Question

A firm requires 96 units of good Q.

The firm can only hire one worker to produce these 96 units.

The firm pays its worker a fixed wage.

The worker’s production function is Q(e) = 24e

The worker’s effort cost function is C(e) = 4e²

The worker gets 0 if unemployed.

Treat the good as continuous.

Assume the firms pay the lowest fixed wage it can. What wage does it pay?

Answer 1

Q = 24e

Differentiating Q wrt e we get

MP_e = 24

Further, Given Q = 96

plug in Q =96 in the production function, we get

96 = 24e

e = 4

Given, C = 4e²

Differentiate TC wrt Q we get

$\large \frac{dC}{dQ} = \frac{d(4e^2)}{dQ}$

$\large MC=8e \frac{de}{dQ}$

$\large MC=\frac{8e }{MP_e}$

$\large MC=\frac{8e }{24}$

$\large MC=\frac{e }{3}$

$\large MC=\frac{4 }{3} = \$ 1.33$

Thus, the firm should pay $ 1.33.