Exercise Set 6.5
Optimal Input

1. A firm's production function is given by Q =3L0.6, where L is labor (number of workers) and Q is output. The firm operates in a perfectly competitive market. It hires workers at a wage w, and sells its output at a price of P.
  1. Does labor exhibit diminishing returns? Explain.
  2. Obtain the expression for MPL.
  3. Noting that the firm's marginal revenue revenue product of labor is the additional revenue obtained by adding 1 more worker, derive the expression for MRPL.
  4. Obtain the firm's profit function in terms of labor, π(L).
  5. Is the profit function (strictly) concave or (strictly) convex?
  6. Use the FOC to obtain the firm's optimal amount of labor in terms of w and P. Call it L*(w,P).
  7. Is the SOC for a maximum satisfied at L*?
  8. At the optimal point, what is the relationship between MPL and real wage? Between MRPL and nominal wage?
  9. Use the partial derivatives of L*(w,P) to show the following:
    1. An increase the the wage rate, ceteris paribus, will lead the firm to hire fewer workers.
    2. An increase the the price of the good, ceteris paribus, will lead the firm to hire more workers.
  10. Select suitable values for the wage rate and price.
    1. Obtain the profit function.
    2. Calculate the optimal value of labor.
    3. Sketch the production function. Indicate L* on the graph.
    4. Show that the slope of the production function at L* equals the real wage (w/P).
    5. At the optimal point, what is the relationship between MPL and real wage? Between MRPL and nominal wage?
    6. Provide an economic interpretation of your finding in (iv).

2. Consider the more general form of the problem in Q. 1. A firm's production function is given by Q = AF(L), where L is labor (number of workers) and Q is output. The firm operates in a perfectly competitive market. It hires workers at a wage w, and sells its output at a price of P.
  1. Obtain the firm's profit function in terms of labor, π(L).
  2. Under what assumptions on the production function will the profit function be strictly concave? Assume that these conditions hold.
  3. Use the FOC to obtain PAF'(L) = w. Call this equation (1).
  4. Check to see that the SOC for a maximum is satisfied.
  5. Equation (1) can be used to solve for the optimal value of labor L* in terms of the parameters P, A and w. Differentiate equation (1) totally and obtain the signs for the following: dL*/dw, dL*/dP, dL*/dA. Interpret each of the signs.
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