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2. Two former college roommates (Jim and Nathan) conduct a blind taste test of
three beverages: T, E, and B. Units of each beverage are 1 ounce.
Their preferences over these three beverages can be represented by the following utility functions:
UJim(T, E, B) = 20T + 16E + 10B UNathan(T, E, B) = 4T + 6E + 8B
a. How would Jim rank these three beverages?
Use appropriate numbers and appropriate economics vocabulary to justify your answer. (2 points)
. How many ounces of T would Nathan be willing to give up to get 1 additional ounce of B?
What is the appropriate economics vocabulary term for this? (2 points)
c. The prices of these three beverages are ordered as follows: pT < pE < pB.
Even though Jim and Nathan have different preferences over these three beverages,
given these prices, an income of $I, and the preferences represented above,
Jim and Nathan would both choose the same optimal consumption bundle.
Clearly explain how many units of each good this optimal consumption bundle contains. That is,
find T*, E*, and B*. (3 points)
Note: Your answers will be functions of I, pT, pE, and/or pB, not numbers.
d. The prices of these three beverages are ordered as follows: pT < pE < pB.
Even though Jim and Nathan have different preferences over these three beverages,
given these prices, an income of $I, and the preferences represented above,
Jim and Nathan would both choose the same optimal consumption bundle.
Clearly explain what this implies about the numerical value of the ratio pB/pT. (3 points)
Hint: Your answer should be expressed as an inequality.
If
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3. Where I grew up in western Ohio, county fairs were a big deal in the summer. Obviously, the main
attractions for kids are the rides (R) and the food (F). My preferences for rides and food at the time
could have been represented by the following utility function: U(R, F) = R⋅F.
a. Assume I had $20 to spend at the county fair. Rides cost $5 per ride, and food cost $1 per unit
(you can think of one “unit” as $1 of food). What was my optimal consumption bundle? (2 points)
b. If my limited budget hadn’t been a constraint, then my limited time would have been.
Assume (for part b only) that I had an unlimited budget, but I only had 8 hours to spend at the fair.
Waiting in line for a ride and riding took one hour per ride.
Waiting in line for food and eating it took one hour per unit of food.
What was my optimal consumption bundle given my limited time to spend at the fair? (2 points)
c. Unfortunately, I was not independently wealthy as a child, so both my limited budget ($20)
and the limited amount of time I had to spend at the fair (8 hours) were constraints.
Given both of these constraints, what was my optimal consumption bundle? (4 points)
d. Illustrate your answer to part c using an appropriate diagram.
Put R on the horizontal axis of your diagram.
Your diagram should clearly illustrate both of the relevant constraints in this problem,
the relevant opportunity set, and the optimal consumption bundle from part c.
Clearly label all intercepts and interesting points with appropriate numerical values. (5 points)
e. If my parents wanted to be really generous, they might have offered me the following choice:
“You can either spend as much time as you want at the fair and spend $20,
or you can spend as much money as you want at the fair and spend 8 hours.”
Which of these options would I have chosen?
How much extra time or extra money would I have gotten?
Justify your answers numerically. (4 points)
2Nd
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4. Consider the following utility function: U(X, Y) = X0.5 + Y0.5
a. Derive the marginal utility of good X (MUX), the marginal utility of good Y (MUY),
and the marginal rate of substitution (MRS) for this utility function. (4 points)
b. This utility function generates smooth, curved indifference curves.
For a generic budget constraint, I = pX∙X + pY∙Y,
derive a formula for the optimal consumption of good X as a function of pX, pY, and I
for a consumer with preferences that can be represented by this utility function. That is,
find X*(pX, pY, I). (4 points)
c. Given the formula you derived in part b,
find the numerical value of X* when I = 150, pX = 1, and pY = 1. (1 point)
d. Given the formula you derived in part b,
find the numerical value of X* when I = 150, pX = 1, and pY = 2. (1 point)
e. Given your answers to parts c and d,
what is the sign of !"
∗
!#"
for a consumer with preferences that can be represented
by the utility function in this problem? In other words, is !"
∗
!#"
positive or negative?
Use appropriate economics vocabulary to explain in words what this means. (3 points)
f. Given your answer to part b,
what is the sign of !"
∗
!$ for a consumer with preferences that can be represented
by the utility function in this problem? In other words, is !"
∗
!$ positive or negative?
Use appropriate economics vocabulary to explain in words what this means. (2 points)
if
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5. A household in northern Ghana has preferences over “Stove goods” (S) and “Nonstove goods” (N)
that can be represented by a utility function that generates smooth, curved indifference curves.
Stove goods (S) are goods like food, tea, heat, and light that require a fuel-burning cookstove. That is,
each unit of good S consumed requires this household to use (and buy!) a certain amount of fuel.
Nonstove goods (N) are “all other goods” (that don’t require fuel or a fuel-burning cookstove).
A development economist is interested in studying the effect of more fuel-efficient cookstoves
on fuel consumption in northern Ghana.
A more fuel-efficient cookstove effectively lowers the price of Stove goods (S). A more fuel-efficient
cookstove doesn’t make food or tea cheaper, nor does it change the price of fuel, but it means that
a household needs to use (and buy!) less fuel for each unit of S, effectively lowering the price of S.
a. Assume that this household faces a typical, generic, linear budget constraint: pS⋅S + pN⋅N = I.
Use an appropriate diagram to clearly illustrate the income and substitution effects
of more fuel-efficient cookstoves (a decrease in the price of S).
Put S on the horizontal axis of your diagram.
Important: Your diagram should clearly indicate both an initial optimal consumption bundle,
and a new optimal consumption bundle consistent with preferences for S and N such that
Nonstove goods (N) are inferior goods for this household. (5 points)
b. Given your answer to part a, are N and S net substitutes? Clearly explain your answer. (2 points)
c. Given your answer to part a, are Stove goods (S) normal or inferior goods? Clearly explain your
answer. (2 points)
d. Given your answer to part a, are N and S gross substitutes or gross complements? Clearly explain
your answer. (2 points)
e. It’s possible that more fuel-efficient cookstoves, which require less fuel for each unit of S,
could actually result in an increase in total household fuel consumption.
Use appropriate economics vocabulary to clearly explain how and why this could happen.
(2 points)
f. Is it more likely that more fuel-efficient cookstoves
will result in an increase in total household fuel consumption
if Nonstove goods (N) are inferior goods (as in this problem),
or if Nonstove goods (N) are normal goods?
Clearly explain your answer. (2 points)
4PM