We assume that the governments have Utilitarian social welfare functions that can be written

$${W}_{j}={\omega}_{j}^{1}{\gamma}_{j}^{1}{U}_{j}^{1}\mathrm{(}{x}_{j}^{1},\text{\hspace{0.17em}}{z}_{j}^{1},\text{\hspace{0.17em}}{h}_{j}\mathrm{)}+{\omega}_{j}^{2}{\gamma}_{j}^{2}{U}_{j}^{2}\mathrm{(}{x}_{j}^{2},\text{\hspace{0.17em}}{z}_{j}^{2},\text{\hspace{0.17em}}{h}_{j}\mathrm{)},\text{\hspace{0.17em}\hspace{0.17em}}j=L,\text{\hspace{0.17em}}H,$$(2)

where ${\omega}_{j}^{i}$ denotes the weights government *j* attaches to the welfare of each ability-type. Note that, in the absence of cross-border health care, both ability types face the same waiting time, meaning that ${h}_{j}^{1}={h}_{j}^{2}={h}_{j}.$ The government budget constraint in region *j* is given by

$$\sum _{i}{\gamma}_{j}^{i}\mathrm{(}{w}^{i}{l}_{j}^{i}-{x}_{j}^{i}\mathrm{)}}=c\mathrm{(}{e}_{j}\mathrm{)}.$$(3)

We make the conventional assumptions about information; governments can observe income, whereas ability is private information. We follow much of the earlier literature in concentrating on a standard case, where the governments want to redistribute from the high-ability to the low-ability type. As a consequence, they would like to prevent the high-ability type from pretending to be a low-ability type, i.e., becoming a mimicker. This is accomplished by imposing a self-selection constraint, implying that the high-ability type (at least weakly) prefers the combination of disposable income and hours of work intended for him/her over the combination intended for the low-ability type. Note that the hours of leisure that the high-ability type enjoys when working just enough to reach the same labor income as the low-ability type is given by ${\widehat{z}}_{j}^{2}=1-{l}_{j}^{1}\mathrm{(}{w}^{1}/{w}^{2}\mathrm{)}$ and that the self-selection constraint can be written

$${U}_{j}^{2}\mathrm{(}{x}_{j}^{2},\text{\hspace{0.17em}}{z}_{j}^{2},\text{\hspace{0.17em}}{h}_{j}\mathrm{)}\ge {\widehat{U}}_{j}^{2}\mathrm{(}{x}_{j}^{1},\text{\hspace{0.17em}}{\widehat{z}}_{j}^{2},\text{\hspace{0.17em}}{h}_{j}\mathrm{)}.$$(4)

Each regional government decides on a non-linear income tax schedule, ${T}_{j}\mathrm{(}{w}^{i}{l}_{j}^{i}\mathrm{)},$ and the number of treatments, *e*_{j}. The choice of *e*_{j} in turn affects the sickness spell *a*_{j}: without cross-border health care *a*_{j}=(1–*e*_{j})/*r*. Since the government has access to a non-linear tax-schedule, we can solve the policy problem as if it directly chooses ${x}_{j}^{1},\text{\hspace{0.17em}}{l}_{j}^{1},\text{\hspace{0.17em}}{x}_{j}^{2},\text{\hspace{0.17em}}{l}_{j}^{2}$ and *e*_{j} to maximize its objective function, subject to the self-selection constraint (4) and the resource constraint. The Lagrangean corresponding to the optimization problem facing the government is written

$$\begin{array}{c}{\pounds}_{j}={\gamma}_{j}^{1}{\omega}_{j}^{1}{U}_{j}^{1}\mathrm{(}{x}_{j}^{1},\text{\hspace{0.17em}}{z}_{j}^{1},\text{\hspace{0.17em}}{h}_{j}\mathrm{)}+{\gamma}_{j}^{2}{\omega}_{j}^{2}{U}_{j}^{2}\mathrm{(}{x}_{j}^{2},\text{\hspace{0.17em}}{z}_{j}^{2},\text{\hspace{0.17em}}{h}_{j}\mathrm{)}\\ +{\lambda}_{j}[{U}_{j}^{2}\mathrm{(}{x}_{j}^{2},\text{\hspace{0.17em}}{z}_{j}^{2},\text{\hspace{0.17em}}{h}_{j}\mathrm{)}-{\widehat{U}}_{j}^{2}\mathrm{(}{x}_{j}^{1},\text{\hspace{0.17em}}{\widehat{z}}_{j}^{2},\text{\hspace{0.17em}}{h}_{j}\mathrm{)}]\\ +{\mu}_{j}[{\gamma}_{j}^{1}{w}^{1}{l}_{j}^{1}+{\gamma}_{j}^{2}{w}^{2}{l}_{j}^{2}-{\gamma}_{j}^{1}{x}_{j}^{1}-{\gamma}_{j}^{2}{x}_{j}^{2}-c\mathrm{(}{e}_{j}\mathrm{)}]\end{array}$$(5)

