Chapter 3 Henderson and Quandt Microeconomics, Questions 3.8-3.11_ Solutions

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3-8 A consumer who conforms to the von Neumann-Morgenstern axioms is faced with four

situations A, B, C, and D. She prefers A to B, B to C, and C to D. Experimentation reveals that

the consumer is indifferent between B and a lottery ticket with probabilities of 0.4 and 0.6 for A

and D respectively, and that she is indifferent between C and a lottery ticket with probabilities of

0.2 and 0.8 for B and D respectively. Construct a set of von Neumann-Morgenstern utility

numbers for the four situations.

Ans. Let u(A) = 1, u(B) = x, u(C) = y, and u(D) = 0, where x and y are unknown values.

From the given information, we know that:

u(A) > u(B) > u(C) > u(D)

u(B) = 0.4u(A) + 0.6u(D)

u(C) = 0.2u(B) + 0.8u(D)

Substituting the given utility values, we get:

1 > x > y > 0

x = 0.4(1) + 0.6(0) = 0.4

y = 0.2(0.4) + 0.8(0) = 0.08

Therefore, the set of von Neumann-Morgenstern utility numbers for the four situations is:

u(A) = 1, u(B) = 0.4, u(C) = 0.08, u(D) = 0

3-9 Show which of the following utility functions exhibit decreasing risk aversion: U(W) =

(W+a)^, a ≥ 0, 0<β<1; U(W) = W; U(W) = In (W+ a), a ≥0; U(W) = W³.

Ans. To determine whether a utility function exhibits decreasing risk aversion, we need to examine the second derivative of the utility function with respect to wealth (W). Specifically, if the second derivative is negative, the function exhibits decreasing risk aversion.

Let's take each of the given utility functions in turn:

U(W) = (W+a)^β, a ≥ 0, 0<β<1:

The first derivative is: U'(W) = β(W+a)^(β-1)

The second derivative is: U''(W) = β(β-1)(W+a)^(β-2)

Since β and (β-1) are both negative, and (W+a)^(β-2) is positive, the second derivative is negative. Therefore, this utility function exhibits decreasing risk aversion.

U(W) = W:

The first derivative is: U'(W) = 1

The second derivative is: U''(W) = 0

Since the second derivative is zero, this utility function does not exhibit decreasing risk aversion.

U(W) = In (W+ a), a ≥0:

The first derivative is: U'(W) = 1 / (W+a)

The second derivative is: U''(W) = -1 / (W+a)^2

Since the second derivative is negative, this utility function exhibits decreasing risk aversion.

U(W) = W³:

The first derivative is: U'(W) = 3W^2

The second derivative is: U''(W) = 6W

Since the second derivative is positive, this utility function exhibits increasing risk aversion.

Therefore, the utility functions that exhibit decreasing risk aversion are: U(W) = (W+a)^β and U(W) = In (W+ a), where a ≥ 0 and 0<β<1.

3-10 A consumer who obeys the von Neumann-Morgenstern axioms and has an initial wealth of

160,000 is subject to a fire risk. There is a 5 percent probability of a major fire with a loss of

70,000 and a 5 percent probability of a disastrous fire with a loss of 120,000. Her utility function is

U = . She is offered an insurance policy with the deductibility provision that she bear the

first 7620 of any fire loss. What is the maximum premium that she is willing to pay for this policy?

Ans. We can begin by calculating the expected utility of the consumer under each of the two possible fire outcomes, as well as the utility of no fire:

U(no fire) = √160,000 = 400

U(major fire) = √(160,000 - 70,000 + 7,380) = √97,380 ≈ 312.15

U(disastrous fire) = √(160,000 - 120,000 + 7,380) = √47,380 ≈ 217.81

Note that in the case of a fire, we have subtracted the deductible of 7,620 from the total loss before taking the square root.

Using these expected utility values, we can calculate the expected utility of the insurance policy as follows:

EU(insurance) = 0.95 U(no fire) + 0.025 U(major fire) + 0.025 U(disastrous fire)

= 0.95(400) + 0.025(312.15) + 0.025(217.81)

≈ 386.73

Next, we can calculate the expected utility of not purchasing the insurance policy, which is simply the expected utility of the no fire outcome:

EU(no insurance) = U(no fire)

= 400

The maximum premium that the consumer is willing to pay for the insurance policy is the difference between the expected utility of the policy and the expected utility of not purchasing the policy, divided by the probability of a fire:

Max premium = [EU(insurance) - EU(no insurance)] / 0.05

= [386.73 - 400] / 0.05

≈ -262.6

Since the maximum premium is negative, the consumer is not willing to pay any premium for the insurance policy with the given deductible. This suggests that the deductible is too low to make the policy attractive to the consumer. Alternatively, if the deductible were increased, the premium that the consumer would be willing to pay might become positive.

3-11 Let a consumer's strictly quasi-concave utility function be U = f(q) + 3M where M is the

quantity of a composite commodity with unit price. Assume that her demand function for Qis

q= where a>0. Determine f(q) by solving a differential equation formed from the

first-order condition for utility maximization.

Ans. To determine f(q), we first need to write down the Lagrangian for the consumer's utility maximization problem:

L(q, λ, M) = f(q) + 3M + λ(pq - M)

where λ is the Lagrange multiplier associated with the budget constraint.

The first-order conditions for utility maximization are:

∂L/∂q = f'(q) + λp = 0 (1)

∂L/∂M = 3 - λ = 0 (2)

pq - M = 0 (3)

Solving equation (2) for λ and substituting into equation (1), we get:

f'(q) + p(3/p)q = 0

f'(q) + 3q = 0

This is a first-order linear homogeneous differential equation, which can be solved using separation of variables:

f'(q) = -3q

∫d f = -3∫q dq

f(q) = -3q^2/2 + C

where C is the constant of integration.

Substituting this expression for f(q) into equation (3), we get:

pq = 3M/2 + C

Substituting the consumer's demand function q = p^(-a), we get:

p^(1-a) = 3M/(2p) + C/p

Taking the natural logarithm of both sides and rearranging, we get:

ln p = (1-a)ln p + ln(3M/2) - ln p + ln C

Solving for ln C, we get:

ln C = ln(3M/2) - ln p

Substituting this expression for ln C into the expression for f(q), we get:

f(q) = -3q^2/2 + ln(3M/2) - ln q

Therefore, the solution for f(q) is:

f(q) = -3q^2/2 + ln(3M/2) - ln q

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