Exercise: Factor Market

1. If a firm has the following production function,

$Q=17L-L^2$

and the firm is a monopoly with the following demand curve,

$P(Q)=150-Q$

find the MRPL equation for this firm.

Step one, solve for the MPL:

$MPL= \frac{dQ}{dL} =17-2L$

Step two, find the revenue function and derive the MR:

$\\ 1. R(Q)= 150Q-Q^2 \\\ \\2. MR= 150-2Q$

Step three, remember the formula for the MRPL:

$\\ 1. MRPL= MPL \cdot MR\\\ \\2. MRPL= (17-2L)\cdot (150-2Q)$

2. the demand and supply for labor in a market are as follows:

$\\ 1. D(w)=70-2w\\\ \\2. S(w)= 5w$

Find the equilibrium wage rate and quantity of labor hired:

$\\ 70-2w=5w\\\ \\ w=10 \\\ \\ 5(10)=50L$

Find the economic rent for workers at the equilibrium wage rate:

$\\ \frac{10 \cdot 50}{2}= \$ 250$

If the workers want to optimize their economic rent, is the wage from part a the one they would choose?

No, the economic rent is optimized by looking at the intersection of the MR and supply curve.

$\\ \$(7.778 \cdot 54.45)>\$ 250$

You can see that if I set my wage at about $7.78, the economic rent for the workers is much bigger than the one found at the equilibrium wage rate.

Source: the problems were inspired by the exercises on pages 297-298 in Study Guide For Microeconomics, 8th edition (2013) by J.Hamilton, and V.Suslow.

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