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

_____ $$ 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 $$I_t equal to $$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 $$k = 10 and $$Y_t-1 = 3000. Suppose in period 1 $$I₀ shifts from $$I₀ to $$I_oo. For example suppose in period 1 $$I_t 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 + 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

Quiz

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