Question

In: Economics

Suppose that Meghan earns $2,000 this month, and $2,200 next month with a utility function over...

Suppose that Meghan earns $2,000 this month, and $2,200 next month
with a utility function over consumption in these two periods given
by U(C1, C2) = C1C2, where MUC1 = C2 and MUC2 = C1. Suppose that
the deposit interest rate is rL = 1%, while the credit interest
rate is rB = 2% + x%, where x is 4.
a) Draw a graph with Meghan’s budget constraint for consumption in this month
vs. next month.

b) Will Meghan borrow or save, or neither?
c) How will your answer change if the interest rates rL and rB both increase by 5
percentage points?

Solutions

Expert Solution

Meghan earns $2,000 this month, and $2,200 next month. Hence, Y1=$2000 and Y2=$2200.

(a) Let's say the consumption in this month is C1 and consumption in next month is C2.

Hence Meghan's consumption in this month is C1 and let's say he saves (Y1-C1) at interesr rate r.

Now, his consumption in next month is

C2=(Y1-C1)(1+r)+Y2

or, C1+C2/(1+r) = Y1+Y2/(1+r)

​​​​​​Now, we can not say right now that the person will borrow or save. The budget constraint will be the same for both of the cases, but we will draw the diagram for rate of interest r in general. The budget constraint of this month versus next month is drawn below.

(b) Now, from the budget line, we can think that the price of C1 is P1=1 and the price of C2 is P2=1/(1+r).

Hence, price ratio P1/P2=(1+r)

Also, the Marginal Rate of Substitution is

MRS = MUC1/MUC2 = C2/C1.

Hence, at optimum consumption, the MRS equals the price ratio.

Hence,

​​​​​​ MRS=P1/P2

or, C2/C1 = (1+r)

or, C2/(1+r) = C1.........(1)

Putting the value of C2/(1+r) in the budget constraint, we get

C1+C2/(1+r)=Y1+Y2/(1+r)

or, C1+C1=Y1+Y2/(1+r)

or, C1* = Y1/2 + Y2/2(1+r).......(2)

and, C2* = C1*(1+r) = Y1(1+r)/2+Y2/2.........(3)

Now, Y1=$2000 and Y2=$2200.

Hence, we can find a range of r for which he will save in this month.

Hence, if he saves in this month then it must ne that,

C1<Y1

or, Y1/2+Y2/2(1+r)<Y1

or, 2000/2+2200/2(1+r)<2000

or, 1100/(1+r)<1000

or, (1+r)>1.1

or, r>0.1

Hence, for any deposit interest rate rL>10%, he will save.

If, we do the same assuming that he borrows, then C1>Y1 gives r<0.1. Means he borrows if credit interest rate is less than 10% or rB<10%.

But he is getting deposit interest rate of 1%. Hence he will not save. He will borrow in this month as rB=6% < 10%.

Hence, r=rB=6% or 0.06

Hence,

C1*=2000/2+2200/2(1+0.06)=$2037.74

and, C2*=C1*(1+0.06)=$2037.74×1.06

or, C2*=$2160.

Hence, Meghan will borrow $(2037.74-2000)=$37.74 in this month.

(c) If, rL increases by 5% then it becomes 6%. Still rL<10%. He could save if it was more than 10%. Hence he will not save.

Also, rB increases by 5% and it becomes 11%. Hence, rB>10% here. He could borrow if it was less than 10%. But here he will not borrow any more.

Hence, he will neither borrow nor save. He will consume his incomes of this month and next month i.e. C1**=$2000 and C2**=$2200.

Hope the solution is clear to you my friend.


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