Question

Consider a consumer who derives utility from three consumption goods: x and y and z, which prices are given, respectively, by px , py and pz . We

let z denote the numeraire good and normalize its prices to unity. The consumer possesses an initial endowment of Z>0 units of z.
The utility function of the consumer is given by:
(1) U(x,y,z)=v(x)+u(y⋅py)+z.

Thus, the consumer is deriving utility from the amount she spends on y and not from the number of units consumed.
The government is seeking to raise funds in the total amount of R>0 (units of z). For this purpose it considers the following two alternative options: (i) levying a lump-sum tax; (ii) imposing an ad-valorem tax on consumption good y.

1. (1) Provide a theoretical explanation for the special form of the utility function we employ.

2. (2) What would be the optimal choice of the government?

Answer 1

I don't have a solid theoretical justification for the the utility function featuring expenditure on Good y (instead of just y like usual), but I can say that it really captures the essence of the modern consumerist idea of "retail therapy": defined on Google as "shopping in order to make oneself feel happier".

It is the value of the shopping bill, the ability to buy expensive things rather than the consumption of the thing itself which brings utility.

^ Another example would be when buying a gift, often people have a level of expenditure in mind. It's reasonable to argue that the buyer and/or intended recipient of the gift may be indifferent between high quantity of a low priced item and low quantity of a high priced item (including the case when the high priced item is highly priced because of tax)

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In the first five pictures, I've discussed lumpsum tax vs consumption tax i.e. per unit tax. The discussion is easily extended to an ad valorem tax in general, as I do in the sixth picture.

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The rest of the logic is exactly as in the case of a per-unit tax:

agent's utility is unaffected as long as total expenditure post tax = optimal alpha-star, which is a function only of price of X and endowment Z

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Optimal choice for the government is to go for a per-unit tax or an ad valorem tax in general (per-unit tax is a special case of ad valorem tax)