1. If the consumer has income y then the budget constraint is. x + F (q) y. where is a variable taking the values 0 or 1, representing the cases not

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1 Chapter 11 Information Exercise 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 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 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. 4. Derive the second-best optimum. Outline Answer 1. If the consumer has income y then the budget constraint is x + F (q) y where is a variable taking the values 0 or 1, representing the cases not buy and buy. 2. Assume that each person s type is common knowledge. (a) If there are two taste types a ; b with The preferences are as shown in Figure a > b (11.1)

2 Microeconomics CHAPTER 11. INFORMATION x τ b τ a quality q Figure 11.1: Preferences: quality F Π 2 = F 2 C(q) increasing profit Π 1 = F 1 C(q) Π 0 = F 0 C(q) quality q Figure 11.2: Isopro t curves: quality cfrank Cowell

3 Microeconomics F τ a q F *a τ b q F *b q *b q *a quality q Figure 11.3: Full-information solution: quality (b) Given that each person s type can be observed the rm can tailor the fee schedule exactly to personal characteristics, charging F a to an a-type and F b to an b-type The rm s pro ts from each of the two groups of consumers are F a C (q a ) (11.2) F b C q b (11.3) and the isopro t curves are as shown in Figure (c) If a person chooses not to buy the good then his utility is just y. So the rm chooses q a and F a to maximise (11.2) subject to the a-type s participation constraint a q a F a + y y or, equivalently a q a F a 0; (11.4) it likewise chooses q b and F b to maximise (11.3) subject to the b- type s participation constraint b q b F b 0: (11.5) The full information solution is found at the tangency of an isopro t curve with a reservation indi erence for each of the two types, as shown in gure 11.3 Clearly, at the optimum (q a ; F a ) and q b ; F b, each of the participation constraints (11.4) and (11.5) is binding and each person just gets his reservation utility y. 3. Now it is no longer possible to condition the fee schedule directly on a person s type. cfrank Cowell

4 Microeconomics CHAPTER 11. INFORMATION F F *a preference τ a q τ b q F *b q *b q *a quality q Figure 11.4: Type a prefers type b contract (a) If the full-information contracts from part 2 were available an a-type person would want to take a b-type contract since his utility would then be a q b F b +y which, in view of the fact that b q b F b = 0, becomes a b q b + y which is strictly greater than y. See Figure (b) We need to nd the second-best contracts that take account of this incentive-compatibility problem. Incentive compatibility requires that, for a: a q a F a a q b F b (11.6) and, for b: b q b F b b q a F a (11.7) Suppose it is known that there is a proportion, 1 of a-types and b-types, respectively. Now the rm s problem is to choose q a, q b, F a and F b to maximise [F a C (q a )] + [1 ] F b C q b (11.8) subject to the participation constraints (11.4) and (11.5) and the incentive-compatibility constraints (11.6) and (11.7). As in the text we can simplify the problem by determining which constraints are slack and which are binding. Note that: (11.6) and (11.1) imply a q a F a a q b F b b q b F b (11.9) This implies that if constraint (11.5) were slack, then constraint (11.4) would also be slack; this cannot be true at the optimum cfrank Cowell

5 Microeconomics since it would then be possible for the rm to increase both F a and F b and increase pro ts. Hence (11.5) must be binding. Given that F b > 0 at the optimum (11.5) then implies that q b > 0: However (11.1). (11.9) and q b > 0 imply a q a F a a q b F b > b q b F b = 0 (11.10) which implies that a q a F a > 0 (11.11) So constraint (11.4) is slack and can be ignored. If (11.6) were slack then, by (11.11) it would be possible to increase F a without violating the constraint. So (11.6) must be binding. a q a F a = a q b F b (11.12) If (11.7) were binding, then this and (11.12) would imply q b a b = q a a b but, given (11.1).this can only be true if q b = q a : But this implies a pooling outcome and we know that the rm can do better than a pooling outcome by forcing the high-value consumers to reveal themselves. This implies that constraint (11.7) is slack and can be ignored. F preference τ a q F a τ b q F b q b q *a quality q Figure 11.5: Second-best solution: quality (c) The Lagrangean is therefore [F a C (q a )] + [1 ] F b C q b + b q b F b + a q a F a a q b + F b (11.13) cfrank Cowell

