Problem Set 3 – Consumer Choice
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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. Suppose Harold’s utility function is given by 𝑈𝑈(𝑥𝑥, 𝑦𝑦) = 6𝑥𝑥
1
4𝑦𝑦
3
4.
a. Write out the lagrangian using 𝑃𝑃𝑥𝑥 = $2, 𝑃𝑃𝑦𝑦 = $6, I = $40.
b. Use the first order conditions to solve for the optimality condition
c. Solve for Harold’s optimal consumption bundle.
d. Draw the budget constraint and optimal bundle on the graph.
e. Calculate Harold’s utility at the optimal bundle.
2. Adam will only drink lattes with a ratio of 1 ounce of espresso (E) and 3 ounces of milk (M).
a. Write the general equation for the utility function.
b. Draw an indifference curve for 𝑈𝑈� = 3. Please put M on the x-axis and E on the y-axis.
c. Find an equation for the optimal consumption ratio.
d. Solve for the optimal bundle if 𝑃𝑃𝑀𝑀 = $1, 𝑃𝑃𝐸𝐸 = $3, I = $12
e. Calculate utility at the optimal bundle.
f. Draw the budget constraint and optimal bundle on the graph.
3. Tacos and hamburgers are perfect substitutes for Sally. She gets the same utility from 2 tacos (T) as
she does from one hamburger (H).
a. Write down an equation that could describe her utility function.
b. Say that the price of tacos is $2, the price of hamburgers is $5, and income equals $20. What is
the optimal consumption bundle for Sally.
c. Draw Sally’s budget constraint, optimal bundle, and indifference curve. Please put H on the xaxis and T on the y-axis.
4. George’s utility is given by 𝑈𝑈(𝑥𝑥1, 𝑥𝑥2) = 6𝑥𝑥1
2𝑥𝑥2.
a. Write out the Lagrangian, using a generic budget constraint (𝐼𝐼 = 𝑝𝑝1𝑥𝑥1 + 𝑝𝑝2𝑥𝑥2).
b. Use the first order conditions to solve for the optimality condition.
c. Combine the optimality condition with the budget constraint to solve for George’s demand
functions 𝑥𝑥1(𝑃𝑃1,𝑃𝑃2,𝐼𝐼) and 𝑥𝑥2(𝑃𝑃1,𝑃𝑃2,𝐼𝐼).
5. Ted’s utility function for goods 1 and 2 is given by 𝑈𝑈(𝑥𝑥1, 𝑥𝑥2) = 3 ln(𝑥𝑥1) + 2𝑥𝑥2.
a. Derive his demand function for good 1.
b. What is the optimal consumption bundle if 𝑃𝑃1 = 2, 𝑃𝑃2 = 20, I = $50?
c. What is the optimal consumption bundle if 𝑃𝑃1 = 2, 𝑃𝑃2 = 20, I = $20?
6. When Lisa cooks a chicken she only eats the chicken wings. Her utility function over whole chickens
and chicken wings is 𝑈𝑈(𝑥𝑥𝑐𝑐, 𝑥𝑥𝑤𝑤) = 2𝑥𝑥𝑐𝑐 + 𝑥𝑥𝑤𝑤, where 𝑥𝑥𝑐𝑐 and 𝑥𝑥𝑤𝑤 are chickens and chicken wings,
respectively. What is her demand function 𝑥𝑥𝑐𝑐(𝑃𝑃𝑤𝑤,𝑃𝑃𝑤𝑤,𝐼𝐼) for chickens?
7. My utility function is 𝑈𝑈(𝑥𝑥1, 𝑥𝑥2) = 2𝑥𝑥1
2𝑥𝑥2 and my utility-maximizing bundle (at existing prices and
my income) consists of 3 units of 𝑥𝑥1 and 2 units of 𝑥𝑥2. If 𝑃𝑃1 = 4, what must 𝑃𝑃2equal?
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