Problem Set 4 – Demand
Instructions: Please write your answers on a separate sheet of paper, showing all necessary work. Upload a
single PDF to the assignments folder in Brightspace. The problem set is worth a total of 10 points – 5 for
completion and 5 for correctness of one (or more) question(s) selected by me.
1. Charlie’s utility function is 𝑈𝑈 = 2𝑥𝑥𝐴𝐴𝑥𝑥𝐵𝐵. The price of apples used to be $1, the price of bananas used to be
$2, and his income used to be $40.
a. Graph the original budget constraint, and find the optimal consumption bundle.
b. Draw the new budget constraint on your graph and find the new optimal consumption bundle if
the price of apples increased to $5.
c. Find the level of income that would be required to purchase the old bundle at the new prices.
d. Calculate the income and substitution effects for apples and bananas.
2. As children, Ariel and Bartholomew spend all of their money on silly putty and gummy worms. Ariel’s
utility function is given by 𝑈𝑈𝐴𝐴 = ln(𝑠𝑠𝐴𝐴) + ln(𝑤𝑤𝐴𝐴), where 𝑠𝑠𝐴𝐴 and 𝑤𝑤𝐴𝐴 are the quantities of silly putty and
gummy worms that she consumes, respectively. Likewise, Bartholomew’s utility function is given by
𝑈𝑈𝐵𝐵 = ln(𝑠𝑠𝐵𝐵) + 2ln(𝑤𝑤𝐵𝐵), where 𝑠𝑠𝐵𝐵 and 𝑤𝑤𝐵𝐵 are defined analogously.
a. Find Ariel’s demand function for silly putty, 𝑠𝑠𝐴𝐴(𝑝𝑝𝑠𝑠, 𝑝𝑝𝑤𝑤,𝐼𝐼𝐴𝐴) , and Bartholomew’s demand function
for silly putty, 𝑠𝑠𝐵𝐵(𝑝𝑝𝑠𝑠, 𝑝𝑝𝑤𝑤,𝐼𝐼𝐵𝐵), given that 𝐼𝐼𝐴𝐴 and 𝐼𝐼𝐵𝐵 are Ariel and Bartholomew’s incomes,
respectively.
b. If Ariel and Bartholomew are the only buyers of silly putty, find the market demand for silly
putty, 𝑆𝑆(𝑝𝑝𝑥𝑥, 𝑝𝑝𝑦𝑦,𝐼𝐼𝐴𝐴,𝐼𝐼𝐵𝐵).
c. What is the price elasticity of this market demand?
3. When they grow up, Ariel and Bartholomew’s preferences change, and now they spend all of their money
on vodka cranberry cocktails, which they always make by mixing four parts cranberry juice (c) with one
part vodka (v). Therefore, Ariel’s new utility function is 𝑈𝑈𝐴𝐴 = 𝑚𝑚𝑚𝑚𝑚𝑚 �
1
4
𝑐𝑐𝐴𝐴, 𝑣𝑣𝐴𝐴�, and Bartholomew’s new
utility function is 𝑈𝑈𝐵𝐵 = 𝑚𝑚𝑚𝑚𝑚𝑚 �
1
4
𝑐𝑐𝐵𝐵, 𝑣𝑣𝐵𝐵�.
a. Find Ariel’s demand function for cranberry juice 𝑐𝑐𝐴𝐴(𝑝𝑝𝑐𝑐, 𝑝𝑝𝑣𝑣,𝐼𝐼𝐴𝐴), and Bartholomew’s demand
function for cranberry juice 𝑐𝑐𝐵𝐵(𝑝𝑝𝑐𝑐, 𝑝𝑝𝑣𝑣,𝐼𝐼𝐵𝐵).
b. Find the market demand for cranberry juice 𝐶𝐶(𝑝𝑝𝑐𝑐, 𝑝𝑝𝑣𝑣,𝐼𝐼𝐴𝐴,𝐼𝐼𝐵𝐵), assuming that Ariel and
Bartholomew are the only consumers.
c. Given that the price of vodka is 𝑝𝑝𝑣𝑣 = 8, find the price elasticity of market demand in terms of 𝑝𝑝𝑐𝑐.
4. Suppose Harriett only consumes two commodities, x1 and x2 , and x2 is an inferior good. If P1 rises, do the
resulting income and substitution effects on good 2 work in the same direction as each other, or in
opposite directions?
5. Peter consumes only beer and pretzels, and he only gets utility if they are combined one-for-one:
𝑈𝑈(𝐵𝐵, 𝑃𝑃) = min (𝐵𝐵, 𝑃𝑃). Initially I=$10, PP = $1, and PB = $1, but then PB increases to $2.
a. Putting beer on the x-axis and pretzels the y-axis, draw the income and substitution effects for
this price change.
b. What are the numerical values for the IE and SE in this case?
6. David consumes pizza and hamburgers and his utility function is given by: 𝑈𝑈(𝑝𝑝, ℎ) = 2𝑝𝑝 + ℎ. Initially,
𝑝𝑝𝑝𝑝 = 6, 𝑝𝑝ℎ = 2, and m = 6 (income). Then the price of pizza falls to 𝑝𝑝𝑝𝑝 = 2, but the price of hamburgers
remains the same as does income (at 𝑝𝑝ℎ = 2 and m = 6). What are the NUMERIC VALUES for the
income and substation effects for PIZZA and HAMBURGERS when the price of pizza falls from 6 to 2?
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