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

the LHS of (12) is given by (ignoring the term π)

^δ = [-^f'(r)] [[       δ(υ)u'(y - t)dF(y, F)

J J(y,fF^p}

- [¹+ t'(p)][[       δ(υ)u'(y - t - p)dF(y,F) (14)

J J{y,l)e^^p)

If we can sign Δ, we also know how the optimal co-payment p changes as
a response to the change in the function Φ(n): If Δ is positive (negative) the
LHS of (12) becomes positive (negative) when Φ(n) is replaced by Φ*(n). To
restore equality, it therefore follows from the second order condition that p
must increase (decline).

It is not possible to unambiguously sign Δ for the general case. It is
therefore useful to consider the special cases in which the heterogeneity in
the population is in either preferences or income, but not both.

4.1 Heterogeneous preferences

In this special case we assume that the income y is the same for everyone, but
that the values of F varies in the population. Expected utility of a person
therefore only depends on this person’s value of F. A person with a value
of F above u(y - t) - u(y - t - p), henceforth denoted L, will choose to be
treated (cf. the discussion of equation (1) in Section 2), giving this person an
expected utility equal to (1 -π)u(y-1)+ πu(y-1-p). A person with a value
of F below L will choose not to be treated, giving this person an expected
utility equal to (1 - π)u(y - t) + π[u(y - t) - F], which is higher the lower is
F. The relationship between υ and F is illustrated in Figure 2.

Restricting ourselves to the case where the function δ(v~) is continuous,
the critical value F* corresponding to υ* must lie below L, i.e. all of the
persons choosing to be treated and some of the persons choosing not to be
treated must get a higher marginal welfare weight as a consequence of the
increased concern for equity (see Figure 2). For this case we can rewrite (14)
as

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