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(501B) Problem Set 5. Bayesian Games Suggested Solutions by Tibor Heumann

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(501B) Problem Set 5. Bayesian Games Suggested Solutions by Tibor Heumann Dirk Bergemann Department of Economics Yale University Microeconomic Theory (501b) Problem Set 5. Bayesian Games Suggested Solutions by Tibor Heumann 1. (Market for Lemons) Here I ask that you work out some of the details in perhaps the most famous of all information economics models. By contrast to discussion in class, we give a complete formulation of the game. A seller is privately informed of the value v of the good that she sells to a buyer. The buyer's prior belief on v is uniformly distributed on [x; y] with 3 0 < x < y: The good is worth 2 v to the buyer. (a) Suppose the buyer proposes a price p and the seller either accepts or rejects p: If she accepts, the seller gets payoff p−v; and the buyer gets 3 2 v − p: If she rejects, the seller gets v; and the buyer gets nothing. Find the optimal offer that the buyer can make as a function of x and y: (b) Show that if the buyer and the seller are symmetrically informed (i.e. either both know v or neither party knows v), then trade takes place with probability 1. (c) Consider a simultaneous acceptance game in the model with private information as in part a, where a price p is announced and then the buyer and the seller simultaneously accept or reject trade at price p: The payoffs are as in part a. Find the p that maximizes the probability of trade. [SOLUTION] (a) We look for the subgame perfect equilibrium, thus we solve by back- ward induction. From the next point it should be clear why there are other Nash equilibrium which are not subgame perfect. If the buyer proposes price p, then the seller accepts if and only if v ≥ p (we assume that, if indifferent the seller accepts). Thus, the payoff of the buyer conditional on offering price p is: 3 fv ≥ pg · [ v − pjv ≥ p]: P E 2 That is, the expected utility of the buyer is the probability of the seller accepting the offer times the expected utility conditional on 1 the seller accepting the offer. Expanding the terms, we get: 3 (p − x) 3 fv ≥ pg · [ v − pjv ≥ p] = ( [ vjv ≤ p] − p) P E 2 y − x E 2 (p − x) 3 x + p = ( − p) y − x 2 2 p − x = (3(x + p) − 4p) 4(y − x) p − x = (3x − p): 4(y − x) One should note that the previous equality holds only for p 2 [x; y]. Obviously for p ≤ x we have that Pfv ≥ pg = 0 and for p ≥ y we have that Pfv ≥ pg = 1. Taking into account the previous observation, we can differentiate with respect to p and get the F.O.C., since the objective function is strictly concave in p we know that the maximum will be achieved at a corner (that is, p 2 fx; yg), if the solution to the F.O.C. yields a p 62 [x; y]. Taking the derivative, we get: @ 3 1 fv ≥ pg · [ v − pjv ≥ p] = (4x − 2p) @pP E 2 4(y − x) @ 3 ) fv ≥ pg · [ v − pjv ≥ p] = 0 () p = 2x @pP E 2 Thus, we have that the buyer proposes the following price depending on the distribution of v. ( 2x 2x ≤ y p∗ = y 2x > y (b) If the buyer knows the value v, then the unique subgame perfect Nash equilibrium is that the buyer offers v and the seller accepts. There are two things to note. First, this is a complete information game, thus the right solution concept is subgame perfect Nash equilibrium. Note that there are other Nash equilibrium which are not sub-game perfect. For example, the buyer offers 0 and the seller rejects all offers. If neither agent know the valuation v, then the game can also be treated as a complete information game. In this case, the buyer offers E[v] = (y + x)=2, and the sellet accepts. Note that without asymmetric information, the game can be treated as complete infor- mation game, in which the value of buyer and seller are 3E[v]=2 and E[v] respectively. (c) From part (a) it is clear that the seller accepts if and only if v ≥ p. On the other hand, the buyer accepts if and only if he gets non-negative 2 profits by accepting. Thus, the buyer accepts if and only if 3 p − x fv ≥ pg · [ v − pjv ≥ p] = (3x − p) ≥ 0: P E 2 4(y − x) To maximize the probability of trade, we just need to find the max- imum p such that the buyer gets non-negative profits. Clearly, we can just equate the previous inequality to 0 to get the price that maximizes trade. That is, p∗ − x (3x − p∗) = 0 4(y − x) The quadratic equation has two solutions, as the buyer also gets 0 profits when there is 0 probability of trade. Thus, we consider the other solution. Thus, the price that maximizes the probability of trade is: ( 3x 3x ≤ y p∗ = ; [y; 3x] 3x ≥ y where we note that whenever 3x ≥ y we have a continuum of prices that ensure trade with probability 1. 