Problems that are on grey shading are optional: you are encouraged to work on them, but donβt need to turn them in (there is no extra credit for doing so)

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

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

Consider the utility function π(π₯, π¦) = π₯π¦, 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 Emiβs 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 Emiβs purchase? Check how much of each good Emi will want to buy now.

e) Solve Emiβs utility maximization problem for general parameters ππ₯, ππ¦ and πΌ, to find the demand functions π₯β(ππ₯, ππ¦, πΌ) and π¦β(ππ₯, ππ¦, πΌ), and indirect utility π(ππ₯, ππ¦, πΌ).

Plug in prices and income from parts (b) and (c) to check your earlier numerical solutions. (Note that you canβt use these functions to answer the question at part (d), since they implicitly assume that prices donβt 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 πΌ=$24, ππ₯=$2, ππ¦=$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 donβt submit:

2) Optimal choice: example #2

Emiβs friend Xindi also likes tacos, but she doesnβt like sandwiches. Instead, she always consumes exactly two tacos (good x) with one glass of horchata (y), and she doesnβt 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 Xindiβs 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?

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 π₯β(ππ₯, ππ¦, πΌ) and π¦β(ππ₯, ππ¦, πΌ), and indirect utility π(ππ₯, ππ¦, πΌ).

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: Albertoβs (x) and Robertoβs (y). But letβs face it, Robertoβs 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 Albertoβs tacos in order for her to give up 2 Robertoβs tacos.) a) Write Aliaβs 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 Aliaβs demand functions π₯β(ππ₯, ππ¦, πΌ) and π¦β(ππ₯, ππ¦, πΌ), 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 π(π₯, π¦) = π₯π¦ + π₯ + π¦

a) Compute the marginal utilities of x and y, and the marginal rate of substitution of x for y (ππ
ππ₯,π¦). Do we have diminishing MRS?

b) Let Amy have budget I=$10, and let prices be Px=$1, Py=$2. Find Amyβs 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 Amyβs 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 Bobβs 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 Bobβs preferences over pet snakes (x) and pet mice (y)? Explain why or why not.

d) Holding πΌ = 60 and ππ₯ = 30, draw the demand curve for good y. (Mark clearly the coordinates at a couple of points β beyond that, the graph doesnβt 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 π π¦

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 Carlaβs preferences over bread (x) and ice cream (y)? Explain why or why not.

d) Solve for the demand functions π₯β(ππ₯, ππ¦, πΌ) and π¦β(ππ₯, ππ¦, πΌ).

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 (πΌ=$24, ππ₯=$2,

ππ¦=$8).

g) Draw the corresponding Engel curves.