Question

EC 501: Problem Set 3

(Due in class on Tuesday, September 27)

1. The following table lists three situations for an individual who consumes two goods X1 and

X2. The table lists the prices of the goods P1 and P2, the quantities consumed of the goods,

x1 and x2, the consumer’s nominal income I, and his utility level, U.

____________________________________________

Situation

P1

P2

x1

x2

I

U

____________________________________________

1

1

1

50

40

90

10

2

1

½

48

84

90

15

3

1

½

25

70

60

10

____________________________________________

(a)

When the price of good 2 drops from $1 to $1/2, what is the change in the quantity

demanded of X2 when nominal income is constant at $90? What part of this change is due

to the substitution effect and what part is due to the income effect? Is X2 a normal good or is

it an inferior good?

(b)

Fill in the blanks in the table and answer the questions in part (a) for this table:

____________________________________________

Situation

P1

P2

x1

x2

I

U

____________________________________________

1

1

1

50

__

70

10

2

1

½

52

__

70

15

3

1

½

__

34

58

10

____________________________________________

2. Todd buys only two types of food: hamburgers (H) and hot dogs (D). He likes them equally

well, except that he feels a hamburger always gives him the same utility as three hot dogs.

(a) Write down Todd’s utility function.

(b) Find Todd’s demand functions for hamburgers and hot dogs.

(c) If Todd’s food budget is $50 per week and the prices for hamburgers and hot dogs

are, respectively, $5 and $2, find how many hamburgers and hot dogs Todd would

buy each week.

(d) Suppose the price of hamburgers went up to $8. What would Todd’s consumption

pattern be now? How much of the change in consumption of each good would you

attribute to the substitution effect and how much to the income effect?

3. The only way Sarah entertains herself is to go to the movies, where she always buys two

bags of popcorn.

(a) Let M and P represent the quantities of movies and popcorn Sarah consumes, let pm and

pc be their prices and let I be the amount that Sarah has budgeted for entertainment.

What is Sarah’s utility function for entertainment?

(b) Find Sarah’s demand functions for movies and popcorn.

(c) If Sarah’s entertainment budget is $60 per month and pm=$10 and pc=$2.50, how many

movies would Sarah go to each month on average and how many bags of popcorn

would she buy?

(d) Suppose the price of popcorn went up to $5 per bag. What would Sarah’s consumption

pattern be now? How much of the change in consumption of each good would you

attribute to the substitution effect and how much to the income effect?

(e) Find Sarah’s compensated demand function for popcorn and draw her ordinary and

compensated demand curves on the same graph.

(f) Starting with I=$60 per month and pm=$10 and pp=$2.50, find the compensating and

equivalent variations of a change in the price of popcorn to $5.

4. John spends $I on bottled water which he can buy in two sizes: 0.75 liter and 2 liter. The

water is identical in the two sizes and John gets no utility from the containers themselves,

only from the water.

(a) Write down a utility function for John in terms of the number of small containers (x)

and the large containers (y) that he consumes.

(b) Find John’s demand function for y.

(c) Find John’s compensated demand function for y.

(d) Suppose initially I=20, px=1 and py=2. Draw John’s ordinary and compensated demand

curves for y in the same diagram. Make sure to label all the relevant points.

5. Adam’s utility function is U = A1/2 B1/2 where A, B represent the number of apples and

bananas respectively that he consumes.

(a) Find Adam’s compensated demand functions.

(b) Write down Adam’s ordinary demand functions (you don’t need to derive them).

Suppose Adam’s income is $100 and the prices of the goods are pa =$2 and pb =$5.

How much A and B would Adam buy?

(c) Suppose pa rises to $5. How much A would Adam buy now? What if his income were

compensated to keep his utility constant? By using your answers to these questions, plot

the ordinary and demand curves for A on the same graph.

(d) Find the compensating and equivalent variations of this price change.

.