Answers to Problem Set 3

Transcription

1 Spring 2007 Economics 425 John Rust University of Maryland Answers to Problem Set 3 0. Jack has a utility function for two perfectly divisible goods, x and y. Jack s utility function is u(x,y) (x + y) 2. Derive Jack s demand function for the two goods as a function of p x (the price of good x), p y (the price of good y), and I, (Jack s total income to be allocated to the 2 goods). Answer As I showed in class the answer to this is easy if you perform a montonic transformation, f (u) u to get v(x,y) f (u(x,y)) u(x,y) (x + y) 2 x + y. he transformed utility function has constant marginal utility for both goods, with marginal utility equal to 1 for both goods. hus, the indifference curves for this consumer are straight lines with a slope of 1. You can see this since the slope of the indifference curve is the negative of the ratio of the marginal utilities of the two good, which are both 1 in this case. Or you can solve for an indifference curve directly, treating the consumption of good y as an implicit function of the consumption of good x, i.e. to solve for y(x) in the equation u(x,y(x)) u x + y(x) (1) so we see that y(x) u x (2) which is a straight line with a slope of 1, as claimed. hen, with such indifference curves, it is easy to see that the demand for the two goods has a bang-bang type of solution, i.e. Jack is almost always at a corner solution where he/she consumes either entirely x or entirely y depending on which one is cheaper. hus, if p y > p x, then the slope of the budget line I p x xp y y is p x /p y > 1 (i.e. less steep than the indifference curves), so that Jack will get the highest utility by spending his entire budget on x and consume none of y as in figure 1 below. We can see from figure 1 that when p y > p x, it is better for Jack to only buy good x, and so with an income of 10, Jack can afford 10 units of x. If p y < p x, then we have the opposite situation where the budget line is steeper than Jack s indifference curves and so Jack only buys good y and none of good x. he knife-edge case occurs when p x p y. hen the indifference curve and the budget line both have the same slope, 1. In this knife-edge case, any combination of x and y on the budget line is utility maximizing, and Jack is indifferent about consuming from any point on the budget line. Writing all of this down mathematically we have x(p x, p y,i) y(p x, p y,i) I/p x if p y > p x 0 if p y < p x [0,I/p x ] if p y p x 0 if p y > p x I/p y if p y < p x (3) [0,I/p y ] if p y p x 1

2 Figure 1: Jack s utility maximization solution when I 10, p x 1 and p y Jacks demand for x and y when I10 and p x 1 and p y 2 Indifference curve Indifference curve Budget line 7 Quantity of Good y Quantity of Good x In the demand equations above, the notation y(p x, p y,i) [0,I/p x ] denotes the knife-edge case where y could be any value between 0 and the number I/p y, which happens when Jack spends his whole budget on good y. In the knife edge case, if Jack consumes y units in this interval, then to satisfy his budget constraint, his consumption of x is given by x (I p y y)/p x, i.e. he spends the rest of his budget on x. It is possible to arrive at the same conclusion by writing down the Lagrangian for this problem as I did in class. We have L(x,y,λ) (x + y) + λ(i p x x p y y) (4) aking the first order conditions (for maximization) with respect to x and y we have L x (x,y,λ ) 1 λ p x 0 L y (x,y,λ ) 1 λ p y (5) he inequalties reflect the possibility of corner solutions: if x 0, (a corner solution for x), then we have L(x,y,λ ) 0. However if x > 0 (and interior solution for x ), then we have L(x,y,λ ) 0. Suppose that p x < p y. hen we conjecture that x > 0, and solving the first first order condition (for an interior solution for x ), L(x,y,λ ) 0, we deduce that λ 1/p x, i.e. the marginal utility of income in this case is 1/p x. From the other first order condition, we have L y (x,y,λ ) 1 λ p y 1 p y < 0 (6) p x 2

3 which implies that we have a corner solution for y, i.e. y 0. Using the budget equation, it follows that when y 0 we have x I/p x (i.e. Jack spends his entire budget on good x, nothing on y), and this is the same solution as we got above. hus, by following through the various cases, we can see that solving the Lagrangian problem gives us the same solution, i.e. the same demand functions for goods x and y as we derived above by intuitive means. 1. Provide an example of a utility function that leads to at least one good being an inferior good. Provide a general proof that all goods cannot be inferior goods. Answer Consider the utility function u(x 1,x 2 ) x 1/2 1 + x 3/2 2. he Lagrangian for the utility maximization problem is L(x 1,x 2,λ) x 1/2 1 + x 3/2 2 + λ(y p 1 x x p 2 x 2 ) (7) he first order conditions for the maximization of the Lagrangian with respect to x 1 and x 2 are Soving these two equations we get L (x x 1,x 2,λ ) [x 1] 1/2 λ p 1 0 L (x 1 x,x 2,λ ) [x 2 ]1/2 λ p 2 0 (8) x 1 1 4λ 2 p 2 1 x λ2 p 2 2 (9) Substituting the above solutions into the budget constraint, y p 1 x 1 + p 2x 2 and solving for λ we get 1 4λ 2 p λ2 p 3 2 y (10) Multiply both sides of this equation by λ 2 and rearrange terms to get 4 9 p3 2λ 4 yλ p 2 0 (11) 2 Let γ λ 2. hen we can rewrite the equation above as a quadratic equation for γ aγ 2 + bγ + c 0 (12) where a 4 9 p3 1 2, b y and c. Recall the quadratic formula: there are two solutions for γ 4p 2 2 from the equation above. hey are γ b ± b 2 4ac 2a Plugging in the formulas for a, b and c above we get λ y γ y ± p3 2 p 3 2 p 1 (13) (14) 3

