Keynesian model with multiplier

The Keynesian Model

the multiplier, the paradox of thrift, savings and
investment, fiscal policy, and the tax multiplier

multiplier – algebra of the model
A simple Keynesian model of the economy with no government or
foreign trade can be represented as:
Y=C+I (1)
where Y is equilibrium output (income), C is aggregate consumption,
and I is aggregate investment. Aggregate consumption, or total
expenditure by households on final goods and services, is
determined by autonomous consumption (a), or the rate of
consumer expenditure independent of disposable income, and the
marginal propensity to consume (b = mpc), which is the part of
each additional dollar of disposable income that is spent on
consumption. Thus, the consumption function is:
C = a + bY (2)

No government, so Y = Yd
Note that since there is no government, taxes are zero, so Y
= Yd, since:
Yd = Y – T
Yd = Y – 0
Yd = Y
Thus while the consumption function is usually C = a + bYd,
here it will simply be:
C = a + bY

Investment
Investment is determined by a complex of factors such as
expectations of investors and lending institutions, business
confidence, political climate, and so on. For present
purposes, what is important is that investment is autonomous
—that is, independent—of income. It is also not a simple
function of the rate of interest, as in neoclassical loanable
funds theory.

solving the equation
Y=C+I (1)
C = a + bY (2)

Substituting equation (2) into equation (1), i.e., replace C
with a + bY, we get:
 Y = a + bY + I (3)
Subtracting bY from both sides:
 Y - bY = a + I (4)

Solving for Y
Y - bY = a + I (4)

Factoring out the Y from the left hand side of equation (4)
 Y (1 - b) = a + I (5)
Dividing both sides by (1 - b):
 Ye = 1 (a + I) (6)
(1-b)

the multiplier
Ye = 1 (a + I)
(1-b)

where 1/(1-b)—or 1 divided by the mps—is the multiplier, or the
feedback mechanism that amplifies any initial increase (injection) or
decrease (withdrawal) in aggregate demand. Therefore, Ye, the
equilibrium level of output (income) is determined by the multiplier
and total injections. The total injections in this simple model are a + I.

Keynesian model - numerical example
Given:
Y=C+I
C = a + bY
Where:
a = 100
b = .75
I = 300

Solving for Ye
Y = 100 + .75Y + 300
 Y - .75Y = 100 + 300
 Y (1 - .75) = 100 + 300
 Y (.25) = 100 + 300
 Y = (1/.25) x 400
 Y = 4 x 400
 Ye = $1600 billion

solving for equilibrium consumption
and savings
Once we have Ye, we can find Ce and Se:
Ce = a + bYe
Ce = 100 + .75 (1600)
Ce = 100 + 1200 = 1300
Se = -a + (1 - b)Ye
Se = -100 + (1 - .75) 1600
Se = -100 + .25 (1600) = 300

double-checking savings
Also:
Ye = Ce + Se
Ye – Ce = Se
1600 – 1300 = 300
Also:
Se = I (savings = investment at equilibrium)
300 = 300

Keynesian Model
45°

Expenditure AS = C + I

C = 100 + .75Y

a+I
S = - 100 + (1 – .75) Y

a I = 300
I

0
Y1 Y* Y
-a
400 1600

the recessionary gap
Assume that this economy, if producing at full employment,
given resources and technology, could produce an aggregate
output of $2000.
We can now calculate the values of consumption and savings
at full employment, as well as aggregate spneding at full
employment, and the recessionary gap.

full employment and the recessionary
gap
Cf = a + bYf = 100 + .75 (2000) = $1600
Sf = -a + (1 - b)Yf = -100 + .25 (2000) = 400
(double-check: Yf – Cf = Sf = 2000 – 1600 = 400)
AS@Yf = Cf + I = 1600 + 300 = 1900
gap = Yf – AS@Yf = (2000 – 1900) = 100 =
= (Sf – I) = (400 – 300) =
= (Yf – Ye)/multiplier = (2000 – 1600)/4 = 100

Keynesian Model
45°


C = a + bY

a+I
S = - a + (1 – b) Y

a I
I

0
Y1 Y* Yf Y
-a
1600 2000

the paradox of thrift
An attempt by the economy as a whole to increase aggregate savings
not only will not succeed, but may lower aggregate output, income
and employment. This is because increased savings at a given
level of aggregate income will mean decreased consumption.
Thus a smaller marginal propensity to consume will reduce
the stimulative effects of investment and other spending.

