Question 4 [20 marks] Analyze if the statements that are presented below are True or False. You MUST justify your answer to get credit. Answers without justification (even if they are correct) will be given zero marks.

(a) In any Pareto-optimal allocation of a two-good economy, each consumer has to consume a positive amount of both goods.

(b) A monopolist never produces on the elastic segment of its average revenue curve.

(c) If a firm’s production exhibits increasing returns to scale, then the firm’s marginal costs are decreasing and below its average costs.

(d) Maroon Theater practices third-degree price discrimination and sells tickets to three groups of customers: students, regular customers and senior citizens. The inverse demand of the three groups is linear. Furthermore, the students’ and senior citizens’ elasticities of demand for tickets are −4 and −3, respectively. Because the price charged to regular customers is greater than the price charged to senior citizens, we know with certainty that the ticket price for students will be lower than the ticket price for regular customers.

Use the six-step hypothesis testing where appropriate:

1. You have been told that the mean price of two movie tickets including online service charges, a large popcorn, and two medium soft drinks is $38. Based on a sample of 10 theater chains and assuming a normal distribution, the sample mean was found to be $36.53 and the standard deviation was $3.38. At the 0.05 level of significance, is there enough evidence to indicate that the average price is now less than $38?

a. What is the appropriate null and alternative hypothesis for problem 1? Use both words and notations

b.What level of significance and sample size is used in this problem?

c.What type of problem is this? What formula will you use? Why?

d. What is the correct critical value for the problem? How did you find it?

e.What is the correct value for the test statistic? Provide the formula used and show work

f. Should you accept or reject the null hypothesis? How much confidence do you have in your decision? Restate the null of alternative hypothesis. What policy decision would you make?

Can you use Twitter activity to forecast box office receipts on the opening weekend? The following data (stored in TwitterMovies indicate the Twitter activity (“want to see” and the receipts ($) per theater on the weekend a movie opened for seven movies. Solve this problem to two significant digits.

1. What is the independent variable (X) in this question?
2. What is the dependent variable (Y) in this question?
3. Create a scatter plot of the X and Y variables.
4. What is the Y intercept when X = 0?
5. What is the slope?
6. What is the correlation between the X and Y variables as measured by the R Square? How strong is this correlation?
7. Predict the mean receipts for a movie that has a Twitter activity of 100,000.
8. Should you use the model to predict the receipts for a movie that has a Twitter activity of 1,000,000? Why or why not?
   

Amazing Productions performs London shows. The average show sells 1,000 tickets at $60 per ticket. There are 120 shows per year. No additional shows can be held as the theater is also used by other production companies. The average show has a cast of 60, each earning a net average of $320 per show. The cast is paid after each show. The other variable cost is program-printing cost of $8 per guest. Annual fixed costs total $459,200.

1)Compute revenue and variable costs for each show.

2)Use the equation approach to compute the number of shows Amazing Productions must perform each year to break even.

3)Use the contribution margin ratio approach to compute the number of shows needed each year to earn a profit of $4,264,000. Is this profit goal realistic? Give your reasoning.

4)Prepare Amazing Production's contribution margin income statement for 120 shows performed in 2016. Report only two categories of costs: variable and fixed.

A company which manufactures compact discs has found that demand for its product has been increasing rapidly over the last 12 months. A decision now has to be made as to how production capacity can be expanded to meet this demand. Three alternatives are available: (i) Expand the existing plant; (ii) Build a new plant in an industrial development area; (iii) Subcontract the extra work to another manufacturer. The returns which would be generated by each alternative over the next 5 years have been estimated using three possible scenarios: (i) Demand rising at a faster rate than the current rate; (ii) Demand continuing to rise at the current rate; (iii) Demand increasing at a slower rate or falling. These estimated returns, which are expressed in terms of net present value, are shown below (net present values in $000s): Scenario Course of action Demand rising faster Demand rising at current rate Demand increasing slowly or is falling Expand 500 400 ?150 Build new plant 700 200 ?300 Subcontract 200 150 ?50 Exercises 239 (a) The company’s marketing manager estimates that there is a 60% chance that demand will rise faster than the current rate, a 30% chance that it will continue to rise at the current rate and a 10% chance that it will increase at a slower rate or fall. Assuming that the company’s objective is to maximize expected net present value, determine (i) The course of action which it should take; (ii) The expected value of perfect information. (b) Before the decision is made, the results of a long-term forecast become available. These suggest that demand will continue to rise at the present rate. Estimates of the reliability of this forecast are given below: p(forecast predicts demand increasing at current rate when actual demand will rise at a faster rate) = 0.3 p(forecast predicts demand increasing at current rate when actual demand will continue to rise at the current rate) = 0.7 p(forecast predicts demand increasing at current rate when actual demand will rise at a slower rate or fall) = 0.4 Determine whether the company should, in the light of the forecast, change from the decision you advised in (a). (c) Discuss the limitations of the analysis you have applied above and suggest ways in which these limitations could be overcome.

