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EC201 Intermediate Macroeconomics
EC201 Intermediate Macroeconomics
Problem set 2 Solution
1) Consider an economy with only two firms, Firm A that produces apples, and Firm J
that produces apple juice. The transactions carried by the two firms in the economy in
2009 are written below:
Firm A
Wages paid to employees
£20000
Profits
£15000
Rent for Land
£10000
Revenues from selling apples to the
£15000
public
Revenues from selling apples to Firm J
£30000
Wages paid to employees
Sales revenues
Apples purchased from Firm A
Juices boxes purchased from abroad
Firm J
£10000
£50000
£30000
£10000
a) Calculate the GDP in this economy in 2009 using the production, the
expenditure and the income method. Show that all methods give the same
answer.
b) Provide some intuition to the fact that the Real GDP may not be a good
measure of overall welfare of a given economy.
Solution
a)
Product approach: in this case the GDP is the defined as the sum of the value of all
goods and services produced minus the value of all intermediate goods used in
production
From the data we have the value of production of Firm A is £15000 + £30000 (its
revenues). That represents the value of the final goods produced and sold by Firm A.
Firm A does not use any intermediate good.
Firm J has a value given by: £50000. Firm J uses intermediate goods for a value of
£30000+£10000=£40000.
Therefore, according to the product approach the GDP is given by: £45000 + £50000
- £40000 = £55000
Expenditure approach: the GDP is defined as the total spending on all final goods
and services produced in the economy in a given period of time.
We know that
Total Expenditure = C + I + G + NX
In our economy, I = 0 (notice that while we have Land in the table we do not have the
purchased value of it but instead only the rental income during a given period of
time), G = 0 and NX = -£10000. We have that total consumption is:
£15000 (what the public consumes from buying the goods from Firm A) + £50000
(consumption of juices) - £10000 (imports) = £55000
Income approach: the GDP is defined as the sum of all income received by economic
agents contributing to production
Here we have: £20000 of wages paid by firm A, £15000 of profits distributed by Firm
A, £10000 of rent paid to the owner of land by Firm A.
Then we have £10000 of wages paid by Firm J.
Summing up all those values we have: £20000 + £15000 + £10000 + £10000 =
£55000.
b)
We know that the GDP measures equivalently the total output produced, the total
income and the total expenditure. Are those things equivalent to a measure of
welfare? When we talk about welfare we think about something like the well-being of
an individual or of a group of individuals.
We need to notice that the GDP measures the value that arises mainly from market
transactions (some elements of the public expenditure that enter in the GDP may be
valued at the cost since a market may not exist, for example, national defense
expenditure). Then we can ask if there are elements that may affect the well being of a
society that do not arise from market transactions. The answer is yes.
For example, the GDP does not measure leisure since we do not have a market for
leisure; nevertheless, leisure tends to provide utility to individuals. Moreover, the
GDP also does not take into account home produced goods and the quality of these
goods. Family care by a mother is normally considered superior to hiring a stranger to
care for your child. Although this service is considered superior for most people they
are not included in GDP.
There are also market transactions that affect the well-being of societies but are not
recorded in the DGP calculations. For example, all the transactions related to the
underground economy (black markets and illegal markets). People can obtain goods
and services (and this affects their well-being) from black markets and in many
countries the underground economy can be as large as the legal economy.
Finally GDP does not take into account income distribution. If a country has a very
high GDP per capita, but all the income is taken by few individuals, the majority of
the population is likely to have a very low level of welfare. For example according to
the IMF, in 2009, the GDP per capita in Qatar was $78260 while the GDP per capita
in UK in the same year was $34388. This would imply that the welfare of an average
citizen in Qatar is more than twice the welfare of an average citizen in UK.
Obviously, a precise measure of welfare is quite difficult to be calculated, while
calculating the GDP is normally quite simple. Therefore, the GDP can be considered
an approximation to the level of welfare (like the consumer surplus is an
approximation of welfare for the single consumer in a market).
2) (Income distribution) Consider an economy where total output is produced
according to the following aggregate production function:
Y = K α Lβ
Where Y denotes output, K is the level of capital and L is the level of labour. The
terms α and β are positive constants. Assume that the input markets (for capital and
labour) and the output market are competitive. Show that when α + β > 1 then the
Product Exhaustion Theorem does not hold. Provide an intuition for your finding.
Solution
The Product Exhaustion Theorem tells that if there are constant returns to scale and all
markets are competitive, then the input factors are paid according to their marginal
productivity and those payments exhaust the value of output.
Here we have the case of a Cobb-Douglas production function that under the
assumption that α + β > 1 displays Increasing Returns to Scale. Then we already
know that the Product Exhaustion Theorem will not work.
To prove it, we need to use the Euler’s Theorem for Homogenous equation.
First we define what a homogeneous function is. A function f ( x1 , x 2 ,..., x n ) (where
we consider a function of n variables just to be general enough, but this will be true
also for a function of a single variable) is homogeneous of degree k if it’s true that:
f (rx1 , rx 2 ,..., rx n ) = r k f ( x1 , x 2 ,..., x n )
Where r>0 is a number (positive) and k is a constant.
Once we know what a homogeneous function is we can apply the Euler’s Theorem.
The Euler’s theorem says that if a function f ( x1 , x 2 ,..., x n ) is homogenous of degree
k, then we can write:
∂f
∂f
∂f
kf ( x1 , x 2 ,... , x n ) = x1
+ x2
+ ...+ x n
∂ x1
∂x2
∂xn
Meaning: the original function (scaled by k) can be written in terms of its partial
derivatives.
