ACME Widgets’ short-term production function is given by q = 75(5l 6)^2/3 , wher

Question

ACME Widgets’ short-term production function is given by q = 75(5l 6)^2/3 ,

where q is ACME’s weekly output (which is equal to the weekly demand for their product) and l is ACME’s labor input, measured in 40-hour work weeks. E.g., if l = 10, then ACME’s workers are working for a total of 10 × 40 = 400 hours a week.

The demand equation for ACME’s product is given in the previous problem. (previous demand equation) q = 1.2(600 4p)^ 3/2

What is ACME’s output and marginal product, dq/dl, when l = 14?

Use your answers to part (a), above, and problem 4(c) to find ACME’s marginal revenue product dr/dl when l = 14. (You may assume that the demand for the firm’s product is equal to their output.)

If ACME hires a new employee to work 20 hours a week, what is the approximate effect on the their revenue?

Explanation / Answer

(1) When l = 14,

From production function: Output (q) = 75 x [(5 x 14) - 6]2/3 = 75 x (70 - 6)2/3 = 75 x (64)2/3 = 75 x 16 = 1,200

MP (dq/dl) = (75 x 5) / (5l - 6)1/3 = 375 / [(5 x 14) - 6]1/3 = 375 / (70 - 6)1/3 = 375 / (64)1/3 = 375 / 4 = 93.75

(2) When l = 14, q = 1,200

From demand function: 1,200 = 1.2 x (600 - 4p)3/2

(600 - 4p)3/2 = 1,000

Taking (2/3)rd root on each side,

600 - 4p = 100

4p = 500

p = 125

Marginal revenue product (MRP) = MP x p = 93.75 x 125 = 11,718.75

(d) 20 hours a week means l = 0.5. When l = 0.5,

MP = 375 / [(5 x 0.5) - 6]1/3 = 375 / (2.5 - 6)1/3 = 375 / (-3.5)1/3 = 375 / (-1.52) = - 247

Assuming output price remains unchanged,

MRP = - 247 x 125 = - 30.875

So revenue will decrease.

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