Transcript CHAPTER 9: The Government and Fiscal Policy

Chapter
9
The Government
and Fiscal Policy
Prepared by:
Fernando & Yvonn Quijano
© 2007 Prentice Hall Business Publishing Principles of Economics 8e by Case and Fair
CHAPTER 9: The Government
and Fiscal Policy
The Government
and Fiscal Policy
9
Chapter Outline
Government in the Economy
Government Purchases (G), Net Taxes (T), and
Disposable income (Yd)
Equilibrium Output: Y = C + I + G
Fiscal Policy at Work: Multiplier Effects
The Government Spending Multiplier
The Tax Multiplier
The Balanced-Budget Multiplier
The Federal Budget
The Budget
The Surplus or Deficit
The Debt
The Economy’s Influence on the Government
Budget
Tax Revenues Depend on the State of the Economy
Some Government Expenditures Depend on the
State of the Economy
Automatic Stabilizers
Fiscal Drag
Full-Employment Budget
Looking Ahead
Appendix A: Deriving the Fiscal Policy Multipliers
Appendix B: The Case in Which Tax Revenues
Depend on Income
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CHAPTER 9: The Government
and Fiscal Policy
THE GOVERNMENT AND FISCAL POLICY
fiscal policy The government’s
spending and taxing policies.
monetary policy The behavior of the
Federal Reserve concerning the nation’s
money supply.
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CHAPTER 9: The Government
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The behavior of the Federal Reserve concerning
the nation’s money supply is called:
a. Discretionary fiscal policy.
b. Automatic fiscal policy.
c. Budgetary policy.
d. Monetary policy.
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CHAPTER 9: The Government
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The behavior of the Federal Reserve concerning
the nation’s money supply is called:
a. Discretionary fiscal policy.
b. Automatic fiscal policy.
c. Budgetary policy.
d. Monetary policy.
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CHAPTER 9: The Government
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GOVERNMENT IN THE ECONOMY
discretionary fiscal policy Changes in
taxes or spending that are the result of
deliberate changes in government policy.
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CHAPTER 9: The Government
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Over which of the following categories does the
government have more control?
a. Tax revenue.
b. Government expenditures.
c. Tax rates.
d. The size of corporate profits.
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CHAPTER 9: The Government
and Fiscal Policy
Over which of the following categories does the
government have more control?
a. Tax revenue.
b. Government expenditures.
c. Tax rates.
d. The size of corporate profits.
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GOVERNMENT IN THE ECONOMY
CHAPTER 9: The Government
and Fiscal Policy
GOVERNMENT PURCHASES (G), NET TAXES (T),
AND DISPOSABLE INCOME (YD)
net taxes (T) Taxes paid by firms and
households to the government minus
transfer payments made to households
by the government.
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CHAPTER 9: The Government
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GOVERNMENT IN THE ECONOMY
FIGURE 9.1 Adding Net Taxes (T) and Government Purchases (G)
to the Circular Flow of Income
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GOVERNMENT IN THE ECONOMY
CHAPTER 9: The Government
and Fiscal Policy
disposable, or after-tax, income (Yd)
Total income minus net taxes: Y − T.
disposable income ≡ total income − net taxes
Yd ≡ Y − T
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CHAPTER 9: The Government
and Fiscal Policy
Select the best answer. Households use their
disposable income (Yd) to do the following:
a. Consume.
b. Consume and save.
c. Consume, save, and pay taxes.
d. Consume, save, pay taxes, and buy imports.
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CHAPTER 9: The Government
and Fiscal Policy
Select the best answer. Households use their
disposable income (Yd) to do the following:
a. Consume.
b. Consume and save.
c. Consume, save, and pay taxes.
d. Consume, save, pay taxes, and buy imports.
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GOVERNMENT IN THE ECONOMY
CHAPTER 9: The Government
and Fiscal Policy
When government enters the picture, the aggregate
income identity gets cut into three pieces:
Yd  Y  T
Yd  C  S
Y  T  C S
Y  C S  T
And aggregate expenditure (AE) equals:
AE  C  I  G
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CHAPTER 9: The Government
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GOVERNMENT IN THE ECONOMY
budget deficit The difference between
what a government spends and what it
collects in taxes in a given period: G − T.
budget deficit ≡ G − T
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GOVERNMENT IN THE ECONOMY
CHAPTER 9: The Government
and Fiscal Policy
Adding Taxes to the Consumption Function
To modify our aggregate consumption function to
incorporate disposable income instead of beforetax income, instead of C = a + bY, we write
C = a + bYd
or
C = a + b(Y − T)
Our consumption function now has consumption
depending on disposable income instead of
before-tax income.
