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The Keynesian Multiplier

U. Michael Bergman

Department of Economics, University of Copenhagen, Studiestrde 6,

DK1455 Copenhagen K, Denmark

September 14, 2005

Introduction

This note explains the main features of the so called Keynesian multiplier, what we call m in

the textbook. We first derive the multiplier within a model with government consumption

(exogenous demand component) and then we show that m measures the total increase in

aggregate goods demand generated by a unit increase in one exogenous demand component.

In our simple version of the model, this is government consumption. The price level and

the real interest rate are constant throughout our discussion.

Let us first define the relationships that are supposed to hold in the simple Keynesian

model:

AE= C+ I+ G (1.a)

C= C0 + mpc Y (1.b)

I= I0 + mpi I (1.c)

where AE is aggregate expenditures, C is private consumption, I is investment, mpc is the

marginal propensity to consume, mpi is the marginal propensity to invest and Y is income.1

c2004 by U. Michael Bergman. This document may be reproduced for educational purposes, so long

as the copies contain this notice and are retained for personal use or is distributed free.

This lecture note is to be used only in conjunction with the course Makrokonomi 2 given at University

of Copenhagen.1The constants C

0and I

0are immaterial for our discussion so we can assume that these are equal to

zero without affecting anything.

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Figure 1: Aggregate expenditures as a function of income.

Note that mpc = CY

and that mpi = IY

using the notation in the textbook and that the

consumption function in (1.b) is the Keynesian consumptions function defined in equation

(1) on page 3 in chapter 16 in the textbook. Aggregate expenditures AE corresponds to

income (or output) Y in chapter 17 where we discuss aggregate demand.

Let us now define autonomous expenditures as the sum of government consumptionand investment, i.e., A = I

0+C

0+G. Insert this, the consumption function (1.b) and the

investment function (1.c) into (1.a) such that

AE= A + CYY + I

YY. (2)

We can now illustrate the model in the following graph where we measure aggregate ex-

penditures AE on the vertical axis and income (or production) Y on the horizontal axis.

Let us also assume that CY

= mpc = 0.65, IY

= mpi = 0.1 and that A = 20. Then the

relationship between AE and Y can be illustrated in Figure 1.

Note that the slope of the AEline is equal to the partial derivative ofAE in equation

(2) with respect to Y, i.e., the slope is equal to CY

+ IY

= mpc + mpi.

We can now derive the equilibrium solution. In equilibrium AE = Y, i.e., the point

where the AEline crosses the 45degree line. We can also solve for equilibrium output

using equation (2):

AE= Y = A + CYY + I

YY

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Y =

Keynesian multiplier

1

1 (CY + IY)A =

1

1DYA = mA (3)

which implies that Y = 80 in equilibrium (as also can be seen in the graph). Note that

the Keynesian multiplier is exactly identical to the one defined on page 7 in chapter 17.

To see that the Keynesian multiplier measures the effects of a unit change in the exoge-

nous demand component (A in our notation) on income (Y), we take the first difference

of (3) such that

Y = mA

where is the first difference operator. A one unit change in A, i.e., A = 1, leads to m

units increase in Y. In numerical terms this corresponds to 4 units increase in Y as canbe verified if we insert the parameter values

Y = mA =1

1 (0.65 + 0.1)= 4.

Alternatively we can compute the new equilibrium value for income when the exogenous

demand component increases from 20 to 21. The new equilibrium solution is then equal

to 84 (21/0.25 = 84) which implies that income has increased by 4 units when we increase

the exogenous demand component by with one unit.

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