Basic Idea
Solve the problem working backward, finding patient choice probabilities based on prices, and then solving for prices. Take as given the employer-insurer relationship, avoiding concerns regarding employers changing insurers or patients choosing different health care plans.
Setup
- Denote consumers by \(i\), insurance companies by \(m\), hospitals by \(j\), and diseases by \(d\)
- Denote by \(f_{mid}\) the probability that patient \(i\) enrolled in MCO \(m\) has disease \(d\) (\(d=0\) implies no disease)
- Denote by \(w_{d}\) the weights reflecting the relative intensity of resource use for illness \(d\), with \(w_{0}\) normalized to 0
- Insurer and hospital negotiate a base price, to which all other payments are adjusted using the relative weight
- Price paid for a given treatment is \(p_{mj} \times w_{d}\).
Patients
- Coinsurance rate, denoted \(c_{mit}\)
- Patient with disease \(d\) chooses the hospital \(j\) in their network (\(N_{m}\)) that maximizes their utility
- Patient utility is given by: \[u_{mijd} = \beta x_{mijd} - \alpha c_{mid} w_{d} p_{mj} + e_{mij}\]
- \(x_{mijd}\) denotes a vector of hospital and patient characteristics, and \(e_{mij}\) is an iid error term assumed to follow a Type I extreme value distribution
Patients
- Probability of choosing hospital \(j\) is given by \[s_{mijd}(N_{m},\vec{p}_{m}) = \frac{\exp (\beta x_{mijd} - \alpha c_{mid} w_{d} p_{mj})}{\sum_{k}\exp (\beta x_{mikd} - \alpha c_{mid} w_{d} p_{mk})},\] where \(\sum_{k}\) denotes the sum over all \(k\) hospitals in the patient’s network as well as their outside option.
- Expected number of patients admitted to hospital \(j\), weighted by the relative intensity of resource use, is \[q_{mj}(N_{m},\vec{p}_{m}) = \sum_{i=1}^{I_{m}} \sum_{d=1}^{D} f_{mid} w_{d} s_{mijd}(N_{m},\vec{p}_{m}).\]
Patients
- Closed form expression for consumer surplus, \[W_{m}(N_{m},\vec{p}_{m})=\frac{1}{\alpha} \sum_{i=1}^{I_{m}} \sum_{d=1}^{D} f_{mid} \ln \left( \sum_{k}\exp (\beta x_{mikd} - \alpha c_{mid} w_{d} p_{mk}) \right)\]
- Difference between \(W_{m}\) with and without hospital \(j\) in the patient’s network, \(W_{m}(N_{m},\vec{p}_{m}) - W_{m}(N_{m,-j},\vec{p}_{m})\), then serves as an estimate of the patient’s willingness to pay for hospital \(j\)
Insurers
- Expected cost to the MCO for a given hospital and associated prices, \[TC_{m}(N_{m},\vec{p}_{m})=\sum_{i=1}^{I_{m}} \sum_{d=1}^{D} (1 - c_{mid}) \times \sum_{j\in N_{m}} p_{mj} f_{mid} s_{mijd}(N_{m},\vec{p}_{m})\]
- Value for the MCO is \[V_{m}(N_{m},\vec{p}_{m}) = \tau W_{m}(N_{m},\vec{p}_{m}) - TC_{m}(N_{m},\vec{p}_{m}),\] where \(\tau\) is the relative weight placed on employee welfare
- Net value that MCO \(m\) receives from including system \(s\) in its network is then \(V_{m}(N_{m},\vec{p}_{m})-V_{m}(N_{m,-j},\vec{p}_{m})\).
Hospitals
- Hospital \(j\)’s marginal cost for services provided to patients in MCO \(m\) is given by \[mc_{mj}=\gamma \nu_{mj} + \varepsilon_{mj},\] where \(\nu_{mj}\) denotes a set of cost shifters, \(\gamma\) are parameters to estimate, and \(\varepsilon\) is an error term
- Profit for hospital \(j\) for a given set of MCO contracts (denoted \(M_{s}\)), is \[\pi_{s}\left(M_{s},\{\vec{p}_{m}\}_{m\in M_{s}},\{N_{m} \}_{m\in M_{s}} \right)=\sum_{m\in M_{s}} \sum_{j \in J_{s}} q_{mj}(N_{m},\vec{p}_{m}) \left[p_{mj} - mc_{mj} \right]\]
- Net value that system \(s\) receives from including MCO \(m\) in its network is \[\sum_{j \in J_{s}} q_{mj}(N_{m},\vec{p}_{m}) \left[p_{mj} - mc_{mj} \right].\]
Nash Bargaining Solution
Nash bargaining solution is the choice of prices maximizing exponentiated product of the net value from agreement:
\[\begin{align*}
NB^{m,s} \left(p_{mj, j\in J_{s}} | \vec{p}_{m,-s}\right) &= \left(\sum_{j\in J_{s}}q_{mj}(N_{m},\vec{p}_{m}) \left[p_{mj} - mc_{mj} \right]\right)^{b_{s(m)}} \\
& \times \left(V_{m}(N_{m},\vec{p}_{m})-V_{m}(N_{m,-j},\vec{p}_{m})\right)^{b_{m(s)}},
\end{align*}\] where \(b_{s(m)}\) is the bargaining weight of system \(s\) when facing MCO \(m\), \(b_{m(s)}\) is the bargaining weight of MCO \(m\) when facing system \(s\), and \(\vec{p}_{m,-s}\) is the vector of prices for MCO \(m\) and hospitals in systems other than \(s\). We can normalize bargaining weight such that \(b_{s(m)} + b_{m(s)} = 1\).