paradox of thrift
For example, suppose an economy is characterized by a
consumption function:
 C = $100 + .8Yd
If autonomous investment is equal to $300 billion then the
equilibrium level of output and income is
 Ye = 5 (100 + 300) = $2000 billion
because the multiplier = 1/(1 – b)
= 1/(1 - .8) = 5.

paradox of thrift
Aggregate consumption is:
 C = $100 + .8 ($2000) = $1700 billion
and aggregate savings is:
 S = -$100 + (1 - .8) ($2000) = $300 bil.
So aggregate savings equals aggregate investment ($300
billion).

paradox of thrift - example
Suppose some political and or business leaders come out and
say we have to save more so the economy can grow. If
people comply in such a way that the mps rises from .2 to .
25, what will be the effect?

paradox of thrift
The new consumption function will be:
 C = $100 + .75Yd
With $300 billion in investment, the new equilibrium will be:
 Ye = 4 (100 + 300) = $1600 billion
because the new multiplier = 1/(1 - .75) = 4.
Aggregate consumption is now:
 C = $100 + .75 ($1600) = $1300 billion
and savings:
 S = -$100 + (1 - .75) ($1600) = $300 billion

paradox of thrift
Thus, savings is still equal to investment at the same level of
$300, but output and employment are much lower.
 So the attempt by the economy as a whole to save more
not only did not result in more savings, but actually lowered
aggregate output and income by $400 billion.

paradox of thrift
 This is the paradox of thrift, and is another example of the
paradoxical nature of macroeconomics. It is rooted in the two-sided
nature of spending and saving. When we just look at one individual firm
or household in isolation, we don't see the impact that our actions have
on other participants in the economy due to the interdependent nature
of economic activity. So while for any one individual, it is wonderful to
save more, for the economy as a whole, it could be a disaster.

paradox of thrift
If, however, the increased saving is the result of higher
incomes, then that is a different story. If income goes up,
consumption and saving both go up. But at a given level of
income, increased aggregate savings can throw the economy
into a recession. Therefore, a policy to increase growth by
increasing savings has it backwards: savings will increase as a
result of growth.

Keynes’s critique of the neoclassical theory
of savings and investment
1. In Keynes, since consumption is a function of disposable income, and
saving is income not spent, saving is also primarily a function of
disposable income. S is a passive residual, determined by disposable
income and the marginal propensity to consume. Keynes did not
believe it was legitimate to hold income constant when analyzing
aggregate saving, as in neoclassical theory. He also disagreed with
the neoclassical belief that saving is primarily a function of the rate of
interest.

Loanable Funds Market

Interest Rate

S (Savings)

i*

I (Investment)

0 S=I S, I

Keynes’s critique of the neoclassical theory
of savings and investment
2. Historical experience of the Great Depression:
 interest rates very low, no investment;
 wages low, no labor demand;
 how long is the long run?.

3. S = I is the macroeconomic equilibrium condition in both
Keynes and neoclassical, but in Keynes I => S through
changes in Y and in neoclassical S => I through changes in i.
In addition, in Keynes the two may be equal at a whole range
of potential levels of output and income, only one of which is
full employment, while in neoclassical the two may be equal
only at full employment.

4. Keynes did not believe it was legitimate to hold the state of
investor expectations constant in analyzing aggregate
investment, as in neoclassical theory. He also disagreed with
the neoclassical view that investment is primarily a function of
the rate of interest. Expected profitability of investors and
lending institutions both required for investment to take
place.

5. Keynes distinguished between risk, which is calculable, and
uncertainty, which is not conducive to statistical probability.
He believed most important determinants of investment
described by uncertainty, not risk. In neoclassical theory,
uncertainty in this sense is not recognized. Also, even under
risk, the confidence of whether one will ‘beat the odds’ is
subject to unpredictable variation. Mass psychology subject to
waves of optimism and pessimism.

6. Business and political climate will influence investment
decisions, as will many other factors, not all of which appear
immediately relevant, at least on the surface.

7. In a modern capitalist economy with high-tech financial
institutions and advanced instruments of credit, a ‘pool’ of
savings is not necessary to finance investment. Banks are
private, profit-maximizing institutions and will not pass up
the chance to make profits if they believe a loan will be
profitable. They will always make a loan and worry about
reserve requirements at the end of the day (often borrowing
themselves to meet their requirements).

8. In Keynes, the rate of interest is not determined by savings and
investment, but by the supply and demand for money. This is
Keynes’s liquidity preference theory (more on this later).

9. Separation of ownership and management means those who own
do not necessarily know the business well, and those who
manage may have different interests and incentives than if they
also owned. Makes investment more unstable.

