374 CHAPTER 11. INFORMATION always necessary to introduce quite strong assumptions about the structure of preferences and technology. In virtually every case we have used the “single- crossing condition” for di¤erent families of indi¤erence curves in order to …nd a tractable solution and to be able to draw interpretable conclusions from th e analysis. Finally, let us remind ourselves of some common curiosities that emerge from imperfect-information models.  The possible multiplicity of equilibria –as in the signalling mo d els (section 11.3). It is not c lear that intellectual devices to reduce this plethora are entirely convincing.  More disturbing perhaps is the possible lack of equilibrium in some cases: see the model of the insurance market (section 11.2.6) and some signalling models (Exercise 11.5).  The use of rationing and price distortions to f orce a second-best solu- tion where imperfect information means that “…rst best“ just can not be implemented. We will see that some of these features will be particularly relevant for our discussion of the problem of economic design. 11.6 Reading notes Good introductions to the economics of information and the theory of contracts are provided in Macho-Stadler and Pérez-Castrillo (1997) and in Salanié (1997). An overview of the issues is provided by Arrow (1986). The classic reference on adverse selection, screening and the economics of insurance markets (on which subsection 11.2.6 is based) is Rothschild and Stiglitz (1976). The classic papers on the economics of signalling are Akerlof (1970) and Spence (1973). The intuitive criterion is attributable to Cho and Kreps (1987). The case of costless signals –so called “cheap-talk”models –is treated in Craw- ford and Sobel (1982). A good introduction is in Salanié (1997), pages 95¤ on which the example in section 11.3.2 is based. For an introduction to the Principal-and-Agent model see Ross (1973) and for a thorough treatment refer to La¤ont and Martimort (2002). The classic pa- pers are Holmström (1979) an d Mirrlees (1999); for the diagrammatic treatment using the Edgeworth box see Ricketts (1986). 11.7 Exercises 11.1 A …rm sells a single good to a group of customers. Each customer either buys zero or exactly one unit of the good; the good cannot be divided or resold. However it can be delivered as either a high-quality or a low-quality good. The quality is characterised by a non-negative number q; the cost of producing one 11.7. EXERCISES 375 unit of good at quality q is C(q) where C is an increasing and strictly convex function. The taste of customer h is  h – the marginal willingness to pay for quality. Utility for h is U h (q; x) =  h q + x where  h is a positive taste parameter and x is the quantity of consumed of all other goods. 1. If F is the fee required as payment for the good write down the budget constraint for the individual customer. 2. If there are two types of customer show that the single-crossing condition is satis…ed and establish the conditions for a full-information solution. 3. Show that the second-best solution must satisfy the no-distortion-at-the-top principle (page 343). 4. Derive the second-best optimum.(Mussa and Rosen 1978) 11.2 An employee’s type can take the value  1 or  2 , where  2 >  1 . The bene…t of the employee’s services to his employer is proportional to z, the amount of education that the employee has received. The cost of obtaining z years of education for an employee of type  is given by C (z; ) = ze  : The employee’s utility function is U(y; z) = e y  C (z; ) where y is the payment received from his employer. The risk-neutral employer designs contracts contingent on the observed gross bene…t, to maximise his ex- pected pro…ts. 1. If the employer knows the employee’s type, what contracts will be o¤ered? If he does not know the employee’s type, which type will self-select the “wrong” contract? 2. Show how to determine the second-best contracts. Which constraints bind? How will the solution to compare with that in part 1? 11.3 A large risk-neutral …rm employs a number of lawyers. For a lawyer of type  the required time to produce an amount x of legal services is given by z = x  The lawyer may be a high-productivity a-type lawyer or a low-productivity b-type:  a >  b > 0. Let y be the payment to the lawyer. The lawyer’s utility function is y 1 2  z: and his reservation level of utility is 0. The lawyer knows his t ype and the …rm cannot observe his action z: The price of legal services are valued is 1. 376 CHAPTER 11. INFORMATION 1. If the …rm knows the lawyer’s type what contract will it o¤ er? Is it e¢ - cient? 