We already know that the Cobb-Douglas production function is homogenous of
degree α + β . To prove that you just need to check that:
F ( rK , rL) = (rK ) ( rL) β = r α K α r β Lβ = ( r )α + β K α Lβ
α
= r (α + β ) F ( K , L )
Using the previous notation we have here k = (α + β )
Using the Euler’s theorem we have then:
∂F ( K , L)
∂F ( K , L)
(α + β ) F ( K , L) =
K+
L
∂K
∂L
That expression is equivalent to:
∂F ( K , L)
∂F ( K , L)
(α + β )Y =
K+
L
∂K
∂L
∂F ( K , L)
∂F ( K , L)
Where
is the marginal productivity of capital while
is the
∂K
∂L
marginal productivity of labour.
∂F ( K , L) r
If all markets are competitive then it must be true that
= , and
∂K
P
∂F ( K , L) w
= , where r is the rental price of capital and w is the price of labour (the
∂L
P
wage).
To see this, suppose that there is a single representative firm in the economy. The
profit of this representative firm is:
π = PY − wL − rK ⇒ PF ( K , L) − wL − rK
If markets are competitive then the firm takes the prices (P, w and r) as given.
The firm maximises profits by choosing the level of inputs (L and K) to be used to
produce the output Y.
By maximising the profit function with respect to K and L and by setting those
derivatives equal to zero we have:
∂π
∂F ( K , L)
=P
−r =0
∂K
∂K
∂π
∂F ( K , L)
=P
−w=0
∂L
∂L
∂F ( K , L) r ∂F ( K , L) w
= ,
= , meaning that if
∂K
P
∂L
P
markets are competitive a firm chooses the level of capital and the level of labour
such that the marginal productivity of each input is equalised to the real price of that
input.
Multiply by P each side of the following equation:
Those two equations imply that:
(α + β )Y =
∂F ( K , L)
∂F ( K , L)
K+
L
∂K
∂L
So we have:
∂F ( K , L)
∂F ( K , L)
K+P
L
1)
∂L
∂K
∂F ( K , L) r
∂F ( K , L)
∂F ( K , L)
Using the fact that
= ⇒r=P
and w = P
into
P
∂K
∂K
∂L
equation 1) we have:
(α + β ) PY = rK + wL
PY is the value of total product, or the nominal GDP. If α + β > 1 , then we have that
(α + β ) PY > PY , meaning that:
PY < rK + wL
When there are increasing returns to scale and each input is paid according to its
marginal productivity, then the payments to the inputs (rK is the payment received by
the owner of capital, while wL is the payment received by the workers) more than
exhaust the total value of output.
This means that IF inputs are paid at their marginal productivity value (P multiplied
the marginal productivity) then the value generated is MORE than the value that will
be distributed (PY). How can it be? A possible explanation is that with increasing
returns to scale there is some sort of positive externality that will create extra value
but nobody (workers and capitalists) can appropriate this extra value.
This also implies that since what is distributed is only PY, at least one input must be
paid LESS than its marginal product. This implies that the respective market
CANNOT be competitive, that is the contrary of what we have assumed.
This result also explains why in economic models the use of constant returns to scale
is normally preferred. This is because by assuming constant returns to scale for the
aggregate production function the problem of income distribution is not an issue
anymore.
(α + β ) PY = P
3) (Twin deficit) From the national accounting identity derive the relationship between
Aggregate Saving, Investment and the Current Account. Using that relationship
explain why a fiscal deficit can be associated with a current account deficit.
Solution
The national accounting identity is:
Y = C + I + G + NX
To derive the link between aggregate saving, investment and current account we need
to define what is aggregate saving. Let’s call it S
Define S = S Pr + S Pub , where S Pr = Y + NFP − T − C is private saving. In particular
Y + NFP is the Gross National Product (i.e. GDP plus Net Factor Payments from
abroad), T are taxes and C is aggregate consumption. In practice aggregate private
saving is just aggregate disposable income (i.e. income after taxes) minus aggregate
consumption. S Pub = T − G is public saving which is just tax revenues minus
government expenditure.
Using the definitions of S Pr and S Pub into the definition of S we get:
S = Y + NFP − T − C + T − G = Y + NFP − C − G
Using the national accounting identity to substitute Y into that equation:
S = C + I + G + NX + NFP − C − G = I + NX + NFP
NX + NFP is the current account and therefore we can write:
S = I + CA
Aggregate domestic saving equals domestic investment plus the current account.
To see how a fiscal deficit can be related to a current account deficit we can do the
following. Write the previous equation as:
S Pr + S Pub = I + CA
Now assume that S Pub = 0 which means that the government is running a balanced
budget ( T = G ) . Moreover assume that S Pr is equal to I . Then the above equation
implies that CA = 0 so the current account is balanced. Now assume that the
government starts running a fiscal deficit, that is S Pub < 0 . If S Pr is still equal to I ,
i.e. they have not changed because of the fiscal deficit, then for the equation above to
hold it must be that CA < 0 . Therefore it is possible that a fiscal deficit can be
associated (in this case it is the cause) of a current account deficit. This is known as
the Twin Deficit Hypothesis. The typical case is the US that in the last 30 years have
experienced high fiscal deficits and high current account deficit. One possible
explanation is that the current account deficits were the consequences of fiscal
deficits.
However for this hypothesis to work (i.e. that fiscal deficits are the cause of current
account deficits) we have to assume that private saving is equal to domestic
investment and they remain constant after the fiscal deficit has been created. Those
assumptions may not hold true in reality (and in general they are not true).