[試題] 110-1 蔡崇聖 個體經濟學上 期中考
課程名稱︰個體經濟學上
課程性質︰必修
課程教師︰蔡崇聖
開課學院：社會科學院
開課系所︰經濟學系
考試日期（年月日）︰2021/11/22
考試時限（分鐘）：
試題 :
1. [20%] In each of the following question, consider a consumer (he) who
consumes two goods, good 1 (x1) and good 2 (x2), whose prices are p1 and p2,
respectively. He has a positive income I. Determine whether each of the
following statements is TRUE or FALSE.
a. [5%] When p1 = 2, p2 = 3, I = 30, he chooses (x1,x2) = (6,5). Then his
preference must violate monotonicity.
b. [5%] Suppose that his preference satisfies strict monotonicity but
violates convexity. Then in the optimum, the marginal rate of substitution
(MRS12) is not equal to the price ration p1/p2.
c. [5%] Suppose that x1 and x2 are perfect complements. Then the
price-comsumption curve are indeed the same.
d. [5%] Suppose that the demand function for good 1 is x1(p1,p2,I) = aI/p1
for any p1, where a is some positive number. Then good 2 must be a normal
good.
2. [25%] There are two kinds of cigarettes in the market, electronic cigarettes
(x1) and regular cigarettes (x2). There are two consumers, Amy and Billy, who
have the following utility functions:
Amy: uA(x1,x2) = √(x1·x2), Billy: uB(x1,x2) = x1 + x2.
Amy considers e-cigarettes and regular cigarettes to be "imperfect"
substitutes, while Billy thinks that they are perfect substitutes. The price
of e-cigarettes and regular cigarettes are p1 and p2 respectively. They each
have the same income I.
a. [8%] For each consumer, set up the maximization problem. Find Amy's demand
functions, x1A(p1,p2,I) and x2A(p1,p2,I), and Billy's demand functions
x1B(p1,p2,I) and x2B(p1,p2,I).
b. [4%] Draw the demand curve for e-cigarettes for each comsumer. Does it
always satisfy the "law of demand"? Explain why or why not.
c. [8%] Recently, there are many cases of lung injuries and deaths associated
with e-cigarettes. Therefore, the Center of Disease Control (CDC) considers
regulating the consumption of e-cigarettes. Suppose that currently, p1 = 1,
p2 = 4 and I = $30. The CDC considers adopting one of the following two
policies:
Policy 1: A limit of consumption. Keeping the prices unchanged, every
consumer is prohibited from buying more than 5 units of e-cigarettes, i.e.,
x1 <= 5. A violator needs to pay a fine of $30.
Policy 2: A "sin tax" on e-cigarettes. The CDC imposes a tax rate of $1 for
each unit of e-cigarettes so that their price becomes p'1 = 2.
d. [5%] Suppose that the CDC maximizes the total utility of the two consumers,
uA + uB, in determining which policy to be adopted. Then which policy is
preferable to te CDC? Does this policy also make the total consumption of
e-cigarettes lower than the other policy? Explain your finding. (Note: you
do not have to consider the tax revenue incurred in Policy 2.)
3. [30%] Fiona has a budget I to be spent at a beauty parlor. She is buying two
products: skin care products (x1) and hair products (x2). She has the
following utility function:
u(x1,x2) = x1 + x2 +x1x2
Suppose that the prices of x1 and x2 are p1 and p2, respectively.
a. [4%] Find the MRS12 of this utility function. Draw some of the indifference
curves.
b. [4%] Determine whether or not the preference represented by this utility
function satisfies strict monotonicity and strict convexity.
c. [7%] Set up her maximization problem. Find the demand functions for x1 and
x2.
d. [4%] Draw a demand curve for x1. Does it always satisfy the "law of demand"
? Explain your finding.
e. [4%] Let p1x1 and p2x2 be her "expenditures" on skin care products and hair
products. Compare p1x1 and p2x2. Explain your finding.
f. [7%] Suppose that currently p1 = 1, p2 = 2 and I = 13. However, the price
of skin care products is about to increase to p'1 = 2. Other things remain
unchanged. Identify the total effect, substitution effect and income effect
in terms of x1 due to the price change. Draw a figure to show them. (Note:
you need to precisely compute the value of each effect.)
4. [25%] David lives for two periods, period 1 and period 2. In the beginning of
period 1, he needs to dicide the consumption in each period, c1 and c2. His
preference can be represented by the following utility function:
u(c1,c2) = 23.1(In c1) + c2.
His incomes in period 1 and period 2 are I1 = $20 and I2 = $22, respectively.
He can save and borrow money from the bank at the same interest rate r = 10%.
a. [7%] Set up his maximization problem. Find the optimum (c*1,c*2). Is he a
saver or a borrower?
For the following questions b.-d., consider the situation where the interest
rate is expected to reduce to r' = 5%. Other things remain unchanged.
b. [3%] Find the new optimum (c'1, c'2) after the change. By comparing with
a., discuss how this change will affect his consumption in each period and
the role of being a saver or a borrower.
c. [7%] Precisely compute the total effect, income effect, and substitution
effect caused by the change in terms of c1. Draw a figure to show these
effects. Provide an economic explanation for each effect.
d. [3%] Will he be better off or worse off due to this change? Explain your
answer.
e. [5%] Suppose that the interst rate remains r = 10%, but his income in the
first period will increase to I'1 = 25. By comparing with (c*1,c*2) that
you found in a., discuss how this change will affect his consumption in
each period and the role of being a saver or borrower. Draw a figure to
show the change. Explain your finding.
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