2. Suppose the …rm believes that the probability that the lawyer has low pro- ductivity is : Assume  b  [1 ]  a : In what way would the …rm then modify the set of contracts on o¤er if it does not know the lawyer’s type and cannot observe his action? 11.4 The analysis of section 11.2.6 was based on the assumption that the in- surance market is competitive. Show how the principles established in section 11.2.4 for a monopolist can be applied to the insurance market: 1. In the case where full information about individuals’risk types is available. 2. Where individuals’risk types are unknown to the monopolist. 11.5 Good second-hand cars are worth  a 1 to the buyer and  a 0 to the seller where  a 1 >  a 0 . Bad cars are worth  b 1 to the buyer and  b 0 to the seller where  b 1 >  b 0 . It is common knowledge that the proportion of bad cars is . There is a …xed stock of cars and e¤ectively an in…nite number of potential buyers 1. If there were perfect information about quality, why would cars be traded in equilibrium? What would be p a and p b , the equilibrium prices of good cars and of bad cars respectively? 2. If neither buyers nor sellers have any information about the quality of an individual car what is p, the equilibrium price of cars? 3. If the seller is perfectly informed about quality and the buyer is uninformed show that good cars are only sold in the market if the equilibrium price is above  a 0 . 4. Show that in the asymmetric-information situation in part 3 there are only two possible equilibria  The case where p b <  a 0 : equilibrium price is p b .  The case where p   a 0 : equilibrium price is p. (This is a version of the “Lemons model” – Akerlof 1970) 11.6 In an economy there are two types of worker: type-a workers have pro- ductivity 2 and type-b workers have productivity 1. Workers productivities are unobservable by …rms but workers can spend their own resources to acquire edu- cational certi…cates in order to signal their productivity. It is common knowledge that the cost of acquiring an education level z equals z for type-b workers and 1 2 z for type-a workers. 1. Find the least-cost separating equilibrium. 11.7. EXERCISES 377 2. Suppose the proportion of type-b workers is . For what values of  will the no-signalling outcome dominate any separating equilibrium? 3. Suppose  = 1 4 . What values of z are consistent with a pooling equilibrium? 11.7 A worker’s productivity is given by an ability parameter  > 0. Firms pay workers on the basis of how much education, z, they have: the wage o¤ered to a person with education z is w (z) and the cost to the worker of acquiring an amount of education z is ze  . 1. Find the …rst-order condition for a type  person and show that it must satisfy  = log  dw (z  ) dz  2. If people come to the labour market having the productivity that the em- ployers expect on the basis of their education show that the optimal wage schedule must satisfy w (z) = log (z + k) where k is a constant. 3. Compare incomes net of educational cost with incomes that would prevail if it were possible to observe  directly. 11.8 The manager of a …rm can exert a high e¤ort level z = 2 or a low e¤ort level z = 1. The gross pro…t of the …rm is either  1 = 16 or  2 = 2. The manager’s choice a¤ects the probability of a particular pro…t outcome occurring. If he chooses z, then  1 occurs with probability  = 3 4 , but if he chooses z then that probability is only  = 1 4 . The risk neutral owner designs contracts which specify a payment y i to the manager contingent on gross pro…t  i . The utility function of the manager is u(y; z) = y 1=2 z, and his reservation utility  = 0. 1. Solve for the full-information contract. 2. Con…rm that the owner would like to induce the manager to take action z. 3. Solve for the second-best contracts in the event that the owner cannot observe the manager’s action. 4. Comment on the implications for risk sharing. 11.9 The manager of a …rm can exert an e¤ort level z = 4 3 or z = 1 and gross pro…ts are either  1 = 3z 2 or  2 = 3z. The outcome  1 occurs with probability  = 2 3 if action z is taken, and with probability  = 1 3 otherwise. The manager’s utility function is u(y; z) = log y z, and his reservation utility is  = 0. The risk neutral owner designs contracts which specify a payment y i to the manager, contingent on obtaining gross pro…ts  i . 378 CHAPTER 11. INFORMATION 1. Solve for the full-information contracts. Which action does the owner wish the manager to take? 2. Solve for the second-best contracts. What is the agency cost of the asym- metric information? 