San José State University
Department of Economics

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 Derivation of the Relationship Between the Income Multiplier and the Employment Multiplier

Let DD stand for Direct Demand and ID for Induced Demand. From the definition of the income multiplier kY

#### ID = (kY−1)DD

Now the direct and induced demands have to be converted into jobs. Let aD and aI be, respectively, the proportions of production revenue that goes for labor. Thus aDDD and aIID will be the payments made for labor. The amount of labor secured by these payments will be (aDDD)/wD and (aIID)/wI. Let these amounts of labor employed be denoted by ED and EI. Thus

#### ED = (aDDD)/wD EI = (aIID)/wI

From the definition of the employment multiplier kE

#### EI = (kE−1)ED

After substitution of the expressions for ED and EI the above equation becomes

#### (aIID)/wI = (kE−1)(aDDD/wD)

This equation may be solved for (kE−1) as

#### (kE−1) = (aI/aD)(wD/wI)(ID/DD)

From the expression for ID it follows that (ID/DD) is equal to (kY−1). Therefore

#### (kE−1) = (kY−1)(aI/aD)(wD/wD)

Finally this reduces to