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

Section 5 we give a discussion of this assumption and of the consequences of
relaxing it.

Denoting the share of the population that is treated by ω(p), the relation-
ship between the size of the co-payment and the tax rate t and is formally
given by

^t(p = π ∙ (c - p⁾ ∙ ω(p)                          (4)

It follows that

t'(p) = -πω(p) + π ∙ (c - p) ∙ ω'(p)                    (5)

which is negative since ω'(p) is negative.

If p is sufficiently small, everyone will choose treatment provided I > 0
for everyone. In other words, ω(p) = 1 and ω'(p) = 0 for sufficiently low
values of p. For sufficiently high values of p, some people will choose not to
be treated, i.e. ω(p) < 1. From (4) and (5) it therefore follows that

—t' (p) = π for p so small that ω(p) = 1

—t' (p) < π for p sufficiently close to c

For intermediate values of p, we may very well have — t' (p) > 1, since there
is nothing that rules out "large" values of — ω'.

We are now ready to see how the expected utility v(y, £,p) of a particular
person depends on the size of the co-payment. To see this, we distinguish
between those who choose treatment and those who don’t: For those who
choose not to be treated, it follows from (3) that

^gv*y^{, f,p)} =      ■      ■ — t) > 0                    (7)

op

For these persons (which include those with “low” income if p initially is large
enough), an increase in co-payments is thus unambiguously desirable. The
interpretation is obvious: Since these persons in any case are choosing not to
be treated, they are not directly affected by the increase in the co-payment.

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