Exercises for Chapter 8 Exercise 8. Consider the following functions: f (x)= e x, (8.) g(x)=ln(x+), (8.2) h(x)= x 2, (8.3) u(x)= x 2, (8.4) v(x)= x, (8.5) w(x)=sin(x). (8.6) In all cases take x>0. (a) Plot each function and determine whether each function is concave, convex or neither. (b) For each function, give the second-order Taylor series expansion. Exercise 8.2 functions, Give the second-order Taylor series expansions for the following f (x) f ( x)+ f ( x)(x x)+ 2 f ( x)(x x) 2 : f (x)= x 3 (8.) f (x)=ln(x) (8.2) f (x)=sin(x) (8.3) 37
CHAPTER 8. TAYLOR SERIES 38 f (x)=cos(x) (8.4) f (x)= e rx (8.5) f (x)= x α (8.6) f (x)= a x. (8.7) Exercise 8.3 Consider the average cost function c(q) = e Q + 2 Q2. Show that the cost function is strictly convex. Find the minimum of the average cost function c: find the value of Q that minimizes c. To do this, first observe that we can write a Taylor series approximation of c: c(q) = c(q 0 )+ c (Q 0 )(Q Q 0 )+ 2 c (Q 0 )(Q Q 0 ) 2. When Q 0 is close to the value of Q that minimizes c, then minimizing c(q) should give approximately the same answer as minimizing c(q 0 )+ c (Q 0 )(Q Q 0 )+ 2 c (Q 0 )(Q Q 0 ) 2. If we minimize the latter expression by differentiating with respect to Q, we get the first-order condition: c (Q 0 )+ c (Q 0 )(Q Q 0 ) = 0. Solving for Q gives Q = c Q 0 (Q 0 ) c (Q 0 ). This is the basis of an iteration: guess an initial Q 0 ; this gives c Q = Q 0 (Q 0 ) c c (Q 0 ). With Q we can obtain another estimate Q 2 = Q (Q ) c (Q ), and with Q 2 we can obtain Q 3, and so on. At the nth iteration Q n+ = Q n. The process converges when the change from Q n to Q n+ becomes small. In this case, we end up with (approximately) Q n = Q n+. Looking at the iteration rule Q n+ = Q n yields 0=Q n+ Q n =. Thus = 0, and since 0, we get =0. Thus, since c is convex, Q n minimizes c. Exercise 8.4 Harveys brewery has a monopoly in the beer market. The demand function is P(Q)=00 ( 3) Q 2 and the cost function is C(Q)=Q 2 + e 0.0Q2. (a) Is the profit function concave? (b) Using numerical optimization, find the profit-maximizing level of output. (Hint: start with Q 0 = 8 and perform four iterations.) Exercise 8.5 function A monopolist is faced with the following demand curve and cost P(Q)=54 2e Q, C(Q)=4Q+6Q 2. (a) Confirm that the profit function is concave. (b) Find the profit-maximizing level of output using numerical optimization. Write down the algorithm. (Start iterations at q 0 = 2 and perform five iterations.)
39 EXERCISES FOR CHAPTER 8 Exercise 8.6 functions: A manufacturer of jackets faces the following demand and cost P(q)=000 0q 2, C(q)=00q+ qln q. (a) Confirm that the profit function is concave. (b) Find the profit-maximizing level of output by numerical optimization. Exercise 8.7 Faced with the following demand and cost functions P(q)=000 q 2, C(q)=25+ q 2 + q(ln q) 2, a widget firm wishes to find the profit-maximizing level of output. Find the profitmaximizing level of output. Exercise 8.8 Suppose a firm s profit function (as a function of output) is given by π(q)=2ln(q+)+ q q. Find the profit-maximizing value of q. Exercise 8.9 Suppose a firm s revenue function (as a function of output) is given by r(q)=3( e q )+3ln(q+) 2q. Find the revenue-maximizing value of q. Exercise 8.0 Consider the function f (x)=+ x+ 4x 2 3x 3. This function has two points at which the first derivative of f is 0. Denote these x and x, so that f (x )= f ( x)=0. It is easy to see that one of these is, say that x=. Find the value of x using a Taylor series expansion. Exercise 8. An individual has utility defined on income according to u( y) = ln(y+ ). The individual has initial income ȳ and accepts a bet on a coin toss: if heads, the individual receives a dollar, if tails the individual pays a dollar. The coin is fair (50/50 chance of heads or tails), thus the average utility from the bet is 2 u(ȳ )+ 2 u(ȳ+). If the individual had not accepted the bet, the utility obtained is u(ȳ) for sure. Compare the average utility of the bet with the sure utility from not betting. Should the individual have accepted the bet? Exercise 8.2 An individual has the utility function u(y) = ln(y+). This gives the utility of having y dollars. The individual faces a random income next period: current income is $00 and next period income will either fall by $5 to $95 or increase by $5 to $05. There is a 50% chance that income will rise and a 50% chance that income will fall, so that average income next period is 00= 2 95+ 2 05. The risk premium, c, is defined as the amount of money less than $00 that makes the individual indifferent between accepting that sum for
