EconS 503 - Microeconomic Theory II Homework #9 - Answer key 1. WEAs with market power. Consider an exchange economy with two consumers, A and B, whose utility functions are u A (x A 1 ; x A 2 ) = x A 1 x A 2 u B (x B 1 ; x B 2 ) = x B 1 (x B 2 ) 2 with endowments e A = (80; 150) and e B = (210; 180) respectively. Assume that consumer A is price setter, i.e., he makes a take-it-or-leave-it price o er to consumer B. (a) Find the Walrasian Equilibrium allocation (WEA) in this economy. Consumer B takes the price ratio announced by consumer A as given, and solves his UMP His Lagrangian is Taking FOCs yields max x B 1 ;xb 2 u B (x B 1 ; x B 2 ) = x B 1 (x B 2 ) 2 subject to p 1 x B 1 + p 2 x B 2 210p 1 + 180p 2 L = x B 1 (x B 2 ) 2 [p 1 x B 1 + p 2 x B 2 210p 1 180p 2 ] dl dx B 1 dl dx B 2 Combining the FOCs, we obtain = (x B 2 ) 2 p 1 = 0 ) = (xb 2 ) 2 p 1 = 2x B 1 x B 2 p 2 = 0 ) = 2xB 1 x B 2 p 2 p 1 p 2 = xb 2 2x B 1 Before plugging this result into consumer s budget constraint, p 1 x B 1 + p 2 x B 2 = 210p 1 + 180p 2 ; we can divide such a constraint by p 2 to obtain p 1 x B 1 + x B 2 = 210 p 1 + 180 ) p 1 (x B 1 210) = 180 x B 2 p 2 p 2 p 2 We can now substitute p 1 p 2 = xb 2 2x B 1 in the left term, x B 2 2x B 1 which, solving for x B 2 ; yields (x B 1 210) = 180 x B 2 x B 2 = 360xB 1 3x B 1 210 which constitutes the o er curve of consumer B. 1
Consumer A anticipates this o er curve of consumer B, along with the following feasibility conditions x B 1 = 290 x A 1 for good 1, and x B 2 = 330 x A 2 for good 2 From our above result of the o er curve of consumer B, x B 2 = 360xB 1 ; the 3x B 1 210 feasibility condition for good 2 can be rewritten as which can be rearranged as 360x B 1 3x B 1 210 = 330 xa 2 360x B 1 990x B 1 + 3x B 1 x A 2 + 69300 210x A 2 = 0 Substituting the feasibility condition for good 1, we obtain, 360(290 x A 1 ) 990(290 x A 1 ) + 3(290 x A 1 )x A 2 + 69300 210x A 2 = 0 and simplifying, yields an expression that is a function of x A 1 and x A 2 alone, that is, 630x A 1 3x A 2 x A 1 + 870x A 2 113; 400 = 0 Hence, consumer A s problem becomes max x A 1 ;xa 2 u A (x A 1 ; x A 2 ) = x A 1 x A 2 subject to 630x A 1 3x A 2 x A 1 + 870x A 2 113; 400 = 0 with associated Lagrangian Taking FOCs yields L = x A 1 x A 2 [630x A 1 3x A 2 x A 1 + 870x A 2 113; 400] dl dx A 1 dl dx A 2 = x A 2 630 + 3x A 2 = 0, = = x A 1 870 + 3x A 1 = 0 () = x A 2 630 3x A 2 x A 1 870 3x A 1 Setting the above FOCs equal to each other, we obtain x A 2 (870 3x A 1 ) = x A 1 (630 3x A 2 ) =) x A 2 = 630 870 xa 1 Plugging this result in the constraint of consumer A, we nd that 630x A 1 2:17(x A 1 ) 2 + 630x A 1 113; 400 = 0 2
Finally, solving for x A 1 yields two roots, x A 1 = 111:356 and x A 1 = 456:23, but the second root is infeasible since it exceeds the total endowment of the good. Hence, x A 1 = 111:356 implying that the amount of good 2 for this consumer is x A 2 = 630 870 xa 1 = 630 111:356 = 80:637 870 Using the feasibility conditions, we can obtain the equilibrium consumption bundle of individual B, x B 1 = 290 x A 1 ) x B 1 = 178:644 and In summary, the WEA is x B 2 = 330 x A 2 = 249:36 x A 1 ; x A 2 ; x B 1 ; x B 2 = (111:356; 80:637; 178:644; 249:36) : (b) Find the Pareto optimal allocation (PEA) in this economy, and check if the WEA from part (a) is a PEA. For a PEA, we need which in this setting entails Using the feasibility conditions, MRS A 1;2 = MRS B 1;2 x A 2 x A 1 = xb 2 2x B 1 x B 1 = 290 x A 1 x B 2 = 330 x A 2 Plugging x B 1 and x B 2 in terms of x A 1 and x A 1 in the MRS A 1;2 = MRS B 1;2 condition, we obtain x A 2 x A 1 = 330 xa 2 2(290 x A 1 ) Rearranging, we nd that the contract curve describing all PEAs is given by 580x A 2 330x A 1 x A 1 x A 2 = 0 Plugging the WEA found in part (a) in this equation, we nd that 580x A 2 330x A 1 x A 1 x A 2 = 1042:57 6= 0 entailing that the WEA is not Pareto optimal, i.e., the WEA does not lie on the contract curve. Hence, the presence of market power (with one individual being the price setter) prevents the First Theorem of Welfare Economics from holding. 3
