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Problem Set 1- Answers
1.1
Question 1
Derive the saving-investment identity, with government and foreign sector.
According to the income-expenditure approach, the total product equals
the expenditures in consumption, investment, government expenditures and
net exports:
Y = C + I + G + NX
Let’s now compute national savings. First, consider the private savings.
Private savings consist of private disposable income minus consumption.
The private disposable income corresponds to the income available to be
used to consume or to save. It is computed by total income minus taxes(T)
plus tranfers from the government (TR), plus interest payments made by the
government (return from government bonds, INT), plus net factor payments
from abroad (NFP).
Therefore, private savings can be expressed in the following equation:
Sprivt = (Y + NF P − T + T R + INT ) − C
Next, we have to compute the government savings, which corresponds to
the government budget surplus. Basically, it consists of government revenues
minus government expenditures and payments made to the public. This is
expressed in the equation below:
Sgovt = (T − T R − INT ) − G
National savings are the sum of private savings and government savings:
S = Sprivt + Sgovt
S = (Y + NF P − T + T R + INT ) − C + (T − T R − INT ) − G
Cancelling the terms involving (T − T R − INT ), we get:
S = Y + NF P − C − G
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By the first expression, for the GDP, we can substitute Y:
S = C + I + G + NX + NF P − C − G = I + (NX + NF P )
The sum of the net factor payments from abroad and the net exports is
called current account (CA) which allows us to rewrite the expression above
as:
S = I + CA
Therefore, national savings can be used to invest or to lend to foreigners.
Using the data from Table 5.1for year 2004 (in billions of dollars), we
obtain the following values:
Gross saving (S)= 1572
Gross private saving =1755.3
Gross government saving =-183.2
Gross domestic investment (I)= 2300.6
Capital account transactions (net)\1\ (NFP)= 1.6
Net lending or net borrowing (-), NIPAs\2\ (NX) =-653.4
Balance of Current Account= -651.8
I+CA= 1648.8
Statistical discrepancy(I+CA-S) =76.8
1.2
Question 2
Information provided in the problem:
There are 2 periods: current and future. Initially, the consumer receives
an income Yc = Yf = 50, 000 in both periods. The real interest rate equals
10%. Also, she wants to save in the current period so as to be able to
make a tuition payment in the future equal to 12,600. Another important
information is that the consumer wants to smooth consumption, which is in
accordance with the life-cycle model, which means that she wants to consume
the same amount in both periods: Cc = Cf = C. Finally, she does not want
to have any debt/asset at the end of the future period.
(a) Under these conditions just described, you are asked to calculate the
amount the consumer wants to consume and to save.
Using the information above, we know that in the first period, the consumer will use her income to consume and to save, what can be expressed
by:
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Cc + Sc = Yc
Note that the consumer has only the income Yc at her disposal since she
has no wealth in the current period.
In the future period, the consumer receives an income Yf ,also receives
the interest payment of her savings in the previous periods and since she
wants to end this period without any asset, she will consume all her savings.
But she also has to make her tuition payment.As a result, we can write:
Cf + tuition = Yf + (1 + r)Sc
Now, using the information that the consumer wants to consume the same
amount in both periods we can rewrite the expressions above:
C + Sc = Yc → Sc = Yc − C
C + tuition = Yf + (1 + r)Sc
Substituting Sc in the second expression, we get:
C + tuition = Yf + (1 + r)(Yc − C)
C + tuition = Yf + (1 + r)Yc − (1 + r)C
(2 + r)C = Yf + (1 + r)Yc − tuition
This is a general expression that helps to solve the other items in the
problem.
For this first item though, remember that: Yf = Yc = Y what simplifies
the last expression:
(2 + r)C = (2 + r)Y − tuition
Isolating consumption:
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C=Y −
tuition
(2 + r)
Substituting the values given:
C = 50, 000 −
12, 600
= 44, 000
(2.1)
Since:
Sc = Yc − C = 6, 000
is the amount she will save.
(b)Now, suppose that Yc = 54, 200.
From the expression derived above:
(2 + r)C = Yf + (1 + r)Yc − tuition
So, the difference now is that the incomes are not equal in both periods.
In this case, we can isolate C and substitute the numerical values.
C=
C=
Yf + (1 + r)Yc − tuition
(2 + r)
50, 000 + (1.1)(54, 200) − 12, 600
= 46, 200
(2.1)
Sc = Yc − C = 54, 200 − 46, 200 = 8, 000
(c) Now, consider if Yf = 54, 200.
Using the expression used above:
C=
Yf + (1 + r)Yc − tuition
54, 200 + (1.1)(50, 000) − 12, 600
=
= 46, 000
(2 + r)
2.1
Sc = Yc − C = 50, 000 − 46, 000 = 4, 000
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(d) Now, consider the consumer receives some wealth in the current period. Then, her budget constraint in the current period becomes:
C + Sc = Yc + Wc
Note that this is equivalent to increasing the current income by this
amount. So, we can consider that Yc is now equal to 51,050.
C=
50, 000 + (1.1)(51, 050) − 12, 600
= 44, 550
(2.1)
Sc = Wc + Yc − C = 51, 050 − 44, 550 = 6, 500
(e) Suppose now that the tuition payment is increased to 14,700. Since
the incomes are equal (we are back to the initial situation, item a), we can
use the following expression to calculate the consumption:
C=Y −
14, 700
tuition
= 50, 000 −
= 43, 000
(2 + r)
2.1
Sc = 7, 000
(f) Now, assume that r=0.25.
Using the same expression as in the previous item, we get that:
C=Y −
12, 600
tuition
= 50, 000 −
= 50, 000 − 5, 600 = 44, 400
(2 + r)
2.25
Sc = 5, 600
1.3
Question 3
Situation described: you are moving to a new city. You have to make a
decision whether to buy a house or to rent one.
House value=ph =200,000
nominal interest rate = 0.10 per year (mortgage or saving)
mortgage= tax deductible
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saving= not tax deductible , t=30% bracket.
expected inflation= 0.05.
cost of maintaining house= 0.06(value of the house)
(a) The expected after-tax interest rate on the home mortgage can be
calculated in two different ways. First, we can think of it as being the opportunity cost of not saving. In this way, if you had decided not to buy
the house and instead save the money, you would earn the nominal interest
i=0.1, discounted by the tax: (1-t)i. Discounting the inflaton during the
period, the after-tax real interest rate = (1 − t)i − π.
Alternatively, if you decide to buy the house, you have to pay the mortgage, which corresponds to i*(value of the house). You also have to pay taxes,
at a rate of 30%. But the mortgage payments are tax deductible, which means
that you can discount from your tax payments the amount corresponding to
this payment: t(i ∗ ph ). As a results, what you are paying for your mortgage
after tax considerations is [(1 − t)i]ph . So, the nominal after-tax mortgage
rate is (1 − t)i. Discounting the inflation gives us:(1 − t)i − π.
ra−t = (0.7)(0.1) − 0.05 = 0.02 = 2%
(b) The user cost of capital is given by:
uc = (ra−t + d)ph = (0.02 + 0.06)(200, 000) = 16, 000
(c) If your only concern is to minimize the living cost of each period, so
that you are indifferent between buying or renting a house, you will need to
be paying the same amount of rent as you would be if you had bought the
house.
Then,your rent level will correspond to the sum of the mortgage payment
and the maintanance expenditures, which is what you spend in the period if
you buy the house. Note that this is exactly the value of the user cost just
derived above.
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