In: Economics
8. Kenny’s intertemporal utility function is U(c1, c2) = 10c1 + 8c1, with time periods 1 and 2 representing consumption today and one year from today, respectively. He earns $100 today and $122 one year from today, and his annual rate of interest for saving and borrowing is 22%. There is no inflation. What values of consumption in each time period are optimal?
Answer 8
First Lets form inter-temporal Budget constraint:
In first period either he will save, borrow or do nothing and let s be saving and if s is negative this means that he will borrow
period 1 : c1 + s = 100 => s = (100 - c1)
period 2 : c2 = s + sr + 122 => c2 = 122 + (100 - c1)(1 + 0.22). Here , r = interest rate = 0.22
=> c2 +1.22c1 = 122 + 122 = 244
=> c2 +1.22c1 = 244
Now we have to Maximize : U = 10c1 + 8c2,
s.t. c2 +1.22c1 = 244
We can see from above that c1 and c2 are perfect substitutes. Hence He he will spend his entire income on either c1 or c2 depending on where he is getting higher utility.
He values c1, 10/8 times that he values c2. Hence If cost of c1 from budget constraint is less that 10/8 times the cost of c2 then he will consume only c1, Hence If cost of c1 from budget constraint is greater than 10/8 times the cost of c2 then he will consume only c2.
Here cost of c1 = 1.22 and cost of c2 = 1. Hence, per unit cost of c1 is 1.25 times cost of c2. Hence using above details he will consume only c2.
Hence c1 = 0
=> c2 +1.22*0 = 244 => c2 = 244
Optimal Value of consumption in period 1 = 0 and Optimal Value of consumption in period 2 = 244