University:
The University of Chicago
Course:
ECON 30200 | Price Theory II
Academic year:

2021
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478

Pages:

39
Author:

pubicswprv

Basic Facts about Risk Aversion Consider an individual with a twice-continuously-differentiable utility function u(C). Suppose this individual has wealth x plus a gamble that will pay a small random amount g, such that E(g) = 0. Let * be the maximum that he would pay to insure against this gamble. So u(x!*) = E(u(x+g)). Assuming that all possible values of g are near 0, Taylor series approximations yield u(x) ! * uN(x) . E[u(x) + uN(x) g + uNN(x) g2'2] = u(x) + uNN(x) Var(g)'2. and then * . ![uNN(x)'uN(x)] Var(g)'2. Fact. There must exist some x1 and x2 in the (small) interval of possible values of x+g such that u(x) ! *uN(x1) = u(x ! *) = Eu(x+g) = u(x) + 0.5 uNN(x2) Var(g), so * = !0.5 Var(g) uNN(x2)'uN(x1), where x2 and x1 are close to x when the range of g is small. The individual's Arrow-Pratt risk-aversion index at wealth x is r(x) = !uNN(x)'uN(x). So we find that the individual's value for a small zero-expected-value gamble is approximately half of the variance of the gamble times the individual's risk-aversion index. Notice that this risk-aversion index r(x) would not change if we changed how i's utility is measured ˆ to some other equivalent scale u(x) = Au(x)+B where A>0 and B are constants. The reciprocal of the risk aversion index J(x) = 1'r(x) is called the risk tolerance index, which has the advantage of being measured in the same units as money (dollars). An individual whose risk aversion is constant, independent of his given wealth x, must have a utility function that satisfies the differential equation !uNN(x)'uN(x) = R, for some constant R. This differential equation is equivalent to d[LN(uN(x))]'dx = !R. For R>0, the solutions to this differential equation with uN>0 are u(x) = B!Ae!Rx, where A>0 and B are arbitary scale constants. An individual whose risk tolerance is proportional to his wealth would have r(x) = "'x with ">0, so u(x) = B+Ax1!"'(1!"), or u(x) = B+A LN(x) if "=1. Suppose an individual with constant risk aversion R (u(x) = !e!Rx) will get a random income Y drawn from a Normal distribution with mean µ and standard deviation F. The certainty equivalent (CE) of this lottery Y for this individual is the sure amount of money W that individual would be willing to accept instead of this lottery Y. Thus, the certainty equivalent W satisfies the equation u(W) = E(u(Y)). Fact: Suppose u(C) is a utility function with constant risk aversion, Y1 and Y2 are independent random variables, and u(Wi) = E(u(Yi)) for each i in {1,2}. Then u(W1+W2) = E(u(Y1+Y2)). Fact: For an individual with constant risk aversion R (risk tolerance J=1'R), a Normal lottery with mean µ and standard deviation F has certainty equivalent W = µ ! 0.5RF2 = µ ! 0.5(F'J)F. 2 That is, !e !R(µ!0.5RF ) ' +4 m!4 2 !e !Ry e !0.5((y!µ)/F) 2B F dy . (Proof: use +4 m!4 2 2 e !0.5((y![µ!RF ])/F) 2B F dy ' 1 .) (This can also be derived from the local approximation formula using the fact that a Normal random variable can be written as the limit of sums of many small independent random variables.) 1 Efficient Risk Sharing in a Syndicate Let N denote the set of members of an investment partnership or syndicate. They hold assets which will yield random returns Y that have some given probability distribution. Each individual i in N has a given utility function for money ui(C). Let xi(y) denote the individual i's planned payoff when the syndicate earns Y=y. To be feasible, we must have 3i0N xi(y) = y, œy0ú. Let us consider an efficient allocation rule (xi(C))i0N that maximizes 3i0N 8i E(ui(xi(Y))), subject to this feasibility constraint, for some given positive utility weights (8i)i0N. To simplify the characterization of the efficient rule, we assume all appropriate differentiability. For each outcome y, the marginal weighted utility 8iuiN(xi(y)) must be equal across all i in N. That is, there must exist some function <(y) such that 8i uiN(xi(y)) = <(y), œi0N, œy0ú (1). Differentiating with respect to y, we get 8i uiNN(xi(y)) Mxi'My = 0 (or ui(x) = LN(x) for "=1) then (1) implies 8i (xi(y))!" = <(y) and so xi(y) = 8i1/" <(y)!1/". Then y = 3j xj(y) = <(y)!1/" 3j (8j)1/", so <(y)!1/" = y'[3j (8j)1/"] and <(y) = [3j (8j)1/"]" y!" . So each i must get a constant fraction of the total y: xi(y) = ki y, where ki = (8i)1/"'[3j (8j)1/"]. Implementation with a price system Suppose that Y has finitely many possible values y, each with some probability p(y). The marginal utility equation (1) 8i uiN(xi(y)) = <(y), œi0N, œy0ú, implies that the efficient allocation can be implemented by a price system, in which each partner i has a budget Ii to spend on acquiring claims on income in each state of the total returns Y, and B(y) is the price of one unit of income when Y=y. In such a price system, i would choose the quantities xi(y) for all possible y to maximize 3y p(y)ui(xi(y)) subject to 3y B(y)xi(y) # Ii. Let µi be the Lagrange multiplier of i's budget constraint (i's marginal utility of budgeted funds). At an optimal solution, we have p(y)uiN(xi(y)) = µiB(y), œi, œy. This coincides with the efficient marginal utility equation (1) iff uiN(xi(y)) = µiB(y)'p(y)= <(y)'8i, œi œy. Thus µi8i = p(y)<(y)'B(y) must be some constant A that does not depend on i or y. That is B(y) = p(y)<(y)'A. This price system allows a risk-free rate of return on budgeted funds 1+D = 1'[3y B(y)] So in terms of the risk-free rate D, we must have (1+D)[3y p(y)<(y)] = A, and so B(y) = p(y)<(y)'[(1+D) 3z p(z)<(z)] œy. The price of future income from returns y depends both on the probability p(y) and on <(y). <(y) measures the marginal social utility of future returns y, and it is generally decreasing in y, because it is the Lagrange multiplier of the total-returns constraint in the problem: choose (xi)i0N to maximize 3i 8iui(xi) subject to 3i xi # y. References: Robert Wilson, Econometrica (1968); Karl Borch, Econometrica (1962). 3 Optimal risk sharing among partners with constant risk tolerance Consider a group of individuals who have formed a partnership to share the risky profits from some joint venture or gamble Suppose each individual j in this group has a constant risk tolerance, denoted by Jj. Let To denote the sum of all the partners' risk tolerances (To = 3j Jj). These partners can maximize the sum of their certainty equivalents by sharing the risky profits among themselves in proportion to their risk tolerances, with each individual j taking the fractional share Jj'To of the risky profits. The maximal sum of the partners' certainty equivalents that can be achieved by such efficient risk sharing is equal to the certainty equivalent of the whole gamble to an individual who has a constant risk tolerance equal to To, the sum of these partners' risk tolerances. Thus, in making decisions about risky investments, the partnership should act as a corporate person with a risk tolerance equal to the sum of its members' risk tolerances. (Coase theorem) If the partners were planning to share risks according to a sharing rule that does not maximize the sum of the partners' certainty equivalents, then any partner j could propose another sharing rule rule that would increase j's own certainty equivalent and would not decrease the certainty equivalents of any other partners. ˜ that is drawn from a Normal Risk-sharing example. Consider a financial asset yielding a return Y distribution with mean µ = $35000, F = $25000. Investor 1 has risk tolerance T1 = 20000, and investor 2 has risk tolerance T2 = $30000. When 1 owns the whole asset, his certainty equivalent is ˜ = 35000!(0.5'20000)*250002 = 35000!15625 = $19375. CE1(Y) If 2 owned the whole asset, her certainty equivalent would be ˜ CE2(Y) = 35000!(0.5'30000)*250002 = 35000!10417 = $24583 ˜ So if 1 sold the asset to 2 for price x then they would get CE1(x) = x, CE2(Y!x) = $24583 ! x. The transaction makes both better off if 19375 # x # 24583. In this range, x=24583 would be best for investor 1. Now consider the possibility that 1 could sell a 50% share to 2 for some price x. Then they get ˜ CE1(0.50Y+x) = 0.50*35000 + x ! (0.5'20000)*(0.50*25000)2 = 17500+x ! 3906 = $13594 + x, ˜ CE2(0.50Y!x) = 0.50*35000 ! (0.5'30000)*(0.50*25000)2 = 17500 ! x ! 2604 = $14896 ! x. This transaction makes both better off if 19375!13594 = 5781 # x # 14896. The sum of their certainty equivalents is 13594+14896 = $28490 which is higher than either could get from the asset alone. When the price x is 14896, investor 1 gets all of this certainty equivalent value. In general, if 1 sells a share 2 to investor 2 for a price x, their certainty equivalents are ˜ CE1((1!2)Y+x) = (1!2)*35000 + x ! (0.5'20000)*[(1!2)*25000]2 ˜ CE2(2Y!x) = 2*35000 ! x ! (0.5'30000)*(2*25000)2 The sum of these is maximal when (1!2)2'T1 + 22'T2 is minimized, at 2 = T2'(T1+T2) = 0.6. When 1 sells the optimal 60% share to investor 2 for a price x, they get ˜ CE1(0.40*Y+x) = 0.40*35000 + x ! (0.5'20000)*(0.40*25000)2 = = 14000 + x ! 2500 = $11500 + x, ˜ CE2(0.60*Y!x) = 0.60*35000 ! (0.5'30000)*(0.60*25000)2 = = 21000 ! x ! 3750 = $17250 ! x. This transaction makes both better off if 19375!11500 = 7875 # x # 17250. The maximized sum of their certainty equivalents is 11500+17250 = $28750. When the price x is 17250, investor 1 gets this maximal certainty-equivalent value. ˜ would be For a fictitious corporate person with risk tolerance To = T1+T2 = 50000, the asset Y ˜ 2 worth CEo(Y) = 35000!(0.5'50000)*25000 = 35000!6250 = $28750. 4 Moral hazard with a linear wage rule, Normal risks, and constant risk tolerance. Suppose that a firm's net profits will be a random variable y drawn from a Normal distribution with a variance F2 but with a mean that depends on the effort of an agent. If the agent's effort is high, then the agent's effort cost is cH and profit has mean E(y*cH) = mH. If the agent's effort is low, then the agent's effort cost is cL and profit has mean E(y*cL) = mL. Suppose mH > mL and cH > cL. Nobody else can observe the agent's effort, but his wage can depend on the observable profit y. For now, let us assume that firm must specify the agent's wage as a linear function of observed profits, according to any linear formula w(y) = "+$y. (This assumption is questionable. We will see soon that the firm might prefer a nonlinear formula. But let's start with this linearity assumption because linearity makes things simpler.) Suppose that the agent has a constant risk tolerance T, but the owners of the firm are risk neutral (meaning that they have linear utility for money or infinite risk tolerance). G elsewhere with zero effort costs. The agent's best outside option is to earn w What linear wage rule (",$) is best for the firm if the firm wants the agent to choose high effort? Under an (",$) linear wage rule, the agent's net certainty equivalent with high effort is E("+$y*cH) ! (0.5'T)Var("+$y*cH) ! cH= " + $mH ! (0.5'T)($F)2 ! cH. G By choosing his outside option, the agent could instead get certainty equivalent w. With low effort, the agent could get the certainty equivalent "+$mL!(0.5'T)($F)2!cL. So for the agent to want to work here and exert high effort, (",$) must satisfy the G participation constraint " + $mH ! (0.5'T)($F)2 ! cH $ w, and the 2 moral-hazard incentive constraint "+$mH!(0.5'T)($F) !cH $ "+$mL!(0.5'T)($F)2!cL. G + c + (0.5'T)($F)2 and $ $ (c !c )'(m !m ). These constraints imply " + $mH $ w H H L H L The risk-neutral owners of the firm want to minimize their expected wage bill "+$mL. So the optimal linear rule should choose " (given any $) to make the participation constraint bind, and should choose $ as small as possible to make the moral-hazard constraint bind. G That is, the optimal rule is $ = (cH!cL)'(mH!mL) and " = w + cH + (0.5'T)($F)2 ! $mH. G + c + (0.5'T)($F)2, and the owner's expected net The firm's minimal expected wage bill is w H G 2 profit is thus mH ! w ! cH ! (0.5'T)($F) . Given that the agent is risk averse and the owners are risk neutral, the optimal sharing rule without G moral hazard would have $=0 (and so "=w+c H), if the agent could be forced to choose high effort. So moral hazard provides a fundamental reason why risk-averse agents cannot insure away their professional risks: because their share of such risks is essential to motivate their efforts. 5 Moral Hazard with constant risk tolerance, monetary effort cost, 2 actions. A risk-neutral principal is designing an incentive plan for a risk-averse agent. The agent must choose among two unobservable actions: aH and aL. G elsewhere. Each action has a cost to the agent, c(aL) = cL < c(aH) = cH. The agent could earn w Suppose the agent's utility from choosing action a and getting wage w would be u(w!c(a)) = !EXP(!