Section II (Answer all in this section) 3) Murray manufacturing company produces

Question

Section II (Answer all in this section) 3) Murray manufacturing company produces pantyhose. The firms production function is given as where Q = pairs ofpantyhose, L = labor measured in person hours, and K-capital measured in machine Murray's labor cost including fringe benefits is $20 per hour, while the firm uses $80 per hour as an impl machine rental charge per hour. Murray's current budget is $64,000 per month to pay labor and capital (20 points) (a) Given the information above, determine Murray's optimal capital/labor ratio. u), 20 (04,000 r 80 MPL 5 k8y 12D 0 80 4

Explanation / Answer

The production function is given as Q = 5L1/4K1/4. The marginal product function of labor is the partial derivative of the production function with respect to labor. This implies

MPL   = dQ/dL

            = d(5LK)/dL

            = 5K and

Similarly, the marginal product function of capital is the partial derivative of the production function with respect to capital

MPK    = dQ/dK

            = d(5LK)/dK

            = 5L

Note that marginal rate of technical substitution RTS is the ratio of marginal products of labor and capital

At equilibrium level, MRTS is equal to w/r

MPL /MPK= w/r

[5K] / [5L] = w/r

K/L = w/r

K/L= (20/80) = 1/4

Thus the optimal capital labor ratio is 1/4

B) Cost structure is given by C = wL + rK. With w = 20 and r = 80, cost function is 64000 = 20L + 80K or 3200 = L + 4K

The equation found above indicates the input mix to be used to produce profit maximizing level of output

To maximize output using the given cost function, the optimal quantities can be found using Lagrangian method:

Maximize Z = 5LK with respect to 3200 = L + 4K

So the Lagrangian equation is:

Z = 5LK + (3200 - L - 4K)

Finding the partial derivatives and setting them equal to zero gives:

Z'(L) = 0

5K - = 0

K = /5

Z'(K) = 0

5L - 4 = 0

L = 4/5

From these two values, we again find that:

K/L = 1/4

K = (1/4)L

This is the same value we find in part A, Use this value of MRTS to move forward.

The third partial derivative is with respect to :

Z'() = 0

3200 = L + 4K

Substitute the value of MRTS as K = (1/4)L in cost equation:

3200 = L + 4(1/4)L

3200 = L + L

2L = 3200

L = 1600 and so K = (1/4)L = 400.

These values of K* = 400 and L* = 1600 are the optimum values that have least cost combinations for the budget $64,000. The optimum quantity produced is Q = 5LK = 5*1600*400 = 3,200,000 units

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