CHAPTER 8. TAYLOR SERIES 40 sure and facing the risk associated with receiving only $95. That is, the individual is indifferent between $00 c and the risky return $95 with 50% chance and $05 with 50% chance. Thus, u(00 c)= 2 u(95)+ 2 u(05), or ln(0 c)= 2 ln(0 5)+ 2 ln(0+5). Find the approximate value of c using Taylor series expansions. Plot the utility function and illustrate how c is calculated on your graph. Exercise 8.3 Repeat the previous question using the utility function u( y) = e y. In this case, note that the utility function is convex (second derivative positive) so that your picture will change significantly! Exercise 8.4 A student has just received a present of a new computer. The student s yearly income is $50 and he student has a utility function u(y) = y 2. Unfortunately, there is a 0% chance that the computer will be stolen and the price of a new computer is $5, so in the event of theft, the replacement will leave the student with only $45 income for the year. Thus, in the absence of theft, the student s income is $50, while if the computer is stolen, the income drops to $45. Computer insurance is available at a cost of $. Use the calculations from a Taylor series expansion to decide whether the student should buy insurance or not. Note u (y)= 2 y 2 and u (y)= 4 y 3 2, so that u (y) u (y) = 3 4 y 2 2 y 2 = 3 4 y 2 2y /2 = 2 y. Exercise 8.5 Consider an individual with the following utility function defined over wealth: u(w) = w. Suppose the person has the opportunity to take the following gamble: draw a ticket from a hat; a blue ticket yields a prize of $200, a yellow ticket yields $50 and an orange ticket yields $00. There are four orange tickets, two blue, and two yellow. (a) Calculate the expected payoff from the bet. (b) Calculate the approximate cost of risk to this individual (using a Taylor series expansion). (c) How much is the person willing to pay to partake in this gamble? (d) Suppose instead the utility function is u(w) = w. Rework your answers to parts (b) and (c). (e) Suppose now the utility function is u(w)=w 2. Rework your answers to parts (b) and (c). (f) Explain intuitively why your answers differ in parts (c), (d) and (e).
4 EXERCISES FOR CHAPTER 8 Exercise 8.6 A risk-averse individual has the following utility function defined over income: u(w) = w. The person is faced with the following lottery with possible outcomes denoted x i and their associated probabilities P(x i ): x 0 5 7 P(x) 2 8 8 4 (a) What is the expected payoff from this lottery? (b) Calculate this person s measure of absolute risk aversion. (c) If tickets for this lottery cost $7.50, will he buy a ticket? (Use a Taylor series expansion.) (d) Another individual has utility function v(x) = kx, k > 0. Will this individual buy a ticket? Exercise 8.7 An individual owns a house worth $200,000 that is located next to a river. There is a % chance that the river will flood in the spring and cause $60,000 worth of damage. The utility function of the home owner is u(w)=ln w. (a) How much is the individual willing to pay for house insurance? (b) Suppose instead that the homeowner s utility function is u = w. How much would he now be willing to pay for insurance? Why is this amount more in part (a)? Exercise 8.8 A group of graduate students have recently pooled together their life s savings and bought a ski cabin. Apart from its inaccessability, the major drawback with this cabin is that it is located near a hill that is somewhat prone to avalanches. The cabin is worth $00,000. There are two types of avalanche. The first, which occurs with 20% probability, will cause $30,000 worth of damage; the second occurs with 2% probability but causes $90,000 worth of damage. Suppose there are three students in this group with utility functions u (w)= w 2, u 2 (w)= w 2, u 3 (w)= w. How much will each student be willing to pay to insure the cabin? What is the relationship between these amounts and how is this related to their attitudes towards risk? Is it true that a risk-loving individual will never purchase insurance?
CHAPTER 8. TAYLOR SERIES 42 Exercise 8.9 According to expected utility theory, faced with uncertain outcomes x with probability p and x 2 with probability ( p), an individual associates utility E(u) = pu(x )+( p)u(x 2 ). The risk premium, c, associated with the risk is defined: u(µ c)=e(u)= pu(x )+( p)u(x 2 ), where µ= px +( p)x 2, the mean of the random variable x. In the case where u(x)= x, x = 90, x 2 = 0 and p = 2, find π. An approximation for c is given by c u (µ) 2 u (µ) σ2, where σ 2 = p(x µ) 2 + ( p)(x 2 µ) 2. Estimate c using this approximation and compare it with the exact solution. Exercise 8.20 (a) Let a risk-averse agent have utility function u(x) = ln(x + ). Suppose that this individual faces the gamble: win $95 with probability 2 and win $05 with probability 2. How much less than $00 would the individual accept instead of the gamble? (b) Let a random variable have two possible outcomes x and x 2, with probabilities p and p 2, respectively. Thus, the mean is µ= p x + p 2 x 2. Let the agent s utility function be u. Suppose that p u(x )+ p 2 u(x 2 )= u(z) (and both x and x 2 are close to z). Show that: µ z p (x z) 2 + p 2 (x 2 z) 2 2 u (z) u (z).