(w!c(a))'T), where T>0 is the agent's constant risk tolerance. [This model differs from standard textbook models in that effort cost here is monetary, subtracted from income before applying the utility function: u(w!c(a)) instead of u(w)!c(a).] The principal can only observe an outcome y that is a random variable which depends on the agent's action according to the conditional probability distribution p(y*aH) or p(y*aL). Let Y denote the set of possible values of y, which we assume here to be a finite set. The principal can promise the agent a wage w(y) that depends on the observable outcome. Suppose that the principal's expected payoff is much higher when the agent chooses aH. So the principal's problem is to design the wage-function w(C) to minimize the expected wage G or the lower expense, subject to the constraints that the agent should not prefer the outside option w action aL: Choose (w(y))y0Y to minimize 3y0Y p(y*aH) w(y) subject to 3y0Y p(y*aH) u(w(y)!c(aH)) $ u(wo) (participation constraint: 8), 3y0Y p(y*aH) u(w(y)!c(aH)) $ 3y0Y p(y*aL) u(w(y)!c(aL)) (moral-hazard constraint: µ). The participation constraint must be binding, or else the principal could reduce all w(y). G If the moral-hazard constraint were not binding, then the optimal solution would be w(y)=w+c H for all outcomes y, but then the agent would prefer the lower-cost action aL (with c(aL) 1+D > pLR + (, so that the project could be worthwhile for risk-neutral investors if the entrepreneur manages it well, but not if he acts badly. If the entrepreneur is charged an interest rate r to borrow I, then the entrepreneur will get the surplus (R!(1+r))I if the project is a success, but he will default and lose the collateral C if the project fails. Including the value of defaulted collateral, the investors' expected return is pH(1+r)I + (1!pH)C. Per unit invested, the investors' rate of return is pH(1+r) + (1!pH)C'I. So the investors' participation constraint is pH(1+r) + (1!pH)C'I $ 1+D, which holds iff 1+r $ (1+D)'pH ! (C'I)(1!pH)'pH. The entrepreneur's participation constraint is pH(R!(1+r))I ! (1!pH)C $ w0. The entrepreneur's participation constraint implies pHR ! w0'I $ pH(1+r) + (1!pH)C'I. The moral-hazard constraint to deter entrepreneurial malfeasance is pH(R!(1+r))I ! (1!pH)C $ (I + pL(R!(1+r))I ! (1!pL)C. The moral-hazard constraint yields a lower bound on an agent's stake C + (R!(1+r))I $ (I'(pH!pL), and an upper bound on investors' rate of return C'I + pHR ! pH('(pH!pL) $ pH(1+r) + (1!pH)C'I. Suppose that C and w0 are small so that C + w0 < pH(I'(pH!pL). [poor agent case] This poor-agent condition implies that C'I + pHR ! pH('(pH!pL) < pHR ! w0'I. Then the upper bound on investors' rate of return is determined by the moral-hazard constraint: pH(1+r) + (1!pH)C'I # C'I + pHR ! pH('(pH!pL). The maximal interest rate r* that can avoid moral hazard is 1+ r* = R + C'I ! ('(pH!pL). Suppose also that R satisfies R $ (1+D)'pH + ('(pH!pL) ! C'(pHI). Then the investors are willing to lend at this rate r*, because pH(1+r*)I + (1!pH)C = pH[R + C'I ! ('(pH!pL)]I + (1!pH)C $ (1+D)I. Even at this maximal interest rate, however, the entrepreneur's expected gain is strictly positive: pH(R!(1+r*))I ! (1!pH)C ! w0 = pH('(pH!pL) ! C ! w0 > 0. The quantity pH('(pH!pL) here may be called the expected moral-hazard rent that the entrepreneur must be allowed in this financial transaction, to deter his malfeasance. Now suppose that there are many such entrepreneurs, but only a limited supply of funds from investors who understand these projects well enough to enter into such financial deals. Then the interest rate for such loans can rise only to the maximal rate r*. If the demand from entrepreneurs at this rate exceeds the supply of funds, there must be credit rationing. 7 Long-term agency relationships Let us consider a version of the above model, but now the projects are available to firms which must hire agents to manage them. As before, R is the return per unit invested if the project succeeds, pH is the probability of success if the manager acts appropriately, but the manager could instead divert a fraction ( of the invest funds and reduce the probability of success to pL. Here pL < pH. Agents' alternative wages and collateral are negligible relative to these investments (so w0 = 0, C=0). An agent can manage a project of various investment sizes I, with an upper bound of I # 1 (billion$) (and with some much smaller positive lower bound that is still large relative to wages and collateral). If firms can profitably hire agents to manage such projects, then many firms will do so, driving down the prices of the goods which are the outputs of these investment projects. So (assuming such a general equilibrium) let us consider the optimal agency contracts for these firms when R is near the lower bound where such contracts are just barely profitable. Let us ignore interest over the periods of the project (D=0). Assuming limited liability for the agents, the terms of an agent's contract specify (I,wS,wF), where I is the investment size handled by the agent, wS is his wage if success, wF is his wage if failure. Assuming risk-neutrality, the agent's participation constraint is pHwS + (1!pH)wF $ 0, the moral-hazard constraint is pHwS + (1!pH)wF $ (I + pLwS + (1!pL)wF, and the limited-liability and investment bound constraints are wS $ 0, wF $ 0, 0 # I # 1. (The actual lower bound on I may be positive, but we allow I=0 to represent not hiring the agent.) The firm wants to maximize its expected profit (pHR!1)I ! [pHwS + (1!pH)wF]. The optimal solution has wF = 0, wS = (I'(pH!pL), giving the firm [pHR ! 1 ! pH('(pH!pL)]I, and so the investment I = 1 is worthwhile as long as R $ 1'pH + ('(pH!pL). Let G = ('(pH!pL) denote the moral-hazard rent per unit invested which a successful agent must get, and let g = pHG = pH('(pH!pL) denote the expected moral-hazard rent per unit invested. So with competitive firms, the output-market equilibrium would yield R at or near R* = 1'pH + G. To verify the optimality of the solution, notice that the optimization is a linear programming problem. The solution is supported by the following Lagrange multipliers (dual variables): 8=0 for the agent's participation constraint, and µ = pH'(pH!pL) for the moral-hazard constraint. These multipliers yield 0 = ML'MwS = !pH + 8pH + (pH!pL)µ, 0 > ML'MwF = !(1!pH) + 8(1!pH) ! (pH!pL)µ, and ML'MI = pHR ! 1 ! µ( $ 0 when R $ 1'pH + G. A surplus G is required here to cover the cost of the agent's moral-hazard rent. Now suppose that an agent can manage one such project in each of T periods in his career, and the agent's responsibilities can depend on his past history of successes or failures. We will now show that the lowest R at which firms can expect nonnegative profits from such multiperiod agency relationships is R* = 1'pH + G'T. Intuitively, the cost of an agent's moral-hazard rent G can now be divided over T periods. The optimal policy at each period t can be characterized in terms of a recursive optimization problem with two new parameters: vt represents the expected monetary rewards that were promised to the agent at the end of the previous period, and 8t+1 denotes the firm's future net marginal cost of fulfilling promises of future monetary rewards. Then the firm's problem at period t is 8 choose (It,wS,t,wF,t,xt) to maximize (pHR!1)It ! 8t+1[pHwS,t + (1!pH)wF,t] ! xt + 8tvt (net profit) subject to xt + pHwS,t + (1!pH)wF,t $ vt (8t, promise-keeping) pHwS,t + (1!pH)wF,t ! [(I + pLwS,t + (1!pL)wF,t] $ 0 (µt, moral hazard) xt $ 0, wS,t $ 0, wF,t $ 0, 0 # It # 1. (bounds) Here xt represents the firm's option of paying its past promises in cash at the start of period t. For the full multi-period problem, we must add the following conditions: the agent's initial credit is v1 = 0 because the firm has no past obligations to a new agent, each later vt is either wS,t!1 or wF,t!1 depending on whether the previous period was a success or failure, the final marginal wage-cost is 8T+1 = 1 because rewards to a retiring agent must be paid in cash, and each earlier period derives its marginal cost of end-of-period wage promises from the Lagrange multiplier of next period's promisekeeping constraint. (The last term in the objective 8tvt is an accounting constant, not affected by current decisions, to let us sum profits across periods without double-counting back-loaded wages.) We will now show that, when R = R* = 1'pH + G'T, the following dynamic policy is optimal but yields zero expected profits for the firm, and so firms cannot make expected profits at any lower R. In the last period, if the agent has managed successful projects in all previous periods, then the agent manages the maximum feasible investment IT = 1 and is paid wS,T = ITG = G if it succeeds. The possibility of this large final payoff motivates the agent at all previous periods. At each period t, if the agent has never failed in the past then he manages a project with It = (pH)T!t. Then his promised reward for success at period t will be wS,t = ItG = (pH)T!tG, which is just the expected value of his potential final reward (G times the probability of T!t future successes). If the agent fails at any period t, he gets wF,t = 0 and does not manage any more projects. The backloading of all rewards is indicated in the solution by xt = 0 œt. Under this plan, the agent invests It = (pH)T!t at time t only if he has never failed before, which has probability (pH)t!1. His expected investment is (pH)t!1(pH)T!t = (pH)T!1 at each period t, and his expected wage bill is (pH)TG, so the firm's expected net profit is T(pH)T!1(pHR!1)!(pH)TG. So at any t>1, as long as the agent has been successful, we get vt = wS,t!1 = (pH)T+1!tG, and so the promise-keeping constraint is satisfied with equality by vt = pHwS,t = pH (pH)T!tG. The moral-hazard constraint is also binding at every t, because wS,t = ItG= It('(pH!pL). To support this as an optimal solution in which investment is positive but the firm's expected net profits are zero, we need Lagrange multipliers µt $ 0 and 8t $ 0 to satisfy the equations 0 = ML'MIt = pHR ! 1 ! µt( and 0 = ML'MwS,t = !8t+1pH + 8tpH + (pH!pL)µt, for all t0{1,...,T}, and we need 81 = 0 because promise-keeping has slack at t=1 when the agent starts with v1=0. For 0 = ML'MIt = pHR!1!µt(, at all t we must have µt = (pHR!1)'(. Then 0=ML'MwS,t = 8tpH +(pH!pL)µt !8t+1pH implies 8t = 8t+1!µt(pH!pL)'pH = 8t+1 ! (pHR!1)'g. (Recall g = pHG = pH('(pH!pL).) Given 8T+1=1, we get 8t = 1!(T+1!t)(pHR!1)'g, œt. Then 81 = 0 is satisfied iff 1 = T(pHR!1)'g, which holds when R = (1+g'T)'pH = 1'pH + G'T. ML'Mxt = 8t ! 1 < 0 and ML'MwF,t = !(8t+1!8t)(1!pH)!(pH!pL)µt < 0 imply xt=0=wF,t is optimal. If R is slightly greater than this R*, this plan can still satisfy all recursive optimality conditions with one change: at T, with IT at the upper bound IT=1, we can have ML'MIT >0 and µT < (pHR!1)'(. Keeping other µt fixed, we get 8t = 1 ! µT(pH!pL)'pH ! (T!t)(pHR!1)'g œt, and we can get 81=0 with µT = [1!(T!1)(pHR!1)'g]pH'(pH!pL). This fails only when a larger R makes µT negative. 9 This model describes a world where productive investments are only possible when agents have longterm relationships with firms which the agents can trust to reliably pay deferred rewards for service. The temptation to deny long-promised wages by finding fault with the agent's late-career performance may become particularly acute when R* < G (which can happen when pH, (, and T are large). So this economy relies heavily on firms' reputations for judging and rewarding their agents. The large moral-hazard rents are also a striking feature of this model. But one might ask whether these are just a result of the limited-liability assumption, which puts a lower bound on agents' payoffs. So let us see how the solution would change if the limited liability constraint were relaxed by allowing punishment in case of failure. A punishment z$0 would reduce the agent's utility to !z but without generating any benefit anyone else. (This is different from taking valuable collateral.) The recursive problem at any period t is now choose (It,wS,t,wF,t,zt) to maximize (pHR!1)It ! 8t+1[pHwS,t + (1!pH)wF,t] +8tvt subject to pHwS,t + (1!pH)(wF,t ! zt) $ vt pHwS,t + (1!pH)(wF,t ! zt) $ (It + pLwS,t + (1!pL)(wF,t ! zt) wS,t $ 0, wF,t $ 0, zt $ 0, 0 # It # 1. with 8T+1 = 1, v1 = 0, and vt+1 = wS,t for agents who succeed through time t. (net profit) (8t, participation) (µt, moral hazard) (bounds) As before, we consider parametric cases where R is at or near the lower bound at which firms can make nonnegative expected profit from agents who could manage a project in up to T periods. Expanding the contract options to include punishment decreases the lowest R at which firms can profitably invest down to R** = 1'pH + G'[T + pH'(1!pH)]. We show now that, at or near this R**, the optimal contract involves no punishment after period 1. As above, consider a plan in which the agent is paid a moral-hazard rent wS,T = G at period T only if he manages successful projects at all previous periods. The prospect of this moral-hazard rent yields expected rewards worth wS,t = G(pH)T!t to the agent when he has been successful through period t. We always have wF,t=0. At any time t>1, the agent can manage It = wS,t'G = (pH)T!t with zt = 0 (no punishment), and the promise-keeping constraint is also binding with vt = wS,t!1 = (pH)T+1!t. With v1=0, the promise-keeping/participation constraint would have slack at t=1 if z1 were zero, but we can make it bind with z1 = wS,1pH'(1!pH) = G(pH)T'(1!pH) Then the moral-hazard constraint will be binding when (I1 = (pH!pL)(wS,1+z1) = (pH!pL)G(pH)T!1'(1!pH) , so I1 = (pH)T!1'(1!pH). The optimality conditions are almost the same as above. The only changes are that 81 can be nonzero but now, for z1>0, we need 0 = ML'Mz1 = µ1(pH!pL) ! 