### Macroeconomic Models : IS-LM

I. The three Keynesian models we will consider are:

IV. IS-LM Model ; IS variables, IS equation(Investment - Saving) ; LM variables, LM equation(Liquidity - Money) ; Solution to the IS-LM model ; 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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Equations:

_____ 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)

where _____ k = 1/(1 - c)

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 Ithrough 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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III. Tax rate model

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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Solution:

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 = k1 (a + I + G)

where k1 = 1/[1 - c(1 - t)] or k1 = 1/[1 - c + ct)]

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

_____ k1 = 1/[1 - (6/7)(1 - 0.3)]

_____ k1 = 1/[1 - (6/7)(0.7)]

_____ k1 = 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?

_____ k1= 1/[1 - (6/7)(5/6)]

_____ k1= 3.5

_____ Y = 1750

_____ T = 291.66

Note: If you have had calculus you will note that

_____ T = tk1(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.

IS: (Investment - Savings)

Variables:

_____ Y real GNP

_____ C consumption

_____ I investment

_____ G government expenditures

_____ T taxes

_____ r interest rate

a, c , I0 ,and b are known constants, and b is -ve

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Equations

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

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

_____ I = I0 + 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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LM: (Liquidity - Money)

Variables:

_____ Md demand for money

_____ Ms supply of money

_____ p price level

_____ Md/p real demand for money

_____ Ms/p real supply of money

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

_____ Md/p=Md , and Ms/p=Ms , 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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Equations:

_____ Ms = Md __________ (4)

_____ Ms = 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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Solution to the IS-LM model

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) + I0 + br + G_________(7)

To simplify things, let

_____ A = a + I0 + 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 = Ms__ ____(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)Ms

Adding up___ [1 + (dkb/e)]Y = k[A + (b/e)Ms]

_____ Y = K[A + (b/e)Ms] ________________(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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Example: Given

_____G = 200 T = 150

_____ Ms = 100 , a = 100 , c = 2/3

_____ I0 = 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 Ms remains constant, by

rearrange terms, ¶(r) = - (d/e)¶(Y)

_____ Now ¶(I) = b ¶(r) from (3)

then,,_¶(I) = $- \left(bd/e\right)$$\left(Y\right)$

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

_____ = -24

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Qd. Now suppose the real money supply Msis increased to compensate the crowding out. What is ¶(Ms) ?

We want to increase Ms enough to lower rate of interest r to the original level.

_____ ¶(Y) = k¶(G)

_________ = 3(40) = 120

_____ 120 = K(40 + (b/e)¶(Ms))

________ = 48 + 2.4¶(Ms)

_____ ¶(Ms) = 30

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