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

Proposition 2 Assume that incomes are identical in the population but pref-
erences, represented by the variable I, differ among persons. Assume also
that the initial optimal co-payment is such that some persons choose treat-
ment while others choose not to be treated. In this case increased concern for
equity reduces the optimal co-payment.

4.2 Heterogeneous income

In this special case we assume that the value of I is the same for everyone, but
that income y varies in the population. Expected utility of a person therefore
only depends on this person’s income. A person with a "high" income (a value
of y satisfying the inequality (1)) will choose to be treated, and a person with
a "low" income (a value of y that does not satisfy the inequality (1)) will
choose not to be treated. Expected utility v will in this case be a strictly
increasing function function of y, and the value of y corresponding to υ* is
denoted by y*. For this case δ(u) > 0 for y < y* and δ(u) < 0 for y > y*. In
words, marginal welfare weights are increased for persons with "low" income
(y < y*) and reduced for persons with "high" income (y > y*).

Use Y to denote the critical value of y giving equality in (1). The expres-
sion (14) can then be written as

- [¹ + t'(p)] J δ(υ)u'(y - t - p)dF(y,I)

Just like for the general case, it is not possible to unambiguously sign Δ
for this special case. However, there exist changes in the function Φ that
make Δ unambiguously positive. Consider first the case where all of those
who initially are treated get a lower marginal welfare weight after the change.
In other words, δ(υ) < 0 for y > Y. In this case the second integral in (17) is
negative. In the first integral there are negative and positive values of δ(v~).

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