Question

4. Graph the budget constraints described in each of these situations and explain whether you think there will be a unique utility maximizing bundle if preferences are rational, strictly convex, and strongly monotone. Be sure to show your calculations!

(a) Suppose you are at a small chocolate shop. There are milk chocolate truffles (x) and dark chocolate truffles (y). You have $20 to spend
and both truffles are $2 each. Graph the budget constraint on the graph below.

B)A stressed out student trying to decide where to take yoga classes.
They have $36 to spend on yoga classes this month. At SuperStrength yoga studio (x), the first 4 yoga classes are free. After that, SuperStrength yoga classes are $12 each. At the university gym (y), yoga classes come out to be about $3 per yoga class. Graph the budget constraint on the graph below.

(c) Let the price of a new magic lotion (x) be $4 per ounce for the first 8 1/3 ounces, and then $0.16 per ounce after that. The corresponding face wash (y) has a flat price of $1 per ounce. The consumer has $40 to spend on their new skincare regimen. Graph their budget constraint below.

(d) If the people in parts a-c all had the utility function

u(x, y) = √xy

do you expect there to be a unique utility maximizing bundle for parts
a-c, respectively? Explain.

Answer 1

The budget constraint will be depicteda as .

(a) In this case, the price of both truflues is $2 each, and have an income of $20. Hence, the budget constraint is . The graph is as below.

(b) The income to spend on yoga, either x or y, is $36. The price of x, for x from 0 to 4. Afterwards, it is $12. The price of y is $3 flat. Hence, the budget constraint is or for . For however, the constraint is , as there is no loss in income due to taking first 4 x. The graph is as below.

(c) In this case, for , and for , . The price of y is $1 flat, and the income to spend for both is $40. The budget constraint is hence for . However, after x is above 25/3, the budget constraint is or for , as after 25/3, the income left is .

(d) The unique solution for an indifference curve which is convex and monotonic, requires the budget line/curve to be either linear or being concave to origin. In this case, the budget is linear in part (a) and concave (not smoothly) in part (b), and there will be a unique solution. But, in part (c), the budget curve is convex to origin (not smoothly), and hence, there might be more than one solution. The MRS might be equal to the slopes of both budget lines in more than one point on same indifference curve, in this case. However, to be certain, it should be checked.