Patients’ Mobility Across Borders: A Welfare Analysis

Abstract

Welfare systems are designed on geographical and membership boundaries. In terms of access to health care this implies that, as a general rule, only individuals residing in their national territory can obtain health care from providers located there. However, in the past few years medical tourism has grown at an explosive pace throughout the world and in Europe. Each year in fact a small, but significant number of European citizens seek medical treatment that is financed by their public insurer in another EU country. From an economic point of view, it is important to distinguish between the two following sources of patients’ mobility: a regulated mobility, where the third payer decides to send patients abroad and patients’ choice, where the patient himself decides to seek care abroad. In this article we show how the combined effect of restrictions to the use of health care, transfer prices, and mobility rules determine social welfare and its allocation between Regions. The results are quite interesting: if the price set for these patients is equal to the marginal cost of the more efficient Region, patients’ mobility should be preferred to patients’ choice. On the other hand, if the price is equal to the marginal cost of the less efficient Region, patient choice should be preferred. The other interesting result is a possible trade off between a static model where each Region chooses its level of cost/effectiveness and a more long term situation, where patient mobility determines a common level for this parameter.

Appendix

1.1 Derivation of the Optimal Quantities

• 1. No Mobility

  The optimal quality $q_{j,/;NM}$ is derived from the maximization of (5) over $q_{j}$. The F.O.C. gives the condition: $- \theta_{j} {\kern 1pt} q_{j} + \frac{1}{2}\left( {1 - c_{j} } \right) = 0$, which gives the result in (6).

• 2. Regulated mobility

  In this case for Region 0 the welfare in (10) has to be maximized for $r$ and $q_{0}$. The function is strictly increasing in the first variable, while the derivative with respect to the second one is the same as for the NM case. Then Region 0 will accept any number of patients as long as $p \ge v_{0}$ without changing the offered quality.

For Region 1 the welfare function changes and can be written as in (12) and the maximization is done on $r$ and $q_{1}$. The F.O.C. gives the following conditions:

$$ \left\{ {\begin{array}{*{20}l} {\left( {v_{1} - p + q_{0} - q_{1} + m\left( {1 - 2r} \right)} \right){\kern 1pt} \left( {1 - c_{1} } \right) = 0,} \\ { - \theta_{1} {\kern 1pt} q_{1} + \left( {1 - c_{1} } \right){\kern 1pt} \left( {1 - r} \right) = 0.} \\ \end{array} } \right. $$

The solutions of the above linear system are reported in (14) and (15). In order for them to be feasible the condition $\frac{1}{2} < r_{RM} < 1$ has to be satisfied. This implies that the following condition has to be verified

$$ 0 < \frac{{q_{0,\;NM} - m + v_{1} - p}}{{1 - c_{1} - 2{\kern 1pt} m{\kern 1pt} \theta_{1} }} < \frac{1}{{2{\kern 1pt} \theta_{1} }}. $$

From the second inequality we get

$$ \frac{{q_{0,NM} + v_{1} - p}}{{1 - c_{1} - 2{\kern 1pt} m{\kern 1pt} \theta_{1} }} < q_{1,NM} \frac{1}{{1 - c_{1} - 2{\kern 1pt} m{\kern 1pt} \theta_{1} }} $$

and since $q_{0,NM} + v_{1} - p > q_{1,NM}$ this is possible only if $1 - c_{1} - 2{\kern 1pt} m{\kern 1pt} \theta_{1} < 0$, which further entails the condition $q_{0,NM} - m + v_{1} - p < 0$. The given optimal value for $r$ is then feasible only if

$$ m > \hbox{max} \left( {\frac{{1 - c_{1} }}{{2{\kern 1pt} \theta_{1} }},\;\frac{{1 - c_{0} }}{{2{\kern 1pt} \theta_{0} }} + v_{1} - p} \right). $$

• 3. Patient choice

  Region 0 maximizes (19) over $q_{0}$; the F.O.C. in this case is

$$ - \frac{1}{{2{\kern 1pt} m}}{\kern 1pt} \left( {v_{0} - p} \right){\kern 1pt} \left( {1 - c_{1} } \right) - \theta_{0} {\kern 1pt} q_{0} + \frac{1}{2}\left( {1 - c_{0} } \right) = 0, $$

which does not depend on the quality chosen in Region 1 and gives the optimal quality $q_{0,PC}$ given in (20).

Region 1 maximizes (21) over $q_{1}$; the F.O.C. is

$$ - \theta_{1} {\kern 1pt} q_{1} + \frac{{1 - c_{1} }}{{2{\kern 1pt} m}}{\kern 1pt} \left( {q_{1} - q_{0} - v_{1} + p + m} \right) = 0 $$

and the optimal quality is thus given by (22).

