D i (w; p) := H i (w; S(w; p)): (1)

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1 EC0 Microeconomic Principles II Outline Answers. (a) Demand for input i can be written D i (w; p) := H i (w; S(w; p)): () where H i is the conditional demand for input i and S is the supply function. From this we nd tha tthe e ect of an output price change i (w; p) (w; q p) @p the e ect of an input price change i (w; p) (w; q ) C iq (w; q )C jq (w; q ) : j C qq (w; q ) where C is the cost function and C iq is short C=@w i. No. Given i (w; q )=@w i must be negative (or zero), equation (3) implies that Di(w; i p) < 0. 3 ii. No. Given i (w; q )=@w j = C ij = C ji j (w; q )=@w i equation (3) implies i =@w j j =@w i. 3 iii. Yes the case of an inferior input in (). (b) Note that the MRS is given by dx = x dx which is decreasing/constant/increasing in x according as < / = / > : so we will only get a conventional interior solution if <

2 i. Given the budget constraint y = px + x the problem is to choose x to maximise x + y px (c) For > consumer maximises utility at either x = 0 or x = 0 (x = y=p): demand is discontinuous. For = any value of x between 0 and y=p is a solution. For < we have the FOC x = p which yields x = [p=] 5 ii. In the case > each individual is indi erent between (0; y) and (y=p; 0). The average behaviour will be indistinguishable from the case = above. 3 i. Relative inequality aversion is constant. 3 ii. The ede income must satisfy which, given =, simpli es to = + 3 = + 3 which implies = :5. 5 (d) See lecture notes, chapter 7. i. Brief explanation of derivation of consumer demand function and supply function of individual rms. Aggregation over agents (point out absence of production externalities) 4 ii. Explain Walras law and the reason for the homogeneity of degree zero of household demand functions and rm supply functions. 4

3 (e) Find all the pure and mixed strategy Nash equilibria of the following game in normal form InII L C R T 3; 0 0; 3 0; M 0; 3 3; 0 0; 0 B ; 0 0; 0 0; 0 Sketched answer: There is one pure strategy equilibrium: (B; R). This involves weakly dominated strategies on the part of both players. Look for mixed strategy equilibria not involving weakly dominated strategies. Let p and p the probabilities for players I on T and M respectively. Let q and q the probabilities for players II on L and C respectively. Player I is indi erent between T and M when 3q = 3( q), q = =. Player II is indi erent between L and C when 3( p) = 3p, p = =. There are no other equilibria. This is because R and B will not be played unless the probability that T and L are played is zero for both (by weak domination). 8 (f) Which strategies survive the process of iterated deletion of strictly dominated strategies in the following game? InII D E F A ; ; 3; B ; ; 4; C 5; ; 7 8; 6 Sketched answer: No strategy is dominated for Player I. Strategy E dominates strategy F for Player II. So, F can be eliminated. After F has been eliminated, both A and B dominate C for Player I. So, C can be eliminated. The strategies A, B, D and E all survive the process of iterated deletion of strictly dominated strategies. Each is a best response (strict) to some strategy of the opponent. 8 (g) Give an example of an extensive form game with complete and perfect information with the following property. If we derive the 3

4 Normal form G corresponding to, there is a Nash equilibrium of G that does not correspond to the backwards induction solution for. Sketched answer: The simplest example would be that of an entry game. In Player I moves rst and can decide to Enter (E), or stay Out (O). If Player I chooses O the payo s are (0; ). If Player I enters, Player II can choose to Fight (F ) or Acquiesce (A). If Player II chooses A then payo s are (; ). If Player II chooses F then payo s are ( ; ). The backwards induction solution is that Player II chooses A and Player I chooses E. The normal form G corresponding to is InII F A E ; ; O 0; 0; Clearly, G has two Nash equilibria: (O; F ) and (E; A). 8 (h) Is the following statement true? Give a reason for your answer. In a duopoly, whether moving rst carries any advantage or not depends on which strategic variables can be chosen by the two rms. Sketched answer: The statement is true. The simplest set-up to refer to is two rms producing a homogeneous product with, say, a linear demand curve, and identical linear cost functions. If the strategic choice variables are quantities clearly there is a rst mover advantage. The Cournot pro t is always smaller than the pro t of a Stackelberg leader. If the strategic choice variables are prices then there is no rstmover advantage. The rst mover can set price equal to marginal cost, in which case both pro ts will be zero. If the rst mover sets any price strictly above marginal cost, then he will be undercut by the second mover. The rst mover s pro t will be zero, while the second mover s pro t will be positive. 8 4

