Economics 121b: Intermediate Microeconomics Final Exam Suggested Solutions

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1 Dirk Bergemann Department of Economics Yale University Economics 121b: Intermediate Microeconomics Final Exam Suggested Solutions 1. Both moral hazard and adverse selection are products of asymmetric information, where one or more of the parties in the transaction involved do not possess some relevant information that the other parties do. Moral hazard refers to hidden actions, while adverse selection refers to hidden information. The example discussed in class demonstrating moral hazard was that of the chef who could produce a high or low quality meal at high or low cost respectively, but some patrons could not observe the quality until after purchasing it, giving him incentive to produce the low quality meal. The example for adverse selection is that of the car dealer who knows the value of the car to be sold and the buyer who does not; due to uncertainty about the value of the car, the buyer does not always make a trade, even when it would benefit him to do so. Constrained optimization is a problem where a party desires to minimize or maximize some objective function, but must satisfy some constraints while doing so. A classic example of constrained optimization is maximizing utility subject to a budget constraint: max x1,x 2 u(x 1, x 2 ) s.t.p 1 x 1 + p 2 x 2 I. First order conditions mark the points at which the derivative of the objective function with respect to each control variable is zero. In the example: u(x 1, x 2 ) x 1 = 0 u(x 1, x 2 ) x 2 = 0. In an unconstrained optimization problem, these points mark potential solutions. In a constrained optimization problem, however, the constraint must be incorporated into the objective function before taking the derivatives. One method for doing so is the method of substitution, which assumes the constraint is binding and substitutes it directly into the objective function, transforming the problem into unconstrained optimization. 1

2 The first order conditions in the example become: x 2 (x 1 ) = I p 1x 1 p 2 u(x 1, x 2 (x 1 )) + u(x 1, x 2 (x 1 )) x 1 x 2 (x 1 ) x 2 (x 1 ) x 1 = 0. A second method is the Lagrangian method, which adds a penalty term that incorporates the constraint into the objective function, transforming the problem into unconstrained optimization. The augmented utility function and the first order conditions in the example become: U(x 1, x 2, λ) = u(x 1, x 2 ) + λ(i p 1 x 1 p 2 x 2 ) U(x 1, x 2, λ) = 0 x 1 U(x 1, x 2, λ) = 0 x 2 U(x 1, x 2, λ) = 0. λ 2. (a) The profit function of firm 1 is The profit function of firm 1 is π 1 (x 1, x 2 ) = (a b(x 1 + x 2 ))x 1. π 2 (x 1, x 2 ) = (a b(x 1 + x 2 ))x 2. Given the quantity x 1 produced by firm 1, firm 2 chooses x 2 to maximize its profit π 2 (x 1, x 2 ): max x 2 (a b(x 1 + x 2 ))x 2 s.t. x 0. The best response function of firm 2 to x 1 is x 2 (x 1 ) = a bx 1. 2b (b) The strategy profile (x 1, x 2) is a Nash equilibrium in pure strategies if given x 2, firm 1 has no incentive to deviate to producing at level other than x 1. In the same way, given x 1, firm 2 has no incentive to deviate. The following conditions characterizes the equilibrium: π 1 (x 1, x 2) π 1 (x 1, x 2) x 1 0 π 2 (x 1, x 2) π 1 (x 1, x 2 ) x

3 (c) From the conditions of Nash equilibrium, we learn that x 1 is firm 1 s best response to x 2. And x 2 is also the best response by firm 2 to x 1. Using the result from part 2.1, we have The solution of the equations is x 2 = a bx 1 2b x 1 = a bx 2. 2b x 2 = a 3b x 1 = a 3b. Correspondingly, the profits of firm 1 and 2 are π 1 = a2 9b π 2 = a2 9b. (d) We need to choose x to maximize the sum of firm 1 and firm 2 s profits, max x (a b(x + x))x + (a b(x + x))x s.t. x 0. The solution is x = a 4b. (e) This level is lower than the equilibrium production level in 2.3. This is because when each firm increases its production, it pushes down the price of the good for both firms. In Nash equilibrium, firms do not take this effect into account and tends to produce more than the collusion level x. i. Given the production level of firm 1 x 1, firm 2 best responds by producing x 2 (x 1 ) = a bx 1 2b Since both firms know that the spy exists, firm 1 knows that his production choice x 1 determines firm 2 s production level and the price as well. In this case, the profit of firm 1 at production level x 1 is π 1 (x 1 ) = (a b(x 1 + x 2 (x 1 )))x 1 = (a b(x 1 + a bx 1 ))x 1. 2b 3

