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

and

t>₂ = In [0.450 + 0.5τ — τ² + 0.05p]                   (22)

It is easily verified that v₁ > v₂ for all τ ∈ [0, 0.25] and p ∈ [0,1] . In
the social welfare function group 2 should therefore be given at least as large
weight as group 1. Since there are only two groups, it is easier to work with
exogenous welfare weights than the concave function used in Sections 3-4.
The social welfare function is thus

W = υ₁ + 7v₂

where 7 ≥ 1.

The optimal value of W is obtained by solving the following equations:

0.045    0.095 ₊ 0.057 = ₀

У1     У1 — P У2

0.9(—0.5 — 2 τ)    0.1(—0.5 — 2 τ)   7 (0.5 — 2 τ)

У1               У1 — P             У2

for p ∈ [0,1], τ ∈ [0, 0.25], given 7. A numerical solution for the optimizing
problem for different values of 7 is found using Maple (s.t. τ ∈ [0, 0.25] and
p ∈ [0,1]):

τ        p        W

7 = 1     .1062872413  .6703546385  -.3575958178

7 = 1.5   .1357928137  .8486112171  -.6707438076

7 = 1.75  .1455947627  .9067929325  -.8227024058

From this table it is clear that as the concern for equity (measured by
7) increases, we get an increase in both the optimal marginal tax and the
optimal co-payment.

18

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