課程名稱︰個體經濟學下
課程性質︰經濟系大二必修
課程教師︰黃貞穎
開課學院：社會科學院
開課系所︰經濟學系
考試日期（年月日）︰04.19.2021
考試時限（分鐘）：150
試題 :
1. Imagine you have 150 minutes to complete an exam with two questions.
Your score on the exam is the sum of your scores in the two questions.
You want to maximize your score.
You produce an output of a score on the exam using inputs of time spent
on each of the questions. Question 1 is harder while question 2 is ea-
sier. These are captured by two coefficients $d_1$ and $d_2$ where
$d_1 > d_2$ as described below. Suppose you allocate $t_1$ minutes on
question 1 and $t_2$ minutes on question 2. By studying hard before
the exam, you can make each minute you spend during the exam more effe-
ctive. This is captured by another coefficient $e$ described below/
Summarizing all above, your optimization problem during the exam is
$$ \limits \mathop{max}_{t_1,t_2} \frac{ln(et_1)}{d_1}+\frac{ln(et_2)}{d_2} $$
$$ subject to t_1 + t_2 = 150
Where $et_1$ represents the effective time you spend on question $i$
and \frac{1}{d_1}ln(et_1) represents your score on question i.
It takes a natural log of your effective time and then multiples by
a factor of \frac{1}{d_1} because (1) your marginal productivity on
each question declines as you spend more time on it and (2) if a qu-
estion is harder, its $d$ is larger, and hence the factor \frac{1}{d}
is smaller, meaning the same effective time will be translated to a
lower score.
As a teacher, I know you are facing this optimization. I know my most
hardworking students have $e=\bar{e}$, and my least hardworking student
have $e=\underline{e}$ with $\bar{e}>\underline{e}>0$. I also know
that my most hardworking students will allocate their time optimally
between these two questions; my least hardworking student will spend
75 minutes on question 1 and 75 minutes on question 2. So I set $d_1$
and $d_2$ to make sure the most hardworkign students will receive the
score of 100 and the least hardworking students 60 in this exam.
(a) Will the most hardworking stidents spend equal minutes of time on
the two questions? How many minutes will they spend on question 1?
(b) Write down the formular=e how 1 would determine $d_1$ and $d_2$?
No need to solve them explicitly.
2. Eva Airlines has two potential types of custimers: businessman and
tourist. Both demands are in discrete amount.
A business customer is willing to pay up to 20 dollars for the first
trip, 20 more dollars for the second trip, 20 more dollars for the
third trip, 10 more dollars for the fourth trip and 10 more dollars
for the fifth trip. For the sixth trip and onward, a business cus-
tomer is not willing to pay anything more.
A tourist customer is willing to pay up to 20 dollars for the first
trip, 16 more dollars for the second trip and 12 more dollars for
the third trip. For the fourth trip and onward, a tourist customer
is not willing to pay anything more. Eva Airline has zero fixed cost
and zero marginal cost.
Suppose there are two business customers and one tourist customer.
When a customer is indifferent between flying or not flying, he goes
for flying.
(a) If Eva Airlines cannot perform any form of price discrimination
(so it only sets a single price per trip). Derive the optimal
price that maximizes Eva's profit. How much profit does Eva Air-
line earn?
(b) If Eva Airline can perform the first-degree price discrimination,
how many trips will each business customer and each tourist cus-
tomer fly respectively? How much in total will it charge each
business customer and each tourist customer respectively?
(c) Suppose Eva Airlines performs the second-degree price discrimina-
tion, so it screens customers by providing two packages.
Each package consists of a number of trips and a
take-it-or-leave-it charge. For instance, if a package $(x,T)$
is offered, then a customer can pay the total amount $T$ to fly
$x$ times. Since EvaAirlines cannot tell apart its two types
customers, it decides to offer one package $(x_B,T_B)$ aimed for
business customers and another package $(x_T,T_T)$ aimed for
tourist customer. Customers will self select. Can Eva Airlines
earn as much as its profit from performing the first-degree price
discrimination? Explain.
(d) Continue from (c). Now use the theory you have learned from the
second-degree price discrimination to derive the optimal amounts
of $T_B$, $T_T$, $x_B$ and $x_T$. How much does Eva Airlines
earn in total?
(e) Suppose Eva Airlines performs the third-degree price discrimina-
tion by checking whether any customer has a tourist vias. It can
tell apart business and tourist customers. How much will it
charge business customers per trip? How much will it charge
tourist customer per trip? How much does it earn in total?
(f) True or False: When a monopolist performs the third-degree price
discrimination, it can tell apart one kind of consumers from the
other kind. On the other hand, when a monopolist performs the
second-degree price discrimination, it cannot tell apart one kind
of consumers from the other kind. Hence the profit of the former
must be higher than the profit of the latter. After stating True
or False, briefly explain your logic.
3. We have two copies of $Intermediate Microeconomics$ textbook to sell
to three enthusiastic students. Each student has a private value for
the textbook. How can we use a seal-bid auction so that the bidders
bid truthfully and thus the two highest values get the books?
Briefly explain.
4. Suppose there is a fall in the demand for shoes, which are provided by
a competitive industry in which the input prices are given.
(a) Does the price of shoes change by more in the short run or in the
long run? Briefly explain.
(b) Does the industry-wide quantity change by more in the short run or
in the long run? Briefly explain.
(c) Does the quantity provided by each individual shoemaker (which still
stays in the industry after the demand falls) change by more in the
short run or in the long run? Briefly explain.
(d) Does the profit of each shoemaker (which still stays in the industry
after the demand falls) change by more in the short run or in the
long run? Briefly explain.
最後附上修正一些錯字過後的 LaTeX 版本：
https://imgur.com/kF3qx3a
https://imgur.com/5WDpstP
https://imgur.com/N3Flx3g