The first order conditions become

$$\frac{\partial {\pounds}_{j}}{\partial {x}_{j}^{1}}={\gamma}_{j}^{1}\left[{\omega}_{j}^{1}\frac{\partial {U}_{j}^{1}}{\partial {x}_{j}^{1}}-{\mu}_{j}\right]-{\lambda}_{j}\frac{\partial {\widehat{U}}_{j}^{2}}{\partial {x}_{j}^{1}}=0$$(6)

$$\frac{\partial {\pounds}_{j}}{\partial {x}_{j}^{2}}={\gamma}_{j}^{2}\left[{\omega}_{j}^{2}\frac{\partial {U}_{j}^{2}}{\partial {x}_{j}^{2}}-{\mu}_{j}\right]+{\lambda}_{j}\frac{\partial {U}_{j}^{2}}{\partial {x}_{j}^{2}}=0$$(7)

$$\frac{\partial {\pounds}_{j}}{\partial {l}_{j}^{1}}=-{\gamma}_{j}^{1}\left[{\omega}_{j}^{1}\frac{\partial {U}_{j}^{1}}{\partial {z}_{j}^{1}}-{\mu}_{j}{w}^{1}\right]+{\lambda}_{j}\frac{{w}^{1}}{{w}^{2}}\frac{\partial {\widehat{U}}_{j}^{2}}{\partial {\widehat{z}}_{j}^{2}}=0$$(8)

$$\frac{\partial {\pounds}_{j}}{\partial {l}_{j}^{2}}=-{\gamma}_{j}^{2}\left[{\omega}_{j}^{2}\frac{\partial {U}_{j}^{2}}{\partial {z}_{j}^{2}}-{\mu}_{j}{w}^{2}\right]-{\lambda}_{j}\frac{\partial {U}_{j}^{2}}{\partial {z}_{j}^{2}}=0$$(9)

$$\frac{\partial {\pounds}_{j}}{\partial {e}_{j}}=\left[{\gamma}_{j}^{1}{\omega}_{j}^{1}\frac{\partial {U}_{j}^{1}}{\partial {h}_{j}}+{\gamma}_{j}^{2}{\omega}_{j}^{2}\frac{\partial {U}_{j}^{2}}{\partial {h}_{j}}+{\lambda}_{j}\mathrm{(}\frac{\partial {U}_{j}^{2}}{\partial {h}_{j}}-\frac{\partial {\widehat{U}}_{j}^{2}}{\partial {h}_{j}}\mathrm{)}\right]\frac{1}{r}-{\mu}_{j}{c}^{\prime}=0$$(10)

Note first, that the first-order conditions for consumption and labor supply (6–9) are standard in the optimal taxation literature. This also implies that the optimal taxation formulas that can be derived follow the standard pattern; see e.g., Aronsson and Blomquist (2008). The term 1/*r* in the first order condition (10) is the negative of $\partial {a}_{j}/\partial {e}_{j}$ and thus describes how the sickness spell is decreased when the number of treatments is increased marginally. Put differently, *r* is the number that treatments must be increase by in order to reduce the sickness spell with one unit. The smaller *r* is, the higher the optimal level of *e*_{j} will be.

Defining $MR{S}_{j}^{i}=\frac{\partial {U}_{j}^{i}/\partial {h}_{j}^{i}}{\partial {U}_{j}^{i}/\partial {x}_{j}^{i}}$ to be the marginal rate of substitution between health and private consumption and combining first order conditions (6), (7) and (10) gives us the following proposition:

**Proposition 1:** *Without cross-border health care, the rule for optimal provision of health care is*

$$r{c}^{\prime}={\displaystyle \sum _{i}{\gamma}_{j}^{i}MR{S}_{j}^{i}}+\frac{{\lambda}_{j}}{{\mu}_{j}}\frac{\partial {\widehat{U}}_{j}^{2}}{\partial {x}_{j}^{1}}[MR{S}_{j}^{1}-{\widehat{MRS}}_{j}^{2}].$$(11)

Note that *rc*′, the marginal cost of additional time as healthy, equals the marginal rate of transformation between health and private consumption, since the price of private consumption is normalized to unity. Thus, the proposition states that the number of treatments, *e*_{j}, should be chosen such that the marginal rate of transformation between health and consumption equals a weighted average of the marginal rates of substitution plus a term associated with the self-selection constraint. As mentioned above, health and labor supply are assumed to be complements, meaning that the term in square brackets is positive since the mimicker supplies less labor than the true low-ability type. This means that increasing the number of treatments slightly (or equivalently reducing the waiting time) relaxes the self-selection constraint. Thus, the proposition implies that reducing the waiting time will allow the governments to redistribute more from the high-ability type to the low-ability type.

