Elements of Economic Analysis I: Problem Set 1

Santiago CaicedoPablo Robles

Quiz on Monday October 5th

We encourage you to work in groups to solve all the problem set. First try them on your own, then discussyour answers with others. The quiz will be a subset of these questions (randomly chosen at the beginningof every TA session).

1 True or False

An answer without a justification will get no points.

1. The infinite divisibility of goods assumption, makes the applications of economics limited since wenever consume 0.2 of a car.

2. A good is defined by the state and date it is consumed.

3. A road is a private good, since it is rival and excludable.

4. Economic models can’t describe the behavior of people who don’t know how to solve constrainedoptimization problems.

5. If the government levies a tax on gasoline, the consumers are going to pay most of the tax.

6. Assume there are lots of substitutes for apples. If the government increases the sales tax on apples,most of the tax burden would go to the consumer in the short term, but to the producer in the longterm (when supply can adjust).

2 Demand and Supply analysis

Suppose both demand and supply of apples is linear,

qd = a − bp (1)

qs = c + dp, (2)

with a > c > 0 and b, d > 0.

1

1. Use the demand qd(p) and supply qs(p) functions to solve for the equilibrium price p∗ and quantityq∗.

2. Calculate the inverse demand pd(q) and inverse supply function ps(q).

3. Calculate the equilibrium price p∗ and the equilibrium quantity q∗. Show that they are the same asthe ones you calculated before.

4. Plot the inverse demand function and the inverse supply function in a graph with p on the y-axis andq on the x-axis.

I. Elasticity and comparative statics

1. Calculate the elasticity of demand εd ∶= dqd

dppq

and the elasticity of supply εs ∶= dqs

dppq. Are they constant

for different values of p and q? What happens to εd when q → 0?

2. Suppose that scientists discover that apples are good for your health and the demand for apples doubles(for a given price there is twice more demand). Which is the new demand function qd(p)? What is thenew equilibrium? Plot the new and the old inverse demand and supply functions showing the changein equilibrium.

II. Taxes

1. Suppose a quantity tax (a tax levied per unit of quantity bought or sold) of t > 0 dollars per apple.Suppose that the tax is imposed to the suppliers such that pd = ps + t. Show that the equilibriumdemand price is pd∗ = a−c+dt

d+band supply price ps∗ = a−c−bt

d+busing the market equilibrium equation

qd(pd) = qs(ps).

2. Show that the equilibrium quantity with tax is q∗t = c + da−c−btd+b

.

3. Would it make any difference if the tax is charged to consumers instead of suppliers?

4. Sketch the demand and supply functions with the tax for the two cases on separate figures.

5. How does the pd∗ and ps∗ depend on b and d? Give some intuition in terms of the elasticity of demandand supply.

III. Welfare

1. Calculate the consumer surplus and producer surplus in equilibrium without tax p∗, q∗.

2. Calculate the consumer surplus and producer surplus in equilibrium without tax pd∗, ps∗, q∗t .

3. What would be the deadweight loss DWL ?

4. Illustrate your results graphically.

3 Short answer questions

1. Suppose that the university is giving free t-shirts to the incoming students, but there is a line to getthem.

2

a. Are the t-shirts really free?

b. Who are the ones that are most likely to get the t-shirts?

c. Is this allocation system Pareto efficient? (Hint: Can there be any gains from trade?)

2. In class we showed f(x) = log(x), f(x) = xα and f(x1, x2) = xα1x

β2 with α,β > 0 and α + β < 1, are

concave functions. Show if the following functions are concave of convex, or none.

a. g(x) = −3 log(x)

b. g(x1, x2) = xα1 − log(x2)

c. g(x1, x2) = log(xα1xβ2 )

4 Constrained optimization

Consider the following optimization problem,

maxx1,x2

x12

1 x12

2 s.t p1x1 + p2x2 =m.

1. Write the lagrangian L(x1, x2, λ), using λ as the lagrange multiplier.

2. Write down the FOC.

3. Solve the FOC, for x∗1, x∗2 and λ∗.

4. Are these FOC also sufficient conditions for a maximum? (Hint: Is L(x1, x2, λ∗) concave?) Fully

justify your answer.

3