Practice exam

Here are some practice problems for the exam, which we’ll also work through as part of our revised Panel Day on 11/17.

1. Expected utility and the risk premium

A person starts with wealth of $100,000. With probability 0.3 they fall ill and pay $40,000 in medical costs. Their utility of wealth is $u(w) = w^{0.2}$.

What is the expected cost of illness, and what is expected wealth?
What is this person’s expected utility?
What is the certainty equivalent, the sure amount of wealth that gives the same utility as facing the risk?
What is the risk premium, and what is the most this person would pay for full insurance?
Suppose the probability of illness falls from 0.3 to 0.2. Does the risk premium rise or fall? Explain in a sentence, without recomputing.

2. Adverse selection

An insurer’s cost of covering $q$ people is $C(q) = 100q - 2q^{2}$, so average cost is $AC = 100 - 2q$ and marginal cost is $MC = 100 - 4q$. Inverse demand for coverage is $P = 110 - 3q$, and there are 20 people in the market.

The insurer enters the market at a price of $65. How many people enroll, and what is the insurer’s profit or loss?
Next year the insurer sets its price equal to last year’s average cost. What is the new price, and what happens to enrollment?
What are the equilibrium price and quantity, where the insurer breaks even?
The government pays insurers a subsidy of $5 per enrollee. Find the new equilibrium and the number of additional enrollees.
Instead of the subsidy, a $10 penalty is charged to anyone without coverage. What happens to enrollment, and how does the penalty differ from the subsidy in who pays for the extra coverage?

3. Physician agency when the physician sets the price

A patient’s benefit from $x$ units of care is $B(x) = 20x - x^{2}$, so marginal benefit is $20 - 2x$. The patient pays the full price $p$ per unit, so net benefit is $B(x) - px$. Care costs the physician $c = 10$ per unit. The physician chooses both $x$ and $p$, subject to leaving the patient at least $NB^{0} = 5$ of net benefit.

If the patient could choose $x$ themselves at a given price $p$, what condition picks their $x$?
Find the physician’s optimal amount of care $x^{\*}$ and the price $p^{\*}$ that goes with it.
At that price $p^{\*}$, how much care would the patient have chosen on their own?
Sketch the marginal benefit line with $p^{\*}$, $c$, $x^{\*}$, and $\bar{x}$ marked, and describe the region that is the surplus transferred from the patient to the physician.
How large is that transfer?

4. Physician agency with an administered price

Now the price is fixed administratively at $\bar{p}$. The patient’s benefit is $B(x) = 4x^{1/2}$, so marginal benefit is $2x^{-1/2}$, and the physician’s cost is $c = 1$ per unit. The physician can push care up to the point where the patient’s net benefit $B(x) - \bar{p}x$ falls to $NB^{0} = 0$.

With $\bar{p} = 2$, how much care does the physician provide, and how much would the patient choose on their own?
The administered price rises to $\bar{p} = 3$. Recompute both amounts.
How did the price increase change the gap between the physician’s amount and the patient’s?
Now the price is cut to $\bar{p} = 1.5$. Recompute both amounts. What happened to overtreatment?
In the usual supply model a higher price brings more output. Why is it the opposite here?

5. Fee-for-service, capitation, and the amount of care

A patient’s net benefit is $NB(x) = B(x) - p_{d}x$, with $B(x) = 16x - 2x^{2}$ and the patient fully insured, so $p_{d} = 0$. The physician’s number of patients rises with the net benefit offered, $n(NB) = NB$. For each patient the physician receives a fixed payment $R$ plus a margin of $(p_{s} - c)$ on each unit of care, and gives every patient the same $x$.

What amount of care would the patient choose for themselves?
Write the physician’s profit as a function of $x$, $R$, and $(p_{s} - c)$.
Pure fee-for-service: $R = 0$ and $p_{s} - c = 1$. Find the physician’s $x$.
A mix: $R = 1$ and $p_{s} - c = 1$. Find $x$.
Pure capitation: $R = 2$ and $p_{s} - c = 0$. Find $x$ and explain the result in a sentence.

6. Hospital pricing with market power

A hospital faces inverse demand $P = 40 - 2q$, so marginal revenue is $40 - 4q$, and has cost $C(q) = 5q + \tfrac{1}{2}q^{2}$, so marginal cost is $5 + q$.

Find the profit-maximizing quantity and price.
What is the markup over marginal cost, in dollars and as a share of marginal cost?
Compute the Lerner index, $(P - MC)/P$. What price elasticity of demand does it imply at the optimum?
A second hospital opens nearby and this hospital’s demand falls to $P = 30 - 2q$ (marginal revenue $30 - 4q$). Recompute quantity, price, and the markup.
The price fell when the competitor entered even though this hospital’s costs did not change. In a sentence or two, why?

7. The two-price market

A provider’s inverse demand from privately insured patients is $P = 16 - q$, so marginal revenue is $16 - 2q$. Cost is $C(q) = 5 + q^{2}$ for $q$ total patients, so marginal cost is $2q$. A public insurer pays a fixed price $\bar{p} = 10$ per patient and will send as many patients as the provider will take.

How many private patients does the provider serve?
How many public patients?
The public price falls to $\bar{p} = 9$. Recompute private, public, and total patients.
What is the private price before and after the cut in the public price? Did the cut push the private price up (cost shifting) or down?
Below what public price does the provider stop taking public patients altogether?

8. Nash bargaining

Two parties split a surplus normalized to 1. With agreement, party 1 gets $x$ and party 2 gets $1 - x$. Without agreement, party 1 gets $t_{1}$ and party 2 gets $t_{2}$. The Nash bargaining solution maximizes $(u_{1} - t_{1})(u_{2} - t_{2})$, the product of each party’s gain over its outside option.

With $t_{1} = 0$ and $t_{2} = a > 0$, find the solution $x$.
How does party 1’s share change as $a$ rises, and why?
Now both have outside options, $t_{1} = b$ and $t_{2} = a$. Find $x$.
With $a = 0.2$ and $b = 0.1$, what does each party get, and how much of the gain from agreeing does each capture?
Let party 1 be a hospital and party 2 an insurer negotiating the hospital’s price. Name one thing that raises the hospital’s outside option and one that raises the insurer’s, and say what each does to the price.

On the exam, you might also be asked about…

why subsidies might lower the cost of individuals newly enrolling in health insurance (and how much those new enrollees cost on average)
discussion of recent payment reforms intended to maintain or improve quality
calculating the optimal amount of care in a physician agency model, or calculating the optimal contract to induce a given amount of care
in a physician agency model, you might be asked to calculate the surplus a price-setting physician takes from the patient and draw this on a graph
short definitions: Medicare Part A, Part B, Medicare Advantage, IPPS, Medicaid
in general, any graph we’ve drawn in class, you should be able to draw on the exam as well
non-linear payoffs in the Nash bargaining model, such as $(u_{1} - t_{1})^{\alpha}$