An alternative way to model merit good arguments

are the tax rates that person h would face under first best.⁹ By premultiplying
(21) through by the inverse of the aggregate Slutsky matrix, we could isolate
the vector of second best tax rates _t^tx , but it is only under Hicksian indepen-
dence between the two non- numéraire commodities ( ^db^h = 0) that this becomes
∂qy

illuminating:

with bhy ^d=^{f dy}b^h yh ^,ya ^d=^f Ph y^h and byay ^d=^f Ph ^dyb^h ya.  Expression (²2) reflects

the second best nature of the policy. As the government cannot individualise
commodity tax rates, it chooses a uniform tax rate on the (de)merit commodity
which is a weighted average of the individual first best rates, where the weight
depend on the social marginal utility of income (β^h), the share in aggregate
consumption ( y_a ) and the sensitivity of individual relative to aggregate demand
bε^h

( α^y) ). A similar expression is true for t_x.

ε_yy

Under first best, the social marginal utilities of income, β^h , should all be equal
to unity. Under uniform commodity taxation, this is no longer necessarily the
case. The first order condition for the lump sum tax on consumer h can now be
written as:
showing that even at an optimal income distribution the social marginal utilities
of income will deviate from one to the extent that (i) the individual first best tax
rates differ form the uniform ones, and (ii) the income effects are different from
zero. The latter case occurs with quasi linear preferences in the numéraire. I
will now give an example where all individual first best tax rates are identical
and used to implement the optimal allocation.

Suppose preferences are quasi-homothetic, meaning that for each agent there
are some ’survival’ quantities z^h,x^h, and y^h such that

u^h(z, x, y) = F^h[v^h(z - z^h, x - x^h, y - y^h)]

with F^h0(∙) > 0 and v^h(∙) homogenous of degree 1 in (z — z^h, x — x^h, y — y^h). We
then have the following

Lemma 1 Suppose that consumers have (i) quasi homothetic preferences that
are (ii) identical, with (iii) both ∂~_h and ∂y_h positive. Suppose that (iv) μ(∙) is
constant and (v) y^g = y^h. Then the vectors of individual first best tax rates are
identical and the first best allocation can be implemented by this common vector
and a set of lump sum taxes that set all β^h equal to unity.

⁹ More correctly would be to say that they characterise the first best tax rates, as the
differences in MRS are all evaluated at the second best solution.

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