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

optimal co-payment, where we use the shorthand notation v for v(y,P,p):

(1 - π)[-t'(p)] [[       Φ'(v)√(y - t)dF(y,P)

√⅜∕)∈[o,i]

+ ^^[-t>(P⁾] [[      ^φ7^v>⅜ ^{- t)dF(}y, ŋ

J JMW_f)

π[f + t'(p)] [[       Φ'(v)√(y - t - p)dF(y,P) = 0 (12)

J J(y,Q∈Ω(p)

The two first terms in this expression are are positive, implying that the
last term is negative, i.e. we must have 1 + t'(p) > 0 at the optimal value
of p. We shall use the expression above to answer the question what is the
effect on the optimal co-payment of an increase in the concern for equity?

4 Co-payments and equity concerns

Consider a change in the function Φ(v) in the direction of stronger prefer-
ences for equity, i.e. a "more concave" function. More precisely, let Φ(v)
be replaced by Φ*(v) ≡ f (Φ(v)) where f' > 0 and f'' < 0. Calling Φ'(v)
and Φ*'(v) the "marginal welfare weights" before and after the change, the
change in marginal welfare weights is given by h(v) = Φ*'(v) - Φ'(v). Since
the level of the function f ' is of no importance, it is convenient to chose this

level so that

It is easy to verify that the definition of the function h(v) implies that
h^,(v) < 0 at the value of v giving h(v) = 0. Denoting this value of v by v* it
therefore follows from (13) that h(v) > 0 for v < v* and h(v) < 0 for v > v*.
In words, marginal welfare weights increase for persons with "low" expected
utility (v < v*) and decline for persons with "high" expected utility (v > v*).

The normalization given by (13) implies that the first of the three terms
in (12) does not change as the function Φ(v) changes. The total change in

<<   6   7   8   9   10   11   12   >>

More intriguing information