5 Section B. (a) The cost-minimisation problem can be represented as minimising the Lagrangean " # mx mx w i z i + log q i log z i (4) i= where w i is the given price of input i, and is the Lagrange multiplier. Given that the isoquant does not touch the axis we must have an interior solution: rst-order conditions are which imply i= w i i z i = 0; i = ; ; ::; m (5) z i = i ; i = ; ; ::; m (6) w i Now solve for. Using the production function and (6) we get q = my i= z i i = z i i = i i ; i = ; ; ::; m (7) w i A m Y i= w i i ; i = ; ; ::; m (8) where := P m j= j and A := [ Q m i= i i ] = are constants, from which we nd = q = A Q m i= w i i = A [qw w :::wm m ] = : (9) Substituting from (9) into (6) we get the conditional demand function: H i (w; q) = zi = i A [qw w :::wm m ] = (0) w i and minimised cost is mx C (w; q) = w i zi = A [qw w :::wm m ] = () i= = Bq = () 5

6 where B := A [w w :::wm m ] =. It is clear from () that cost is increasing in q and increasing in w i if i > 0 (it is always nondecreasing in w i ). Di erentiating () with respect to q marginal cost is C q (w; q) = Bq (3) Clearly marginal cost falls/stays constant/rises with q as T. 8 (b) Suppose, without loss of generality, that in the short run inputs ; :::; k (k m) remain variable and that the remaining inputs are xed. In the short-run the production function can be written as log q = kx i log z i + log k (4) i= where k := exp mx i=k+ i log z i! (5) and z i is the arbitrary value at which input i is xed; note that B is xed in the short run. The general form of the Lagrangean (4) remains unchanged, but with q replaced by q= k and m replaced by k. So the rst-order conditions and their corollaries (5)-(9) are essentially as before, but and A are replaced by k := kx j (6) j= and A k := is h Qk i =k i= i i. Hence short-run conditional demand ~H i (w; q; z k+ ; :::; z m ) = =k i q A k w w :::w k k (7) w i k and minimised cost in the short run is ~C (w; q; z k+ ; :::; z m ) = kx w i zi + c k i= = k A k q k w w :::w k k =k + c k (8) = k B k q = k + ck (9) 6

7 where c k := mx i=k+ w i z i (0) is the xed-cost component in the short run and B k := A k [w w :::w k k = k] = k. Di erentiating (9) we nd that short-run marginal cost is ~C q (w; q; z k+ ; :::; z m ) = B k q k k () (c) Using the Marginal cost=price condition we nd 6 B k q k k = p () where p is the price of output so that, rearranging () the supply function is p k k q = S (w; p; z k+ ; :::; z m ) = (3) wherever MCAC. The elasticity of (3) is given by B log S (w; p; z k+ ; :::; z m log p = k k > 0 (4) It is clear from (6) that k k k ::: and so the positive supply elasticity in (4) must fall as k falls (a) i. Budget constraint is x = w` + y ii. Substitute the budget constraint into the utility function and transform it by taking U (). Then the problem is equivalent to maximising i h[w` + y] + [ `] First-order condition for an interior maximum is: w [w` + y] [ `] = 0 7

8 and so w` + y w ` = w` + y ` = w where := Therefore we have ` = w y w + w It is clear that this expression is less than given w > 0 and y 0. However, if y were large enough then ` in the above formula would become negative, which is logically impossible. Hence the result. iii. From part (ii) the individual would choose not to work if If ` > 0 then w w := = w + w < 0 So leisure ( `) is increasing in y. 6 (b) If w were higher, would ` and x w w + w = w w y w w + w [w + w] + = w w + w [w + w] w y [w + w] w w y [w + w] = w w + y [w + w] w y [w + w] = w + w y w + y [w + w] = w [ ] + y [w + ] [w + w] 8