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CHAPTER 9: The Government
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When government enters the circular flow of
income, which of the following is an
expression for planned aggregate
expenditure?
a. Y − T
b. C + S + T
c. C + I + G
d. G – T
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CHAPTER 9: The Government
and Fiscal Policy
When government enters the circular flow of
income, which of the following is an
expression for planned aggregate
expenditure?
a. Y − T
b. C + S + T
c. C + I + G
d. G – T
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GOVERNMENT IN THE ECONOMY
CHAPTER 9: The Government
and Fiscal Policy
Investment
The government can affect investment behavior
through its tax treatment of depreciation and other
tax policies.
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GOVERNMENT IN THE ECONOMY
EQUILIBRIUM OUTPUT: Y = C + I + G
CHAPTER 9: The Government
and Fiscal Policy
equilibrium condition: Y = C + I + G
TABLE 9.1 Finding Equilibrium for I = 100, G = 100, and T = 100
(All Figures in Billions of Dollars)
(1)
(2)
OUTPUT
NET
(INCOME) TAXES
Y
T
(3)
(4)
DISPOSABLE
INCOME
Yd / Y  T
CONSUMPTION
SPENDING
(C = 100 + .75 Yd)
(5)
(6)
(7)
(8)
(9)
(10)
PLANNED
PLANNED
UNPLANNED
SAVING INVESTMENT GOVERNMENT AGGREGATE INVENTORY
ADJUSTMENT
S
SPENDING PURCHASES EXPENDITURE
CHANGE
TO
(Yd – C)
I
G
C+I+G
Y  (C + I + G) DISEQUILIBRIUM
300
100
200
250
 50
100
100
450
 150
Output8
500
100
400
400
0
100
100
600
 100
Output8
700
100
600
550
50
100
100
750
 50
900
1,100
100
100
800
1,000
700
850
100
150
100
100
100
100
900
1,050
0
+ 50
Output8
Equilibrium
1,300
100
1,200
1,000
200
100
100
1,200
+ 100
Output9
1,500
100
1,400
1,150
250
100
100
1,350
+ 150
Output9
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Output9
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CHAPTER 9: The Government
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GOVERNMENT IN THE ECONOMY
FIGURE 9.2 Finding Equilibrium Output/Income Graphically
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GOVERNMENT IN THE ECONOMY
CHAPTER 9: The Government
and Fiscal Policy
The Leakages/Injections Approach to
Equilibrium
Taxes (T) are a leakage from the flow of income. Saving
(S) is also a leakage.
In equilibrium, aggregate output (income) (Y) equals
planned aggregate expenditure (AE), and leakages (S + T)
must equal planned injections (I + G). Algebraically,
AE  C  I  G
Y  C S  T
C S  T  C I  G
leakages/injections approach to equilibrium: S + T = I + G
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CHAPTER 9: The Government
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In the circular flow that includes households,
firms, and government, which of the following
expressions is the leakages/injections
approach to equilibrium?
a. Y = C + I + G.
b. C + S = I + G.
c. Y = a + bT + I + G.
d. S + T = I + G.
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CHAPTER 9: The Government
and Fiscal Policy
In the circular flow that includes households,
firms, and government, which of the following
expressions is the leakages/injections
approach to equilibrium?
a. Y = C + I + G.
b. C + S = I + G.
c. Y = a + bT + I + G.
d. S + T = I + G.
