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Economics 302 Menzie D. ChinnSpring 2012 Social Sciences 7418University of Wisconsin-Madison

The Keynesian Model of Income Determination

This set of notes outlines the Keynesian model of national income determination in closed andopen economy. It then shows how to solve for multipliers.

1. An Expanded Model and Equilibrium

Eq.No. Equation Description

(1) ZY= Output equals aggregate demand, an equilibrium condition(2) IMXGICZ +++ Definition of aggregate demand

(3) Do YccC 1+= Consumption function,1c is the mpc

(4) TYYD Definition of disposable income

(5) YttT 10 += Tax function; 0t is lump sum taxes, 1t is marginal tax rate.

(6) 0bI= Investment function, exogenous

(7) 0GOG = Government spending on goods and services, exoagenous

(8) 0xX = Exports, exogenous

(9) 0mIM= Imports, exogenous

Note: the marginal propensity to consume out of disposable income is)(

1TY

Cc

= .

To simplify matters, for the moment, close the economy so that there are no imports andexports, 000 == mx . Then (8) and (9) are irrelevant.

Substitute (3)-(7) into (2), and substitute (2) into (1):

(10) )()()( 010 oD GObYccZY +++==

)()()))((( 01010 oGObYttYccZY ++++==

Collect up terms:

(11) 011 )1( += YtcY where oGObtcc ++ 00100 )(

Shift Y terms to the left hand side:

(12) 011 )1( = YtcY 011 )]1(1[ = tcY

Divide both sides by the term in the square bracket to obtain equilibrium income, Y0:

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(13) 00 = Y let)]1(1[

1

11 tc =

Where the 0 subscript denotes the equilibrium value of output. Interpretation of (13):equilibrium income is a multiple of the amounts of autonomous spending. The higher the levelof autonomous spending, the higher the equilibrium level of income. Notice also that lump sumtaxes enter in negatively, so the higher lump sum taxes, the lower equilibrium income is.

Figure 1: Equilibrium in the Keynesian Cross

2. Effects of Changes in Autonomous Spending and Multipliers

To think about how changes in autonomous spending the constant in consumption ( 0c ),

government spending ( 0GO ), investment spending ( 0b ), affect equilibrium income, consider a

change of income (Y ) as being attributable to changes in each of those autonomous spending

components. Take equation (13):

(14) = Y

(holding constant the tax rate). Remember, however, that oGObtcc ++ 00100 )( . So if, for

instance, the only autonomous spending component that changes is government spending (so)GO= , then:

Y

Y=Z

Z=c1(1-t1)Y+0

Z

45

Y0

0

Slope = c1(1-t1)

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(15) GOY =

Notice that a similar expression occurs if one holds government spending, lump sum taxes,

exports and imports constant, then:

(16) 0bY =

Returning to (15), the increase in government spending can interpreted in the following figure.

Figure 2: Change in income due to a change in autonomous spending

Note that Y1 = Y0 + Y. You should also understand that 1 = 0 + .

Returning to (15), consider what the change in income for a change in government spending is.That can be obtained by dividing both sides by GO :

(17))1(1

1

11 tcGO

Y

=

Why is the change in income greater than the change in government spending? Consider a $1increase in government spending, and for simplicity set the marginal tax rate (t1) to zero. The $1in spending is income for others; of that $1, $c1 is spent. That spending becomes income forsomebody else, of which (c1)( $c1). Adding up the entire sequence of spending, one obtains:

1

1

3

1

2

111

1...1

ccccc n

=+++++ forn= .

Y

Y=Z

Z=c1(1-t1)Y+0

Z

45

Y0

1

Y1

GOY =

Z=c1(1-t1)Y+1

0

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See if you can solve for the lump sum tax multiplier. How does it differ from the governmentspending multiplier?

3. An Alternative Approach

(18) CTYCYS D

In a closed economy:

(19) GICY ++

Bringing C to the left hand side, and subtracting T from both sides:

(20) TGICTY +

Notice the left hand side of (20) is the same as (18)

(21a) TGIS + (21b) )( GTSI +

The term in the parentheses is the government budget balance. (21b) indicates investment equalsthe sum of private saving and public saving.

An interesting aspect of the equilibrium condition that saving equals investment is Paradox of

Thrift, which states that individual consumers attempts to increase saving will end updecreasing overall saving. To see this, assume no government sector (so T, G both equal zero),and restate (18):

(22) )( 10 YccYCYS +=

(23) YccYccYS +== )1( 1010

An increase in saving is represented by a decrease in autonomous consumption 00

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(25) 000 =+= ccS

In other words, aggregate saving remains unchanged, despite the fact that individual householdsattempt to save more. At the same time, output is lower.

4. The Open Economy Multiplier

One important point to realize is that there are different multipliers corresponding to differentvariables changing (consider the government spending vs. tax lump multipliers). Moreimportantly, the multiplier for the same variable will be different if the modelchanges. In theprevious section, net exports depended upon an exogenously determined amount. Suppose weassume net exports, the difference between exports and imports, depend upon income. Inparticular, suppose when income rises, consumption rises, and some of that consumption falls onimported goods. Hence, imports (which enter in negatively in the calculation of net exports) rise

with income. This assumption can be incorporated into the export equation thus:

(9) YmmIM 10 +=

Note thatY

importsm

=1 . If one goes through the same steps as in Section 1, one obtains:

(26) )()()()()( 100010 YmmxGObYccZY oD +++++==

(27) )()()()()))((( 00011010 mxGObYmYttYccZY o ++++++==

Solving for equilibrium, one obtains:

(28) 0111

0)1(1

1

+=

mtcY where 0000100 )( mxGObtcc o +++

In this case the, government spending multiplier is:

(29)111 )1(1

1

mtcGO

Y

+=

In this open economy model, the government spending multiplier is less than that in the closedeconomy.

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