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Unformatted text preview: Microeconomics, Fall 2007, Dr. Susan Laury Problem Set 5 — Show any work needed to get answers to these questions. Please turn in your
answers using your own paper. (You do not have to include a copy of the questions - just be clear about what question you are answering!) 1. Suppose your utility is given by the equation U = 0.5XY. a. If your income is 120, Px = 10 and Py = 10, what is your utility-maximizing bundle of goods X
and Y? MRS = P—“’
1 = 19
Substitute this into the budget constraint:
I = PXX + PYY 120=10X+10Y
120=20X X = 6, Y: 6 b. What is the level of utility that is associated with the optimal bundle you found in part (a)?
(Hint: take your answer from (a) and substitute it into the Utility function) U = .SXY
U = .5(6)(6)
U = 18 c. If Px decreases to 5, what is your utility-maximizing bundle goods X and Y? If the PX decreases, this does not change the MRS, so you must set the MRS equal
to the new ratio of prices and go from there as in part (a). Remember also that the budget constraint has changed! Y _ 5 X _ E 10Y= 5X Y=.5X 120= 5X+10Y
120 = 5X +10(.5X)
120= 10X X = 12, Y= .5(12)=6 d. Are goods X and Y substitutes, complements, or independent over this range of prices? Please
explain your answer. Independent — when the price of X changed, there was no effect on the quantity of Y consumed. e. How much did your optimal consumption of Good X increase when the price fell from 10 to 5? It increased by 6 units f. How much of this total change (from part e) is a result of the substitution effect? (Hint: you
must ﬁnd the amount of goods X and Y that would be consumed at the original level of utility
— from part b — at the new prices. I will provide an example with a different utility function below.) At the new prices, we saw in part (c) that optimal consumption occurs when y =
.5x. We substitute this into the utility ﬁinction, holding utility at its original level of 18 (from part b).
18 = .5X(.5X)
18 = .25X2
X = 8.48 The substitution effect is the increase caused by the change in relative prices, holding utility constant. So the increase in X is 2.48 (from 6 to 8.48).
g. How much of this total change (from part e) is a result of the income effect? The total effect is the sum of the income and substitution effects. The total
increase is 6 (from part c), so we have: 6 = IE + 2.48
IE = 3.52
h. Is Good X a normal good or an inferior good? Explain your answer. Normal — the income effect is positive. 2. The table below gives information on the individual demand for Julie, Maggie, and Nichole: Price of Julie’s Maggie’s Nichole’s Market
Margaritas Demand Demand Demand Demand
$4 5 5 12 5+2+5= 3+1+4= 1+o+2=
a. If Julie, Maggie, and Nichole are the only three consumers in this market, what is the market
demand for margaritas? Please give me the answer in a table and then draw a graph that shows the market demand curve. See table above. Graph is on the next page. g A ID IL b. If the price of margaritas is $6.50, what is the consumer surplus in the market? When the price if $6.50, one margarita is purchased. The value of this margarita if
$7, so the consumer surplus is 7 — 6.50 = .50 ...
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