A consumer lives for two periods. Her first period income is zero and second period income is y' > 0. The initial real interest rate is r_1. Her first period income consumption is c and second period consumption is c'. There are no taxes in this economy.

(A) Write down the consumer's lifetime budget constraint

(B) Suppose that she considers c and c' as perfect complements and also prefers equal consumption in the two periods. Derive an algebraic expression for her optimal consumption in the two periods. Draw her budget constraint and indifference curve in a diagram. Show the endowment and equilibrium points in the diagram.

(C) Now suppose that the real interest rate decreases from r₁ to r₂. In a new diagram, show her initial and new equilibria. Show the income and substitution effects on c and c' in the diagram

(D) for this part, suppose that she considers c and c' as perfect substitutes and her utility function is u = ac + c' , where a is a parameter greater than 1 + r₂ and smaller than 1 + r₁. Once again, consider a decreases in the real interest rate from r₁ to r_2. In a new diagram, show her initial and new equilibria. Show the income and substitution effects on c and c' in the diagram? Briefly explain.