Samantha Smoothie's utility function is U(c₁, c₂) = c₁c₂, where c₁ is her consumption in period 1 and c₂ is her consumption in period 2. She earns $200 in period 1 and $220 in period 2. Samantha can borrow and lend at an interest rate of 10% and there is no inflation. The number of dollars that Samantha spends in the second period must be A) more than 200 but less than 220. B) exactly 220. C) more than 220. D) exactly 200. E) less than 200.

Question

Quizplus · Accepted Answer

To maximize her utility, Samantha will want to consume equal amounts in both periods. Let x be the amount of dollars she spends in each period. Then her utility function can be written as U(x, x) = x*x. 
In period 1, she earns $200 and can borrow at an interest rate of 10%, so she can consume 1.1*200 = $220. Let C1 be her consumption in period 1. 
In period 2, she earns $220 and has to repay the loan plus interest, which is 1.1C1. So her budget constraint in period 2 is: 
C2 = 220 - 1.1C1 
To maximize her utility, she wants to set C1 = C2 = x. So: 
x = 220 - 1.1x 
2.1x = 220 
x = 104.76 
Therefore, Samantha will spend $104.76 in each period, which is more than 220. So the answer is C.

Samantha Smoothie's Utility Function Is U(c1, C2) = C1c2, Where

Samantha Smoothie's Utility Function Is U(c₁, C₂) = C₁c₂, Where