Macroeconomics M.Com Class Lecture University of Calcutta

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2.12 Equilibrium Income in SKM with government (Saving—Investment Approach):
An alternative approach to equilibrium is the saving--investment approach. Since equilibrium
in the SKM requires that the supply of real national output, Y, equals the quantity of national output
which people wish to buy, E, the condition for static equilibrium is:
Y = E (or, AS = AD)
Since E = AD = C + I + G and Y = C + S +T (which follow from NI accounting), the
following condition must hold:
C + S +T = C + I + G
By canceling C from both sides of the equation, we get:
S + T ≡ I + G …..….(21)
The condition implies that savings plus taxes (leakages) is just equal to investment plus
government spending (injections). This equation is true all the time at every level of GNP. Also, this
equation is true in realized or actual sense. This equation is called accounting identities.
Here, we note that realized investment (I) consists of planned (or intended) investment (Ip)
and unplanned (or, unintended) investment (Iu) i.e.
I = Ip + Iu ……..(22)
Intended or planned or ex ante investment (Ip) is the amount of investment which firms
intended or planned to invest; its magnitude is indicated by the investment function. Unintended
investment (Iu) is the change in business inventories due to a discrepancy between aggregate supply
and aggregate demand. Unintended investment (Iu) is also known as ∆inv. which means change of
inventories. ∆inv and hence Iu could be positive, negative or zero depending on whether supply is
greater, smaller or equal to demand.
By putting the Eq.22 into Eq.21, we get:
S + T = Ip + Iu + G ……..(23)
To derive equilibrium income in the SKM with government, we need to put Iu = 0 i.e. realized
investment is just equal to planned investment. Thus, the alternate equilibrium condition in the SKM
with government is:
S + T = Ip + G ……..(24)
2.13 Fallacy relating to saving-investment equality:
Confusion sometimes arises with regard to two equations:
Y = C + I
and Y = C+ S
These two equations follow from NI accounting and, hence, called accounting identities.
Equilibrium Income: Cases of Open and Closed Economies, Multiplier theory-
Keynesian Approach

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The former implies that investment is the difference between income and consumption ( I =
Y – C); the latter implies that saving is the difference between income and consumption ( S = Y –C).
Thus, in NI accounting sense, investment equals saving, by definition i.e. I ≡ S. This equation is true
all the time at every level of GNP. Hence, in effect this equation is also accounting identities.
But the above equation does not suggest that investment is always equal to saving in an
economy. To understand why we need to distinguish between intended investment and realized
investment. Intended or ex ante investment is the amount of investment which firms intended or
planned to invest; its magnitude is indicated by the investment function, I = 𝐼̅. Realized or ex post
investment is the sum of intended and unintended investment. Unintended investment is the change
in business inventories due to a discrepancy between aggregate supply and aggregate demand. If
aggregate supply exceeds aggregate demand, production exceeds sales and business inventories
increase. The increase in inventories is treated as investment in the national income accounts.
Geometrically, unintended investment is represented by the vertical distance between the 45° line
(aggregate supply) and the C + I line (aggregate demand).
Realised saving (the amount of income people actually save), on the other hand, is always
equal to intended saving (the amount of income people intend to save, as given by the saving
function).
Thus, when we say I ≡ S, we mean realized investment is equal to realized saving. This
follows from NI accounting and it is true for every level of income. But when we say I = S as an
equilibrium condition, we mean intended investment is equal to realized saving.
Summing up, realized investment equals saving (realized = intended) regardless of the level
of income. Intended investment equals saving only at the equilibrium level of income. Intended
investment is either greater than or less than saving at all other income levels. Consequently, I = S
can serve as an equilibrium condition if I represents intended investment.
2.14 Saving Function:
Since the decision on how much income to consume implies a decision on how much to
save, a saving function may be derived with the aid of the consumption function. With no government
and foreign trade sectors, income equals, by definition, consumption C plus saving, S:
Y = C + S …..(25)
Since C = C0 + c.Y , from the above equation we get saving function as:
S = -- C0 + (1--c)Y {0 < (1--c) < 1} …..(26)
where S and Y represent real saving and real income, respectively.
The parameter (1--c), referred to as the marginal propensity to save or MPS, is the slope of
the saving function. If ∆Y denotes a change in income and ∆S denotes the change in saving associated
with the change in income, (1--c), the MPS, equals ∆S/∆Y. For example, if income increases by
Rs.200 Crore and, as a consequence, saving increases by Rs.50 Crore, the MPS is 50/200= 0.25.
Since c, the MPC, is assumed to be between 0 and 1, (1—c), the MPS, lies between 0 and 1, which
implies that saving increases as income increases, but by a smaller amount.
The saving function may be plotted in the same manner as the consumption function. Since
saving is the difference between income and consumption, saving is positive (negative) when income

