1. The company Ducati produces sport motorcycles which are sold in Italy (I) and Spain (S). The company faces different demand functions in the two countries:

Q_I = 200 – P_I; Q_S  = 600 – 2P_S

The company’s total cost function depends on the total output Q as follows:

                TC = 10Q         with      Q = Q_I  + Q_S

Write down the firm’s total profit function.         [2 marks]                                           

Find the profit maximizing quantity sold in each country and check that it is really a maximum.    [7 marks]

Define accurately the price elasticity of demand (PED) and calculate it for the two markets in equilibrium. Interpret the results. [6 marks]

[15 marks Total]

2. Consider the utility function   and a point (81, 27) – good  on the horizontal axis.

a) Determine the value of the marginal utilities of  and  at this point. [5 marks]

b) Find the equation of the line tangent to  at this point. [6 marks]

c) Define, calculate and then comment the marginal rate of commodity substitution (MRCS) at this point. [4 marks]

d) Estimate the change in utility when  and  both increase by 4 units. [3 marks] 

[18 marks Total] 

3. 

a) Two bags contain red and yellow balls. Bag A contains five red and three yellow balls, bag B has one red and two yellow balls. A ball is drawn at random from one bag and turns out to be red. Which bag is more likely to be the source of the drawn ball? Explain why by using proper calculations.  [6 marks]

b) Twelve taxi arrive on average at a certain street corner for every hour. Four people are waiting at the street corner for taxi (assuming they do not know each other and each one will have his own taxi). Each person will be late for work if he does not catch a taxi within the next 15 minutes. What is the probability that at least three people will make it to work on time? [6 marks]

[12 marks Total]

4. 

A. In how many different ways can the letters of the word 'CORPORATION' be arranged so that the vowels always come together? [7 marks]

B. The Consob audits 10% of all companies every year. The companies selected for auditing in any one year are independent of the previous year’s selection. What is the probability that the company will be audited exactly twice in the next 3 years? [4 marks] 

C. Consider the values of the joint probability distribution of X and Y shown in table below:

1. Find the cumulative probability F(1,2) (the exact value is required for each result); [2 marks]

2. Find the probability distribution of X alone (the exact value is required for each result). [2 marks]

[15 marks Total] 

5. The company General Electric claims that a certain brand of its flashlight battery lasts, on average, 400 hours of use, and the standard deviation is 80. You suspect that the population of batteries average fewer than 400 hours. You select a random sample of 100 batteries and obtain a sample mean of 390.

Perform a hypothesis test for your concerns. Use a level of significance of 8%.                      [8 marks]

What distribution did you need to use to determine the critical value and why? [2 marks]

Calculate the p-value of the test statistic. Interpret. [5 marks]

Define the Type 2 Error and calculate the value of β if the true mean is 385. [8 marks]

Define and calculate the power of the test in this case. [2 marks]

[25 marks Total]