Econ 100A, Fall 2019 Problem Set #5
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Problem Set #5.
Please clearly show all necessary math, and explain your answers with enough (and no more than enough)
words to clearly explicate the relevant economic concepts.
Problem 1. Consider a market with three consumers, 1; 2; and 3, each of whom have identical Cobb-
Douglas preferences for some good, X, and all other goods, Y (i.e. 1 = 2 = 3), and each of whom has
income, M1; M2; and M3; respectively.
(a) Suppose M1 = M2 = M3 = 10. Compute the aggregate demand curve for this distribution of incomes.
(b) Suppose, instead, that M1 = 5; M2 = 10; and M3 = 15. Compute the aggregate demand curve for
this distribution of incomes.
(c) Now suppose we don’t know each consumer’s income, so we have to just work with the variables,
M1; M2; and M3. Compute the aggregate demand curve, simplifying as much as possible.
[Hint: a
sign will be helpful.]
(d) Explain what we mean when we say that the market demand curve can be thought of as the demand
curve of a representative consumer only if it can be written as a function of prices and aggregate
income. [See Varian, page 271.]
Problem 2. Consider a simple, generalizable model of the market for some good, X, with supply represented
by Ps(X) = F +cX and demand represented by Pd(X) = A􀀀bX, and a tax of $t per unit of X.1 Throughout
this problem, your answers should be entirely in terms of A; b; F; c and t. You may want to review parts
of chapter 15 in Varian to help you understand the concepts in this problem.
(a) Compute the pre-tax equilibrium price and quantity. Graph the market, and the pre-tax equilibrium,
and label your graph clearly. [Make your graph large and neat enough to be able to add things.]
(b) Compute the after-tax equilibrium quantity, the price paid by consumers, and the price received by
rms. Add them to your graph and label them.
(c) Compute the total revenue raised by the tax, and compute the proportion that is paid by consumers
and the proportion that is paid by rms. Add these to your graph and label them.
(d) Compute the deadweight loss, add it to your graph, and label it.
1You may feel that a model that uses linear supply and demand is ridiculously unrealistic, but in fact this is a very common
approach in a lot of policy analysis.
Econ 100A, Fall 2019 Problem Set #5
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(e) Now let’s consider using this model to do some actual policy analysis. Imagine that you have been
hired to give advice to the government of a medium-sized city (such as Oakland or San Francisco).
The city council needs to raise some revenue. They are considering three possibilities:
ˆ Tax customers at restaurants that serve lunch in the business district.
ˆ Tax customers at fast-food restaurants in residential neighborhoods.
ˆ Tax customers at bars and nightclubs.
The city council has two main policy priorities: create as little deadweight loss as possible, put as little
of the tax-revenue burden on consumers as possible. They have hired you to analyze the three tax
options and tell them how the three options compare in terms of these two policy priorities. Assume
that the supply curves are the same for the three types of establishment. Using the results you derived
in parts (a) through (d), and the concepts you have learned since the midterm, write a short memo
presenting your analysis.
[Your answer will depend on what you believe to be true about the demand curves in each of the three
markets. There is no one right answer, but whatever answer you give must be carefully supported by
economic reasoning, using the concepts we’ve learned recently. As in problem set 3, you can assume
that your clients are familiar with the concepts and terminology we have learned in this class.]
Problem 3. Imagine that you are consulting the government of a developing country which, until recently,
has had a state-controlled economy, which means that the government owns and controls all rms. The
government is in the process of privatizing as much of the economy as they can, but they want to make
sure that they do so wisely. In particular, they are concerned that if they privatize certain industries, those
industries may be taken over by a monopoly or a cartel2 which will cut back market production in order
to drive up the market price and increase market revenue. The government wants your advice about which
industries they should worry about, and which ones they should not worry about.
Over the years, the government has charged di erent prices for goods produced in various industries, and
they now have a large amount of data on how much consumers demand at di erent prices for various goods.
They have given you data for two di erent goods, A and B, which are shown in the two tables below. Based
on these data, which market should they worry about possibly being taken over by a monopoly or a cartel,
and which market should they not worry about? [Assume that income and preferences have been constant
over the time that these data were accumulated, and that cross-price e ects and in ation are negligible.]
[Hint 1: What would happen if you took the logs of prices and quantities and graphed them? Review section
15.8 carefully. Hint 2: If hint 1 makes no sense, you can ignore it and still gure out the answer, but if you
can gure out how to make use of hint 1 it will make your life a lot easier.]
Table 1: Industry A
3.75 25,819
2.25 33,333
1.50 40,824
1.25 44,721
3.50 26,726
2.00 35,355
2.75 30,151
3.00 28,867
Table 2: Industry B
2.50 25,298
3.00 19,245
1.75 43,195
3.75 13,770
2.25 29,629
1.50 54,433
4.00 12,500
3.25 17,067
2A cartel is when a group of rms colludes to x the quantities that they will each produce (and hence the total quantity
produced in the market) and the price at which they will all sell.