6 Microeconomics CHAPTER 11. INFORMATION where and are the Lagrange multipliers for the constraints (11.5, 11.6) respectively. The rst-order conditions are C q (q a ) + a = 0 (11.14) [1 ] C q q b + b a = 0 (11.15) = 0 (11.16) 1 + = 0 (11.17) From (11.16) and (11.17) and we have = and = 1. Using these values in (11.14) and (11.15) we have C q (q a ) = a (11.18) C q q b = b a b < b (11.19) 1 Clearly the a-type s consumption is at the point where marginal cost - marginal willingness to pay for quality see Fig cfrank Cowell

7 Microeconomics Exercise 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 The employee s utility function is C (z; ) = ze : 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 expected 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 the second-best problem compare with that in part 1? Outline Answer This is a problem of hidden information: The type of the employee is exogenously given, but private information. The problem for the employer is to design a contract that leads the employee to reveal his true type. The employer is interested in truthful revelation since it is less costly for employees of type 2 to attain a given level of education. Hence, the employer would have to reward 2 types by less to get the same level of education. However, type 2 would like to pretend to be of type 1 to receive the higher payment. The reservation utility of the employee corresponds to the case where the employee receives 0 years of education, for which he receives 0 from the employer: = 1: This yields participation constraints for each type: 1 e y ze 0 For further reference, we can deduce the shape of the indi erence curves for a given type. The rst and second derivatives, respectively, are: dy = e y dz i > 0 (11.20) d 2 y dz 2 = e y dy i dz = e2[y i] > 0 See Figure 11.6 for an illustration of the reservation indi erence curves for the two types, which go through the origin. The shaded areas are the acceptance sets for the two types of individuals. Note that as dy dz is independent of z, the indi erence curves for a given type must be horizontally parallel in (z; y) space. Also note that the single crossing property is satis ed. cfrank Cowell

8 Microeconomics CHAPTER 11. INFORMATION y υ_υ 1 υ_υ 2 y * 2 y * 1 0 z * 1 z * 2 z Figure 11.6: Full-information contracts 1. If the employer knows the employee s type, then, as before, the employer can implement full-information contracts, maximising the return from each type separately. Let be the type. Then the problem of the employer is to max z;y := z y subject to the participation constraint 1 e y ze 0 (11.21) We can now set up the Lagrangean associated with this maximization problem L (z; y; ) := z y + 1 e y ze where is the Lagrange multiplier for the constraint (11.21). The = 1 e = 0 (11.22) = 1 + e y = 0 (11.23) = 1 e y ze 0 (11.24) Combining equations (11.22) and (11.23), we see that e = e y and hence that y = Those equations also imply that > 0, and hence that the participation constraint (11.24) must bind. Hence cfrank Cowell z = 1 e y e = e 1 > 0

9 Microeconomics Thus the employer o ers two di erent contracts to the two types, where the level of education is zi = e i 1, and the compensation is yi = i. Since the slope of the indi erence curve in Figure 11.6 was given by equation 11.20, we see that the contracts are located where the slope of the reservation indi erence curves are equal to Now assume that the employer cannot observe the type of the employee. (a) Given the contracts found in part 1 an employee of type 2 will selfselect the wrong contract. To see this, compare the utility of type 2 when choosing the type-2 contract 2 = e 2 [e 2 1] e 2 to that when choosing the type-1 contract: 2 = e 1 [e 1 1] e 2 Since 1 < 2 it is clear that 2 > 2. Hence, type 2 would self-select the type 1 contract, and hence incentive compatibility is violated. (b) If there is an incentive for type 2 to self-select the wrong contract, then any second-best contract has to insure that the incentive compatibility constraint for type 2 is not violated. Type 1 will never have an incentive to select the type 2 contract, hence the incentive compatibility constraint for type 1 will not be binding. Furthermore, since any non-trivial contract for type 1 would enable type 2 to achieve a utility level greater than his reservation utility level, the participation constraint for type 2 cannot be binding. However, we can keep type 1 on his reservation utility. Hence, we expect that the two constraints which will be binding in the second-best contract will be the participation constraint for type 1 1 e y1 z 1 e 1 0 (11.25) and the incentive-compatibility constraint for type 2: e y2 z 2 e 2 e y1 z 1 e 2 (11.26) This is as in the standard adverse selection model: (c) Assume that the probability of encountering a type 1 employee is 2 (0; 1). We may now set up the maximization problem for the monopolist, which consists of choosing z 1 ; y 1 ; z 2 ; y 2 to maximise : := [z 1 y 1 ] + [1 ] [z 2 y 2 ] subject to (11.25) and (11.26). The Lagrangean is L (z 1 ; y 1 ; z 2 ; y 2 ; ; ) := [z 1 y 1 ] + [1 ] [z 2 y 2 ] + [1 e y1 z 1 e 1 ] + [e y1 e y2 + z 1 e 2 z 2 e 2 ] cfrank Cowell

