### Macroeconomic Models

I. The two Keynesian models presented herein are:

a. Two equation static

b. Two equation dynamic

These two models deal exclusively with the real side of the economy, that is with real variables. The monetary effects of policy are ignored. These models are simplifications of actual economic conditions. Their purpose is primarily to illustrate economic concepts to students.

II. Two equation model.

_____ variables:

_____ C consumption

_____ Y real GNP

_____ I Investment

_____  a,b known coefficients

equations: _____  Y = C + I _____ (1)

_____________  C = a + bY _____(2)

Note: The first equation is an identity, that is it is true by definition and the second is the Keynesian consumption function. In this model the variables Y and C are endogenous, the variable I is exogenous, and the coefficients a and b are known. The purpose of the model is to explain the level of Y and C as a function of the level of I model solution

Substitute (2) into (1)

_____ Y = a + bY + I

Subtract bY from both sides

_____ Y - bY = a + I

Noting that Y = 1Y and a + I = 1 (a + Y)

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

Dividing both sides by (1 - b)

_____ Y = k(a + I) where k = 1/(1 - b)

example: a = 100, b = .75 and I = 300 what is Y?

solution: k = 1/(1-.75) = 1/.25 = 100/25 = 4  Y = 4(100 + 300) = 1600

Effect of a change in level of investment: What is the effect of an increase in the investment from I0 to I1?

_____  Y1 = k(a + I1)

_____  Y0 = k(a + I0)
___ _________________

_____  Þ(Y) = kÞ(I)

where Þ(Y) = Y1-Y0 and Þ(I) = I1-I0

Example: Suppose I increases by 10 then the impact on Y is 10k or 40 for b = .75 . The other question which policy makers are interested is how much would I have to increase to increase Y by 100 (associated with this increase is a decrease in unemployment, an important reelection variable). In this case 100 = Þ(k)I or Þ(I) = 25.

#### Two equation dynamic model

_____  Yt = Ct + It _______ (1)

_____  Ct = a + bYt-1 _____(2)

Note: The subscript t indicates the time period. Currently most econometric models are quarterly. Ct is a flow variable. If the time periods were quarters it measures the amount of consumption which would occur in a year if the level which occurred in the particular quarter occurred all year. The first equation is a dynamic version of the GNP identity. The second equation is a much simplified equation representing the fact there is a lag from the time people earn their income and the time they spend it.

Solution to the dynamic model

Substitute 2 into 1

_____  Yt = a + bYt-1 + It _____(3)

Since the subscripts on the variable Y are not the same we can not solve (3) for Yt. What this model is used for is to indicate the time path which results from a shift in the level of investment. The problem formulation has three parts: the economy is initially in equilibrium with It equal to I0. The initial equilibrium Y0 is found by applying the solution to the static model.

_____  Y0 = k(a + I0)

For example if b = .9, a = 100 and I0 = 200, then k = 10 and Yt-1 = 3000. Suppose in period 1 I0 shifts from I0 to Ioo. For example suppose in period 1 It shifts from 200 to 210 and remains at 210 for all t. The time path is found by organizing 3 into a table:

_____ t _____ a + It_____ bYt-1 _____ Yt
________________________________________

_____  0______ 300 ______ 2700______ 3000

_____  1______ 310______ 2700 ______ 3010

_____  2______ 310______ 2709 ______ 3019

and so on

Note: Column 2 is obtain from column 3 one line up. The final equilibrium is obtained from applying the static formula to the new level of investment. That is

_____  Yoo = k(a + Ioo )

_____  Yoo = 10(310) = 3100