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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