Problem Set 3 – Econ 203 Intermediate Microeconomics
Please show all the work you do to solve a problem. (Total 40 Points)
1. The figure below shows a firm’s isoquants for Q = 1, 2, 3, & 4. Labor is on the
horizontal axis and capital is on the vertical axis. The firm’s production function
is Q = f(K,L) = K0.25L0.25. While working, you can fill out the table on the next
page to help yourself, but it will not be graded. (12 pts)
a. Show that the marginal rate of technical substitution is !
!
. (1 pt)
b. Suppose that the wage rate is w = 10 and the rental price of capital is r = 10.
Graph the four isocost lines tangent to the four isoquants. (1 pt)
c. Set the MRTS equal to the input price ratio to find the long run output expansion
path (K* as a function of L*
) and plot it on the graph. (2 pts)
d. What are the cost-minimizing K* & L*
, as functions of Q? (1 pt)
e. What is the long run total cost function? (Hint: wL* + rK*
) (1 pt)
f. Suppose in the short run, the firm is stuck at K0 = 9 units of capital. Graph the
short-run expansion path. (1 pt)
g. What is the firm’s short run production function? (1 pt)
h. Rearrange the short run production function to find L as a function of Q. (1 pts)
i. Find the firm’s fixed costs, variable costs, and short run total costs as functions of
Q. (3 pts)
Long Run Short Run
Q K* L* LRTC K0 L FC VC SRTC
0 9
1 9
2 9 16/9 90 160/9 90 + 160/9
3 9
4 16 16 320 9
2. A firm’s short run marginal cost, average total cost, and average variable cost
curves are graphed below. (16 pts)
a. The firm’s short run total cost function is ???? ? = !
!”# ?! ? ?! + 50? +
1000. Find SRMC, ATC, and AVC as functions of Q. (Use a graphing calculator,
Grapher, Wolfram Alpha, etc. to verify your equations match the graph.) (3 pts)
b. Suppose the perfectly competitive market price for the firm’s good is ?!
? = 50.
Find the firm’s profit-maximizing quantity Q* and mark it on the graph. (2 pts)
c. Calculate the firm’s profit, and shade in the rectangle with area representing the
firm’s profit on the graph. (Hint: ? ?? = ?? ? ?? = ?!
??? ? ???? ?? =
?!
? ? ??? ?? ? ??, so you want the rectangle with height ?!
? ? ??? ?? and
width ??. You should also estimate the area of the rectangle to check your work.)
(2 pts)
d. Suppose the perfectly competitive market price for the firm’s good is ?!
? = 32.5.
Find the firm’s profit-maximizing quantity Q* and mark it on the graph. (2 pts)
e. Is the firm making positive or negative profit? Calculate it and shade in the
rectangle representing the firm’s “profit”. (2 pts)
f. Calculate what the firm’s profit would be if it produced Q = 0. (1 pt)
g. In general, what condition must hold for a firm to produce output even if it is
losing money? What is the lowest price at which this firm will choose to operate?
(2 pts)
h. Would a firm ever operate at a loss in the long run? Why or why not? (2 pts)
3. Suppose a firm’s long run marginal cost and long run average cost curves are as
shown on the following graph. (12 pts)
a. Suppose the perfectly competitive market price for the firm’s good is ?!
? = 4. Is
the firm’s profit positive or negative? (1 pt)
b. Assuming free entry and exit, what happens over time? (Hint: Think about what
incentives this creates for other entrepreneurs & investors.) What does this do to
the market supply curve? How does this change ?!
?? (2 pts)
c. Suppose the perfectly competitive market price for the firm’s good is ?!
? = 1.75.
Is the firm’s profit positive or negative? (1 pt)
d. Assuming free entry and exit, what happens over time? What does this do to the
market supply curve? How does this change ?!
?? (2 pts)
e. Assuming all firms have this identical production function, what value does ?!
?
approach in the long run? What will the firm’s profit be? (2 pts)
f. Does this imply there is no incentive to innovate? What happens to the average
cost curve of a firm that invests to develop some new production process? What
happens to its profits? What do other firms do in the long run? (4 pts)
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