4 Regardless of whether we take the + or root in the formula above, λ is an increasing function of y, as we would expect. But since x 1 1, it follows that x 4λ 2 p 2 1 is a decreasing function of y, i.e. 1 x 1 is an inferior good. Another example of a utility function that leads to an inferior good is u(x 1,x 2 ) log(x 1 ) + e x 2 (15) Guess which of the two goods is inferior. Yes, you re right, it is x 1, the good with the diminishing marginal utility. he other good, x 2 has increasing marginal utility, just like the previous example. Setting up the Lagrangian as in the previous case and taking first order conditions, you should be able to show that x 1 1 λp 1 e x 2 λp 2 (16) So we just need to show that λ is an increasing function of y in this case and we will have another example of a utility function for which one of the goods is an inferior good. Plugging into the budget constraint to solve for λ we get y p 1 x 1 + p 2 x 2 1 λ + p 2 log(λp 2 ) (17) Rearranging this equation, we get { } y p2 log(p 2 ) 1 λ λ exp p 2 (18) Now, by totally differentiating the above equation (more exactly, using the implicit function theorem), we get λ y A 1 + A > 0 (19) where A 1 p 2 exp { y p2 log(p 2 ) 1 λ p 2 } (20) hus, we see that λ is an increasing function of y, so that x 1 1 and is thus and inferior good. λp 1 is a decreasing function of y 2. Define mathematically what a homothetic utility function is. a. Show that homothetic utility functions lead to demand functions that are linear in income, y. Answer A homothetic function is defined to be a monotonic transformation of a homogeneous of degree 1 function. hus, a utility function is homothetic if it can be represented as u(x) f (l(x)) (21) where f : R R is a monotonic increasing function (i.e. f (u) > 0 for all u R) and l : R n R is a linearly homogenous function of the vector x (i.e. for any positive scalar λ > 0 we have 4

5 l(λx) λl(x)). Since we showed in class that the gradient of a linearly homogeneous function is homogenous of degree zero, this implies that the slopes of the indifference curves of a homothetic utility function are the same on any ray through the origin. o see this, recall that the slope of an indifference curve the marginal rate of substitution between two goods x i and x j is give by u x MRS i j i (x) u x j (x) f (l(x)) l l x i (x) f (l(x)) l x j (x) x i (x) l x j (x) (22) But since l x j is homogenous of degree 0 we have for any λ > 0 l (λx) l (x) (23) x j x j his implies that the MRS i j is constant at any point on the ray throught the origin, R {y R y λx for some λ > 0}. his property provides the intuitive basis for the argument that a homothetic utility function has a demand function of the form x(p, y) x(p, 1)y (24) where x(p,1) is the demand for the good when y 1. he reason that this is the case is that if x(p,1) maximizes the utility function when y 1, this means that the MRS i j equals the price ratio p i p j for every pair of goods {i, j}. But then if we consider demand for some other income y, we know that as long as the prices are fixed, the budget hyperplane for income y will be parallel to the budget hyperplane (or budget line in the two good case) when y 1. But the property of homothetic functions implies that the MRS i j is constant along any ray through the origin, this implies that by scaling up the demand bundle x(p,1) which makes the budget plane tangent to the indifference surface when y 1, then by scaling x(p,1) by the factor y, the bundle x(p,1)y (which is on a ray from the origin that contains the point x(p,1)) will be optimal at income y. It is also budget-feasible since x(p,1) is budget-feasible at income y 1, i.e. p x(p,1) 1 so then we must have y p x(p,1) y 1 y, so that x(p,1) y is also budget-feasible at income y. o prove this rigorously, we use a proof by contradiction. Let x(p, 1) maximize the utility function when y 1. hat is, x(p, 1) argmax l(x) subject to: p x 1 (25) x Notice that we have used the property of homotheticity and the fact that the solution to a utility maximization problem is unchanged if you take a monotonic transformation of the utility function. So we have conveniently chosen a transformation to make the utility function u(x) a linear homogeneous utility function l(x). Now, we claim that for any y 1, the utility maximizing bundle must be x(p,y) x(p,1) y. Suppose this is not the case. hen there is some bundle ˆx which is budget-feasible, i.e. p ˆx y and for which we have l( ˆx) > l(x(p,1) y). However since l is linear homogeneous we have l( ˆx/y) > l(x(p, 1)) (26) Notice that if p ˆx y then p ˆx/y 1 so the bundle ˆx/y is budget-feasible for an income of y 1. hen the equation above tells us that this alternative bundle produces higher utility than the supposed utility maximizing bundle x(p, 1). his is a contradiction. 5

