Read through the attachments below and answer the questions accordingly.

Currently there is a debate about cigarette advertising. Some anti-smoking advocates would like to ban cigarette advertising. The tobacco companies agree that advertising merely affects people's choice of brands, not the overall level of smoking, and hence should not be banned. The following simplistic model is intended to explore the issues raised by this controversy. There are two firms that play the following 2 stage game. In stage 1, they simultaneously choose the amount of money each will spend on advertising. In stage 2, having observed the advertising levels, the firms choose the quantities that they will sell. 49 Let us assume that there is no discounting between periods, and no further activity after stage 2. The demand curve faced by each firm in stage 2 is given by; P, = 100 + A, - A, - (Q, + 0,) Where; A, + A, are firms i's and J's advertising presence and Q, & Q, , their outputs respectively The cost of a given level of advertising presence is C(A) =0.5A. The marginal cost of producing cigarettes is zero. a) Find the Nash equilibrium of this game, i.e. show, i. The advertising levels by each firm in the first stage. ii. The quantities sold by each firm in the second stage. iii. The profit earned by each firm overall. b) Would these firms oppose a government ban on advertising? Explain fully.A random variable X has a Gamma distribution with parameters o and 1. (i) Show that the moment generating function of X, Mx (t) , is given by: [3] (ii) Use this moment generating function to show that the mean and variance of X are given by and . respectively. [2] 12 (iii) In the case when 2c is an integer, prove that 21X is a 27 random variable. [3] [Total 8]