Thanks a lot

3. Consider the following mechanism design problem. Let x E [0, 1] denote the level of a public good that affects the welfare of two individuals. Suppose both individuals receive utility v; from public good x and monetary transfer m = (m1, m2). Specifically, let v1(x, m1, 01) = 01 In(x) +m where 01 = 2 with probability 1/2 and 61 = 4 with probability 1/2, while v2(x, m2, 02) = 02a1/2 - x + m2 where 02 = 1 for sure. (a) Identify all possible states (i.e., type profiles) and find the first-best decision rule. (b) Find the pivotal mechanism transfer rule and verify that budget balancedness is not satisfied. (c) Find the expected externality transfer rule and show that budget balancedness is satisfied. (d) Verify that the expected externality mechanism you found above is incentive com- patible. (e) Verify that the expected externality mechanism you found above is individually rational