6 b. Is the converse true, i.e. if a utility function leads to demand functions that are linear in income, must the utility function be homothetic? Provide a proof if you answer yes, or a counterexample if you answer no. Answer he converse is true. Once again we will prove this by contradiction, but first let s understand the intuition. If the demand function has the form x(p,y) x(p,1) y (27) his means that holding prices fixed as we vary income, the demanded bundle lies on a ray from the origin going through the vector x(p,1), which is the demanded bundle when y 1. his implies that the slopes of the indifference curves along this ray through the origin must be the same at every point on the ray, otherwise we can find a point on this ray, call it x(p,1) y for some y > 0, where the slope of the indifference curve is different than the slope of the budget hyperplane. But if this is the case, x(p, 1) y cannot satisfy the necessary condition for utility maximization, and thus cannot be the utility-maximizing (i.e. demanded) bundle, contradicting the assumption that x(p,1) y is the demanded bundle for every p and y. his is the basic idea of the proof. o make it completely rigorous we need to show a) a function is homothetic if and only if the slopes of its indifference curves are constant along every ray through the origin, b) the unit demand function x(p,1) varies sufficiently as p varies such that it crosses any ray from the origin in the positive orthant, i.e. given any strictly positive vector x there is a p R n and an income y such that x x(p,1) y. We are going to assume that the latter condition holds, so we only need to prove part a). We do this by contradiction. So suppose we have a utility function u(x) that has indifference curves whose slopes are the same along every ray from the origin (in the positive orthant) but which is not homothetic. his means that u(x) cannot be written as a monotonic transformation of a linear homogeneous function l(x). I leave it to you to complete the proof that this implies that along some ray through the origin the indifference curves for u(x) do not have the same slope for every x along this ray. his is a contradiction. 3. Prove that if (x,λ ) are a saddlepoint pair for the Lagrangian function then x is a solution to the constrained optimization problem L(x,λ) u(x) + λ[k f (x)] (28) max u(x) subject to: f (x) K (29) x Answer Suppose x does not solve the constrained optimization problem. hen either x is not feasible (i.e. f (x ) > K), or there is some other ˆx which is feasible that gives a higher objective value, (i.e. ˆx such that u( ˆx) > u(x ) and f ( ˆx) K). We show that in either case, there is a contradiction of the hypothesis that (x,λ ) is a saddlepoint. If x is not feasible there is a contradiction since then we would have that K f (x ) < 0 but in this case we can drive the Lagrangian to by driving λ +. his contradicts the assumption that (x,λ ) is a solution to the Lagrangian saddlepoint problem, i.e. that x, λ and L(x,λ ) are finite quantities. Now suppose that x is feasible but there is a feasible ˆx for which u( ˆx) > u(x ). hen we also have a contradiction since by the complimentary slackness condition λ [K f (x )] 0 we have L(x,λ ) u(x ) (30) 6

7 Now consider the value of the Lagrangian at the point ( ˆx, ˆλ) where ˆλ 0. We have L( ˆx, ˆλ) u( ˆx) > u(x ) L(x,λ ) (31) But this contradicts the assumption that (x,λ ) is a saddlepoint solution to the Lagrangian, i.e. that it maximizes the Lagrangian in x and minimizes it in λ. 1. Intertemporal utility maximization with certain lifetimes. Suppose a person has an additively separate, discounted utility function of the form V(c 1,...,c ) β t su(c t ) (32) where β s is a subjective discount factor and u(c t ) is an increasing utility function of consumption c t in period t. Let the market discount factor is β m 1/(1 + r) where r is the market interest rate. a. If β s β m show that the optimal consumption plan in a market where there are no borrowing constraints (i.e. the consumer has unlimited ability to borrow and lend subject to an intertemporal budget constraint) is to have a constant consumption stream over time, i.e. c 1 c 2 c t c t+1 c. b. If β s < β m will the optimal consumption stream be flat, increasing over time, or decreasing over time, or can t you tell from the information given? c. How does your answer to part b change if I tell you that the utility function u(c) is convex in c? Answers: We did this in class, and it is also in the lecture notes. See pages 23 onward in the lecture notes on intertemporal choice. For part c, note that if the utility function is convex, then u (c) > 0 and the answers to parts b is reversed, iv β s < β m, then optimal consumption will be increasing over time, the opposite of the case if utility is concave (diminishing marginal utility), in which case consumption would be decreasing over time. 2. Expected Discounted Utility with Uncertain Lifetimes Consider the intertemporal utility maximization problem, but extended to allow for uncertain lifetimes. Let denote the (random) lifetime of a person, in years. Let f (t) denote the probability density function for the person s lifetime. hus, we have f (t) Pr{ t}, (33) i.e. f (t) is the probability that the person lives for t 1 years and dies when they reach t years old. a. What does the sum f (t) equal? b. Show that the person s expected discounted lifetime utility, allowing for the possibility of dying, is given by [ ] t E{U} β s u(c s ) f (t) (34) s1 7

8 c. Show that the person s expected discounted lifetime utility can also be written as E{U} [1 F(t 1)]β t u(c t ), (35) where F(t) is the cumulative probability distribution corresponding to the probability density f (t), i.e. F(t) Pr{ t} t s1 d. In words, what is the interpretation of the quantity [1 F(t 1)]? f (s). (36) e. Suppose that the person s random age of death is geometrically distributed, i.e. f (t) p t (1 p), t 1,2,... (37) where p (0,1) is the probability of surviving in any given year. Show that the expected discounted lifetime utility in this case is 0 E{U} 0 f (t) t s1 β s u(c s ) [pβ] t u(c t ). (38) Hint: Use the rules from calculus on interchanging the orders of integration of an integral over a triangular region, Z [ Z x ] Z [ Z ] f (x,y)dy dx f (x,y)dx dy (39) and show that the same reasoning leads to the following analogous formula for interchanging the order of summations in a summation over a triangular region [ ] t ] f (t,s) f (t,s). (40) s1 s1[ ts Answers: a. f (t) 1 since it is a probability distribution so it must sum to 1. b. Show that the person s expected discounted lifetime utility, allowing for the possibility of dying, is given by [ ] t E{U} β s u(c s ) f (t) (41) s1 0 he expected lifetime utility is just the weighted summation of the probability of living to age t, f (t), times the discounted utility of living t years, t s1 βs u(c s ). We assume that person dies at the end of their t th year of life, so they actually get to enjoy the consumption c t in their last year before they die. y 8

