Graded Problem Set 1
Deadline: Oct 2nd, 2019
(Please hand in your answers to Antony)
Please be sure to seek help from the me or Antony at our o¢ ce hours if you
…nd any question challenging. We’re happy to provide guidance.
1. Suppose Bob has a budget of $132 dollar tomorrow to spend on three
commodities, namely, bottled water, sandwiches, and chocolate bars. Let the
prices of these commodities be $5 per bottle of water, $20 per sandwich, and
$8 per chocolate bar. For breakfast, Bob plans to purchase one bottle of water,
one sandwich, and no chocolate. For lunch, he plans to purchase two bottles of
water, one sandwich and two chocolate bars. For dinner, he plans to purchase
one bottle of water, two sandwiches and one chocolate bar.
a) Plese calculate Bob total planned consumption bundle for tomorrow.
(Please use vector x to denote Bob’s total consumption of the day, and xB ,
xL , and xD to denote, respectively, his consumption bundles for Breakfast,
Lunch, and Dinner.)
b) Please write down Bob’s budget set per day and whether the planned
spending for tomorrow in his budget set.
c) Suppose Bob’s utility is determined by his total consumption of the day
x and not by his separate consumption during the three meals. In particular,
his utility function is
u (x) = min fx1 ; x2 ; x3 g
where x1 , x2 , and x3 are his total consumption of bottle water, sandwiches,
and chocolate bars for the day. How should Bob adjust his consumption if his
objective is to maximize his utility given his budget?
2. Draw a set of indi¤erence curves for the following pairs of goods:
a) Hamburgers and carrots for a vegetarian who neither likes nor dislikes
meat. (Vegetarians do not eat meat.)
b) Peanut butter and jelly for an individual that will not eat peanut butter
sandwiches or jelly sandwiches, but loves peanut butter and jelly sandwiches
made with two parts peanut butter and one part jelly.
c) Tickets for Knott’s Berry Farm (KBF) and Universal Studios (US) for a
tourist that believes that KBF and US are perfect substitutes.
d) Bob loves hamburgers and hates soft drinks.
3. Connie has a monthly income of $200, which she allocates between two
goods: meat and potatoes.
a) Suppose meat costs $4 per pound and potatoes cost $2 per pound. Draw
her budget constraint.
b) Suppose also that her utility function is given by the equation u(M, P) =
2M + P. What combination of meat and potatoes should she buy to maximize
her utility? (Hint: Meat and potatoes are perfect substitutes.)
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c) Connie’s supermarket has a special promotion. If she buys 20 pounds of
potatoes (at $2 per pound), she gets the next 10 pounds for free. This o¤er
applies only to the …rst 20 pounds she buys. All potatoes in excess of the …rst
20 pounds (excluding bonus potatoes) are still $2 per pound. Draw her budget
constraint.
d) An outbreak of potato rot raises the price of potatoes to $4 per pound.
The supermarket ends its promotion. What does her budget constraint look
like now? What combination of meat and potatoes maximizes her utility?
4. Clara’s utility function is U(X, Y) = (X + 2)(Y + 1), where X is her
consumption of good X and Y is her consumption of good Y.
a) Write an equation for Clara’s indi¤erence curve that goes through the
point (X, Y) = (2,8). On a graph below, sketch Clara’s indi¤erence curve for U
= 36.
b) Suppose that the price of each good is 1 and that Clara has an income of
11. Draw in her budget line. Can Clara achieve a utility of 36 with this budget?
c) At the commodity bundle, (X, Y), …nd the Clara’s marginal rate of substitution of Y for X, as a function of X and Y.
d) Solve for these two equations for the two unknowns, X and Y.
5.Donald Fribble is a stamp collector. The only things other than stamps
that Fribble consumes are Hostess Twinkies. It turns out that Fribble’s preferences are represented by the utility function u(s, t) = s + ln t where s is the
number of stamps he collects and t is the number of Twinkies he consumes. The
price of stamps is ps and the price of Twinkies is pt. Donald’s income is m.
a) Write down the optimal condition of this consumer:
b) Using the equation you found in the last part to show that if he buys
both goods, Find the Donald’s demand function for Twinkies.
c) Notice that for this special utility function, if Fribble buys both goods,
then the total amount of money that he spends on Twinkies has the peculiar
property that it depends on only one of the three variables m, ps and pt. Write
down the variable without explaining.
d) Since there are only two goods, any money that is not spent on Twinkies
must be spent on stamps. Find an expression for the number of stamps he will
buy if his income is m, the price of stamps is ps and the price of Twinkies is pt
.
e) Donald’s wife complains that whenever Donald gets an extra dollar, he
always spends it all on stamps. Is she always right? Explain brie‡y.
6. David goes to the supermarket. He sees that bags of potato chips are
labeled “buy 3 get 1 free.” Assume that he has an income of $300 to spend on
potato chips and other goods, the price of other goods is $10, and the price of
a bag of potato chips is $20.
a) Draw David’s budget constraint.
b) Suppose that David would purchase 2 bags of potato chips if there was
no “buy 3 get 1 free.”Would he buy more than 2 bags when there is “buy 3 get
2
1 free?” Is it possible that he would buy less than 2 bags when there is “buy 3
get 1 free?” Explain.
c) Suppose that David would purchase 4 bags of potato chips if there was
no “buy 3 get 1 free.”Would he buy more than 4 bags when there is “buy 3 get
1 free?” Is it possible that he would buy less than 4 bags when there is “buy 3
get 1 free?” Explain.
7. Suppose a new technique for studying economics is invented, and that it
really works; that is, it really does increase a student’s economics exam score
for any given number of hours of studying.Assuming that Peter studies only
two subjects: economics and physics. Is it possible that Peter’s economics exam
score decreases despite using the new technique? Explain.
(Hint: To answer this question, assume the student has preference over
his “economics score” and “physics score” and that the score of each subject
increases linearly in the hours he spends on that subject. His budget constraint
captures his allocation of a …xed amount of time spent on these two subjects.)
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