An alternative way to model merit good arguments

approach the government’s MWP for tobacco will exceed that of the smoking
agent.⁶

4 An alternative way of modeling merit good
arguments

The previous analysis indicates the need for an approach that ties down in a
more robust way the relationship between the government’s MWP and that of the
consumer. I propose to model this relationship by means of the total willingness-
to-pay (TWP) function in terms of the numéraire commodity:

F(x,y,u).                                 (11)

This function gives the amount of the numéraire required to bring the con-
sumer at the utility level u when consuming x and y units of the other two
commodities; its graph is the indifference surface. The marginal willingness to
pay (MWP) for the (de)merit good is then

- ^xyu    -F2(     ).                    (12)

^y

If the government is of the opinion that consumers appreciate the (de)merit
good too (much) little, then a natural way of proceeding is to attribute to the
government the MWP function

^-F2^g (x^,y^,u) = ^-F2(^χ,y^,u) + μ(y)                   (¹3)

where μ(y) is (negative) positive for a (de)merit good.⁷ Notice that (13) does
not directly impose any single crossing in the commodity space because it is
conditional on the utility level u: the amount of z the consumer believes is
necessary to reach that utility level need not be the same as the amount the
government believes is required. Indeed, integrating this MWP function to a
TWP function gives

^y

F^g(^χ,y^,u) = F(^χ,y^,u) ^-    μ(χ)^dχ^{,                  (14)}

⅛

where y^g can be thought of as the level of consumption above which the govern-
ment’s marginal evaluation starts to deviate from the consumer’s.

⁶ With additive preferences quasi-linear in the numéraire, the (own) elasticity of the inverse
demand schedule is exactly the reciprocal of the (own) Marshallian price elasticity.

⁷ More complicated modifications of the MWP function are possible, but one should make
sure that for a corresponding TWP function to exist, the cross partial derivatives should be
symmetric: F₂^g₁ = F₁^g₂ (Frobenius theorem).

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