Exclamation  General  Instructions

• You should read the Individual Coursework 2 Assessment Brief posted to Moodle before reading this document.

•  For answers that include codes, remember to make sure that your codes are readable. Hard-to-read codes may lead to a loss of marks.  Beware that both excessive and insufficient comments hurt readability.

•  Submit your work in a PDF file to the designated submission box on Moodle by 10am on 29 November 2023.

• If you encounter any difficulty with submission, reach out to the Programme Adminis- trator as soon as possible.

Question 1

Load the tuna. csv file from Moodle into R. The dataset was included in the package bayesm. You could read the dataset’s documentation here. The documentation includes the description of each column, which is necessary for the following questions.

a. (5 marks) Regress the logarithmic demand for Geisha tuna on the logarithmic price of all seven brands with an intercept. Show your R codes and the regression summary.

b. (3 marks) What is the self-price elasticity of Geisha tuna? How to interpret the it?

c.  (3 marks) How do you interpret the coefficient for the price of Bumble Bee Solid?

d. (5 marks) Assume that our regression specification is correct and that our estimation is unbiased. Deduce the average cost of goods for each of the seven brands assuming every brand/product is applying the optimal markup rule. Report your results in a named vector or a data frame so that we know which number corresponds to which brand/product. You could include the product index rather than its full name in the name.

Question 2

A national fashion retailer has many stores across the country.  The company has spent years trying to furnish its brand premium, and would like to evaluate its branding strategy by estimating how price sensitive its customers are. Given different market conditions in different parts of the country, the company delegates the task of pricing to its store managers, with some guidance from the headquarter. Consequently, there are price variations across different stores. The company is interested in estimating the average price elasticity of demand of its customers through this price variation.

The company holds alarge dataset recording each transaction at every store. It decided to focus on the best-selling garment the company has been known for since its beginning. Specifically, for a fixed calendar month, for each store, the company computes the aggregate sales volume of the garment and the average price of the garment in that month. Suppose the true monthly store-level demand function looks like the following:

q k = a + βklog pk+ ek,

where k is the index for the store and βk is the price coefficient for the store.

The company has collected {(qk, pk)} K stores. The company is interested in the average elasticity of demand across all stores, that is, E(βk).  To do so, it is thinking about running the following regression:

qk = a + β log pk+ ek.

(16 marks) Will the proposed regression yield an unbiased estimate of E(βk)? If it is biased, what is the direction of the bias? Explain your answer.

Question 3

Consider the gym example in the lecture about price discrimination. You may use the notations from the example in the lecture slides. Note that in the questions below, we refer the enthusiasts as the high-type customers and the casual goers as the low-type customers.

a. (10 marks) Show why it is impossible for the gym to design a menu with plans of different hours quota such that the high-type customers buy the low-quota plan and the low-type customers buy the high-quota plan. That is, show why the gym cannot incentivize the high- type customers to choose the low-quota plan and simultaneously incentivize the low-type customers to choose the high-quota plan.

b.  (5 marks) Suppose the gym offers a plan with hours quota xl  and charges it at pl in the optimal menu. Assume the plan targets the low-type customers. What is the optimal price pl in the optimal menu?

c. (10 marks) Now the gym is thinking about adding a new plan to target the high-type customers. Write an R function with minimal signature  high_plan(xl,  pl,  mc) that takes the low-quota plan (xl, pl) as inputs and outputs the optimal high-quota plan (xh, ph) given the low-quota plan. Specifically, your function’s output should be a data.frame or a data.frame-based object such as data.table. The output should have one column named xh and one column named ph. You can include other columns to suit your needs. Feel free to define any intermediate utility functions.

d.  (10 marks) Write an R function with minimal signature two_item_profit(xl,  high_prop, mc), where xl is the hours quota for the low-quota plan, high_prop is the proportion of high-type customers, and mc is the marginal cost of serving a customer for an additional hour.

For a fixed xl, use the functions you have defined to find the optimal price pl and the optimal high-quota plan (xh, ph) conditional on the low-quota plan. Calculate the profit of the menu and draw the plot of profit against different values of xl. The upper bound of the range of xl in plotdata should be appropriately chosen based on the insights from the lecture. An upper bound that is too low or unnecessarily high will lead to mark deductions. Report the optimal two-plan menu.

Question 4

Consider the cinema example in the lecture about price discrimination. Suppose the demand curve for students is now q1(p) = max{0, 21 − p} and that for adults remains q2(p) = max{0, 20 − p/2}. Marginal cost is c1  = 5 for students and c2  = 4 for adults.