Albert's expected utility function is pc^1/2₁  (1  p) c^1/2₂, 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.

To determine Albert's expected utility from the lottery, we need to calculate the expected value of his utility function. Let Z = the sure payment, $400 be represented as x, and $2,500 be represented as y. Then the expected utility from the lottery is:

(.3)(pc^1/2₁  (1 F1F1F1p)x) + (.7)(pc^1/2₂  (1 F1F1F1p)y)
Simplifying, we get:

.3(.6(Z)1/2 + .4(1-Z)1/2) + .7(.6(Z/2500)1/2 + .4(1-Z/2500)1/2)
To determine the value of Z at which Albert is indifferent between the lottery and the sure payment, we can apply the expected utility function to each option and set them equal:

pc^1/2₁  (1 F1F1F1p)Z = .3(.6(Z)1/2 + .4(1-Z)1/2) + .7(.6(Z/2500)1/2 + .4(1-Z/2500)1/2)
To solve for Z, we can use trial and error or an iterative method. One way to guess a solution is to use the midpoint of the range of possible values for Z, which is between 0 and 2500. In this case, the midpoint is 1250. Plugging this value into the equation, we get:

pc^1/2₁  (1 F1F1F11,250) = .3(.6(1,250)1/2 + .4(1-1,250)1/2) + .7(.6(1,250/2500)1/2 + .4(1-1,250/2500)1/2)
Simplifying, we get:

pc^1/2₁ F1F1F11,250) = .3(235.7) + .7(154.4)
pc^1/2₁ F1F1F11,250) = 193.6
To solve for p, we can use trial and error or an iterative method. One way to guess a solution is to use the midpoint of the range of possible values for p, which is between 0 and 1. In this case, the midpoint is 0.5. Plugging this value into the equation, we get:

pc^1/2₁ F1F1F11,250) = 193.6
p*0.5*(1/2+1*(1-p))+(1-p)*0.5*(1/2+1*(p)) = 193.6
p = 0.117

Now we can calculate the expected utility of the lottery and the sure payment at this value of p:

Expected utility of the lottery = .3(pc^1/2₁  (1-0.117)400) + .7(pc^1/2₂  (0.117/2500)1,250) = 1,785.03
Expected utility of the sure payment = pc^1/2₁ (1-0.117)Z = 1,677.24Z1/2
Setting these two equal and solving for Z, we get:

1,677.24Z1/2 = 1,785.03
Z = 1,681

Therefore, Albert would choose the sure payment if Z ≥ $1,681 and the lottery if Z < $1,681. Answer choice D is the only option that matches this result.