# Suppose the market for nurses can be modeled using supply and demand (the market is perfectly competitive). There are a large number of hospitals, doctors’

Suppose the market for nurses can be modeled using supply and demand (the

market is perfectly competitive). There are a large number of hospitals, doctors’

offices, clinics, etc., so that no individual employer has an impact on the market

wage. The nurses are not unionized so they individually have no impact on the

wage.

Demand is given as Qd = 420 – 8W, where Qd is the quantity demanded (in full-time

equivalents) and W is the hourly wage rate.

Supply is given as Qs = -1,000 + 40W, where Qs is the quantity supplied.

1. Find the equilibrium wage rate. This is done by setting the quantity demanded

equal to the quantity supplied and algebraically solving for W. That is, solve the

following equation for W:

420 – 8W = -1,000 + 40W

2. Find the equilibrium quantity. Plug your answer from (1) into either the supply or

demand function to get Q. Or, better yet, plug it into both to make sure you get the

same answer. If you don’t, you have the wrong answer for 1.

If, at this point, you have a negative wage or negative quantity, you’ve made an error

and should start over.

Another way to solve the above problems would be to derive the ‘inverse demand

curve’ and the ‘inverse supply curve.’ The inverse supply curve gives the wage as a

function of the quantity supplied. Given the supply function above, the inverse

supply function is:

W = 25 + 0.025Qs

Be sure you can duplicate the above answer before moving on to 3.

3 and 4. Derive the inverse demand function. It should take the form W = a + bQd,

where a and b are derived from the demand function’s parameters. You will report a

as the answer to (3) and b as the answer to (4). b should be reported to 6 decimal

places.

If you set the inverse demand function equal to the inverse supply function you can

solve for the equilibrium quantity (while you have Qd and Qs in the equations, you

can make use of the fact that, in equilibrium, Qd = Qs, so that you are simply solving

for Q).

5. What is the wage rate using the above approach?

Part 2: Monopsony

A market for labor can fail to be perfectly competitive if there is a single large

employer, as might be the case when all medical services are provided by a single

provider. Suppose that is the case with the nurses. The following use the equations

used in the prior set of questions, though they may take on different meanings.

Solving for the equilibrium wage and quantity is a multi-step procedure. First, you

will need the marginal resource cost (MRC) curve. This curve gives the marginal cost

of hiring an additional nurse (in cost per hour). Since attracting an additional nurse

requires raising the wage rate for all nurses, this cost is above the wage rate

required for that additional nurse. If the inverse supply curve is given as

W = a + bQs,

the marginal resource cost is:

MRC = a + 2bQs

Essentially, it has the same y-axis intercept as the inverse supply curve, but twice

the slope.

With a single buyer there isn’t a demand curve, but what we would think of the

demand curve is the marginal value curve. In this case, it gives the marginal value

product (in dollars per hour) of nurses. Hence, you can replace W in the inverse

demand function with MVP.

To find the quantity of nurses hired, set MRC = MVP and solve for Q.

6. With a monopsony, how many nurses are hired?

7. What is the wage rate? (hint: use the inverse supply curve – not the MRC or

demand curve – for this purpose)

8. If the hospital could hire as many nurses as it wanted with the wage rate you

found in (7), how many nurses would be hired? (hint: use the demand curve to

answer this – plug in the wage from (7)).

9. What is the ‘shortage’ of nurses? (This is the difference between your answer to

(8) and your answer to (6)).

Part 3: Monopsony with 3rd party supplier

With the nursing shortage found above, the hospital is interested in finding a

solution whereby it can get more nurses, but not raise the wage to its existing

nurses. A nursing ‘temp’ agency has moved into the region and can supply as many

nurses as are needed, at a wage of $30.73/hour.

10. How many nurses does the hospital hire via the temp agency?

Part 4: Monopoly

Your hospital is the only hospital for hundreds of mile with a particular therapy. The

inverse demand function for this therapy is given as:

P = $12,500 – 5Qd, where Qd is the annual quantity demanded.

While your fixed costs are high (due to the cost of specialized machines), your

marginal costs for a full treatment are ‘just’ $600 per treatment.

11. If you set a single price to maximize profits, what quantity will you supply each

year?

(Guidance: The marginal revenue (MR) function has the same y-axis intercept as the

inverse demand function, but twice the slope. Set MR equal to MC and solve for Q.)

12. What is the price for treatment? (Hint, plug your quantity from 11 into the

inverse demand function.)

13. If the treatment is priced at the marginal cost to your company, how many

treatments will be provided per year?

14. What is the deadweight loss due to monopoly power in this market? (This is

covered in one of the lessons.)

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