Fin 2005 (Fall 2023)
Problem Set 1
Instructions
• Due date: Tuesday, September 26, 2023, 23:59:59
• Please answer all questions by writing complete English sentences as appropriate.
• Please show all your work for each question.
• Please feel free to collaborate with your classmates on the homework. However, you
must hand in your own assignment.
• Please submit your assignment as an attachment through NTU COOL. The attachment
can be in any easily readable format, such as PDF, Word, JPEG, or PNG.
1
Rational Preferences
Charlie, an experimental economist, wants to investigate whether people’s actual choices
are consistent with the axioms of ”rational preferences” (completeness, reflexivity, and transitivity). Therefore, he conducts a simple experiment in which participants are endowed
with a fixed amount of money, namely $50, and are asked to select a consumption bundle
of cookies (x1 ) and candies (x2 ). Throughout this experiment, Charlie proposes a series of
hypothetical prices (p1 , p2 ) to the participants and observes the choices they make.
The choice profiles of one participant, Bob, are provided below:
Proposed prices: (p1 , p2 )
Choices: (x1 , x2 )
(5,5)
(3,8)
(7,2)
(6,4)
(4,4.75)
(7,0.5)
Focusing on transitivity, discuss whether or not Bob’s preferences are rational.
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2
Cardinality Doesn’t Matter
At McDonald’s, Albert has a tendency to choose between a Big Mac, x1 , and a chicken
sandwich, x2 . The price of a Big Mac is p1 = N T $80, and the price of a chicken sandwich
is p2 = N T $50. He usually spends N T $800. Suppose his preferences over fast food can be
represented by the following utility function:
U (x1 , x2 ) = x61 x42
(1) If Albert seeks to maximize his utility given his budget constraint, write down the corresponding Lagrangian function.
(2) Derive the first-order conditions and compute the optimal consumption bundle that
maximizes Albert’s utility.
(3) What is Albert’s utility?
Lizzy, Albert’s friend, comes to the same shop. She also chooses between a Big Mac and a
chicken sandwich and spends N T $800. Her utility function is:
V (x1 , x2 ) = 1.5 ln x1 + ln x2
(4) Does Lizzy choose the same consumption bundle as Albert? What about her utility?
(5) Briefly discuss the reasons for your findings in (4).
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3
Promoting Electric Vehicle Adoption can be Eco-Unfriendly
Many policymakers worldwide subsidize electric vehicles (EV, henceforth) buyers to electrify
the vehicle fleet and reduce carbon emissions from the transportation sector. According to
recent research, a typical combustion vehicle emits 4.6 tons of CO2 per year, while an EV
emits zero CO2 .
Suppose a representative citizen in Taipei chooses between a combustion vehicle, x1 , and
an EV, x2 . Her budget is 120. The price of a combustion vehicle is p1 = 4, and the price of
an EV is p2 = 16. A representative citizen’s utility function is:
U (x1 , x2 ) =
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(x1 + x2 )
(1) Write down the Lagrangian function corresponding to the citizen’s utility maximization
problem.
(2) Derive the optimal consumption bundle of combustion vehicles and EVs.
(3) Calculate the amount of carbon emissions from the vehicle fleet (4.6 ⇥ x1 ).
The Taipei city council seeks to reduce total carbon emissions through an ad-valorem subsidy for EVs: p2 = 16 ! p2 (1
) = 9. The council believes that carbon emissions will
decrease due to this subsidy as the demand for EVs would rise.
(4) Find the optimal consumption bundle under this subsidy scheme.
(5) Do the quantities demanded for EVs increase? Does this increase result in a reduction
in carbon emissions?
(6) Discuss the reasons for your findings in (5) (Hint: Look into the relation between a
combustion vehicle and an EV).
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4
Food Stamp Subsidy
Recall the 1964 Food Stamp Program in the U.S., discussed in class. Suppose a representative U.S. household with an income of U S$300 chooses between food, x1 , and composite
commodities, x2 . Assume that the prices of both commodities are equal: p1 = p2 = 1. The
household’s utility is given by:
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U (x1 , x2 ) = 30 x1 + x2
(1) Consider a household without food stamps. What bundle of food and composite commodities should the household purchase to maximize its utility?
(2) Now, the household has food stamps. What is the utility-maximizing bundle of food and
composite commodities?
(3) Suppose policymakers of this program were only interested in increasing the food consumption of low-income households (not the household’s overall utility). Based on your
answers to (1) and (2), how would you evaluate the effectiveness of this program?
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