10 Microeconomics CHAPTER 11. INFORMATION where and are the Lagrange multipliers associated with the constraints (11.25) and (11.26), respectively. Maximising this yields 1 = e 1 + e 2 = 1 = + [ ] e y1 = 2 = [1 ] e 2 = 2 = [1 ] + e y2 = 0 (11.30) As 0 by Kuhn-Tucker conditions, and > 0, equation implies that > 0. Hence, constraint (11.25) will bind. Considering equation 11.29, we see similarly that > 0, so that constraint (11.26) will bind. Since both constraints will bind, and the rst order conditions are satis ed, we have a system with six equations and six variables. Call the solutions to this system (^z 1 ; ^y 1 ) and (^z 2 ; ^y 2 ). (d) Since the incentive-compatibility constraint (11.26) is binding, these two contracts will lie on the same type 2 indi erence curve. But the type-1 incentive compatibility constraint is slack, so type 1 strictly prefers (^z 1 ; ^y 1 ) to (^z 2 ; ^y 2 ). Given the Single Crossing Property and the fact that the type 2 indi erence curves are atter that of type 1, we must have ^z 1 < ^z 2 and ^y 1 < ^y 2. Consider equations and We nd that ^y 2 = 2, the no distortion at the top result. Clearly, the marginal rate of substitution for type 2 at this point is e 2 =e y2 = 1; which is the same as the slope of the isopro t contour. Now compare the second-best contracts with the full-information solution. Type 2 was on his reservation indi erence curve 2 under the full-information contract. But under the second-best, the participation constraint for type 2 is slack, and hence he is now on an indi erence curve I 2 above 2. But since ^y 2 = y 2 = 2 ; this must mean that ^z 2 < z 2. (e) Finally, how far has the solution moved from 2? There are two possibilities: The new (^z 1 ; ^y 1 ) could be below and to the left of the full-information (z1; y1), or above and to the right. We will show that ^z 1 < z1 and ^y 1 < y1 (recall that (11.25) is still binding, so the new solution will be on the reservation indi erence curve for type 1). The slope of the indi erence curve 1 at (^z 1 ; ^y 1 ) is e 1 e ^y1 : From equations (11.27) and (11.28) we have cfrank Cowell e ^y1 = + e ^y1 e 1 = + e 2

11 Microeconomics hence the slope of the indi erence curve is + e 2 < 1 (11.31) ^y1 + e This follows because ^y 1 < ^y 2 = 2 which implies e 2 < e ^y1. We found that the slope of the type 1 indi erence curve at (z1; y1) was 1, so condition (11.31) implies that ^z 1 < z1 and ^y 1 < y1. See gure y υ_υ 1 υ_υ 2 ^y2 ^y1 0 ^z1 ^z2 z Figure 11.7: Second-best contracts cfrank Cowell

12 Microeconomics CHAPTER 11. INFORMATION Exercise 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 z y 1 2 and his reservation level of utility is 0. The lawyer knows his type and the rm cannot observe his action z: The price of legal services is 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 productivity 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? Outline Answer. The problem is one of adverse selection with hidden information. y υ b _υ b _υ a y *a = 1 slope = 1 y *b = ¼ slope = 1 0 x *b = ½ x x *a = 2 Figure 11.8: Full-information contracts 1. Full-information. (a) The principal knows the type and so maximises x where p y x = 0 y subject to cfrank Cowell

