### Macroeconomic Models II

I. The three Keynesian models we will consider in this section are:

a. Two equation with government and taxes.

b. Fractional tax model.

c. IS-LM model.

The first two models were presented in Model I and the last three are presented herein. The first four 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. Their purpose is primarily to illustrate economic concepts to students. The last model the IS-LM model integrates money with the real sector. This model is used to illustrate policy issues of the Keynesian paradigm.

II. Two equation static model with government expenditures and taxes.

_____ Variables :

_____ C consumption

_____ I investment

_____ G government expenditures

_____ T taxes

_____ Y real GNP

_____ a,b 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.

Equations:

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

_____  C = a + b(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 + b(Y -T) + I + G

_____  Y = a + bY - bT + I + G

Subtract bY from both sides

_____  Y - bY = a + I - bT + G

Collect terms

_____  (1 - b)Y = 1(a + I - bT + G)

_____  Y = k(a + I - bT + G)

where

_____ k = 1/(1 - b)

Note: The solution to this model has the same form as the simple two equation model. In policy work the analyst is interested in considering the impact of a change in G of T. Using the same type of algebra as for the simple two equation model we can obtain the following equations.

_____  Þ(Y) = kÞ(G)

_____  Þ(Y) = kÞ(I)

_____  Þ(Y) = kÞ(a)

_____  Þ(Y) = -kbÞ(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. The government has direct control over G and T and indirect influence over a and I through incentives and its policies.

Example: _____ a = 100, b = 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

Suppose the government wished to raise Y by 100 to reduce unemployment how could it accomplish this objective? If the government raised G by 50 it would accomplish the objective (100 = 2Þ(G)). The government could also lower taxes by 100 (100 = -2(1/2)Þ(T)). If the government wished to maintain a balanced budget they could simultaneously raise G by 100 and raise T by 100 (100 = 2Þ(G) - 2(1/2)Þ(T)) and Þ(G) = Þ(T).

III. Tax rate model

Variables:

all the variables for previous model plus t the tax rate

Equations

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

_____  C = a + b(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, the rate people actually pay, is approximately constant.

Solution: Remember the principle of substituting up the stack of equations.

Substitute 3 into 2

_____ C = a + b(Y - tY)

substitute (2) into (1)

_____  Y = a + b(Y - tY) + I + G

Transfer b(Y - tY) from right to left

_____ Y-b(Y - tY) = a + I + G

Collecting terms

_____ (1 - b(1 - t))Y = a + I + G

_____  Y = k1 (a + I + G)

where

_____ k1 = 1/[1 - b(1 - t)]

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

_____ k1 = 1/[1 - (6/7)(1-.3)]

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

_____  k1 = 1/(4/10) = 2.5

_____  Y = 1250

Note: The fundamental issue for analyzing Reagan's economic policy is can the government ever increase the tax revenues by decreasing taxes as claimed by the devout 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 > b > t > 0

This means that if you decrease taxes Keynesian theory indicates that tax revenues must fall. (You do not need to know any calculus to provide an excellent answer to the test question on the subject. You do need to integrate the math with the opinions of both the supply sider and the fiscal conservatives.)

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

_____ i interest rate

_____ a, b , I0 ,and c are known constants

Equations

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

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

_____  I = I0 - ci ____________ (3)

Note: Equation 3 is the investment function which says that the higher the rate of interest the more attractive are risk free government securities versus risky private 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.

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

_____ e,f are known constants
_____ (e is the Cambridge k)

Equations:

_____  Ms = Md __________ (4)

_____  Ms/p = eY-fi _______ (5)

Note: Equation 4 is the Keynesian money market equilibrium equation . The first term on the right is the transactions demand for money. The second term is the speculative demand for money which 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.

Solution to the IS-LM model

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

Substitute (2) into (1)

_____ Y = a + b(Y - T) + I + G_____________(6)

Substitute (3) into above

_____ Y = a + b(Y - T) + I0 - ci + G_________(7)

To simplify things define

_____ A = a + I0 + G - bT

_____ Y = bY + A - ci_____________________(8)

The IS curve is

_____  Y = k(A - ci) where k = 1/(1 - b) ______ (9)

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

_____ Y + kci = kA

_____ eY - fi = Ms/p______________________(10)

Divide both sides of 2nd eqn by f and multiply both sides by kc

_____ Y + kci = kA

_____  (ekc/f)Y - kci = (kc/f)Ms/p

_____ [1 + (ekc/f)]Y = k[A + (c/f)Ms/p]

_____  Y = K[A + (c/f)Ms/p] ________________(11)

_____ where K = k/[1 + (ekc/f)]

Example: Given

_____G = 200 T = 150

_____ Ms = 100 p = 1.0 a = 100 b = 2/3

_____ I0 = 600 c = 2500 e = 0.25

_____ and f = 1250 which implies K = 6/5 or 1.2

a. What is the equilibrium Y?

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

 Y = 1200

b. Suppose to reduce unemployment the government desire to raise Y by 48 what is the required Þ(G)

 Þ(Y) = KÞ(G)

48 = 1.2Þ(G)

 Þ(G) = 40

c. If G is increased by the amount in b above, how much I is crowded out (Assuming p remains constant)?

We must determine the effect of the Þ(G) upon the Þ(Y) first and then on Þ(i) and finally on Þ(I)

Consider (5)

_____ eÞ(Y) - fÞ(i) = 0 if Ms/p remains constant

_____ Þ(i)=(e/f)Þ(Y)

_____ Now Þ(I) = -cÞ(i) from (3)

_____  -(ce/f)Þ(Y)

_____  = -(2500(.25)/1250)48

_____  = -24

d. Now suppose the real money supply Ms/p is increased to compensate the crowding out. What is Þ(Ms)/p?

We want to increase Ms/p enough to lower i to the original level.

_____ Þ(Y) = kÞ(G)

_________  = 3(40) = 120

_____  120 = K(40 + (c/f)Þ(Ms)/p)

________  = 48 + 2.4Þ(Ms)/p

_____  Þ(Ms)/p = 30