**I.
****The three Keynesian models
we will consider are: **

**II.**** ****Two equation
static model with government expenditures and taxes**** ; ****Equations**** ; ****k = 1/(1 - c) the symbol ¶ means change****;**** ****Example**** ; ****More
example**

**III.**** Tax rate model**** ; ****k**_{1}**=
1/[1 - c + ct)]**** ; ****Example**** ; ****Can ever
increase the tax revenues by decreasing taxes as analysed by the model?**

**IV. ****IS-LM Model**** ; ****IS ****variables****, IS ****equation****(Investment
- Saving) ; LM ****variables****, LM ****equation****(Liquidity
- Money) ; ****Solution
to the IS-LM mode****l ; ****eq. in both real good market
and money market**** ;
Examples : ****Given
Data**** ; ****Qa and Qb**** ; ****Qc**** ; ****Qd**

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**I. ****The three Keynesian models we will consider are:
**

**a. equation with government
and taxes. **

**b. Fractional tax model. **

**c. IS-LM model. **

**The Simple Keynesian Models deal
exclusively with the real side of the economy, that is with real
variables. The monetary effects of policy are ignored. All of
these models are simplifications of actual economic conditions. **

**The last model the ****IS-LM ****model integrates money with the real sector.
This model is used to illustrate Keynesian policy issues .**

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**II.**** static model with government expenditures and
taxes: **

**_____**** Variables : **

**_____**** ****C**** consumption **

**_____**** ****I**** investment **

**_____**** ****G**** government expenditures **

**_____**** ****T**** taxes **

**_____**** ****Y**** real GNP **

**_____**** ****a,c**** known constants **

**Note: In this model the level of government
expenditures, taxes and investment are fixed. **

**The purpose of this model is to study
the fiscal policy options of government, that is the effect of ****G
****and ****T****
on ****Y****
and ****C****.
This model is the simplest model of this type. **

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**_____**** ****Y = C
+ I + G ****________**** (1) **** **

**_____**** ****C = a
+ c(Y - T) ****______**** (2) **** **

**Note: This model is slightly more
realistic than the two equation model in that it contains a
government and consumption is based on disposable income.**

**As models get bigger they attempt to
capture more of the behavior of the economy. These simple models
are solely for instructional purposes. **

**Solution: **

**Substitute ****2****
into ****1**** **

**_____**** ****Y = a
+ c(Y -T) + I + G**** **

**_____**** ****Y = a
+ cY - cT + I + G**** **

**Subtract cY from both sides **

**_____**** ****Y - cY
= a + I - cT + G **** **

**Collect terms **

**_____**** ****(1 -
c)Y = 1(a + I - cT + G) **** **

**_____**** ****Y = k(a
+ I - cT + G) **** **

**Note : In policy work the analyst is interested in considering
the impact of a change in ****G**** (or
a change in autonomous spending) on ****Y****.**** **

**Using the same type of algebra as for the simple two
equation model we can obtain the following equations.****
(****the symbol ¶ means change ****)**

**_____**** ****¶(Y)
= k¶(G) **** **

**_____**** ****¶(Y)
= k¶(I) **** **

**_____**** ****¶(Y)
= k¶(a) **** **

**_____**** ****¶(Y)
= -kc¶(T) **** **

**The first shows the impact of a change
in government expenditures, the second the impact of private
investment, the third shows a shift in consumer confidence, and
the last indicates a shift in tax policy. **

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**The government has direct control over
****G**** and ****T ****and
indirect influence over ****a**** and ****I****through
incentives and its policies. **

**Example : ****_****a = 100, c = 1/2, I = 200, G = 200, T = 200****
**

**Find the equilibrium ****Y**** and
****C****? **

**_____**** ****k = 1/(1
- 1/2) = 1/(1/2) = 2**** **

**_____**** ****Y = 2(100
+ 200 + 200 - 100)**** **

**_____**** Y = 800**** **

**_____**** ****C = 100
+ (1/2)(800 - 200) = 400**** **

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**More example: Suppose the government
wished to raise ****Y**** by ****100**** to reduce unemployment how could it accomplish this objective?
**

**Solution:**

**If the government raised ****G**** by ****50**** it would accomplish the objective ****(100 = 2¶(G))****. **

**However, the government could also lower taxes by ****100
(100 = -2(1/2)¶(T))****. **

**Also, if the government wished to maintain a balanced
budget they could simultaneously raise ****G ****by ****100**** and raise ****T ****by ****100 **

