1 An intertemporal household problem In order to prepare for this question, revisit Question 4.5 from the Calculus exercises dis- tributed at the beginning of the course. The problem is also covered in Section 16.2 of the textbook which we will study soon. Consider a household problem that lives in two periods. In period 1, the household starts with wealth W, and it chooses how to divide it between consumption C and saving S. The saving earn an interest rate R, so that at the beginning of period 2, the agents owns (1+ R) S. He then consumes his saving completely in period 2. Denote C2 his consumption in period 2. Notice that this problem implies two budget constraints C + S C2 = = W S (1 + R) The first constraint states that wealth is split between consumption in period 1 and saving. The second constraint states that consumption in period 2 is given by savings, including the interest earned. Question 1.1 Express S from the second equation and use it to substitute S out of the first equation to obtain C1 + C2 1+ R = W. This equation is called the intertemporal budget constraint. Observe that on the left-hand side, we have the present discounted value of consumption. The intertemporal budget constraint states that the present discounted value of consumption (i.e., all your spending) must be equal to your wealth. Question 1.2 Now consider the utility of the household. Let us assume that the utility takes the form Uln C+ln C2 where is the time preference coefficient. Use the intertemporal budget constraint to substitute out C from the utility U. Then write down the first-order condition for maximization of U with respect to C2. , which is equal to C. Notice that the first-order condition contains the term W Substitute C back and show that you can write the first-order condition as 1 C' = (1 + R) 1 This equation is called the consumption Euler equation. (1) Question 1.3 Argue that when consumption C2 is kept fixed (for reasons outside of this model), then consumption C in period 1 is a decreasing function of the interest rate. Observe that in Question 1.3, we proceeded differently than in Question 4.5 of the Calculus exercises. There, we found a full solution for C and C2 and actually inferred that C = 1W, which is independent of the interest rate. The reason was that when interest rate changed, consumption C2 did not stay the same and actually increased. == Why we assume here that C2 is assumed to be kept fixed will be explained in class. What you should take away is that the consumption Euler equation in this case implies (asuming that C2 is fixed) that household demand for consumption is a decreasing function of the real interest rate