Econ Macroeconomics I Optimizing Consumption with Uncertainty

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Mabel Sparks
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1 Econ 07 - Macroeconomics I Optimizing Consumption with Uncertainty Prof. David Weil September 16, 005 Overall, this is one of the most important sections of this course. The key to not getting lost is to understand that any consumption optimization is set up identically. Uncertainty only adds to how you form the utility function, and to how many budget constraints you must face. This handout is just designed to solve a few of these problems more formally, so that you can see how they fit in with the typical utility maximization. Also, this may give you a structure around which you can solve other problems with similar set-ups. 1 Uncertainty About Wages Given: You live two periods. You get a wage in period one that is fixed at W 1. In period two, you receive wage of W A with probability p, and wage W B with probability (1-p. You find out after period 1, and before period which wage you will receive. First note that you must consume the same amount in period 1 no matter which wage you might receive. But in period two, you can consume whatever your new wage is plus whatever you saved in period one. Your formal optimization is: max L = U(C 1 + pu C A + (1 p U C B + λ A (W 1 + W A C 1 C A + λ B (W 1 + W B C 1 C B You have an expected utility that you try to maximize. The probabilities are used to weight the 1

2 potential amount of utility you get in your life. The probabilities are NOT used to weight the budget constraints. You have two budget constraints. There are two unique constraints that your life must satisfy when you plan consumption, even though you ll end up actually only facing one. In these kinds of problems, you generally will have multiple budget constraints. We don t always see them explicitly, because we embed them directly in the expected utility equation. Solve this over all choice variables ( C 1, C A, CB. The FOC are: U (C 1 λ A λ B = 0 (1 pu ( C A (1 pu ( C B λ A = 0 ( λ B = 0 (3 W 1 + W A C 1 C A = 0 (4 W 1 + W B C 1 C B = 0 (5 Notice that we can solve the final two FOC for C A = W 1 + W A C 1 (6 C B = W 1 + W B C 1 (7 Now utilizing the second and third FOC: pu W 1 + W A C 1 = λ A (1 p U W 1 + W B C 1 = λ B (8 Putting this all into the first FOC: U (C 1 pu ( W 1 + W A C 1 (1 p U ( W 1 + W B C 1 = 0 Which we can now solve out for C 1. Notice that this is exactly what we would have gotten if we had embedded the budget constraints in the expected utility form and then taken a derivative with respect to C 1

3 E [U] = U (C 1 + pu (W 1 + W A C 1 + (1 p U (W 1 + W B C 1 Notice that once we ve chosen C 1, we ve effectively chosen both C A actual choice to make is first period consumption. and CB. In essence, our only Uncertainty about Lifespan Given: You have a wage in period one of W 1. You get no other wages. You live for two periods at least, and live for a third period with probability p.(call this case A. You therefore only live for two periods with probability (1-p. (Call this case B You find out after the first period how long you ll live. What is your optimal consumption choice? [ ] max L = U(C 1 + p U C A + U C3 A + (1 p U C B + +λ A (W 1 C 1 C A C A 3 + λ B (W 1 C 1 C B Again notice we have multiple budget constraints. Also note that there are two distinct choices for C. That is because you find out after period one how long you ll live, and you ll have different options for C then. Take FOC: U (C 1 λ A λ B = 0 (9 pu ( C A (1 p U ( C B pu ( C A 3 λ A = 0 (10 λ B = 0 (11 λ A = 0 (1 W 1 C 1 C A C A 3 = 0 (13 W 1 C 1 C B = 0 (14 3

4 Solve the nd and 4th together: pu C A = pu C3 A U C A = U C3 A (15 (16 C A = C A 3 = C A (17 This simplifies the budget constraints: C A = W 1 C 1 (18 C B = W 1 C 1 (19 Plug these into the 3rd and nd FOC: pu W1 C 1 = λ A (0 (1 p U (W 1 C 1 = λ B (1 Put in all in the first FOC: U (C 1 pu W1 C 1 (1 p U (W 1 C 1 = 0 Which we can again solve for optimal C 1 and which again is exactly what we get if we take shortcuts right to the expected utility function. 3 What if you find out how long you live after the nd period? In this case you have two periods of consumption to choose, and then you find out how long you ll live. max L = U(C 1 +U (C +pu C3 A +(1 pu C3 B +λ A (W 1 C 1 C C3 A +λ B (W 1 C 1 C C3 B 4

