Intermediate Microeconomics Problem Set #4:
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1. A poncho company has the following short-run production function:
Q (L) = 100L+ 5L^2
The cost of capital in the short-run is $1,000 per day and the cost of labor is $100
(a) What is the marginal product of labor whenL = 50?
(b) Invert the production function to find the number of workers needed per day
to produce Q ponchos. (Note: you will need the quadratic equation here. Be
(c) Use this to derive the formula for total cost (as a function of output
Q). What is the fixed cost? What is the variable cost?
(d) Find the marginal cost of producing a poncho.
(e) If L = 50, what is the level of production? What is marginal cost?
(f) Verify that, at L = 50, MC = w/MPL and explain in words what this means.
2. Consider the supply decisions of a firm with SR cost function
C(q) = 1200 +(1/25)q^4
(a) Write out equations for marginal cost (MC), average total cost (ATC), and
average variable cost (AVC), and show these on a properly-labeled graph.
(b) What is the lowest possible ATC? At what level of production (q) does the
lowest ATC occur?
(c) What is the lowest possible AVC? At what level of production (q) does the
lowest AVC occur?
(d) At P = 250, calculate how much this firm will supply in the short run by
equating price marginal cost and solving for quantity q. Will the firm make a
profit, a loss, or neither? (If a profit or a loss, calculate its size.) Show this
point on the graph of the average and marginal cost curves, and indicate the
area corresponding to the firm’s profit or loss.
(e) Write down the equation for the firm’s short-run supply function in the form
q(P): i.e., for every price, the quantity that the firm will supply as a function
of the market price.
3. A raincoat producer has short-run cost function
C(q) = 50 +q+(1/10)*q^2
(a) Show the firm’s marginal and average cost curves on a properly-labeled graph.
(b) If the price of a raincoat is $4, how much will the firm supply? Will it make
a profit, a loss, or neither? (If a profit or a loss, calculate its size.)
(c) Is there a price below which the firm won’t supply any quantity even in the
short run? Explain.
(d) Equate price and marginal cost to find the firm’s short run supply function
q(p) and graph it on a well-labeled graph.
(e) If the market consists of 100 firms just like this one, find and graph the short-
run industry supply curve.
(f) Suppose that this firm has constant returns to scale; what does this tell you
about the shape of its LRAC curve? What about the firm’s LR supply curve?
(g) Is the industry in long-run equilibrium? How do you know? If yes, explain
why. If no, find the price associated with long-run equilibrium.
(h) In the long-run equilibrium, will there be more or fewer than the current 100
firms in the market? Explain.
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