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

However, they gain from the tax reduction that is implied by the increase in
the co-payment.

For those who choose to be treated, it follows from (3) that

— (1 ^{— π})[^-t^,(p)]^u'(y — ^t) ⁺ A^t>(p) — ¹¼^,(^{y — t} — ^{p)     (8)}

op

In this expression, the first term is positive. We cannot unambiguously sign
the second term without knowing more about t(p) than t'(p) < 0. However,
the concavity of и implies that u'(y — t) < u'(y — t — p) when p is positive.
It therefore follows from (8) that

^ev(y'^f-^p) < [-f(p) — π]u'(y — t — p)
op

From (6) it follows that the LHS of (9) is negative for p sufficiently close to
0, and for p sufficiently close to c. From (7) and (9) we therefore have the
following proposition:

Proposition 1 If the initial co-payment is sufficiently low, everyone chooses
to be treated, and a small increase in the co-payment will make everyone
worse off. If the initial co-payment is sufficiently close to the treatment cost
and some people choose to be untreated at this co-payment, those who have
chosen not to be treated gain from an increase in the co-payment, while those
who have chosen to be treated lose from an increase in the co-payment.

Figure 1 illustrates how v will depend on the size of the co-payment for
a typical person. If p is so low that everyone chooses treatment (below p⁰ in
Figure 1), it follows from (6) that (8) can be rewritten as

°^υ⅛,t,^p) — (ι — ^[^(y — t) — u(y — t — p)] < 0
op

if ω(p) — 1

where the inequality sign follows from the concavity of u. A positive co-
payment can thus only be optimal if it is set so high that it makes some
persons choose not to be treated.

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