Multiplier Theory-Keynesian Approach-Lecture-3

SM/CU/MCOM/FIRST YEAR/MEBE/MOD-I/2017-18 MOOCS:subirmaitra.wixsite.com/moocs 1
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Equilibrium Income: Cases of Open and Closed Economies, Multiplier theory-Keynesian Approach
Comparative Static Results in the Simple Keynesian Model:
2.19 Multiplier in the SKM:
An important comparative static result of the SKM is that a change in autonomous
expenditure causes larger change in equilibrium income. This is commonly known as ‘multiplier
effect’. The change in equilibrium Y due to one unit change in autonomous (or exogenous) aggregate
expenditure is known as multipliers. The multiplier is defined as dY/d𝐴̅ which represents the change
in equilibrium Y due to one unit change in autonomous aggregate expenditure where 𝐴̅ is
autonomous aggregate expenditure.
Autonomous Expenditure Multiplier:
Autonomous expenditure multiplier shows how an unit increase in autonomous expenditure
cause an increase in equilibrium income. In the previous lecture, we have derived:
YE = [ 1 / {1 – c(1-- t)}] { C0 + I0 + G0}= [ 1 / {1 – c(1-- t)}] 𝐴̅
where 𝐴̅ = { C0 + I0 + G0}
It can be obtained by differentiating Y w.r.t 𝐴̅ i.e.
∂Y/∂𝐴̅ = [ 1 / {1 – c(1-- t)}] …….(30)
Since 0< c < 1 and 0 < t < 1, ∂Y/∂𝐴̅ = [ 1 / {1 – c(1-- t)}] >1
2.20 Economic interpretation:
∂Y/∂𝐴̅ > 1
 ∂Y > ∂𝐴̅
Thus, a change in 𝐴̅ causes a larger change in Y. Actually, the change in Y
∂Y = [ 1 / {1 – c(1-- t)}]. ∂𝐴̅
Thus ∂Y depends on both the value of the multiplier and ∂A. The value of the multiplier depends
on c and t. As c(=MPC) increases, the value of the multiplier increases and vice versa. As t (=tax
rate) increases, the value of the multiplier decreases and vice versa.
2.21 Graphical Representation:
Initial AD = C + I +G
New AD after an ↑ in𝐴̅: AD′ parallel to AD.
Change in AD = ∂𝐴̅ = E2F
Change in AS = ∂Y = E1H
∂Y = E1H = E2H = E2F (E2H / E2F)
Therefore, ∂Y = E2F {1 / (E2F / E2H)}

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= E2F {1 / (E2H - FH / E2H)}
= E2F [1 / {1 – (FH / E2H)}]
Since, E2F = ∂𝐴̅, FH / E2H = c,
∂Y = ∂𝐴̅ { 1 / (1- c)}
 ∂Y/ ∂𝐴̅ ={1/(1- c)} = Autonomous Expenditure Multiplier
2.22 Lump-sum Tax Multiplier:
Lump-sum tax multiplier shows the effect of a change in lump-sum tax on equilibrium income in
the SKM. To derive this, we begin with the equilibrium condition:
Y = C + I + G
Since C = C + c (Y + TR – T)
TR = TR0 (exogenously given transfer payment)
T = T0 (lump-sum tax)
I = I0 (exogenously given investment)
G = G0 (exogenously given government expenditure)
By substituting these in equilibrium condition, we get:
Y = C +c(Y + TR0 – T0) + I0 + G0
 Y –cY = C + c.TR0 – c.T0 + I0 + G0
 YE = {1/ (1-c)}(C +c.TR0 – c.T0 + I0 + G0)
 YE is the equilibrium income in the SKM with lump-sum taxes.
To derive lump-sum tax multiplier, differentiate YE partially with respect to T:
∂YE / ∂T0 = ( - c / 1- c) < 0 ………(31)
Hence, an increase in lump-sum tax leads to a reduction in YE.
2.23 Graphical Representation:
Initial AD = C + I +G
New AD after an ↑ in T: AD′ parallel to AD.