**(100 = 2¶(G) - 2(1/2)¶(T))**** and ****¶(G) = ¶(T)****. **

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**Variables: **

**all the variables for previous model plus ****t**** the tax rate **

**Equations **

**_____**** ****Y = C
+ I + G ****________**** (1)**** **

**_____**** ****C = a
+ c(Y - T) ****______**** (2)**** **

**_____**** ****T = tY
****_______________**** (3) **** **

**Note: This model adds a fixed tax rate
to determine the amount of revenue the government will receive. (When
all the deductions and tax shelters are considered, the effective
tax rate, i.e. the rate that people actually pay, is
approximately constant.) **

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**Substitute ****(3)**** into**** (2)**

**_____**** ****C = a
+ c(Y - tY)**** **

**substitute ****(2)**** into ****(1)**** **

**_____**** ****Y = a
+ c(Y - tY) + I + G**** **

**Transfer ****c(Y - tY)**** from right to left **

**_____**** ****Y-c(Y
- tY) = a + I + G**** **

**Collecting terms **

**_____**** ****[1 -
c(1 - t)]Y = a + I + G**** **

**_____**** ****Y = k**_{1}**
(a + I + G)**** **

**where ****k**_{1}**
= 1/[1 - c(1 - t)]**** or ****k**_{1}**
= 1/[1 - c + ct)]**

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**Example: ****a = 100 I = 200 G = 200 and t = 0.3 c = 6/7****What
is ****Y****? **

**_____**** ****k**_{1}**
= 1/[1 - (6/7)(1 - 0.3)]**** **

**_____**** ****k**_{1}**
= 1/[1 - (6/7)(0.7)]**** **

**_____**** ****k**_{1}**
= 1/(4/10) = 2.5**** **

**_____**** ****Y = 1250****
**

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**Note: The fundamental issue for analyzing
economic policy is can the government ever increase the tax revenues by decreasing
taxes as claimed by the eager supply-siders. What are the tax revenues? ****T = .3(1250) = 375.**** Suppose ****t**** is cut to ****1/6****, what happens to ****T****? **

**_____**** ****k**_{1}**=
1/[1 - (6/7)(5/6)]**** **

**_____**** ****k**_{1}**=
3.5**** **

**_____**** ****Y = 1750****
**

**_____**** ****T = 291.66****
**

**Note: If you have had calculus you will
note that **

**_____**** ****T = tk**_{1}**(a
+ I + G)**** **

**_____**** ****dT/dt
> 0 for 1 > c > t > 0 **** **

**This means that if you decrease taxes
Keynesian theory indicates that tax revenues must fall. **

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**IV. ****IS-LM Model :**** This model integrates the real sector with the
money sector. The model can be used to analyze ****crowding out**** or ****the displacement of private finance by the sale
of government securities****. **

**Variables: **

**_____**** ****Y**** real GNP **

**_____**** ****C**** consumption **

**_____**** ****I**** investment **

**_____**** ****G**** government expenditures **

**_____**** ****T**** taxes **

**_____**** ****r**** interest rate **

**a****, ****c**** , ****I**_{0}** ,and ****b**** are known constants,
and ****b is -ve**

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**_____**** ****Y = C
+ I + G ****________**** (1) **** **

**_____**** ****C = a
+ c(Y - T) ****______**** (2) **** **

**_____**** ****I = I**_{0}**
+ br ****____________**** (3) **** **

**Note: Equation 3 is the investment
function which says that the higher the rate of interest the more
attractive are risk free investments. **

**Thus the amount of private investment
decreases as the interest rate increases. **

**To invest in a private project, the
rate of return must equal the market interest rate plus a premium
for the higher risk. **

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**Variables: **

**_____**** ****M****d**** demand for money **

**_____**** ****M****s**** supply of money **

**_____**** ****p**** price level **

**_____**** ****M****d****/p**** real
demand for money **

**_____**** ****M****s****/p**** real
supply of money **

**_____**** Suppose ****p**** price level constant
and equals to ****1, ****then**

**_____**** ****M****d****/p****=****M****d**** , and ****M****s****/p****=****M****s**** , and **

**_____**** ****d****,**** e**** are known as the ****income elasticity of money demand (+ve value)
****and ****interest elasticity of money demand (-ve value)
****respectively, which are constants .**

**_____**** (****d**** is the Cambridge ****k****) **

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**_____**** ****M****s**** = M****d**** ****__________**** (4) **** **