13 Microeconomics for each individual type. We know that the participation constraint binds and that there is no distortion. So = p y Di erentiate to nd the slope of the indi erence curve: dy dx = 2p y = x (11.32) (11.33) Since there is no distortion (11.33) must be equal to 1 which implies y = : (11.34) Using the fact that = and substituting (11.34) into (11.32) we get x = : (11.35) In this case the optimal contracts are (x a ; y a ) and shown in Figure x b ; y b as (b) Since there is no distortion the solution is e cient. There is no change which could make one person better o without making the other worse o. 2. Types unknown. (a) If it is impossible to monitor the lawyer s type it is no longer viable to o er the e cient contracts (x a ; y a ) and x b ; y b from part 1. If a type-a lawyer accepts the e cient contract meant for him he gets utility p y a x a a = 1 2 a 1 2 [ a ] 2 a = 0 But if a type-a lawyer were to get a type-b contract he would get utility p y b x b a = 1 2 b 1 2 b 2 a = 1 2 b 1 So a type a would prefer to take a type-b contract. b a > 0: (b) Given this problem the best that the rm can do is to maximise expected pro ts subject to an incentive-compatibility constraint for the a types: p y a x a a p y b x b a Let be the the probability that the lawyer is of type a. expected pro ts are cfrank Cowell [x a y a ] + [1 ] x b y b Then

14 Microeconomics CHAPTER 11. INFORMATION In the problem of maximising expected pro ts under these conditions we know from previous exercises (such as Exercise 11.2) that the participation constraint (11.32) for type b will be binding p y b x b b = b (11.36) as is the incentive-compatibility constraint for type: p y a So the relevant Lagrangean is x a a = p y b x b a : (11.37) L x a ; y a ; x b ; y b ; ; := [x a y a ] + [1 ] x b y b 9 hp i >= + y b x b h py b p i + a x a a yb + >; xb a (11.38) where and are the Lagrange multipliers for the constraints (11.36) and (11.37) respectively. (c) Let the solution values for maximising (11.38) be denoted ^x a, ^y a, ^x b, ^y b. Di erentiating (11.38) the FOC are: From (11.41) and (11.42) we have + 2 p^y a = 0; (11.39) [1 ] + 2 p^y b 2 p^y = 0; (11.40) b a = 0; (11.41) 1 b + a = 0: (11.42) = a = b ; and substituting these values in (11.39) and (11.40) we get [1 ] + b + a 2 p y a = 0 2 p^y b a 2 p^y b = 0 which implies ^y a = 1 4 [ a ] 2 cfrank Cowell ^y b b a = 2 [1 ] 2

15 Microeconomics And so, using the participation constraint (11.32) we have ^x b = bp^y b = b b 2 [1 ] a Finally, from the incentive- which is non-negative by assumption. compatibility constraint we get ^x a = [ a ] 2 2 a b b ^x b : y υ b _υ b _υ a ^ya ^yb 0 ^xb ^xa x Figure 11.9: The second-best solution in the adverse selection problem (d) Note that ^x a < x a ^x b < x b ^y a = y a ^y b < y b see Figure This is the no-distortion-at-the-top result. Note that indi erence curves for a given type are horizontally parallel because utility is linear in x. So ^y a = y a is immediate. cfrank Cowell

16 Microeconomics CHAPTER 11. INFORMATION Exercise 11.4 The analysis of insurance in the text (section ) was based on the assumption that the insurance market is competitive. Show how the principles established in section 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. Outline Answer 1. See Figure x BLUE profits, π b _ y a profits, π b y L κ a 0 _ y a y x RED Figure 11.10: Insurance: iso-pro t (a) The endowment point for both types of individual is at (y; y L). Given that the probability of an accident for high-risk type a is a an insurance rm would break even if it sold insurance against the loss L for a premium a where De ne a = a L: (11.43) y a := y a L = a : (11.44) The line with slope 1 a passing through the endowment point and a the point (y a ; y a ) is an isopro t contour for the rm when dealing with the high-risk types. Pro ts must increase to the South-West (consider the impact on pro ts if the rm were able to charge a higher premium, all other things being equal). cfrank Cowell