Nash Bargaining Solution
Note that taking the natural log of the objective does not change the maximum, since the natural log is a ``monotonic transformation.’’ In other words, the log will change the value of some function \(f(x)\), but it will not change the order, so that if \(f(x_{1})>f(x_{2})\), it follows that \(\ln (f(x_{1})) > \ln (f(x_{2}))\).
The resulting first order condition yields: \[\begin{align*}
\frac{d \ln (NB^{m,s})}{p_{mj}} &= b_{s(m)} \frac{d q_{mj} + \sum_{k\in J_{s}} \frac{d q_{mk}}{d p_{mj}} \left[p_{mk}-mc_{mk}\right]}{\sum_{k\in J_{s}} q_{mk}\left[p_{mk}-mc_{mk}\right]} + b_{m(s)} \frac{\frac{d V_{m}}{d p_{mj}}}{V_{m}(N_{m},\vec{p}_{m})-V_{m}(N_{m,-j},\vec{p}_{m})} \\
&= 0 \end{align*}\]
Nash Bargaining Solution
Rearrange this equation to write: \[q_{mj} + \sum_{k\in J_{s}} \frac{d q_{mk}}{d p_{mj}} \left[p_{mk}-mc_{mk}\right] = -\frac{b_{m(s)}}{b_{s(m)}} \frac{\frac{d V_{m}}{d p_{mj}} \sum_{k\in J_{s}} q_{mk}\left[p_{mk}-mc_{mk}\right]}{V_{m}(N_{m},\vec{p}_{m})-V_{m}(N_{m,-j},\vec{p}_{m})}\]
By assumption, the first order conditions are separable across insurers. Combining all first order conditions therefore yields the following system of equations: \[q + \omega (p-mc) = -\lambda (p-mc),\] where \(\omega\) and \(\lambda\) are each \(J_{s} \times J_{s}\) matrices, while \(q\) and \(p-mc\) are \(J_{s} \times 1\) vectors
We can then solve for prices: \[p = mc - (\omega + \lambda)^{-1} q\]
Insurer Steering
“Reasonable assumptions” such that \[\frac{d V_{m}}{d p_{mj}}=-q_{mj}-\alpha \sum_{i}\sum_{d}\gamma_{id}c_{id}(1-c_{id}) \left(\sum_{k\in N_{m}} p_{mk}s_{ikd} - p_{mj}\right),\] where \(\gamma_{id}\) includes several terms including disease weights and probability of disease
- \(c_{id}\times (1-c_{id})\) captures role of insurer coinsurance rates in steering patients
- If \(p_{mj}\) is high relative to the weighted average price of other hospitals, then \(\frac{d V_{m}}{d p_{mj}}\) is less negative due to the term in parenthesis, and thus it is optimal (in the sense of maximizing joint surplus) to allow \(p_{mj}\) to increase so as to channel customers to the low price hospital.
- In the extreme cases, at \(c_{id}=1\) the insurer wouldn’t care about negotiating price with physicians since all of the extra cost is borne by patients. Conversely, at \(c_{id}=0\) the term in parenthesis is again irrelevant as the patients do not switch to lower price hospitals.
Bargaining and Prices
Consider the special case of a single-hospital system, \[p_{mj} - mc_{mj} = -q_{mj} \left(\frac{d q_{mj}}{d p_{mj}} + q_{mj} \times \frac{b_{m(j)}}{b_{j(m)}} \times \frac{\frac{d V_{m}}{d p_{mj}}}{\triangle V_{m}} \right)^{-1}\]
- The term \(\triangle V_{m}\) is positive by construction. With some work, we can find that \(\frac{d V_{m}}{d p_{mj}}<0\) under most conditions. This means that the presence of bargaining tends to increase the ``effective’’ price sensitivity and reduce hospital margins relative to standard pricing conditions.
- However, note that this result does not always hold. In cases where some hospitals are particularly higher priced than others, then the insurer may allow a given hospital to increase its margins (relative to standard pricing conditions) so as to steer patients away from the very high priced hospital.