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) + At>(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
(9)
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
(10)
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.