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
-F2g (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
Fg(χ,y,u) = F(χ,y,u) - μ(χ)dχ, (14)
⅛
where yg 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: F2g1 = F1g2 (Frobenius theorem).