3. In part 1, the manager’s action can be observed. Are the full-information contracts equivalent to contracts which specify payments contingent on ef- fort? 11.10 A risk-neutral …rm can undertake one of two investment projects each requiring an investment of z. The outcome of project i is x i with probability  i and 0 otherwise, where  1 x 1 >  2 x 2 > z x 2 > x 1 > 0  1 >  2 > 0: The project requires credit from a monopolistic, risk-neutral bank. There is limited liability, so that the bank gets nothing if the project fails. 1. If the bank stipulates repayment y from any successful project what is the expected payo¤ to the …rm and to the bank if the …rm selects project i? 2. What would be the outcome if there were perfect information? 3. Now assume that the bank cannot monitor which project the …rm chooses. Show that the …rm will choose project 1 if y  y where y :=  1 x 1   2 x 2  1   2 4. Plot the graph of the bank’s expected pro…ts against y. Show that the bank will set y = y if  1 y >  2 x 2 and y = x 2 otherwise. 5. Suppose there are N such …rms and that the bank has a …xed amount M available to fund credit to the …rms where z < M < Nz Show that if  1 y >  2 x 2 there will be credit rationing but no credit ra- tioning otherwise (Macho-Stadl er and Pérez-Castrillo 1997). 11.11 The tax authority employs an inspector to audit tax returns. The dollar amount of tax evasion revealed by the audit is x 2 fx 1 ; x 2 g. It depends on the inspector’s e¤ort level z and the random complexity of the tax return. The probability that x = x i conditional on e¤ort z is  i (z) > 0 i = 1; 2. The tax 11.7. EXERCISES 379 authority o¤ers the inspector a wage rate w i = w(x), contingent on the result achieved and obtains the bene…t B(x w). The inspector’s utility function is U(w; z) = u(w)  v(z) and his reservation level of utility is . Assume B 0 () > 0; B 00 ()  0; u 0 () > 0; u 00 ()  0; v 0 () > 0; v 00 ()  0: Information is symmetric unless otherwise speci…ed. 1. For each possible e¤ort level …nd the …rst-order conditions characterising the optimal contract w i i = 1; :::; n. 2. What is the form of the optimal contract when the tax-authority is risk- neutral and the inspector is risk-averse? Comment on your solution and illustrate it in a box diagram. 3. How does this optimal contract change if the inspector is risk-neutral and the tax-authority is risk-averse? Characterise the e¤ort level that the tax authority will induce. State clearly any additional assumptions you wish to make. 4. As in part 2 assume that the tax authority is risk-neutral and the tax inspector is risk-averse. E¤ort can only take two possible values z or z with z > z. The e¤ort level is no longer veri…able. Because the agency cost of enforcing z is too high the tax authority is content to induce z. What is the optimal contract? 380 CHAPTER 11. INFORMATION Chapter 12 Design The ill designed is most ill for the designer –Hesiod, Opera et dies 12.1 Introduction The topic of design is not really new to our discussion of microeconomic princi- ples and analysis. We have already seen examples of design in chapter 11 when we considered the rôle that participation and incentive-compatibility constraints play in shaping fee schedules and wage schedules. We have alluded to the design problem in chapter 9 when we mentioned the implementation problem associ- ated with e¢ ciency and other welfare criteria. Here we will focus more precisely on the issues that we glimpsed in those contexts. The purpose of the discussion in this chapter is to understand the principles that apply to the design of systems that are intended to implement a particular allocation or social state. The design issue could be precisely focused on a very narrow context (a single market?) or implemented at the level of the whole economy. The “designer”–the economic actor undertaking the design problem – could be just one …rm or one person endowed with the appropriate amount of power, or “the government” as a representative agent for all the persons in the economy under consideration. We will …nd that a lot of headway can be made by reusing concepts and methods from chapters 9–11. Indeed some of the analysis can be seen as an extension and generalisation of ideas that were introduced in the discussion of Principal and Agent. The key problem can be summarised thus. In most of our previous work we have assumed the existence of an economic institution that sets and administers the rules of economic transactions: usually this was the market in some form. Occasionally we have noted cases where the shortcomings of the institution are evident –for example in the allocation of goods characterised by “nonrivalness” or in the presence of externalities (see pages 245¤). Now we want to turn this mental experiment around. Can we establish the principles which would 381 382 CHAPTER 12. DESIGN underpin a well-functioning economic system and thereby provide guidelines for designing such a system? 