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is greater (less) than consumption. When Y= 0, S = --C0, which is represented by the negative
intercept. Saving is negative at income levels less than Y1 since consumption exceeds income. The
constant C0 is called autonomous consumption or ‘subsistence consumption’. It is that level of
consumption which people must have in order to subsist even if income level falls to zero and it is
exogenously given. Thus, when Y = 0, people dissave by an amount equal to C0 (or, save by an
amount equal to –C0).
If there is a change in C0, the saving function will shift so that the new function is parallel to
the old. If there is a change in c => a change in (1—c) => a change in MPS, the function will rotate
about the intercept, --C0 and will be either steeper or flatter.
The saving function would shift up parallely upward if C0 decreases. C0 may decrease
because of (i) forced saving such as to gain tax benefits and/or (ii) people becoming more thrifty i.e.
careful about spending money. In such situation, we will have a new saving function above and
parallel to the earlier one.
Figure: 6 Figure: 7
With government imposing taxes and making transfer payments saving would depend on
disposable income rather than on the income. Thus, saving function would be
S = -- C0 +(1—c).YD where YD = ( Y + TR –T);
TR = Transfer payments; T = Taxes.
2.15 Equilibrium Income in Simple Keynesian Model with Government (S—I Approach):
The economy in the SKM is said to be in equilibrium when saving plus taxes equals
intended investment and government purchases i.e.
S + T = Ip + G
Now, S = -- C0 + (1—c) YD (savings function)
YD = (Y + TR –T) (disposable income)
TR = 0 (no transfer payment)
T = t. Y 0 < t < 1 (assuming proportional tax)
I = Ip = I0 (exogenously given intended investment)
G = G0 (exogenously given government expenditure)
By substituting these in to Eq. we get:
 -- C0 + (1—c) ( Y – t.Y) + t.Y = I0 + G0
 -- C0 + (1—c) .Y – (1—c) t.Y + t.Y = I0+ G0

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 Y. [(1—c) – (1—c) t + t] = C0 + I0 + G0
 Y. [1—c – t + c.t + t] = C0 + I0 + G0
 Y. {1—c(1—t)} = C0 + I0 + G0
 YE = [ 1 / {1 – c(1-- t)}] { C0 + I0 + G0}
where YE is the equilibrium level of income—same as that derived earlier but now with TR=0.
2.16 Graphical Illustration:
The equilibrium level of income can be determined geometrically with the investment-saving
approach. The relationship between the aggregate supply-aggregate demand and investment-saving
approaches is illustrated in Figure:8.
If Ip = S, AS = AD, income is at its equilibrium level. With AS = AD, there is
no change in inventories.
If Ip < S, AS > AD, and income is at a disequilibrium level. With AS > AD,
inventories accumulate and income tends to fall.
If Ip > S, AD > AS, and income is at a disequilibrium level. With AS < AD,
inventories are depleted and income tends to rise.
Thus, when S > Ip, Y tends to fall, and when Ip > S, Y tends to rise. Under these
circumstances, Y will eventually gravitate to YE where Ip = S
Figure: 8
2.17 Paradox of Thrift (POT):
An increase in thriftiness of people may give two paradoxical results in SKM:
(i) aggregate savings may remain same (Weak Version of POT) and
(ii) aggregate saving may even fall (Strong Version of POT)
Why paradox? People becoming more thrifty i.e. careful about spending money should lead to
increased aggregate savings in the economy. But that is not the case in SKM.
(i) Weak Version of POT:
Weak version of POT holds if we assume investment as autonomous i.e. exogenously given:
I = I0
This is the crucial assumption we need for weak version of POT.
YE

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The government expenditure is as usual exogenously given:
G = G0
The savings function is
S = -- C0 + (1—c) YD =-- C0 + (1—c) (Y—T)
Taxes are lump-sum:
T = T0
Initial equilibrium income is at Y1 (Fig: 9) where
I + G = --C + (1—c) (Y—T) + T
 C0 + I + G0 = (1—c)Y –T + c.T + T
 (1—c) Y = { C0 + I + G0—c.T}
 Y1= { C0 + I0 + G0—c.T} / (1—c)
Now, people become more thrifty => C0 decreases to C1 => savings function shifts up parallely
and new equilibrium is attained at income level Y2 where
I + G = -- C1 + (1—c) (Y—T) + T
 C1 + I0+ G0 = (1—c)Y –T + c.T + T
 (1—c) Y = { C1 + I0 + G0—c.T}
 Y2 = {C1 + I0 + G0 —c.T} / (1—c)
At both the equilibrium level of income, S+T equals constant I+G. Hence, aggregate savings remains
same. Thus, although people are more thrifty now, aggregate saving remains the same. We can show
this algebraically.
At Y1, S + T = -- C0 + (1—c). [Y1 –T] + T
= -- C0 + (1—c). [{C0 + I0 + G0—c.T0} / (1—c) –T0]+ T0
= -- C0 + (1—c). [{C0 + I0 + G0—c.T—T + c.T}/ (1—c)]+T
= -- C0 + C0 + I0 + G0 —T + T
= I0 + G0
At Y2, S + T = -- C1+ (1—c). [Y2 –T] + T
= -- C ' + (1—c). [{C ' + I0 + G0—c.T} / (1—c) –T]+ T
= -- C1 + (1—c). [{C0+ I0 + G0—c.T—T + c.T}/ (1—c)]+ T
= -- C1+ C1 + I0 + G0—T + T
= I0 + G0
Thus, S + T| Y1 = S + T| Y2
(ii) Strong Version of POT:
Strong version of POT holds if we assume investment as induced i.e. investment is function
of income: I = I0 + i .Y I > 0; i > 0 ……(27)
This is the crucial assumption we need for strong version of POT.
The government expenditure is as usual exogenously given:
G = G0
The savings function is
S = -- C0 + (1—c) YD =-- C0 + (1—c) (Y—T)

