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) = σ^{2}
T doesn't like a working and incurs a utility cost of effort: c (e) = ce^{2} = 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, σ^{2}_{x })
We cannot observe x. However, we can observe y. y is an informative signal of z, where y _{˜} N(0, σ^{2}_{y})
Further assume that Aís cost of e§ort is c (e) = e^{2} = 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) + γ^{2}Var (y)
A) Show formally that the optimal weight, γ*, on signal y is such that:
γ* = - cov (x, y)/δ^{2}_{y}
B) What is the sign of γ if cov (x, y) < 0? Why?
C) How does γ vary with δ^{2}_{y}? Why?