9 Work may decrease with the wage rate if < (equivalently < 0) and y is small. Also note that, for given y if some people choose not to work it must be the low-wage people for whom w w. ii. From the budget constraint and part (b): 8 w+y < if y < w +w x = : y otherwise and so, if y < w = [ + w ] [w + y] w [ ] [ + w ] = [ yw ] + w + yw [ + w ] which must be positive. Also note that (c) We know from part y 0 and y = 0, w + y + w > y if w > w. High-ability people have higher money income. 8 < 0 and from part (b), given > 0. Introduction of the bene t is pure income e ect and so decreases `; the tax will decrease the e ective wage and so, for small B it will decrease `. All those with positive hours of work will work less. 6 (a) See chapter 8 of notes. (b) Consider the person s wealth after taking out (partial) insurance cover using the two-state model (no loss;loss). If the person remained uninsured it would be (y 0 ; y 0 L); if he ensures fully it is (y 0 ; y 0 ). So if he ensures a proportion t for the pro-rata premium wealth in the two states will be which becomes ([ t] y 0 + t [y 0 ] ; [ t] [y 0 L] + t [y 0 ]) (y 0 t; y 0 t [ t] L) 9

10 So expected utility is given Eu = [ ] u (y 0 t) + u (y 0 t [ t] L) = [ ] u y (y 0 t) + [L ] u y (y 0 t [ t] L) Consider what happens in the neighbourhood of t = (full insurance). t= = [ ] u y (y 0 ) + [L ] u y (y 0 ) = [L ] u y (y 0 ) We know that u y (y 0 ) > 0 (positive marginal utility of wealth) and, by assumption, L <. Therefore this expression is strictly negative which means that in the neighbourhood of full insurance (t = ) the individual could increase expected utility by cutting down on the insurance cover. 8 (c) For an interior maximum = 0 which means that the optimal t is given as the solution to the equation [ ] u y (y 0 t ) + [L ] u y (y 0 t [ t ] L) = 0 (d) Di erentiating the above equation with respect to y 0 we get [ ] u yy (y 0 0 which gives 6 +[L ] u yy (y 0 t [ t ] = [ ] u yy (y 0 t ) + [L ] u yy (y 0 t [ t ] 0 [ ] u yy (y 0 t ) + [L ] u yy (y 0 t [ t ] L) The denominator of this must be negative: u yy () is everywhere negative and the other terms are positive. The numerator is positive if DARA holds: therefore an increase in wealth reduces the demand for insurance coverage. 6 0

11 5. (a) Using standard notation for net output pro ts are given by = 3X p i y i (5) i= If pro ts are maximised then production must take place on the transformation curve. Therefore, substituting from the production constraint into (5) we get: = p y + p y [y ] + [y ] A (6) To maximise just maximise (6) with respect to y and y. This gives the FOCs: p i A y i = 0 (7) i = ;, and so from which, using (6), pro ts are y = A p (8) y = A p (9) = A [p ] + [p ] : (30) 4 (b) Given that the capitalist utility function is 6 x c x c it is immediate that in the optimum the capitalists spend an equal share of their income on the two consumption goods and so x c i = p i : Worker utility is x w [ x w 3 ] (3)

12 and the budget constraint is p x w x w 3 (3) Maximising (3) subject to (3) is equivalent to maximising The FOC is which gives optimal labour supply as: p [ x w 3 ] [ x w 3 ] : (33) p [ x w 3 ] = 0 (34) x w 3 = p (35) and, from (3), the workers optimal consumption of good is (c) x w = [p ] : (36) 6 i. The economy has no stock of good or good ; workers do not consume good ; so excess demand for the two goods is, respectively: x c + x w y = p + [p ] A p (37) x c y = p A p (38) ii. To nd the equilibrium set each of (37) and (38) equal to zero. This gives p + = A [p ] 3 (39) = A [p ] (40) Substituting in for pro ts from (30) in (40) we have [p ] + [p ] 4 = [p ]