12.2 Social choice If we are to consider the problem of economic design from scratch then we had better be clear about the objectives of the exercise. What is it that the economic system is supposed to achieve? We need a representation of the workings of the economy that it is su¢ ciently ‡exible to permit general modelling of a variety of individual and social objectives. We can do this simply and powerfully by revisiting the ideas that underlay the concepts of social welfare discussed in chapter 9. First we will reuse the very general description of a social state  and the concept of a “pro…le” of preferences de…ned over , the set of all possible social states: remember that a pro…le is just an ordered list of preference relations, one for each household in the economy under consideration (see page 228). However, we will …nd it more convenient to work with the notation of utility functions rather than with the weak preference symbol  h as in chapter 9, although this tweak is little more than cosmetic. In particular let us use the “reduced-form” representation of the utility function that expresses utility of household (agent) h as a direct function of the social state, v h () (see page 234). So in th is notation a pro…le of preferences is an ordered list of utility functions,  v 1 ; v 2 ; v 3 ; :::  ; (12.1) one for each member of the population; as a shorthand for a particular pro…le (12.1) we will again use the symbol [v] and as a shorthand for the set of all possible pro…les [v] we use the symbol V. Two other key concepts from chapter 9 are relevant here: the constitution and the social welfare function. To these we need to add one new concept that …ts neatly into the language of social choice, but that has wider applicability. De…nition 12.1 A social choice function is a mapping from the set of prefer- ence pro…les V to the set of social states . So, using the utility representation agent h’s preferences, v h (), the social- choice function in de…nition 12.1 can be written as:   =   v 1 ; v 2 ; :::  (12.2) A few points to note about the social-choice function :  As a true function (rather than a correspond enc e) it selects a single mem- ber of  once a given pro…le of preferences is plugged in.  The arguments of  are utility functions, not utility levels: this is like the constitution  that we de…ned in chapter 9 (page 228). 12.2. SOCIAL CHOICE 383  social state  set of all so cial states v h () “reduced-form”utility function for agent h [v] =  v 1 ; v 2 ; v 3 ; :::  pro…le of utility functions V set of all p ossib le pro…les  social-choice function Table 12.1: Social-choice functions: Notation   subsumes technology, markets, and the distribution of property in a summary of the process that transforms pro…les of preferences into social states. So the expression (12.2) says “you tell me what people’s preferences are – the collection of their indi¤erence maps – and then I will tell you what the social state should be.”  Because its speci…cation is similar in spirit to that of the constitution it inherits some of the di¢ culties that we have come to associate with the constitution –see the discussion on pages 229–234. On a grand scale we can consider the social choice function as a kind of black box that transforms a pro…le of preferences into a social state. It is an intellectual device that focuses attention on consumer sovereignty as a principle governing the workings of the economy: it is as though the social choice function lies ready for the collection of consumers to express their wishes and then brings forth an outcome  in accordance with those wishes. On a smaller scale we can think of this apparatus as a convenient abstraction for describing a class of design problems that a¤ect …rms and other decision makers. To pave the way for a more detailed analysis let us consider some possible properties of . First we pick up on some essential concepts from the funda- mental aggregation problem in social-welfare analysis (it is useful to compare these with the four axioms on page 229). De…nition 12.2 Suppose there is some   such that for all h and all  2  : v h (  )  v h (). Then the social-choice function  is Paretian if   =   v 1 ; v 2 ; :::  (12.3) De…nition 12.3 Suppose there are two pro…les [v] and [~v] such that   =   v 1 ; v 2 ; :::  and, for all h : v h (  )  v h () ) ~v h (  )  ~v h () : (12.4) [...]