EC401 Money and Banking II

Problem 1

Suppose an agent has an exponential utility function u(w) = −e−ρw, where w denotes the agent’s final

wealth and ρ is a positive constant. The agent has a project that yields return r(θ), which follows a

normal distribution of mean θ and variance σ2.

a. derive , which is the coefficient of absolute risk aversion for the utility function u(w).

b. show that , where W0 is a positive constant.

hint 1. the definition of expectation:

hint 2.

c. show that E [u(W0 + r(θ))] is decreasing in variance σ2.

Problem 2 Coalition of borrowers.

Recall the binomial case (θ1,θ2) discussed in the lecture. Suppose there are N type θ1 entrepreneurs

and N type θ2 entrepreneurs. Returns of individual entrepreneurs’ projects are independently

distributed.

a. Suppose N type θ1 entrepreneurs form a coalition, which means they equally share the

proceeds of equity issuance and the final returns. Derive the expected return and the variance of

return for each entrepreneur. (It is insufficient to simply give the final answer.)

b. Suppose N type θ1 entrepreneurs form a coalition and type θ2 entrepreneurs don’t do so.

Derive the informational cost of capital for a type θ2 entrepreneur given the minimum possible value

of α.

(There is no need to solve for α explicitly. It is enough to derive the second order polynomial that α

must satisfy.) Compare the type θ2 entrepreneur’s utility in this case to the one in lecture.

hint the information cost of capital for the example discussed in the lecture is .

c. Suppose N type θ2 entrepreneurs form a coalition and type θ1 entrepreneurs don’t do so.

Answer the same two questions in part b. (a type θ2 entrepreneur’s informational cost of capital and

utility.)

d. Suppose N type θ1 entrepreneurs form a coalition. So do N type θ2 entrepreneurs. Answer the

same two questions in part b. (a type θ2 entrepreneur’s informational cost of capital and utility.)