9 c. Show that the person s expected discounted lifetime utility can also be written as E{U} [1 F(t 1)]β t u(c t ), (42) where F(t) is the cumulative probability distribution corresponding to the probability density f (t), i.e. F(t) Pr{ t} t s1 f (s). (43) Using the Hint above, equation (40), we reverse the order of the summations to get [ ] t E{U} β s u(c s ) f (t) s1 his last equation follows because we have or since by definition we have ts 1 s1ts s1[ ts s1 f (t)[β s u(c s )] ] f (t) [β s u(c s )] [1 F(s 1)][β s u(c s )] (44) f (t) s 1 f (t) + ts f (t) (45) s 1 1 f (t) 1 F(s 1), (46) F(s 1) Pr { s 1 } s 1 f (t). (47) d. In words, what is the interpretation of the quantity [1 F(t 1)]? Answer: this is the survival probability, i.e. the probability a person will live to at least age t or longer. e. Suppose that the person s random age of death is geometrically distributed, i.e. f (t) p t 1 (1 p), t 1,2,... (48) where p (0,1) is the probability of surviving in any given year. Show that the expected discounted lifetime utility in this case is E{U} f (t) t s1 β s u(c s ) [pβ] t u(c t ). (49) 9

10 Answer: Using the result from part c, we just need to figure out what F(s 1) is for the geometric distribution. We have F(s 1) s 1 p t 1 (1 p) s p j (1 p) j0 (1 p) s p j j0 hus we have 1 F(s 1) p s and using formula (44) above we get (1 p) 1 ps (1 p) 1 p s (50) E{U} s1 s1 s1 [1 F(s 1)][β s u(c s )] [p s ][β s u(c s )] [pβ] s u(c s ) [pβ] t u(c t ). (51) 3. Recursive Representation of Lifetime Utilities Consider a discounted sum of utilities for a person with a known lifespan of years V 0 t0 β t u(c t ) u(c 0 ) + βu(c 1 ) + β u(c ). (52) hus, V 0 represents the discounted utility of a person at age t 0, looking ahead over the rest of their life. Now let V t denote the discounted utility of an age t person, looking forward from age t onwards. a. Write a formula for V t. What is V? V u(c ) V 1 u(c 1 ) + βv u(c 1 ) + βu(c ) V t u(c t ) + βv t+1 V 1 u(c 1 ) + βv 2 V 0 u(c 0 ) + βv 1 10 st s1 s0 β s t u(c s ) β s 1 u(c s ) β s u(c s )

11 t0 β t u(c t ). (53) b. Show that the utilities V t and V t+1 are connected recursively via the formula Answer: see the answer to part a above. c. Show that by age t 0, the recursive representation of V 0, i.e. V t u(c t ) + βv t+1 (54) V 0 u(c 0 ) + βv 1 (55) gives the same value V 0 as the non-recursive representation of discounted lifetime utility as in the original formula, (52). Answer: see the answer to part a above. d. Now let s extend this recursive way of thinking about discounted utilities to expected discounted utilities when there are uncertain lifetimes. Let V 0 be given by V 0 t0 [1 F(t 1)]β t u(c t ), (56) the same formula for expected lifetime utility when there is uncertain mortality as you derived in equation (42) above, where F(t) Pr{ t} t s0 f (t) is the cumulative probability of dying on or before age t and f (t) Pr{ t} is the probability of dying exactly on age t. Define V t analogously to V t in the case where there is no uncertainty about age of death, i.e. V t is the expected discounted utility from age t onwards, to whatever random age the person dies. Show that the appropriate form for the recursive representation for expected discounted utilities is in this case V t u(c t ) + β 1 F(t) 1 F(t 1) V t+1 (57) Answer. If the oldest any person can possibly live, the maximal lifespan is, then if you survive to this age, it must be your last year of life. hus V u(c ) (58) since there is zero probability that the person could live to age + 1 or longer. Now consider someone who survived to age 1. I claim that the value function at this age, V 1, is given by 1 F( 1) V 1 u(c 1 ) + β 1 F( 2 V. (59) hus, V 1 equals the sum of the utility at time 1, u(c 1 ), plus the expected discounted utility in the last period,. his is the product of the value in the last period, V (which equals u(c )), the discount factor β, and the conditional probability of surviving to age given that the person survived to age 1. his probability is [1 F( 1)]/[1 F( 2)]. o see why this is, note that there are two possibilities: either this person dies at the end of their 1 st year, or 11