2. (Akerlof's Lemon in a General Equilibrium Model) Suppose there is a continuum of buyers with a characteristic vb 2 [0; 1] distributed ac- cording to a distribution function G(vb). There is also a continuum of sellers with mass one, all with the same characteristic vs 2 (0; 1). Each seller owns an object of value θ 2 [0; 1] which is distributed according to a distribution function F (θ) on the unit interval, independent across sell- ers. The value θ is known to the seller and unknown to the buyer. The valuations for the object are us = vsθ and ub = vbθ: (a) Determine the efficient volume of trade and the efficient mix of sellers (in terms of types of θ) and buyer (in terms of types of vb). (b) Suppose for the moment that the characteristic of each object is observable and hence that information is symmetric across sellers and buyers. What would be the equilibrium price for the object with value θ be and would trade be efficient. (c) Under asymmetric information, compute the demand and supply functions. Explain what the demand is not necessarily downward sloping (despite the absence of income effects.) 3 (d) Solve for the competitive equilibrium for f and g uniform on [0; 1]. Verify that trade is suboptimal. (e) Argue that, in general, there can exist multiple equilibria. Then as- sume that it is the case. Show that a higher price equilibrium Pareto dominates a lower price one. Compare with the first fundamental theorem of welfare economics. (f) Show that a minimum quality standard s0 > 0, if enforceable, may improve welfare. [SOLUTION] (a) If you are the benevolent social planner, you will want trade to occur only when ub ≥ us; that is, when vb ≥ vs: Thus, the probability of trade is Pr (Trade) = Pr (vb > vs) = 1−G (vs) : In other words, you are only going to match high-value buyers to some sellers. To which sellers? To the ones that allow you (the planner) to maximize gains from trade θ (vb − vs) : Therefore, you will choose the highest-valuation sellers (the best objects) to sell. The set of matched sellers must have the same measure of the set of buyers. Let θ∗ be the lowest seller that gets matched. Then, ∗ 1 − F (θ ) = 1 − G (vs) ∗ −1 θ = F G (vs) : So the efficient mix of sellers and buyers (in terms of maximizing social ∗ ∗ welfare) involves buyers with vb ≥ vs and sellers with θ ≥ θ , with θ as defined above. (b) Let p(θ) be the price of an object of value θ (since θ is observable we can condition on it). In equilibrium, prices p(θ) must be such that market clears. We expect p(θ) to be strictly increasing in θ. Suppose not, say there are two sellers with v2 > v1 and p(v2) < p(v1), then clearly good 2 will \dominate" good 1 (2 has a higher quality which is valued by consumers and a lower price). Also note that, the higher vb buyers have higher valuation for quality and hence are willing to pay more for quality. Therefore, we expect high θ goods, which have higher prices, to be matched with high vb buyers. By the market clearing condition we require that for all items sold in equilibrium G(vb) = F (θ) which in turn implies the matching between sellers and buyers −1 vb(θ) = G (F (θ)) (1) −1 That is, a good with quality θ is matched with a buyer with vb = G (F (θ)). From the participation conditions (i.e., matched buyer and seller both will- ing to trade), we can immediately see that, if θ is sold in equilibrium, p(θ) 4 should satisfy −1 vsθ ≤ p(θ) ≤ G (F (θ))θ (2) But not all functions satisfying (2) will be equilibrium prices1 because, to support the matching, buyers have to be willing to buy the θ corresponding to them and not other. Type-vb buyers, if in equilibrium buying, are going to choose a good so that max vbθ − p(θ) θ From the FOC we have, 0 p (θ) = vb (3) Combining conditions (1) and (3) we get that, in equilibrium, p(θ) should satisfy the following condition, p0(θ) = G−1(F (θ)): (4) Can we further characterize p(θ)? Yes, we can put additional restrictions on the last unit sold. Let θ be the last unit sold, that is, for θ < θ @p(θ) such that (2) and (4) hold.
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