Change in AD = - c. ∂T = E2F
Change in AS = ∂Y = E1H
∂Y = E1H = E2H = E2F (E2H / E2F)
Therefore, ∂Y = E2F {1 / (E2F / E2H)}
= E2F {1 / (E2H - FH / E2H)}
= E2F [1 / {1 – (FH / E2H)}]
Since, E2F = - c. ∂T, FH / E2H = FH / E1H= c,
∂Y = - c. ∂T{ 1 / (1- c)}
 ∂Y/ ∂T ={ - c / (1- c)} = Lump-sum Tax Multiplier
2.24 Tax Rate Multiplier:

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Tax rate multiplier measures the effect of a change in tax rate on
equilibrium income in the SKM. To derive this, we begin with equilibrium condition:
Y = C + I + G
Since C = C + c (Y + TR – T)
TR = TR0 (exogenously given transfer payment)
T = t.Y, 0 < t < 1 (proportional taxes; t = tax rate)
I = I0 (exogenously given investment)
G = G0 (exogenously given government expenditure)
By substituting these in equilibrium condition, we get:
Y = C +c(Y + TR0 – t.Y) + I0 + G0
 Y – c(Y-- t.Y) = C + c.TR0 + I0 + G0
 Y[1 – c(1-- t)] = C + c.TR0 + I0 + G0
 YE = (C +c.TR0 + I0 + G0) / [1-- c(1--t)]
 YE is the equilibrium income in the SKM with proportional taxes.
To derive tax rate multiplier, we take total derivative of above Equation.
dY = dC +c(dY + dTR0 – t dY --Ydt) + dI0 + dG0
Letting dC = dTR0 = dI0 = dG0 =0, we get
∂Y = c(∂Y – t ∂Y --Y∂t)
 ∂Y -- c(∂Y – t ∂Y) = -- c.Y∂t
 [1—c(1—t)] ∂Y = -- c.Y∂t
 ∂Y / ∂t = (- c.Y) / [1-- c(1—t)] < 0 …….(32)
Hence, an increase in tax rate leads to a reduction in Y.
2.25 Transfer Payment Multiplier:
Transfer payment multiplier shows how an unit change in TR causes a change in Y. To derive this,
we set up a SKM with proportional taxes as shown above. The equilibrium income in such a SKM is
given above in Eq.
 YE = (C +c.TR0 + I0 + G0) / [1-- c(1--t)]
To derive transfer payment multiplier, we take partial differentiation of Y w.r.t TR:
∂Y / ∂TR = c / [1-- c(1—t)] = Transfer Payment Multiplier..(33)
∂Y/∂TR > 0 => an ↑ in transfer payment causes an ↑ in Y.
2.26 Subsistence Consumption Multiplier:
Subsistence consumption multiplier shows how an unit change in C causes a change in Y. To derive
this, we set up a SKM with proportional taxes as shown above. The equilibrium income in such a
SKM is given above as: YE = (C0 + I0 + G0) / [1-- c(1--t)] ……..….(34)
To derive subsistence consumption multiplier, we take partial differentiation of Y w.r.t C:
∂Y / ∂C0 = 1 / [1-- c(1—t)] = Subsistence Consumption Multiplier
∂Y/∂C0 > 0 => an ↑ in subsistence consumption causes an ↑ in Y.
2.27 Investment Multiplier:
Investment multiplier shows how an unit change in autonomous investment causes a change in Y.
To derive this, we set up a SKM with proportional taxes as shown above. The equilibrium income
in such a SKM is given above in Eq. 34.
 YE = (C0 + I0 + G0) / [1-- c(1--t)]
To derive Investment multiplier, we take partial differentiation of Y w.r.t I:
∂Y / ∂I0 = 1 / [1-- c(1—t)] = Investment Multiplier ……..(35)
∂Y/∂I0 > 0 => an ↑ in Investment causes an ↑ in Y.

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2.28 Government Expenditure Multiplier:
Government Expenditure multiplier shows how an unit change in autonomous government
expenditure causes a change in Y. To derive this, we set up a SKM with proportional taxes as
shown above. The equilibrium income in such a SKM is given above in Eq.
