1) [20 points] Consumer theory

The n-good Cobb-Douglas utility function is:

n

u(x) = A

!xi

?i

,

i=1

where A > 0 and "n

i=1 ?i = 1.

a) Derive the Marshallian demand functions.

b) Derive the indirect utility function.

c) Compute the expenditure function.

d) Compute the Hicksian demands.

2) [15 points] Consumer theory

a) Suppose that u : Rn Rk is a continuously differentiable function, and consider

the problem:

max u(z,? ) z?Rn

where ? ? Rk. Think of Rk as the space of parameters and Rn as the space of

choice variables. State and prove the envelope theorem. Give an intuition for the

theorem.

b) Now suppose we are given g : Rn Rm ? and suppose that our problem is

max u(z,? ) z?Rn

subject to g(z) = 0.

State the envelope theorem for this constrained maximization problem. (Note:

you do not need to prove it)

c) State and prove Roy's identity.

3) [25 points] Producer theory

Consider the profit maximization problem:

max pF(k, l) ? rk ? wl

k,l

where F(k, l) is the production function, r is the rental rate of capital, w is the wage

rate of labor.

a) Suppose that F is not differentiable. When are k and l are substitutes in production? Provide an example of a production function with this property (note:

example can be differentiable).

b) Demonstrate that if k and l are substitutes, then the optimal capital choice as a

function of exogenous parameters, k?(r, w), is weakly increasing in w and likewise

l

?(r, w) is weakly increasing in r. Show this without assuming F is differentiable

if you can.

2

c) Suppose that the initial wage is w0. When wages change to w, write the optimization problems that the short-run demand for labor solves and the long-run

demand for labor solves.

d) How the short-run demand compare to the long-run demand when k and l are

substitutes? Provide a proof and give economic intuition.

e) Explain how the results of part d) changes if k and l are complements? What

if k and l are complements when both are small, and substitutes when both are

large?

4) [25 points] General equilibrium

Suppose there are I consumers, and xi denotes the demand of consumer i at prices p

and wealth wi (obtained by utility maximization). Let x = "

i?I xi be the aggregate

demand. All consumers face the same prices p, although their wealth and preferences

could differ.

A benevolent planner has the power to re-allocate x among the consumers but does

not have additional resources. That is, he can change the original allocation, (xi)i?I ,

to some re-allocation (xi

#

)i?I provided that "

i xi

# = x. His aim is to find a feasible

re-allocation (x#

i) such that "

i?I xi

# = x and ui(x#

i) > ui(xi) for all i ? I, a weak

Pareto improvement.

a) Prove that there does not exist a weak Pareto improvement. Please highlight

clearly what assumptions on preferences you use in the argument.

A feasible re-allocation, (x#

i) with "

i x#

i = x, is said to be a Pareto improvement if

ui(xi

#

) ? ui(xi) for all i ? I and uj (xj

#) > uj (xj ) for at least one j ? I.

b) Prove that there does not exist a Pareto improvement over (xi)i?I . Please highlight what assumptions on preferences you use in the argument.

c) Do there exist preferences where there exists a Pareto improvement, but there

does not exist a weak Pareto improvement? If so, provide an example. If not,

explain your reasoning.

5) [15 points] Externalities

Assume the cost of commuting to work for an individual with wage w is wf(n) by car

and wx + t by subway where n is the number of cars on the road, and x and t are

constants. Let there be a total of N commuters. Assume that f(0) = 0 and f#

, f## > 0.

a) Assume everyone makes the same wage. What will be the equilibrium number of

drivers? How will the equilibrium number change with the wage?

b) What is the socially efficient number of drivers? How does this number change

with wage?

c) What are three ways to enforce the social optimum? Make a case for the most

plausible one.

and quantity (0""). (5 points) Elasticity Use your answers in the supply and demand question above, to calculate the mid- point formula for the price elasticity of demand (two decimal places). (4 points) b. Using your answer in part a, is the price elasticity of demand elastic, inelastic or unit elastic? (3 points) P1 = 200, 0 1 = 300 P2 = 185, Q2 =330 State how you were able to determine if the price elasticity of demand is elastic, inelastic or unit elastic. (3 points) 2 decimalPrice (dollars per Supply. S pound) 52.00 1,50 Demand, D 1.300 1,400 1.500 Quantity of coffee (thousands of sacks) Figure 2.2.1 No answer text provided.Question 1 Not yet answered Points out of 1 Flag question (This question refers to the MRU video 'The Demand Curve Shifts'.) If, at a certain price, buyers suddenly demand a greater quantity of a good than they did previously, there has been: Select one: a. movement along a demand curve, from right to left. b. movement along a demand curve, from left to right. c. a decrease in demand d. an increase in demand. Question 2 Not yet answered Points out of 1 Flag question The government plans to increase cigarette taxes in six months. Since consumers should expect the future price of cigarettes to increase, the current demand for cigarettes will increase, and the price of cigarettes will rise even before the tax is implemented. Select one: True False AQUESTION 13 A perfectly inelastic demand curve exhibits O a change in quantity demanded that is proportional to the change in price. O zero quantity demanded when there is a slight change in price. O a change in quantity demanded that is always twenty percent of the change in price. O zero responsiveness to changes in price. QUESTION 14 The price elasticity of demand measures how responsive consumers are to a change in income. how responsive market prices are to a change in demand. how responsive consumers are to a change in price. O changes in demand. QUESTION 15 If demand is inelastic and the price of a product decreases by 100 percent, then O the change in quantity demanded is equal to 100 percent. O the decrease in quantity demanded is greater than 0 percent. the change in quantity demanded is less than 100 percent. the change in quantity demanded is greater than 100 percent.4. Draw Figure for Comparative Static Result for a $500.00 increase in I. How would an economist describe the relationship between widgets and income? 5. Calculate the income elasticity of demand for widgets. Interpret your results. 6. Calculate the price elasticity of supply for widgets. Interpret your results