Solve.the following questions as attached below.

Problem 4: presidential predictions (6 Pts.) An Individual has preferences over "contingent" consumption in two possible altua- tions: consumption -in dollars- if Biden wins ro and consumption -In dollars- if Trump wins TT, given by U(TB, IT) = PBIB + (1- PB)IT where PB is the probability, according to the agent, of a Biden win, her belief so to speak. Thus we can interpret this agent as someone that evaluates consumption in the two situations using as weights the probabilities that each take place (since the probability of a Trump win is 1 - PB)- Her endowment is (10, 10), namely she has 10 dollars in her pocket, which remain 10 dollars whether Biden or Trump win. a) our agent has acces to a prediction market where she can buy or sell bets on a Biden win. Let a ( [0. I] be the price of a 1 dollar bet on a Biden win, namely a bet that repays $ 1 if Biden wins, and zero otherwise. To understand how this works, immagine the agent buys a 2 dollar bet on Biden winning. Then she doles out a2 dollars, and, if Biden wins, she earns $ 2. This allows her to move, in our contingent consumption space, from (10, 10) to (12-02, 10-a2). Draw a graph of her budget set for a generic a e (0, 1). How would the budget set look like if the agent could only buy, and not sell, bets on Biden? b) assume the agent believes the probability of a Bidon win is 0.6. Find the agent's net demand and net supply for bets on Biden. At which price a* is the agent Indifferent between buying and selling bets on Biden? c) Suppose now that the agents utility is described by u(FB, IT) = 0.6u(FB) + 0.4u(IT) where u is increasing, continuous and differentiable, and such that the agent's pref- crences are strictly convex (and as in b) her belief of a Biden win is 0.6). Will the1. Question 1: the Real Business Cycle Model Consider an economy consisting of a constant population of infinitely-lived in- dividuals. The representative individual maximises the expected value The instantaneous utility function u(Ct) = C; 6 (Ct + 02, 6 > 0. Assume that C is always in the range where u' (C) is positive, and that Q's are i.i.d. taste shocks with mean zero. Output is linear in capital: Y; = AKt. There is no depreciation; thus Kt\" = Kt + Y; Ct, and the interest rate is A. Assume A = p. (a) Find the rst-order condition (Euler equation) relating Cr and the expecta- tions of CH1 and explain its meaning. (20 points, 200 words) (b) Guess that consumption takes the form Ct = a + Kt + 75.5. Given this guess, What is K+1 as a function of Kt and Q? Interpret your results. (20 points, 200 words) (c) What values must the parameters a, ti, and 7 have for the rst-order con- dition in part (a) to be satisfied for all values of Kt and Q? Interpret your results. (20 points, 200 words) (d) What are the effects of a one-time positive shock to (t equal to (1 + A) on the paths of Yt, Kr, and Ct? Interpret your results. (20 points, 200 words) (8) Interpret the Covid-19 Pandemic as a (negative) taste shock, i.e. people have a lower appetite for consumption. How would you adjust your answer in part (d)? (20 points, 200 words) Show that (inx-p)- [4] 12no- A general insurance company writes claims, whose amounts have a lognormal distribution, with mean 300 and standard deviation 400. The insurance company purchases excess of loss reinsurance with retention 500 per claim. (ii) Calculate the average expected claim size payable by the insurance company. [6] Next year, claim inflation is 10%, but the retention amount remains the same. (iii) Explain whether the average expected claim size payable by the insurance company next year would increase by 10%. [2] ['Total 12] Consider the following time series model: Y, = 1+0.6Y,_1+0.167,-2+ 5, where & is a white noise process with variance of. Determine whether Y, is stationary and identify it as an ARMA(p,q) process. [3] (ii) Calculate E(F). [2] (iii) Calculate for the first four lags: the autocorrelation values P1. P2- P3. P, and the partial autocorrelation values V1, Vy, (;, W4. 171 [Total 12]Show that (inx-p)- [4] 12no- A general insurance company writes claims, whose amounts have a lognormal distribution, with mean 300 and standard deviation 400. The insurance company purchases excess of loss reinsurance with retention 500 per claim. (ii) Calculate the average expected claim size payable by the insurance company. [6] Next year, claim inflation is 10%, but the retention amount remains the same. (iii) Explain whether the average expected claim size payable by the insurance company next year would increase by 10%. [2] ['Total 12] Consider the following time series model: Y, = 1+0.6Y,_1+0.167,-2+ 5, where & is a white noise process with variance of. Determine whether Y, is stationary and identify it as an ARMA(p,q) process. [3] (ii) Calculate E(F). [2] (iii) Calculate for the first four lags: the autocorrelation values P1. P2- P3. P, and the partial autocorrelation values V1, Vy, (;, W4. 171 [Total 12]There are two segments of the market represented by demand relationships: Q = 90 3p and Q = 180 6p. There's a constant marginal cost of $10 to serve the market. There are 5 consumers in each of the two demand segments. In other words, there are 5 people whose individual demand is represented by Q = 30 3p and 5 people whose demand is represented by Q = 180 6p. (a) Find the optimal two-part tariff scheme if you are required to charge a per-unit price of $10, but must offer the same xed price to both types of demanders. Report both prices and prots. (b) 12 points Find the optimal twopart tariff scheme if you are free to set any per unit price but must offer the same xed price to both types of demanders. Report both prices and prots. ((3) Find the optimal twopart tariff scheme if you are able to offer a different xed price to both types of demanders (and any per-unit price). Report both prices and prots