Economics 207
Intermediate Microeconomics
Spring 2007
Dr. Léonie L. Stone
Homework 4
1. Based on the following statement, draw the short-run marginal product, average product, and total product curves
AND the total cost, variable cost, marginal cost, average variable cost and average total cost curve.
“The firm initially experiences increasing returns to the variable factor. After the 50th worker, the marginal product
of labor is constant until the firm reaches the 100th worker. After that point, there are diminishing returns.”
SEE GRAPH SHEET.
2. A firm has the production function Q = K1/2L1/2, and is currently using 100 units of labor and 100 units of capital.
The wage rate and the rental rate on capital are both $10. The expansion path for this firm is K=L.
a. Draw an isoquant map showing the current production of the firm and the isocost curve on which the
firm is currently producing. Carefully label EVERYTHING. What quantity is the firm producing, and
what is the total cost of the firm at this moment?
SEE GRAPH SHEET.
The firm is currently producing Q = 1001/21001/2 = 100 units.
TC = wL + rK = 10L + 10K = 10(100) + 10(100) = $2000
b. Suppose that in the short run, the firm wishes to increase its production to 225. Draw a NEW isoquant
map showing (1) the position you showed in (a), (2) how the firm would expand production in the short
run, and the related cost, and (3) how the firm would expand production in the long run, and the related
cost. (Note: you should be able to figure out the exact point for part (3), but not necessarily for part (2).)
SEE GRAPH SHEET.
In the short run, capital is fixed at 100. So to produce more, the firm can ONLY use labor.
Q = K1/2L1/2 = 1001/2L1/2 = 10 L1/2
Set Q=225, and then solve for L. 225 =10 L1/2 , so L = 506.25 (round to nearest whole number)
TC = 10(506.25) + 10(100) = $6,062.50
(If you did not use exact numbers here, that’s ok, but your graph must show that you would need to
use a LOT more labor, and that the isocost curve you would be on would be far higher than the one
in the long-run part of the problem, below. For the exam, you should be able to find the short-run
point numerically.)
In the long run, the firm will increase output along the expansion path, K=L. Thus to produce 225
units, you need 225K and 225L, which will cost $4,500, far less than the short-run cost. (You can find
the quantities of K and L by setting K=L in the production function, setting Q = 225, and solving for
either K or L.)
NOTE: The endpoints of the isocosts should really be labeled; I just ran out of room!
c. Using your results for b., draw a graph of the approximate shape of the firm’s short and long-run total
cost curves as a function of Q. (Hint: This is a constant returns to scale production function, and thus the
long run cost curve must be a straight line.)
SEE GRAPH SHEET.
This is a constant returns to scale function, so the long-run total cost curve must be a straight line.
Since in the long run, K=L along the expansion path, for every additional unit you produce, you will
need one more K and one more L. Since the cost of each is $10, each additional unit will cost you
$20. Thus LTC = 20Q (and LMC = LAC = $20).
In the short run, there is a fixed cost. For the original point, K is fixed at 100, so there is a fixed cost
of $1000. Your short-run curve should start at this point. As you can see below, the shape of the
STC curve is actually quadratic, but if you drew it as a cubic function, that’s ok, too, since the
problem just says, approximate shape. Either way, the short-run curve is tangent to the long run
curve at the optimal long-run points. Each level of fixed cost is associated with a different short-run
total cost curve.
How to find the short-run total cost curve mathematically:
Q = 10 L1/2 (from b. above), so L = Q2/10
TC = 10 L + 10 K = 10L + $1000
Substitute from above.
TC = 10 (Q2/10) + 1000 = Q2 + 1000
(This is the STC curve that corresponds with K=100).
d. Use your graph for c. to explain why we say that the long run cost curve is the envelope of the short run
cost curves.
See the graphs drawn for c. The long-run total cost curve is tangent to the short run total cost curves
at optimal points, and these points are the lowest costs of producing that particular output. Thus the
LR curve envelops the short-run curve. You can have higher costs in the short run, but you could
never have lower costs than you would in the long run (when you have adjusted everything
optimally).
3. For each of the following statement, explain or demonstrate whether it is true, false, or uncertain.
a. If marginal product is increasing, marginal cost must be increasing.
FALSE. MC = w/MP, so if MP is rising, MC must be falling.
b. If marginal cost is increasing, average total cost must be increasing.
FALSE. SEE GRAPH. As long as MC is less than ATC, ATC is falling, even when MC is upwardsloping.
c. If a firm has diminishing returns to the variable factor in the short run, it will have diseconomies of scale
in the long run.
FALSE OR UNCERTAIN. These concepts are unrelated. Diminishing returns refers to declining
MP in the short run, the result of the increasing scarcity of the fixed input (capital). Diseconomies of
scale refers to rising long-run average cost, due to either decreasing returns to scale or other
coordination problems. All firms have diminishing returns in the short run. This is unconnected to
whether there are diseconomies in the long run.
Chapter 6 and 7 concepts that you should know for the exam: formulas for total, marginal, average product;
formulas for total, marginal, average variable and average total cost; how to find all of those things from a table; the
relationship between marginal things and average things; the relationship between product curves and cost curves;
how to draw an isoquant map and show the cost-minimizing point; the effect of a change in the cost of factors of
production; the expansion path; returns to scale; diminishing marginal returns; economies of scale; economies of
scope; the difference between returns to scale, economies of scale, marginal returns, and economies of scope; the
relationship between short and long run cost curves; the difference between minimizing cost in the short run and
minimizing cost in the long run.