Question

Jimmy was given $30, 000 on his 16th birthday and will be given $20, 000 when he turns 17. He has a credit card that charges 100% interest. He can use the credit card this year, but must pay the balance in full (plus interest) next year. He cannot use the credit card next year. Aaron is also able to save this year (for next year) in a savings account that pays 50% interest.

a) Plot and label Aaron’s endowment (what he starts off with) in the space of consumption this year (C16) and consumption next year (C17). Put C16 on the horizontal axis.

(b) On a separate graph from (a), plot and label the point indicating the maximum amount of consumption that Aaron can have this year. Plot and label the maximum amount of consumption that he can have next year. Use these points, along with the endowment from (a), to draw the budget line and label the slope(s).

(c) Suppose that preferences are given by the following utility function: U(C16, C17) = C16 + C17 What is his optimal consumption bundle? Does Aaron use his credit card? If so, how much does he charge? (d) (7.5) Suppose that preferences are given by the following utility function: U(C16, C17) = 3C16 + C17 What is his optimal consumption bundle? Does Aaron use his credit card? If so, how much does he charge?

Answer 1

Consider the given problem here in 16^th year, “Aaron” has “$30,000” and in the 17^th year he will have “$20,000”, => “Y16=$30,000” and “Y17=$20,000”. The rate of interest to borrow is “100%” and the saving interest rate is “50%”.

a).

So, here the endowment point is given by. “(C16, C17)=(Y16, Y17) = ($30,000, $20,000).

Consider the following fig.

b).

The current income is “Y16=$30,000”, => the current maximum consumption is “Y16 + Y17/(1+1)”.

=> C16 = 30,000 + 20,000/2 = 40,000. So, the current maximum consumption is “40,000”.

The maximum amount of consumption for the next year is given by, “Y16*(1 + 0.5) + Y17”.

=> C17 = 30,000*(1+5) + 20,000 = 45,000 + 20,000 = $65,000.

Consider the following fig of the budget line.

So, in the above fig. point "A" show the maximum future consumption and "B" shows the maximum present consumption. So, the budget line have to slope, from “AW” it’s slope is “(-1.5)” and the slope of “WB” is “(-2)”.

c)

Suppose the preference of the consumer is given by, “U=C16+C17”. Now there are 3 possibilities either the consumer can consumer today and nothing in the future, => “C16 = Y16 + Y17/2 and C17 = 0, => the corresponding utility is, “30,000+20,000/2 = 40,000”, or can consume totally is future and nothing in the correct year, => “C16 =0 and C17 = Y17+Y16*1.5, => the corresponding utility is, “30,000*1.5 + 20,000 = 65,000, or can consume at the point “W”, => C16=Y16 and C17=Y17, => the corresponding utility is “30,000 + 20,000 = 50,000”.

=> If we compare all these possibility we will get “65,000” is maximum, => the consumer will prefer to consume only in the future and not in the current year. “Aaron” will not use his credit card rather he will save totally of his current income.

d).

Now, let’s assume that his preference is given by, “U = 3*C16 + C17”.

So, given this preference the utility derive from the 1^st possibility is “U=3*45,000 + 0 = 135,000”, under 2^nd possibility is “U=3*0 + 65,000” = 65,000 and under 3^rd possibility is “U=3*30,000 + 20,000 = 110,000.

So, we can see that the utility is maximum under the 1^st possibility, => the consumer will consumer totally at the present and the future consumption is “zero”. So, here he will use his credit card to barrow from future.