13 and so p p = p 3: (4) Substituting for and p from (30) and (4) into (39) we get p A [p ] + 3 [p ] 4 and, on rearranging, this gives + = A [p ] 3 p = =3 3 (4) A Using (4) for p completes the answer. 4 (d) Using (30), (4) and (4) pro ts in equilibrium are A [p ] = A 3 [p ] = A 3 =3 3 = A =3 A =3 : 3 Given that the price of good 3 is normalised to, using (35)and (4) total labour income in equilibrium is = =3 3 = p A =3 A =3 3 So, workers and capitalists get the same money income in equilibrium. Note that this is una ected by the value of A; so the income distribution remains unchanged by technical progress. 5 3

14 6. Section C Consider the following Stackelberg Duopoly Game. Two quantity-setting rms, and, produce a homogeneous product. Let q and q denote the output of rm and rm respectively, with Q = q + q total output.. The two rms face a (linear) demand curve given by p = a Q, with a >. The cost function of rm is given by C (q ) = q, with C (q ) the total cost of producing output q. The cost function for rm is given by C (q ) = q, with C (q ) the total cost of producing output q. As is standard in a Stackelberg Duopoly Game, rst Firm sets q, then Firm observes q, and then sets its own output q. (a) Find the (unique) Subgame Perfect Nash Equilibrium of the game. That is, nd the (unique) quantities q and q that constitute a Subgame Perfect Nash Equilibrium of the Stackelberg Duopoly Game. Sketched answer: The pro t of Firm can be written as q (a q q ) q Therefore (di erentiate wrt q and set equal to zero) Firm s best response function is q = a q Plugging this into the pro t function for Firm we get a q q a q q Di erentiating and setting equal to zero gives and hence q = a q = a 4 4

15 (b) Now suppose that before playing the Stackelberg game the two rms have to decide whether to enter the market or not. Their decisions to enter are taken simultaneously. Entering is an irreversible decision. Before playing the Stackelberg game both rms observe whether the other has entered or not. Entering the Stackelberg game costs F > 0 to Firm and F > 0 to rm. The entry costs are sunk once incurred. For what range of costs F and F will it be the case that both rms strictly prefer to enter? Sketched answer: Plugging q and q into the pro t function of Firm gives (a ) = 8 Plugging q and q into the pro t function of Firm gives = (a ) 6 So both Firms will strictly prefer to enter if F < (a ) 8 and F < (a ) 6 7. [not for resit] A Seller, S, has a an object for sale. The object costs c to the seller. A potential buyer B has a value of v for the object. There is potential surplus from the exchange in the sense that v > c. S and B have to agree on a price p at which to exchange, and will do so according to the Nash Bargaining Solution. S s utility if the object is sold at price p is (p c) with <. B s utility if the object is sold at price p is v p. If no transaction takes place, both S and B have a utility of 0. 0 (a) Find the price at which the object will be traded according to the Nash Bargaining Solution. Call this price p. Sketched answer: The Nash product is (p c) (v p) Di erentiating and setting equal to 0 gives (p c) (v p) = (p c) 5

16 Rearranging and simplifying gives p = + v + + c 5 (b) Is the price p that you found in (a) an increasing or decreasing function of? 5 Sketched answer: Since v > c, clearly p is increasing in 8. [not for resit] Consider the following auction game. A single indivisible object is for sale. There are two bidders, and. Each bidder i = ; can have a valuation of either or 0 for the object. Each bidder s valuation is or 0 with probability =, and their valuations are independent. The bidders are only allowed to bid either or 0. (a) Suppose the auction is a rst-price one. The bidders are only allowed to bid either or 0. The highest bidder wins and pays her bid. Ties are broken randomly, by ipping a fair coin. Show that there is no equilibrium of the game in which both bidders bid when their value is and bid 0 when their value is 0. Sketched answer: Suppose bidder bids if v = and bids 0 if v = 0. Suppose v =. If bidder sets b = he gets an expected payo of ( ) + ) ( ) = 0 This is because of the following. With probability = bidder s valuation is, in which case b =, the bids are tied and wins with probability =. If he wins, he pays, while his value is. With probability = bidder s valuation is 0, in which case b = 0, and wins for sure. If he wins, he pays and his value is. Still assuming that v =, suppose that bidder sets b = 0. Then he gets an expected payo of 0 + ( 0) = 4 This is because of the following. With probability = bidder s valuation is, in which case b =, wins for sure and gets a payo of 0. With probability = bidders s valuation is 0, in which case b = 0, the bids are tied and wins with probability =. If he wins, he pays 0 and his value is. 6

