A firm produces a product with labor and capital, and its production function is described by Q = LK. Suppose that the price of labor equals 2 and the price of capital equals 1. Derive the equations for the long-run total cost curve and the long-run average cost curve.

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Joshua StredderLv10

28 Sep 2019

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A fi? rm produces a product with labor and capital. Its production function
is described by Q = 2L + 3K: Let w and r be the prices of labor and capital,
respectively.

(a) Find the equation for the? firm's long-run total cost curves as a function
of quantity Q, for input prices, w = 1 and r = 1.

(b) Find the equation for the firm's long-run total cost curves as a function of
quantity Q and input prices, w and r (and so as a function of three parameters).

c) Find the solution to the ?firm's short-run cost-minimizationproblem when
capital is fi?xed at a quantity of 5 units (i.e., K = 5), and w = 1and r = 1.
Derive the equation for the? firm's short-run total cost curve as a function of
quantity Q. Graph this curve together with the long-run total cost curve for
the same input prices.

(d) Redo parts a) and c) for w = 1 and r = 2:

peachhound332

The daily demand for travel at a small regional airport is initially QD = 1000 - p; where quantity is in passengers and price is in dollars per passenger. There are initially 10 price-taking firms serving this market and each of them has the production function: q = sqrt (lk) where q is in passengers, and labor and capital are in some appropriately specified units. The price of a unit of labor is w = 100 per day; the price of a unit of capital is v = 100 per day, and this industry has constant costs. Each firm initially has a capital stock of k0 = 80. day; and this industry has constant costs. Each firm initially has a capital stock of k0 = 80.

(a) Derive the short-run cost function, SC(q), for each firm in the market.

(b) Derive the short-run market supply function, QS.

(c) Find the short-run competitive equilibrium price, p*, and quantity, Q*. How much labor, L*, does the industry employ at the competitive equilibrium?

(d) Derive the long-run cost function, C(q), of each firm in the industry.

(e) Describe the long-run market supply curve. Does the short-run competitive equilibrium at (Q*; p*) also correspond to a long-run competitive equilibrium? Explain why or why not.

Now, imagine that a catastrophic event causes the daily demand for air travel in this market to fall unexpectedly to ~QD = 750 - p Continue to assume that each of the 10 firms has a capital of k0 = 80 in the short run.

(f) Find the new short-run competitive equilibrium price, p**, and quantity, Q**, in this market. How much labor, L**, does the industry employ in the new short-run competitive equilibrium?

(g) What is the new long-run competitive equilibrium price, p***, and quantity, Q***? Describe what must happen as the industry moves from the short-run competitive equilibrium at (Q**; p**) to the long-run competitive equilibrium at (Q***; p***). How much labor L*** does the industry employ at the new long-run equilibrium?