5 But notice that C B 3 = 0, so max L = U(C 1 + U (C + pu C3 A + (1 pu (0 + λ A (W 1 C 1 C C3 A + λ B (W 1 C 1 C FOC: U (C 1 λ A λ B = 0 ( U (C λ A λ B = 0 (3 pu ( C A 3 λ A = 0 (4 (1 pu (0 λ B = 0 (5 W 1 C 1 C A C A 3 = 0 (6 W 1 C 1 C = 0 (7 Notice that the fourth FOC involves the term U (0. What is this equal to? U (0 = 0. If there is no consumption, there is no marginal utility. This implies that λ B = 0. In other words, the multiplier on the second budget constraint is zero, which implies that the second budget constraint doesn t bind, so that W 1 C 1 C > 0. Think about this. It means that if you die after the second period, you won t have spent all your wealth. Why not? You will have saved some for the third period just in case. But once you find out you will die, it s too late to change your decision. The first two FOC imply: C 1 = C = C So from the budget constraint: C A 3 = W 1 C The third FOC and first FOC imply U ( C pu ( W 1 C = 0 5

6 Which can be solved for the optimal consumption in the first two periods. If you set up a problem like this and find that you have an extra multiplier floating around like λ B, set it to zero and see if things work out. Or add in a dummy term like (1 pu (0 to help you keep track of what is happening, and then see that λ B = 0 4 What if I m uncertain about two things? Given: A wage in period one. A wage in period two. After the first period, you find out if you live for two periods or three periods with probabilities p and (1-p. If you live for three periods, then after the second period, you find out whether your wage in period 3 will be either W H 3 probabilities q and (1-q. So how in the world do we set this up? or W L 3, with Lets map out our possible worlds first World A: I live for two periods only. P(A=p World B: I live for three periods and get the high wage. P(B=(1-pq World C: I live for three periods and get the low wage. P(C=(1-p(1-q max L = U(C 1 +pu [ ( C A +(1 p q U C BC ] [ ( + U C3 B +(1 p(1 q U C BC ] + U C3 C + +λ A (W 1 +W C 1 C A +λ B (W 1 +W +W H 3 C 1 C BC C B 3 +λ C (W 1 +W +W L 3 C 1 C BC C C 3 Where C BC is the amount I consume in period if I know I ll live for three periods, but I don t know yet what my wage in period three will be. FOC are many: (1 p qu C BC U (C 1 λ A λ B λ C = 0 (8 λ A = 0 (9 ( pu C A + (1 p(1 qu C BC λ B λ C = 0 (30 (1 p qu C3 B λ B = 0 (31 6

7 (1 p(1 qu C3 C λ C = 0 (3 W 1 + W C 1 C A = 0 (33 W 1 + W + W H 3 C 1 C BC C B 3 = 0 (34 W 1 + W + W L 3 C 1 C BC C C 3 = 0 (35 When solving this, notice that I really only have two choice variables. C 1 and C BC. All the other consumption amounts will be fixed by picking those two. It s a very ugly representation with all these FOC, but it can be solved. It s more important to pick up on how to set up the initial maximization and find the FOC themselves. You can work these out into two equations in two unknowns. U (C 1 pu (W 1 + W C 1 (1 p qu ( W 1 + W + W H 3 (1 p(1 qu W 1 + W + W3 L C 1 C BC = 0 C 1 C BC And (1 p qu C BC ( + (1 p(1 qu C BC (1 p qu W 1 + W + W3 H (1 p(1 qu W 1 + W + W3 L C 1 C BC = 0 C 1 C BC 5 What about time discounting and the interest rate? You can obviously think of extending these problems to include those factors. They will complicate the notation, but won t make any serious difference. Take the first example given in this handout. Let s assume they have some discount rate and some interest rate, both of which are positive. max L = U(C 1 + p U C A + (1 p U C B + λ A (W 1 + W A 1 + r C 1 CA r The FOC will look similar to before: ( +λ B W 1 + W B 1 + r C 1 CB 1 + r 7

8 d dc B d L = U (C 1 λ A λ B = 0 (36 dc 1 d L = p U C A λ A 1 + r = 0 (37 dc A ( L = (1 p U C B λ B 1 + r = 0 (38 d dλ A L = W 1 + W A 1 + r C 1 CA 1 + r = 0 (39 d dλ B L = W 1 + W B 1 + r C 1 CB 1 + r = 0 (40 Put the first three together: U (C 1 p U C A (1 + r (1 p U C B (1 + r = 0 Use the budget conditions: C A = (1 + r W 1 + W A (1 + r C 1 (41 C B = (1 + r W 1 + W B (1 + r C 1 (4 So we have: U (C 1 p U (1 + r W 1 + W A (1 + r C 1 (1 + r [ ] (1 p U (1 + r W 1 + W B (1 + r C 1 (1 + r = 0 We can solve this for the optimal value of first period consumption. It looks more complicated, but it ll just end up depending on the interest rate and discount rate as well as the probabilities and the wages. 8

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