12 they live to enjoy their final year too. he conditional probability that a person who survived to age 1 will survive to age is defined as Pr { > 1 1 } 1 F( 1) 1 F( 2). (60) his is a specific example of the general formula for conditional probability given by Pr {A B} Pr{A B} Pr{B}. (61) Where A B is the event that A and B occurs. So for this example, event A is the event that the person lives to age, i.e. A { } and event B is the event that the person lives to at least age 1, i.e. B { 1}. Notice that since is the maximal possible lifespan, we can write A { > 1} and similarly we can write B { > 2}. Now the event A B is clearly the same as event A since A B (i.e. the event that someone lives to at least age is a subset of the event that someone lives to at least age 1). hus we have in this case Pr {A B} Pr{A B} Pr{B} Pr{A} Pr{B}. (62) But if A { > 1}, clearly Pr{A} Pr{ > 1} 1 F( 1), since F( 1) is the cumulative probability that a persion will die at age 1 or before, 1 F( 1) f (t). (63) Similarly, we have Pr{B} Pr{ > 2} 1 F( 2). hus, the conditional probability of living to age given that you have lived to age 1 is Pr {A B} Pr { > 1 > 2 } Pr { > 1 } Pr { > 2 } 1 F( 1) 1 F( 2) More generally, we can see that if V t is the expected utility from age t onwards, we have (64) V t u(c t ) + β 1 F(t) 1 F(t 1) V t+1 (65) since [1 F(t)]/[1 F(t 1)] is the conditional probability of surviving to age t + 1 or longer, given that the person already survived to age t. Now let s see why, when we work backward to period t 0 and write V 0 u(c 0 ) + β 1 F(0) 1 F( 1) V 1 (66) we get the same answer as the direct way of writing the expected utility, i.e. V 0 t0 [1 F(t 1)]β t u(c t ), (67) 12

13 Note first that F( 1) 0, i.e. there is zero chance of dying in year t 1, the year before the person is born. hus we can write the recursive formula for V 0 as V 0 u(c 0 ) + β[1 F(0)]V 1. (68) But notice that by the backward recursion process, V 1 is the expected discounted utility of a person who survived birth and made it to age t 1. his utility would be V 1 u(c 1 ) + β 1 F(1) 1 F(0) V 2 t 1 1 F(t 1) β u(c t ), (69) 1 F(0) since [1 F(t 1)]/[1 F(0)] is the conditional probability of surviving to age t given that the person survived to age 1. So when be multiply V 1 by [1 F(0)] we get so we have e. Show that [1 F(0)]V 1 V 0 u(c 0 ) + β 1 F(0) 1 F( 1) V 1 u(c 0 ) + β[1 F(0)]V 1 u(c 0 ) + t0 β t 1 [1 F(t 1)]u(c t ), (70) β t [1 F(t 1)]u(c t ) β t [1 F(t 1)]u(c t ). (71) 1 F(t) 1 h(t) (72) 1 F(t 1) where h(t) is the hazard rate, i.e. the conditional probability of dying at age t given that one has survived to age t 1: h(t) f (t) 1 F(t 1) f (t) f (0) + f (1) + f (t 1) (73) and thus, 1 h(t) is the survival rate, i.e. the conditional probability that a person who lives to age t 1 will survive another year, to be at least age t or older before they die. Answer: Note that So we have F(t) F(t 1) + f (t) (74) 1 F(t) 1 F(t 1) f (t) (75) 13

14 It follows that 1 F(t) 1 F(t 1) 1 F(t 1) f (t) 1 F(t 1) f (t) 1 1 F(t 1) 1 h(t). (76) 4. Annuities with Uncertain Lifetimes a. Suppose that a person s lifetime is uncertain, so that the random variable denotes the random age of death, but that we know that the probability distribution of is geometric with parameter p (0,1). hat is, as noted above, f (t) p t (1 p), t 0,1,... If a person consumes a flat amount of $10000 per year until they die, and if the discount factor is β (0,1), write a formula for the expected discounted amount that this person will consume over their lifetime. Answer: Let V a denote the expected value of an annuity that pays the holder an annual amount a until they die, with a discount factor of β 1/(1 + r). Using the ideas from problem 2, we can either write a direct formula for V a, or an indirect or recursive formula for V a. he recursive formula is much easier, so let s start with this. We have V a a + pβv a. (77) hat is, the expected value of an annuity equals the current payment, a, plus the expected discounted value of the annuity from tomorrow onward, pβv a. his latter term is the product of the probability you survive p, the discount factor β times the expected discounted value of the annuity from tomorrow onward, V a. he expected value of the annuity does not change over time due to the particular memoryless property of the geometric distribution. hat is, the survival probability of a person with a geometric distribution of lifetimes is always p regardless of how old they currently are. his means that we do not have to keep track of the person s age to compute the expected value of the annuity, and this makes the problem stationary. he stationarity results in a the simple equation above, (77) for the expected present value of the annuity. Now if your initial wealth is W, we equate the value W you hand over to the annuity company to the expected value of the annuity and solve for a V a a 1 βp βp. (78) Now consider the direct way to compute V a. We have V a t0 f (t) s0ts s0 s0 t s0 β s a f (t)β s a [1 F(s 1)]β s a p s β s a a 1 βp, (79) 14

