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
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
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
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
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
(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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