A family farms corn on their land. They must decide how to produce corn: they can harvest the crop entirely with hours of their own labor (z1), they can harvest entirely by machine (z2), or they can use a combination of labor hours and machine hours. Using only labor is very time-consuming and results in waste because it is impossible to harvest all the corn by hand before it begins to spoil. Using only machine is also problematic because the machines can’t access some parts of the farm. The production function relates the amount of each input used to total corn production y. The production function is y = f(z1, z2) = z 1^1/2 z2^1/2 . The firm faces the following prices. The price of corn p is $3/pound. The price of an hour of labor is $w1 = 1. The price of a machine hour is $w2 = 4.
(a) Does the firm experience diminishing marginal product of labor? Solve for the expression for marginal product of labor.
(b) In the short run, the farm’s number of machine hours available is fixed. The farm rents machine time from an equipment rental company and has signed a contract for z2 = 64 machine hours. How many pounds of corn y should the farm produce and how many hours of labor z1 should the farm use to produce it?
(c) Graph the firm’s short-run production function and isoprofit lines. Show the profit-maximizing production point on the graph.
(d) What are the slopes of the isoprofit lines associated with this problem?
(e) How would an increase in the output price p affect the optimal choice of input use in the short run? Explain and show with a graph