The accompanying data set provides the closing prices for four stocks and the stock exchange over 12 days:

With the help of the Excel Exponential Smoothing tool, I was able to forecast each of the stock prices using simple exponential smoothing with a smoothing constant of 0.3 (ie, damping factor of 0.7).

I was also able to calculate the Mean Absolute Deviation (MAD) of each of the stocks: MAD of Stock A = 1.32 MAD of Stock B = 0.37 MAD of Stock C = 0.41 MAD of Stock D = 0.26 MAD of Stock Exchange = 83.85.

The Mean Square Error (MSE) of the stocks: MSE of Stock A = 2.22, MSE of Stock B = 0.17, MSE of Stock C = 0.21, MSE of Stock D = 0.08, MSE of Stock Exchange = 7963.44.

Help me to understand the concept of Mean Absolute Percentage Error (MAPE). I realize that MAPE is the average of absolute errors divided by actual observation values. I'm wondering if this is just the MAD divided by the total observation values for a particular stock. For example, for Stock A, If my understanding is correct (which I don't think it is), the MAPE of Stock A would be 1.32 / each of the observation values individually. Or, would it be [(127.15 - 127.37) / 127.15]. Or, do I need to add up all the absolute errors for Stock A and all the actual observation values for Stock A and divide the former by the latter and then multiply by 100. As you can see, I'm confused. Please help.

Use the above information from the tables to work out the following missing entries, and then calculate the company’s return on equity. Note: Turnover and the average collection period are calculated using start-of-year, not average, values. (Enter your answers in millions. Round intermediate calculations and final answers to 2 decimal places.)

PROBABILITY

2. Given the following table

answer each of the questions.

Find µ_X and σ_X .

Given that

                          Y = -X+2

find µ_Y and σ_Y .

2B. Suppose that in the game of AFL the mean total number of points per match is 185 with a standard deviation of 33. Suppose that a season involves 202 matches. If the total points are normally distributed then answer the following questions.

1. How many games do you expect to have less than 100 points scored in them in a given season?

2. How many games do you expect to have between 200 and 220 points scored in them in a season?

3. How many points should be needed for a match to be in the top 5% of high scoring matches?

4. If a match is one of the bottom 5 lowest scoring matches of the season, what is an upper bound on what you expect the score to be?

2C. Suppose that asteroid impacts on the earth are modelled with a Poi-son distribution. Asteroids of diameter 4m are estimated to enter the Earth’s atmosphere once per year. Asteroids of diameter 1km are estimated to impact the Earth once every 500,000 years.

1. What is the probability of exactly 2 asteroids of 4m diameter entering the atmosphere in a 6 month period?

2. How many years are required for the probability of at least one 1km diameter impact to exceed 1%?

Use the above information from the tables to work out the following missing entries, and then calculate the company’s return on equity. Note: Turnover and the average collection period are calculated using start-of-year, not average, values. (Enter your answers in millions. Round intermediate calculations and final answers to 2 decimal places.)

3/26

1 mo 0.01

6 mo 0.04

1 yr 0.13

2 yr 0.3

3 yr 0.36

5 yr 0.51

3/27

1 mo 0.01

6 mo 0.02

1 yr 0.11

2 yr 0.25

3 yr 0.3

5 yr 0.41

1. It is March 26 and you hold a $1mm (market value) long position in the 1-yr zero-coupon bond. Using modied durations, determine how much of the 5-yr zero-coupon bond you need to short so that your portfolio remains approximately unchanged if the 1-yr and 5-yr zero rates move in parallel.

2. What would the change in your portfolio value be if both the 1-yr rate and the 5-yr rate go up by 2 basis points?

3. What would the change in your portfolio value be if both the 1-yr rate and the 5-yr rate go down by 2 basis points?

4. What would the change in your portfolio value be if the 1-yr rate goes down by 2 basis points but the 5-yr rate stays the same?

5. What actually happens the following day? Calculate the value of your portfolio. Why did your portfolio value change from before?