課程名稱︰個體經濟學一
課程性質︰
課程教師︰Patrick Dejarnette (狄萊)
開課學院：社會科學院
開課系所︰經濟系
考試日期（年月日）︰2017.11.13
考試時限（分鐘）：150分鐘
試題 :
1 Short Questions - 25 points total
EACH SHORT QUESTION IS WORTH 5 POINTS. ALL SHORT ANSWER QUESTIONS ARE SCORED.
S.Q.1 Can a good be "Giffen good" for all positive prices? For example, at
p=1, at p=10, at p=100, and so on. Why or why not?
S.Q.2 True, False, or Not Enough Information + Why? Suppose Pat has a utility
for a water bottle = 30 (U(1waterbottle)=30) and a utility for bubble tea = 90
(U1bubbletea)=90). Then Pat would be willing to pay at most three times more
for a bubble tea than he is willing to pay for a waterbottle.
S.Q.3 True, False, or Not Enough Information + Why? If individual demand is
a linear function (for positive quantities) for all individuals in a market,
then market demand will also be a linear function (for positive quantities).
S.Q.4 True, False or Not Enough Information + Why? Although it cannot solve
every probelm, if the Subsitution Method provides a solution, it will be a
maximum and not a minimum; whereas the Lagrangian approach can provide either
a maximum or a minimum. (Note: By the Substitution method, I mean the
Substitution method learned in class to solve constrained optimization
problem, for example utility given a budget constrain.)
S.Q.5 True, False, or Not Enough Information + Why? If there are only two
goods in the market, Apples and Bananas, then for an individual who enjoys
both, Apples and Bananas must be substitutes.
2 Long Questions - 75 points total
EACH LONG QUESTION IS WORTH 25 POINTS. REMEMBER - ONLY CHOOSE 3 OF 4. SEE
INSTRUCTION.
L.Q.1 Pat is considering which snacks to buy: Apple pies (A) or Biscuits (B).
Suppose his utility function to these snacks is
U(A,B)=A^(1/3)+2*B^(1/3)
A. Suppose that the prices of A, and B are pA=1/9, pB=1/8,
respectively, and Pat's allowance ( income) from his mom is I=10.
Please state Pat's constraint. (1 point)
B. Please find A and B that maximize Pat's utility subject to the
budget constraint. (4 points)
For part C and D and E, assume now that the price of apple pies is fixed to 1,
so pA=1, but both pB and I are allowed to vary.
C. What is the demand function for A as a function of just pB and I
(hold pA=1)? You only need to find the demand function for Apple pies,
not B. (4 points)
D. Holding the price of Apple pies to 1, are Apple pies a (cross-price)
complement for Biscuits or a (cross-price) substitute for Biscuits?
Prove it. (2 points)
E. Are Apple pies a normal or inferior food? Prove it. (1 point)
Now Pat has a new choice: Cranberries (C). His utility function to these
snacks becomes
U(A,B,C)=A^(1/3)+2*B^(1/3)+C
Suppose the prices are now as follows, pA=1/27, and pB=1/24 and the new good
C has cost pC=1/4.
F. Suppose Pat's Mom on;y gives him 1 dollar for allowance now, I=1.
Please find the new utility-maximizing A, B and C subject to the new
budget constraint. (6 points)
G. If Pat's mom wanted Pat to buy at least 3 Cranberries, would she
need to change his allowance I? If so, what level of I would cause Pat
to buy at least 3 Cranberries per day? If not, prove that the current
allowance I=1 is sufficient for Pat to want to purchase at least 3
Cranberries at the above prices. (Assume Pat's ,mom cannot actually
choose what Pat consumes, and only alter I.) (3 points)
Suppose now that the store has a new rule. Each Apple pie must be sold in a
set with one Biscuit, and vice versa. In other words, the store only sell
Apple pies and Biscuits in equal quantities (A=B). Assume Pat's utility
function unchanged, and that the price of each Apple pie and Biscuits inchanged
, so that the price of a set is 1/27+1/24. Pat's allowance is I=1.
H. What happens to the optimal quantity of Cranberries that Pat
purchases relative topart F? Why? For full points, prove it
mathematically. But adequate reasoning will help toward partial credit.
(2 points)
I. What happens to Pat's optimal utility relative to Part F? Why? For
full points, prove it mathematically. But adequate reasoning will
help toward partial credit. (2 points)
L.Q.2 Suppose we have a world with two dish, Salmon (S) and Tuna (T). In
addition, we have two markets in the US, New York (NY) and California (C).
Consumers from California have the following utility:
U_C(S,T)=4 * S^(1/3) * T^(2/3)
While consumers from New York have the following utility:
U_NY(S,T)=3 * S^(2/3) * T^(1/3)
A. Can we infer anything about consumption or consumer surplus from
the fact that the constant in U_C is a 4 but the constant in front of
U_NY is only 3? If so, what can we infer? If not, why not? (2 points)
B. Given the budget constraint pS * S + pT * T = I, please derive the
individual demand function of S and T for California consumers.
(3 points)
C. If there were 200 consumers in California all with equal income
I=50, please state the market demand functions for S and T. (1 point)
D. Given the budget constrain pS * S + pT * T = I, please derive the
individual demand function of S and T for New York consumers.
(3 points)
E. If there were 100 consumers in New York all with equal income
I=50, please state the market demand functions for S and T. (1 point)
Suppose the supply functions of S and T for California are
Q_S(pS)=100 * pS
Q_T(pT)=400 * pT
F. Please find the equilibrium quantity and price of S and T in
California (assuming 200 people with equal income = 50) (2 points)
Suppose the supply function of S and T for New York are
Q_S(pS)=200 * pS
Q_T(pT)=200 * pT
G. Please funcd the equilibrium quantity and price of S and T in
New York (assuming 100 people with equal income = 50) (2 points)
H. Compare the prices and quantities of the goods in New York and
California. Which goods are higher prices in NY? Which in California?