81(1!pH). Also, to justify zt=0 for all t>1, we need 0 $ ML'Mzt = µt(pH!pL) ! 8t(1!pH). From 0=ML'MIt in the zero-expected-profit solution at R=R**, we again get µt = (pHR!1)'( œt. From 0=ML'Mz1 = µ1(pH!pL) ! 81(1!pH), we get 81 = µ1(pH!pL)'(1!pH). As before, 0 = ML'MwS,t = !8t+1pH + 8tpH + (pH!pL)µt implies 8t+1 = 8t + µt(pH!pL)'pH at all t, and so 8t = [pH'(1!pH) + t!1]µt(pH!pL)'pH = [pH'(1!pH) + t!1](pHR!1)'g for all t. Then we get 1 = 8T+1 = [pH'(1!pH) + T](pHR!1)'g when R = R** = 1'pH + G'[T+pH'(1!pH)]. Finally, to justify zt=0 for t>1, we have ML'Mzt = µt(pH!pL) ! 8t(1!pH) # 0 for all t because ML'Mz1=0, µt is a constant over t, and 8t is increasing in t. This solution could fail only if I1>1, but then we can proportionally decrease all It and wS,t until I1=1. When R is slightly more than R**, we can still show optimality by reducing µT if IT=1, or µ1 if I1=1. 10 Becker-Stigler-Shapiro-Stiglitz efficiency wages Consider a similar moral-hazard problem where the agent is risk neutral, and there are only two possible observations: y=1 denotes normal business, and y=0 denotes an accident occurring. We consider a short interval of time g, in which the probability of an accident is "g if the agent chooses to be diligent aH, and $g if the agent chooses to shirk aL, where $ > ". Choosing aL also yields a hidden benefit of Dg to the agent in this period. Suppose that participation constraints apply ex-post: after the outcome is observed, the agent cannot G which is the present-discounted value of his lifetime be made worse off than his outside option of v, income in the competitive labor market. The principal wants to minimize the expected cost subject to the ex-post participation constraints and the moral-hazard incentive constraint that the agent should not shirk. Let V1 and V0 denote the agent's expected total payoff after observing y=1 or y=0 respectively. So the principal's problem is choose (V1,V0) to minimize (1!"g)V1 + "gV0 subject to G G V $ v. (1!"g)V1 + "gV0 $ Dg + (1!$g)V1 + $gV0, V1 $ v, 0 G G that We could add an ex-ante participation constraint (1!"g)V1 + "gV0 $ w, but if the payoff w must be promised to recruit the agent is the same as the payoff vG that he can get by quitting later, then G this constraint is redundant with the ex-post participation constraints V1 $ vG and V0 $ v. The moral-hazard constraint implies V1 ! V0 $ D'($!"). G + D'($!"). G V =v So the optimal solution is V0 = v, 1 So the agent's expected reward (1!"g)V1 + "gV0 = vG + (1!"g)D'($!") for a short g-period of service must be greater than the outside option vG by a positive bonus even as g!0. In a dynamic model, if the problem is repeated with a different agent every g-period then the principal's cost becomes huge, but the cost can be reduced by using benefits of future employment be part of current incentive-pay. Consider a stationary solution: if y=1 this period then the agent will be paid gw and rehired for next G period, but if the y=0 this period then the agent is dismissed to the outside option v. G So with discount rate r, we get the recursion equation: V1 = wg + (1!rg)((1!"g)V1 + "gv). With V1 = vG + D'($!"), this implies w = rvG + (r + " ! gr")D'($!"). G Here rvG is the outside wage rate corresponding to the present-discounted value v. As g60, this optimal stationary incentive plan pays the agent an efficiency wage rate that exceeds the outside wage rate by (r+")D'($!"), but dismisses the agent when an accident occurs. This g60 is a problem of controlling a Poisson process where accidents occur as a Poisson process with the low rate " when the agent is diligent, but the high rate $ when the agent shirks. ˜ ˜ When X is a Poisson random variable with mean 8, X can be any nonnegative integer, ˜ !µ k ˜ ˜ ˜ P(X=k) = e (µ) 'k! for any k 0{0,1,2,...}, E(X) = µ, Var(X) = µ, Stdev(X) = µ0.5. When accidents occur in a Poisson process with rate 8, the number of accidents between any two times t and t+* (*>0) is a Poisson random variable with mean 8*, and it is independent of the number of accidents before time t. In any short time interval of length g, the probability of an accident is approximately 8g, the probability of no accidents is approximately 1!8g, and the probability of two or more accidents is vanishingly smaller of order g2. 11 Holmstrom-Milgrom Control of Brownian Motion (Econometrica 55(2):303-328, 1987): Suppose the agent is risk-averse with constant risk tolerance T, subject to an ex-ante participation constraint with the alternative reservation wage wo. But now suppose that the observable is a Normal random variable with mean µ(a) that depends on his action, and variance F2 that does not depend on his action. For two Normals with different means and the same variance, the likelihood ratios go to infinity in the tails, and so the multiplier of our moral-hazard constraint must be µ=0: it is costless! The optimal solution achieves first-best nonlinearly: pay wo except for an infinite punishment in an event that has infinitesimal probability, infinitesimally smaller when he chooses high µ. Something seems wrong in this solution. Holmstrom-Milgrom changed the problem to allow the agent to get feedback as the Normal outcome evolves, by considering the problem of controlling the drift of a Brownian motion. For such a problem, they showed that linear incentive pay becomes optimal. For important applications, they explicitly considered multidimensional Brownian motion. We may (w.l.o.g.) restrict attention to Brownian motion (Y1(t),...,Yn(t)) where the various components Yi(t) are independent and have the same volatility: In any time interval from t to t+* (*>0), the changes Yi(t+*)!Yi(t) are Normal random variables, with mean µi(a)* and variance F2*, independent across i, and independent of the path to time t. Here the drifts µi(a) are functions of the agent's hidden action (a) during this interval. The agent also pays a hidden cost of effort at rate c(a) over time. The agent's choices of action after time t can depend on the past (Y1(t),...,Yn(t)). We may assume that the process begins at (Y1(0),...,Yn(0)) = (0,...,0). A linear incentive plan for the agent over the period [0,S] specifies constants (A,B1,...,Bn) and promises to pay the agent w(Y) = A + 3i BiYi(S). The agent's final utility payoff is u(w(Y)!C(a)), where C(a) = IS0 c(a(t))dt, and u(C) is a utility function with constant-risk-tolerance T. We may let the ending time be S=1, so that, for any constant action a, the Yi(S) are independent Normals with mean µi(a) and variance F2. Holmstrom and Milgrom show that such simple linear incentive plans are optimal for the principal to maximize any linear function of the final (Y1(S),...,Yn(S)) subject to an ex-ante participation constraint at time 0 and a moral-hazard constraint that the constantrisk-tolerant agent always chooses an action that maximizes his expected utility: Choose w(C) and a* to maximize E(3i Bi Yi(S) ! w(Y) *a*) subject to a* 0 argmaxa E(u(w(Y)!C(a))*a), and E(u(w(Y)!C(a*))*a*) $ u(wo). Linearity makes w(Y) = A + 3i BiYi(S) a Normal random variable, so the agent's constraints can be rewritten with the formula for certainty equivalents of Normals with constant risk tolerance: a* 0 argmaxa A+3i Biµi(a)!C(a)!(0.5'T)F23i Bi2, A+3i Biµi(a*)!C(a*)!(0.5'T)F23i Bi2 $ wo. The binding participation constraint yields A = wo + C(a*) + (0.5'T)F23i Bi2 ! 3i Biµi(a*). Then the principal's expected profit is E(3i BiYi(S)! w(Y)*a*) = 3i Biµi(a*) ! 3i Biµi(a*) ! A = 3i Bi µi(a*) ! C(a*) ! (0.5'T)F23i Bi2 ! wo, which we want to maximize subject to the incentive constraint, which reduces to: a* 0 argmaxa 3i Biµi(a) ! C(a). For each component ak of vector a, first-order conditions are 3i Bi Mµi(a*)'Mak = MC(a*)'Mak or, with nonnegativity constraints ak $ 0, we may have 3i Bi Mµi(a*)'Mak # MC(a*)'Mak if ak=0. 12 Technical key to their proof: Such a Brownian motion can be viewed as the limit of a linear function of multidimensional Poisson process. Large-mean Poissons are approximately Normal. Consider some small g>0. For any action a, let the Poisson processes start at xi(0*a,g)=0 for all i. For any time interval t to t+*, let xi(t+**a,g)!xi(t*a,g) (= arrivals of type i) be a Poisson random variable with mean *7i(a,g) = *(F2+gµi(a))'g2, independently of the past before t, and independently across all i=1,...,n. Recall that the variance of a Poisson random variable is equal to its mean or expected value. Now let yi(t*a,g) = g xi(t*a,g) ! t F2'g. So yi(t*",g) has frequent discontinuous upward jumps of size g, but between these jumps it is decreasing at the rate !F2'g. Then E(yi(t+**a,g)!yi(t*a,g)) = g * (F2+gµi(a))'g2 ! * F2'g = µi(a)*, and Var(yi(t+**a,g)!yi(t*a,g)) = g2 * (F2+gµi(a))'g2 = F2* + gµi(a)* 6 F2* as g60. So as g60, this y process converges to the Brownian motion Y that we wanted to study. So a wage that is linear in the yi, say A+3ni=1 Biyi, would also be linear in the xi, with coefficients: A+3ni=1 Bi yi(t*a,g) = A+3ni=1 Bi [g xi(t*a,g)!tF2'g] = (A!ntF2'g) + 3ni=1 (Big) xi(t*a,g) Consider a discrete-time approximation to the multidimensonal Poisson process problem (g>0). In any short time interval of length * (much less than g2, say *=g3), there are n+1 events that could occur: no arrival, or one arrival of some type i0{1,..,n}, each with probability *F2'g2+*µi(a)'g. Two or more arrivals in a short time interval has vanishingly small probability. The optimal incentive plan in this short time interval, given any participation constraint, would pay some amount for each of these possible events. Changing the participation constraint would only add a constant to all payments, because of constant risk tolerance. At any point in time, past payments would not affect the agent's preferences over gambles for additional income and effort-cost, again because of constant risk tolerance. Thus, the optimal incentive plan for the dynamic model can be decomposed to identical incentive problems in each short time interval, plus a constant to meet the ex-ante participation constraint. But xi(S) measures the number of times that a type-i arrival occurred, for each of which the agent is paid the same amount. So the optimal final payment to the agent is a linear function of (x1(S*a,g),...,xn(S*a,g)), and so it is also a linear function of (y1(S*a,g),...,yn(S*a,g)). Now take the limit as g!0, and the limit of the optimal plans pays the agent as a linear function of the Brownian-motion endpoint (Y1(S),...,Yn(S)). 13 Controlling a Poisson processes with constant risk tolerance, no liability limits ˜ Suppose that an agent with constant risk tolerance T is to be paid $X, ˜ is a Poisson random variable with some mean 8. where X Then the agent's expected utility is !8 k !$'T k 8 ) 4 e !k$'T !8 4 (8e !8T (1!e !B'T)'T (!e 3 ) !e 3 EU = k=0 = = !e k=0 k! k! ˜ is 8T(1!e!$'T). and so the agent's certainty equivalent of this income $X Now suppose that the agent chooses an action a 0 A (where A is his set of feasible actions), and this ˜ ,...,X ˜ . action affects the means 71(a),...,7n(a) of n independent Poisson random variables X 1 n The agent also pays a personal cost C(a) for his action, and has constant risk tolerance T. The a action and cost C(a) cannot be observed by anybody except the agent, but the principal can ˜ ˜ observe the Poisson random variables X 1,...,Xn. ˜ ˜ A risk neutral principal gets revenue X 1B1+...XnBn from these Poisson random variables. Consider incentive plans, where the principal pays the agent a wage w according to a linear formula ˜ ˜ ˜ ˜ w(X 1,...,Xn) = " + X1$1 + ... + Xn$n. With constant risk tolerance, the agent's certainty equivalent for the sum of n independent random payments is equal to the sum of the certainty equivalents of these n random payments (this additivity condition only holds with constant risk tolerance). G denote the best alternative wage that the agent could earn elsewhere. Let w So the principal's problem can be written: choose (",$1,...,$n, a*) to maximize 71(a*)B1+...+ 7n(a*)Bn ! [" + 71(a*)$1 + ... +7n(a*)$n] !$1'T !$ 'T ) %...% 7n(a) T (1!e n ) ! C(a) subject to maxa0A " % 71(a) T (1!e * = " % 71(a ) T (1!e !$1'T ) %...% 7n(a *) T (1!e !$n'T G . ) ! C(a *) $ w !$ 'T * * i ) from the participation constraint, Substituting " ' w̄ + C(a ) !ji 7i(a ) T (1!e !