Econ. 410
Spring 2008
Tauchen/Biglaiser
Practice Problems on Long Run Costs -- Answers
Unless otherwise mentioned, assume that a firm’s technology satisfies the usual assumptions. Specifically, the marginal product for each input is positive and satisfies diminishing marginal returns. Also, the production function satisfies diminishing marginal rate of technical substitution. As usual, w denotes the price of a unit of labor and r denotes the price of a unit of capital.
1. The graph below shows the isoquants for output levels q=10, 20, 30, and 40 units of output with the isoquants for higher outputs farther from the origin. The price of labor is $2 and the price of capital is $8.
a. Construct the isocost lines and identify the cost-minimizing input combinations for producing 10, 20, 30, and 40 units of output (approximately). What is the marginal rate of technical substitution between labor and capital at each of the cost-minimizing input combinations?
Ans. The straight lines are the isocost lines. The cost minimizing input combination for each isoquant is marked with a dot. The cost minimizing input combinations are:
10 units output 2L and .5 K
20 units output 4L and 1K
30 units output 6L and 1.5 K
40 units output 8L and 2 K
The marginal rate of technical substitution between labor and capital equals minus the slope of the isoquant. At a cost-minimizing input combination, the slope of the isoquant equals the slope of the isocost line. Thus, the marginal rate of technical substitution between labor and capital at a cost minimizing input combination equals w/r which is .25 in this example.
b. What is the minimum cost at which the firm can produce 10 units, 20 units, 30 units, and 40 units of output? Ans. There are two equivalent ways to determine the LR cost. First, we could compute the cost of purchasing each of the cost minimizing bundles listed above. The LR costs for 10, 20, 30, and 40 units are $8, 16, 24, and 32 respectively. Equivalently, we compute the cost level associated with the isocost line that is tangent to the isoquant. For example, the isoquant for 10 units is tangent to the isocost line with a labor intercept of 4 and a capital intercept of 1, which is the isocost line for $8.
c. What is the LRAC for producing 10, 20, 30, and 40 units of output? Ans. The LRAC for each of these quantities is $.80 .
d. The isoquant map is consistent with diminishing marginal returns to labor. Suppose, for example, that the amount of capital is fixed at one unit.
Increasing output by 10 units from 10 to 20 units requires about 3 more units of labor (from 1 to 4 units). Increasing output by 10 units from 20 to 30 units requires about 5 more units of labor.
Increasing output by 10 units from 30 to 40 units requires about 7 more units of labor.
Thus, as labor use increases, the amount of labor required to increase output by 10 units increases.
The same reasoning applies for capital fixed at some other value.
2. Suppose now that the firm’s capital is fixed at the optimal level of capital for producing 30 units of output. The input prices are again w=$2 and r=$8. What is the short-run cost of producing 10, 20, 30, and 40 units of output with the fixed capital stock?
Ans. The cost-minimizing input combination for producing 30 units of output is 6 labor and 1.5 capital. We can read the isoquant graph to determine (approximately) the amount of capital required to produce each quantity with the amount of capital fixed at 1 unit.
With 1.5 units of capital, 2/3 of a unit of labor is required to produce 10 units of output. The cost of this input combination is $13.33. The SR average total cost for producing 10 units of output is $1.33
With 1.5 units of capital, 2 2/3 units of labor are required to produce 20 units of output. The cost of this input combination is $17.33. The SR average total cost for producing 20 units of output is $.87
With 1.5 units of capital, 6 units of labor are required to produce 30 units of output. The cost of this input combination is $24. The SR average total cost is $.80
With 1.5 units of capital, 10 2/3 units of labor are required to produce 40 units of output. The cost of this input combination is $33.33. The SR average total cost is $.83
3 Graph the LRAC from question #1 and the short-run ATC for a firm that has the amount of capital that is optimal for producing 20 units of output.