17 So, when v = bidder would like to deviate from the putative equilibrium in which he bids. He gains by bidding 0 in this case. 0 (b) Now consider the same auction game but with a cap on the bids, so that the bidders are only allowed to bid either = or 0. Show that there is an equilibrium for the game in which both bidders bid = when their value is and 0 when their value is 0. Sketched answer: All checks are done from the point of view of bidder. All calculations for bidder are symmetric. Suppose bidder bids = if v = and bids 0 if v = 0. Suppose v =. If bidder sets b = = he gets an expected payo of + = 3 8 This is because of the following. With probability = bidder s valuation is, in which case b = =, the bids are tied and wins with probability =. If he wins, he pays =, while his value is. With probability = bidder s valuation is 0, in which case b = 0, and wins for sure. If he wins, he pays = and his value is. Still assuming that v =, suppose that bidder sets b = 0. Then he gets an expected payo of 0 + ( 0) = 4 This is because of the following. With probability = bidder s valuation is, in which case b = =, wins for sure and gets a payo of 0. With probability = bidders s valuation is 0, in which case b = 0, the bids are tied and wins with probability =. If he wins, he pays 0 and his value is. So, contingent on v = bidder does not want to deviate from the proposed equilibrium. Suppose now that v = 0. If bidder sets b = = he gets an expected payo of = 3 8 This is because of the following. With probability = bidder s valuation is, in which case b = =, the bids are tied and wins with probability =. If he wins, he pays =, while his value is 0. 7

18 With probability = bidder s valuation is 0, in which case b = 0, and wins for sure. If he wins, he pays = and his value is 0. Still assuming that v = 0, suppose that bidder sets b = 0. Then he gets and expected payo of 0 + (0 0) = 0 This is because of the following. With probability = bidder s valuation is, in which case b = =, wins for sure and gets a payo of 0. With probability = bidders s valuation is 0, in which case b = 0, the bids are tied and wins with probability =. If he wins, he pays 0 and his value is 0. So, contingent on v = 0 bidder does not want to deviate from the proposed equilibrium [not for resit] A principal P hires an agent A to carry out a task that requires unobservable non-contractible e ort e [0; ]. A s e ort determines the probability that the task is successful in generating output. Output equals with probability e and 0 with probability e. Output is observable and contractible. First, P o ers a contract to A, then A accepts or rejects it. After a contract is signed, A chooses e. (a) A contract is a pair of reals (w ; w 0 ), with the rst being the wage (in units of output) that P pays A if output is, and the second being the wage if output is 0. Assume that A cannot be paid a negative wage, regardless of output. Both P and A are risk-neutral, and A dislikes e ort which generates disutility ke =, with k > =. Denote by e, w and w 0 the equilibrium values of e, w and w 0. Compute e, w and w 0. Sketched answer: Given w, w 0 and e, the payo to A is e w + ( e) w 0 k e Given w and w 0 he will choose an e that maximizes this. Di erentiating wrt e and setting equal to 0 gives the IC constraint for A. This is w 0 = k e w 8

19 Hence, it must be that w 0 = 0. This is because otherwise we could decrease both w and w 0, keep e constant and increase P s expected pro t. So we know that w 0 = 0 and w = k e Therefore P s expected pro t can be written as e( w ) + ( e)( w 0 ) = e( w ) = e( k e) Di erentiating wrt e and setting equal to 0 gives e = k Substituting back into the IC gives w = 0 7 [resit]a rm is the only employer in a city. This employer can hire workers on the local labor market, where there are two types of workers, low skill and high skill workers. Workers are expected to perform tasks for the employer. Utility of the workers of type i depend on their income, y i, and on the number of tasks they perform, t i, according to the following utility function: u i (y i ; t i ) = y i i d (t i ) ; where d (t i ) denotes the disutility of performing tasks and i is the skill parameter, with ` > h (the disutility to work is multiplied by a larger factor for the low skill agents). If they are not hired by the employer, workers have a reservation income of 0. We also assume that d (t i ) = t i, and that each task yields a pro t of to the rm, that ` = > h = ; and, nally, that there are 400 high skill workers and 00 low skill workers. (a) Suppose that the employer proposes the following contract. Each hired worker can decide the number of tasks shenhe performs and gets a payment of per task shenhe performs. Would workers of type ` accept to be hired? If so, how many tasks would they choose to perform? Would workers of type h accept to be hired? If so, how many tasks would they choose to perform? Is there an incentive compatibility problem? Outline of solution: Workers try to maximize u i (y i ; t i ) = y i i t i under the constraint that y i = t i : Substituting for y i in the utility 9

