Consider a simple two-period model where a consumer's utility is a function of consumption in both periods. Let the consumer's utility function be U(x1,x)) = x, 12x23/2 (1) where X7 is consumption in period 1 and X2 is consumption in period 2. The consumer is also endowed with a budget B and the beginning of period 1. Let I denote a market interest rate at which the consumer can choose to borrow or lend across the two periods. The consumer's intertemporal budget constraint is that x, and the present value of x, add up to B. Thus, X x x + = B (2). 1 1+r This maximization problem can be written as 1/2 3/2 X max s. t. i x2 x + = B {x,x) 1+r 1 1. Write down the first order conditions. 2. Calculate the optimal consumption in each period. 3. Are the second order condition(s) satisfied? 4. Calculate the maximum value function of this problem, and all its partial derivatives. 5. What is the economic interpretation of the Lagrange multiplier in this