4. The graph below shows the same isoquant map as for the previous problems. The isoquants are for output levels q=10, 20, 30, and 40 units of output with the isoquants for higher outputs farther from the origin. Construct the isocost line that allows you to identify the cost minimizing input combination for producing 20 units of output with the input prices of w=$2 and r=$8. What is the cost associated with this isocost line? Let’s refer to this cost as C2.
Now suppose that the price of labor increases to $8. Construct the isocost line for the new input price and for cost C2 . Is it possible to produce 20 units of output with any of the input combinations on this isocost line? Does the cost or producing 20 units of output necessarily increase as the price of one input increases (assuming that the firm uses both inputs)?
Ans. Initially the input prices are w=$2 and r=$8. The graph below shows the lowest isocost line for which it is possible to produce 20 units with w=$2 and r=$8. This is the isocost line with a labor intercept of 8 and a capital intercept of 2. It is the isocost line for $16 at the initial input prices..
The price of labor then increases to w=$8. The isocost line for $16 with w=$8 and r=$8 has a labor intercept of 2 and a capital intercept of 2. The firm cannot produce 20 units of output using any of the input combinations on this line. Keep in mind that there is an entire family of isocost lines for the new input prices. In order to produce 20 units of output at the new input prices, the firm must move to an isocost line for a cost level greater than $16.
5. In the previous problem, you used a specific isoquant map and numerical values for the input prices to show that the cost of producing 20 units of output increases if the price of one of the inputs increases (assuming that the firm uses both inputs). Use the same line of reasoning to show that the cost of producing any output level increases if one of the input prices increases (assuming that the firm uses both inputs).
Ans. We can use the same type graph to reason that an increase in one of the input prices necessarily increases the cost of producing output Q even without using specific numerical values. (We are assuming the case in which the firm uses positive amounts of both inputs.) The graph below shows the isoquant for output Q. The graph also chows the lowest attainable isocost line at the initial inputs prices. The cost level for this isocost is denoted c.
Now suppose that the price of labor increases. The price of capital remains constant. The new isocost line for cost level c has the same vertical intercept as the initial isocost line for cost level c, but the new isocost line is steeper. This isocost line is shown as the dotted isocost line on the graph. With the new higher price of labor, none of the input combinations on the isocost line for cost level c (dotted line) allows the firm to produce output Q. The firm must move to a higher isocost line in order to produce output Q.
6. The graph below shows the same isoquant map as for problem #1.
a. Show the expansion path for w=$1 and r=$8.
b. Show the expansion path for w=$4 and r=$32.
c. Show the expansion path for w=$2 and r=$1.
d. Provide an intuitive explanation for the differences and similarities in the expansion paths for parts a., b., and c.
Ans. a. and b. The graph below shows the isoquant map with isoquants for output 10, 20, 30, and 40. The isocost lines are the straight lines with slope = -1/8 for both a. and b. Although the slopes of the isocost lines are the same for a. and b., the cost level associated with each of the isocost lines shown on the graph differs for a. and b. For example, the straight line with a labor intercept of 16 and a capital intercept of 2 is an isocost line for both a and b. The line is the isocost line for cost level $16 for a., and the isocost line for cost level $64 for b.
The cost-minimizing input combinations for output 10, 20, 30, and 40 are marked with dots. If we had the isoquants for more output levels (11, 12, 13,…..) we could construct the entire expansion line shown.
c. The expansion path for w=$2 and r=$1 is shown below.
d. Since the isocost lines for a. and b. are the same, the expansion path for a. and b. are the same. The relative price of labor is higher in part c. than for parts a. and b. The bundles on the expansion path for part c. involve relatively more capital and less labor than those on the expansion path for parts a. and b.
7. A firm’s production function is Q=K1/3 L1/3 . The price of a unit of capital is r and of a unit of labor is w.
a. Determine the cost-minimizing input combination for producing Q units of output.
