Concerns for Equity and the Optimal Co-Payments for Publicly Provided Health Care

However, since δ(v)u'(y — t) is lower the higher is y, it follows from (13) that
the positive values dominate, i.e. that the first integral in (17) is positive.
Since the terms [—t'(p~)] and [1 + t'(p)] are both positive, it follows that Δ > 0
in this case.

The case in which δ(v) < 0 for all who initially choose to be treated is
obviously rather restrictive. However, even if some of those who choose to
be treated get increased marginal welfare weights, the negative δ-terms in
the second integral may dominate the positive δ-terms, so that the second
integral remains negative. And even if so many of those treated get increased
marginal welfare weights that the second integral becomes positive, Δ may
still be positive since the first term in (17) is always positive. This leads to
the following proposition:

Proposition 3 Assume that the value of £ is the same for everyone, but
that income y varies among persons. Assume also that the initial optimal
co-payment is such that some persons choose treatment while others choose
not to be treated. In this case increased concern for equity increases the op-
timal co-payment provided a sufficiently large number of persons who choose
treatment get reduced marginal welfare weights.

5 Interaction with a progressive tax system

So far, we have only considered tax changes that were identical for everyone.
This assumption may be justified as follows: Distributional considerations
are achieved through distortionary taxation. The optimal design of a distor-
tionary tax system implies that redistribution has been taken to the point
where the social gain from further redistribution is exactly offset by the in-
cremental distortion of higher rates of taxation. For an optimally designed
tax system of this type, social welfare cannot be increased by increasing or
reducing a tax component which is equal for all (and thus non-distortionary)
and adjusting the distortionary part of the tax system so that total revenue
is unchanged. On the margin, it therefore makes no difference whether an

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