20 function and taking the derivative yields t i = i so that t ` = and t h = ; and, consequently, y` = and y h = : Workers of both types would accept to be hired at a wage rate of per task. There is no incentive problem, as u` (y ` ; t `) = ` = 4 > u` (yh; t h) = ` () = 0 and u h (y h; t h) = h () = > u h (y ` ; t `) = h = 3 8 : (b) Suppose the employer observes the skill of the workers. What is the pro t maximizing pair of contracts shenhe will propose to low skill and high skill workers, that is, how many tasks will each of them be asked to perform and how much will they be paid? Is there an incentive compatibility problem? Outline of solution: The employer proposes a contract (t h ; y h ) to type-h workers so as to maximize t h y h under the participation constraint u h (y h ; t h ) 0; and proposes a contract (t`; y`) to type-` workers so as to maximize t` y` under the participation constraint u` (y`; t`) 0: This gives us t i = i so that t ` = and t h = ; and, given that the participation constraints are binding, y ` = 4 and yh = : There is now an incentive compatibility constraint, as u h (y h; t h) = h () = 0 < u h (y ` ; t `) = 4 h = 8 : Workers of type h would like to masquerade workers of type ` : this contract can only be enforced if the employer observes the types. (c) Suppose the employer does not observe the skill of the workers. What is the pro t maximizing pair of contracts shenhe will propose to low skill and high skill workers, that is, how many tasks will each of them be asked to perform and how much will they be paid? Who will be better o, who will be worse o, compared to the complete information situation (part (b) of this question)? Outline of solution: Given what we have deduced above, we know that we have to take the incentive constraint of agents of type 0

21 h into account, as well as the participation constraint of agents of type `: As long as those constraints are met, the other two constraints (participation constraint of workers of type h and incentive constraints of agents of type `) are automatically met. The problem becomes that of under the constraints max 400 (t h y h ) + 00 (t` y`) u` (y`; t`) = y` `t ` 0; u h (y h ; t h ) = y h h t h u h (y`; t`) = y` h t `: Deriving with respect to t h ; t`; y h ; y` h = 400 ic ~t h = 00 p ~t` + ic ~t` h = ic = 00 + p ic = 0 where p is the multiplier associated to the participation constraint, and ic that associated to the incentive compatibility constraint and a tilda means an optimal value of the variable. We get ic = 400; therefore, ~t h = ; p = 500; ~t` = : Deriving with 6 respect to p and ic yields the participation constraints (which must be binding), so that ~y` = and 36 ~y` = 37 : Workers of type ` 7 have the same utilitynlevel as before, 0, whereas workers of type h have a strictly positive utility level, 37 7 = and the employer has a lower pro t both in the contract with a worker of type 7 37 h (a pro t of = 35 instead of ) and in the contract with a 7 7 worker of type ` (a pro t of = 5 instead of = = 9 ) [resit] The manager of a rm can exert a high e ort level E h = or a low e ort level E l =. The gross pro t of the rm is either P = 6 or P = P. The manager s choice a ects the probability of a particular pro t outcome occurring. If he chooses E h, then P occurs with probability h = 3, but if he chooses E 4 l then that probability is only l =. The risk neutral owner designs contracts which specify a 4