The number of moving patients is given by

$$ \begin{aligned} z & = \frac{1}{2} + \frac{1}{{2{\kern 1pt} m}}\left( {q_{0,PC} - q_{1,PC} } \right) \\ & = \frac{1}{2} + q_{0,NM} \frac{{\theta_{1} }}{{ - 1 + c_{1} + 2m\theta_{1} }} + \frac{1}{2}\frac{{\left( {1 - c_{1} } \right)\left( {\theta_{1} \left( {p - v_{0} } \right) + \theta_{0} \left( {v_{1} - p - m} \right)} \right)}}{{\left( { - 1 + c_{1} + 2m\theta_{1} } \right)\theta_{0} m}} \\ \end{aligned} $$

which is greater than $\frac{1}{2}$ because $q_{0,PC} \ge q_{0,NM} > q_{1,\;NM} > q_{1,PC}$ and under (13) is also less than 1. In fact $z < 1$ iff $q_{0,PC} - q_{1,PC} < m$ and substituting in the inequalities the values in (20) and (22) the condition is equivalent to require that

$$ \frac{{2{\kern 1pt} m^{2} {\kern 1pt} \theta_{1} {\kern 1pt} \theta_{0} - \left( {1 - c_{0} } \right){\kern 1pt} \theta_{1} {\kern 1pt} m - \left( {\theta_{0} {\kern 1pt} (v_{1} - p} \right) - \theta_{1} \left. {{\kern 1pt} \left( {v_{0} - p} \right)} \right){\kern 1pt} \left( {1 - c_{1} } \right)}}{{2{\kern 1pt} m{\kern 1pt} \theta_{1} {\kern 1pt} \theta_{0} }} > 0. $$

The ensuing condition

$$ m \ge \frac{{1 - c_{0} }}{{4{\kern 1pt} \theta_{0} }} + \frac{1}{4}\sqrt {\frac{{\left( {1 - c_{0} } \right)^{2} }}{{\theta_{0}^{2} }} + 8{\kern 1pt} \left( {\frac{{v_{1} - p}}{{\theta_{1} }} + \frac{{p - v_{0} }}{{\theta_{0} }}} \right){\kern 1pt} \left( {1 - c_{1} } \right)} $$

(27)

is always verified under hypothesis (13) because

$$ \frac{{1 - c_{0} }}{{2{\kern 1pt} \theta_{0} }} + v_{1} - v_{0} > \frac{{1 - c_{0} }}{{4{\kern 1pt} \theta_{0} }} + \frac{1}{4}\sqrt {\frac{{(1 - c_{0} )^{2} }}{{\theta_{0}^{2} }} + 8{\kern 1pt} \frac{{v_{1} - v_{0} }}{{\theta_{0} }}{\kern 1pt} \left( {1 - c_{1} } \right)} $$

and since $\theta_{0} < \theta_{1}$ the right-hand side is greater than the lower bound in (27).

1.2 Welfare Analysis for the Patient Choice Case

The welfare difference between the PC and the RM case is given in Eq. (25) and by substitution of the optimal values $q_{0,PC}$, $q_{1,PC}$ and $r_{RM}$ it can be written as

$$ \begin{aligned} \frac{1}{8}{\kern 1pt} \frac{{(1 - c_{1} ){\kern 1pt} (p - v_{0} )}}{{m^{2} {\kern 1pt} \theta_{0} {\kern 1pt} ( - 1 + c_{1} + 2{\kern 1pt} m{\kern 1pt} \theta_{1} )}} & \left[ {\left( {8\theta_{0} m^{2} \theta_{1} - 4m(1 - c_{1} )\left( {\theta_{0} - \frac{1}{2}\theta_{1} } \right) + \left( {1 - c_{1} } \right)^{2} } \right)} \right.p \\ & - 8\theta_{0} m^{2} \theta_{1} v_{1} + \left( {4\left( {1 - c_{1} } \right)} \right)\left( {\theta_{0} - \frac{1}{2}\theta_{1} } \right)v_{1} m \\ & - \left( {1 - c_{1} } \right)^{2} v_{0} + 2\left. {\left( {1 - c_{1} } \right)} \right)\theta_{1} \left. {\left( {v_{1} - v_{0} } \right)m} \right]. \\ \end{aligned}$$

The term outside the square bracket is positive and by algebraic tools it is easy to show that the term multiplying $p$ inside the square bracket is positive for $\theta_{0} < \theta_{1}$. Thus the welfare difference is positive whenever $p$ is greater than the bound in (26).

For Region 1 the best way to proceed is to analyze the behavior wrt $p$ of the two optimal welfare levels $W_{1,PC}$ and $W_{1,RM}$ obtained by substitution of the optimal quantities in (21) and (12) resp. Since

$$ \frac{{\partial^{2} W_{1,RM} }}{{\partial p^{2} }} = \frac{{\left( {1 - c_{1} } \right)\theta_{1} }}{{ - 1 + c_{1} + 2{\kern 1pt} m{\kern 1pt} \theta_{1} }} $$

$W_{1,RM}$ is convex in $p$. As for $W_{1,PC}$ we have:

$$ \frac{{\partial^{2} W_{1,PC} }}{{\partial p^{2} }} = \frac{{\left( {1 - c_{1} } \right)^{2} }}{{\left( { - 1 + c_{1} + 2{\kern 1pt} m{\kern 1pt} \theta_{1} } \right)^{2} }}\left( {2\theta_{1} {\kern 1pt} A{\kern 1pt} B - \theta_{1} {\kern 1pt} B^{2} - \frac{{1 - c_{1} }}{{2{\kern 1pt} m}}A^{2} } \right) $$

with

$$ A = \frac{{\theta_{1} - \theta_{0} }}{{\theta_{0} }},\quad \quad B = - 1 + \frac{{1 - c_{1} }}{{2{\kern 1pt} m{\kern 1pt} \theta_{0} }}. $$

