An alternative way to model merit good arguments

The interpretation of (14) is as follows. Suppose the agent has (x, y) avail-
able. Then in order to reach a utility level U, she believes F(x, y,u) units of the
numéraire are required, while the government, convinced of the merit properties of
the third commodity, believes F^g(x,y,u) units are sufficient. What (13) then tells
is that the slope of the government’s indifference curve through (F^g(x, y, U), x, y)
differs from the slope of the consumer’s indifference curve through (F(x, y, U), x, y)
with a factor μ(y).

To this TWP function corresponds a utility function u^g (∙), defined as

u^g (F^g(x, y, ν),x,y,ν) ≡ ν (all ν).                     (15)

It is easy to show that

(see appendix). A sufficient condition for u^g(∙) to be strongly quasi-concave is
that μ⁰(∙) ≤ 0.

The two MRS expressions necessary to compute the optimal tax rates are
therefore

Again, the reason (17b) is not exactly μ(y) is that the evaluation here is at the
same bundle, not at the same utility level as in (13). A sufficient condition for
single crossing of indifference curves (in the sense that sig n(MRS_y^g_z - MRS_yz)=
sign(μ(y')')} is that the (de)merit good is a normal good (in the sense that ∂∂Z ( u³ ) >
0).

Inserting (17) in (6) and using a Taylor expansion of ui(z + J^ μ(χ)dχ,x,y)
around u- (z,x,y) gives

(∂ ^u2 (z,x,y)∖ ^yy

tχ  '  -Ç u¹ _dZ , J y₉μ(χ)dχ                    (18a)

∂∂ ^u3 (z,x,y)∖ [У

ty ` ^- u¹ _dz— J y μ(x)^dx ^- μ(y)           ^(18b)

The round bracket terms on the rhs denotes the uncompensated effect of
a marginal increase in z on the demand price for each of the non-numéraire
commodities . The signs of these effects are related to the normality of these

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