Question

Suppose a hospital only needs one input, nurses, to treat their patients. A public insurer reimburses each treatment at $10 and suppose that the hospital needs to hire L nurses to provide T = 5L − 1/20 ∗L^2 treatments. Hospitals pay their nurses a wage W and suppose that 5 + W/2 nurses are willing to work at the hospital for a wage W.

In perfect competition, what would be the equilibrium wage and employment level?

Answer 1

Under perfect competition,

equilibrium exits at price (P) = Marginal cost (MC)

According to question, P= 10

number of nurses required(Q): T= 5L- 1/20 *L^2

So, total cost(TC) = MC * Q (MC= 10, because, at equilibrium, P= MC)

= 10( 5L -1/20 * L^2)

TC = 50L - 1/2 * L^2

Differentiating TC with respect to L, we get

MC= 50- L

10 = 50-L

This implies, L= 40  

Equilibrium employment level = 40.

To find equilibrium wage,

number of nurses demanded= number of nurses supplied

It is given,

Number of nurses demanded: T= 5L -1/20 * L^2

Number of nurses supplied = 5 + W/2

At equilibrium, 5L -1/20 * L^2 = 5 + W/2

By substituting the value of L=40, we get,

5(40) -1/20 * (40)^2 = 5 + W/2

200 - 80 = 5 + W/2

This implies, W= 230.

Equilibrium wage: W = 230