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FISCAL POLICY AT WORK:
MULTIPLIER EFFECTS
CHAPTER 9: The Government
and Fiscal Policy
THE GOVERNMENT SPENDING MULTIPLIER
TABLE 9.2 Finding Equilibrium After a $50 Billion Government Spending
Increase (All Figures in Billions of Dollars; G Has Increased from 100
in Table 9.1 to 150 Here)
(1)
OUTPUT
(INCOME)
Y
(2)
(3)
(4)
DISPOSABLE CONSUMPTION
NET
INCOME
TAXES
SPENDING
Yd / Y  T (C = 100 + .75 Yd)
T
(5)
(6)
(7)
(8)
(9)
(10)
PLANNED
PLANNED
UNPLANNED
SAVING INVESTMENT GOVERNMENT AGGREGATE INVENTORY
ADJUSTMENT
S
SPENDING
PURCHASES EXPENDITURE
CHANGE
TO
(Yd – C)
I
G
C+I+G
Y  (C + I + G) DISEQUILIBRIUM
300
100
200
250
 50
100
150
500
 200
Output8
500
100
400
400
0
100
150
650
 150
Output8
700
100
600
550
50
100
150
800
 100
Output8
900
100
800
700
100
100
150
950
 50
Output8
1,100
100
1,000
850
150
100
150
1,100
0
1,300
100
1,200
1,000
200
100
150
1,250
+ 50
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Equilibrium
Output9
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CHAPTER 9: The Government
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FISCAL POLICY AT WORK:
MULTIPLIER EFFECTS
1
government spending multiplier 
MPS
government spending multiplier
The ratio of the change in the
equilibrium level of output to a change
in government spending.
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CHAPTER 9: The Government
and Fiscal Policy
How much of an increase in government
spending would be required to generate a
$200 billion increase in the equilibrium level
of output?
a. An amount less than $200 billion in
government spending.
b. An amount greater than $200 billion in
government spending.
c. Exactly $200 billion in government spending.
d. None of the above. Equilibrium output does
not change with changes in government
spending.
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CHAPTER 9: The Government
and Fiscal Policy
How much of an increase in government
spending would be required to generate a
$200 billion increase in the equilibrium level
of output?
a. An amount less than $200 billion in
government spending.
b. An amount greater than $200 billion in
government spending.
c. Exactly $200 billion in government spending.
d. None of the above. Equilibrium output does
not change with changes in government
spending.
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CHAPTER 9: The Government
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FISCAL POLICY AT WORK:
MULTIPLIER EFFECTS
FIGURE 9.3 The Government Spending Multiplier
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FISCAL POLICY AT WORK:
MULTIPLIER EFFECTS
CHAPTER 9: The Government
and Fiscal Policy
THE TAX MULTIPLIER
The multiplier for a change in taxes is not the same as the multiplier for a change in
government spending.
tax multiplier The ratio of change in
the equilibrium level of output to a
change in taxes.
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CHAPTER 9: The Government
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FISCAL POLICY AT WORK:
MULTIPLIER EFFECTS
 1 
 Y  (initial increase in aggregate expenditure)  

 MPS 
 1 
 MPC 
 Y  (   T  MPC )  
   T  

 MPS 
 MPS 
 MPC 
tax multiplier  

 MPS 
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CHAPTER 9: The Government
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Which of the following formulas shows the impact
of a change in taxes on equilibrium income?
a. Y = a + b(Y – T) + I + G
b. Y = 1/(1 – b) * (a – bT + I + G)
c. S + T = I + G
d. – ∆T * (b/1 – b)
e. C + S = I + G
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CHAPTER 9: The Government
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Which of the following formulas shows the impact
of a change in taxes on equilibrium income?
a. Y = a + b(Y – T) + I + G
b. Y = 1/(1 – b) * (a – bT + I + G)
c. S + T = I + G
d. – ∆T * (b/1 – b)
e. C + S = I + G
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FISCAL POLICY AT WORK:
MULTIPLIER EFFECTS
CHAPTER 9: The Government
and Fiscal Policy
THE BALANCED-BUDGET MULTIPLIER
balanced-budget multiplier The ratio
of change in the equilibrium level of
output to a change in government
spending where the change in
government spending is balanced by a
change in taxes so as not to create
any deficit. The balanced-budget
multiplier is equal to 1: The change in
Y resulting from the change in G and
the equal change in T is exactly the
same size as the initial change in G or
T itself.
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CHAPTER 9: The Government
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FISCAL POLICY AT WORK:
MULTIPLIER EFFECTS
balanced - budget multiplier  1
An increase in government spending has a direct initial effect on planned aggregate
expenditure; a tax increase does not. The initial effect of the tax increase is that
households cut consumption by the MPC times the change in taxes. This change in
consumption is less than the change in taxes, because the MPC is less than 1. The
positive stimulus from the government spending increase is thus greater than the
negative stimulus from the tax increase. The net effect is that the balanced-budget
multiplier is 1.
