1. Size matters
Solve Exercise 1 in Chapter 12 of Ray's book (p. 478f)
2. Share cropping with limited liability
Consider the model of share cropping studied in class and in Appendix 1 to Ray ch. 12. Assume now that the tenant is risk neutral, so his utility function can be written u(c)=c.
- Find the new incentive constraint
- Set up the landlord's maximization problem as a Kuhn-Tucker problem and find the first order conditions. Show that the incentive constraint does not bind.
- Show that one solution to the landlord's problem is to charge a fixed rent, and compute the fixed rent.
- Assume now that the bad outcome L is very bad so L=0. Moreover, the tenant has no funds of his own, so it is impossible for him to pay a higher rent than his earnings (limited liability). Solve the landlord's problem in this setting and show that the solution in 3. not longer holds.
3. Linear contracts
Consider the model of share cropping studied in class and in Appendix 1 to Ray ch. 12. However, now there are 3 possible outcomes, L, M and H. The probability of each outcome with effort 0 and 1 is given in the following table:
| Outcome | Probability with e=0 | Probability with e=1 |
|---|---|---|
| L | 0.5 | 0.5 |
| M | 0.4 | 0.1 |
| H | 0.1 | 0.4 |
- Solve the landlord's problem in this case (with a risk averse tenant)
- Show that the optimal wage schedule you get is not of the form w=α+βY where Y is output.
- Discuss whether the wage structure found in 1. or the simple linear contract from 2. is more reasonable when:
- The landlord has good overview over the tenant's production and only rents out land to one tenant
- The landlord lives in the city and rents out land to many tenants