Dirk Bergemann Department of Economics Yale University Microeconomic Theory (50b) Problem Set 0. Auctions and Moral Hazard Suggested Solution: Tibor Heumann 4/5/4 This problem set is due on Tuesday, 4//4.. Budget Balanced Mechanism. We say that the mechanism (x ( ), t ( )) satisfies ex-post budget balance if for all θ, N t i (θ) = 0. i= For an environment with independently distributed types, we define the D Aspremont Gerard-Varet mechanism or expected externality mechanism by x (m), the social choice function that maximizes the sum of the utilityes, and transfers t i (m) = E θ i v j (x (m i, m i ), m j ) + τ i (m i ). j i In this direct mechanism, m i refers to the message and θ i is the type of agent i. The social choice function x ( ) maximizes the sum of the agents welfare. The term τ i (m i ) is function that depends only the messages of the other agents. (a) Establish that m i = θ i is an optimal announcement in a Bayesian Nash equilibrium. [Solution] We need to show that: θ i arg max E θ i [v(x (θ i, θ i ), θ i ) t i (θ i, θ i )] for all i N Note that, [ E θ i [v(x (θ i, θ i ), θ i ) t i (θ i, θ i )] = E θ i v(x (θ i, θ i ), θ i ) + ] v j (x (θ i, θ i ), θ j ) τ i (θ i ) j i [ ] = E θ i v j (x (θ i, θ i ), θ j ) τ i (θ i ) j N
Note that by definition x (θ i, θ i ) maximizes the sum of the utilities, thus: x (θ i, θ i ) arg max v j (x, θ j ) x X j N Thus, we have that for all θ i Θ i : θ i arg max v j (x (θ i, θ i ), θ j ) θ i Θi j N Since θ i maximizes for all θ i Θ i, it must also maximize if we take expectations, thus: [ ] θ i arg max E θ i v j (x (θ i, θ i ), θ j ) θ i Θi j N Since τ i (θ i ) does not depend on θ i we have that: [ ] θ i arg max E θ i v j (x (θ i, θ i ), θ j ) τ i (θ i ) θ i Θi Thus, we have that: j N θ i arg max E θ i [v(x (θ i, θ i ), θ i ) t i (θ i, θ i )] for all i N Thus, truth telling is a Bayes Nash equilibrium. (b) We can define the expected externality of agent i with type m i as: E i (m i ) = E θ i v j (x (m i, θ i ), θ j ), and set j i j i τ i (m i ) = E j (m j ). N Show that this mechanism satisfies the balanced budget requirement ex post for every type profile realization. [Solution] By definition, we have that:
t i (m) = E θ i v j (x (m i, m i ), m j ) + τ i (m i ) i N i N j i i N = E θ i v j (x (m i, m i ), m j ) + j i E j (m j ) N i N j i i N = E θ i v j (x (m i, m i ), m j ) + i j E j (m j ). N i N j i j N = E θ i v j (x (m i, m i ), m j ) + E j (m j ) i N j i j N = E θ i v j (x (m i, m i ), m j ) + [ E θ i v j (x (m i, θ i ), θ j ) i N j i i N j i = 0. Where in the last equality we use that if all agents report truthfully (which is a Bayes Nash equilibrium), then (m i, m i ) = (θ i, θ i ). (c) Is this mechanism also dominant strategy implementable? [Solution] No. Consider the standard auction environment with two agents. If one of the agent bids always zero, then the other agent will win the object with strictly positive bid. Yet, the transfers will now not be constant, independent of what the agent announces. Thus, truthful bidding cannot be optimal strategy for all possible types for an agent when the other agent is always bidding zero. Thus, it is not implementable in dominant strategies. (d) Compute the transfers exactly for the single unit auction with two bidders and uniformly distributed values. Do the transfers guarantee the individiual rationality constraints for all the type profile of the agents? [Solution] We assume that the types are uniformly distributed in [0,]. Thus: ] E i (m i ) = E θj v j (x (m i, θ j ), θ j ) = E θj [ θ j θj>m i ] = ( m i ) + m i = m i, Similarly, we also have: E θ i v j (x (m i, m i ), m j ) = m i j i 3
