Many estimates of “waste” are after-the-fact. It’s actually very hard to identify waste before-hand. Report on End-of-life Spending
With this framework, how much care will be provided? What is the optimal value of \(x\)?
\[\begin{align} \frac{\partial \mathcal{L}}{\partial p} & = x - \lambda x = 0 \\ \frac{\partial \mathcal{L}}{\partial x} & = p - c + \lambda B'(x) - \lambda p = 0 \\ \frac{\partial \mathcal{L}}{\partial \lambda} &= B(x) - p x - NB^{0} = 0 \end{align}\]
\[\begin{align} \frac{\partial \mathcal{L}}{\partial p} & = x - \lambda x = 0 \\ \frac{\partial \mathcal{L}}{\partial x} & = p - c + \lambda B'(x) - \lambda p = 0 \\ \frac{\partial \mathcal{L}}{\partial \lambda} & = B(x) - p x - NB^{0} = 0 \end{align}\]
Maximizing the profit function yields: \[n'(NB)(B'(x) - p_{d}) \left[R + (p_{s} - c)x \right] + n(NB)(p_{s}-c) = 0\]
Rearranging terms and multiplying both sides by \(\frac{1}{NB}\), we get: \[\frac{B'(x) - p_{d}}{NB} \frac{R + (p_{s} - c)x}{p_{s}-c} = - \frac{1}{\varepsilon_{n,NB}}\]
Physicians aim to maximize: \[u_{ij} = \alpha B_{S}(x_{i}) + \beta \pi(x_{i}) - \delta_{d} (x_{i} - x_{i}^{D}) - \delta_{O}(x_{i} - x_{i}^{O}),\]
First order condition for \(x_{i}\) yields: \[\alpha B{'}_{S}(x_{i}) = - \beta \pi{'}(x_{i}) + \delta_{d}{'} + \delta_{O}{'}\]
\[\alpha B{'}_{S}(x_{i}) = - \beta \pi{'}(x_{i}) + \delta_{d}{'} + \delta_{O}{'}\]