15 where the latter formula is a consequence of the formula for an infinite geometric series, j0 r j 1/(1 r) when r (0,1). b. An annuity is a contract such that if a person pays a given amount W up front at the start of their lifetime, the annuity company will in return provide that person with a constant payment of $c per year over their entire lifetime. Using the result from part 1 above, if a person has endowment of $1,000,000 when they are born and an annuity is purchased for them, how much will this annuity pay the person if their probability of survival is p.98 and the discount factor is β.95? Answer: he maximal annuity that would be paid in a competitive annuity market will satisfy W V a (80) hat is, competition by different annuity companies will result in bidding up the the present value they offer the person until the present value, V a, equals the amount of the person s wealth W to be invested in the annuity. Solving for a we get or W V a a 1 βp (81) a W(1 βp) ( ) (82) his annuity enables the annuity company to just break even, not earning a profit but also not making any losses. c. Suppose there are no annuity markets and that a person has a lifetime utility function (conditional on living years) equal to U β t log(c t ) (83) Describe the optimal consumption strategy for this person using dynamic programming, assuming that they are born with an initial endowment of W 1,000,000 and β.95 and p.98. Use the hint (below) that the maximized expected discounted utility function V (W) max {c t } t0 satisfies the Bellman equation p t (1 p) t s0 V (W) max 0 c W β t log(c s ) subject to: initial wealth W (84) [ log(c) + βpv and the additional hint (conjecture) that V(W ) takes the form ( )] W c β (85) V (W ) a + blog(w ) (86) for coefficients (a, b) to be determined. Plugging this conjecture on both sides of the Bellman equation (85) we get [ ( ) ] W c a + blog(w) max log(c) + βp[a + blog ] (87) 0 c W β 15

16 aking first order conditions for c on the right hand side of this equation, and using the fact that log(x/y) log(x) log(y) we get 0 1 c βpb W c. (88) Solving this for c we get c (W ) W 1 + βpb. (89) Now, substitute this expression for c back into the Bellman equation (85) to get ( ) [ ( )] W pbw a + blog(w ) log + βp a + blog, (90) 1 + βpb 1 + βpb Where we used the fact that W c (W) β W W 1+βpb β W pb 1 + βpb. (91) Now consider the left and right hand sides of the substituted version of the Bellman equation, (90). If this equation is to hold for all (positive) values of W then we need the coefficient of log(w ) on the left hand side, b, to equal the coefficients of log(w) on the right hand side. Using log(x/y) log(x) log(y) and gathering the coefficients of log(w ) on the right hand side of (90), we get the following equation for b b 1 + βpb (92) or b 1 1 βp. (93) Substituting this into the equation for the optimal consumption function c (W ) in equation (89), we get c W (W ) 1 + βp W(1 βp). (94) 1 βp his is a version of the permanent income hypothesis. hat is, the person consumes a constant fraction of their wealth, where the fraction is a mortality adjusted interest earnings. Now we need to solve for the a coefficient to complete the problem. Using the finalized version of the consumption function above, and using the formula for the b coefficient, we get another expression for the Bellman equation a + log(w) 1 βp [ log(w) + log(1 βp) + βp a + ] log(w ) + log(p). (95) 1 βp Gathering the terms that are constants (i.e. do not have log(w) in them) on the left and right hand sides of equation (95), we get the following equation for a [ a log(1 βp) + βp a + log(p) ], (96) 1 βp where we used the fact that W c (W ) β W W(1 βp) β pw. (97) 16

17 Solving the equation for a above, we bet a log(1 βp) 1 βp + log(p) (1 βp) 2. (98) We conclude the the utility of a person who gradually consumes their initial endowment (as opposed to buying an annuity with it) is V(W ) log(1 βp) 1 βp + log(p) (1 βp) 2 + log(w) 1 βp. (99) d. Now suppose that there are annuity markets. he person now has the option, at the start of their life, to exchange their entire initial endowment of wealth W for a lifetime annuity. Which option would the person prefer: 1) to exchange W and take the annuity, or 2) not buy the annuity and follow the optimal consumption plan described in part 3 above? Answer: Let V a (W ) be the expected discounted utility to the person if he/she chooses the annuity, a W(1 βp), that they can afford by converting all of their initial wealth W into the annuity. hen we have the following recursive formula for V a (W ) or solving, we get V a (W ) log(a) + βpv a (W ) (100) V a (W ) log(a) 1 βp log(w) + log(1 βp). (101) 1 βp Comparing this formula with the formula for expected discounted utility associated with not buying an annuity, (99), we get V a (W ) V(W) log(p) > 0, (102) (1 βp) 2 since log(p) < 0 since p (0,1). hus, we conclude that the person is better off taking the annuity. Why is this? It might appear that the person should be indifferent between taking the annuity and managing their own consumption saving since in the first period of their life the optimal consumption (no annuity) is c (W ) W (1 βp) a, (103) i.e. the optimal consumption in the first period is the same as the annuity payment. However consider the second period. he person s wealth in the second period is W 2 W c (W ) W W(1 βp) pw < W. (104) hus, the consumption for the person in period 2 is c (W 2 ) W 2 (1 βp) pw(1 βp) < a. (105) hus, the second period consumption is less than the annuity. Continuing forward we see that in period t we have c (W t ) W t (1 βp) p t 1 W (1 βp) < a. (106) 17

