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INTERMEDIATE MICROECEONOMICS

BA (HONOURS) ECONOMICS, MIRANDA HOUSE

SEMESTER 3, 2013-14

PRACTICE QUESTIONS

True/False/Uncertain

1. If you know the slope of the budget constraint for two goods, X and Y, then you know the

prices of the two goods, X and Y. Explain.

2. A person who gives money away to people on the street does not have preferences that can be

represented by a utility function.

3. If Patrick's utility function is U ( x , y ) = x y and Julie's utility function is U(x,y) = x1/2 y1/2,

then Patrick will always derive more happiness than Julie does from any combination of x

and y .

4. A monotonic transformation of a utility function does not change the marginal rate of

substitution at any point.

5. If Lori has downward-sloping indifference curves, then this is the same thing as saying that

she prefers averages to extremes.

6. If Steve is maximizing U(x,y) subject to a budget constraint, then at the maximal point, the

marginal utilities of x and y will be equal.

Short Answer

1. Keith's preferences over cars are described as follows: One car is preferred to another if its fuel

efficiency is lower by 2 km/lt. Otherwise Keith is indifferent between the two. Are Keith's

preferences complete? Are they transitive? Can they be represented by a utility function.

2. Mr. Smith likes cashews better than almonds and likes almonds better than walnuts. He likes

pecans equally well as macadamia nuts and prefers macadamia nuts to almonds. Assuming his

preferences are transitive, which does he prefer –

i. pecans or walnuts?

ii. cashew or macadamia?

3. Picabo, an aggressive skier, spends her entire income on skis and bindings. (Binding are the

mechanism by which skiers attach their boots to the skis.)

a. If Picabo wears out one pair of bindings for every one pair of skis, graph her indifference

curves for skis and bindings, illustrating bindings on the horizontal axis and skis on the

vertical axis.

b. If Picabo wears out two pairs of bindings for every one pair of skis, graph her indifference

curves for skis and bindings, illustrating bindings on the horizontal axis and skis on the

vertical axis.

c. Now assume that Picabo has $5,760 in income to spend on binding and skis each year. Skis

cost $480 per pair, and bindings cost $240 per pair.

i. Graph Picabo's optimal consumption bundle for skis and bindings under the

assumptions in part a).

ii. Graph Picabo's optimal consumption bundle for skis and bindings under the

assumptions in part b).

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4. Paula, a former actress, spends all her income attending plays and movies. She likes plays

exactly three times as much as she likes movies.

a. Graph Paula's indifference curves, illustrating plays on the horizontal axis and movies on the

vertical axis.

b. Paula earns $120 per week. If tickets to plays cost $12 each and tickets to movies cost $5

each, graph her optimal consumption bundle, illustrating plays on the horizontal axis and

movies on the vertical axis.

5. Sally likes peppermint candy canes in her hot chocolate. Specifically, she will only drink hot

chocolate with 2 candy canes in each cup. Sally has a weekly income of $15 to spend on hot

chocolate and candy canes. Hot chocolate costs $1.50 per cup.

For the purposes of deriving Sally's demand curve for candy canes, use the following three prices

for candy canes: P1=$0.25, P2=$0.50, and P3=$0.75. Draw two separate graphs. On the top

graph, illustrate the three optimal consumption points. On the bottom graph, illustrate Sally's

demand curve.

6. Consumer A has utility function u(x,y)=x1/3y2/3. Provide an expression that describes the

consumer maximization problem. Derive the optimal bundle when px=10, py=5 and I=60. How

does the bundle change if py=10?

7. Consumer B's utility for goods x1 and x2 is described by u(x1,x2)=x1ax2

b. Prices and income are

given by p1; p2 and I. Derive consumer A's demand for goods x1 and x2. What are the shares of

income spent on goods x1 and x2, respectively? How do the shares change if the price of good x1

doubles?

8. Consumer C's preferences are described by the utility function u(x,y)=sqrt(xy). Her desired utility

level is given by the parameter u*. Given prices px and py, derive the expenditure-minimizing

bundle of goods that provides this level of utility u*. Describe how the consumption of goods x

and y changes in response to changes in u*, px and py.

9. Maximize u(x,y)=x1/2+y subject to pxx+pyy=I. You may need to consider different solutions for

different values of the parameters. Briefly explain your solution.

