Question

Rosie makes fudge using two ingredients, sugar and chocolate. She prefers to have more of both goods, but she also prefers similar quantities of each; otherwise the fudge is too sweet or too bitter. In fact, her preferences are given by the utility function U(S,C) = S 1/2 + C 1/2 .

A. What is Rosie’s preference between the bundle (4,16) and (16,4)? (Note that the consumption bundle is (# units of sugar, # units of chocolate)). What is Rosie’s preference between the bundle (4,16) and (9,9)? What is Rosie’s preference between the bundle (4,16) and (10,10)? Explain your choice in each case.

B. On a graph with sugar on the horizontal axis and chocolate on the vertical axis, draw the indifference curve that gives Rosie 6 units of utility (i.e., U = 6).

C. Find Rosie’s marginal utilities of sugar and chocolate. Does she have diminishing marginal utilities for them? How does that relate to the shape of the indifference curve in part (B)?

D. Write Rosie’s utility-maximizing tangency condition and solve for C. (You do not know prices so you will need to use PS and PC for the prices of sugar and chocolate, respectively.)

E. Write Rosie’s budget constraint. Use this and your work in part (D) to determine the demand for sugar. Is sugar a normal or inferior good for Rosie?

F. Assume that PS = PC = $ 2. What bundle will Rosie consume with an income of $ 20? What bundle will Rosie consume with an income of $ 40? What bundle will Rosie consume with an income of $ 60? Draw an income consumption curve (ICC) based on this data.

G. If war breaks out in the Middle East and sugar is rationed so that Rosie is limited to purchasing only 8 units of sugar, what bundles will she consume with an income of $ 20? Did the restriction alter her consumption? Explain. What bundles will she consume with an income of $ 40? Did the restriction alter her consumption? Explain. (2)

Answer 1

U= S(1/2)+C(1/2)

A. For the bundle (4,16) & (16,4)

(4,16) U= 4/2+16/2

= 2+8

=10

(16,4) U= 16/2+4/2

= 8+2

=10

In both cases the utility is same, Rosie will remain "Indifferent".

For bundle (4,16) & (9,9)

(4,16) U= 4/2+16/2

= 2+8

=10

(9,9) U= 9/2+9/2

= 4.5+4.5   

= 9

As the utility is higher in first case but the amount of chocolate is more. And in second case the utility is 9( which is less than 10) but here the quantity of both ingrediants is same. Therefore, Rosie should choose 2nd bundle given her preference.

For bundle (4,16) & (10,10)

(4,16) U= 4/2+16/2

= 2+8

=10

(10,10) U= 10/2+10/2

= 5+5

= 10

In this case also she'll remain indifferent as the utility is same.But given that she wants equal proportion of both ingredients, she should go for the bundle (10,10)

B. See photo attached for graph

Given, U=S1/2+C1/2, following bundles should give utility = 6 for which Rosie will be indifferent

Sugar (S)	4	8	0
Chocolate (C)	8	4	12