22 payment y i to the manager contingent on gross pro t P i. The utility function of the manager is u(y; E) = y = E, and his reservation utility u = 0. (a) Solve for the full information rst best contract, rst, if it is optimal to induce the manager to exert a high e ort, and second if it is optimal to induce hernhim to exert a low e ort. Outline of solution: Under the full information, rst best (FIFB) contract, both manager and owner can observe the action of the manager. Thus, we can solve the optimization problem for the owner for both high and low e ort levels separately, and then study which one yields higher expected pro ts. Formally, denoting the expected pro t to the owner under high and low e ort levels by the manager as P h and P l respectively, we have the following optimization problem for the owner, where i fh; lg: max Pi = i (P y ) + ( i )(P y ) y ;y s.t. i u(y ; E i ) + ( i )u(y ; E i ) u = 0 The participation constraint will be binding at the optimum. Furthermore, since the owner can observe the manager s action, and since the manager is risk-averse, the owner will set y = y = yi. Hence, we can solve for yh and y l simply by setting u(y i ; E i ) = 0 y i E i = 0 Using the numerical values given in the question, we obtain y h = 4 y l = which yield expected pro ts to the owner of P h = 8 + P 4 P l = 3 + 3P 4 : (b) Find the threshold value of P above which high e ort is optimal and below which low e ort is optimal. Outline of solution: Thus, the owner would like to induce the manager to take action E h if 8 + P 3 + 3P ; that is, if P

23 (c) Solve for the second-best contracts in the event that the owner cannot observe the manager s action in both cases where either high e ort or low e ort is optimal. Again, nd the threshold value of P: Outline of solution: However, if the manager could convince the owner that he was using E h, and get the owner to pay him yh, while in fact only using E l, his payo would be u(y h; E l ) = y h E l = > 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. 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 rst best solution continues to hold. The interesting case is inducing the manager to take the high e ort level with a second-best contract. Under the secondbest contract, the participation constraint continues to hold. In addition, however, we have an incentive compatibility constraint that guarantees that the manager would choose E h over E l, given the contract. Thus, the owner has to solve the problem such that max Ph = h (P y ) + ( h )(P y ) y ;y h y + ( h )y E h 0; that is, 3y + y 8 h y + ( h )y E h l y + ( l )y E l ; that is, y y Setting up a Lagrangean L in the usual manner, assigning Lagrange multipliers p and ic to the participation and the incentive compatibility constraint respectively, we nd from the rst-order conditions that (6 p + ic )y = 3 (43) ( p ic )y = (44) 3

24 We rst have to determine whether the constraints will be binding, using Kuhn-Tucker conditions 0, 0 and L = L = 0. Consider = 0. By equations 43 and 44, this implies that y = y = and hence that y = y. But this violates the incentive compatibility constraint, as before. Thus, > 0, and, by L = 0, the IC h must bind. But equation 44 implies that >, and hence R h must bind as well. We can now solve for y and y using IC h and R h constraints, and obtain that y = 5 4 y = 4 The expected pro ts associated with this contract are P h = P 4 P l = 3 + 3P 4 : (note that the pro t in the case of a low e ort remains unchanged, as there is no IC problem in that case). High e ort is optimal when 9 + P > 3 + 3P ; that is, when P < 8:5. 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 secondbest 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. (d) Compare the two threshold values. What can you deduce? Outline of solution: For values of P in the interval [8:5; 0] ; it is optimal to induce the manager to choose a high e ort level if e ort is observable, and it is optimal to induce him to take a low e ort level when e ort is not observable. This is an illustration of the agency cost: it is so costly to enforce a high level of e ort when it is not observable that the owner of the rl has an interest to let the manager choose the low e ort level when e ort is no longer observable. 9 [resit] Answer any two questions from (a)-(c). 4

