class: center, middle, inverse, title-slide # Module 3: Hospital Pricing and Competition ## Part 1: Single and Two-price Market ### Ian McCarthy | Emory University ### Econ 372 --- <!-- Adjust some CSS code for font size and maintain R code font size --> <style type="text/css"> .remark-slide-content { font-size: 30px; padding: 1em 2em 1em 2em; } .remark-code, .remark-inline-code { font-size: 20px; } </style> <!-- Set R options for how code chunks are displayed and load packages --> # Table of contents 1. [Single Prices](#single-prices) 2. [Two Prices](#twoprice) <!-- New Section --> --- class: inverse, center, middle name: single-prices # Unilateral Pricing (depends on the objective) <html><div style='float:left'></div><hr color='#EB811B' size=1px width=1055px></html> --- # Pricing for NFP hospitals Objective is to maximize some function of profits and quantity of care provided, denoted by<br> `\(U\left( \pi_{j} = \pi_{i,j} + \pi_{g,j},D_{i,j}, D_{g,j} \right)\)`<br> where `\(\pi_{j}\)` denotes total profits for hospital `\(j\)` and `\(D_{i,j}\)` denotes hospital demand from insurer `\(i\)`. We assume that `\(p_{j}\)` is exogenous and determined by a public payer, so the hospital need only set its price for private insurance customers, `\(p_{i}\)`. --- # Solution for NFP hospital The hospital chooses `\(p_{i}\)` such that<br> `\(\frac{\mathrm{d}U}{\mathrm{d}p_{i}} = U_{1} \pi_{1}^{i} + U_{2} \frac{\mathrm{d}D_{i}}{\mathrm{d}p_{i}}=0\)`,<br> where `\(U_{1}\)` denotes the derivative of `\(U(\cdot)\)` with respect to its first argument and similarly for `\(U_{2}\)`.<br> -- In general, we can't solve this directly without knowing the hospital's utility function. --- # Assuming pure profit maximization <img src="03-pricing1_files/figure-html/tikz-fp-pricing-1.png" style="display: block; margin: auto;" /> --- # Example Consider the firm's demand curve, `\(d=16-q\)`, and cost curve, `\(c(q)=5+q^{2}\)`. Where will the firm produce and at what price? What is the firm's markup over marginal cost? -- The profit function is, `\(\pi = (16-q)q - 5 - q^{2}\)`. Differentiating with respect to quantity yields `\(-q + 16 - q - 2q= 16-4q=0\)`, or `\(q=4\)`. At this quantity, the price is `\(p=12\)`, which is a markup of 4 over the marginal cost (or 50% markup). --- # In-class problem (unilateral pricing) Consider the firm's demand curve, `\(d=40-2q\)`, and cost curve, `\(c(q)=5q+\frac{1}{2}q^{2}\)`. 1. What is the firm's profit maximizing choice of quantity and price? 2. What is the markup over marginal cost? <!-- New Section --> --- class: inverse, center, middle name: twoprice # Two-price Market <html><div style='float:left'></div><hr color='#EB811B' size=1px width=1055px></html> --- # Relationship between prices In health care, providers usually face two prices: 1. A price fixed by Medicare and Medicaid, `\(p_{m}\)`. 2. A price that is negotiated with insurers, `\(p_{n}\)`. How does `\(p_{m}\)` affect `\(p_{n}\)`?<br> --- # 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...<br> -- `$$\frac{\mathrm{d}p_{i}}{\mathrm{d}p_{j}} = - \frac{U_{11}\pi_{1}^{i}\pi_{1}^{j} + \frac{\mathrm{d}D_{i}}{\mathrm{d}p_{i}}U_{12}\pi_{1}^{j}}{\frac{\mathrm{d}^{2}U}{\mathrm{d}p_{i}^{2}}}$$` --- # Two price market and FP .pull-left[ - 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 ] .pull-right[ <img src="03-pricing1_files/figure-html/tikz-fp-twoprice-1.png" style="display: block; margin: auto;" /> ] --- # 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 `\(\bar{p}=10\)`.<br> 1. How many private patients will the provider serve? 2. How many public patients? 3. What if `\(\bar{p}\)` drops to <span>$</span>9. --- # 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?<br> -- The idea that hospitals will increase private prices following a decrease in the public price is called <b>cost shifting</b>. --- # Cost-shifting But how could it happen?<br> -- 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<br> -- but economists usually see this as a smaller effect than most policy makers.