18 hus, since 0 < p < 1, the consumption of the person who did not choose the annuity is tending to zero over time, whereas the consumption of the person who chose the annuity is alway constant and equal to the initial consumption W(1 βp) of the person who did not choose the annuity. hus, since the consumption stream under the annuity clearly dominates the consumption stream a person can get from self-insuring their own mortality risk by only gradually consuming their wealth as they age, it is clear that in this example it would always be a better choice to purchase the annuity rather than to self-insure mortality risk. Hint: o solve part 3, use the method of dynamic programming with the following Bellman equation Conjecture that V(W ) is of the form V (W) max [log(c) + pβv (W c)] (107) 0 c W V(W ) a + blog(w ) (108) and using the Bellman equation above, solve for the coefficients a and b so that the Bellman equation will hold. From this solution you should be able to derive the associated optimal consumption function, c(w ), which specifies how much the person will consume each year given that they start that year with total savings of W. 5. Consumption and axes Suppose a consumer has a utility function u(x 1,x 2 ) log(x 1 )+log(x 2 ) and an income of y 100 and the prices of the two goods are p 1 2 and p 2 3. a. In a world with no sales or income taxes, tell me how much of goods x 1 and x 2 this consumer will purchase. b. Now suppose there is a 10% a sales tax on good 1. hat is, for every unit of good 1 the person buys, he/she has to pay a price of p 1 (1 +.1) 2.2, where the 10% of the price, or 20 cents, goes to the government as sales tax. How much of goods 1 and 2 does this person buy now? c. Suppose instead there is a 5% income tax, so that the consumer must pay 5% of his/her income to the government. If there is no sales tax but a 5% income tax, how much of goods 1 and 2 will the consumer consume? d. Which would the consumer prefer, a 10% sales tax on good 1, or a 5% income tax? Explain your reasoning for full credit. e. How big would the sales tax on good 1 have to be for the government to get the same revenue as a 5% income tax? Which of the two taxes would the consumer prefer in this case, or is the consumer indifferent because the consumer has to pay a total tax of $5 (5% of $100) in either case? Answer I answer each part separately below. Notice that the utility function is a monotonic transformation of a Cobb-Douglas utility function l(x 1,x 2 ) x 1/2 1 x 1/2 2, so demands are x 1 (p 1, p 2,y) y/2p 1 and x 2 (p 1, p 2,y) y/2p 2. With these, it is very easy to answer this question. a. x 1 100/(2 2) 25 and x 2 100/(2 3)

19 b. With the tax in place, the price of good 1 increases to 2.2 so quantities demanded are x 1 100/(2 2.2) and x 2 100/(2 3) he total taxes the person pays are.2x /(2 2.2) c. With a 5% income tax, the consumer has after-tax income equal to $95 (100(1 τ) where τ.05). So the consumption of goods 1 and 2 is given by x 1 95/(2 2) and x 2 95/(2 3) d. With the sales tax, the consumer consumes less of good 1 and more of good 2, and pays less in tax overall. With the income tax the consumer consumes more of good 1 but less of good 2 and pays more overall in tax ($5.00 versus $4.54). But the only way to see which alternative the consumer prefers is to plug the consumption bundles into his/her utility function and see which one give more utility. he utility under the sales tax is log( ) + log( ) he consumer s utility under the income tax is log(23.75) + log( ) so the consumer prefers the sales tax to the income tax. e. Now we want to set the sale tax rate α so that we raise tax revenue of $5, the same revenue that we collect under an income tax of 5%. he equation for the necessary tax rate is 5 α 100 2(2 + α) (109) Solving this for α we get α 2/ Under this tax rate, consumption of good 1 falls to x (2+α) 22.5 and the tax revenue collected is /9 5. Now the person s utility under the sales tax is log(22.5) + log( ) , so that now, the consumer slightly prefers to have the income tax over the sales tax. 6. Risk Neutrality and Risk Aversion An person is said to be risk neutral if when offered a gamble, their maximum willingness to pay to undertake the gamble equals the expected value of the gamble. hat is, if G denotes a random payoff from a gamble, the maximum entry fee F that a risk neutral person would be willing to pay to get the gamble payoff G is A person is risk averse if F < E{ G} and risk loving if F > E{ G}. F E{ G}. (110) a. Suppose a person has a utility function u(w ) W, and suppose that initially (before considering taking the gamble) the person has W of wealth. Suppose the gamble under consideration is to flip a coin, and if it lands heads the person wins $100, and if tails the person gets nothing. What it is the maximum amount F this person would be willing to pay for this gamble? Is this person risk neutral, risk loving, or risk averse? A risk neutral person will be willing to pay up to the expected value of the gamble, i.e. F E{ G}. In this case it is F E{ G} p100 + (1 p)0 p (111) b. Suppose a person has a utility function u(w ) log(w) and this person also has W in initial wealth before considering taking the gamble. What is the maximum amount F this person would be willing to pay for the gamble? 19

