Two-Price Market
Motivation
In health care, providers usually face two prices:
- A price fixed by Medicare and Medicaid, \(p_{m}\).
- A price that is negotiated with insurers, \(p_{n}\).
How does \(p_{m}\) affect \(p_{n}\)?
Profit maximizing hospital
Two price market and FP
Because the hospital is a profit maximizing firm, it will choose the quantity that maximizes profits, \(q^{*}\), where:
- Sell to “private” market as long as marginal revenue exceeds the public price
- Switch to “public’” market otherwise, and sell to the point where price equals marginal cost
Two-price market in pictures
In-class problem (two-price market)
Consider the firm’s demand curve in the private insurance market, \(d=16-q\), and costs, \(c(q)=5+q^{2}\). Assume that there exists a public insurer that pays a fixed price of \(\bar{p}=10\).
- How many private patients will the provider serve?
- How many public patients?
- What if \(\bar{p}\) drops to $9.
Not-for-profit objectives
Two price market and NFP
Although we don’t know the general solution for the private price, we can find how it varies with the public price…
\[\frac{\mathrm{d}p_{i}}{\mathrm{d}p_{j}} = - \frac{U_{11}\pi_{1}^{i}\pi_{1}^{j} + \frac{\mathrm{d}D_{i}}{\mathrm{d}p_{i}}U_{12}\pi_{1}^{j}}{\frac{\mathrm{d}^{2}U}{\mathrm{d}p_{i}^{2}}}\]
Cost-shifting
- Relationship between public and private price is important
- Speaks to anticipated effects of a change in Medicaid or Medicare rates
- Do hospitals “make up” the difference?
The idea that hospitals will increase private prices following a decrease in the public price is called cost shifting.
How does cost-shifting work?
Assumes that hospitals could have increased private prices earlier but chose not too. This is technically possible if, for example:
- Hospital has very low margins (maybe negative with a lower public price)
- Insurer wants to prop up the hospital for competitive reasons
- Hospital has diminishing returns to profits
but economists usually see this as a smaller effect than most policy makers