2. Excess demand and stability of equilibria. Consider a two-commodity economy where the price of commodity 1 is normalized in terms of commodity 2, whereby p 1 p 2 = p. Suppose the excess demand function for commodity 1 is given by (a) How many equilibria can you nd? z 1 (p) = 1 4p + 5p 2 2p 3 The excess demand for commdity 1 at relative price can be written z 1 (p) = 1 4p + 5p 2 2p 3 = (1 p) 2 (1 2p) so that z 1 (p) = 0 holds at two equilibrium prices, p 1 = 1 and p 2 = 1, i.e., the 2 two roots of (1 p) 2 (1 2p) = 0. Figure 6.17 plots z 1 (p) as a function of price ratio p. Figure 6.17. Excess demand z 1 (p). (b) Which of the equilibrium price ratios you found are stable? In order to test whether an equilibrium price p is stable, we need that @z 1 (p) < 0 so the excess demand function crosses the horizontal axis from @p above. Intuitively, for p < p i there is excess demand while for p > p there is excess supply. Since @z 1(p) = 4 + 10p 6p 2, evaluating this derivative at @p each of the equilibrium prices p 1 and p 2 we obtain that At p 1 = 1, @z 1(p) @p = 4 + 10 6 = 0. As depicted in the gure of part p1 =1 (a), this equilibrium is unstable. (To be precise, it is only locally stable from above.) At p 2 = 1, @z 1 (p) = 4 + 10 1 6 1 2 2 @p 2 2 = 1 < 0; and thus 2 p2 = 1 2 the equilibrium is stable, i.e., the excess demand function crosses the horizontal axis from above, as depicted in the gure. (c) Consider now that the aggregate endowment of good 1 increases. How are your results from parts (a) and (b) a ected? 4
An increase in the aggregate endowment of good 1,! 1, reduces the excess demand of this good, z 1 (p) = x 1 (p)! 1, where x 1 (p) denotes aggregate demand for good 1. Hence, z 1 (p) experiences a downward shift, as depicted in Figure 6.18. While equilibrium price p 2 = 1 moves leftward, the equilibrium 2 price p 1 = 1 is absent. Therefore, the only equilibrium is stable. 3. Exercises from MWG: Figure 6.18. Excess demand and Stability. (a) Chapter 23 (mechanism design): Exercise 23.C.10. [See last pages of his handout for an answer key to this exercise.] 4. Public Good Provision - Di erent mechanisms. Imagine that you and your colleagues want to buy a co ee machine for your o ce. Suppose that some of you may be heavily addicted to co ee and are willing to pay more for the machine than the others. However, you do not know your colleagues willingness to pay for the machine. The cost of the machine is C. We would like to nd a decision rule in which (i) each individual reports a valuation (i.e., direct mechanism), and (ii) the co ee maker is purchased if and only if it is e cient to do so. Let us next analyze if it is possible to nd a cost-sharing rule which gives incentive for everyone to report his valuation truthfully. In particular, assume n individuals, each of them with private valuation i U(0; 1). The allocation function is binary y 2 f0; 1g, i.e., the co ee machine is purchased or not. Let t i be the transfer from individual i, implying a utility of u i (y; i ; t i ) = y i t i Let i 2 f1; :::; ng denote the individuals, and let i = 0 denote the original owner of the good. (a) What is the e cient assignment rule, y ( 1 ; :::; n )? 5
The e cient assignment rule is P 1 if n y j=1 ( 1 ; :::; n ) = j C 0 otherwise In words, the co ee machine is purchased if and only if the sum of all valuations exceeds its total cost. (b) Equal-share rule. Consider the following equal-share rule: When the public good is provided, the cost is equally divided by all n individuals. 1. Before starting any computation, what would you expect - whether each individual would overstate or understate their valuation? Because of free-rider incentives, each individual may have an incentive to understate his valuation. The equal-share payment rule, however, makes transfers independent of his report. 2. Con rm that the transfer rule is written by: t i () = C n y () By the equal-share rule, each individual will pay C n and 0 otherwise. Hence, the transfer rule is if the project happens, t i () = C n y () 3. Let V i ( ~ i j i ; i ) be individual i s payo when i reports ~ i instead of his true valuation i, while the others truthfully report their valuations i. Show that V i ( ~ C i j i ; i ) = i y ( n ~ i ; 1 ) Using the de nition of player i s utility function, we can intert in the above equal-share transfer rule to obtain V i ( ~ i j i ; i ) = i y ( ~ i ; 1 ) t ~i i ; i = i y ( ~ i ; 1 ) = i C n C n y ( ~ i ; 1 ) y ( ~ i ; 1 ) 4. Let U ~i i j i be individual i s expected payof when he reports ~ i instead of the true valuation i. Show that U ~i C h i j i = i E n i y ( ~ i i ; 1 ) 6