$i'T * )] ! w̄ ! C(a *) . the principal's expected profit becomes ji 7i(a ) [Bi ! $i + T (1!e If we can differentiate with respect to the action a, first-order conditions of the incentive constraint !$i'T * ) ' CN(a *) . are ji 7iN(a ) T (1!e Extending the Poisson process over time, the number of i-arrivals in a time period of length * may be ˜ denoted by X i(*), and it is a Poisson random variable with mean *7i(a) (=variance). Results of separate time intervals are independent. With constant risk tolerance, they can be analyzed ˜ ˜ independently, so the optimal wage is linear in X 1(*),...,Xn(*) over the whole period. Now fix µi(a) and F2, and consider small positive numbers g. ˜ (*) = gX ˜ (*) ! *F2'g. Given any g>0, let 7i(a) = [F2 + gµi(a)]'g2, and let Y i i ˜ (*)) = *µ (a), and Var(Y ˜ (*)) = g2Var(X ˜ (*)) = *[F2 + gµ (a)] 6 *F2 as g60. Then E(Y i i i i i 7i(a) here becomes large as g60, and Poisson random variables with large means are approximately ˜ (*) approaches a Normal random variable, Normal. So as g60, Y i ˜ becomes a Brownian-motion process with drift µ (a) and volatility F. and Y i i ˜ n ˜ ˜ Wages that are linear in Y (*), say A*+3 B Y (*), are also linear in X i i=1 i i i(*), with coefficients: n ˜ n ˜ 2 2 n ˜ A*+3 i=1 BiYi(*) = A*+3 i=1 Bi [g Xi(*)!*F 'g] = (A*!n*F 'g) + 3 i=1 (Big) X i(*) 14 First adverse-selection problem: one privately-informed agent with two possible types An agent (seller) is one of two types, 2H or 2L where 2H>2L. His type is his private information. A principal (buyer) thinks the probability of 2H is pH, and the probability of 2L is pL = 1!pH. The agent's effort q and wage w are both observable numbers which may depend on the agent's reported type, but the agent can misrepresent his type. Effort must be nonnegative q $ 0. When the agent's type is 2, any wage w and effort q$0 yield payoff w!2q for the agent and yield profit B(q*2)!w for the principal. The agent's payoff is his gain from trade over alternatives worth 0. With prime N for M'Mq, suppose B(0*2)=0, BN(q*2)>0, BNN(q*2)#0, œ2, œq, and BN(0*2L)>2L. The principal's problem is to choose a contract-menu (qH,wH,qL,wL) to maximize pH(B(qH*2H)!wH) + pL(B(qL*2L)!wL) subject to wL 0 ú, wH 0 ú, qL $ 0, qH $ 0, wL ! 2L qL $ 0, [L-participation, 8L] wH ! 2H qH $ 0, [H-participation, 8H] wL ! 2L qL $ wH ! 2L qH, [*L-informational incentive, "*L] wH ! 2H qH $ wL ! 2H qL. [*H-informational incentive, "*H] (If there were no incentive constraints, the solution would have BN(qi*2i)=2i, wi = 2iqi, œi.) The incentive constraints (*H first, then *L) imply: 2H(qL!qH) $ wL!wH $ 2L(qL!qH). With 2H>2L, this implies qL!qH $ 0 and wL!wH $ 0, so qL $ qH and wL $ wH. If both incentive constraints were binding, these would all be equalities. So both incentive constraints can bind only for a pooling plan that satisfies qL=qH and wL=wH. Note: If qL > qH then: wH ! t qH $ wL ! t qL <=> t $ (wL!wH)'(qL!qH). So any cost type t >2H would prefer (qH,wH) over (qL,wL), while any type t < 2L would prefer (qL,wL). The Lagrangean can be written: L(w,q;8,") = = pH(B(qH*2H)!wH) + pL(B(qL*2L)!wL) + 8L[wL!2LqL] + 8H[wH!2HqH] +"*L[wL!2LqL!wH+2LqH] + "*H[wH!2HqH!wL+2HqL] = pHB(qH*2H) ! qH [(8H+"*H)2H ! "*L2L] + pLB(qL*2L) ! qL [(8L+"*L)2L ! "*H2H] + wH [!pH + 8H + "*H ! "*L] + wL [!(1!pH) + 8L + "*L ! "*H]. Lagrange multipliers must be nonnegative 8H$0, 8L$0, "*H$0, "*L$0, and satisfy complementary slackness. The first-order Lagrange optimality conditions for wL 0 ú and wH 0 ú are 0 = ML'MwL = !pL + 8L + "*L ! "*H, and so 8L + "*L = pL + "*H; 0 = ML'MwH = !pH + 8H + "*H ! "*L, and so 8H + "*H = pH + "*L. So we must have 8L+8H = pL+pH = 1. With these "balance" equations, the Lagrangean simplifies to: L = pH{B(qH*2H) ! qH [2H + (2H!2L)"*L'pH]} + pL{B(qL*2L) ! qL[2L + (2L!2H)"*H'pL]}. The 8's have dropped out, but we still need 8L = pL + "*H ! "*L $ 0 and 8H = pH + "*L !"*H $ 0, as well as "*H $ 0 and "*L $ 0 and complementary slackness with the corresponding constraints. Note: So far, the analysis would be exactly the same if we had no participation constraints but took 8L and 8H as parametric weights for the payoffs of the two types of agent in a social-welfare function [pH(B(qH*2H)!wH) + pL(B(qL*2L)!wL)] + 8L[wL!2LqL] + 8H[wH!2HqH]. 15 With 2L < 2H and qH $ 0, the L-participation constraint is implied by the H-participation and *L-incentive constraints. So L-participation is a redundant constraint, and its multiplier is 8L = 0. So "*L = pL + "*H > 0. If the optimal solution is not a pooling plan, then at most one incentive constraint can bind, and so from "*L>0 we get "*H=0. Nonpooling also implies qL > qH $ 0. Thus, either (case 2) the optimal solution is a pooling plan with "*H > 0, or (case 1) the Lagrange multipliers for the optimal solution must be 8L = 0, "*H=0, "*L = pL, 8H = pH + "*L = 1. In the nonpooling case 1, we have 8L = 0, "*H = 0, "*L = pL, 8H = 1. The optimal efforts must maximize the Lagrangean subject only to qL $ 0 and qH $ 0: BN(qL*2L) = 2L, BN(qH*2H) # 2H+(2H!2L)pL'pH and qH $ 0 with at least one equality. Wages are determined by the constraints that must bind: wH = 2H qH (for 8H>0), and wL = 2LqL + wH ! 2LqH (for "*L>0). So when qH>0, the binding *L-incentive constraint induces ex-post inefficiency for type H: BN(qH*2H) = 2H+(2H!2L)pL'pH > 2H (undersupply of qH), and it induces a positive payoff or information rent for type L: wL!2LqL = (2H!2L)qH > 0. This solution is feasible if it satisfies the *H-incentive constraint, which will be satisfied if and only if our computed quantities satisfy qH # qL. Otherwise, we have the pooling case 2 where both incentive constraints bind, and qH = qL = q* > 0. The pooling q* satisfies BN(q**2H) = 2H + (2H!2L)"*L'pH and BN(q**2L) = 2L ! (2H!2L)"*H'pL. So q* can be computed from pHBN(q**2H) + pLBN(q**2L) = 2H, because "*L!"*H = pL = 1!pH. For the H-participation constraint to bind, total wages must be wH = wL = 2Hq*. The Lagrange multipliers for this pooling solution are "*H = pL[2L ! BN(q**2L)]'(2H!2L), "*L = "*H+pL = pL[2H ! BN(q**2L)]'(2H!2L), 8L=0, 8H=1. "*H > 0 requires BN(q**2L) < 2L for this pooling solution (oversupply by type L at q*). Example Suppose the agent has 2 possible cost-types: 2L=1 or 2H=2, each with probability 1/2. The principal's value of effort q from an agent of type 2 is B(q*2) = (1+2)q0.5, so the principal's gain from trade when paying w for effort q is 2 q & w if 2=2L, 3 q & w if 2=2H. (Without incentive constraints, the solution would be qL=1, wL=1, qH=9/16=0.5625, wH=1.125.) We can apply case 1 from our previous analysis of adverse-selection problem with two types. The Lagrange multiplier of the *L-incentive constraint is "*L = pL = 0.5, and the Lagrangean is: L = 0.5{3qH0.5 ! qH [2+(2!1)0.5'0.5]} + 0.5{2qL0.5 ! qL[1]}. To maximize this over (qH,qL), we need 0 = 3(0.5)qH!0.5 ! 3 and 0 = 2(0.5)qL!0.5 ! 1, and so qH = 0.25 and qL = 1. Because qH < qL, we know that the case 1 assumptions about binding constraints are OK here. The values of wH and wL are determined by the binding constraint equations: To make H-participation binding, wH = 2HqH = 2×0.25 = 0.5 . To make *L-incentive binding, wL = 2LqL + wH!2LqH = 1×1 + 0.5 ! 1×0.25 = 1.25. Then type L gets information rent UL = wL!2LqL = 0.25, but UH = wH!2HqH = 0. If we had changed B(qH*2H) to be AH qH with some AH > 6, then the above analysis would have yielded qH > qL, and so the pooling case 2 would have applied. 16 A trading example where sellers with a single asset have two possible types To be specific, let us consider an example where the seller's cost type is either 2L=20 or 2H=40, and the buyer's value of the seller's asset is BL=30 if the seller's type is 2L, but the buyer's value is BH=50 if the seller is 2H. So the asset is always worth 10 more to the buyer than to the seller. (This is MWG's example 23.F.2.) Our analysis uses the inequalities BL > 2L and BH > 2H > 2L. For now, let's keep the probability of the high type pH as a parameter, with pL = 1!pH. In trading plan (w,q), for each seller-type t, qt is t's probability of selling, wt is t's expected revenue. In such a plan (w,q), a high-type seller's expected gain is U1(w,q*H) = wH!2HqH, a low-type seller's expected gain is U1(w,q*L) = wL!2LqL, and the buyer's expected gain is U2(w,q) = pH(BHqH!wH) + pL(BLqL!wL). A plan (w,q) is incentive-compatible iff wH!2HqH $ wL!2HqL and wL!2LqL $ wH!2LqH. To be feasible here, (w,q) must be incentive compatible and have 0 # qH # 1, 0 # qL # 1. A trading plan is (strongly) interim (Pareto-)dominated if there is some other feasible plan that would yield higher expected gains to each possible type of each individual (given only his type information). Here, the seller has two possible types and the buyer has only one possible type, ˆ ˆ iff and so a plan (w,q) would be interim dominated by some other plan (w,q) ˆ ˆ ˆ ˆ ˆ ˆ U1(w,q*H) < U1(w,q*H), U1(w,q*L) < U1(w,q*L), and U2(w,q) < U2(w,q). A plan is (weakly) incentive efficient if it is feasible and is not interim dominated by any other feasible plan. (See B. Holmstrom and R.Myerson, Econometrica, 1983.) G G is incentive efficient iff there exist nonnegative Fact An incentive-compatible trading plan (w,q) G G solves the problem weights (8H, 8L, 82) such that (w,q) maximize 8HU1(w,q*H) + 8LU1(w,q*L) + 82U2(w,q) subject to (w,q) being feasible (incentive compatible with qt0[0,1] œt). Without loss of generality, we may let 82 = 1 = 8L+8H. The Lagrangean for such an optimization problem in this case can be written: L(w,q;8,") = 8H(wH!2HqH) + 8L(wL!2LqL) + (pH(BHqH!wH) + pL(BLqL!wL)) + "*H(wH ! 2HqH ! wL + 2HqL) + "*L(wL ! 2LqL ! wH + 2LqH). For an optimum with finite payments wH and wL, we must have: 0 = ML'MwH = 8H + "*H ! pH ! "*L and 0 = ML'MwL = 8L + "*L ! pL ! "*H. These "balance" equations imply 8L+8H=pL+pH=1, and they simplify the Lagrangean to L = pHqH{BH ! [2H + (2H!2L)"*L'pH]} + pLqL{BL ! [2L + (2L!2H)"*H'pL]}. This Lagrangean must be maximized over 0 # qH # 1, 0 # qL # 1, and we can only have positive multipliers " for incentive constraints that bind at the solution These "|H and "|L must also yield pH + "|L ! "|H = 8H $ 0 and pL + "|H ! "|L = 8L $ 0. For this example, all the incentive-efficient trading plans can be supported by one (8,") vector. (The incentive-efficient frontier is flat.) Let us now see how to construct this "universal" (8,"). 17 Any feasible trading plan must have qH # qL, and qH < 1 can be a useful signal of the high type. But 0 2H > 2L, this equality can be achieved by letting [*] "*L = pH(BH!2H)'(2H!2L) = pH(50!40)'(40!20) = 0.5pH > 0. For any separating plan, only one incentive constraint can bind, so we should try "*H = 0; then the balance equations yield 8H = pH + "*L = 1.5pH and 8L = pL!"*L = 1!pH!"*L = 1!1.5pH. These formulas yield valid nonnegative Lagrangean parameters with 8L$0 if and only if pH # (2H!2L)'(BH!2L) = (40!20)'(50!20) = 2/3 here, So in the low-pH case when pH # (2H!2L)'(BH!2L)=2/3, the Lagrangean with this (8,") reduces to L = pHqH{BH!BH} + pLqL{BL!2L} = pLqL(30!20), which is maximized by qL=1. So when pH#2/3, these (8,") values verify the incentive-efficiency of any feasible plan (w,q) such that qL = 1 and the *L-incentive constraint is binding wL!2LqL = wH!2LqH. These are all the incentive-efficient trading plans when pH#2/3, and their expected payoffs satisfy (pH+"*L)U1(w,q*H) + (pL!"*L)U1(w,q*L) + U2(w,q) = pL(BL!2L) = (1!pH)10. But when pH > (2H!2L)'(BH!2L) = 2/3 here, this attempt to find incentive-efficient separating plans fails, because it yields would yield 8L = pL!"*L # 0. In this high-pH case when pH>2/3, the "*L formula [*] can satisfy the Lagrangean conditions for incentive-efficiency only if we also have "*H>0, to get 8L$0, and so we can only find incentiveefficient plan that are pooling, with both incentive constraints binding and with qH=qL. So in this case the Lagrangean conditions for incentive-efficiency can be satisfied by "*L = pH(BH!2H)'(2H!2L) = 0.5pH, 8L=0, "|H = "*L!pL = 1.5pH!1, and 8H=1. Then the Lagrangean simplifies to L = pHqH{BH!BH} + pLqL{BL![2L+(2L!2H)"*H'pL]} = qL{pHBH+pLBL!2H} = qL{20pH!10}, which is maximized by qL=1. So when pH>2/3, the incentive efficient plans are the pooling plans with qL=1, and their expected payoffs satisfy U1(w,q*H) + U2(w,q) = pHBH + pLBL ! 2H = 20pH!10. In such bilateral trading problems with a single asset and one-sided private information, the uninformed buyer's optimal trading plan is always to offer a fixed take-it-or-leave it bid that is equal to (or slightly more than) one of the possible cost-types of the seller. That is, the buyer's optimal incentive-compatible plan could be either to bid 2L=20, so that qL=1, wL=20, qH=0, wH=0 (separating); or to bid 2H=40, so that qL=1=qH, wL=40=wH (pooling). The pooling offer 2H is better than the separating offer 2L for the buyer when pHBH + pLBL ! 