects make the analysis much harder even in two-type models. Those income effects were avoided in the standard model of this chapter since the principal’s marginal utility of income was one under com- plete and asymmetric information. Doing so may be justified in partial equilibrium settings where income effects may be negligible.

Green and Kahn (1983) and Chari (1983) have also highlighted the incentives for overemployment in similar models but with a contin-uum of types. Neglecting the impact on the marginal utility of income, all types are induced to expand outputs when the local incentive constraints have a positive multiplier. The normality of leisure can be shown to be a sufficient condition for this property. For further references on this topic, the interested reader can look at Hart and Holmström (1987) and the references therein.

Appendix 2.1: Proof of Proposition 2.5 

Let us form the following Lagrangian for the principal’s problem

where λ is the multiplier of (2.21) and µ is the multiplier of (2.65).

Optimizing  w.r.t.  U  and  U¯ yields  respectively

Summing (2.134) and (2.135), we obtain

and thus µ > 0. Using (2.136) and inserting it into (2.134) yields

Moreover,  (2.21)  implies  that    and  thus  λ ≥ 0,  with  λ > 0  for  a positive output y.

Optimizing with respect to outputs yields respectively


Simplifying by using (2.137) yields (2.66).

Source: Laffont Jean-Jacques, Martimort David (2002), The Theory of Incentives: The Principal-Agent Model, Princeton University Press.

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