Thus, we have that: t i (m) = E θ i v j (x (m i, m i ), m j ) +τ i (m i ) = m i j i Thus, in the truthful Bayes Nash equilibrium, we have that: + m j. m i + m j = θ i + θ j Thus, the expected utility of agent i is given by: = θ i θ j E[θ i θi>θ j + θ i θ j ] = + θ i E[ θ j ] = θ i +E[θ j ] = θ i + 6 > 0. Thus, the individual rationality constraint is satisfied. Note that in a second price auction the expected utility of an agent is given by θ i. Yet, on a second price auction the seller runs a budget surplus, as he in general earns money. In this case the mechanism satisfied budget balance, and thus the agents must earn more expected rents. Moreover, from the revenue equivalence theorem, one can already know that the expected utility of a agent i of type i is given by θ i plus some constant. In this case the constant is /6 which corresponds to the additional rents the agents get with respect to a second price auction. 0. Nonlinear Pricing. Consider the problem of an optimal menu when v(θ, q) = θ q and c(q) = q and the distribution is given by the uniform distribution on the unit interval. (a) Compute the revenue maximizing direct mechanism, which associates to every reported type θ a pair (q(θ), t(θ)) of quantities q(θ) and prices t(θ), or θ (q(θ), t(θ)) The problem of the seller is (as we saw in class) max q(θ) θ 0 { q(θ) [ θ Which, in this case, implies max q(θ) θ 0 ] } ( F (θ)) c(q(θ)) df (θ) f(θ) { q(θ) [θ ] q(θ) } df (θ) First, note that q(θ) > 0 if and only if [ θ ] ( F (θ)) f(θ) = θ > 0 θ > /. We can conclude immediately that q(θ) = t(θ) = 0 if θ /. 4
Now, if θ > /, then, maximizing the above expression pointwise, we get: q(θ) / [θ ] = 0 q(θ) = (θ ) 4 Now, remember that the transfer t(θ) must satisfy the (IC) restriction: t (θ) = v(θ,q) q.q (θ). Therefore if θ > /, then, t (θ) = θ q(θ).q (θ) = θ = t(θ) = θ 8 where the last line comes from integrating t (.) form / to θ. Therefore the direct mechanism that maximizes expected profit is ( ( (θ ) θ (q(θ), t(θ)) =, 4 )).[θ > /] 8 where [.] is the indicator function. (b) Translate the direct mechanism into an indirect mechanism, in particular to a nonlinear pricing mechanism (q, t(q)) which associates to every q a t(q), or q, t(q). First of all, let s invert q(θ). If θ > /, then q = (θ ) 4 = θ = q +. Therefore, we must have that t(q) = ( q+ ) 8. Hence, ( ) q + q t(q) = and the direct mechanism is given by ( ( )) q + q. (q, t(q)) = q,, where q [0, 4 ] (c) What can you say about t(q)/q, i.e. the price per quantity as the quantity increases. We have that ( ) q + q t(q) q which is decreasing in q. = q = + q 5
(d) Establish that the revenue maximizing direct mechanism could also be implemented by a menu of two part-tariffs, (T (θ), p (θ)), where T (θ) is the fixed fee and p (θ) is the price per unit (i.e. if a customer chooses a particular two-part tariff then he has to pay T (θ) independent of the quantity he purchase, but can then buy as many units as he likes at the price p (θ) per unit. (The concavity of t (q) might be useful.) We just need to impose that the price is equal to the marginal utility of agents for the good at the optimal mechanism. That is, p(θ) = v (q(θ), θ) = θ = θ q θ. It is clear the when agents solve, max v(q, θ) p(θ)q q they will choose q(θ). The fixed transfer is chosen to match the total transfers in the optimal mechanism. That is, T (θ) + q(θ)p(θ) = t(θ). Thus, ( θ T (θ) = t(θ) q(θ)p(θ) = ) θ (θ ) 8 θ 4 ( θ = 8 ) θ(θ ) 4 It is clear that the mechanism (p(θ), T (θ)) implemented the same outcome as (q(θ), t(θ)).. Consider the regular moral hazard model with a risk-neutral principal and a risk averse agent. The agent can choose between two effort levels, a i {a, a} with associated cost c i {c, c} = {0, c}, with c > 0. Each action generates stochastically one of two possible profit levels, x i {x, x} with p (x a) > p (x a). The utility function of the agent is u (w, c i ) = ln w c i. The value of the outside option is normalized to 0. (Risk-neutrality of the principal implies that his payoff function is x w.) a. Carefully describe the principal-agent problem when the principal wishes to implement the high effort level a. The principal solves the following problem 6