10. Speed of asset revaluation increasingly faster and faster. Assets
are revalued within the space of seconds, and ability to react
immediately, without having to wait to see if a change is a
temporary deviation, creates instability. Self-fulfilling
prophecies become a characteristic of the system (for
example, people think an asset’s value is going to go down, so
they sell and because people sell, the value goes down).

fiscal policy for full employment:
eliminating a recessionary gap
Keynes’s demonstration of the possibility of the economy
being in macroequilibrium, with S = I, below full
employment provides a theoretical justification for more
interventionist policies by the government.
Fiscal policy: the attempt to affect macroeconomic variables
(such as C, I, Y) through government spending and tax
policies.

Increasing G
If Yf = $2000 billion and Ye is $1600, how much does the
government have to increase spending to push the economy
to Yf?

Increasing G
If Yf = $2000 billion and Ye is $1600, how much does the government
have to increase spending to push the economy to Yf? Not $400.

Increasing G
have to increase spending to push the economy to Yf? Not $400. If G
increased by $400 then:
Y=C+I+G
C = a + bY

Increasing G
have to increase spending to push the economy to Yf? Not $400. If G
increased by $400 then:
Y=C+I+G
C = a + bY
Ye = 1/(1 - .75) * 100 + 300 + 400
= 4 (800)
= $3200
Way past Yf—impossible, so inflation will occur. What happened?

closing the recessionary gap
The government spending of 400, like all other autonomous
expenditures, had a multiplier effect, in this case of 4, and so increased
total output and income not by 400 but by 1600.
How much do we need to increase G by to just get the economy to full
employment?
By the size of the gap, or the amount we need to increase total spending
(Yf – Ye) divided by 4.
gap = Yf – AS@Yf = Sf – I = (Yf – Ye)/mult.
= 100

Keynesian Model
45°

C+I+G


C

a+I+G

a+I
S = - a + (1 – b) Y
I+G I+G
a I
I

0
Y1 Y* Yf Y
-a

full employment
Notice that at Yf, S = I + G; (I + G can be thought of as private and
public investment)
Notice the aggregate spending function with government (still no
foreign trade), or the C + I + G line:
has a y-intercept of (a + I + G);
has a slope = mpc; and
intersects the 45 degree line at Yf

Multiplier with Taxes
Let's add taxes and government spending into the multiplier
formula!
First, we begin with:
Y = C + I + G (1)
Then we take our consumption function:
C = a + bYd (2)
Only now we have to account for the fact that Y and Yd are not
equal
Yd = Y - T (3)

Yd = Y - T (3)
because disposable income is aggregate income less taxes.
Since taxes can be determined by the tax rate times aggregate income:
T = tY (4)
Then:
Yd = Y - tY (5)

Substituting equation (5) into the consumption function:
C = a + b(Y - tY) (6)
And substituting equation (6) into equation (1):
Y = a + b(Y - tY) + I + G (7)

Y = a + b(Y - tY) + I + G (7)

We then solve for Y:

Y = a + bY - btY + I + G (8)

Y - bY + btY = a + I + G (9)

Y(1 - b + bt) = a + I + G (10)

Y(1 - b + bt) = a + I + G (10)

Y = 1 * (a + I + G) (11)
1- b + bt

So the multiplier with taxes is:
1/(1 - b + bt) (12)
And the multiplier times total injections (a + I + G) will give us the
equilibrium level of output and income.

Multiplier with taxes – numerical
example
Given:
C = $100 + .8Yd
I = $50
G = $350
t = .25
Find: Ye, value of mult., T, Yd, C, & S
Does S = I? Why or Why Not?

example
Ye = 1/(1-b+b[t]) (a + I + G )
= 1/.4 (100 + 50 + 350)
= 2.5 (500)
= 1250

example
Value of the multiplier
= 1/(1-b+b[t])
= 1/(1-.8+.8[.25])
= 1/.4 = 2.5

[since (.8 x .25) =.2]

example
 T = .25 (1250) = 312.50
 Yd = Y – T = 1250 – 312.50 = 937.50
 C = a + bYd = 100 + .8(937.50)
= 100 + 750 = 850
 S = -a + (1-b)Yd = -100 + (1-.8)937.50
= -100 + 187.50 = 87.50

example
Does S = I? Why or why not?

example
No, total injections = total withdrawals
I+G=S+T
= 50 + 350 = 87.50 + 312.50 = 400 = 400

example
I+G=S+T=
50 + 350 = 87.50 + 312.50 = 400 = 400
and the:
public deficit = private sector surplus
G – T = S – I = 350 – 312.50 = 87.50 – 50
= 37.50 = 37.50

Keynesian model with multiplier

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