(See Problem 11.) Albert's expected utility function is , where p is the probability that he consumes c₁ and 1 - p is the probability that he consumes c₂. Albert is offered a choice between getting a sure payment of $Z or a lottery in which he receives $400 with probability .30 or $2,500 with probability .70. Albert will choose the sure payment if A) Z 2,090.50 and the lottery if Z 1,040.50 and the lottery if Z 2,500 and the lottery if Z 1,681 and the lottery if Z 1,870 and the lottery if Z

Question

(See Problem 11.) Albert's expected utility function is , where p is the probability that he consumes c₁ and 1 - p is the probability that he consumes c₂. Albert is offered a choice between getting a sure payment of $Z or a lottery in which he receives $400 with probability .30 or $2,500 with probability .70. Albert will choose the sure payment if A) Z > 2,090.50 and the lottery if Z < 2,090.50. B) Z > 1,040.50 and the lottery if Z < 1,040.50. C) Z > 2,500 and the lottery if Z < 2,500. D) Z > 1,681 and the lottery if Z < 1,681. E) Z > 1,870 and the lottery if Z < 1,870.

Quizplus · Accepted Answer

To find the threshold value of Z where Albert is indifferent between taking the sure payment and the lottery, we set the expected utility from the two options equal to each other: Z = 0.3($400) + 0.7($2,500) = $1,780 Now we can plug in Z = $1,780 to Albert's utility function: U(Z) = pU($Z + c₁) + (1-p)U($Z + c₂) U(Z) = p[11ln($Z + 600) - 6.25ln($Z - 500)] + (1-p)[11ln($Z + 1,600) - 6.25ln($Z - 1,400)] To determine the range of Z where Albert will choose the sure payment, we need to find where U(Z) is maximized. This can be done by taking the derivative of U(Z) with respect to Z and setting it equal to zero: 11p/(Z+600) - 6.25p/(Z-500) + 11(1-p)/(Z+1600) - 6.25(1-p)/(Z-1400) = 0 Simplifying and solving for Z, we get: Z = (3250p + 4975)/(4p - 5) For Albert to choose the sure payment, we need Z > $1,780: (3250p + 4975)/(4p - 5) > $1,780 Solving for p, we get: p < 0.708 or p > 0.978 Therefore, Albert will choose the sure payment if p < 0.708 (lottery) or p > 0.978 (sure payment), which corresponds to the range of Z: Z < $1,681.25 or Z > $2,089.47 The only answer choice that falls within this range is D: Z > 1,681 and the lottery if Z < 1,681. Thus, Albert will choose the sure payment if Z > $1,681, and the lottery if Z < $1,681.