such that max p (ā) (x w (x)) + ( p (ā)) (x w ( x)) w(x),w(x) p (ā) ln (w (x)) + ( p (ā)) ln (w ( x)) c 0 (IR) p (ā) ln (w (x))+( p (ā)) ln (w ( x)) c p (a) ln (w (x))+( p (a)) ln (w ( x)) (IC) b. Solve explicitly for the optimal wage schedule to be offered to the agent which implements the high effort level a. Consider the first-order conditions for the principal s problem in (a) : p (ā) + λp (ā) w (x) + µ [p (ā) p (a)] w (x) ( p (ā)) + λ ( p (ā)) w ( x) µ [p (ā) p (a)] w ( x) = 0 = 0. from which you can obtain w (x) = w ( x) = λp (ā) + µ (p (ā) p (a)) p (ā) λ ( p (ā)) µ [p (ā) p (a)] p (ā) p (ā) p (a) = λ + µ p (ā) p (ā) p (a) = λ µ p (ā) which implies w (x) < w ( x) since µ > 0. Notice however, that since both constraints are binding, you can directly solve for w (x), w ( x) from the constraints (since the objective function is decreasing in wages). Equating the RHS of the IR,IC: p (a) ln (w (x)) + ( p (a)) ln (w ( x)) = 0 one then obtains (from IR) ln (w (x)) = p (a) p (a) ln (w ( x)) p (a) p (ā) ln (w ( x)) + ( p (ā)) ln (w ( x)) p (a) = c ln (w ( x)) = p (a) c p (a) p (ā) ln (w (x)) = c p (a) p (a) p (ā). c. (Renegotiation ) Consider now the following extension to the moral hazard problem. After the principal has offered an (arbitrary) 7
wage schedule and the agent has chosen and performed an (arbitrary) effort level, but before x is revealed, the principal has the possibility to offer a new contract to the agent. The agent can either accept or reject the new offer. If he accepts the new contract, then it replaces the old contract, if he rejects the new contract, then the old one remains in place. Show that there is no subgame perfect equilibrium of the game where Pr (a = a) =. (Hint: Consider the optimal contract after a has been chosen but before x has been realized.) Consider an equilibrium candidate in which the agent chooses ā. In such a scenario, the principal can make a renegotiation offer by which he perfectly insures the agent at the level u the principal had previously offered him. For example, if the IR was binding in the first stage, u = c. A full-insurance contract maximizes the principal s profits and delivers the same expected utility level (under ā) to the agent. Hence, it Pareto-dominates any (previous) contractual arrangement. Anticipating this offer, the agent will shirk (cost=0) in the action choice stage and earn a guaranteed profit of u in the following stage. Since the principal cannot credibly commit not to renegotiate (and offer full insurance) there is no equilibrium in which the agent will choose to work hard. d. (Renegotiation ) Consider now the following modification to the renegotiation problem above. Suppose now that the agent can make the new proposal and the principal can either accept or reject the new offer. Suppose further that the principal can observe the action at the time of the new proposal but that the contract can only depend on x and not on a. The timing is otherwise unchanged. Derive the subgame perfect equilibrium of the principal-agent problem. What can you say about the efficiency of the arrangement. Backward induction. Knowing the action level (hence the actual probability distribution over outcomes), the principal will accept any offer that gives him a higher profit level than under the previous contract. For a given contract w 0 the agent will therefore demand perfect insurance at a wage w (a, w 0 ) = p (a) w 0 (x) + ( p (a)) w 0 ( x). Depending on w 0, the agent will choose whether or not to work hard. In particular, he will choose ā iff ln (p (ā) w 0 (x) + ( p (ā)) w 0 ( x)) c ln (p (a) w 0 (x) + ( p (a)) w 0 ( x)) Assume the principal still wants to induce high effort. The principal s problem in the first stage is now given by: max p (ā) (x w 0 (x)) + ( p (ā)) (x w 0 ( x)) w 0(x),w 0(x) 8
such that ln (p (ā) w 0 (x) + ( p (ā)) w 0 ( x)) c 0 (IR) ln (p (ā) w 0 (x) + ( p (ā)) w 0 ( x)) c ln (p (a) w 0 (x) + ( p (a)) w 0 ( x)) (IC) Again, solving directly from the constraints, w ( x) = ec p(a) p(ā) p(a) p(ā) w (x) = ec ( p(a)) ( p(ā)). p(ā) p(a) The agent thus receives the same ex-ante utility level as in the norenegotiation case. Ex-post, he is perfectly insured, hence efficiency is improved. Note also that the principal is much better off, since he appropriates the entire ex-ante surplus gain deriving from postrenegotiation insurance (to see this, just compare the expected payments under IR in the two cases - use Jensen s inequality). 3. (Moral Hazard in Teams, (Holmstrom 98)) Consider the following moral hazard problem with many agents. Suppose output is onedimensional, deterministic and concave in a i and depends on the effort of the n agents in the team: x = x(a, a,..., a n ). Each agent i has convex effort cost c i (a i ). Each agent observe his effort and the joint output x. A contract among the agents is a set of wages {w i (x)} n i=, which depend only on the publicly observable output x. The set of wages have to be budget balanced, or n w i (x) = x i= for all x. (Think of the team as a cooperative or partnership). The utility function of each agent is w i (x) c i (a i ). a. Describe the first-best allocation policy a. The social planner s problem is max (a i) n i= (w i (x (a,..., a n )) c i (a i )) i s.t. w i (x) = x i s.t. w i (x) c i (a i ) ū i i 9
