Suppose the process of producing lightweight parkas by Polly’s Parkas is describ

Question

Suppose the process of producing lightweight parkas by Polly’s Parkas is described by the function:

q = 40K0.6 (L-40)0.4

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.

Note that q/K = 40(0.6) K -0.40 (L-40)0.4   and   q/L = 40K 0.6 (0.4) (L-40) -0.60

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 of machines (r).

The cost-minimizing demands for K and L are:

______________________________________

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. The total cost function (TC) is:

________________________________________

2. This process requires skilled workers, who earn $64 per hour. The rental rate on the machines used in the process is $16 per hour. At these factor prices what are total costs as a function of q.

Total costs (TC) are:

__________________________________________

Does the technology exhibit decreasing, constant, or increasing returns to scale?

3. 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?

The firm should use ___ worker hours.

The firm should use ___ machine hours.

What are the marginal and average costs at this level of production?

Marginal cost is $___ per unit of output.

Average cost is $___ .

Explanation / Answer

(a)

Q = 40K0.6 (L - 40)0.4

We have to minimize C = 10 + wL + rK

Subject to Q = 40K0.6 (L - 40)0.4

We form the Lagrangian function:

Z = 10 + wL + rK + [Q - 40K0.6 (L - 40)0.4]

The first order conditions for this minimization problem is:

Z / L = w - 40K 0.6 (0.4) (L-40) -0.60 = 0

Z / K = r - 40(0.6) K -0.40 (L-40)0.4 = 0

Z / = Q - 40K0.6 (L - 40)0.4 = 0

Or,

w = . MPL

r = . MPK

Q = 40K0.6 (L - 40)0.4

Or,

(w / r) = (MPL / MPK)

(w / r) = (0.4 / 0.6) x [K / (L – 40)]

Or,

K = (3 / 2) x (w / r) x (L – 40)   ..... (1)

And

(L – 40) = (2 / 3) x (r / w) x K   ...... (2)

(b)

TC = 10 + wL + rK

= 10 + wL + [r x 1.5 x (w / r) x (L – 40)]                                

TC = 10 + wL + 1.5wL – 60w

TC = 10 + 2.5w (L – 24)

(2)

W = 64, r = 16

K = (3 / 2) x (w / r) x (L – 40)

Or, K = 1.5 x 4 x (L – 40)

Or, K = 6 x (L – 40)

Or, (L – 40) = K / 6

Substituting in production function, we get

Q = 40K0.6 (L - 40)0.4

= 40K0.6 (K / 6)0.4

= 40 x K / [(6)0.4] = 25.56K

Or, K = 0.04Q

Then, L = (K / 6) + 40 = (0.04Q / 6) + 40

L = 0.006Q + 40

So, total cost TC = 10 + wL + rK

= 10 + [64 x (0.006Q + 40)] + (16 x 0.04Q)

= 10 + 0.42Q + 2,560 + 0.64Q

TC = 2,570 + 1.06Q

NOTE: Out of 3 questions, the 1st two have been answered.

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