Macroeconomic Theory and Policy – Midterm Exam | © Sanjay K. Chugh 2 Problem 1: Two-Period Consumption-Savings Framework (20 points). Consider the two-period economy (with zero government spending and zero taxation), in which the representative consumer has no control over his real income (y1in period 1 and y2in period 2). The lifetime utility function of the representative consumer is ()2211,lnlnu c ccc=+. The lifetime budget constraint (in real terms) of the consumer is, as usual, 22110(1)11cycyr arr+=+++++. Suppose the consumer begins period 1 with zero net assets (a0= 0), and as per the notation in Chapters 3 and 4, r denotes the real interest rate. For use below, it is convenient to define the gross real interest rate as R= 1+r(as a point of terminology, “r” is the netreal interest rate). a.(4 points) Set up a lifetimeLagrangian formulation for the representative consumer’s lifetime utility maximization problem. Define any new notation you introduce. Solution: The lifetime Lagrangian is 2212101(111ln)lnycr accycrrλ+++++−++−, which contains Lagrange multiplier λ.