Microeconomics Problem Set 4 Solutions PDF

Micro II Solutions to exercises Problem Set 4: Solutions Chapter 6: Externalities Exercise 6.1 Theorem 21 Su...

Micro II Solutions to exercises Problem Set 4: Solutions Chapter 6: Externalities Exercise 6.1 Theorem 21 Suppose that two agents are deciding how fast to drive their cars. Agent i chooses speed xi and gets utility ui (xi ) from this choice. Assume that ui (xi ) > 0 and decreasing rate ui (xi ) < 0. The faster the agents drive, the more likely it is that they are involved in a mutual accident. Let p(x1 , x2 ) be the probability of an accident, and assume 80->90 > 120->130 it is increasing in each argument. Let ci > 0 be the cost that the accident imposes on agent i. Assume that each agent’s utility is linear in money. 1. Show that each agent has an incentive to drive too fast from the social point of view. 2. If agent i is fined an amount ti in the case of an accident, how large should ti be optimally? 3. If the optimal fines are being used, what are the total costs, including fines, paid by the agents? How does this compare to the total cost of the accident? 4. Suppose now that the agent i gets utility ui (xi ) only if there is no accident. What is the appropriate fine in this case? Part 1 Agent i solves max ui (xi ) − p(xi , xj )ci. xi The optimal xi has to satisfy ∂p(xi , xj ) ui (xi ) − ci = 0. ∂xi The social problem is we sum the uitilities of the two speeds max ui (xi ) + uj (xj ) − p(xi , xj )(ci + cj ). xi ,xj cost brought by each speed The optimal xi has to satisfy ∂p(xi , xj ) ∂p(xi , xj ) ui (xi ) − ci − cj = 0 ∂xi ∂xi       in private optimum = 0 b2 > 0. 1. Find the efficient amount of public good G∗. 2. Now suppose that agents are privately deciding how much to contribute to the public good. Find the reaction functions of the agents, Ri (gj ). That is, for any contribution gj , find the optimal contribution of agent i. 3. What are the (Nash) equilibrium contributions to the public good? Is the private provision of the public good efficient? 4. In a graph, show the reaction functions and the equilibrium point. Part 1 We want to maximize the sum of utilities max u1 (G, x1 ) + u2 (G, x2 ) g1 ,g2 ,x1 ,x2 s.t. G = g1 + g2 , g1 ≥ 0 g 1 + x1 = ω1 , g2 ≥ 0 g 2 + x2 = ω2. We can equivalently write substituted x_1 and x_2 with ω_1-g_1 and ω_2-g_2,respectively max u1 (g1 + g2 , ω1 − g1 ) + u2 (g1 + g2 , ω2 − g2 ) g1 ,g2 max b1 ln(g1 + g2 ) + ω1 − g1 + b2 ln(g1 + g2 ) + ω2 − g2 g1 ,g2 b1 b2 ! FOCs −1+ =0 g1 + g2 g1 + g2 b1 b2 ! suppose and must verify + −1=0 g1 + g2 g1 + g2 b1 + b2 = g 1 + g 2 G ∗ = b 1 + b2. 32 Micro II Solutions to exercises Part 2: Suppose that agent i contributes gi : The problem of agent i is then max bi ln(gi + gj ) + ωi − gi gi s.t. gi ≥ 0 bi ! FOC: −1=0 gi + gj bi = g i + g j g i = bi − g j. Since gi cannot be negative the reaction function is given by Ri (gj ) = max {0, bi − gj }. Part 3: In a Nash equilibrium, both agents have to be best-responding to each other’s actions. That is g1∗ and g2∗ are such that R1 (g2∗ ) = g1∗ R2 (g1∗ ) = g2∗. First, suppose that g1∗ > 0 and g2∗ > 0. Then g1∗ = b1 − g2∗ and g2∗ = b2 − g1∗. Combining: g1∗ = b1 − b2 + g1∗ 0 = b 1 − b2 , which is never the case. Next, observe that g1∗ = g2∗ = 0 is also never an equilibrium. Next, suppose g1∗ = 0, g2∗ > 0. Then g2∗ = R2 (0) = b2 − 0 = b2. But R1 (b2 ) = b1 − b2 > 0. Hence, this is also not an equilibrium. Finally, suppose g2∗ = 0 and g1∗ > 0. Then, g1∗ = R1 (0) = b1. Observe that R2 (b1 ) = max{0, b2 − b1 } = 0. Thus, the equilibrium si given by g1∗ = b1 g2∗ = 0. Private provision is not efficient and agent 2 completely free-rides. 33 Micro II Solutions to exercises Part 4: Chapter 8: Auctions Exercise 8.1 Theorem 23 Suppose that there are N bidders, and each bidder’s value is uniformly distributed on [0, 1]. That is, F (v) = v. 1. Derive the equilibrium bidding function in the first-price auction. Is it in- creasing in v? 2. What is the bidding function for N = 2? And N = 3? 3. Fix some v. How does the bid at v change as the number of bidders increase from N to N  ? 4. What is the optimal bid as N → ∞? 1) Applying Proposition 9.1 we get  v N −1 b(v) = N −1 xf (x)F N −2 (x) dx F (v) 0 34 Micro II Solutions to exercises Setting F (v) = v and f (x) = 1,  N −1 v b(v) = N −1 x · xN −2 dx v 0   N −1 1 N v = N −1 x v N 0 N −1 b(v) = v. N Take the derivative of b(v) with respect to v ∂b(v) = (N − 1)/N > 0 ∂v The bidding function is increasing in v. 2) 1 b2 (v) = v 2 2 b3 (v) = v 3 3) The bid will increase because N − 1 1 1 N −1  =1−  >1− =. N N N N We could also take the derivative with respect to N ∂b(v) 1 = 2v > 0 ∂N N 4) As N → ∞, b(v) → v. Exercise 8.2 Theorem 24 Suppose that there are 2 bidders, and each bidder’s value is uniformly distributed on [0, 1]. That is, F (v) = v. Recall that the expected revenues in the first-price auction and in the second-price auction are given by  1 RF P A = N b(v)f (v)F N −1 (v)dv 0  1 RSP A = N (N − 1) vf (v)(1 − F (v))F N −2 (v)dv. 0 Calculate the expected revenues from the FPA and the SPA. 35 Micro II Solutions to exercises From the previous exercise we know that b(v) = 12 v. We have:    1 1 1 3 1 1 RF P A = 2 v · v dv = v = 0 2 3 0 3  1 RSP A = 2 v(1 − v) dv 0  1  1 =2 v dv − 2 v 2 dv 0 0 2 1 1 1 = v 0 −2 =. 3 3 Exercise 8.3 Theorem 25 Show that RF P A = RSP A.  1 RF P A = N b(v)f (v)F N −1 (v) dv 0  1  v  N −1 N −2 =N xf (x)F (x) dx f (v)F N −1 (v) dv 0 F N −1 (v) 0  1  v  N −2 = N (N − 1) xf (x)F (x) dx f (v) dv 0 0   1  v  N −2 = N (N − 1) xf (x)F (x) · f (v) dx dv (by changing the order of integration) 0 0  1 1 = N (N − 1) xf (x)F N −2 (x)f (v) dv dx (only f (v) depends on v) 0 x  1 = N (N − 1) xf (x)F N −2 (x) 1 − F (x) dx 0 = RSP A 36 Micro II Solutions to exercises 37