25 (a) A set of three agents, f; ; 3g, have to choose an alternative in a set of possible options. The problem can be represented as the choice of a point x in the interval [0,]. Agents have singlepeaked preferences over the set of alternatives (that is, each agent i f; ; 3g has a top-ranked alternative in the interval, say t i ; and moving away from that alternative makes hernhim worse o ), but the mechanism designer does not know the preferences of the agents and will have to design a direct revelation mechanism. Show that the following rule is manipulable: let a be a precise point in the interior of the interval: 0<a<. Given the preferences announced by the three agents, if a is e cient, then it is selected, and if a is not e cient, then the selected alternative is the minimum between the three alternatives top-ranked by the agents, min i t i. Outline of solution: In the following graph, the preferences of three agents are represented. Let us assume that these are the true preferences. Given that a is not between the peaks, it is not e cient. Consequently, the rule will select t = min i t i : u i u 3 (a) 6 u 3 (t ) 0 t t t 3 a - But given that u 3 (a) > u 3 (t ) ; agent 3 has an interest to misrepresent his preferences, for instance by pretending that t 0 3 = a: In this case, a becomes e cient, and the rule selects a: (b) A set of two voters, f; g, have to choose an alternative among fx; y; zg : They have strict preferences over the alternatives, but any ranking is possible. Show that the following rule is manipulable. If alternative x is e cient, then it is selected. If alternative x is not e cient, but alternative y is, then y is selected. If neither x nor y is e cient, then z is selected. Outline of solution: One way to solve the problem is by de ning the voting rule extensively. This gives us the following table. 5

26 n xyz xzy yxz yzx zxy zyx xyz x x x x x x xzy x x x x x x yxz x x y y x y yzx x x y y y y zxy x x x y z z zyx x x y y z z One can see that agent, for instance, when she is of type zxy; and is of type yzx; the selected alternative is y; whereas it would be x if she says she has preferences xyz: Therefore, she has an incentive to misrepresent her preferences in that case, and truthtelling is not a dominant strategy. Also, when she is of type yxz; and is of type zxy; the selected alternative is x; whereas it would be y if she says she has preferences yzx: Another way of solving the problem is by thinking directly to the possible ways or pro tably misreprenting one s preferences. If x is agent s top alternative, then it is e cient, and then clearly she does not have any incentive to misrepresent her preferences as she will get x: If y is agent s preferred alternative, then, either x is selected, if it is e cient, or y is selected. If y is selected, then clearly agent has an incentive to tell the truth. If x is e cient, is it possible to still obtain y by misrepresenting her preferences? It must be the case that by announcing x lower in her ranking than what is true, but still having y on top, makes x becomes ine cient, that is, by announcing yzx instead of yxz; x is no longer e cient. Can it be possible? Yes, provided agent has z on top and x is not dominated by y; that is, if agent is of type zxy, so that, by announcing yzx instead of yxz; agent makes x dominated by z: As x is no longer e cient, but y is, y is the new selection. So, agent has an interest to say she is of type yzx when she is actually of type yxz and agent is of type zxy: (c) Two rms, and, have to decide whether or not to undertake some joint investment. The cost of the investment is 0. The bene ts of the investment for each rm, v and v ; are private knowledge. Pro ts are simply bene ts minus payments. The possible values for v are and 7, those for v are 4 and 7. The rms would like to undertake the project if the sum of their values exceed the cost of the project. They agree to use the pivotal mechanism. Compute the amount each rm will have to pay as a function of their announcement. Check that the mechanism 6

27 cannot be manipulated. Check that there will always be enough money if they decide to undertake the project. Outline of solution: The project will be undertaken if (v ; v ) f(7; 4) ; (7; 7)g : Agent is pivotal in the case (; 7) : indeed, agent s valuation is larger than what is, on average, necessary to undertake the project, whereas the project is not undertaken; in this case agent pays P j6=i v j (n ) c =7-5=. She is also pivotal in the case (7; 4) : indeed, agent s valuation is lower than the average cost, whereas P the project is undertaken. In this case, agent pays nc j6=i v j = 0 4 = 6: In the other cases, she pays the average cost. Agent is never pivotal. To sum up, the payments are n 4 7 (0; 0) (; 0) 7 (6; 5) (5; 5) The mechanism cannot be manipulated. Indeed, when agent has valuation, her payo s are n so that telling the truth is dominant. When she has valuation 7, her payo s are n so that truthtelling is, again, a dominant strategy. 7 The relevant tables for agent are n and n when she has valuation 4 and 7 respectively, so that one sees again that truthtelling is a dominant strategy. Finally, the total payments by the agents are in the (7; 4) case, and 0 in the (7; 7) case, so that the cost is always covered. 7