17 Microeconomics (b) For the low-risk types the isopro t contours are a family of lines with slope 1 b b. (c) Clearly the reservation indi erence curve for each type of person is given as the contour passing through the endowment point. It has slope 1 h ; h = a; b where it intersects the 45 line see Figure h x BLUE (y a, y a ) (y b, y b ) (y, y L) 0 x RED Figure 11.11: Insurance: monopoly with full information (d) So the full-information outcome is where the high-risk types are located at (y a ; y a ) and the low-risk types at y b ; y b : the monopoly insurance rm rationally o ers better insurance terms to the lowrisk. 2. Take the case where the individual s risk type is unknown to the insurer. (a) Given that there is imperfect information it is clear that a high-risk type would like to masquerade as a low-risk type and so take advantage of the more favourable terms. In Figure see the a-type indi erence curve passing through the point y b ; y b. The monopolist must take account of this possibility in setting up the second-best optimisation problem. (b) The solution to this problem will be of the following form: restrict the low-risk b-types in the amount of insurance that they can purchase so that they choose the prospect ~ P which lies on the b-type reservation indi erence curve in Figure 11.12; o er full insurance at point (~y a ; ~y a ) cfrank Cowell

18 Microeconomics CHAPTER 11. INFORMATION x BLUE (y b, y b ) P ~ b ~ ~ (y a, y a ) (y, y L) 0 x RED Figure 11.12: Insurance: monopoly second-best solution only the a-types will wish to take up this o er. The formal analysis closely follows that of section cfrank Cowell

19 Microeconomics Exercise 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 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. Outline Answer. 1. If there is perfect information about quality then: (a) Cars of whatever quality will always be traded if the value of the buyer is greater than that to the seller, as in the question. (b) Equilibrium prices are p a = a 1 p b = b 1 2. Given that neither party can verify the quality of a car ex ante, but both know that the probability of a bad car is : (a) The expected value of a car to the buyer is and the expected value of a car to the seller is [1 ] a 1 + b 1 (11.45) [1 ] a 0 + b 0 (11.46) (b) Given that (11.45) is greater than (11.46) the equilibrium price is p = [1 ] a 1 + b 1 (11.47) 3. Sellers will only be willing to supply good cars to the market if the price is at least as great as their private valuation a 0. cfrank Cowell

20 Microeconomics CHAPTER 11. INFORMATION 4. Consider two cases. If the price is less than a 0 then, from part 3, the sellers will only supply bad cars and the buyers will be well aware of this. The fact that, after purchase, the quality is revealed as bad con rms the buyers beliefs. Hence we have an equilibrium with price set at the buyers valuation of bad cars: p = b 1 If the price is not less than a 0 then both types of car may be supplied to the market. Once again the buyers will be aware of this and, in the absence of further information will estimate the value as given by (11.45). We have an equilibrium with price p = p where p is given by (11.47) if this price is at least as high as the sellers valuation of good cars [1 ] a 1 + b 1 a 0: (11.48) cfrank Cowell

21 Microeconomics Exercise 11.6 In an economy there are two types of worker: type-a workers have productivity 2 and type-b workers have productivity 1. Workers productivities are unobservable by rms but workers can spend their own resources to acquire educational 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 2z for type-a workers. 1. Find the least-cost separating equilibrium. 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? Outline Answer 1. There is some z such that rms believe a worker to be of type a if z z and type b otherwise. Beliefs are self-con rming if type-a workers choose z = z and type-b workers choose z = 0. This is satis ed if in other words z z 1 z 2: Any z in [1; 2] does the job so that there is a continuum of separating equilibria. z = 1 is the least-cost separating equilibrium. The net payo to an a-type in the least-cost separating equilibrium is z = Given the de nition of the expected productivity of a randomly chosen worker is + 2 [1 ] = 2 : This would give a better payo to a-types in the separating equilibrium if so that < > If z is education in the pooling equilibrium then the net payo to a-type workers is z = z and for b-type workers is 2 z = z The b-type workers are better o in the pooling equilibrium if z > 1 cfrank Cowell z < 3 4

22 Microeconomics CHAPTER 11. INFORMATION The a-type workers are better o in the pooling equilibrium if z > 11 2 z < 1 2 cfrank Cowell