2H $ pL(BL!2L), but this inequality is equivalent to pH $ (2H!2L)'(BH!2L) = 2/3, which defines the high-pH case. To verify optimality for the buyer of the plan to 2H bid in the high-pH case, notice that trading at the pooling bid 2H satisfies the conditions for incentive efficiency for this case (pooling and qL=1), and it 18 achieves the maximum of U1(w,q*H) + U2(w,q) with U1(w,q)*H) = 0. So it maximizes U2(w,q) over all feasible plans that satisfy the participation constraints (even while allowing U1(L)>0). To verify optimalize for the buyer of the plan to bid 2L in the low-pH case, notice that trading at the separating bid 2L satisfies the conditions for optimality in this case (*L-incentive binding and qL=1), and it maximizes (pH+"*L)U1(w,q*H) + (pL!"*L)U1(w,q*L) + U2(w,q) with U1(w,q*H) = 0 = U1(w,q*L). So it maximizes U2(w,q) over all feasible plans that satisfy the participation constraints. But what trading plan might be used when the informed sellers have market power? This would happen when there are many buyers and each buyer could trade with as many sellers as are available. So let us now assume such a market, with many buyers, each of whom could buy from as many sellers as are available. We might start by considering Walras's concept of market equilibrium, defined for a model in which trade can only occur at a uniform price for all trades that is posted by a neutral Walrasian auctioneer. With uninformed buyers who can trade with as many sellers as available, there would be excess demand if the uninformed buyers could get positive expected profits at the announced prices. So in equilibrium, the buyers must just get zero expected profit from trading with the types of sellers who are willing to trade at the given price. In this example (with BL<2H), for any pH there can be such a Walrasian equilibrium with a positive supply at the price w'q = BL = 30, where supply comes only from L-types. This Walrasian equilibrium has no trade for H types and so is not incentive efficient. When pH$ (2H!BL)'(BH!BL) = (40!30)'(50!30) = 0.5, there is another Walrasian posted-price equilibrum at the price w'q = pHBH+(1!pH)BL = pH50+pL30, where both types sell. When it exists, the higher price equilibrium is better for sellers and yields the same expected payoff for buyers. So we may find a Pareto ranking among Walrasian equilibria with adverse selection. We might also ask: what trading plan maximizes the seller's ex-ante expected payoff? That is, let us choose (w,q) to maximize pHU1(w,q*H)+pLU1(w,q*L) subject to feasibility plus interim participation constraints U1(w,q*H)$0, U1(w,q*L)$0, and U2(w,q)$0? This ex-ante seller's optimum depends on pH. If pH $ 0.5 then it is qL = 1 = qH, wL = wH = pHBH+pLBL = pH50+pL30 $ 40; but if pH # 0.5 then it is qL= 1, wH = 40qH, wL = 30+10qHpH'pL, qH = 1'(2!pH'pL), to satisfy the equations wH = 2HqH, wL!2L = qH(2H!2L), and pL(BL!wL)+pHqH(BH!2H) = 0. For example, with pH = 0.2, we get qL=1, qH = 4/7, wH=40qH, qL = 31 3/7 = 31.43. (L-multipliers with 8H=0.2, 8L=0.8: U2$0 has (=8/7, U1(H)$0 has $H=1/7, "|L = 8/70, "|H=0.) When pH is small, this ex-ante optimal plan is bad for an H-type seller (as U1(w,q*H)=0), but it would involve losses for the buyer (30qL!wL < 0) if the seller's type were not H. So if the seller, knowing his type, were to select this trading plan, then the buyer might reasonably infer that the seller's type is L and thus reject this plan! 19 In such problems (where sellers have private information, and where buyers have no private information but will participate in the market only if expected gains from trade are nonnegative), we may say that a trading plan is separable iff the plan is incentive-compatible and yields a nonnegative expected profit for the buyer with each type of seller. So a separable trading plan (w,q) is incentive-compatible and satisfies the separate break-even constraints: BHqH ! wH $ 0, BLqL ! wL $ 0. Fact. All types of the informed seller can agree on which separable plan is best. Proof Suppose to the contrary that the best separable plan for type H was (w,q) but the best separable ˆ ˆ G G ˆ ˆ plan for L was (w,q). Then let the plan (w,q) coincide with (w,q) for type H but coincide with (w,q) G ,q G G G ˆ ˆ for type L. That is, let (w H H,wL,qL) = (wH,qH,wL,qL). Then G G G G w H!2HqH = wH!2HqH (by definition of the (w,q) plan when the type is H) ˆ !2 qˆ ˆ ˆ $w H H H (because H prefers (w,q) over (w,q)) ˆ ˆ ˆ ˆ $ w L!2HqL (because (w,q) is incentive compatible) G !2 q G G G =w L H L (by definition of the (w,q) plan when the type is L) G !2 q G ˆ ˆ G G G G Also, w L L L = wL!2LqL $ wL!2LqL $ wH!2LqH = wH!2LqL. So (w,q) is incentive compatible. G G is separable because B q G G G G ˆ ˆ (w,q) H H!wH = BHqH!wH $ 0 and BLqL!wL = BLqL!wL $ 0. G G as the other plan that was assumed But each type of seller gets the same expected payoff from (w,q) G G is best for both types of seller among all separable plans. best for it, and so (w,q) So we can talk about the best separable plan for the sellers. In our example, the best separable plan is qL=1, wL = BL = 30, qH = (BL!2L)'(BH!2L) = 1/3, wH = BHqH = 50/3, with U1(w,q*L) = 30!20 = 10, U1(w,q*H) = (50!40)'3 = 3.333, U2(w,q) = 0. Having qL=1 and wL=30 gives type L the highest expected profits subject to the constraint that the buyer cannot pay more than BL = 30. Then the best separable plan for H must have (wH,qH) that maximizes wH!40qH subject to 50qH!wH$0 and wH!20qH # 30!20. The solution has wH=50qH and (50!20)qH = 30!20 so qH = 1/3. Notice that the best separable plan does not depend on the probability of the high type pH. In the low-pH case when pH < 2/3, the best separable plan here is interim incentive efficient, because the *L-incentive constraint is binding and qL=1 in this best separable plan. When the best separable plan is incentive efficient, it can be arguably considered an equilibrium for the adverse-selection market with competitive uninformed buyers, because of the following fact. Fact. If the best separable plan is incentive efficient then, for any other incentive-compatible plan, the buyers must expect losses from the seller-types that would prefer this plan to the best separable plan. If not, then letting each type of seller choose its preference among this alternative plan and the best separable plan would also be an incentive-compatible plan that would interim Pareto-dominate the best separable plan, contradicting our assumption that the best separable plan is incentive-efficient. 20 In the high-pH case when pH > 2/3, however, the best separable plan for this example can be interim ˆ =w ˆ = 50p + 30(1!p ) = 30+20p > 43.33. dominated by a pooling plan such as qˆ L = qˆ H = 1, w L H H H H For example, when pH=0.8, a pooling price 46 yields U1(L) = 26 and U1(H) = 6 with U2 = 0. In this market with many competitive buyers, any one of whom could use all the sellers' supply, we may anticipate that buyers' competition should drive their expected profits to 0 in equilibrium. If the buyers' expected profits were positive, then one buyer could deviate and increase his price offers (wH,wL) slightly, to make greater profit by attracting all the sellers. In the low-pH case when pH # 2/3, the seller's best separable plan (qL=1, wL=30, qH=1/3, wH=50/3) is incentive-efficient and we have seen that it may then be considered a competitive equilibrium here. In high-pH cases with pH>2/3, however, it is harder to say what a market equilibrium should be. In these cases, the only incentive-efficient plans are pooling plans in which the buyers profit from trading with H-sellers but suffer losses from trading with L-sellers. If the trading plan that is offered by the buyers in the market is not incentive-efficient, then a buyer could gain by a dominating deviation, that is, by deviating to offer an interim Pareto-dominating plan that attracts all types of sellers and gives the buyer greater expected profits, when we average over all types according to their given (pH,pL) distribution. On the other hand, if the trading plan in the market is a pooling plan or any other plan in which buyers lose from trading with L-types, then a deviating buyer could do better by making an offer that is slightly better for H-sellers and slightly worse for L-sellers, thus only attracting the profitable Htypes. That is, if the other buyers are offering an incentive-compatible plan with 50qH!wH > 0 and 30qL!wL<0, then a deviating buyer could do better by offering to buy some quantity qˆ = qH!g for a ˆ = w !* such that g is small and positive and 40g>*>20g. payment w H ˆ ˆ > w !40q , the H-types will prefer the deviator's offer; and with g small, the HThen with w!40q H H ˆ ˆ ˆ ˆ < w !20q # w !20q , the L-types types will still yield positive profits 50q!w>0. But with w!20q L L H H will prefer to trade with other buyers, and the deviator will gain by avoiding the unprofitable L-trades. This tactic of leaving unprofitable types to trade with others in the market is called cream-skimming. Another approach to this problem of defining equilibrium with competitive uninformed buyers is to model it as a game where each of two buyers independently announces an incentive-compatible plan ˜ ˜ ˜ (q˜ H,w H,qL,wL), and then each informed seller chooses the plan that he prefers (given his type). The above argument shows that this game has no pure-strategy equilibrium when pH>2/3. A symmetric randomized equilibrium can be found. Let us consider the case of pH = 0.8. In this equilibrium, each buyer independently chooses his q˜ H from a uniform distribution on [1/3,1], ˜ = 2+44q˜ , q˜ = 1, and w ˜ = 20q˜ + w ˜ ! 20q˜ = 22 + 24q˜ . and then lets w H H L L L H H H The low end q˜ H=1/3 is best separable plan where L-sellers get U1(L)=10, H-sellers get U1(H)=10/3. ˜ = p B +p B = 46, U (L)=26, U (H)=6. The high end q˜ H=1 is the pooling plan with w H H L L 1 1 21 All sellers choose whichever buyer offers the higher q˜ H, and this buyer gets zero expected profits, ˜ ) + 0.8(50q˜ !w ˜ ) = 0.2(8!24q˜ ) + 0.8(6q˜ !2) = 0. because these plans all satisfy 0.2(30q˜ L!w L H H H H The winning buyer gains from H-sellers but loses from L-sellers. When the other buyer's offer is uncertain, no alternative offer could expect to earn positive profits from the types that would prefer it. But ex post, for any plan that buyer 1 might offer, when buyer 2 observes it then buyer 2 could do better either by deviating either to a profitable dominating plan that attracts all types, or to a creamskimming plan that attracts only H-types and leaves buyer 1 with the unprofitable L-types. So this approach seems unsatisfactory, as it relies too heavily on any assumption that buyers must commit themselves simultaneously to offers that they then cannot revise. To formulate a concept of market equilibrium for which general existence can be proven, we must find some way for equilibria to deter some of these dominating or cream-skimming deviations. Two general ways have been suggested. One is to allow buyers some ability to react to each others' deviation from the anticipated equilibrium. The other is to allow that the self-selected sellers who accept a deviator's offer might not be representative of the overall population of sellers. We consider each in turn. Threats of competitive reactions could make any market uncompetitive. Buyers could sustain monopsonistic collusion if they could threaten to increase their offers against any buyer's deviation. But such collusion could not be sustained if buyers were only allowed to withdraw their offers in response to a deviation. In a standard market without adverse selection, a buyer's profit could never be reduced by other buyers withdrawing some competitive offers. But in a market with adverse selection, seemingly profitable cream-skimming deviations could be made unprofitable by withdrawing competitive offers that the deviation itself has made unprofitable. For example, with pH=0.8, the efficient pooling plan with zero expected profit for buyers has q=1 and w = pHBH+pLBL = 46. A cream-skimming challenge that attracts the profitable H-types but not the ˆ ˆ unprofitable L-types must have q=1!g and w=46!* where 40g > * > 20g. But then the average price ˆ ˆ > (46!40g)'(1!g) > 46, and so the deviation would become unprofitable if the per unit must be w'q other buyers withdrew their offers that were supposed to attract the unprofitable L-types. The problem is that, by treating the type-L sellers so well, the efficient pooling plans actually make it easier for a competitive challenger to design terms of trade that appeal only to profitable H types. Thus, the possibility of defensive withdrawal of offers could deter cream-skimming deviations and thus could sustain the efficient 0-profit pooling plan as a market equilibrium. The other way of sustaining an equilibrium is by dropping the assumption that a deviating buyer can confidently expect to attract all sellers of any type that prefer the deviator's offer. We normally think of a competitive equilibrium as being characterized by many small competitors, each of whom serves only a small fraction of the market. (If the buyers were few, each with a large share of the market, it would be hard to justify an assumption that they could not react to each other.) 