Monopsony described a single big employer that must raise pay for everyone to attract one more worker. With an upward-sloping supply curve $w(L)$, a marginal revenue product from Production Functions and Marginal Products, and the The Product Rule, we can find its hiring and its wage.
Key idea
A monopsony employer pays $w(L)$ to each of $L$ workers, so its wage bill is $w(L)\,L$ and the marginal cost of labor is $$MCL = w(L) + L\,w'(L) > w(L).$$ It hires where $MRP = MCL$ and pays the wage on the supply curve at that $L$.
Why it works
The product rule splits the cost of one more worker in two: her own wage, $w(L)$, and the raise everyone else gets, $L\,w'(L)$. Hiring stops where the value of one more worker's output (MRP) equals that full cost. But the wage itself is whatever the supply curve says is needed to attract $L$ workers, which is less than MCL. The result is fewer workers and a lower wage than in a competitive labor market, where hiring would continue until $MRP = w(L)$.
Worked example: Hiring and the wage
$w(L) = 7 + 2L$ and $MRP = 67 - L$.
- Wage bill $= 7L + 2L^2$, so $MCL = 7 + 4L$.
- $67 - L = 7 + 4L$ gives $L = 12$.
- Wage from the supply curve: $7 + 24 = 31$. (MRP there is 55; MCL is 55 too.)
- Competitive benchmark: $67 - L = 7 + 2L$ gives $L = 20$ and $w = 47$.
Answer: 12 workers at \$31.
Worked example: A minimum wage in the model
Same employer, with a minimum wage of \$40.
- Up to the point where the supply curve reaches \$40 ($7 + 2L = 40$, so $L = 16.5$), every worker costs exactly \$40, so $MCL = 40$ there.
- $MRP = 67 - L$ is above 40 until $L = 27$, so the employer hires all 16.5 workers willing to work at \$40.
- Employment rises from 12 to 16.5 and the wage from \$31 to \$40.
Answer: 16.5 workers (in this model).
Common mistake
Reading the wage off the marginal cost of labor at the chosen $L$. MCL at $L = 12$ is \$55, but no one is paid that: the employer pays the lowest wage that attracts 12 workers, \$31, from the supply curve. Check: the wage you report must satisfy $w = w(L)$ for the $L$ you found.
What economists broadly agree on and what they debate: in the competitive model a binding minimum wage reduces employment; in the monopsony model a moderate one can raise it, as above. Which model better describes a given labor market, and the measured employment effects of particular minimum wages, are studied and debated.