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FISCAL POLICY AT WORK:
MULTIPLIER EFFECTS
CHAPTER 9: The Government
and Fiscal Policy
TABLE 9.3 Finding Equilibrium After a $200-Billion Balanced-Budget Increase
in G and T (All Figures in Billions of Dollars; Both G and T Have
Increased from 100 in Table 9.1 to 300 Here)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
OUTPUT
(INCOME)
Y
NET
TAXES
T
DISPOSABLE
INCOME
Yd / Y  T
CONSUMPTION
SPENDING
(C = 100 + .75 Yd)
PLANNED
AGGREGATE
EXPENDITURE
C+I+G
UNPLANNED
INVENTORY
CHANGE
Y  (C + I + G)
ADJUSTMENT
TO
DISEQUILIBRIUM
500
300
200
250
100
300
650
 150
Output8
700
300
400
400
100
300
800
 100
Output8
900
300
600
550
100
300
950
 50
Output8
1,100
300
800
700
100
300
1,100
0
1,300
300
1,000
850
100
300
1,250
+ 50
Output9
1,500
300
1,200
1,000
100
300
1,400
+ 100
Output9
PLANNED
INVESTMENT GOVERNMENT
SPENDING
PURCHASES
I
G
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Equilibrium
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CHAPTER 9: The Government
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What happens when there is a simultaneous
increase in government spending of $100 and
a lump-sum tax of $100?
a. Equilibrium income would increase by $100,
or the amount of increase in G.
b. Equilibrium income would decrease by $100,
or the amount of increase in T.
c. Equilibrium income would decrease by $200,
or double the amount of the increase in T.
d. Nothing happens. Equilibrium income
remains the same because the amount of
government spending (G) is compensated by
the amount of taxation (T).
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CHAPTER 9: The Government
and Fiscal Policy
What happens when there is a simultaneous
increase in government spending of $100 and
a lump-sum tax of $100?
a. Equilibrium income would increase by
$100, or the amount of increase in G.
b. Equilibrium income would decrease by $100,
or the amount of increase in T.
c. Equilibrium income would decrease by $200,
or double the amount of the increase in T.
d. Nothing happens. Equilibrium income
remains the same because the amount of
government spending (G) is compensated by
the amount of taxation (T).
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FISCAL POLICY AT WORK:
MULTIPLIER EFFECTS
CHAPTER 9: The Government
and Fiscal Policy
TABLE 9.4 Summary of Fiscal Policy Multipliers
POLICY STIMULUS
MULTIPLIER
Governmentspending
multiplier
Increase or decrease in the
level of government
purchases:  G
1
MPS
Tax multiplier
Increase or decrease in the
level of net taxes:  T
 MPC
MPS
Balancedbudget
multiplier
Simultaneous balanced-budget
increase or decrease in the
level of government purchases
and net taxes:  G   T
FINAL IMPACT ON
EQUILIBRIUM Y
G
T 
1
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1
MPS
 MPC
MPS
G
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CHAPTER 9: The Government
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THE FEDERAL BUDGET
federal budget The budget of the
federal government.
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THE FEDERAL BUDGET
THE BUDGET
CHAPTER 9: The Government
and Fiscal Policy
TABLE 9.5 Federal Government Receipts and Expenditures, 2004 (Billions of
Dollars)
AMOUNT
Receipts
Personal income taxes
Excise taxes and custom duties
Corporate income taxes
Taxes from the rest of the world
Contributions for social insurance
Interest receipts and rents and royalties
Current transfer receipts from business and persons
Current surplus of government enterprises
Total
Current Expenditures
Consumption expenditures
Transfer payments to persons
Transfer payments to the rest of the world
Grants-in-aid to state and local governments
Interest payments
Subsidies
Total
PERCENTAGE
OF TOTAL
801.8
94.0
217.4
9.2
802.5
21.9
28.6
− 0.5
1,974.8
40.6
4.8
11.0
0.5
40.6
1.1
1.4
0.0
100.0
725.7
1,014.0
28.9
348.3
221.5
43.0
2,381.3
30.5
42.6
1.2
14.6
9.3
1.8
100.0
Net federal government saving—surplus (+) or deficit (−)
(total current receipts − total current expenditures)
− 406.5
Source: U.S. Department of Commerce, Bureau of Economic Analysis.
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THE FEDERAL BUDGET
CHAPTER 9: The Government
and Fiscal Policy
THE SURPLUS OR DEFICIT
federal surplus (+) or deficit (−) Federal
government receipts minus expenditures.