22 For a small buyer, a deviation could be worthwhile if it profitably increased the buyer's share of the market, even if it could not hope to suddenly recruit all sellers in the market. After such a deviation, a small buyer would recruit a self-selected sample of sellers who are attracted by the new offer, and their types might not be representative of the overall type distribution. Of course, types that prefer the market equilibrium will not do business with the deviator; but other types could also be under- or over-represented in the sample drawn by the deviator. For a deviation that would attract two or more types, if any of them would be unprofitable for the deviator, then the deviation could be deterred by fear of over-sampling the unprofitable types. So a dominating deviation by a pooling plan against an inefficient separable plan (in the high-pH case) could be deterred by fear of L-types being adversely over-represented in the deviator's sample. Thus, the possibility of such adverse selection against a deviator could deter dominating deviations and thus could sustain the best separable plan at an equilibrium even when it is inefficient. Consider a dynamic version of this market into which a flow of informed sellers continuously enter, search for a suitable buyer, and then exit when they find an offer that seems competitively optimal. In any time interval, suppose that 80% of the entering sellers are H-types, 20% are L-types. Suppose that all buyers' offers are very close to some part of the best separable plan, but most offer only (qH,wH) = (1/3!g,50(1/3!g)) for H-types, and L types must search longer to find (qL,wL)=(1,30). So at any point in time, the stock of searching sellers, from whom a deviating buyer must sample, might consist of mostly L-types, and then a short-term deviator could not gain by a pooling plan. Thus, the best separable plan could be sustained as an equilibrium by a version of Gresham's law: that the bad (L) types circulate more than the good (H) types. Fact. If both defensive withdrawals and adverse selection are possible against deviators, then any feasible plan that (weakly) interim dominates the best separable plan is sustainable. For such a sustainable equilibrium, buyers could also offer the best separable plan, which no sellers would be expected to accept (because the given feasible plan interim Pareto-dominates it). But if any buyer deviated to another offer, then the other buyers could withdraw to the best separable plan, and then the deviator could not earn profits from its self-selected sellers if the fraction of Htypes among them were in the low range (below 2/3, where this separable plan cannot be dominated). [Some parametric variants. If the parameters were 2L=30, BL=20, 2H=40, BH=50, then the best separable plan for the sellers would have no trade at all, even with high pH: 0=qL=qH, 0=wL=wH. Here buyers fear to trade with L sellers, with whom they cannot profitable trade. If the parameters were 2L=20, BL=50, 2H=40, BH=30, then the best separable plan for the sellers would be qL=1, wL=40, 0=qH=wH, which lets buyers earn strictly positive profits. Here fear of trading with H-sellers could deter buyers from offering more than 40 for L-sellers who are worth 50.] 23 w, expected payment 50 40 30 20 10 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 q, probability of trade, or fraction traded Diagram of the MWG example 23.F.3, with 2L=20, BL=30, 2H=40, BH=50, pH=0.8 . The lowest dashed line (blue) is where the buyer's expected profit is zero with low type, the highest dashed line (green) is where the buyer's expected profit is zero with high type, and the middle dashed line (grey) is where the buyer's expected profit is zero with both types pooled, given fractions pH and pL=1!pH. The solid lines with lower slope (blue) are lines of constant profit to low-type sellers, and the solid lines with higher slope (green) are lines of constant profit to high-type sellers. The seller's best separable plan is indicated by the two large squares, the low type at the blue square getting qL=1 and wL=30, the high type at the green square getting qH=1/3, wH=50/3. In this case, because pH > 2/3, the best separable plan can be interim pareto-dominated by pooling plans. The large grey triangle at q=1, w=46 denotes the best pooling plan for the sellers. But if sellers expected get this pooling plan with buyers, then a buyers who offered the smaller green triangle (at q = 0.86, w = 41.8 = 48.60q) could hope to make a profit by attracting high sellers but not low sellers. Notice that this small green triangle is wedged in between two short lines that are parts of the indifference curves through the pooling plan (q=1,w=46) for the two types of seller, above the green indifference curve of the high type, below the blue indifference curve of the low type, and it is profitable for the buyer with high-type sellers, as it is below the green (highest) dotted line. But it is above the grey dotted line, so it would become unprofitable if it attracted both types, as would happen if the grey-triangle offer were withdrawn. 24 Signalling. In the above example, suppose now that there is a signal that sellers could make which would cost cL>0 for L-types but would cost cH>0 for H-types. Let us apply sequential equilibrium to a market game in which the informed sellers must make their choices about whether to get the costly signal before the buyers compete for them. Suppose there are excess buyers in the market, and the terms of trade that these competitive buyers offer to any seller will depend only on what the buyers observe about the seller's signal. So for each observable signal-value, the competitive buyers should just break even, paying exactly the value of the seller's asset conditional on his signal. For what values of cH and cL could there be an equilibrium where the seller would choose to signal when his type is H but not when his type is L? When the low type is separated, its competitive bids from buyers will be BL = 30, and so type-L sellers will get expected gain BL!2L = 30!20 = 10 from their nonsignaling separation. But signaling in this separating scenario becomes evidence of type H, and so those sellers who signal can get competitive bids of BH = 50 from the buyers. So for the low types to not signal, we must have BH ! 2L ! cL # BL!2L. For the high types to signal, we must have BH!2H!cH $ max{0, BL!2H}. So this separating scenario is an equilibrium when 50!20!cL # 30!20 and 50!40!cH $ 0, that is, when cL $ 20 and cH # 10. For this equilibrium to be better for the H-sellers than the best separable plan that we found previously, we also need 50!40!cH > (50!40)'3, and so cH < 6.67 (with cL $ 20). Lagrangean analysis explains why the costly signal can change the sellers' best separable plan only when cL'cH > 3. The best separable plan here maximizes a Lagrangean (from the low-pH case with "|L = 0.5pH) in which the H-type's value of his asset is transformed from 2H to the virtual value 2H+(2H!2L)"*L'pH = 1.52H!0.52L. That is, in this Lagrangean, an H-type's virtual utility is 1.5 times the actual H utility minus 0.5 times the utility that an L type would get. So the signal's virtual cost for H is 1.5cH!0.5cL, which becomes a positive virtual benefit (a negative virtual cost) when cL > 3cH > 0. For what values of cH and cL could there be an equilibrium where both types choose to signal? If both types are expected to signal, then at best they could both sell the asset for pHBH+pLBL. The worst that buyers could infer about a seller is that his type is L, in which case they would pay up to BL=30 for his object. So a seller always has two alternatives to signaling: he can sell it for BL=30, or he can keep the asset himself and get zero gains from trade. Thus, for both types of seller to actually want to use this signal in an equilibrium, we must have (pHBH+pLBL) ! 2H ! cH $ max{BL!2H, 0} and (pHBH+pLBL) ! 2L ! cL $ max{BL!2L, 0}. In this example, this becomes pH50+(1!pH)30 ! 40 ! cH $ 0, pH50+(1!pH)30 ! 20 ! cL $ 30!20, and so we must have 20pH ! 10 $ cH and 20pH $ cL. The former condition cannot be satisfied by any positive cH unless pH > 1/2. But when pH = 0.8, for example, this pooling scenario is an equilibrium if cH # 6 and cL # 16. This may be called a "rat-race equilibrium," as the costly signal does nobody any good. 25 A general model of markets with adverse selection Consider a large market where sellers have private information, and each seller must be matched with one buyer, but each buyer can be matched with any number of sellers. (We may think of the sellers as workers, and buyers as employers.) Each seller privately knows his own type t, which is in some finite set of possible types T, but a seller can misrepresent his type if he not given an incentive for honesty. Let p(t) be the fraction of type-t sellers among new arrivals into the market over any interval of time. Let M denote the set of possible terms of trade or transactions between a matched seller and buyer. Each transaction m in M includes a full specification of the quantity sold and the price paid (and any required signals by the seller), but for simplicity we treat M here as a finite set (possibly very large). Let u1(m,t) and u2(m,t) denote the utility values to the seller and buyer respectively of transaction m when the seller's type is t. Suppose that M includes the null transaction m0 for which u1(m0,t) = 0 = u2(m0,t) œt0T. A transaction plan is a function from types to probability distributions over transactions, µ:T6)(M). That is, to describe the anticipated plan of transactions in such a market, we may let µ(m*t) denote the probability that a seller of type t will get a transaction m in the market. So transaction plan µ must satisfy the probability constraints: [1] µ(m*t) $ 0, œm0M, œt0T; and 3m0M µ(m*t) =1, œt0T. The expected utility for a type-t seller in the plan µ may be written U1(µ,t) = 3m0M µ(m*t)u1(m,t). Because customers can misrepresent their types, such a market plan must be incentive compatible, that is, it must satisfy the informational incentive constraints: [2] 3m0M µ(m*t)u1(m,t) $ 3m0M µ(m*s)u1(m,t), œt0T, œs0T. Buyers should not expect to lose from these transactions. In the overall market, the p-pooled break-even constraint says that aggregate net profit of all buyers must be nonnegative: [3] U2(µ,p) = 3t0T 3m0M p(t)µ(m*t)u2(m,t) $ 0. We say that a plan µ is a feasible with the type-distribution p (or p-feasible) iff µ satisfies [1]-[3]. Such a trading plan could be derived from informed sellers' choosing among a menu of buyers' offers. In a competitive market, the buyers could announce offers specifying the transactions that they are willing to make with sellers, and then each seller could choose the offer that is best for his type. For convexity, let us assume buyers can offer lotteries that randomly assign sellers to transactions in M according to any specified probability distribution in )(M) = {T0úM| T(m)$0, 3m0M T(m)=1}. Then the menu of offers available in the market would be some a subset of )(M), say S f )(M). The market trading plan would then be derived from the sellers' choices among the buyers' offers. Sellers' type-dependent selections in a menu S would yield an incentive-compatible plan µ such that µ(C*t) 0 argmaxT0S 3m0M T(m)u1(m,t), œt0T. Here µ(C|t) denotes the vector µ(C|t) = (µ(m|t))m0M . We will consider the possibility that buyers might withdraw some offers, in response to a deviation. If buyers were to withdraw some offers from the menu S, then the smaller menu S' f S would yield a 26 new incentive-compatible plan µ' that cannot be better for any type: U1(µ',t) # U1(µ,t) œt0T. So a fallback from µ could be any incentive-compatible plan µ' such that U1(µ',t) # U1(µ,t) œt0T. Any such fallback could be implemented with menus S' = {µ'(C|t)| t0T} and S = S'c{µ(C|t)| t0T}. If buyers lose money trading with some types, then individual buyers would want to find ways of avoiding those types that yield negative profits for buyers. Such incentives to manipulate the selection process will exist unless the market plan satisfies the following separate break-even constraints, which assert that buyers do not expect to lose with any type: [4] U2(µ,t) = 3m0M µ(m*t)u2(m,t) $ 0, œt0T. We may say that a transaction plan µ is separable iff µ satisfies [1], [2], and [4]. Condition [4] implies that the p-pooled break-even constraint [3] is satisfied for any distribution p. G is best G A separable plan µ for the sellers iff, for every other separable plan µ, U1(µ,t) $ U1(µ,t) œt0T. G Fact 1. There exists a separable plan µ that is best for the sellers of all types. Proof. With M and T being finite sets, the set of separable plans that satisfy [1], [2], and [4] is closed, bounded, and convex. It is also nonempty because the plan µ0 that always puts probability one on m0 (µ0(m0*t)=1 œt) satisfies all the constraints. Now pick any vector of positive weights 8(t)>0 for all types t. (We could let 8(t)=p(t), for example.) G will be an optimal solution to the problem The plan µ [5] choose µ to maximize 3t0T 8(t) U1(µ,t) subject to the constraints [1], [2], and [4]. Now to verify that an optimal solution of [5] is best for all types of seller, suppose to the contrary that there is some other separable plan µˆ that some types prefer. ˆ denote the set of types that prefer this plan, so S(µ) ˆ = {s0T: U (µ*s) ˆ G Let S(µ) > U1(µ*s)}. 