Player i s expected payo for misreporting ~ i 6= i is just the expected value of the utility found above, that is U ~i i j i = E i hv i ( ~ i C h i j i ; i ) = i E n i y ( ~ i i ; 1 ) 5. Suppose that i s private valuation i satis es i > C. Assuming that the n others are telling the truth, what is the best response for i? What if i < C? Is this mechanism strategy-proof? Is this mechanism Bayesian incentive n compatible? If player i s valuation i satis es i > C, U ~i n i j i is maximized when E i hy ( ~ i i ; 1 ) is maximized. Hence, individual i would report ~ i as large as possible, i.e., ~ i = 1. In contrast, if i satis es i < C n, individual i would report ~ i as small as possible, i.e., ~ i = 0. The mechanism is neither strategy-proof, nor Bayesian incentive compatible. (c) Proportional payment rule. Consider now the proportional payment rule: t i () = P ic j y () j where every individual i pays a share of the total cost equal to the proportion that his reported valuation signi es out of the total reported valuations. 1. Under this rule, what would you expect - whether each individual would overstate or understate the valuation? Now the payment is a function of the report. Notice that this costsharing rule is balanced-budget. Hence, you may expect that the agents have incentive to free-ride. 2. Show that the utility of reporting ~ i is now! V i ( ~ ~ i C i j i ; i ) = i ~ i + P j6=i y ( ~ i ; 1 ) j The payo to each individual will be their actual valuation, less the amount the have to pay based on what they report if the project happens, and 0 otherwise. That is, V i ( ~ i j i ; i ) = i y ( ~ i ; 1 ) t ~i i ; i = i y ( ~ ~ i C i ; 1 ) ~ i + P j6=i y ( ~ i ; 1 ) j! ~ i C = i ~ i + P j6=i y ( ~ i ; 1 ) j 3. For simplicity, suppose two individuals, n = 2 and a total cost of C = 1. Show that U ~i i j i = ~ i i log ~i + 1 7
Again, by de nition, the expected utility of misreporting ~ i is "! # ~ i U ~i C i j i = E i i ~ i + P j6=i y ( ~ i ; 1 ) j Suppose now that n = 2 and C = 1. Then the above expression becomes "! # ~ i U ~i i j i = E i i y ~ ( ~ i ; 1 ) i + j Z! 1 ~ i = i d 1 ~ i ~ j i + j h = i j ~ i log( ~ i 1 i + j ) 1 ~ i = i ~ i log( ~ h i + 1) i (1 ~ i ) ~ i log( ~ i + (1 i ~ i )) = ~ i i log( ~ i + 1) 4. Is this mechanism strategy-proof? Is it Bayesian incentive compatible? It is straightforward to show that the expected utility of reporting ~ i is decreasing in player i s report ~ i, since @ @ ~ i U i ~i j i ~i = i = 2 i 1 + i log( 1 + 1) < 0 for all i 2 (0; 1] implying that every player i has incentives to underreport his true valuation i as much as possible, i.e., ~ i = 0. Hence, this mechanism is neither strategy-proof nor Bayesian incentive compatible. 5. Which way is everyone biased, overstate or understate? What is the intuition? The negative sign in part (iv) suggests that U i ~i j i is maximized at ~ i smaller than i. Each individual has an incentive to understate the valuation. (d) VCG mechanism. Let us consider now the VCG mechanism. Recall that the e cient rule y () determines that the co ee machine is bought if and only if total valuations satisfy P i i C. Remember that we need to include the original owner of the public good; i = 0. Then, the total surplus when the valuation of individual i is considered in = ( 1 ; 2 ; :::; n ) is X P v j (y j6=i (); j ) = j if P j j C C if P j j < C j6=i while total surplus when the valuation of individual i is ignored, i, is X P v j (y j6=i ( i ); j ) = j if P j6=i j C C if P j6=i j < C j6=i 8
