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 $$**I _{0}** to $$

_____ $$** Y _{1} = k(a + I_{1})**

_____ $$** Y _{0} = k(a + I_{0})**

___ _________________

_____ $$** Þ(Y) = kÞ(I) **

where$$** Þ(Y) = Y _{1}-Y_{0}**
and $$

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**.

_____ $$** C _{t} = a + bY_{t-1} _____(2) **

Note: The subscript $$**t** indicates the time period. Currently most econometric models are quarterly. $$**C _{t}** 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

_____ $$** Y _{t} = a + bY_{t-1} + I_{t} _____(3) **

Since the subscripts on the variable $$**Y** are not the same we can not solve (3) for $$**Y _{t}**. 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 $$

_____ $$** Y _{0} = k(a + I_{0}) **

For example if $$**b = .9, a = 100** and $$**I _{0} = 200**, then $$

_____
$$**t _____ a + I _{t}_____ bY_{t-1} _____ Y_{t}**

________________________________________

_____
$$** 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

_____ $$** Y _{oo} = k(a + I_{oo} ) **

_____ $$** Y _{oo} = 10(310) = 3100 **