The two conditions for the cost minimizing input combination (L,K) for producing output Q at input prices w and r are that
(i) it is possible to produce Q units of output with the input combination (L,K) or Q=F(L,K)
(ii) the MRTS between labor and capital at the input combination (L,K) equals w/r. Since the MRTS between labor and capital at the input combination (L,K) equals the ratio of the MP of labor to the MP of capital at (L,K) the conditions can be written as MPL(L,K)/MPK(L,K)=w/r.
We need to compute both the MP of labor and of capital. To find the MP of labor, we take the derivative of K1/3 L1/3 with respect to L. In taking the derivative with respect to L, we treat the term involving capital as a constant. Thus the MP of labor is (1/3)*K1/3 L-2/3 . Similarly, the MP of capital is (1/3)*K-2/3 L1/3 . The MRTS between labor and capital is
(1/3)*K1/3 L-2/3 / = / K1/3 L-2/3 / = / K(1/3)*K-2/3 L1/3 / K-2/3 L1/3 / L
For the production function given in this problem, the two conditions for identifying the cost-minimizing input combination for producing Q units of output at input prices w and r can be written as
(i) Q=K1/3 L1/3
(ii) K/L=w/r
We need to use any correct algebraic steps in order to solve for K and L. Let’s begin by solving the second equation for K to find K=wL/r. Then substitute this expression for K into the first equation to obtain
Q= (w/r) 1/3 L1/3 L1/3 .
Combining the two L terms (and remembering that xb xa = xa+b ) ,
Q= (w/r) 1/3 L2/3 .
Now solve for L. The first step is that
L2/3 = Q/(w/r) 1/3
or
L2/3 = Q(r/w) 1/3
Raising both sides of the equality to the 3/2 power yields
L= Q3/2 (r/w) 1/2.
Since K= wL/r,
K= Q3/2 (w/r) 1/2.
b. Determine the firm’s long run cost function. Also determine the firm’s long run average and marginal cost curves
The firm’s long run cost for producing output Q with input price w and r is
C(Q,w,r)= w * cost minimizing amount of L for producing output Q at input price w and r
+ r * cost minimizing amount of K for producing output Q at input price w and r.
Substitute the cost minimizing amount of labor and capital derived in part a. to obtain.
C(Q,w,r) = w * Q3/2 (r/w) 1/2 + r* Q3/2 (w/r) 1/2 ,
which equals 2 (wr) 1/2 Q3/2 .
To find the LRAC divide the cost by Q to obtain LRAC(Q,w,r)= 2 (wr) 1/2 Q1/2 . To find the LRMC take the derivative of LR cost with respect to Q to obtain
LRMC(Q,w,r) = 3 (wr) 1/2 Q1/2 .
Notice that the LRAC is increasing with Q. And, the LRMC for any Q is a larger number than LRAC. Or, in other words, the LRMC curve lies above the LRAC curve.
8. A firm selects its inputs to minimize the cost of producing its output. Recently, the firm increased each of its inputs by 10 percent. As a result, its output increased from 100 units to 120 units. Indicate whether each of the following statements is true, false, or uncertain. Explain your answer.
a. At q=100, there are increasing returns to scale.
Ans. True. The firm increased its inputs by 10 percent and output increased by 20 percent. The firm’s production function exhibits increasing returns.
b. At q=100, the LRAC curve is downward sloping.
Ans. True. If the production function exhibits increasing returns, the LRAC is downward sloping.
c. At q=100, LRMC exceeds LRAC.
Ans. False. If the LRAC curve is downward sloping, then the LRMC curve lies below it.
9. A firm uses two inputs – capital and labor. When w=$10 and r=$10, the cost-minimizing input combination for producing 100 units of output is 20 units of labor and 20 units of capital. When w=$8 and r=$10, the cost-minimizing bundle for 100 units of output is 28 units of labor and 19 units of capital. Indicate whether each of the following statements is true, false, or uncertain. Explain your answer.