Why? (1 point)
Suppose New York and California just signed a trade agreement that the
consumers can buy Salmon and Tuna from either state. Therefore, the consumers
and sellers in New York and California are now combined into a single market
economy, the US.
I. Please derive the new market demand functions of Salmon and Tuna
for the combined US economy. (as before, assume 200 people in
California and 100 people in New York all with equal income) (4 points)
In the combined ecnonmy, the supply function are now
Q_S(pS)=300 * pS
Q_T(pT)=600 * pT
J. Please derive the new wquilibrium pricess and quantities for both
Salmon and Tuna in this combined economy. (4 points)
K. With these new prices, focus on consumer in California. Are they
better off than before as a result of opening trade, in utility terms?
Why or why not? (2 points)
L.Q.3 We now focus on a single consumer, Tom, who is choosing how much to
consume and how much to work. Suppose Tom's preference over daily Consumption
(C units of goods) and Leisure time (L hours) can be presented as a utility
function U(C,L). While we don't know Tom's utility function yet, we do know
that the sum of Leisure (L) and working hours (h) cannot be more than 24 hours
a day. The hourly wage of work is w USD, and the pricce of a unit of goods is
p USD.
A. Please state all budget constraints for this utility maximization
problem. Then create a single budget constraint from these budget
constraints without h. (4 points)
B. Suppose w=10 and p=30. Please depict the combined budget constraint
in a graph with C as the vertical axis (y-axis) and L as the horizontal
axis (x-axis). Be sure ti label the intercepts where the budget
constraint intercpts the axis. (1 point)
C. Suppose the hourly wage has increased to w=20. Please depict the
new budget constraint in a new graph. Be sure to label the intercpts
where the budget constraint intercpets tha axis. Explain how and why
the graph changed. (2 points)
Now we know that Tom's utility function is:
U(C,L)=-5 * C^2 + 5*C*L + 4(L-10)^2
D. Pretend for this part that Tom owns a time machine and has infinite
money. Thus, L and C can be any positive number - they do not have to
fit the budget constraints. Find the unconstrained utility-maximizing
Consumption (C) and Leisure time (L) for Tom. (5 points)
E. Tom stps daydreaming about time machines and infinite money. Instead
, suppose his w=2, and the price of C is p=5. Please find the utility
maximizing C and L under the budget constraint. (5 points)
F. Is Tom's utility higher at the unconstrained optimum or at the
constrained optimum. Prove it mathematically AND provide intuition.
(2 points)
Suppose that the US government implements a minimum income policy (also know as
"basic income"). Under this policy, every citizen (including Tom) can get an
extra 5 USD in daily income from the government. This additional income is
provided regardless of how many hours Tom works. We do not consider how the
government financially supports this policy for now.
G. Please state the new budget constraints with the additional 5 USD
"basic income", as well as the budget constraint removing h the hours
worked. (2 points)
H. Suppose w=2, and p=5. Please find the utility maximizing C and L
with this additional basic income. What happened to C and L compared to
before? Provide intuition why this happened and relate it to part D.
(4 points)
L.Q.4 Suppose Pat's mom offers him up to 5 cookies to take to Taiwan. But
Pat prfers to make decision one at a time, so she offers him just two options
at a time, such as "Would you like to bring 4 cookies or 5 cookies?" For any
two cookie options x and y, suppose Pat has the following preferences.
⊙If x=y, Pat is indifferentm that is x cookies～ y cookies.
⊙If either x or y is odd, and x>y, then x cookies is strictly
preferred to y cookies.
⊙If both x and y are even, and x>y, then y cookies is strictly
preferred to x cookies.
For example, for 5 and 4, let x=5, y=4 and you can see that 5 cookies is
strictly preferred to 4 cookies. FOR PARTS A to D, COOKIES MUST REMAIN WHOLE.
A. Are Pat's preference complete (for 5 cookies or less)? If so,
prove it. If not, provide an example that shows they are incomplete.
(2 points)
B. Are Pat's preference transitive (for 5 cookies or less)? If so,
prove it. If not, provide an example that shows they are intransitive.
C. Is there a utility function that can accurately describe these
preferences? If so, create it. If not, prove why not. (3 points)
D. Is there an option that Pat chooses in the end (one option that is
strictly preferred to all other options)? (2 points)
Now consider the broader market for cookies. Cookies are now allowed to be
divisible (4.5555 cookies is fine). Suppose the demand function and the
supply function of cookies are
Q_D(p) = 150 - 2p
Q_S(p) = p^2 - 45
E. Please find the market equilibrium price and quantity. (3 points)
F. Please state the inverse demand function. (2 points)
G. Please calculate the consumer surplus at the equilibrium price.
(2 points)
H. Suppose the sellers provide Q_S=30. What is the market's willingness
to pay at this quantity? If the pay this price, what is the consumer
surplus now? (2 points)
Suppose the government implements price floor of 15 dollars as a reult of
lobbying.
I. What is the quantity of cookies the sellers want to supply now?
(1 point)
J. Is there excess demand(shortage) or excess supply? If so, what is
size of the excess? (2 points)
K. If the suppliers wanted to increase the price above equilibrium,
what other tool could they use that would ensure no excess supply?
What specifically could they do if they wanted to achieve a price of
15? (2 points)