FIGURE 9.4 The Federal Government Surplus (+) or Deficit (−) as a
Percentage of GDP, 1970 I–2005 II
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CHAPTER 9: The Government
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The federal budget can be conceived as:
a. A political document that dispenses favors to
some groups and places burdens on others.
b. A reflection of goals the government wants to
achieve.
c. An embodiment of some beliefs about how (if
at all) the government should manage the
macroeconomy.
d. All of the above.
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CHAPTER 9: The Government
and Fiscal Policy
The federal budget can be conceived as:
a. A political document that dispenses favors to
some groups and places burdens on others.
b. A reflection of goals the government wants to
achieve.
c. An embodiment of some beliefs about how (if
at all) the government should manage the
macroeconomy.
d. All of the above.
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THE FEDERAL BUDGET
CHAPTER 9: The Government
and Fiscal Policy
THE DEBT
federal debt The total amount owed
by the federal government.
privately held federal debt The
privately held (nongovernmentowned) debt of the U.S. government.
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CHAPTER 9: The Government
and Fiscal Policy
THE FEDERAL BUDGET
FIGURE 9.5 The Federal Government Debt as a Percentage of GDP, 1970 I–2005 II
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THE ECONOMY’S INFLUENCE
ON THE GOVERNMENT BUDGET
CHAPTER 9: The Government
and Fiscal Policy
TAX REVENUES DEPEND ON THE STATE OF THE
ECONOMY
Tax revenue depends on taxable income, and
income depends on the state of the economy,
which the government does not control.
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CHAPTER 9: The Government
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After a large deficit buildup in the in the 1980s,
the federal government deficit:
a. Continued to worsen steadily throughout the
1990s and into the 2000s.
b. Turned into a surplus during the two Clinton
administrations.
c. Was vastly diminished during the G.W. Bush
administration.
d. Was an even larger deficit, as a percent of
GDP, in 2003 than it was in 1983.
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CHAPTER 9: The Government
and Fiscal Policy
After a large deficit buildup in the in the 1980s,
the federal government deficit:
a. Continued to worsen steadily throughout the
1990s and into the 2000s.
b. Turned into a surplus during the two
Clinton administrations.
c. Was vastly diminished during the G.W. Bush
administration.
d. Was an even larger deficit, as a percent of
GDP, in 2003 than it was in 1983.
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THE ECONOMY’S INFLUENCE
ON THE GOVERNMENT BUDGET
CHAPTER 9: The Government
and Fiscal Policy
SOME GOVERNMENT EXPENDITURES DEPEND
ON THE STATE OF THE ECONOMY
Transfer payments tend to go down
automatically during an expansion.
Inflation often picks up when the economy
is expanding. This can lead the government
to spend more than it had planned to
spend.
Any change in the interest rate changes
government interest payments.
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THE ECONOMY’S INFLUENCE
ON THE GOVERNMENT BUDGET
CHAPTER 9: The Government
and Fiscal Policy
AUTOMATIC STABILIZERS
automatic stabilizers Revenue
and expenditure items in the federal
budget that automatically change
with the state of the economy in
such a way as to stabilize GDP.
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CHAPTER 9: The Government
and Fiscal Policy
Which of the following statements is correct about
the government’s control over its budget?
a. The government has complete control over
the revenue side of the budget, but not
complete control over the expenditure side.
b. The government has complete control over
the expenditure side of the budget, but not
complete control over the revenue side.
c. The government does not have complete
control of either the revenue side or the
expenditure side of the budget.
d. The size of the government budget, and
whether it is in surplus or deficit, is controlled
entirely by Congress, not by the economy.
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CHAPTER 9: The Government
and Fiscal Policy
Which of the following statements is correct about
the government’s control over its budget?
a. The government has complete control over
the revenue side of the budget, but not
complete control over the expenditure side.
b. The government has complete control over
the expenditure side of the budget, but not
complete control over the revenue side.
c. The government does not have complete
control of either the revenue side or the
expenditure side of the budget.
d. The size of the government budget, and
whether it is in surplus or deficit, is controlled
entirely by Congress, not by the economy.
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THE ECONOMY’S INFLUENCE
ON THE GOVERNMENT BUDGET
CHAPTER 9: The Government
and Fiscal Policy
FISCAL DRAG
fiscal drag The negative effect on
the economy that occurs when
average tax rates increase because
taxpayers have moved into higher
income brackets during an
expansion.
