A consumer has an income 𝑚 > 0 to spend on good 1 and good 2. The price of good 1 is 𝑝1 and the price
of good 2 is 𝑝2. The consumer’s utility function is given by
𝑈(𝑥1, 𝑥2) = 𝑥1
1/5 𝑥2
4/5,
where 𝑥1 > 0 is the amount of good 1 consumed, and 𝑥2 > 0 is the amount of good 2 consumed.
(a) Find the consumer’s optimal consumption bundle.
(b) Find the consumer’s indirect utility function.
(c) How does the value of the indirect utility function change as the price of good 1 increases?
(d) How does the value of the indirect utility function change as the price of good 2 increases?
(e) How does the value of the indirect utility function change as the consumer’s income increases?
For (c), (d), and (e), show your calculations and determine whether the change in the optimal value is
an increase or a decrease.
2. Consider the expenditure minimization problem for the consumer in Problem 1.
(a) Find the consumer’s expenditure function. What is the relationship between the expenditure function
and the indirect utility function you found in Problem 1(b)? Explain.
(b) Show that the expenditure function is homogenous of degree 1 in prices.
(c) Show that the expenditure function is nondecreasing in each of the prices.
3. Jaden’s mom gives him $12 every day to buy lunch at school. Jaden only likes fried chicken and frozen
yogurt, so he buys these two items every day. His utility function from eating lunch is
𝑈(𝑥1, 𝑥2) = 𝑥1
1/2𝑥2
1/2,
where 𝑥1 > 0 is the amount of fried chicken he consumes (in pounds) and 𝑥2 > 0 is the amount of
frozen yogurt he consumes (in ounces).
(a) Suppose that fried chicken costs $3 per pound and frozen yogurt costs $2 per ounce. Calculate
Jaden’s optimal amounts of fried chicken and frozen yogurt.