Problem 1. a) Derive analytically a solution to the intertemporal utility maximization problem to obtain c₁ and c. maxu (c1, c2) = c₁c C1,C2 s.t. C2 m2 C₁+ ≤m₁+ 1+r 1+r C1, C2 >0 b) Show the solution graphically for two cases: i) when m₁ is small and m2 is large and the agent is a borrower at t = 1 and ii) when m₁ is large and m2 is small and the agent is a lender at t = 1. c) Assume that interest rate increases: Show how the graphical solution in b) will change in both cases. d) Compute the savings s = m₁ - c. Analyze how the savings change with increase in r, m₁, m2 as well as the parameters a and b (this can be done by computing partial derivatives with respect to these parameters). Does this agree with common sense? e) The Social Security System taxes working population in order to pay the retirement benefits. Assume that the Social Security administration im- poses lump sum tax 7 on income m₁, saves in on the behalf of agent in period t = 1 and returns it with the interest in period t = 2, so that the agent has income m₁-T and m2 + (1+r) when young and old respectively. How the Social Security affect the intertemporal choice of the agents?