Recall Bob and Ray in Problem 4. They are thinking of buying a sofa. Bob's utility function is U_B(S, M_B) = (1 + S)M_B and Ray's utility function is U_R(S, M_R) = (2 + S)M_R, where S = 0 if they don't get the sofa and S = 1 if they do and where M_B and M_R are the amounts of money they have respectively to spend on their private consumptions. Bob has a total of $1,200 to spend on the sofa and other stuff. Ray has a total of $1,200 to spend on the sofa and other stuff. The maximum amount that they could pay for the sofa and still arrange to both be better off than without it is

Question

Quizplus · Accepted Answer

To find the maximum amount they can pay for the sofa and still both be better off, we need to find the range of prices where both Bob and Ray's utility increases from their utility without the sofa. First, we need to determine the utility that Bob and Ray each get without the sofa. Bob's utility without the sofa is: US1U1B1B0S10MS1U1B1B0 = (1 + 0)MS1U1B1B0 = MS1U1B1B0. Similarly, Ray's utility without the sofa is: US1U1B1R0S10MS1U1B1R0 = (2 + 0)MS1U1B1R0 = 2MS1U1B1R0. Now, we need to find the range of prices where both Bob and Ray's utility increases from the respective values above. For Bob: US1U1B1B1S1U1B0 > MS1U1B1B0 => (1 + 1)MS1U1B1B0S1U1B0 > MS1U1B1B0 => 2MS1U1B1B0S1U1B0 > MS1U1B1B0 => MS1U1B1B0(S1U1B0 - 2) < 0 => S1U1B0 < 2. This means that for Bob to be better off with the sofa, he needs to have at least $2 left over after buying the sofa (i.e. he can spend at most $1,200 - $2 = $1,198 on the sofa). For Ray: US1U1B1R1S1U1B0 > 2MS1U1B1R0 => (2 + 1)MS1U1B1R0S1U1B0 > 2MS1U1B1R0 => 3MS1U1B1R0S1U1B0 > 2MS1U1B1R0 => MS1U1B1R0(S1U1B0 - 2/3) < 0 => S1U1B0 < 2/3. This means that for Ray to be better off with the sofa, he needs to have at least $2/3 left over after buying the sofa (i.e. he can spend at most $1,200 - $800/3 = $400/3 ≈ $133.33 on the sofa). Since they both need to be better off with the sofa, the maximum they can spend on the sofa is the minimum of the amounts calculated above: Max amount = min($1,198, $400/3) ≈ $399.33 ≈ $400. Therefore, the maximum amount that they could pay for the sofa and still arrange to both be better off than without it is $400. Hence, the correct choice is (C).