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The Expanded Model
of Income
Determination
Expanded model of income
determination

In chapter 14, a very basic Keynesian
model of income determination was
introduced
 This
model serves as an introduction to
income determination and capacity
utilisation in the economy
 If
is far to simple to be of any use in the
real world, but it establishes some
important points nevertheless
Keynes, John Maynard, 1st Baron
Keynes of Tilton (1883-1946)
Expanded model of income
determination

Recall when Keynes was writing – mid
thirties with massive unemployment
 Established
theory until then had assumed
that this would be a temporary
phenomenon
 In
a world with flexible prices, in the long
run equilibrium will exist in all markets
 Keynes:
In the long run, we are all dead
Expanded model of income
determination

Keynes gave politicians theoretically
sound arguments for intervening in the
economy
 Keynes
in particular focused on how the
authorities could affect aggregate demand
through fiscal policy, i.e. government
purchases of goods and services and
taxes
 In
chapter 15, this is incorporated into the
basic model of income determination.
Expanded Model of Income
Determination

We introduce a public sector, with government
purchases of goods and services G and taxes
T. This model could be labelled a Keynes
model for a closed economy with a public
sector

Later in the chapter, another sector is
introduced – the foreign sector. Only goods
transactions takes place, exports (X) and
imports (Z)

This chapter also provides a more satisfactory
explanation of investment demand
Investment demand

Demand for investment goods (I) very much
depends on the outlook for the economy

Profitability depends on:

Investment outlay

Increased income due to the investment

Costs of financing the investment

Increased income – cost of investment = MEI
(marginal efficiency of investment)

Cost of financing: R
Time value of money

The investment outlay is paid for ”today”

Income will accrue in the future, and
value may be reduced due to:
 impatience
and postponement of demand
 risk
 inflation

Income must be discounted by an
interest rate R
Net Present Value

Example:
 Investment

outlay = 10 000
 Income
year 1: 6 000
 Income
year 2: 2: 6 000
 Interest
rate (R) = 5 % (0,05)
What is the PV of the income?
6000 6000
PV 

 11156
2
1,05 1,05
Marginal Efficiency of Investment (MEI)
I0
I1
I2
Rate of return (R)
R1
Marginal efficiency of
investment
R2
Expectations change
I2
I1
I0
Rate of return (R)
Marginal efficiency of
investment
R0
Keynesian business cycle

The accelerator
 changes
in national income and induced
investment

 the
accelerator coefficient
 the
instability of investment
The multiplier / accelerator interaction
GDP, Investment (% annual change)
Fluctuations in UK real GDP and
20
investment:
1978-2002
18
16
14
12
10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
-14
1978 1980
1982 1984 1986 1988
1990 1992 1994 1996
1998 2000 2002
GDP, Investment (% annual change)
Fluctuations in UK real GDP and
20
investment:
1978-2002
18
16
14
12
10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
-14
1978 1980
GDP
1982 1984 1986 1988
1990 1992 1994 1996
1998 2000 2002
GDP, Investment (% annual change)
Fluctuations in UK real GDP and
20
investment:
1978-2002
18
16
14
12
10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
-14
1978 1980
Investment
GDP
1982 1984 1986 1988
1990 1992 1994 1996
1998 2000 2002
Accelerator 1970-1999 in Norway
30,0 %
GDP
Investment
20,0 %
10,0 %
0,0 %
1974
-10,0 %
-20,0 %
-30,0 %
1978
1982
1986
1990
1994
1998
Accelerator theory
capital output ratio = 2
Year
1
2
3
4
5
Sales
1 mill
1 mill
1,2 mill
1,5 mill
1,6 mill
Required
Net
capital stock investment
2,0 mill
2,0 mill
0
2,4 mill
0,4 mill
3,0 mill
0,6 mill
3,2 mill
0,2 mill
Accelerator theory

Investments are dependent on expected
changes in GDP or I =  Y
– a small change in income
gives a large change in induced investment
 Accelerator
 This
depends on the marginal ratio
between capital and production
 In
addition, we will have multiplier effects
between I and Y
Introducing the public sector

Taxes T represent a withdrawal from the
economic circulation (like savings S)

The Governments demand for goods
and services G represent an injection
(like investments I)

Equilibrium when realised withdrawals =
realised injections
S
+T=I+G
Keynes expanded model - 1

The public sectors demand for goods
and services G is always exogenous
Taxes (T)
 Version
 Version
1: Lump sum taxes T = T
2: Income taxes T = tY, where t is
the (average) tax rate
The model version 1
Y  CIG
C  bYd
II
GG
TT
Equilibrium
Y  CIG
Y  b(Y  T)  I  G
Y  bY  bT  I  G
Y  bY  I  G  bT
Y(1  b)  I  G  bT
1
Y
(I  G  bT)
1 b
An example

Assume we have the following:
C
I
= 0,8Yd
= 60
G
= 50
T
= 50
1
Y
(60  50  40)  350
1  0,8
The multipliers
1
Assume : G  10
Y 
G
1
1 b
Y 
 10  50
0,2
1
Y 
I
1 b
b
Y 
T
1 b
The model
Example
a
I
G
T
b = MPC
Y
0
60
50
50
0,8
350
Consumption
Government
Investment
Y = GNP
240
50
60
350
Multiplier
5,00
Haavelmos theorem

What happens if an increase in public
spending is financed by an equivalent tax
increase, i.e. G= T?
1
b
G 
T
1 b
1 b
By assumption G  T, this gives
Y 
1
b
G 
G
1 b
1 b
1 b
Y 
G
1 b
Y  G, the multiplier is 1
Y 
The Model Version 2
Y  CIG
C  bY(1  t)
II
GG
T  tY
Equilibrium
Y  CIG
Y  bY(1  t)  I  G
Y  bY  bYt  I  G
Y  bY  bYt  I  G
Y(1  b(1  t)  I  G
1
Y
(I  G)
1  b(1  t)
The multipliers
Assume : G  10, t  0,2 (20 %)
1
Y 
G
1  b(1  t)
1
Y 

1
1

0,8(1

0
,
2
)
Y 
I
1  b(1  t)
2,78  10  27,8
The multiplier is reduced due to increased
leakages in the form of taxes
The model
Example
a
I
G
t
b = MPC
Y
0
60
50
0,3
0,8
250,00
Consumption
Government
Investment
Y = GNP
140
50
60
250,00
Multiplier
2,27
Built in stabilisers
T
T
G
Government
expenditure and Taxes
G
Yb
Introducing the foreign sector

Imports: Z and Exports: X

Equilibrium when leakages = injections
S+T+Z=I+G+X

It is assumed that imports are
endogenous and dependent on income

Exports are exogenous
Economic circulation
The model
Y  CIGXZ
C  bY(1  t)
II
GG
T  tY
XX
Z  nY
The multipliers
1
(I  G  X)
Y
1  b(1  t)  n
1
I
Y 
1  b(1  t)  n
1
G
Y 
1  b(1  t)  n
1
X
Y 
1  b(1  t)  n
The open economy model
Input
G
I
X
t
n
b
Y=
G-T
X- Z
20
16
30
0,2
0,3
0,8
100,00
0,00
0,00