1. Suppose that we have an economy with two goods, a private good x and a public good y. There are I consumers in this economy, each with an initial endowment of the private good ex(i)  ≥ 0 for all i = 1, ...I. You can denote E = Pi(I)=1 ex(i) . Consumers’ utility function is given by

ui (xi ,y) = ln xi +lny,

where xi  is the consumption of private good and y is the level of the public good. There is a firm producing the public good from the private good using the production function y = ^z, where z is the total amount of the private good allocated to produce the public good. The price of the private good is 1.

(i) Find the consumers’ demand for the private good and the public good [8 marks]

(ii) Find the firm’s supply of the public good as a function of the personalized prices vector [8 marks]

(iii) Use your responses to (i) and (ii) to show that the vector of personalized prices that constitutes the Lindahl equilibrium of this economy is such that pi  =  for all i = 1, ...,I.

[24 marks]

2. Consider a pure exchange economy with three consumers, Alice,Bob and Carol, and three goods, 1, 2 and 3.  Consumers have the     utility functions

u (xi1(i),x2(i),x3(i)) = ↵ ln x1(i)+β ln x2(i)+(1−↵−β)ln x3(i) ,

for i = A,B,C

Endowments are eA = (0, 1, 1), eB  = (0, 1, 1) and eC  = (2, 0, 0). Find the Walrasian equilibrium and the Walrasian  Equilibrium Allocation (WEA) of this economy. [40 marks]

3. An Agent (A) needs to complete a task for a Principal (P).  A can choose between working hard, e = 1 or shirking e = 0. If A accepts the task, he is paid a wage w and his utility is ln(w)−e. A can always reject the task and obtain an outside option u. The only thing P can observe is whether A succeeds or fails his task. He can make the wage w contingent on that observation (i.e. have wages wS  and wF ).  If e = 1, the probability of success is pH .  If e = 0, the probability of success is pL .  The pay-of to P is xS  − w in case of success and xF  − w in case of failure. Assume xS  > xF  and pH  > pL .

(i) Suppose P wants A to work hard, write down the incentive compatibility constraint  (IC) and  participation constraint (PC) of the Agent. [13 Marks]

(ii) Argue that the  optimal wage  schedule  for the  Principal (wS ,wF ) implies both constraints holding with equality. [14 Marks]

(iii) Let W1 denote P’s expected payof if he makes A work hard. Of course, it might also be optimal for P to have e = 0, in which case the wage is w and his expected pay-of is W0 . When is W1  > W0 ? Express this inequality only in terms of pH ,pL ,xS ,xF  and u. [13 Marks]