1 G that is: Then let µ be the plan that gives each type its preference among µˆ and µ; ˆ ˆ G µ(m*t) = µ(m*t) if t 0 S(µ), and µ(m*t) = µ(m*t) otherwise. It is straightforward to check that this µ is also a separable plan, and so it would be a strictly better ˆ were nonempty. QED solution to problem [5] if S(µ) Let )(T) be the set of probability distributions over T, )(T) = {r0úT*r(t)$0 œt0T, 3t0T r(t)=1}. For any r in )(T), a transaction plan µ is r-feasible iff it satisfies [1]-[3] with the type-distribution r replacing p in the buyers' break-even constraint [3]: U2(µ,r) = 3t0T 3m0M r(t)µ(m*t)u2(m,t) $ 0. G Being separable, the plan µ is r-feasible with any other type-distribution r in )(T). An r-feasible plan µ is r-efficient for the informed sellers iff there exists no r-feasible plan µ* such that U1(µ*,t) $ U1(µ.t) œt0T with a strict inequality U1(µ*,t) > U1(µ,t) for some type t. G is Gr-efficient for Fact 2. There exists a type-distribution Gr in )(T) such that the best separable plan µ the informed sellers. G is a linear programming problem. Proof: The above problem [5] (with 8) that characterizes µ G So the optimal solution µ of [5] is supported by nonnegative Lagrange multipliers (",() such that G µ maximizes the following Lagrangean subject to only the probability constraints [1] 27 L(µ;",() = 3t0T 8(t)U1(µ,t) + 3t0T 3s0T "(s*t) 3m0M [µ(m*t)!µ(m*s)]u1(m,t) + 3t0T ((t) 3m0M µ(m*t)u2(m,t), and we have the following complementary slackness conditions G G "(s*t) $ 0 and "(s*t) 3m0M [µ(m*t) !µ(m*s)]u 1(m,t) = 0, œt0T, œs0T, G ((t) $ 0 and ((t) [3m0M µ(m*t)u2(m,t)] = 0, œt0T. The complementary slackness conditions imply that the maximized value of the Lagrangean is just G G L(µ;",() = 3t0T 8(t) U1(µ,t). Let ' = 3t0T ((t) $ 0. Choose Gr in )(T) so that r(t)' = ((t), œt0T. (If ' is 0 then let Gr be any distribution in )(T); otherwise, with '=/0, let Gr(t) = ((t)'' œt.) If µ is feasible with the type-distribution Gr, then we have 3m0M [µ(m*t)!µ(m*s)]u1(m,t) $ 0 œt œs, and 3t0T ((t) 3m0M µ(m*t)u2(m,t) = ' 3t0T Gr(t)3m0M µ(m*t)u2(m,t) $ 0. G implies 3 G G So Lagrangean-optimality of µ t0T 8(t)U1(µ,t) = L(µ;",() $ L(µ;",() $ 3t0T 8(t)U1(µ,t). G G With all 8(t)>0, if any s has U1(µ,s)>U1(µ,t), then another type t must have U1(µ,t) U1(µ,t). In response to this deviation, the other buyers could withdraw some of their offers so that their reduced menu would yield a fallback plan µ' such that U1(µ',t) # U1(µ,t) œt0T. Seller types t with U1(0,t) # U1(µ',t) would have no incentive to select into the deviation 0, and the deviating buyer could send any such types away to trade with the other buyers who offer µ'. So the types selecting 0 could have a distribution r such that r(t)=0 unless U1(0,t) > U1(µ',t). A plan µ is sustainable with adverse selection iff µ is feasible with the given type-distribution p, and, for every incentive-compatible deviation 0 such that {t0T| U1(0,t) > U1(µ,t)} =/ O ' , there exists a p-feasible fallback plan µ', satisfying U1(µ'|t) # U1(µ|t) œt0T, and there exists a type-distribution r such that {t0T| r(t)>0} f {t0T| U1(0,t) > U1(µ',t)} and 0 is not r-feasible. When 0 is incentive compatible, not being r-feasible means U2(0,r) < 0. G G Fact 3. A p-feasible plan µ is sustainable with adverse selection iff U1(µ,t) $ U1(µ,t) œt0T, where µ G is the best separable plan. Any such µ can be sustained against a deviation 0 by the fallback µ' = µ G and distribution r such that any t in S = {s0T* U1(0,s)>U1(µ,s)} has probability r(t) = Gr(t)'3s0S Gr(s). 28 G Proof. If there is any type t such that U1(µ,t) < U1(µ,t), then µ cannot be sustainable, because the best G separable plan µ is a deviation that would be feasible with any type-distribution r. G G Now consider any µ such that U1(µ,t) $ U1(µ,t) œt0T, so that µ can be a fallback from µ. Consider any incentive-compatible deviation 0 such that {s0T* U1(0,s)>U1(µ,s)} =/ O '. G Let S = {s0T* U1(0,s)>U1(µ,s)}. ˜ ˜ G Define the plan 0˜ such that 0(m*t) = 0(m*t) if t0S, and otherwise 0(m*t) = µ(m*t). It is straightforward to verify that 0˜ must be incentive compatible for the sellers, because it could be G implemented by letting each type of seller choose between the incentive-compatible plans 0 and µ. G ˜ G ˜ We have U1(0,t) > U1(µ,t) $ U1(µ,t) for all t in S, U1(0,t) = U1(µ,t) for all other t. ˜ G So 0 is Pareto-superior to the best separable plan µ for the informed sellers. Thus, by Fact 2, 0˜ cannot be feasible with the type distribution Gr. ˜ That is, we must have 3t0T Gr(t) 3m0M 0(m*t)u 2(m,t) < 0. G are all nonnegative, and the other types are all In this negative sum, the terms for types that choose µ in S where 0˜ is the same as 0. Thus, 3t0S Gr(t) 3m0M 0(m*t)u2(m,t) < 0. So some t in S must have Gr(t) > 0. Then we can let r(t) = Gr(t)'3s0S Gr(s) for all t in S, and let r(t) = 0 for all other t. So U2(0,r)<0 and 0 is not r-feasible, as required for µ to be sustained against 0. QED. G Reaction to deviations may not actually be needed to sustain a plan µ such that U1(µ,t) $ U1(µ,t) œt. We are assuming that sellers are continuously flowing into the market with the type-distribution p. But when sellers apply for a transaction in µ, a small (infinitesimal) fraction of them could be put into G a special wait list from which they will be matched according to the best separable plan µ. G Because U1(µ,t) $ U1(µ,t) for all t, sellers in this wait list would be more eager than others of the same type to trade with any buyer who deviates from the µ system. Different types could have different probabilities of entering the wait list (after revealing types while expecting the incentive-compatible µ); so the type-distribution in the wait list could be Gr from Fact 2. So if all sellers are matched with buyers quickly except for those going on the wait list, then the stock of sellers who are available in the market at any point in time may consist almost entirely of the wait G and type-distribution Gr, from which no deviation 0 can profitable sample (by Fact 2). list with plan µ The key idea here, that the stock of traders who are actually available at any point in time may differ adversely from the flow of traders who enter the market over any interval of time, is really a version of Gresham's law: that bad types may circulate more than good types. See also R. Myerson, "Sustainable matching plans with adverse selection," Games and Economic Behavior 9:35-65 (1995); or Game Theory section 10.9. 29 Finding the distribution Gr for an example. Recall again the example where the seller's cost-types are 2L=20 and 2H=40, the corresponding buyer's values are BL=30 and BH=50. The set of possible transactions is the set of all (q,w) in M = [0,1]×ú. This is an infinite set, but the linearity of utility functions will make the problem [5] that characterizes the best separable plan a linear programming problem as follows: Choose (qH,wH,qL,wL) to maximize 8H(wH!40qH) + 8L(wL!20qL) subject to wH!40qH ! (wL!40qL) $ 0 ["*H] wL!20qL ! (wH!20qH) $ 0 ["*L] 50qH!wH $ 0 [(H] 30qL!wL $ 0 [(L] 1$qH$0, 1$qL$0. Lagrangean conditions for wH and wL yield 0 = 8H + "*H ! "*L ! (H, 0 = 8L + "*L !"*H !(L. To support a separable plan with 1 > qH > 0 and slack in the *H-incentive constraint, we must have 0 = 50(H + 20"*L ! 40(8H+"*H) and "*H = 0. These four equations give us "*L = (1/3)8H, (H = 8H!"*L = (2/3)8H, (L = 8L + "*L = 8L+(1/3)8H. Then qL has Lagrangean cocfficient 30(L + 40"*H ! 20(8L+"*L) = 108L + (10/3)8H > 0, for qL=1. From above, these Lagrange multipliers yield an Gr distribution that sustains the best separable plan: Gr = ( '(( +( ) = (2/3)'(8 '8 + 1), Gr = ( '(( +( ) = 1!r G . H H H L L H L L H L H As the given 8L'8H can range from 0 to infinity, this can yield any GrH between 0 and 2/3. So we confirm that the seller's best separable plan becomes incentive-efficient when the fraction of H-sellers is less than 2/3. 30 Motivating an agent with a linear type drawn from a continuous distribution on an interval. Suppose that the agent's type t is a random variable drawn from an interval [A,B]. The agent's type t is his cost of effort, and his utility for income w and effort q is w ! t q. Consider any contract (w(C),q(C)) where the terms of trade for each type 2 would be (w(t),q(t)). Let U(w,q*t) = w(t) ! tq(t) denote the expected utility of type t under this contract. For any pair of possible types t and s in [A,B], the (s*t)-informational incentive constraint says U(w,q*t) = w(t) ! t q(t) $ w(s) ! t q(s) = U(w,q*s) + (s!t)q(s). Similarly, the (t*s)-incentive constraint implies U(w,q*s) $ U(w,q*t) + (t!s)q(t) So the (t*s) and (s*t) constraints together imply (s!t)q(t) $ U(w,q*t)!U(w,q*s) $ (s!t)q(s). So when s > t we must have q(t) $ q(s), and so q(t) is a decreasing function of the cost-type t. These inequalities over many small steps from t up to B yield the information-rent equation: U(w,q*t) = U(w,q*B) + IBt q(r) dr. (Seller type B has the least motivation to trade.) The expected income of any type t is then q(t) !1 w(t) = U(w,q*t) + tq(t) = U(w,q*B) + IBt q(r) dr + tq(t) = U(w,q*B) + q(B)B + Iq(B) q (() d( . With this w(C), we get U(w,q*t)![w(s)!tq(s)] = U(w,q|t)!U(w,q|s)!(s!t)q(s) = Ist [q(r)!q(s)]dr, which is always nonnegative (verifying incentive compatibility) if q(s) is weakly decreasing in s. Suppose the principal's beliefs about the agent's type are described by the cumulative distribution F(t) = P(t˜ # t), and f(t) = FN(t) is the continuous probability density of this distribution, with f(t)>0 for all t in [A,B]. Here F(B)=1, F(A)=0, and P(a#t#b) = F(b)!F(a)=Iba f(t) dt whenever a # b. Then the expected wage bill is B B IB A w(t) f(t) dt = IA [U(w,q*B) + tq(t) + I t q(r) dr]f(t) dt B r = U(w,q*B) + IB A t q(t) f(t) dt + IA IA f(t) dt q(r) dr B B = U(w,q*B) + IB A t q(t) f(t) dt + IA F(r) q(r) dr = U(w,q*B) + IA q(t) [t + F(t)'f(t)] f(t) dt . So the incentive-compatible expected wage E(w(t)) looks like what the principal would have to pay without incentive constraints if the cost of each type t were increased to a virtual cost t+F(t)'f(t). This virtual-cost formula expresses the fact that, when we ask more effort from any type t, we increase the amount that we must pay all types below t, because of incentive constraints. Example: Akerlof's Lemons. The "agent" is the seller of a unique object, of which the "principal" is the only potential buyer. The seller's type is the value of the object to him, which depends on his unverifiable private information about its quality. Then q(t) can be reinterpreted as the probability of his selling the good if he acts like type t, which must satisfy 0 # q(t) # 1, and w(t) is his expected revenue from selling if he acts like type t. Suppose t is drawn from a Uniform distribution on the interval from 0 to 100, but the value of the object to the buyer also depends on the quality (which the buyer would learn only after the transaction) and would be 1.5t. So the object would always be worth 50% more to the buyer. If (w,q) satisfies the incentive constraints and U(w,q*t)$0, the buyer's expected gain from trade is I0100 [1.5tq(t)!w(t)]f(t)dt = I100 0 [1.5t!t!F(t)'f(t)]q(t)f(t)dt!U(w,q*100) 100 = I0 [1.5t!2t]q(t)dt'100 ! U(w,q*100) # 0. The buyer can only expect to lose if any q(t)>0. 31 A continuous example: Now suppose that 2 is drawn from a Uniform distribution on [1,2], keeping everything else the same as in the previous example, with B(q*2) = (1+2)q0.5. So F(2) = (2!1)'(2!1), f(2) = 1'(2!1) =1, F(2)'f(2) = 2!1, for any 2 in [1,2]. Then the principal's expected gains from trade are I12 [B(q(2)*2) ! w(2)] f(2) d2 = I21 {B(q(2)*2) ! q(2)[2+F(2)'f(2)]}f(2)d2 ! U(w,q*2) = I12 [(2+1)q(2)0.5 ! q(2)(22!1)] d2 ! [w(2)!2q(2)] . To maximize the integrand at every 2, we want 0 = 0.5(2+1)q!0.5 ! (22!1), which yields q(2) = [0.5(2+1)'(22!1)]2. Because this q(2) is monotone decreasing in 2, we know that it is actually feasible. [If this q(2) were increasing over any part of the interval, then it would not be feasible, and we would have an "irregular" case which is more complicated to solve (e.g: Myerson, 1981).] At the extreme types, we get q(1) = 1, q(2) = 0.25. To get U(w,q*2) = 0, the optimal solution must have w(2) = 2q(2) = 2×0.25 = 0.5. Then the information-rent equations give us U(w,q*2) = I22 q(t) dt = 0.25[0.25t + 0.75LN(2t!1) ! 