Hoel and Saether (2003) derived a different result, stating that longer waiting times can serve as a means of redistribution. The intuition behind their result was that patients with low waiting cost are better off waiting, since waiting time makes patients with high waiting costs choose private treatment which reduces the cost of publicly financed health care. The difference in results is explained by that they performed a partial equilibrium analysis, which did not take into account how waiting times affect labor supply and tax revenues. In our general equilibrium model, longer waiting times affect mimickers the least, since health and labor supply are complements and a high-ability type mimicking a low-ability type work the fewest hours. Therefore, if waiting times were exogenously increased and health thus decreased, it would become more tempting for high-ability workers to reduce their hours of work to mimic the income of low-ability workers. To prevent this, the government would need to reduce the relative tax payment of the high income earners so much that the total redistribution is less than with shorter waiting times.

Note that our result that shorter waiting times will allow the governments to redistribute more holds also if private health care without waiting times are introduced in this model. With private health care bought by the true high-ability type, so that publicly provided health care is only directed towards the low-income consumers, equation (11) would change to

$$r{c}^{\prime}=MR{S}_{j}^{1}+\frac{{\lambda}_{j}}{{\gamma}_{j}^{1}{\mu}_{j}}\frac{\partial {\widehat{U}}_{j}^{2}}{\partial {x}_{j}^{1}}[MR{S}_{j}^{1}-{\widehat{MRS}}_{j}^{2}].$$(12)

If the government, starting in a first-best situation where $r{c}^{\prime}=MR{S}_{j}^{1},$ raised taxes on low incomes to finance shorter waiting times and thus better health for low income earners, this would reduce welfare for potential mimickers since they work less and hence value health improvements less. As a consequence, mimicking would become less attractive and the government could therefore raise taxes for high income earners and lower taxes for low-income earners without high-ability workers choosing to mimic low-ability workers.

It should, however, be made clear that waiting times can serve as a means of redistribution in the way explained by Hoel and Saether if the government knows that the self-selection constraint is not binding and prefers to redistribute more but is unable to do so using efficient means. This follows from the results of Besley and Coate (1991) who showed that a low quality of a publicly provided private good might facilitate redistribution if the government lacks effective means of redistribution such as non-linear income taxation. Absence of non-linear income taxation is, however, not a sufficient condition for waiting times to facilitate redistribution and increase welfare. In a model where public health care was financed through a linear income tax and where private treatment could be bought without delay, Fossati and Levaggi (2008) showed that delaying treatments in public health care do not increase welfare.

Let us now return to the model without private health care. Using that ${\gamma}_{L}^{1}-1=-{\gamma}_{L}^{2},$ equation (11) can be rewritten as

$$r{c}^{\prime}-MR{S}_{j}^{1}={\gamma}_{j}^{2}\mathrm{(}MR{S}_{j}^{2}-MR{S}_{j}^{1}\mathrm{)}+\frac{{\lambda}_{j}}{{\mu}_{j}}\frac{\partial {\widehat{U}}_{j}^{2}}{\partial {x}_{j}^{1}}[MR{S}_{j}^{1}-{\widehat{MRS}}_{j}^{2}].$$(13)

The left hand side of equation (13) shows for the low-ability type the difference between the marginal rate of transformation and the marginal rate of substitution, a difference that in a first best scenario would be zero. As discussed above, the term in square brackets is positive, thus showing one incentive to over-provide health care to the low-ability type, i.e., to have shorter waiting times for them than would be optimal in a first best scenario. Assuming that ${l}_{j}^{2}\ge {l}_{j}^{1},$ the first term on the right hand side of equation (13) is also positive,^{5} indicating a second reason for over-provision of health care to the low-ability type. That is, the waiting time for the low-ability type will be held down by the fact that the first best waiting time for the high-ability type is lower than the first best waiting time for the low-ability type.

In region *L* where ${\gamma}_{j}^{2}$ is lower than in region *H*, the first term on the right hand side of equation (13) will be smaller than in region *H*, i.e., health care will be less over-provided to the low-ability type in region *L*. This is one important reason to why, without cross border health care, *e*_{j} is likely decreasing in ${\gamma}_{j}^{1}$ meaning that *e*_{L} likely is lower than *e*_{H}. As discussed in the Appendix, the sign of $d{e}_{j}/d{\gamma}_{j}^{1}$ is also affected by indirect effects and is generally not possible to determine, but in the following we assume that $d{e}_{j}/d{\gamma}_{j}^{1}<0.$

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