20 Answer: he general equation for the maximum amount someone would be willing to pay is u(w ) E{u(W F + G)} (112) since the left hand side is the utility of not participating in the gamble, and the right hand side is the expected utility of paying the fee F and then getting the gamble payoff G. Note that if you have a linear utility function u(w ) a + bw with b > 0, then a + bw E{a + b(w F + G)} a + bw bf + be{ G}, (113) using the properties of expectations (i.e. that for a linear function E{a + b X} a + be{ X}). We see that the value of F that equates both sides of the above equation is F E{ G}, which was the answer we got in part a for a risk neutral person. Now for a person with log utility, we get log(w ) E { log ( W F + G )} plog(w F + 100) + (1 p)log(w F). (114) his is the equation that needs to be solved for F to determine the person s maximal willingness to pay. Note that in general F is a function of W so we can write F(W ) as the amount a person of wealth level W would be willing to pay. It is not difficult to show that in general for this person, F(W ) < E{ G}, (115) so that the person is risk averse since their maximal willingness to pay is less than the expected value of the gamble. c. Suppose a third person has a utility function u(w ) W 2 and also has initial wealth W before considering the gamble. What is the maximum amount this third person would pay for the gamble? Answer: In this case the equation for F is W 2 E { (W F + G ) 2 } p(w F + 100) 2 + (1 p)(w F) 2. (116) In this case it turns out that the solution F(W) satisfies and so this person is risk loving. F(W ) > E{ G} (117) d. Are the persons in cases b and c above risk neutral, risk averse or risk loving? Answer: In part a the person is risk neutral, in part b the person is risk averse, and in part c the person is risk loving. e. Prove that a person is risk neutral if their utility function is linear, u(w ) a + bw for b > 0, and risk averse if their utility function is concave, u (W) > 0 and u (W) < 0, and risk loving if their utility function is convex, u (W ) > 0 and u (W) > 0. Hint: If a person does not take the gamble, they will have utility u(w ) from consuming their wealth W. If the person pays an amount F for a gamble G, their expected utility would be E{u(W F + G)}. he 20

21 maximum willingness to pay for the gamble G would be the amount F that makes the person indifferent between paying F for the gamble and not taking the gamble, i.e. it is the solution to U(W) E{u(W F + G)}. (118) You can use Jensen s Inequality which states that for a concave function u and any random variable X we have E{u( X)} u(e{ X}). (119) You should be able to use Jensen s inequality to show that people with concave utility functions are risk averse. Answer: We already showed above that a person with a linear utility function is risk neutral, i.e. they would be willing to pay the expected value of the gamble, F E{ G}. Now consider a risk averse person. Jensen s inequality tells us that net of the fee F we have for a risk averse person E { u(w F + G) } u(w F + E{ G}). (120) It follows that the fee F(W ) must be less than or equal to E{ G} to induce the risk averse person to take the gamble since if F > E{ G} we have u(w ) > u(w F + E{ G}) E{u(W F + G)} (121) and this says that at the fee F the person would not want to take the gamble. For a risk loving person, Jensen s inequality is reversed, E { u(w F + G) } u(w F + E{ G}). (122) hus, for this person the fee F(W) must be greater than E{ G} since otherwise if F < E{ G} we have u(w ) < u(w F + E{ G}) E{u(W F + G)} (123) and thus the person would be strictly better off taking the gamble. he fee can therefore be raised above the expected payoff E{ G} and still induce the risk loving person to take the gamble. he specific answers for F( ) in the logarithmic utility case are F( ) , which is just slightly less than the expected value of the gamble, and in the quadratic utility case, F( ) , which is just slightly more than the expected value. his is because at such high wealth levels, both the logarithmic and quadratic utility functions look pretty linear and thus individuals with these utility functions at high wealth levels act approximately risk neutrally, even though they are technically risk-averse and risk-loving, respectively. If you recalculate the maximal willingness to pay at lower levels of wealth, you get bigger deviations from the expected value. For example when W 1000, F(1000) for the logarithmic utility function, and F(1000) for the quadratic utility function. 7. St. Petersburg Paradox Consider the following gamble G. You flip a fair coin until it lands on tails. Let h denote the number of heads obtained until the first tail occurs and the game stops. Your payoff from playing this game is G 2 h (124) 21

22 a. Suppose you are risk neutral. What is the maximum amount F that you would be willing to pay to play this game? Answer: A risk neutral person would be willing to pay up to the the expected value E{ G} h02 h [ ]h h0 1, (125) since the probability of getting h heads until the first tail is geometrically distributed with density f (h) given by [ ] 1 h 1 f (h) 2 2. (126) b. Suppose you have a utility function u(w ) log(w) and W in initial wealth. What is the maximum amount you would be willing to pay to play this gamble in this case? Answer: As we saw above, the logarithmic utility has a negative second derivative (diminishing marginal utility of wealth) and is thus concave, and therefore the person is risk averse. he maximal willingness to pay for a gamble by a risk averse person is less than the expected value of the gamble. In this case, the willingness to pay satisfies log(w ) E { log(w F + G) } h0 log(w F + 2 h )[ 1 2 ]h 1 2. (127) Figure 1 plots the maximal willingness to pay function calculated by the Matlab function certaintyequiv.m which I have posted also with these answers. his program finds a zero of the Matlab function eu.m which is defined by eu(f) h0 log(w F + 2 h )[ ]h log(w ), (128) 2 where I numerically approximate the infinite summation in the equation for eu(f) above. he program certaintyequiv.m uses Matlab s fsolve routine to find an F such athat eu(f) 0. Figure 1 plots F(w) for W [100,1000]. We see that the amount that this person is willing to pay for this gamble is faily small: less than $7 for wealth up to $1000. If I solve for the certainty equivalent (i.e. F(W )) when W , using certaintyequiv.m I get F( ) hus, even a very rich person with a logarithmic utility function would not be willing to pay very much to undertake this gamble. In this sense, risk aversion is the resolution of the St. Petersburg Paradox. 22

23 6 Certainty Equivalents for Log Utility, Bernoulli Paradox Certainty Equivalent, C Initial Wealth, W Figure 1 F(W): Maximal Willingness to Pay for the Gamble G 23