2.25'(4t!2)]*22, and w(2) = 2q(2) + I22 q(t) dt. In particular, we get U(w,q*1) = 0.456 and w(1) = 1.456. (Recall: In the discrete case where types 2L=1 and 2H=2 each had probability 1/2, the optimal solution also had q(1)=1, q(2)=0.25, and w(2)=0.5, but it had w(1)=1.25, so type 1 got a smaller information rent.) Now let's do the analogous result for the case where the agent is a buyer of some object. Here the agent's type t is interpreted as his valuation of the object. Now let q(t) denote the probability of the agent buying the object if her type is t, and let w(t) denote expected amount that the agent will have to pay if her type is t. ˆ (If w(t) denotes the price that the type-t agent will pay if she buys, and if she would pay nothing if she ˆ does not buy the object, then our w(t) is equal to q(t) w(t).) So the expected gains from trade for a type-t buyer are U(w,q*t) = t q(t) ! w(t). An incentive compatible trading plan must satisfy, for all types s and t in the interval [A,B], U(w,q*t) = t q(t) ! w(t) $ t q(s) ! w(s) = U(w,q*s) + (t!s)q(s). Assuming differentiability, 0 = M/Ms [tq(s)!w(s)]*s=t , so wN(t) = tqN(t), and wNN(t) = tqNN(t)+qN(t). Then 0 $ M2/Ms2 [tq(s)!w(s)]*s=t = tqNN(t)!wNN(t) = !qN(t), and so a buyer's q(C) must be increasing. Notice M/Ms [tq(s)!w(s)] = tqN(s)!wN(s) = (t!s)qN(s). With qN$0, this is #0 if s>t, $0 if s2) = (2+B)'2, 2 + F(2)'f(2) = 2 + (2!A) = 22!A, 2 ! (1!F(2))'f(2) = 2 ! (B!2) = 22!B. Revenue equivalence in auctions Let {1,2,...,n} be n bidders to buy an object, and assume that each bidder i has an independent private value for this object ti that is drawn from a Uniform distribution on an interval Ai to Bi (Ai < Bi). Let unsubscripted t = (t1,...,tn) denote the profile of n bidders' types. Let Qi(t) = Qi(t1,...,tn) denote the conditional probability of i winning the object given that the type-profile t is equal to t, and let Wi(t) denote the conditional expected payment from i given the same type-profile t. Let qi(ti) = E(Qi(t!i,ti)) = E(Qi(t)|ti=ti) and wi(ti) = E(Wi(t!i,ti)) = E(Wi(t)|ti=ti) denote the conditional probability of i winning and i's conditional expected payment given i's own type ti, but not knowing the others' independent types. With the Uniform [Ai,Bi] distribution, ti ! (1!Fi(ti))'fi(ti) = 2ti!Bi. The seller's total expected revenue from the auction is: 3i E(Wi(t)) = 3i E(wi(ti)) = 3i E{qi(ti)[ti!(1!Fi(ti))'fi(ti)]} ! 3i Ui(W,Q*Ai) = 3i E[qi(ti) (2ti!Bi)] ! 3i Ui(W,Q*Ai) = E [3i Qi(t) (2ti!Bi)] ! 3i Ui(W,Q*Ai). Consider a symmetric case where all Ai=A and all Bi=B. Any auction that always delivers the object to the bidder who actually values it most would yield qi(ti) = ((ti!A)'(B!A))n!1 for all ti in [A,B]. If it allows no profit to the minimal-value type-A bidders Ui(W,Q|A)=0, then these facts determine the seller's expected revenue, regardless of the other details of the auction. But the formula for expected revenue that we have derived above would be maximized, subject to the constraints that 3i Qi(t) = 1 œt and all Qi(t)$0, by the seller keeping the object when maxi 2ti!B < 0, but otherwise giving it to the bidder with highest ti. Bilateral bargaining between a seller and a buyer for a single object The seller is i=1, the buyer is i=2. Each individual i has an independent private value ti for the object that is drawn from some probability distribution with cumulative probability distribution Fi(ti) and positive density fi(ti) on the interval [Ai,Bi]. Let Q(t) = Q(t1,t2) denote the conditional probability of trade given their types t1 and t2 Let qi(ti) denote the conditional probability of trade given that i's type is ti. Here q1(t1) = E(Q(t1,t2)) must be decreasing in t1, and q2(t2) = E(Q(t1,t2)) must be increasing in t2. Let w1(t1) be the conditional expected payment to the seller given that his type is t1. Let w2(t2) be the conditional expected payment from the buyer given that her type is t2. Incentive compatibility implies that E(w1(t1)) = E(q1(t1)[t1+F1(t1)'f1(t1)]) + U1(B1), and E(w2(t2)) = E(q2(t1)[t2!(1!F2(t2))'f2(t2)]) ! U2(A2). The difference ) = E[w1(t1)) ! E(w2(t2))] is the expected net subsidy (if any). So interim participation constraints yield the following inequality, from Myerson-Satterthwaite: 0 # U1(B1) + U2(A2) = ) + E{Q(t1,t2)([t2!(1!F2(t2))'f2(t2)] ! [t1+F1(t1)'f1(t1)]} When each ti is Uniform [Ai,Bi], for no subsidy ()=0) we need 0 # E{Q(t1,t2)[2(t2!t1)!(B2!A1)]} 33 A Uniform bilateral trading problem like Akerlof's Let's consider a bilateral trading problem where agent 1 is the seller of some unique object which he owns, and agent 2 is the only possible buyer of this object. Depending on the object's quality, it may be worth as little as $40 to agent 1 and $60 to agent 2 (if its quality is low) or as much as $100 to agent 1 and $120 to agent 2 (if its quality is high). Player 1 knows the quality of the object. Let 1's cost type ˜t1 is his value of keeping the object. With any quality, the object would be worth $20 more to player 2 than to player 1. That is, given 1's type ˜t1, the value of the object to player 2 would be g(t1) = ˜t1+20. Player 2's belief about ˜t1 is described by a Uniform distribution on the interval $40 to $100. Game where buyer bids Suppose first that agent 2 can offer to buy for any positive price r, and then agent 1 will accept or reject the offer. If the offer is rejected then they each get profit 0. If the offer is accepted then 1's profit is r!t˜1 and 2's profit is g(t˜1)!r. In a subgame-perfect equilibrium, agent 1 will accept if ˜t1 < r, but agent 1 will reject if ˜t1 > r. Agent 2's expected profit from offering any price r is Y(r) = P(t˜12) ! 100 = E(t˜1+20*t˜1>2) ! 100 = (2+100)'2 + 20 ! 100, and so 2 = 60. For agent 1 to offer $70 below when his type is below 2 but $100 when his type is above 2, we need that 70!t1 $ q(100!t1) when t1<2, and 70!t1 # q(100!t1) when t1>2. These inequalities imply 70!2 = q(100!2), and so q = (70!60)'(100!60) = 1/4. In this equilibrium, Pr(trade) = Pr(t˜1<2) + Pr(t˜1>2)q = (20/60) + (40/60)(1/4) = 1/2. This is also a market equilibrium, with no excess demand at either price: E(g(t˜1)*t˜1#2) = 70. There is a separating equilibrium in which each possible type t1 of agent 1 would offer to sell for r(t1) = t1+20, and the probability of agent 2 accepting would depend on the offer r according to the formula Q(r) = e!(r!60)'20, for any r $ 60. The derivation is as follows: For r = t1+20 to maximize Q(r)(r!t1), we need 0 = QN(r)(r!t1) + Q(r) when r=t1+20, and so !1'20 = QN(r)'Q(r) = d/dr LN(Q(r)). In our usual notation for incentive-compatible trading plans, this separating equilibrium has, for any seller's type t, the type-conditional probability of trade q(t) = Q(t+20) = e!(t!40)'20 and the type-conditional expected payment w(t) = (t+20)q(t) = (t+20)e!(t!40)'20. This function q() could also be derived from the incentive constraints, for all t in [40,100]: w(t)!tq(t) = maxs w(s)!tq(s) = maxs q(s)(s+20!t), which yields 0 = qN(t)(t+20!t) ! q(t)1. The low-end boundary condition q(40)=1 identifies the best separating equilibrium. If we compare different types expected gains from trade in these three equilibria, we find that low types (t1#73) prefer the pooling equilibrium where all trade occurs at the price 80, middle types (74#t1#94) prefer the two-price equilibrium where trade occurs at 70 or 100, and high type (t1$95) prefer the separating equilibrium. But the separating equilibrium is actually interim pareto-dominated by the following semi-separating equilibrium: Low types in [40,60] are pooled together and sell for $70 with probability q=1, but any higher type t>60 separates and sells with a probability q(t)<1 at price t+20. For t=60 to be indifferent between selling for price 70 with probability 1 or selling for price 60+20 with a lower probability, that probability must be q(60)=0.5. Then as before, we get qN(t)'q(t) = !1'20 and so q(t) = 0.5e!(t!60)'20. (Compare: e!(t!40)'20 = 0.368 < 0.5.) 35 Mechanism design with informational incentive constraints: a general framework i,j 0 {1,...,I} = {agents}. x 0 X = {feasible allocations}. si,2i 0 1i = {possible types of agent i}. 2 = (21,...,2I) = (2!i,2i) 0 1 = 11×...×1I. 2!i 0 1!i = ×j=/i 1j. We restrict our attention to the case where agents' types are independent random variables. Let pi be the probability distribution of i's type in 1i. p(2) = Ji0N pi(2i). p!i(2!i) = Jj=/i pj(2j). ui:X×1!ú is i's utility function. A direct-revelation mechanism is any mapping from type-profiles to allocations µ:16X. ˆ (µ,s *2 ) = 3 Ui(µ*2i) = 32!i01!i p!i(2!i) ui(µ(2),2) . U 2!i01!i p!i(2!i) u i(µ(2!i,s i),2) . i i i ˆ (µ,s *2 ). µ is incentive compatible iff œi0{1,...,I}, œ2i01i, œsi01i, Ui(µ*2i) $ U i i i A generalized mechanism is of the form (:S1×...×SI6X. An equilibrium of such ( is of the form (F1:116S1,...,FI:1I6SI). For any equilibrium F of any generalized mechanism (, µ(2) = ((F1(21),...,FI(2I)) defines an equivalent incentive-compatible direct-revelation mechanism. So without loss of generality, we may restrict our attention to incentive compatible mechanisms (revelation principle). A mechanism µ is (strongly) interim (Pareto-)dominated by another mechanism µˆ iff ˆ Ui(µ*2 i) > Ui(µ*2i), œi0{1,...,I}, œ2i01i. µ:16X is (weakly) (interim) incentive efficient iff µ is incentive compatible and µ is not interim ˆ Pareto-dominated by any other incentive-compatible mechanism µ:16X. G is interim incentive-efficient iff we can find nonnegative weights 8 (2 ) for all types 2 of Fact µ i i i all agents i such that: at least some 8i(2i)>0, and µ is an optimal solution to the problem maximizeµ:16X 3i0{1,...,I} 32 01 8i(2i) Ui(µ*2i) ˆ i i subject to Ui(µ*2i) $ Ui(µ,si*2i) œi0{1,...,I}, œ2i01i, œsi01i. (si*2i incentive constraint) The Lagrangean of this problem (with constraint multipliers "i(si*2i)$0, œi, œ2i, œsi) is L(µ;") = 3i0{1,...,I} 32i01i 8i(2i) Ui(µ*2i) + 3i0{1,...,I} 32i01i 3s i01i "i(si*2i) (U i(µ*2i)!Û i(µ,s i*2i)) = 32 p(2) 3i [8i(2i) ui(µ(2),2) + 3si "i(si*2i)(ui(µ(2),2) ! ui(µ(2!i,si),2)]'pi(2i) = 32 p(2){3i [8i(2i)+3si "i(si*2i)]ui(µ(2),2) ! 3si "i(si*2i)ui(µ(2!i,si),2)}'pi(2i) = 32 p(2){3i [8i(2i)+3si "i(si*2i)]ui(µ(2),2) ! 3si "i(2i*si)ui(µ(2),(2!i,si))}'pi(2i) = 32 p(2) 3i vi(µ(2),2,8,"), where vi is the (8,")-virtual utility function for agent i, defined to be vi(x,2,8,") = [(8i(2i)+3si "i(si*2i))ui(x,2) ! 3si "i(2i*si) ui(x,(2!i,si)]'pi(2i). Lagrange multipliers can be positive "i(2i*si)>0 only for binding constraints Ui(µ*si)=Ui(µ,2i*si). With such an incentive-efficient µ, we say that type si jeopardizes 2i iff the incentive constraint ˆ (µ,2 *s ) is binding and has positive multiplier " (2 *s )>0 in the Lagrangean for µ. Ui(µ*si) $ U i i i i i i Virtual utility of any type 2i of agent i differs from the actual utility of type 2i by exaggerating the differences from i's other types si that jeopardize 2i. The incentive-efficient mechanism µ appears ex-post efficient in terms of such virtual utilities. Now suppose agents have separable linear utility for money wi, that is, any social choice x can be decomposed into a pair x = (qi,wi) such that i's utility is ui(x,2) = ui(qi,wi,2) = ui(qi,0,2) + wi, and these money transfers wi are not bounded (no liquidity constraints). 36 Then all types of all agents must have the same marginal virtual-utility of money (or else Lagrangean optimality would call for infinite transfers from agents with lower marginal utility of money to those with higher marginal utility for money). We may take this constant marginal virtual-utility for money to equal 1, without loss of generality (if not, just rescale all the 8weights). Then the conditions Mvi((qi,wi),2,8,")'Mwi = 1 give us the "balance" equations: 8i(2i) + 3si "i(si*2i) ! 3si "i(2i*si) = pi(2i), œi, œ2i. Substituting these equations into the formula for the (8,")-virtual utility function, it simplifies to vi(x,2,8,") = vi(x,2,") = ui(x,2) + 3si [ui(x,2) ! ui(x,(2!i,si))]"i(2i*si)'pi(2i). Fact. With independent types and linear utility for money, an incentive-compatible mechanism µ is incentive-efficient iff there exists some vector " of nonnegative Lagrange multipliers "i(si*2i) $ 0, œsi, œ2i, œi that have complementary slackness with the corresponding constraints ˆ (µ,s *2 )] = 0, œs , œ2 , œi "i(si*2i)[Ui(µ*2i)!U i i i i i and yield nonnegative 8 weights for all possible types of all individuals 8i(2i) = pi(2i) + 3si ("i(2i*si) !"i(si*2i)) $ 0, œ2i, œi such that the mechanism µ maximizes the sum of virtual-utility payoffs for each profile of types 3i vi(µ(2),2,") = maxx 3i vi(x,2,"), œ2. [There are examples where player i is a seller with an unknown cost type, and the possible cost types are {A, A+*, A+2*,..., B}. If the question is how to maximize the buyer's expected payoff, then we find that the participation constraint is binding only for i's highest possible type B, and so only that highest type has positive 8i; that is 8i(B)=1 and all other 8i(ti)=0. In such examples, we often find that the incentive constraints that have positive Lagrange multiplier are "i(ti*ti!*) = 3s