# Math – PROBLEM SET 4

Spring 2021, Tocoian

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PROBLEM SET 4

– out of 16 points, due Monday May 3rd at 11pm –

(Part 2 was added to Part 1, which had been posted last week as “PracticeProblems.pdf”)

Problems that are on grey shading are optional: you are encouraged to work on them, but don’t need

to turn them in (there is no extra credit for doing so)

Please turn in problems 1, 3, 5, and 6.

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1) (4 pts) Optimal choice: example #1

Consider the utility function 𝑈(𝑥, 𝑦) = 𝑥𝑦, which describes the enjoyment Emi gets from consuming

tacos (x) and sandwiches (y) over a period of 1 week.

a) Does Emi like both tacos and sandwiches? Does she like variety?

b) Let Emi have budget I=$24, and let prices be Px=$2, Py=$4. Find Emi’s optimal basket of goods x

and y. Is this an interior or a corner solution?

c) What will happen if tacos go on sale for $1? Find the new optimal bundle.

d) Continuing from these prices, suppose that, in order to boost salles, the sandwich vendor

introduces a discount card. Each week, every sandwich a consumer buys after the first 3 will be

on a 50% off sale. (Purchase records are not transferred from week to week.) Draw the new

budget constraint and express it algebraically.

Will this change have an effect on Emi’s purchase? Check how much of each good Emi will want

to buy now.

e) Solve Emi’s utility maximization problem for general parameters 𝑝𝑥, 𝑝𝑦 and 𝐼, to find the demand

functions 𝑥

∗

(𝑝𝑥, 𝑝𝑦,𝐼) and 𝑦

∗

(𝑝𝑥, 𝑝𝑦,𝐼), and indirect utility 𝑉(𝑝𝑥, 𝑝𝑦,𝐼).

Plug in prices and income from parts (b) and (c) to check your earlier numerical solutions. (Note

that you can’t use these functions to answer the question at part (d), since they implicitly assume

that prices don’t vary with the quantity bought)

f) Calculate the income and own-price elasticity of demand for good x, and well as the cross-price

elasticity of demand for good x with respect to the price of good y.

Starting from 𝐼=$24, 𝑝𝑥=$2, 𝑝𝑦=$4:

g) Draw the income consumption curve. (You have to figure out what axes you need and which

parameters can change.)

h) Draw the corresponding Engel curves for goods x and y (Once again, pay attention to the axes.)

EXTRA: Think about but don’t submit:

2) Optimal choice: example #2

Emi’s friend Xindi also likes tacos, but she doesn’t like sandwiches. Instead, she always consumes

exactly two tacos (good x) with one glass of horchata (y), and she doesn’t enjoy either good without

the other. Suppose prices are the same as initially in problem 1, so that horchata is twice as expensive

as tacos (Px=$2, Py=$4), and Xindi has the same budget as Emi: I=$24

a) How can we write Xindi’s utility function?

b) What basket of goods does Xindi purchase?

c) What if, just as in problem 1, tacos go on sale for $1 each?

1 Each problem is worth 4 points, for simplicity. The grader will exercise judgement on the number of points to assign, in the

case of problems that are incomplete.

Spring 2021, Tocoian

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d) Next, horchata goes on a 50% off sale for glasses past the first 3. Will Xindi change what she

does?

e) Solve the general problem to find the demand functions 𝑥

∗

(𝑝𝑥, 𝑝𝑦,𝐼) and 𝑦

∗

(𝑝𝑥, 𝑝𝑦,𝐼), and

indirect utility 𝑉(𝑝𝑥, 𝑝𝑦,𝐼).

f) Draw the income consumption curve, starting from the initial parameters.

g) What will the Engel curves look like for these goods, and what axis labels do you need?

h) Draw the price consumption curve for good x, starting from initial parameters, then do the same

for good y.

i) What do the demand curves for these goods look like? Will the demand curve for x shift if the

price of y increases?

3) (4 pts) Optimal choice: example #3

Emi and Xindi have another friend, Alia, who only likes tacos. She only buys tacos from two local

restaurants: Alberto’s (x) and Roberto’s (y). But let’s face it, Roberto’s is the best, and she likes their

tacos 1.5 times as much, regardless of how many she is currently consuming of each. (In other words,

we would have to offer Alia at least 3 Alberto’s tacos in order for her to give up 2 Roberto’s tacos.)

a) Write Alia’s utility function and draw some of her indifference curves.

b-d) Go through cases b-d, same as in problems 1 and 2.

e) Find and write down formally Alia’s demand functions 𝑥

∗

(𝑝𝑥, 𝑝𝑦,𝐼) and 𝑦

∗

(𝑝𝑥, 𝑝𝑦,𝐼), including

for the threshold case. Can you find a compact way of expressing her indirect utility function?

Holding I=$24, Py=$4:

f) draw the price consumption curve for x;

g) draw the demand curve for x

4) Optimal choice: example #4

Consider the utility function 𝑈(𝑥, 𝑦) = 𝑥𝑦 + 𝑥 + 𝑦

a) Compute the marginal utilities of x and y, and the marginal rate of substitution of x for y (𝑀𝑅𝑆𝑥,𝑦).

Do we have diminishing MRS?

b) Let Amy have budget I=$10, and let prices be Px=$1, Py=$2. Find Amy’s optimal basket of goods x

and y. Is this an interior or a corner solution?

c) Now suppose good x becomes extremely expensive: Px=$15. Find Amy’s optimal basket now. Is

this an interior or corner solution?

d) Do you think we could use this U(x,y) function to describe the utility from consuming tacos (x)

and sandwiches (y) over a period of 1 week? Explain why or why not.

5) (4 pts) Optimal choice, example #5

Consider the utility function U(x,y)=x2+y

a) Do we have a name for this type of utility function? Compute the marginal utilities of x and y, and

the marginal rate of substitution of x for y (MRSx,y). Do we have diminishing MRS?

b) Let Bob have budget I=$60, and let prices be Px=$30, Py=$10. Find Bob’s optimal basket of goods

x and y. Is this an interior or corner solution?

c) Do you think we can use this utility function to describe Bob’s preferences over pet snakes (x)

and pet mice (y)? Explain why or why not.

Spring 2021, Tocoian

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d) Holding 𝐼 = 60 and 𝑝𝑥 = 30, draw the demand curve for good y. (Mark clearly the coordinates

at a couple of points – beyond that, the graph doesn’t have to be especially precise.)

e) What is the income elasticity of demand for good y?

6) (4 pts) Optimal choice, example #6

Consider the utility function 𝑈(𝑥, 𝑦) = √𝑥 + 2𝑦

a) Do we have a name for this type of utility function? Compute the marginal utilities of x and y, and

the marginal rate of substitution of x for y (MRSx,y). Do we have diminishing MRS?

b) Let Carla have budget I=$40, and let prices be Px=$2, Py=$8. Find her optimal basket of goods x

and y. Is this an interior or corner solution?

c) Do you think we can use this utility function to describe Carla’s preferences over bread (x) and

ice cream (y)? Explain why or why not.

d) Solve for the demand functions 𝑥

∗

(𝑝𝑥, 𝑝𝑦,𝐼) and 𝑦

∗

(𝑝𝑥, 𝑝𝑦,𝐼).

e) Calculate the income and own-price elasticity of demand for good x, and well as the cross-price

elasticity of demand for good x with respect to the price of good y.

f) Draw the income consumption curve, starting from the given parameter values (𝐼=$24, 𝑝𝑥=$2,

𝑝𝑦=$8).

g) Draw the corresponding Engel curves.