Question

ISLM-Model

Y=C(Y-T)+I(r)+G

How can I analytically (and step by step) derive from this equation that the IS curve slopes downwards? Please explain thoroughly. TIA

Answer 1

The given economic relationship is:

Y=c(Y-T)+I(r)+G

Where,

Y=National Output

c=Marginal Propensity to Consume

T=Taxes

I(r)= Investment function with respect to interest rate, r

G=Government final consumption expenditure

Three are 3 pointers that is to be kept in mind while deriving IS curve and showing that it slopes downward. They are:

1) Investment demand is inversely related to interest rate, that is, lower the interest rate, higher the investment demand

2) When Investment increases, national output increases by multiplier times

3) Following from 1) and 2), interest rate is also inversely related national output

Consider I(r) as I=a-br

This implies,

Y=c(Y-T)+I(r)+G

Y=c(Y-T)+a-br+G

br=cY-cT+a+G-Y

br=(a+G-cT)-(1-c)Y

r= [(a+G-cT)/b]-(1-c)Y

Thus, the equation form of IS curve is: r= [(a+G-cT)/b]-(1-c)Y

r = z0 - z1(Y), where, z0=Intercept = [(a+G-cT)/b]

z1=(1-c)

Given, 0<c<1, z1 >0 , thus from r=z0-z1(Y), one could see that r is inversely related to Y