课程名称︰个体经济学下 课程性质︰经济系大二必修 课程教师︰黄贞颖 开课学院：社会科学院 开课系所︰经济学系 考试日期（年月日）︰04.19.2021 考试时限（分钟）：150 试题 : 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. 最後附上修正一些错字过後的 LaTeX 版本： https://imgur.com/kF3qx3a https://imgur.com/5WDpstP https://imgur.com/N3Flx3g --

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