23 Microeconomics Exercise 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 dw (z ) = log (11.49) dz 2. If people come to the labour market having the productivity that the employers expect on the basis of their education show that the optimal wage schedule must satisfy w (z) = log (z + k) (11.50) where k is a constant. 3. Compare incomes net of educational cost with incomes that would prevail if it were possible to observe directly. Outline Answer 1. Individual income, net of educational costs is w (z) ze : Maximising this with respect to z gives the FOC w z (z ) e = 0 from which (11.49) immediately follows. 2. If the employer s expectations are ful lled then the revealed marginal product equals the wage paid w (z). Using this result in (11.49) we obtain dw (z) log = w (z) (11.51) dz which is a rst-order di erential equation in z. Rearranging we have Integrating over z we get w(z) dw (z) e = 1 (11.52) dz e w(z) = z + k (11.53) where k is a constant of integration. From this (11.50) follows. 3. In this model the costs of education ze are a net loss to the workers who would have been paid according to their type anyway, if only could have been observed. cfrank Cowell

24 Microeconomics CHAPTER 11. INFORMATION Exercise 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 = 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. Outline Answer. 1. Under the full-information contract, both manager and owner can observe the action of the manager. So we can solve the optimization problem for the owner for both high and low e ort levels separately, and then let the owner choose the one that yields higher expected pro ts. (a) Formally, denoting the expected pro t to the owner under high and low e ort levels by the manager as and respectively, we have the following pair of optimisation problems for the owner: subject to in the high-e ort case and subject to max = [ 1 y 1 ] + [1 ] [ 2 y 2 ] (11.54) y 1;y 2 u (y 1 ; z) + [1 ] u (y 2 ; z) 0 (11.55) max y 1;y 2 = i [ 1 y 1 ] + [1 ] [ 2 y 2 ] (11.56) u (y 1 ; z) + [1 ] u (y 2 ; z) 0 (11.57) in the low-e ort case, where (11.55) and (11.57) are the participation constraints that will be binding at the optimum. (b) Since the owner can observe the manager s action, and since the manager is risk-averse, the owner will set y 1 = y 2 = y in the solution to (11.54, 11.55) and we can solve for y simply by setting which implies u (y; z) = 0 y 1=2 z = 0 Likewise in the solution to (11.56,11.57) we can solve for y from the equation y 1=2 z = 0 cfrank Cowell

25 Microeconomics (c) Using the numerical values given in the question, we obtain y = 4 y = 1 2. From part 1 the optimal payments in the case of high and low e ort yield expected pro ts to the owner of = 8:5 = 4:5 Thus, the owner would indeed like to induce the manager to take action z. By construction, the manager is indi erent between taking z or z, since he receives his reservation utility in either case. 3. However, if the manager could convince the owner that he was using z, and get the owner to pay him y, while in fact only using z, his payo would be u(y; z) = y 1=2 z = 1 > 0 and hence the incentive-compatibility constraint would be violated. Thus, we now consider the second-best contract, where the owner cannot observe the action of the manager, but can induce him to take the right e ort level. (a) Under the second-best contract, the owner has the choice of inducing the manager to choose either high or low e ort levels. Since there is no incentive compatibility problem with the low e ort level, the full-information solution continues to hold. The interesting case is inducing the manager to take the high e ort level with a second-best contract. (b) Under the second-best contract, the participation constraint continues to hold. In addition, however, we have an incentive compatibility constraint that guarantees that the manager would choose z over z, given the contract. Thus, the owner has to solve the problem max = [ 1 y 1 ] + [1 ] [ 2 y 2 ] y 1;y 2 subject to the participation constraint y [1 ] y z 0 which becomes 3y y (11.58) and the incentive-compatibility constraint y [1 ] y z y [1 ] y z which becomes y y (11.59) cfrank Cowell