 YE = (C0 + I0 + G0) / [1-- c(1--t)]
To derive Government Expenditure multiplier, we take partial differentiation of Y w.r.t G:
∂Y/∂G0 = 1 / [1-- c(1—t)] =Government Expenditure Multiplier ....(36)
∂Y/∂G0 > 0 => an ↑ in Government Expenditure causes an ↑ in Y.
2.29 Comparison between Tax Multiplier and Government Expenditure Multiplier:
An increase in G or a reduction in T (or, ’t’), both the policies are ‘expansionary’ in the sense
that both help raise Y through multiplier process. Yet there are considerable differences. First, the
effect of an increase in G is more certain as it directly adds to the AD. The impact of a cut in T is less
certain as it raises Yd first. One cannot be sure whether an increase in Yd would increase C or S. In
some situations a cut in T results in more savings. This considerably affects the effectiveness of tax
multiplier.
Secondly, in absolute terms, tax multiplier is less than government expenditure multiplier.
To show this we write tax multiplier as
∂Y/∂T = ( - c / 1- c)
 |∂Y/∂T | = c / 1-- c
The government expenditure multiplier (with lump-sum tax) is
|∂Y/∂G| = 1 / (1—c)
The denominator of both the multiplier is same (1—c), but the numerator of the former is c (< 1),
while the latter is 1.Therefore, we can write
|∂Y/∂T | < |∂Y/∂G |
Thus, decreasing T by Rs.1 leads to a smaller increase in Y than increasing G by Rs.1. This may seem
puzzling, but here's the explanation. If T decreases by Rs.1, this is a decrease in withdrawals from
the circular flow. The initial effect is that disposable income (Y - T) increases by Rs.1. Households
now want to increase consumption by Rs.0.75 if MPC is 0.75. Firms produce an additional Rs.0.75
worth of goods, so output increases by Rs.0.75, and hence so will Y
Tracing through the circular flow, the increase in Y due to decreasing T by Rs.1 will be:
Rs.0.75 + (0.75)(Rs.0.75) + (0.75)(0.75)(Rs.75) + ... = Rs.3.
As in the case of a Rs.1 increase in G, C has increased by Rs.3, but there is no increase in G
when T is decreased. The decrease in taxes increases aggregate demand only through its effects on
consumption.
2.30 Super Multiplier when investment is partly induced:
If investment is partly induced, I = I (Y) = I0 + i Y where i = marginal propensity to invest > 0
The equilibrium condition: Y = C0+c.(Y—T) +I0 + i. Y + G0
Taking total derivatives of the above equation:
dY = dC0+c.(dY—dT) + dI0 + i.dY + dG0
 dY –c dY –i.dY = dC0 – c dT + dI0+dG0
 dY(1—c -- i) = dC0 – c dT + dI0+dG0
By putting dC0 = d I0 = dT = 0, we get the multiplier as
∂Y/ ∂G0 = 1 / (1—c -- i) > 0 if c + i < 1
∂Y/ ∂G0 = 1 / (1—c -- i) < 0 if c + i > 1
2.31 Comparison between usual Multiplier and Super Multiplier:
When investment is autonomous,
∂Y/ ∂G0 = 1 / (1—c) = Government Expenditure Multiplier
When investment is partly induced,

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∂Y/ ∂G0 = 1 / (1—c--i) = Government Expenditure Super Multiplier
Since i > 0, ∂Y/∂G0 = 1 / (1—c--i) > ∂Y/ ∂G0 = 1 / (1—c)
 super multiplier is stronger than usual multiplier.
The economic reason behind this is as follows:
If investment is partly induced, an increase in G would lead to increase in AD leading to an increase
in Y. This increase in Y now causes both C and I to rise, which increases AD by larger amount, giving
a stronger multiplier effect to the economy.
2.32 The Importance of the Multiplier:
The idea of the fiscal multiplier was introduced by John Maynard Keynes in the 1930's.
