Question. Why is a Broadly-Targeted Tax Better than a Narrowly-Targeted Tax?
We will consider a setting with just one consumer, but everything would be exactly the same if there were lots of consumers with identical preferences and wealth.
Suppose there are two goods in our economy, beer (x) and pizza (y). Our consumer has Cobb-Douglas utility: u(x,y) = x0.5 y0.5 where x is beer consumed and y is pizza consumed. And he has $60.
(a) Given a price p per pint of beer, and a price q per slice of pizza, calculate how much of each the consumer will demand.
(Maximizing x0.5 y0.5 is the same as maximizing xy, and the consumer's budget constraint is px + qy = 60; the easiest way to solve this is to solve the budget constraint for y, giving y = (60 - px)/q, and plug this into the utility function, maximizing xy = x * (60 - px)/q.)
The markets for both beer and pizza are perfectly competitive, and both beer and pizza can be produced at a marginal cost of $1 per unit, so in the absence of government intervention, both beer and pizza are priced at $1.
(b) Calculate the consumer's demand, and his utility, in the absence of government intervention.
Now suppose the government decides to raise revenue by imposing a $0.50 tax on each pint of beer purchased. This effectively increases the price of beer to $1.50, while leaving the price of pizza unchanged at $1.
(c) Calculate the consumer's demand, and his utility, when beer is taxed at $0.50 per pint.
(d) How much revenue does the government raise with this tax?
Now suppose instead that the government decides to tax both goods at $0.20 per unit, effectively raising the prices of both beer and pizza to $1.20.
(e) Calculate the consumer's demand, and his utility, when beer and pizza are each taxed at $0.20 per unit.
(f) How much revenue does the government raise with this tax?
(g) Given your answers to parts c, d, e, and f, which is the better way for the government to raise revenue, taxing one good or taxing both?