Harvey Habit has a utility function U(c₁, c₂) = min{c₁, c₂}, where c₁ and c₂ are his consumption in periods 1 and 2 respectively. Harvey earns $147 in period 1 and he will earn $63 in period 2. Harvey can borrow or lend at an interest rate of 10%. There is no inflation. A) Harvey will save $102. B) Harvey will borrow $40. C) Harvey will neither borrow nor lend. D) Harvey will save $40. E) None of the above.

Question

Quizplus · Accepted Answer

To maximize his utility, Harvey needs to choose his consumption levels in both periods such that U(c₁, c₂) is as high as possible. Given that his utility function is U(c₁, c₂) = min{c₁, c₂}, the maximum utility he can get is when he consumes an equal amount in both periods, i.e., c₁ = c₂. Let his common consumption level be denoted by c. Then, his total income over the two periods is $210. If he consumes an amount c in both periods, he will have a total of $210 - 1.1c in period 1 (i.e., $147 - c borrowed or saved at 10% interest) and $1.1c + $63 in period 2 (i.e., the borrowed or saved amount at 10% interest plus his income in period 2). We can solve for the value of c that maximizes his utility by considering two cases: 1. If c ≤ $63/1.1 ≈ $57.27, then he would borrow to finance his consumption in period 1 and save in period 2. This is because the future value of the savings at 10% interest is greater than the amount he would need to repay the loan (which also accrues interest at 10%). However, this would result in him not being able to consume an equal amount in both periods, which would reduce his utility. Therefore, this case is not optimal. 2. If c > $63/1.1 ≈ $57.27, then he would save in period 1 and consume the saved amount in period 2. This is the only choice that allows him to consume an equal amount in both periods, while also maximizing his utility. The amount he would save is $147 - c, which would accrue interest at a rate of 10% and become $1.1(147 - c) = $161.7 - $1.1c in period 2. He would consume this amount in addition to his period 2 income of $63, for a total consumption of $161.7 - $0.1c. To maximize his utility, he would choose the value c that makes his total consumption $2c (i.e., the total consumption in both periods when he consumes an equal amount) as close to $161.7 - $0.1c as possible. Setting $2c = $161.7 - $0.1c and solving for c, we get c ≈ $40. Therefore, Harvey would save $147 - $40 = $107 in period 1 and consume $107 + $1.1($40) = $151 in period 2. His total utility would be U($40, $40) = $40, which is the maximum possible.

Harvey Habit Has a Utility Function U(c1, C2) = Min{c1

Harvey Habit Has a Utility Function U(c₁, C₂) = Min{c₁