2. A local government is trying to decide whether to build a school that will jointly serve two neighborhoods. The school will cost one million dollars to build. Neighborhood 1 values the school at value In dollars and neighborhood 2 values the school at 12 dollars. The government does not know these valuations, though it knows that each is somewhere between 0 and one million, that is, 0 1, 000, 000, the school will be built with neighborhood 1 paying $600,000 and neighborhood 2 paying $400,000. If the two statements sum to one million or less, the school will not be built. i. Suppose players are willing to lie if it could benefit them. That is, assume that each neighborhood will say that its valuation is 1 million if it wants the school to be built and will say it is 0 if it doesn't want the school to be built. For which pairs of true valuations, I'm and 1/2 , will the school be built? ii. Suppose that the true valuations are Vi = 300, 000 and 1/2 = 800, 000. Each neighborhood does not know how much the other neighborhood values the school. What statement Si would neighborhood 1 submit, and what statement S2 would the neighborhood 2 submit? iii. Given the statements S, and $2 submitted by the neighborhoods identified in part ii, would the school be built? Is this an efficient or inefficient outcome? Clos iv. Draw a picture that shows for which valuations ', and 1/2 the proposed mechanism will lead to an inefficient outcome. Clearly identify the set of such (VI, V2 )c. Suppose the government proposes the following mechanism. Each neighborhood will state the amount that it values the school. The neighborhoods can say any number between 0 and one million, and they need not tell the truth. If the sum of the two statements is greater than one million, the school will be built with neighborhood 1 paying $600,000 and neighborhood 2 paying $400,000. If the two statements sum to less than or is equal to one million, the school will not be built. In addition, the government will set up an incentive payment to each neighborhood t1 and t2 as a function of the stated valuations (S1 and $2) Let us consider how the payments can align the incentives of a utilitarian government and the neighborhoods. The government will build the school if S1 + $2 -1, 000, 000 > 0 which is the same as S1 > 1, 000, 000 - S2 Neighborhood 1 would like the school to be built if V1 - 600, 000 + (1 > 0 which is the same as V1 > 600,000 - 1 Comparing these two inequalities, one might conjecture that neighborhood 1 cannot benefit from lying if the two right-hand sides are the same-i.e.,. If (1 = $2 - 400, 000 so that 100, 000 - $2 = 600, 000 - 11 . If this is the case, the argument goes, then if player 1 tells the truth, the government will build exactly when player 1 wants the government to build. Show that this conjecture is correct. That is, show that if /1 = $2 - 400, 000 , then for any V'i and any statement of neighborhood 2, $2 , neighborhood 1 cannot benefit from lying to the government about its valuation