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Taxes are lump-sum:
T = T0
Initial equilibrium income is at Y1 (Fig:10) where
I0 + i .Y + G0 = -- C0 + (1—c) (Y—T) + T
 C0 + I0+ G0 = (1—c)Y –T + c.T + T—i.Y
 (1—c—i) Y = { C0 + I0 + G0—c.T}
 Y1= { C0 + I0 + G0—c.T} / (1—c—i) …….(28)
Again, people becoming more thrifty => C0 decreasing to C1 => savings function shifts up parallely
and new equilibrium is attained at income level Y2 where
I0 + i .Y + G0 = -- C1 + (1—c) (Y—T) + T
 C1 + I0 + G0 = (1—c)Y –T + c.T + T-- i.Y
 (1—c—i) Y = { C1 + I0 + G0—c.T}
 Y2 = { C1 + I0 + G0 — c.T} / (1— c— i) ……(29)
Comparing Y1 and Y2 we see that Y2 < Y1. Although at both the equilibrium level of income, S + T
equals I + G0, but investment being induced here, at lower income I + G is less and so is S + T.
Hence, aggregate saving falls. Thus, although people are more thrifty now, aggregate saving in the
economy has in fact fallen. We can show this algebraically.
At Y1, S + T = -- C0 + (1—c). [Y1 –T] + T
= -- C0 + (1—c). [{C0 + I0 + G0—c.T} / (1—c—i) –T]+ T
= -- C0 + (1—c). [{C0 + I0 + G0—c.T—T + c.T+ i.T}/ (1—c—i)]+ T
= -- C0 + (1—c). [{C0 + I + G0 —T + i.T}/ (1—c—i)] + T
At Y2, S + T = -- C1 + (1—c). [Y2 –T] + T
= -- C1 + (1—c). [{C ' + I0 + G0—c.T} / (1—c—i) –T]+ T
= -- C1 + (1—c). [{C ' + I0 + G0—c.T—T + c.T+ i.T}/ (1—c—i)]+ T
= -- C1 + (1—c). [{C1 + I0 + G0 —T + i.T}/ (1—c—i)] + T
Now,
S + T |Y2 -- S + T |Y1 = -- C1 + (1—c). [{C1 + I0 + G0 —T + i.T}/ (1—c—i)] + T
+ C0 -- (1—c). [{C0 + I0 + G0 —T + i.T}/ (1—c—i)] – T
= --( C1 -- C0 ) + {(1—c)/ (1—c—i)} [C1 + I0 + G0 —T + i.T--
C0 -- I0 -- G0 + T -- i.T] + T --T
= --( C1 -- C0) + {(1—c)/ (1—c—i)} (C1— C0)
= (C1— C0) [(1—c)/ (1—c—i) – 1]
= (C1— C0) [(1—c –- 1 + c + i)/ (1—c—i)]
= (C1— C0) [ i / (1—c—i)]
Since C has decreased C1 < C0 => (C1— C0) < 0,
Again, i > 0, and (c + i) < 1 being a stability condition in the SKM ,
[ i / (1—c—i)] > 0
 (C1— C0) [ i / (1—c—i)] < 0
 [S + T |Y2 -- S + T |Y1 ] < 0
 (S + T |Y2 ) < ( S + T |Y1)

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 The aggregate saving has fallen even though people are more
thrifty now.
Figure: 9
Figure:10
2.18 Economic Explanation of Paradox of Thrift:
In a situation of widespread unemployment of resources, a reduction in consumption due to
say, an increase in thriftiness or forced savings, adversely affects the economy. With exogenously
given investment (Weak version of POT), a cutback in aggregate consumption reduces the aggregate
demand. Industries producing consumer goods reduce production. Thus, a fall in AD causes output
to fall. Aggregate saving, however, remains unchanged as increased thriftiness prevents aggregate
savings from falling in spite of a fall in Y.
In case of induced investment (Strong version of POT), Y falls not only due to a reduction in
consumption but also due to a fall in investment induced by the fall in Y. As Y decreases by a larger
amount in this case, even an increase in thriftiness can not prevent aggregate savings from falling.