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THE ECONOMY’S INFLUENCE
ON THE GOVERNMENT BUDGET
CHAPTER 9: The Government
and Fiscal Policy
FULL-EMPLOYMENT BUDGET
full-employment budget What the
federal budget would be if the
economy were producing at a fullemployment level of output.
structural deficit The deficit that
remains at full employment.
cyclical deficit The deficit that
occurs because of a downturn in the
business cycle.
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CHAPTER 9: The Government
and Fiscal Policy
When the economy reaches full employment, the
budget deficit is:
a. A combination of cyclical and structural
deficits.
b. Zero.
c. Equal to the cyclical deficit.
d. Equal to the structural deficit.
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CHAPTER 9: The Government
and Fiscal Policy
When the economy reaches full employment, the
budget deficit is:
a. A combination of cyclical and structural
deficits.
b. Zero.
c. Equal to the cyclical deficit.
d. Equal to the structural deficit.
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CHAPTER 9: The Government
and Fiscal Policy
REVIEW TERMS AND CONCEPTS
automatic stabilizers
balanced-budget
multiplier
budget deficit
cyclical deficit
discretionary fiscal policy
disposable, or after-tax,
income (Yd)
federal budget
federal debt
federal surplus (+) or
deficit (−)
fiscal drag
fiscal policy
full-employment budget
government spending
multiplier
monetary policy
net taxes (T)
privately held federal debt
structural deficit
tax multiplier
1. Disposable income Yd ≡ Y − T
2. AE ≡ C + I + G
3. Government budget deficit ≡ G − T
4. Equilibrium in an economy with
government: Y = C + I + G
5. Leakages/injections approach to
equilibrium in an economy with
government: S + T = I + G
1
6. Government spending multiplier ≡
MPS
7. Tax multiplier ≡ 
MPC
MPS
8. Balanced-budget multiplier ≡ 1
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Appendix A
CHAPTER 9: The Government
and Fiscal Policy
DERIVING THE FISCAL POLICY MULTIPLIERS
THE GOVERNMENT SPENDING AND TAX
MULTIPLIERS
C  a  b(Y  T )
Y  C I  G
Y  a  b(Y  T )  I  G
Y  a  bY  bT  I  G
Y  bY  a  I  G  bT
Y (1  b)  a  I  G  bT
1
Y 
(a  I  G  bT )
1  b
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Appendix A
CHAPTER 9: The Government
and Fiscal Policy
THE BALANCED-BUDGET MULTIPLIER
The balanced-budget multiplier is found by combining
the effects of government spending and taxes:
increase in spending:
- decrease in spending:
= net increase in spending
G
C  T ( MPC )
G  T ( MPC )
In a balanced-budget increase, ΔG = ΔT, so we can
substitute:
net initial increase in spending:
ΔG − ΔG (MPC) = ΔG (1 − MPC)
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Appendix A
CHAPTER 9: The Government
and Fiscal Policy
Because MPS = (1 − MPC), the net initial increase in
spending is:
ΔG (MPS)
 1 
We can now apply the expenditure multiplier 

MPS


to this net initial increase in spending:
 1 
Y  G ( MPS ) 
  G
 MPS 
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Appendix B
CHAPTER 9: The Government
and Fiscal Policy
THE CASE IN WHICH TAX REVENUES DEPEND
ON INCOME
Yd  Y  T
Yd  Y  (200  1 / 3Y )
Yd  Y  200  1 / 3Y
C  100  .75Yd
C  100  .75(Y  200  1 / 3Y )
FIGURE 9B.1 The Tax Function
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CHAPTER 9: The Government
and Fiscal Policy
Appendix B
Y  C  I G
Y  100  .75(Y  200  1 / 3Y )  100
 100

 
I
G
C
Y  100  .75Y  150  .25Y  100  100
Y  450  .5Y
.5Y  450
FIGURE 9B.2 Different Tax Systems
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Appendix B
CHAPTER 9: The Government
and Fiscal Policy
THE GOVERNMENT SPENDING AND TAX
MULTIPLIERS ALGEBRAICALLY
C  a  b(Y  T )
C  a  b(Y  T0  tY )
C  a  bY  bT0  btY
Y  a  bY  bT0  btY  I  G
1
Y
(a  I  G  bT0 )
1  b  bt
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