Download Chapter 7. National Income

Chapter 7. National Income
National Income – effect of changes in government spending, private
investment, the marginal propensity to consume and the tax rate.
In a closed economy, given the consumption function, the average tax rate, the level of
private investment and the level of government spending, calculate the equilibrium level of
income and output in the economy.
Calculate the expenditure multiplier and show the effect of
a) an increase in government spending and
b) an increase in private investment.
Consumption function: C = 0.8(1 – 0.25)Y + 12
(1)
Investment: I = 26
Government spending: G = 14.
Solution:
The general equation for the consumption function can be represented by
C = c(1 – t)Y + k
where C is consumption by households, c is the marginal propensity to consume, t is the tax
rate, Y is national income and k is a constant (autonomous consumption).
Note: Do not confuse consumption “C” (upper case) with the marginal propensity to consume
“c” (lower case). The marginal propensity to consume (MPC) is the increase in consumption
per unit increase in income. The value of c is between 0 and 1. Any increase in income that is
not consumed is savings. The values of c and t determine the slope of the consumption
function.
In a Closed Economy (no imports or exports), aggregate demand is the sum of household
consumption, C, private investment, I, and government spending, G.
AD = C + I + G.
At equilibrium, aggregate demand equals income and output.
Y=C+I+G
Y = c(1 – t)Y + k + I + G
Y – c(1 – t)Y = k + I + G
Y(1 –c + ct) = k + I + G
Y=
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Chapter 7. National Income
Given C = 0.8(1 – 0.25)Y + 12 (MPC = 0.8, average tax rate is 25% and autonomous
consumption is 12).
Y = 0.8(1-0.25)Y + 12 + I + G
Y = 0.8 (0.75)Y + 12 + I + G
Y = 0.6Y + 12 + I + G
0.4Y = 12 + I + G
Y=
Y = 30 + 2.5I + 2.5G
If private investment, I = 26 and government spending, G = 14, the economy is in equilibrium
when
Y = 30 + 2.5(26) + 2.5(14)
(2)
Y = 30 + 65 + 35
Y = 130.
Demand equals income and output.
Table 7.1 Calculate points for Consumption and Agg. Demand functions
Y
C = 0.6Y + 12
AD = C + 26 +
14
0
12
40
36
80
60
120
84
160
108
200
132
52
76
100
124
148
172
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Chapter 7. National Income
200
180
160
E
140
D
120
e
m
100
a
n 80
d
AD= 0.6Y+12+26+14
60
C=0.6Y+12
40
20
0
0
20
40
60
80
100
120
Income / Output
140
160
180
200
Figure 7.1C. Consumption Function and Aggregate
Demand
Equilibrium occurs where the 45° line intersects the AD line.
Demand is a function of Y, C, c, t, I and G.
In economics we often want to see the effect of a change in one variable while holding other
variables constant.
Given the consumption function, the effect on equilibrium income of a change in government
spending, G, while holding private investment, I, constant is given by ∂Y/∂G. This is a partial
derivative
Y = 30 + 2.5I + 2.5G
∂Y/∂G = 2.5
An increase in government spending results in 2.5 times that increase in national income. If
government spending increases by 8, from 14 to 22, the equilibrium level of income increases
by 2.5 × 8 = 20. The new equilibrium level of income and output is 150.
With government spending increased by 8, the new aggregate demand schedule, AD’ = 0.6Y
+ 12 + 26 + 22.
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Chapter 7. National Income
Table 7.2
Calculate points for function Y = 0.6Y + 12 + 26 + 14
and Y = 0.6Y + 12 + 26 + 22
Y
C = 0.6Y + 12
AD = C + 26 +
14
AD' = C + 26 +
22
0
12
40
36
80
60
120
84
160
108
200
132
52
76
100
124
148
172
60
84
108
132
156
180
200
180
160
140
D
e 120
m
100
a
n 80
d
60
AD'=0.6Y+12+26+22
AD=0.6Y+12+26+14
C=0.6Y+12
40
20
0
0
20
40
60
80
100
120
140
160
180
200
Income / Output
Figure 7.2C. Aggregate Demand – Effect of increase in government spending.
An increase in government spending shifts the aggregate demand schedule upwards, parallel
to the original aggregate demand schedule.
Similarly, an increase in private investment I, with government spending held constant, will
result in a 2.5 times increase in national income.
Y = 30 + 2.5I + 2.5G
∂Y/∂I = 2.5
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Chapter 7. National Income
This effect is known as the multiplier. The slope of the consumption function determines the
value of the multiplier. The value of the multiplier is given by
When c = 0.8 and t = 0.25
the multiplier M =
(
)
=
= 2.5
Calculate the effect of a change in a) the marginal propensity to consume and b) the tax
rate on the value of the multiplier.
Solution a)
If the tax rate remains constant, the effect of a change in the marginal propensity to consume,
c, on the value of the multiplier is given by ∂M/∂c.
M =
=
when t = 25%
=
∂M/∂c =
(
=
=
)
(
(
)
(
(
)
)
)
when c = 0.8
=
=
= 4.69
When the marginal propensity to consume, c, = 0.8, a small change in c will change the value
of the multiplier by 4.69 times that change.
M=
When c = 0.8 and t = 0.25, M = 2.5 (see above)
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Chapter 7. National Income
If c increases to 0.81,
M =
(
)
=
=
= 2.547
An increase of 0.01 in the marginal propensity to consume (from 0.8 to 0.81) caused the
multiplier to increase by 0.047 (from 2.5 to 2.547), confirming the value of ∂M/∂c above.
Solution b)
If the marginal propensity to consume remains constant, the effect of a change in the tax rate,
t, on the value of the multiplier is given by ∂M/∂t.
M =
=
when c = 0.8
=
∂M/∂t =
∂M/∂t =
(
)
(
(
(
)
)
when t = 25%
)
=
=
= -5
When t = 0.25, a small change in t will change the value of the multiplier by 5 times that
change, in the opposite direction (indicated by the - sign).
M=
When c = 0.8 and t = 0.25, M = 2.5 (see above)
If t increases to 0.26,
M =
(
)
=
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Chapter 7. National Income
=
= 2.45
An increase of 0.01 in the tax rate (from 0.25 to 0.26) caused the multiplier to decrease by
0.05 (from 2.5 to 2.45), confirming the value of ∂M/∂t above.
Summary
A change in either government spending or private investment is subject to the multiplier.
Any increase (decrease) results in the equilibrium level of income increasing (decreasing) by
a multiple of the change.
The value of the multiplier depends on the slope of the consumption function.
The slope of the consumption function depends on the marginal propensity to consume and
on the tax rate.
An increase in the MPC increases the slope of the consumption function and increases the
value of the multiplier.
An increase in the tax rate reduces the slope of the consumption function and reduces the
value of the multiplier.
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