26 Microeconomics CHAPTER 11. INFORMATION (c) We may set up the appropriate Lagrangean as: L(y 1 ; y 2 ; ; ) = 3 4 [16 y 1]+ 1 h i h i 4 [2 y 2]+ 3y y y y where and are the Lagrange multipliers on constraints (11.58) and (11.59) respectively. Using the Kuhn-Tucker conditions the FOC for a maximum we = y y = 0 if y 1 > 0 = y 1 = h 3y y y = 0 if y 2 > 0 (11.61) i = = h y y From (11.60) and (11.61) we have i (11.63) [6 + 2] y = 3 (11.64) 2 [ ] y = 1 (11.65) We need to determine whether the constraints will be binding. Consider = 0. By equations (11.64) and (11.65), this implies that y = y = 1 2 and hence that y 1 = y 2. But this would violates the incentivecompatibility constraint, as before. Thus, > 0, and, by (11.62) the incentive-compatibility constraint (11.59) must bind. But equation (11.65) implies that >, and hence (11.58) must bind as well. We can now solve for y 1 and y 2 using the participation and incentive-compatibility constraints, and obtain that y 1 = 25 4 y 2 = 1 4 It can be shown that the expected pro ts associated with this contract are higher than those under a rst best contract with low e ort level, but lower than those under rst best contract with high e ort level. 4. We observe that y 1 > y > y 2. The manager receives more than under rst best in the good outcome, but less in the bad outcome. While it was rational for the owner to bear all the risk under rst best, given that he was risk-neutral while the manager was risk-averse, under the second-best contract the risk is shared between the owner and manager. This induces the manager to take the action which yields a higher probability of a good outcome. cfrank Cowell

27 Microeconomics Exercise 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. 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 asymmetric 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 effort? Outline Answer 1. The structure of the problem is identical to that of Exercise 11.8 so we may follow the same logic. (a) Since the owner can observe the manager s action, and since the manager is risk-averse, the owner will set y 1 = y 2 = y in the solution to the counterpart to (11.54, 11.55) and we can solve for y by setting In the present case, this implies u (y; z) = 0: log y z = 0 Likewise we can solve for y from the equation log y z = 0 Substituting in the speci c values given in the questions, we immediately obtain that the full-information contract puts y = e 4=3 (11.66) and y = e: (11.67) (b) To ascertain which action the owner would like the manager to take, compare expected pro ts under the two actions: 2 = e 4=3 (11.68) and = e : (11.69) cfrank Cowell

28 Microeconomics CHAPTER 11. INFORMATION Clearly = 1 3 [ 1 2 ] h e 4=3 i e which is positive. Hence, the owner would like the manager to take z. 2. To solve for the second-best contract, we again use the structure developed in the Exercise (a) The goal is to nd a contract that induces the manager to take the high e ort level. From the Exercise 11.8, we know that both the participation and incentive-compatibility constraints will be binding. Hence, denoting v(y) = log(y), we have the participation constraint: log (y 1 ) + [1 ] log (y 2 ) z = 0 (11.70) and the incentive-compatibility constraint: log (y 1 ) + [1 ] log (y 2 ) z = log (y 1 ) + [1 ] log (y 2 ) z (11.71) Rearranging equations (11.70) and (11.71), we nd log (y 1 ) log (y 2 ) = z z (11.72) Substituting in the values given in the question, the right-hand side of (11.72) is 1 and so we have Hence, using (11.70) we have log (y 1 ) = 1 + log (y 2 ) (11.73) [1 + log (y 2 )] + [1 ] log (y 2 ) z = 0 Substituting in the values given in the question we have Using this value in (11.73) we get y 2 = e 2 3 (11.74) y 1 = e 5 3 (11.75) [ 1 y1] + [1 ] [ 2 y2)] 2 h i 3z 2 e h i 3z e (b) The agency cost of asymmetric information is given by the di erence of pro ts under a full information contract, given by (11.68) and those under a second-best contract. So the agency cost is [ 1 y] + [1 ] [ 2 y] {z } [under full information] > 0 [ 1 y 1] + [1 ] [ 2 y 2] {z } [under second-best] cfrank Cowell

29 Microeconomics which becomes [y 1 y] + [1 ] [y 2 y] substituting in from (11.66),(11.67), (11.74) and (11.75) we get 2 h i e 5 3 e h i e 2 3 e 4 3 = e e 2 3 e 4 3 which is positive. Hence, the agency cost of asymmetric information is greater than zero, as expected. The owner incurs an agency cost, which arises from delegating the decision making to the manager and not observing the manager s e ort level. 3. Full-information contracts are equivalent to contracts which specify payments contingent on e ort, since in both kinds of contracts, the owner can induce the manager to implement the high e ort level, and expected gross pro ts are the same. However, the high-e ort manager extracts a positive rent, so we do not have equivalence in payo structure. cfrank Cowell