4 ii. The first order condition with respect to x 1 is which gives a 2 b 2 x 1 b 2 x 1 = 0, x spy 1 = a 2b. Substituting this quantity into firm 2 s best response function, we have x spy 2 = a 4b. iii. Firm 1 produces more than in the Nash equilibrium and firm 2 produces less. The profits of firm 1 and firm 2 are 3. (a) See figure 3a. π 1 = a2 8b π 2 = a2 16b. In this situation, firm 1 benefits from firm 2 s spy and firm 2 loses. This is because spy enables firm 1 to credibly commit to a relatively high production level. This is as if firm 1 could move first. By confirming quantity early, firm 1 can force firm 2 to produce less by producing a relatively large number of units first. (b) See figure 3b. As δ B increases, the indifference curves shift downwards, as shown in the diagram. As p 2 increases, the budget line rotates around the endowment (1,1), becoming steeper. (c) Setting Bill s marginal rate of substitution equal to the ratio of prices yields: 1 x b 1 δ B x B 2 = 1 p 2 Solving yields δ B x B 1 = p 2 x B 2 Plugging this into the budget constraint yields (1 + δ B )x B 1 = 1 + p 2 4

5 Q3 Figure 1.jpg Solving yields x B 1 = 1 + p δ B A similar calculation for x B 2 yields We have x B 1 = x B 2 if δ B = p 2. (d) See figure 3d x B 2 = 1 + p 2 p 2 δ B 1 + δ B 5

6 Q3 Figure 2.jpg The endowment point is (1, 1). The x-axis is Bill s consumption in period 1 and the y-axis is Bill s consumption in period 2. There is a total of 2 units of consumption available in each period. (e) Bill s demand for consumption in period 1 is and Jane s demand is x B 1 = 1 + p δ Zero excess demand implies that x J 1 = 1 + p δ 6

7 Q3 Figure 3.jpg 1 + p δ p δ = 2 Solving yields p 2 = δ. At this price, the equilibrium allocations are x B 1 = x B 2 = x J 1 = x J 2 = 1 So both agents consume their endowments and there is no trade. (f) Bill cares more about consumption this year than Jane. This implies that Bill s consumption this year should increase relative to the allocation in the previous part, and his consumption next year should decrease. Jane s consumption will change in the opposite way. 7

8 Q3 Figure 4.jpg (g) Zero excess demand implies that Solving yields 1 + p δ B p δ J = 2 The equilibrium allocations are: p 2 = δ J + δ B + 2δ J δ B 2 + δ J + δ B 8

9 x B 1 = 2 + 2δ J 2 + δ B + δ J x J 1 = 2 + 2δ B 2 + δ B + δ J x B 2 = 2δ B + 2δ B δ J δ J + δ B + 2δ B δ J x J 2 = 2δ J + 2δ B δ J δ J + δ B + 2δ B δ J Jane sells consumption in period 1 to Bill and buys consumption in period 2. The price is between δ B and δ J. 4. (a) Welfare maximizing problem: Hence the F.O.C. is given by, max q(θ) 2θ q(θ) c.q(θ) θ 1 q(θ) c = 0 (b) The monopolist solves: q (θ) = θ2 c 2 max t c q sub. to 2θ q(θ) t 0 Now given that the monopolist can observe θ he will charge as much as he can satisfying the participation constraint. Hence, t(θ) = 2θ q(θ) Therefore the maximization problem for the monopolist is the same as the welfare maximization problem. Therefore, q(θ) = q (θ) = θ2 c 2 t(θ) = 2θ2 c (c) Both q(θ) and t(θ) are increasing in θ. So higher type buyers get more good and pay higher amounts for it. The per unit price is therefore, t(θ) q(θ) = 2c Hence the per unit price is constant. 9

10 (d) i. Consumer s problem: max 2θ q p q q ii. Now the monopolist s problem: q(θ, p) = θ2 p 2 max p q(θ, p) c q(θ, p) p max p θ 2 p cθ2 p 2 p(θ) = 2c iii. p(θ) does not vary with θ as it is a constant. iv. q(θ, p(θ)) = q(θ, 2c) = θ2 4c 2 = 1 4 q(θ) p(θ) = 2c = t(θ) q(θ) So the per unit price doesn t change compared to [4.2] but the quantity traded goes down significantly. This is because in the first case the monopolist offered a take-it-or-leave-it contract to the consumer and hence was able to extract all the consumer surplus. But in the second case the buyer has more flexibility and he can change his demand as a function of price charged. Since the optimal per unit price was constant in the first case, it remains the same when the monopolist can only charge per unit price and not a lump-sum transfer. But in the first case the monopolist was able to sell a lot more at that price because of the nature of contract. Now in the second case given the same price the buyer reduces his demand as that maximizes his utility. 10