The only di erence in total surplus arises from the allocation rule which speci es that, when i is considered, the good is purchased if and only if P j j C, whereas when i is ignored, the good is bought if and only if P j6=i j C. Hence, the VCG transfer is! X X t i () = v j (y (); j ) v j (y ( i ); j ) = C j6=i j6=i P j6=i j if P j6=i j < C P j j 0 otherwise Intuitively, player i pays the di erence between everyone else s valuations, P j6=i j, and the total cost of the good, C. Such a payment, however, only occurs when aggregate valuations exceed the total cost, P j j C, and thus the good is purchased, and when the valuations of all other players do not yet exceed the total cost of the good, P j6=i P j < C, so the di erence C j6=i j is paid by player i in his transfer. 1. Show that in this mechanism player i s utility from reporting a valuation ~ i 6= i is V i ( ~ i j i ; i ) = v i y ~i ; i ; ~i i ; i = 8 < : t i 0 if ~ i + P j6=i j < C P j P j C if j6=i j < C ~ i + P j6=i i i if C P j6=i j This is just the de nition of the payo function for the VCG. 2. Is this mechanism strategy-proof? Is this Bayesian incentive compatible? In order to test if this direct revelation mechanism is strategy-proof, Suppose that C P j6=i j, i.e., the public good will be purchased regardless of individual i s reported valuation. Then V i ( ~ i j i ; i ) = i, which is independent of player i s reported valuation, ~ i. Hence, telling the truth is player i s best response. Now suppose that P j6=i j < C P j j, i.e., individual i s valuation is pivotal. Then by reporting a valutation ~ i such that ~ i C P j6=i j, his utility becomes V i ( ~ i j i ; i ) = P j j C 0. This includes the case of telling the truth; ~ i = i C P j6=i j. If, instead, individual i lies by reporting ~ i < C P j6=i j, then his utility becomes V i ( ~ i j i ; i ) = 0 since the good is not purchased given that ~ P i < C j6=i j entails ~ i + P i6=j j < C. Hence, misreporting his valuation cannot be pro table. Finally, suppose that P j j < C, i.e., the public good will not be purchased regardless of individual i s valuation. Then, by honestly revealing his valuation, ~ i = i < C P j6=i j, his payo is V i ( ~ i j i ; i ) = 0 since the good is not purchased. By lying, ~ P i C j6=i j, his payo is V i ( ~ i j i ; i ) = P j j C < 0. Telling a lie is then not pro table. Hence, 9
truth-telling is the best strategy for i, regardless of the values of i. The VCG mechanism is thus strategy-proof, and also Bayesian incentive compatible. 3. For simplicity, suppose two individuals, n = 2, and a total cost of C = 0:5. Compute y, t 1 and t 2 for the following ( 1 ; 2 ) pairs. 1 2 0.1 0.3 0.3 0.3 0.3 0.8 0.8 0.8 For the case of 1 = 0:1 and 2 = 0:3, we have that VCG transfers become P t j6=i i () = j + C if P j6=i j < C P j j 0 otherwise implying that the transfer player 1 pays is 0:3 + 0:5 if 0:3 < 0:5 0:4 t 1 () = 0 otherwise and the transfer that player 2 pays is 0:1 + 0:5 if 0:1 < 0:5 0:4 t 2 () = 0 otherwise As we can see, the upper inequality does not hold, and thus the good is not purchased, y () = 0, and transfers are zero, t 1() = t 2() = 0. Following the same steps, the results for valuation pairs (0:3; 0:3), (0:3; 0:8), and (0:8; 0:8) are presented in the following table 1 2 y () t 1() t 2() 0.1 0.3 0 0 0 0.3 0.3 1 0.2 0.2 0.3 0.8 1 0 0.2 0.8 0.8 1 0 0 4. Show that the expected revenue from this mechanism is E [t 1( 1 ; 2 ) + t 2( 1 ; 2 )] = ' 0:167. Based on what you calculated in part (iii), is this problematic? 1 6 If 2 C, then player 1 doesn t need to pay anything t 1 = 0. If 2 < C, then player 1 s transfer is t 1 = 2 + C if and only if 1 + 2 C. Hence, player 1 s expected transfer is Z E [t 1( 1 ; 2 )] = ( 2 + C)d 1 d 2 = 10 f( 1 ; 2 )j 1 + 2 Cg Z C Z 1 0 2 +C( 2 + C)d 1 d 2 = 1 12
By symmetry, E [t 2( 1 ; 2 )] = 1, entailing that expected revenue becomes 12 E [t 1( 1 ; 2 ) + t 2( 1 ; 2 )] = 1 6 ' 0:167 This is problematic, because the expected revenue, 0.167, is smaller than the total cost, 0.5, implying a budget de cit. The VCG mechanism has two nice properties: e ciency and incentive compatibility. However, balanced budget condition and participation constraint are not necessarily satis ed. 11