**_____**** ****M****s**** = dY+ er ****_____ _**** (5) **** **

**Note: Equation****(4)**** is the Keynesian money market equilibrium
equation. The first term of Equation ****(5)**** on the right is the ****transactions demand for money****. The second term is the ****asset or speculative demand for
money****. **

**It can be explained as follows:
individuals hold a portfolio of assets such as money, stocks,
bonds, real estate, etc. **

**The individual holds money in the
portfolio to take advantage of opportunities which may present
themselves. As the interest rate rises the opportunity cost of
holding money increases. Therefore he shifts from money to other
assets with a rising interest rate. **

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**The strategy is to condense ****(1) to (3) ****into a single IS equation and then to solve the
IS and LM equations simultaneously. **

**Substitute (2) into (1) **

**_____**** ****Y = a
+ c(Y - T) + I + G****_____________****(6)**** **

**Substitute (3) into above **

**_____**** ****Y = a
+ c(Y - T) + I**_{0}** + br + G****_________****(7)**** **

**To simplify things, let**

**_____**** ****A = a
+ I**_{0}** + G - cT**** **

**_**** then**** ****Y = cY + A + br****___________ ____ ___ ****(8)**

**The IS curve is **

**_____**** ****Y = k(A
+ br) where k = 1/(1 - c) ****______**** (9) **** **

**To obtain the solution to the entire
model rewrite (9) as **

¡@**_****eq. in real good market****_**** ****Y
- kbr = kA**** **

**=__**** eq. in money market****_ ****dY + er = M****s****__ ____****(10)**** **

**Divide both sides of eqn ****(10) ****by ****e**** and multiply both sides
by ****kb**** **

**from (9)****_**** ¡@¡@****Y
- kbr = kA**** **

**from (10)****____**** ****(dkb/e)Y
+ kbr = (kb/e)M****s**** **

**Adding up****___**** ****[1 +
(dkb/e)]Y = k[A + (b/e)M****s****]**** **

**_____**** ****Y = K[A
+ (b/e)M****s****] ****________________****(11)**** **

**_____**** where ****K = k/[1 + (dkb/e)]**** **

**Or,**

**K = 1/[(1- c)+ (db/e)]**** ,**** K****
is +ve, and ****K < k, as k=1/(1-c)**

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**_____****G = 200 T = 150**** **

**_____**** M****s**** = 100 , a = 100 , c = 2/3**** **

**_____**** I**_{0}**
= 600 , b = - 2500 , d = 0.25**** **

**_____**** and e = - 1250 which implies K = 6/5 or 1.2****
**

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**Qa.**** What is the equilibrium ****Y****? **

**solution:**

**Y = 1.2(100 + 600 + 200 - 100 + (2500/1250)100)**** **

**Y = 1200**** **

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**Qb.**** Suppose to reduce unemployment the government desire to raise
****Y ****by ****48**** what is the required
****¶(G)**** **

**solution:**

**¶(Y) = K****¶****(G)**** **

**48 = 1.2¶(G)**** **

**¶(G) = 40**** **

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**Qc. ****If ****G**** is increased by the
amount of 40 above, how much ****I**** is crowded
out (Assuming price index ****p ****remains
constant)? **

**We must determine the effect of the ****¶(G)**** upon the ****¶(Y)**** first and then
on ****¶(r)**** and finally on ****¶(I)**** **

**Consider (5) **

**_____**** ****d¶(Y)
+ e¶(r) = 0**** if ****M****s**** remains constant, by**

**rearrange terms, ****¶(r) = - (d/e)¶(Y)**** **

**_____**** Now ****¶(I) = b ¶(r)**** from (3) **

**then,****,_$$****¶(I) = $-\; (bd/e)$¶$(Y)$**** **

**_____**** ****= - (2500(0.25)/1250)48****
**

**_____**** ****= -24****
**

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**Qd. ****Now suppose the real money supply ****M****s****is increased to compensate
the crowding out. What is ****¶(M****s****)**** ? **

**We want to increase ****M****s**** enough to lower rate of
interest ****r**** to the
original level. **

**_____**** ****¶(Y)
= k¶(G)**** **

**_________**** ****= 3(40)
= 120**** **

**_____**** ****120 =
K(40 + (b/e)¶(M****s****))**** **

**________**** ****= 48
+ 2.4¶(M****s****)**** **

**_____**** ****¶(M****s****) = 30**** **

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