4. Consider the case of a production cooperative in which there are only two farmers. Each independently chooses independently the amount of labor he will give to the cooperative. Each unit of labor is offered with an opportunity cost of w. The production function is realized with a function of diminishing returns where the production factor is the aggregate quantity of output offered by both farmers. There is no other variable factor.

a. Plot the output and total cost as a function of the labor factor. Find graphically the amount of labor that maximizes the cooperative's profit.

b. Now, return to the problem in which labor is offered on an individual basis. Show that, if the output is shared equally between the two farmers, the output will be less than that obtained in a), show graphically.

c. Show what happens if farmer 1 gets a fraction s greater than or equal to .5, while farmer 1 gets 1-s. Everything else remains the same as before. Describe what happens to production and effort (i.e. the L each farmer contributes) as s varies between .5 and 1.

d. Now assume that each farmer faces an increasing cost g(L) instead of a constant marginal cost w. Now explain how you would find the surplus that maximizes profits and output, as you did in a), show graphically.

e. Suppose you were to redo part c) under the conditions described in d). How would your answer change?

f. Try to relate the situations posed in this problem to the Marshallian inefficiency problem of the sharecropping system.