30 Microeconomics CHAPTER 11. INFORMATION Exercise 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. limited liability, so that the bank gets nothing if the project fails. There is 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 := 1x 1 2 x 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 rationing otherwise. Outline Answer. 1. The expected payo to the rm if project i is selected is given by i [x i y] (11.76) The expected pro t for the bank is i y z 2. If there were perfect information then the bank can observe which project is carried out and whether or not it succeeds. Only project 1 (with the higher probability of success) will be funded and carried out and the bank will require a repayment y = x 1. The rm is e ectively forced on to its reservation indi erence curve so that no rm gets less in the absence of credit than with credit; there is no credit rationing. cfrank Cowell

31 Microeconomics Π Π (y) 0 y x 2 y π 1 π 2 Figure 11.13: Credit rationing 3. The rm will choose the project that gives the greatest pro ts so that, given z, project 1 is chosen if and only if: 1 [x 1 y] 2 [x 2 y] which is equivalent to the condition y y where y := 1x 1 2 x The bank s pro ts are given by 8 < 1 y z if y y (y) := : 2 y z if y < y x 2 see Figure There are clearly two local maxima for and the bank would set y = y if 1 y > 2 x 2 (11.77) and set y = x 2 if 1 < 2 x 2 : (11.78) 5. Distinguish between the situations at the two local maxima Take the case (11.78). From (11.76) the rm s expected payo from the loan is 2 [x 2 x 2 ] = 0: So rms are indi erent between There is no credit rationing. taking and not taking the loan. cfrank Cowell

32 Microeconomics CHAPTER 11. INFORMATION Take the case (11.77). From (11.76) the rm s expected payo from the loan is 1 [x 1 y] > 0: All rms would wish to apply for the loan, so that the total demand is Nz. However, the amount available is M so that there is credit rationing. cfrank Cowell

33 Microeconomics Exercise 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 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: where primes denote derivatives. speci ed. Information is symmetric unless otherwise 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 riskneutral 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? Outline Answer. 1. The problem is max 2X i (z)b(x i w i ) i=1 over z and w i such that X i (z)u(w i ) v(z) : First x z, write down the Lagrangian, and di erentiate with respect to w i to obtain i (z)b 0 (x i w i ) + i (z)u 0 (w i ) = 0 so 8i : = B0 (x i w i ) u 0 (w i ) so that the participation constraint binds. cfrank Cowell > 0

34 Microeconomics CHAPTER 11. INFORMATION x 2 x 1 w 1 Authority w 2 x 2 w 2 υ Inspector 45 o w 1 x 1 Figure 11.14: Inspector-authority equilibrium 2. Risk neutrality implies that B 0 () is constant and so u 0 (w i ) is constant. As u 00 () 6= 0 this implies w(x 1 ) = w(x 2 ) = w, say. The tax authority bears all the risk see Figure Therefore X i (z)u(w) v(z) = w = u 1 ( + v(z)) Of course the wage rate does depend on the e ort demanded. 3. In this case we have so B 0 (x i which implies u 0 () = constant w i ) is constant 8i. Therefore x i w i = constant > 0 w(x i ) = x i This is a franchise contract where the inspector keeps the result x i but pays a xed amount k for the privilege independent of the results. The participation constraint implies Thus k(z) = X i (z)x i v(z) X i (z)b(x i w i ) = B(k(z)) X i (z) cfrank Cowell k = B(k(z))

35 Microeconomics x 2 x 1 w 1 Authority w 2 x 2 w 2 45 o υ Inspector w 1 x 1 Figure 11.15: Inspector is risk-neutral As B 0 () > 0 the tax authority seeks to induce an e ort level which maximises k. This implies that it chooses z to maximise X i (z)x i v(z) so i X 0 i(z)x i v 0 (z) = 0 i which has the interpretation expected marginal payo = marginal cost. The second-order condition is X 00 i (z)x i v 00 (z) 0 and so a su cient condition is X 00 i (z)x i 0 i Note that this is trivially satis ed if z is discrete. See O er w L = u 1 ( +v(z L )) irrespective of x i :This guarantees the inspector his reservation level of utility. As u(w L ) v(z L ) u(w L ) v(z H ) the incentive compatibility constraint is satis ed. hazard problem here. There is no moral cfrank Cowell