problem 1: A landlord, L, decides on a sharecropping arrangement with a tenant, T. L decides on the proportion, b, of output and the fixed wage, a to pay the tenant::

w = a + bY

Output, Y , depends on the level of effort, e, and also on the weather, ε:

Y = e + ε where var (ε) = σ²

T doesn't like a working and incurs a utility cost of effort: c (e) = ce² = 2.

T has an outside option, u.

Furthermore, assume that L is risk neutral and therefore has the utility function: U (m) = m. T's utility has the functional form:

U (m) =  -e^-rm

T's utility can also be written in terms of its certainty equivalent:

m - r
. var (m)

where r is the degree of relative risk aversion.

Show all your workings with the following problems:

A) prepare down the L's and T's objective functions, in terms of their certainty equivalent.

B) What is Tís optimal level of effort, as a function of a and b?

C) What is the minimum level of a that T is willing to accept, given b?

D) prepare down the incentive compatibility and participation constraints.

E) find out the optimal level of a and b.

F) How should the output be shared between tenant and landlord if r = 0? What kind of contract is this? and if r →∞?

problem 2: A principal, P, pays an agent, A, the following wage:

w = α + β (z + γy)

z is a noisy measure of effort

z = e + x where x _˜ N (0, σ²_x )


We cannot observe x. However, we can observe y. y is an informative signal of z, where y _˜ N(0, σ²_y)

Further assume that Aís cost of e§ort is c (e) = e² = 2; A has an outside option of zero; and has the utility function:

U (m) = -e^-rm

Recall that, with normally distributed error terms:

Var (x + γy) = Var (x) + 2γCov (x, y) + γ²Var (y)

A) Show formally that the optimal weight, γ*, on signal y is such that:

γ* = - cov (x, y)/δ²_y

B) What is the sign of γ if cov (x, y) < 0? Why?

C) How does γ vary with δ²_y? Why?