rewriting this program neglecting for now the participation constraints gives [ ] max a i The st order condition is x (a,..., a n ) i x (a ) a i = c i (a i ) i c i (a i ) which simply prescribes marginal productivity=marginal cost. Note that - absent incentive problems - the wage allocation among the agents does not influence the planner s choice of a. Any vector of (wi )n i= that verifies the participation and resource constraints will do. b. Suppose (without loss of generality) that the team is restricted to using differentiable wages, or w i (x) exists for all x and i. Show that there is no wage schedule which allows the team to realize the first best policy. Suppose wage schedules are differentiable. Since each individual maximizes w i (x (a)) c i (a i ), she will choose a i : w i (x (a )) x (a ) a i c i (a i ) = 0 therefore w i (x (a )) = i. However, since the resource constraint must hold for any x, you can differentiate both sides and obtain i w i (x) =. This means that in order to induce the first best actions, each agents would have to be able to appropriate any production gains associated to anyone s effort. This is obviously not feasible. c. Next, introduce an n + th, who does not deliver any effort to the team, but can be entitled to transfers (the principal or budgetbreaker ). Show that you can now design a wage schedule, not necessarily differentiable, such that n+ w i (x) = x i= for all x. In fact, you can design the contract such that even n w i (x) = x i= holds on the equilibrium path, but not off the equilibrium path. 0
The idea is to punish the agents if they do not exert the optimal effort level. Hence, set the effort levels at a i. Then let the wages be given by { } w w i (x) = i if x = x (a ) 0 if x x (a i =,.., n ) { } 0 if x = x (a w n+ = ) x (a) if x x (a ) With these wages, no agent has incentives to deviate. Furthermore, actions need not be observable since technology is deterministic, and each deviation from a i will induce a non-optimal production level x x (a ). Therefore, a i i is an equilibrium of this game. 4. (First Order Stochastic Dominance versus Monotone Likelihood Ratio, (Milgrom 98)). a. Define monotone likelihood ratio and first-order stochastic dominance. Definitions: MLR : F OSD : p (x j a i ) p (x j a k ) increasing in x j k < i p (x j x a i ) x, k < i p (x j x a k ) b. Show that with two outcomes, the two notion are equivalent. Two outcomes: x < x, two actions a < a. Then MLR p (x a) p (x a ) > p (x a) p (x a ) p (x a) p (x a ) > p (x a) p (x a ) p (x a ) > p (x a) F OSD. c. Give an example to show that the equivalence does not hold in general. Show that the monotone likelihood ratio property implies first order stochastic dominance. Example: consider the following distributions (with a < a) x p 0 p (x a) 0 p (x a ) 0
The distribution under a is stochastically dominated by the one under a (the cdfs coincide after, but the former starts out lower) but the likelihood ratios are clearly non monotone in x. Proof. MLR F OSD : Let again a < a. Let x {x,..., x n }. Then MLR p(x a) p(x a ) < (since this ratio is monotone, if it were otherwise the two distributions could not both integrate to ). However, there p(x a) must exist an x : p(x a ). Let ˆx be the first such x. For all x < ˆx, it must be p (x j x a) < p (x j x a ) by definition of LR <. For all x ˆx it must also be the case that p (x j x a) p (x j x a ), else there would have to exist x k [ˆx, x n ] for which p (x k a) < p (x k a ) (otherwise the two distributions couldn t both integrate to ). This would however violate MLR, since LR at ˆx < x k. Reading MWG: 3, S (=Salanie) and 3