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

Appendix: A numerical example

Consider an economy where the average gross income is 1. There are two
equally sized groups. Group 1 has income 1.5, while group 2 has income 0.5.
The probability of illness is 0.1, and the treatment cost is 1. The utility loss
in case of illness for group 1 is so large that treatment is chosen no matter
how high the co-payment is, while there is no utility loss for group 2.

There are two tax parameters: The tax t that is equal for everyone, and
a proportional income tax τ. There is a deadweight loss associated with τ,
implying that the tax function (4) now becomes

t = 0.1 ∙ (1 — p) ∙ 0.5 — [τ — τ²]                      (18)

The term in square brackets is the income revenue (per capita) due to the
proportional tax (since average income is 1). The last term (τ²) represents
the deadweight loss due to the distortion implied by the income tax. This
distortion implies that the top of the "Laffer curve" for this tax component
is at τ = 0.5.

Net incomes for the two groups are (using (18))⁴

y₁ = 1.5 — 1.5τ — t = 1.450 — 0.5τ — τ² + 0.05p           (19)

and

У2 = 0.5 — 0.5τ — t = 0.450 + 0.5τ — τ² + 0.05p           (20)

Notice that whatever p is, y₁ declines and y₂ rises as τ increases, as long
as τ < 0.25. However, for τ > 0.25 both net incomes decline as τ increases.
In a social optimum it therefore must be the case that τ < 0.25.

The utility function is u(x) = In x for both groups. Expected utility levels
for the two groups are

u₁ = 0.9 In [1.450 — 0.5τ — τ² + 0.05p] + 0.1 In [1.450 — 0.5τ — τ² — 0.95p]

(21)

⁴Notice that we have subtracted also the constant tax t in this definition of net income,
unlike what we did in the main text.

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