During this time, the United States, as well as most of Europe, was experiencing a severe economic
downturn known as the Great Depression. In the United States, the unemployment rate peaked at
over 25% while extreme poverty spread throughout the population. Given these conditions, CY,
people's marginal propensity to consume, was probably very large, which implied that the multiplier
was very large. You can see the enormous appeal that this proposition had: by spending one dollar,
the government could increase aggregate demand by factors many times more than one dollar.
However, economists today do not believe that fiscal multipliers are very large. In an
economy that is far below its potential level of output, which was the case during the Depression,
expansions in aggregate demand do lead to large increases in output, with little or no effect on
prices and inflation. That is, in Depression-like times, the SRAS curve is close to horizontal, which is
the implicit assumption behind the multiplier analysis we just discussed. However, if the economy
operates very near the full-employment level, the fiscal multiplier is quite small, even in the short
run. Increases in government spending, for example, will be partially "crowded out" by decreases in
other components of aggregate demand, especially investment.
2.33 Features of Multiplier:
(i) A higher MPC (lower MPS) raises the value of the multiplier. If MPC is high, the rise in
demand at each round of the process will be high and the ultimate rise in output will also
be high. Saving is a leakage from demand and therefore, higher MPS depresses the
multiplier. Consumption is virtue when the economy is in recession.
(ii) The multiplier will be in operation if any autonomous component of AD, and not just
autonomous investment, changes.
(iii) The multiplier cuts both ways. If instead of rising, autonomous demand falls, income will
fall to a multiplied extent. Consumption spending is a relatively stable component of AD.
Planned private investment is much more volatile and unpredictable, chiefly because it
is swayed by expectations of profit in future periods.
(iv) The operation of the multiplier is crucially dependent on the assumption that unutilized
resources are available in abundance, so that output can smoothly increase in response
to rise in demand without any upward push on the price level. If, however, some crucial
input is in limited supply then the process of income expansion will come to a halt as
soon as that supply is exhausted. Further increase in demand will lead only to a rise in
price. Unable to raise real income and employment, the multiplier process will work in
nominal terms. This will also be the outcome, if, in the initial situation the economy was
at or close to its potential output YF. A rise in AD in this case will cause prices to rise.
(v) If MPC > 1, then income will rise (fall) without limit following a rise (fall) in autonomous
demand and the expression (1/ 1—c ) cannot be used as the value of the multiplier. The
value (1/1—c ) is valid only for 0 < c < 1.
2.34 Balanced Budget Theorem:
The balanced budget theorem states that, in the context of a closed economy (with
Government ) having a considerable amount of unutilised or underutilised recourses, a balanced

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(i.e. equal) increase in Government expenditure and taxes -- assuming no shift in the functional
relationship of consumption to disposable income and no change in private investment will result
in an exactly equivalent increase in net national product i.e.
if ∂G = ∂T , then ∂Y = ∂G = ∂T where ∂G, ∂T and ∂Y represents changes in
government expenditure, taxes and national income respectively.
To show the result mathematically, we use a simple one commodity income expenditure
model of a closed economy with government that collects only direct taxes. The equilibrium
condition of this economy is given by the equality between aggregate supply and aggregate demand,
i.e.
Y = C + I + G, where C, I & G have their usual meaning …. (37)
Consumption depends on disposable income which equals national income plus transfer earnings
(TR) minus direct taxes (T.) i.e.
C = C (Y + TR – T) , 0 < C´ < 1 ...... (38)
Assuming a lump sum tax for simplicity and no transfer earnings give:
T = T0, T0 >0, a constant ..…. (39)
TR = 0 …… (40)
So disposable income now becomes YD=(Y – T0)
Assuming a linear consumption function C = C0 + c (Y – T0) ….... (41)
where C is the real autonomous consumption expenditure and 0 < c < 1 is the marginal propensity
to consume.
Finally, investment expenditure and government expenditure are, for the time being, taken as
constant i.e.
