Econ 522 – Economics of Law – Spring 2010 – Dan Quint
In lecture on January 25, we said that one of the defenses for wanting the law to be efficient is that, even
if society has other goals (such as concerns over the distribution of wealth), these are better handled
through taxes rather than through the legal systems. One of the reasons we gave for this was that laws
designed to redistribute wealth would function like “narrowly-targeted taxes” – penalizing certain
activities correlated with wealth – whereas an actual tax could target wealth or income, and therefore be
more broad. In this question, we explore why broadly-targeted taxes tend to cause less distortion, and
therefore less deadweight loss, than narrowly-targeted ones.
(You are not required to solve this problem. It will not be graded. This is the example we will do in
lecture on Wednesday – I’m distributing it ahead of time in case you want to look at it beforehand.)
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 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?
And the answers, as solved in lecture…
Question a. Given prices (p,q) and wealth 60, the consumer’s problem is
max u(x,y) subject to the budget constraint px + qy <= 60
or
max x0.5 y0.5 subject to px + qy <= 60
To simplify this, note that maximizing x0.5 y0.5 is the same as maximizing xy; and that, since more is
better, he will spend every dollar available, so the budget constraint will hold with equality, so now the
problem is
max xy subject to px + qy = 60
We can solve the budget constraint for y, giving y = (60 – px)/q, and plug this in, so now the problem is
max x (60 – px)/q
We can multiply by q (doesn’t change which x maximizes this), then take the derivative and set it equal to
zero:
60 – px – px = 0
px = 30, or x = 30/p
We then get y from the budget constraint: y = (60 – px)/q = (60 – 30)/q = 30/q
So given prices (p,q), the consumer demands (x,y) = ( 30/p, 30/q )
Question b. At prices (p, q) = (1,1), demand is (30, 30), so utility is 300.5 300.5 = 30.
Question c. Now, the effective prices are (p,q) = (1.5, 1), so demand is (30/1.5, 30/1) = (20, 30). This
gives utility of 200.5 300.5 = 600.5, which is approximately 24.49.
Question d. Since the consumer demands 20 pints of beer and the government raises $0.50 per pint,
government revenue is 20 X 0.5 = $10.
Question e. Now, effective prices are (p,q) = (1.2, 1.2), so demand is (30/1.2, 30/1.2) = (25, 25). This
gives utility of utility of 250.5 250.5 = 25.
Question f. Demand is (25, 25), and the government raises $0.20 per unit of both beer and pizza, so
revenue is 25 X $0.20 + 25 X $0.20 = $5 + $5 = $10.
Question g. The two policies (a $0.50 tax on beer, or $0.20 taxes on both products) raise the same
amount of revenue, but the second one leaves the consumer with higher utility. So we conclude that a
“broad” tax (taxing all goods in the economy equally) is a better way to raise revenue than a “narrow” tax
(taxing just one good).