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

the population. For each person there is a probability π (same for everyone)
that the person becomes ill. In this case the welfare level of the person is
u(y — t) — £ if untreated, where £ may vary among persons. The illness can
be completely cured by a treatment that costs c (same for all treatments).²
The price that a patient must pay for this treatment is p ∈ [0, c].

If a person becomes ill he or she will choose treatment if and only if

u(y — t — p) ≥ u(y — t) — £                      (1)

Obviously, a person with £ > 0 will choose to be treated if p is zero or
sufficiently close to zero, no matter how low y is. For higher values of p,
some or even all persons may choose not to be treated. A person is more
likely to choose treatment the higher are y and £ and the lower is p. It is
useful to denote the set of people who choose treatment by Ω(p), i.e.

ОД = {y^,£ I ^u(y ^{— t —} p) ≥ ^u(y ^— £) ^{— £}             (2)}

The set Ω(p) obviously depends on how y and £ are distributed in the popu-
lation. We assume that the joint distribution function is F(y, £), and without
loss of generality we assume that all values of y and £ in the population are
in the range [0,1].³

Denoting the expected utility of a person by v(y,£,p), we have

v(y,£,p) = (1 — π)u(y — t) + π max[u(y — t) — £,u(y — t — p)]    (3)

The size of the co-payment will affect the revenue requirement of the
government. There are two reasons for this. First, the higher is the co-
payment, the lower is the cost paid by the government per treatment. Second,
the higher is the co-payment, the fewer persons will choose to be treated. We
shall assume that any change in the government’s revenue requirement is met
by a corresponding change in the tax rate t that is the same for everyone. In

²The assumptions of a separable utility function and that the illness can be completely
cured simplify the formal analysis, but are not essential for the main results.

³Formally, we assume F(y, 0) = F(0,t) = 0 V (y,t) ∈ [0,1] and F(1, 1) = 1.

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