1. Are each of the following assertions TRUE, FALSE, or UNCERTAIN?
Explain your answers with the broadest possible exception (if any), using graphs if possible.
a) If a firm is operating with constant marginal cost, it will also have constant average cost.
b) If marginal cost is increasing, average cost will also be increasing.
c) If a product requires two inputs for its production, and if the prices of the two inputs are
equal, profit maximization requires that these inputs be used in equal amounts.
d) If a firm is operating at minimum short-run average cost, it is also operating at a point on
its long-run average cost curve.
e) Long-run average cost can never exceed short-run average cost.
f) Long-run marginal cost can never exceed short-run marginal cost.
2. A firm has the following production function: Q = K1/2L1/2
It pays wages @ $5/hour, rents capital @ $10/hour, has 25 units of capital, and can sell any
amount of output $4/unit. How much output should the firm produce, with what combination
of inputs, and how much profit will the firm earn?
3. A firm has a short-run production function defined by: Q = -.02L2 + 8L.
What is the short run demand curve for labor (L) in terms of the market wage rate (w),
if the firm can sell all its output at $5 per unit?
4. A firm’s technology is defined by the production function: Q = 9K2/3L1/3.
It pays wages @ $18/hour (w) and rents capital @ $36/hour (r).
The demand for its goods is given by the demand function: Q = 240 - 10P.
a) What is the firm’s marginal rate of technical substitution (MRTSKL),
optimal condition in the product market, optimal inputs in terms of its optimal output,
cost constraint in terms of its optimal output (TC), and marginal cost function (MC)?
What is its demand function in terms of Q, its total revenue (TR) and marginal revenue
(MR) functions?
b) What is the firm’s profit-maximizing levels of price (P*), output (Q*), capital (K*),
labor (L*), and profit (?*)?
c) What is the firm’s revenue-maximizing levels of price (P), output (Q), and profit (?)?
Which of (b) or (c) is better?
5. A firm’s production function is: Q = K1/2L1/2
and the demand for its output is: Q = 300 - 10P.
a) If the firm’s wage rate is $10/hour and capital rental rate is $2.50/hour, what are its
optimal output, inputs, price, and profit?
b) If now the wage rate rises to $22.50, what will be its output, inputs, price, and profit?
c) Explain this change in terms of output and factor substitution effects.
d) Are these normal or inferior inputs?
6. A firm in a perfectly competitive industry has this cost function: TC = XXXXXXXXXX3q2
a) If market demand is QD = 1500 – 5P, what is an individual firm’s optimal price, output,
and profit? What is the industry’s total price and output? How many firms are there in
the industry?
b) Now, if demand increases to QD = XXXXXXXXXX5P, what is the short-run optimal price, output
and profit per firm, total industry output and number of firms in the industry?
c) What happens in the long-run?
Graduate-level question (to be answered by all students taking the course for graduate credit):
Show the step-by-step derivation of all results.
7. Using the Lagrange Method, minimize C = wL + rK subject to Q = A Ka Lß.
(This is a Cobb-Douglas production function, where A, a, ß are constants; this is used to
estimate demand functions and demand elasticities, and, when a + ß = 1, long-run costs.
See Ch. 7 App. in Pindyck & Rubinfeld or Ch. 6 App. in Besanko & Braeutigam for details.)
a) Find the cost-minimizing levels of capital (K*), labor (L*).
b) If there are constant returns to scale (a + ß = 1), what is the cost function?
c) What is the elasticity of substitution along a Cobb-Douglas production function?
d) Often it is impossible to estimate the production function from engineering or process
data, but economic data – costs (C), input prices (w, r) and output (Q) – is easily
available. How can we estimate the production function from this economic data?
8. What are some of the similarities between the *economic models* of the Theory
of the Consumer and the Theory of the Firm? What is the primary difference and
how do we resolve it?
9. What did I discuss about the “back office” and “front office” operations and
what results do they give? Specifically, where does each of the operations
begin, and what end-result do they give? What then must the “owner” do?