Consider the case of a firm that acts as a monopsonist in the local labor market.
As the sole employer in town, it can push down wages and increase profit by reducing the
employment level. The firm's goal is to maximize profit:
$$\Pi (L)=p\cdot Q(L)w(L)\cdot L$$
with respect to labor,
$L$
.
For simplicity, assume the production function exhibits
diminishing marginal returns and is given by:
$$Q(L)=100\cdot \sqrt{L}$$
assume that the quantity of labor supplied is an increasing function of the wage rate
and the inverse of that relationship is given by:
$$w(L)=L\cdot \sqrt{L}$$
and assume that the firm sells its output in a perfectly competitive market (so that the
market price of output is exogenous to the firm) and that the price is fixed at:
$$p=5$$
◆ ◆ ◆
NOTE: It may be helpful to write the monopsonist's profit as:
$$\begin{array}{ccccc}\Pi (L)& =& \mathrm{TR}\left(L\right)& & \mathrm{TC}\left(L\right)\\ & =& p\cdot Q(L)& & w(L)\cdot L\\ & =& 5\cdot 100\cdot \sqrt{L}& & L\cdot \sqrt{L}\cdot L\\ & =& 500\cdot \sqrt{L}& & {L}^{2.5}\end{array}$$
◆ ◆ ◆
 Derive the marginal revenue from increasing employment,
$L$
.
$$\begin{array}{ccccc}& & \mathrm{TR}\left(L\right)& =& 500\cdot {L}^{0.5}\\ \frac{d\mathrm{TR}\left(L\right)}{dL}& \equiv & \mathrm{MR}\left(L\right)& =& 0.5\cdot 500\cdot {L}^{0.5}\\ & & \mathrm{MR}\left(L\right)& =& 250\cdot {L}^{0.5}\end{array}$$
 Derive the marginal cost of increasing employment,
$L$
.
$$\begin{array}{ccccc}& & \mathrm{TC}\left(L\right)& =& {L}^{2.5}\\ \frac{d\mathrm{TC}\left(L\right)}{dL}& \equiv & \mathrm{MC}\left(L\right)& =& 2.5\cdot {L}^{1.5}\end{array}$$
 What is the necessary condition for maximizing
$\Pi \left(L\right)$
with respect to
$L$
?
$$\mathrm{MR}\left({L}^{*}\right)=\mathrm{MC}\left({L}^{*}\right)$$
 What is the sufficient condition for maximizing
$\Pi \left(L\right)$
?
$$\frac{d\mathrm{MR}\left({L}^{*}\right)}{dL}<\frac{d\mathrm{MC}\left({L}^{*}\right)}{dL}$$
 What is the value of
${L}^{*}$
that maximizes
$\Pi \left(L\right)$
?
$$\mathrm{MR}\left({L}^{*}\right)=\mathrm{MC}\left({L}^{*}\right)$$
$$250\cdot {{L}^{*}}^{0.5}=2.5\cdot {{L}^{*}}^{1.5}$$
$$100={{L}^{*}}^{2}$$
$${L}^{*}=10$$

what is the profitmaximizing wage at that value of
${L}^{*}$
?
$$w\left(10\right)={10}^{1.5}$$
$$w\left(10\right)=31.62$$
 what is the monopsonist's profit at that value of
${L}^{*}$
?
$$\Pi \left(10\right)=500\cdot \sqrt{10}{10}^{2.5}$$
$$\Pi \left(10\right)=1264.91$$
◆ ◆ ◆
SUMMARY: In the absence of a minimum wage:
$${L}^{*}=10$$
$$w\left(10\right)=31.62$$
$$\Pi \left(10\right)=1264.91$$
◆ ◆ ◆
Now, assume that the government imposes a minimum wage:
$${w}_{min}=65$$
 Derive the marginal revenue from increasing employment,
$L$
.
Imposing a minimum wage only affects cost, so marginal revenue remains unchanged:
$$MR\left(L\right)=250\cdot {L}^{0.5}$$
 Derive the marginal cost of increasing employment,
$L$
.
After imposing a minimum wage, the marginal cost of increasing employment
is the cost of hiring at the minimum wage:
$$MC\left(L\right)=65$$
 What is the necessary condition for maximizing
$\Pi \left(L\right)$
with respect to
$L$
?
$$\mathrm{MR}\left({L}^{*}\right)=\mathrm{MC}\left({L}^{*}\right)$$
 What is the sufficient condition for maximizing
$\Pi \left(L\right)$
?
$$\frac{d\mathrm{MR}\left({L}^{*}\right)}{dL}<\frac{d\mathrm{MC}\left({L}^{*}\right)}{dL}$$

What is the new value of
${L}^{*}$
that maximizes
$\Pi \left(L\right)$
?
$$\mathrm{MR}\left({L}^{*}\right)=\mathrm{MC}\left({L}^{*}\right)$$
$$\frac{250}{\sqrt{{L}^{*}}}=65$$
$${L}^{*}=14.79$$

what is the new equilibrium profit at that value of
${L}^{*}$
?
$$\Pi \left(14.79\right)=500\cdot \sqrt{14.79}65\cdot 14.79$$
$$\Pi \left(14.79\right)=961.54$$
Effect of the Minimum Wage
 What happened to total employment in the town after the minimum wage was imposed?
Imposing the minimum wage increased total employment from 10 to 14.79.
 What happened to the monopsonist's profit after the minimum wage was imposed?
Imposing the minimum wage decreased the monopsonist's profit from 1264.91 to 961.54.
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