... appropriate will depend on the timing and information structure built into the model and any restrictions that we may want to introduce on admissible strategies The standard model paradigm is the Bayesian game of incomplete information (see section 10 .7. 1 on page 311) that formed the basis of most of chapter 11 and we will need to use both the conventional Nash equilibrium and also the more restrictive equilibrium... status quo and apply the mirror-image criterion for “go/no-go”of a switch back to , with a mirror-image penalty for any “pivotal” household Then the complete contingent penalty system can be summarised as in Table 12 .7, and it yields the following important result (For proof see Appendix C) Theorem 12 .7 (Clark-Groves) A scheme which (a) approves a project if and only if (12.29) is non-negative, and (b)... can be designed if there is less than full information 12.3 Markets and manipulation To illustrate the power of misrepresentation and manipulation in a familiar setting let us rework the standard model of an exchange economy 12.3.1 Markets: another look Take the particularly interesting example of a social-choice function from chapter 7 Specify the details of the following: The technology of the …rms;... expressions (12.5)-(12 .7) 386 CHAPTER 12 DESIGN h = a; b i = 1; 2 xh i h Ri households goods consumption by h of i endowment of h with i Table 12.2: The Trading Game Then we appear to have almost all the ingredients needed to construct the economy’ excess-demand function (7. 16); all that is missing is the pro…le of s preferences represented by the list of utility functions (7. 1) Once we plug those... functions) Suppose the number of social states is more than two and the social-choice function is de…ned for all logically possible utility functions Then, if is Paretian and monotonic, it must be dictatorial The ‡ avour of Theorem 12.1 is similar to Theorem 9.1 and, indeed, the proof is similar (check the reading notes to this chapter and Appendix C) But its implication may not be immediately striking... extract payment from one or more potential buyers of an object, a collection of goods, ownership rights, How do the mechanics work? How can the principles of design help us to understand the rules and likely outcomes? Of course the problem that makes the analysis of auctions economically interesting is the nature of the concealed information: the seller usually does not know the characteristics of... appropriate to formulate the problem in terms of a Bayesian game and to use the revelation principle to simplify the analysis There is a great variety of types of auction that di¤er in terms of the information available to participants, the timing, and the rules of conduct of the auction We will …rst discuss the informational issues and then the rules 396 CHAPTER 12 DESIGN The informational set-up... agents let us take a simple example Alf and Bill are a pair of risk-neutral agents who take part in a sealed bid, …rst-price auction They have private values a and b , respectively, drawn from a distribution F on the support [0; 1]: i.e the minimum possible value that either could place on the good is 0 and the maximum is 1 The problem is symmetric in that, although Alf and Bill may well have di¤erent realisations... reasoning follows for Bill just by interchanging the a and b superscripts.14 To illustrate this, suppose that tastes are distributed according to the beta distribution with parameters (2; 7) : the density function for this is in Figure 12.3 and the formal de…nition is given in Appendix A (page 519) Then the equilibrium bid function ( ) in equation (12.15) and the resulting probability of winning (12.9) as... might it actually harm you to submit a bid p greater than ? 1 6 Why might the English open-bid auction and the second price sealed bid auction not be equivalent if one were selling o¤ the mineral rights on a plot of land? 1 7 In the independent private value model suppose there is a …xed number of bidders and that the private valuations for the good are distributed on the interval [0; 1] Consider the following . introductions to the economics of information and the theory of contracts are provided in Macho-Stadler and Pérez-Castrillo (19 97) and in Salanié (19 97) . An overview of the issues is provided by. screening and the economics of insurance markets (on which subsection 11.2.6 is based) is Rothschild and Stiglitz (1 976 ). The classic papers on the economics of signalling are Akerlof (1 970 ) and Spence. Salanié (19 97) , pages 95¤ on which the example in section 11.3.2 is based. For an introduction to the Principal -and- Agent model see Ross (1 973 ) and for a thorough treatment refer to La¤ont and Martimort