ECON 474
Economic Development
Name:
Quiz:
Suppose there is a risk-neutral farmer whose output is either zero or one. The probability
that the farmer gets an output of one is e, where e is a number between zero and one that
represents the effort level chosen by the farmer. The farmer’s cost of providing effort e is
1
given by e 2 .
2
A) Find the efficient level of effort for the farmer to supply. The efficient level of effort
maximizes the expected output minus the cost of providing effort. (5 points)
Solution:
Probability of high output = e
Probability of low output = 1 – e
Expected output = e*1 + (1 – e)*0 = e
1
Expected gain = Expected output – cost of providing effort = e  e 2
2
Take the derivative with respect to e and set equal to zero to find the optimal effort:
1–e=0  e=1
B) Suppose that the farmer enters into a sharecropping contract in which she keeps a
fraction s of the output. What effort level does she choose? (10 points)
Solution:
Since the farmer only keeps a fraction s of output:
1
Expected gain = se  e 2
2
Take the derivative with respect to e and set equal to zero to find the optimal effort:
s–e=0  e=s
C) Now suppose there are two contractual options for the tenant to choose from:
1) Rental contract: Keep all of the output, pay fixed rent of ¼
2) Sharecropping contract: Keep ½ of the output (no additional rent)
Suppose that there is no possibility of default and that the tenant has zero nonlabor
income. Which option will this risk-neutral tenant prefer? Carefully justify your answer
by finding the tenant’s utility under each contract. (10 points)
Solution:
You need to find the tenant’s utility under the two contracts
Rental contract:
Sharecropping contract (s = ½):
1
1
Expected gain = e  e 2 
2
4
1
1
1
Expected gain = se  e 2  e  e 2
2
2
2
You found in part A) that e = 1 when
the tenant keeps all the ouput.
You found in part B) that e = ½ if the
tenant keeps ½ of the output.
1
1
1 1 1
Exp. gain = 1  12   1   
2
4
2 4 4
11 11
1 1 1
Expected gain =        
22 22
4 8 8
2
1
1
under the rental contract and only under the sharecropping contract.
4
8
Therefore, the tenant will choose the rental contract.
So utility is
D) Describe how your answer to part C) might change if the tenant was risk-averse
rather than risk-neutral. Consider a more realistic environment in which the probability
of the harvest being a success is not entirely determined by the tenant’s effort, but that
there are other factors (such as the weather) beyond her control. (5 points)
Solution:
A risk-averse tenant would be more likely to choose the sharecropping contract. The
sharecropping contract ensures that the tenant will not incur a loss of ¼ when the bad
outcome of zero output occurs.