I = I0 , I0 > 0, (exogenously given) ……(42)
G = G0 G0 > 0, (exogenously given) ..….(43)
Putting (5), (6) and (7) in (1) we get
Y = C + c (Y – T0) + I0 + G0 ……(44)
Now, taking total differentiation of Eq.(8), we can write:
∂Y= ∂C0 + c (∂Y –∂T0) + ∂I0 + ∂G0 …….(45)
Now, assuming equal changes, in G and T => ∂ G0 = ∂T0 and also assuming ∂I0 = ∂C0 = 0 from equation
(45), we can find the effect of this fiscal policy changes on the level of income:
∂Y = c (∂Y – ∂G0) + ∂G0 = c∂Y – c∂G0 + ∂G0
 ∂Y – c ∂Y = ∂G0 –c ∂G0
 ∂Y (1 – c) = ∂G0 (1– c)
 ∂Y/∂G0 = 1
 ∂Y = ∂G0 ……..(46)
In words, the ratio of the change in real income to the change in government expenditure
when government expenditure and tax revenues are changed by equal amount is unity in this simple
income--expenditure model.
2.35 Explanations:
For an intuitive explanation of the balanced budget theorem, given by P.A. Samuelson in his
the ‘Simple Mathematics of Income Determination’, we subtract T from both sides of equation (1)
and get,
Y – T = C (Y – T) + I + (G – T) ……..(9)
where Yd disposable Income (assuming TR = 0).
The equation (9) determines Yd uniquely. Therefore, so long as (G – T) and I remain constant,
Yd must remain constant. An increase in G and T by the same amount, with an unchanged I,
therefore leaves Yd unchanged. Hence Y must increase by the same amount of increase in T. Thus
the balanced budget multiplier is exactly unity.

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Another explanation of the process of the balanced budget multiplier was given by W.A.
Salant. He pointed out that in case of the increase in government purchases, ∂G raises national
income by the amount of ∂G directly and then indirectly via the series of increase in consumption
spending giving in total effect of : ∂Y = ∂G + c ∂G + c2 ∂G + …….
where c is the marginal propensity to consume.
But the tax increase by an amount ∂T does not enter directly into the national income on
the very first round. It enters when the cut in disposable income by ∂T reduces consumption
expenditure by c ∂T and then via the series of further reductions in consumption spending.
Thus Y = --c ∂T – c2 ∂T --………………
The sum of these two gives the net effect on Y and is equal to ∂G = ∂T
The balanced budged theorem indicated that a balance budget increases in the government
sector will be expansionary, while a balanced budget reduction in the government sector will be
concretionary.
2.36 Limitations:
The 1 : 1 version of the balanced budget multiplier rests on several simplifying assumptions,
the most important of which are the following :
1. Identical MPC: The marginal propensity to consume are identical for all people. If the
recipients of the government expenditure have different marginal propensities to consume from
the payers of the taxes, the two effects would fail to cancel out. For example, if the government
expenditure go to the individuals with high MPCs, the net expansionary impact will be greater than
one, since the proportion of the expenditure that enters the income stream directly is greater than
the proportion of the funds removed from the income stream by those paying the taxes.
2. Differential tax treatment for different groups: If the government charges different taxes
to different sections of the people, them BBM may not be unity.
Let there be two sections of people: Rich and Poor. The Rich is taxed (T) while the Poor is
not.
Let z fraction of NI (Y) going to the Rich while (1—z) fraction going to the Poor.
Thus, YR = z Y and YP = (1—z) Y (z > 0.5)
Consumption function of the Rich: CR = CR0 + cR(YR –T)
Consumption function of the Poor: CP = CP0 + cP(YP )
The equilibrium condition: Y = C + I + G = CR + CP + I + G
 Y = CR0 + cR(YR –T) + CP0 + cP(YP ) + I0 + G
 Y = C0 + cR (zY –T) + cP(1—z)Y + I0 + G ( where C0 = CR0 + CP0)
 dY = dC0 + cR (zdY –dT) + cP(1—z)dY + dI0 + dG
(Putting dI0= dC0 = 0 and dT=dG)
 dY = cR (z dY –dG) + cP (1—z) dY+ dG
 dY -- cR .z dY -- cP (1—z) dY = --cR dG + dG
 dY [1-- cR .z -- cP (1—z)] = dG (1—cR)
 dY / dG = (1—cR) / [1-- cR .z -- cP (1—z)]
Thus, if cR = cP, BBM = 1. If cR > cP, BBM < 1. If if cR < cP, BBM > 1.