The term $A$ is always positive, while $B > 0$ iff $m < \frac{{1 - c_{1} }}{{2{\kern 1pt} \theta_{0} }}$. Since $1 - c_{0} > 1 - c_{1}$, by (13) $m > \frac{{1 - c_{0} }}{{2{\kern 1pt} \theta_{0} }} + v_{1} - v_{0} > \frac{{1 - c_{1} }}{{2{\kern 1pt} \theta_{0} }}$, thus $B$ is always negative and $W_{1,PC}$ is concave. Thus the difference $W_{1,PC} - W_{1,RM}$ is concave. Since for $p = v_{0}$ we have $W_{1,PC} < W_{1,RM}$, while for $p = v_{1}$ it is $W_{1,PC} > W_{1,RM}$, it can be concluded that for $p \in (v_{0} ,\;v_{1} )$ a unique value $p^{*}$ exists and such that for $p < p^{*}$ the welfare level in RM is greater than in PC, while the contrary is true for $p > p^{*}$.

The welfare difference for Region 1 between PC and NM is given by

$$ \begin{aligned} W_{1,PC} - W_{1,NM} =\; & \;\left( {v_{1} - p} \right){\kern 1pt} \left( {1 - c_{1} } \right){\kern 1pt} \left( {z - \frac{1}{2}} \right) - \frac{1}{2}{\kern 1pt} \theta_{1} \left( {q_{1,PC}^{2} - q_{1,RM}^{2} } \right) \\ & + \left( {1 - c_{1} } \right)\left( {q_{0,PC} \left( {z - \frac{1}{2}} \right) - \frac{1}{2}{\kern 1pt} m\left( {z^{2} - \frac{1}{4}} \right) + q_{1,PC} {\kern 1pt} \left( {1 - z} \right)} \right. \\ & \left. { - \frac{1}{2}{\kern 1pt} m{\kern 1pt} \left( {1 - z} \right)^{2} - \frac{1}{2}{\kern 1pt} q_{1,RM} + \frac{1}{8}{\kern 1pt} m} \right). \\ \end{aligned}. $$

As shown above, $W_{1,PC}$ is concave. Also we have

$$ \begin{aligned} \left( {\frac{{\partial W_{1,PC} }}{\partial p}} \right)_{{p\; = \;v_{0} }} =\; & \;\frac{{1 - c_{1} }}{{4\theta_{0}^{2} {\kern 1pt} m{\kern 1pt} (2\theta_{1} {\kern 1pt} m - 1 + c_{1} )}}\left( {2\theta_{0}^{2} {\kern 1pt} \theta_{1} \left( { - \frac{{1 - c_{0} }}{{\theta_{0} }} + \frac{{1 - c_{1} }}{{\theta_{1} }}} \right)m} \right. \\ & - \left( {1 - c_{1} } \right)\left. {\left( { - \theta_{0} {\kern 1pt} c_{1} - 2\theta_{0}^{2} v_{0} + \theta_{1} c_{0} + 2v_{1} \theta_{0}^{2} \theta_{1} + \theta_{0} - 2v_{1} \theta_{0} \theta_{1} + 2\theta_{0} v_{0} \theta_{1} } \right)} \right) \\ \end{aligned} $$

which, since $- \frac{{1 - c_{0} }}{{\theta_{0} }} + \frac{{1 - c_{1} }}{{\theta_{1} }} < 0$, is negative if

$$ m > \frac{{1 - c_{1} }}{{2{\kern 1pt} \theta_{0} }}\left( {\frac{{2(v_{1} - v_{0} )\;(\theta_{0} - \theta_{1} )}}{{\theta_{1} \left( { - \frac{{1 - c_{0} }}{{\theta_{0} }} + \frac{{1 - c_{1} }}{{\theta_{1} }}} \right)}} + 1} \right) $$

and by (13) this condition is always verified. Thus we can conclude that $W_{1,PC}$ is strictly decreasing. Since

$$ \begin{aligned} \left( {W_{1,PC} - W_{1,NM} } \right) & |_{{p\; = \;v_{1} }} = \frac{{1 - c_{1} }}{{8{\kern 1pt} \theta_{0}^{2} \theta_{1} m^{2} \left( {2m\theta_{1} - 1 + c_{1} } \right)}} \\ & \cdot \left( {2m\theta_{1} {\kern 1pt} q_{0,\;NM} \theta_{0} + \theta_{1} v_{1} - v_{0} \theta_{1} - \theta_{0} m - \theta_{1} c_{1} v_{1} + c_{1} v_{0} \theta_{1} + \theta_{0} mc_{1} } \right)^{2} \\ \end{aligned} $$

is positive, it holds that $W_{1,PC} > W_{1,NM}$ for all $p \in \left[ {v_{0} ,\;v_{1} } \right]$.