CHAPTER 7 APPENDIX
PRODUCTION AND COST THEORY—
A MATHEMATICAL TREATMENT EXERCISES
1. Of the following production functions, which exhibit increasing, constant, or decreasing returns to scale?
a. F(K, L) = KL
b. F(K, L) = 10K + 5L
c. F(K, L) = (KL)
Returns to scale refer to the relationship between output and proportional increases in
all inputs. This is represented in the following manner (with λ > 0):
F(λK , λL ) > λF(K, L) implies increasing returns to scale;
F(λK , λL ) = λF(K, L) implies constant returns to scale; and
F(λK , λL )
a. Applying this to F(K, L) = K L ,
F(λK , λL ) = (λK ) (λL ) = λK L = λF(K, L).
This is greater than λF(K, L); therefore, this production function exhibits increasing returns to scale.
b. Applying the same technique to F(K, L) = 10K + 5L ,
F(λK , λL ) = 10λK + 5λL = λF(K, L).
This production function exhibits constant returns to scale.
c. Applying the same technique to F(K, L ) = (KL ) ,
F(λK , λL ) = (λK λL ) = (λ) (KL ) = λ(KL ) = λF(K, L).
This production function exhibits constant returns to scale.
2. The production function for a product is given by q = 100KL. If the price of capital is $120 per day and the price of labor $30 per day, what is the minimum cost of producing 1000 units of output?
The cost-minimizing combination of capital and labor is the one where
MRTS =MP L w =. K .52.5 .5.5.52 3232.52
The marginal product of labor is
∂q =100L . Therefore, the marginal rate of technical substitution is ∂K
100K K =. ∂q =100K . The marginal product of capital is ∂L
To determine the optimal capital-labor ratio set the marginal rate of technical substitution equal to the ratio of the wage rate to the rental rate of capital:
K 30=, or L = 4K . L 120
Substitute for L in the production function and solve for K when output is 1000 units:
1000 = (100)(K )(4K ), or K = 1.58.
Because L equals 4K this means L equals 6.32.
With these levels of the two inputs, total cost is:
TC = wL + rK, or
TC = (30)(6.32) + (120)(1.58) = $379.20.
3. Suppose a production function is given by F(K, L) = KL; the price of capital is $10 and the price of labor $15. What combination of labor and capital minimizes the cost of producing any given output? The cost-minimizing combination of capital and labor is the one where
MRTS =MP L w =. MP K r 2
The marginal product of labor is
∂q =L 2. K ∂q =2KL . The marginal product of capital is ∂L
Set the marginal rate of technical substitution equal to the input price ratio to determine the optimal capital-labor ratio:
2KL
L =15, or K/L = 0.75. 10
Therefore, the capital-labor ratio should be 0.75 to minimize the cost of producing any given output.
4. Suppose the process of producing lightweight parkas by Polly’s Parkas is described by the function
q = 10K (L – 40)
where q is the number of parkas produced, K the number of computerized stitching-machine hours, and L the number of person-hours of labor. In addition to capital and labor, $10 worth of raw materials is used in the production of each parka. .8.2
We are given the production function: q = F(K,L) = 10K(L – 40)
We also know that the cost of production, in addition to the cost of capital and labor, includes $10 of raw material per unit of output. This yields the following total cost function:
TC(q) = wL + rK + 10q .8.2
a. By minimizing cost subject to the production function, derive the cost-minimizing
demands for K and L as a function of output (q ), wage rates (w ), and rental rates on machines (r ). Use these results to derive the total cost function: that is, costs as a function of q , r , w , and the constant $10 per unit materials cost.
We need to find the combinations of K and L that will minimize this cost function for
any given level of output q and factor prices r and w . To do this, we set up the Lagrangian:
Φ = wL + rK + 10q – λ[10K (L – 40) – q]
Differentiating with respect to K, L, and λ, and setting the derivatives equal to zero:
(1)
(2)
(3) .8.2∂Φ−.2.2=r −10λ(.8)K (L −40) =0 ∂K .8−.8=w −10λK (.2)(L −40) =0 ∂L ∂Φ=10K .8(L −40) .2−q =0. ∂Note that (3) has been multiplied by –1. The first 2 equations imply:
r =10λ(.8)K −.2(L −40) .2 and w =10λK .8(.2)(L −40) −.8.
or
r 4(L −40) =. w K
This further implies:
K =4w (L −40) rK and L-40=. r 4w
Substituting the above equations for K and L–40 into equation (3) yields solutions for K and L:
⎛ 4w ⎞ ⎛ rK ⎞ q =10(L −40) .8(L −40) .2 and q=10K .8. ⎝r ⎠ ⎝4w
or .8.2
r . 8q w . 2q L =+40 and K =. 30. 3w 7. 6r We can now obtain the total cost function in terms of only r, w, and q by substituting these cost-minimizing values for K and L into the total cost function:
TC (q ) =wL +rK +10q
wr . 8q rw . 2q TC (q ) =+40w ++10q 30. 3w . 87. 6r . 2 . 2. 8. 8. 2w r q r w q TC (q ) =+40w ++10q . 30. 37. 6
b. This process requires skilled workers, who earn $32 per hour. The rental rate on
the machines used in the process is $64 per hour. At these factor prices, what are total costs as a function of q ? Does this technology exhibit decreasing, constant, or increasing returns to scale?
Given the values w = 32 and r = 64, the total cost function becomes:
TC(q) = 19.2q + 1280.
The average cost function is then given by
AC(q) = 19.2 + 1280/q.
To find returns to scale, choose an input combination and find the level of output, and then double all inputs and compare the new and old output levels. Assume K = 50 and L = 60. Then q1 = 10(50)0.8(60 – 40)0.2 = 416.3. When K = 100 and L = 120, q2
= 10(100)0.8(120 – 40)0.2 = 956.4. Since q2/q1 > 2, the production function exhibits
increasing returns to scale.
c. Polly’s Parkas plans to produce 2000 parkas per week. At the factor prices given
above, how many workers should the firm hire (at 40 hours per week) and how many machines should it rent (at 40 machine-hours per week)? What are the marginal and average costs at this level of production?
Given q = 2000 per week, we can calculate the required amount of inputs K and L using the formulas derived in part a: w . 2q r . 8q L =+40 and K =. 7. 6r 30. 3w Thus L = 154.9 worker hours and K = 229.1 machine hours. Assuming a 40 hour week, L = 154.9/40 = 3.87 workers per week, and K = 229.1/40 = 5.73 machines per week. Polly’s Parkas should hire 4 workers and rent 6 machines per week, assuming she cannot hire fractional workers and machines.
We know that the total cost and average cost functions are given by:
TC(q) = 19.2q + 1280 AC(q) = 19.2 + 1280/q ,
so the marginal cost function is
MC(q) = dTC (q ) = 19.2. dq
Marginal costs are constant at $19.20 per parka and average costs are 19.2 + 1280/2000 or $19.84 per parka.