The Maxwell School
Syracuse University
Syracuse University
The first step in computing Ada's consumption is to calculate the present value of her income:
PVI = I0 + I1/(1+r)
PVI = 100k + 500k/1.2 = $516.7
Inserting PVI into her demand equations gives C0 and C1:
C0 = 0.4*PVI = 0.4*516.7k = $206.7k
C1 = 0.6*(1+r)*PVI = 0.6*1.2*$516.7 = $372k
Her utility will be the following, where U1 indicated that it is her utility from question 1:
U1 = ($206.7k)^0.4 * ($372k)^0.6
U1 = 294.1k
Since her consumption in period 0 is greater than her income, she borrows. The amount will be:
Borrowing = C0 - I0 = $206.7k - $100k = $106.7k
If Ada had been unable to borrow or lend, she would have had to consume at her endowment point: C0=I0 and C1=I1. In that case, her utility would have been the following, where U3 indicates that it is the hypothetical utility associated with question 3:
U3 = (100k)^0.4 * (500k)^0.6
U3 = 262.7k
U3 is lower than U1 because without the ability to borrow, she must consume less in period 0 and more in period 1 than she would like.
Her intertemporal expenditure function can be derived by inserting her demand equations into her utility function and solving for PVI:
U = (0.4*PVI)^0.4 * (0.6*(1+r)*PVI)^0.6
U = 0.4^0.4 * (0.6*(1+r))^0.6 * PVI
PVI = U /( 0.4^0.4 * (0.6*(1+r))^0.6 )
The PVI needed for Ada to have had utility U3 is straightforward to calculate by evaluating the expenditure function with U=U3:
PVI3 = U3 /( 0.4^0.4 * (0.6*(1+r))^0.6 )
PVI3 = 262.7k /( 0.4^0.4 * (0.6*1.2)^0.6 )
PVI3 = $461.5k
The difference between PVI3 and Ada's actual PVI of $516.7k is $55.2k. That says that taking away her ability to borrow would lower her utility by the same amount as reducing her PVI by $55.2k (for example, by taking $55.2k away from her in period 0). That is, either of the following would give her U3: