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pset_3_consumer_theory.pdf

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Shenoy
ARE 100A PS 3
Due electronically: 10/26/2017, 6:10pm
Jordan is planning a party and deciding how much of the budget to devote to Beverages (B) and how much
to comestibles (C). Guests at the party will have the utility function
U (B, C) = 3.45B 3 C
(a) Using monotonic transformations, rewrite the utility function in Cobb-Douglas form.
(b) Write Jordan’s maximization problem symbolically as a function of prices PB , PC , and total budget
I. That is, write an expression of the form max utility s.t. budget constraint.
(c) Using either the graphical method (MRS = MRT) or the Lagrange method, solve for the quantity
demanded of B and C as a function of prices PB , PC , and total budget I. Solve for indirect utility as
a function of prices and budget.
(d) To maximize utility, what fraction of the budget will Jordan spend on B, and what fraction on C?
Verify that this is consistent with your answer in (c).
We will first derive the Engel curve:
(e) Fix the prices at PB = 5 and PC = 2. What is the quantity demanded of beverages and comestibles
when the total budget for the party I is 60, 80,100, and 120. (Hint: Start by computing total expenditure on each good.)
(f) On a single graph with B on the x-axis and C on the y-axis, plot the budget set, optimal demand, and
the indi↵erence curve the optimal bundle lies on for each of the budgets listed above.
(g) On a second graph directly below the first one, with B on the x-axis and I on the y-axis, plot the
Engel curve for B. Show how the two graphs relate to each other as we did in class.
(h) Suppose Jordan shops at SecurePath, a grocery store that carries a cheap store-brand variety of beverage called Victor’s Mug that doesn’t t taste very good. Would you expect Victor’s Mug to be a normal
or an inferior good? Explain what this means and the reasoning for your answer.
Next, we will derive the demand curve:
(i) Now suppose the total budget for the party is I = 80 and the price of comestibles is PC = 2. What
is the quantity demanded of beverages when the price of beverages is PB = 2? What about PB = 4,
PB = 6, PB = 8, and PB = 10?
(j) On a single graph with B on the x-axis and C on the y-axis, plot the budget set, optimal demand, and
the indi↵erence curve the optimal bundle lies on for each of the prices listed above.
(k) On a second graph directly below the previous one, with B on the x-axis and PB on the y-axis, plot
the demand curve for B. Show how the two graphs relate to each other as we did in class.
(l) Would you consider beverages and comestibles to be complements, substitutes, or neither? Explain
your reasoning. (Hint: What is the quantity demanded of comestibles at each of the prices above?)
Finally, we will decompose a demand change into income and substitution e↵ects:
(m) What is the optimal consumption bundle when PB = 6, PC = 2, and I = 80? What is Jordan’s utility
at this consumption bundle?
(n) Now let the price of beverages increase to 8. What is the new consumption bundle? What is Jordan’s
utility at this new bundle?
(o) Compute the compensated budget set: Set indirect utility equal to the old utility at the new prices,
and solve for the total budget that would be required to reach the old utility given the new prices.
(p) How much of the change in consumption of B and C between your answers to (m) and (n) is due to
substitution e↵ects? How much is due to income e↵ects? (Hint: what is the compensated demand?)
(q) Depict your answer to (p) graphically as we did in class.
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