.Kindly solve the following as given in the attachment.

QUESTION 1: (worth 15%) Consider an economy with two agents, Anna (A) and Bill (B), and two goods z and z, which are public and private goods, respectively. The utilities of two agents are UA = 21CAI UB = 21CBS The economy is endowed with 15 units of the private good .The production technology in the economy is such that one unit of private good can be transformed into one unit of public good; that is, the MRT = Or/0= = 1. (a) Find a Pareto optimal allocation where Anna has a utility level UA = 5 . (b) Suppose that Anna is endowed with 7 units of private good (CA ) while Bill is endowed with 8 units of private good (xp). What is the allocation obtained in this economy through private provision (voluntary contributions), where Anna's contribution (cA ) plus Bob's contribution (cs ) renders an amount of the public good (z) that they will both enjoy? (c) Is the allocation found in (b) Pareto efficient? Explain. (d) Graph Anna's indifference curve and the PPF (resembling the graphs for George's situa- tion in the "Samuelson Condition" lecture video). Graph them over values of = = [0, 15] on the horizontal axis. If you wish to make the graphing easier, for the indifference curve UA = 5, you may use integers for z = {1, 2, 3..., 1 1}3. (a) Each year an insurance company issues a number of household con- tents insurance policies, for each of which the annual premium is f80. The aggregate annual claims for all policies is modeled by a compound Poisson dis- tribution parameter 0.4n, where n is the number of the policies in the portfolio. The individual claim amounts have gamma distribution with parameters a and A. The amount of the individual claims is independent from the number of claims. Let the r.v. S represent the total aggregate claims and expenses in one year from this portfolio. (i) Write down the form of S. (ii) Find the mean and the variance of S. (iii) Assume that a = 1 and A = 0.01. Find the number of the policies, n, that the firm must cell in a year to be 99% sure that the premium income will exceed the claims. [You may use a normal approximation.] [You are given that if X ~ Gamma(a, A), then mx(=) = E(e *) = ()". If X ~ Poi(A), then E(X) = Var(X) = A. Also, you are given that P(Z