10. Suppose there are three commodities, all of which can be consumed in any non-negative amount,

and a consumer has a utility function u ( x 1 , x 2 , x 3 ). The consumer's choice of a commodity bundle

must satisfy the budget inequality:

P1 + P2 + P3 < m,

where P1 , P2 , P3 and m denote the market prices and income, respectively (all assumed to be

strictly positive). If the marginal utility for each commodity is positive (all commodities are

goods), is it possible that a utility maximizing consumer will choose a commodity bundle which

costs less than m? Explain your answer. Prove that if m were to increase, while prices remain

unchanged, the demand for at least one commodity will increase, i.e., at least one commodity is a

normal good.

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11. Suppose a consumer has an income of $25 and buys 10 apples and 5 oranges when the price of

apples is $2 and that of oranges is $1. At another time, the same consumer is observed consuming

5 apples and 10 oranges when the price of apples is $1 and that of oranges is $2. Is this behavior

consistent with utility maximization? Explain. (A diagram will help). You may assume that the

consumer's income and tastes remain unchanged during the period these observations were made.

Also assume that both commodities are goods.

12. Consumer 1 has a utility function u(x1, x2) = x1 x2 and consumer 2 has a utility function u(x1, x2) =

min(x1, x2). Derive the demand functions for each consumer.

a. Suppose each consumer has income of $90 and the market prices are p1 = 1, p2 =1. How

much does each consumer demand of each commodity?

b. Suppose the government needs to raise revenues through taxation the two proposals being

considered are the following:

i. a specific tax of $ 1 per unit will be imposed on commodity 2. This means that

the consumers will face a new price of $2 for commodity 2. For every unit of

commodity 2 that a consumer buys the government collects $1.

ii. a lump-sum tax of $30 on each consumer. This means that the prices remain

(1,1) but each consumers disposable incomes goes down from $90 to $60.

How much total revenue does the government collect from policy (A)? How much does it

collect from policy (B)? Does consumer 1 prefer one tax policy to the other? If so, which

one and why? Make the same comparison for consumer 2. Based on your answers can you

conclude that one kind of tax policy is better than the other? Explain.

13. Derive the demand functions for the utility function

U(x1 , x2) = x11/2 + x2

1/2

a. Assume that the consumption of both commodities in non-negative and that

prices and income are strictly positive. You do not need to explicitly use the Lagrangian

for the derivation but state clearly why the solution to the consumer's problem is unique.

For example, if you get a tangency solution state clearly why there is no corner solution.

b. What are the demands when p1 = p2 = 1 and m = 4? Suppose the price of

commodity 1 goes up to 2 (but p2 remains unchanged). Of course, if income remains

unchanged, the consumer is made worse-off. Suppose the consumer succeeds in

convincing her employer that her income should be raised to m' so that even after p1 goes

up from 1 to 2, she can still just afford to buy her original demand bundle. What is the

value of m'?

c. Sketch the original budget line, the original demand, and the new budget line.

Although the consumer can now continue to consume the original commodity bundle

(before the change in p1 and m), is it rational to do so? From your figure can you argue

that the consumer is actually better-off with this new budget line?

d. Use your derivation of the demand functions to compute the demands on the

new budget line, and use these to prove that the consumer is better-off on the new line.

(Use the utility function to check whether or not one bundle is better than another).

e. Calculate the value of income, m*, which would leave the utility of the

consumer unchanged, after p1 goes up to 2 but p2 remains unchanged. In other words,

find the value of m* such that when the budget line is 2x1 + x2 = m*, the consumer's

utility remains the same as it was with the original demand bundle.

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14. (Special credit) Consider the consumption maximization problem of young John D. Rockefeller

in 1855. He chooses to work hours L for wage w, since he is only a clerk in the

merchant house of Hewitt & Tuttle. He must choose how to spend his income on

two goods: clothing C and housing H, from which he derives utility via utility

function U = Ca + Ha — L where a ϵ (0,1). Suppose that prices of one year of

rent for housing and on year's worth of clothing are r and p, respectively.

1. Solve for Rockefeller's optimal choice of L1 , C1, H1.

2. It is now 1872, and John D. Rockefeller has just finished consolidating his competitors in oil

refining to form Standard Oil. His hourly wage has now jumped to W> w . He still lives in

the same small house in Cleveland as he did in 1855, though, since he still has time left on his

lease. Solve for his optimal choice of C2 and L2.

3. At the end of the year, Rockefeller's lease is finally up, so he can choose housing freely. Solve

for his new choices of L3,C3, and H3.

4. In which period did Rockefeller work the most?