3. Constant prices: Prices are assumed to be constant. Thus there is no difference between
changes in real and money income.
4. Autonomous investment: Investment has been taken as constant, or at least,
independent of income, consumption government expenditure and the tax functions. If investment
is a rising function of income, then the resulting increase will be even larger.
5. Monetary repercussions: Monetary repercussions have been ruled out. The introduction
of the interest sensitive investment along with the effects of money market reduces the value of
the multiplier. To see this, we take the help of the IS –LM analysis, with a constant money supply

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M0 and a constant price level P0. Investment is assumed to be inversely related to the rate of interest
i.e.
I = I (r), I´= dI/ dr < 0
The equations of this extended model are given by
IS: Y = C ( Y-- T) + I (r) + G, 0 < C´ < 1, I´ < 0 ..…(47)
LM: M0/ P0 = K (Y) + L (r) , K´ > 0 , L´ < 0 ..…(48)
Differentiating (10) and (11) totally we get
dY = C ´( dY – dT) + I´dr + dG …… (49)
From (8) dr = -- K´/ L´ dY
Putting dG = dT and = - K`/ L` dr in (12), we get
dY = (1 – C´) / (1 -- C´ + I´K´/L) dG
The value of dY according to the sign specifications of the model is less than unity. This is because,
as income rises with a balanced change in the government purchases and revenues, the demand for
money for transaction purposes goes up, raising the interest rate and reducing investment. This
partially offsets the initial increase in Y, giving a final expansion of Y that is less than the initial dG =
dT. Only in the liquidity trap region (L´ = ∞ ) or in case of perfectly interest inelastic investment (I΄
= 0) the value of the multiplier will be unity. Also if a compensatory monetary policy takes place to
ensure that the rate of interest remains constant, the balanced budget theorem will hold good.
6. Growth ignored: The simple analysis ignores any growth considerations and is static in
the sense that it is solely concerned with comparing two levels of income under two different
assumption s about the behavior of the government.
2.37 Balanced Budget Multiplier with induced investment:
If investment is induced, I = I (Y) = I0 + i. Y where i = marginal propensity to invest > 0
The equilibrium condition: Y = C0+c.(Y—T) + i. Y + G0
dY = dC0+c.(dY—dT) + dI0 + i. dY + dG0
 dY –c dY –i.dY = dC0 – c dT + dI0+dG0
 dY(1—c -- i) = dC0 – c dT + dI0+dG0
By putting dC0 =d I0 = 0, and dT = dG we get the BBM as
∂Y/ ∂G = ( 1—c) / (1—c -- i) > 1 …….(50)
2.38 Balanced Budget Multiplier for an open economy:
For an open economy, the equilibrium condition is
Y = C0+c(Y—T) + I0+G0 + X0 –M(Y) …….(51)
where I0 ,G0 and X0 are exogenously given investment, government purchase and exports. M(Y) is
the import as a function of income with M' >0. More specifically, M(Y) = m.Y (m > 0) where m is
marginal propensity to import (MPI). Thus, equilibrium condition is:
Y = C0+c(Y—T) + I0+G0 + X0 – m.Y
dY = dC0+c(dY—dT) + dI0+dG0 + dX0 – m.dY
 dY –c dY + mdY = dC0 – c dT + dI0+dG0 + dX0
 dY(1—c +m) = dC0 – c dT + dI0+dG0 + dX0
Putting dC0 = dI0 = dX0 = 0, and dT = dG we get the BBM as
∂Y/ ∂G = ( 1—c) / (1—c + m) < 1 ……(52)
Thus, BBM in an open economy with positive marginal propensity to import not unity. If, however,
m = 0 i.e. import is not sensitive to income, then only BBM = 1.

Multiplier Theory-Keynesian Approach-Lecture-3

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