the multiplier, the paradox of thrift, savings and investment, fiscal policy,

and the tax multiplier

The Keynesian Model

multiplier – algebra of the model

A simple Keynesian model of the economy with no government or foreign trade can be represented as:

Y = C + I (1)where Y is equilibrium output (income), C is

aggregate consumption, and I is aggregate investment. Aggregate consumption, or total expenditure by households on final goods and services, is determined by autonomous consumption (a), or the rate of consumer expenditure independent of disposable income, and the marginal propensity to consume (b = mpc), which is the part of each additional dollar of disposable income that is spent on consumption. Thus, the consumption function is:

C = a + bY (2)

No government, so Y = YdNote that since there is no government,

taxes are zero, so Y = Yd, since:Yd = Y – TYd = Y – 0Yd = Y

Thus while the consumption function is usually C = a + bYd, here it will simply be:

C = a + bY

InvestmentInvestment is determined by a complex of

factors such as expectations of investors and lending institutions, business confidence, political climate, and so on. For present purposes, what is important is that investment is autonomous—that is, independent—of income. It is also not a simple function of the rate of interest, as in neoclassical loanable funds theory.

solving the equationY = C + I (1)C = a + bY (2)

Substituting equation (2) into equation (1), i.e., replace C with a + bY, we get:

Y = a + bY + I (3)Subtracting bY from both sides: Y - bY = a + I (4)

Solving for YY - bY = a + I (4)

Factoring out the Y from the left hand side of equation (4)

Y (1 - b) = a + I (5)Dividing both sides by (1 - b): Ye = 1 (a + I) (6)

(1-b)

the multiplier

Ye = 1 (a + I)(1-b)

where 1/(1-b)—or 1 divided by the mps—is the multiplier, or the feedback mechanism that amplifies any initial increase (injection) or decrease (withdrawal) in aggregate demand. Therefore, Ye, the equilibrium level of output (income) is determined by the multiplier and total injections. The total injections in this simple model are a + I.

Keynesian model - numerical exampleGiven:

Y = C + IC = a + bY

Where:a = 100b = .75I = 300

Solving for YeY = 100 + .75Y + 300 Y - .75Y = 100 + 300 Y (1 - .75) = 100 + 300 Y (.25) = 100 + 300 Y = (1/.25) x 400 Y = 4 x 400 Ye = $1600 billion

solving for equilibrium consumption and savingsOnce we have Ye, we can find Ce and Se:Ce = a + bYeCe = 100 + .75 (1600)Ce = 100 + 1200 = 1300Se = -a + (1 - b)YeSe = -100 + (1 - .75) 1600Se = -100 + .25 (1600) = 300

double-checking savings Also:

Ye = Ce + SeYe – Ce = Se1600 – 1300 = 300

Also:Se = I (savings = investment at equilibrium)300 = 300

Keynesian Model

Expenditure

Y

45°

C = 100 + .75Y

0

a

AS = C + I

S = - 100 + (1 – .75) Y

I = 300

Y*Y1

a + I

- a

I

1600400

the recessionary gapAssume that this economy, if producing at

full employment, given resources and technology, could produce an aggregate output of $2000.

We can now calculate the values of consumption and savings at full employment, as well as aggregate spneding at full employment, and the recessionary gap.

full employment and the recessionary gap

Cf = a + bYf = 100 + .75 (2000) = $1600Sf = -a + (1 - b)Yf = -100 + .25 (2000) = 400(double-check: Yf – Cf = Sf = 2000 – 1600 = 400)AS@Yf = Cf + I = 1600 + 300 = 1900gap = Yf – AS@Yf = (2000 – 1900) = 100 =

= (Sf – I) = (400 – 300) = = (Yf – Ye)/multiplier = (2000 – 1600)/4 = 100

Keynesian Model

Expenditure

Y

45°

C = a + bY

0

a

AS = C + I

S = - a + (1 – b) Y

I

YfY*Y1

a + I

- a

I

1600 2000

the paradox of thriftAn attempt by the economy as a whole to

increase aggregate savings not only will not succeed, but may lower aggregate output, income and employment. This is because increased savings at a given level of aggregate income will mean decreased consumption. Thus a smaller marginal propensity to consume will reduce the stimulative effects of investment and other spending.

paradox of thriftFor example, suppose an economy is

characterized by a consumption function: C = $100 + .8YdIf autonomous investment is equal to $300

billion then the equilibrium level of output and income is

Ye = 5 (100 + 300) = $2000 billionbecause the multiplier = 1/(1 – b)

= 1/(1 - .8) = 5.

paradox of thriftAggregate consumption is: C = $100 + .8 ($2000) = $1700

billion and aggregate savings is: S = -$100 + (1 - .8) ($2000) = $300

bil.So aggregate savings equals aggregate

investment ($300 billion).

paradox of thrift - exampleSuppose some political and or business

leaders come out and say we have to save more so the economy can grow. If people comply in such a way that the mps rises from .2 to .25, what will be the effect?

paradox of thriftThe new consumption function will be: C = $100 + .75YdWith $300 billion in investment, the new equilibrium

will be: Ye = 4 (100 + 300) = $1600 billionbecause the new multiplier = 1/(1 - .75) = 4.

Aggregate consumption is now: C = $100 + .75 ($1600) = $1300 billionand savings: S = -$100 + (1 - .75) ($1600) = $300 billion

paradox of thriftThus, savings is still equal to investment at

the same level of $300, but output and employment are much lower.

So the attempt by the economy as a whole to save more not only did not result in more savings, but actually lowered aggregate output and income by $400 billion.

paradox of thrift This is the paradox of thrift, and is another

example of the paradoxical nature of macroeconomics. It is rooted in the two-sided nature of spending and saving. When we just look at one individual firm or household in isolation, we don't see the impact that our actions have on other participants in the economy due to the interdependent nature of economic activity. So while for any one individual, it is wonderful to save more, for the economy as a whole, it could be a disaster.

paradox of thriftIf, however, the increased saving is the

result of higher incomes, then that is a different story. If income goes up, consumption and saving both go up. But at a given level of income, increased aggregate savings can throw the economy into a recession. Therefore, a policy to increase growth by increasing savings has it backwards: savings will increase as a result of growth.

Keynes’s critique of the neoclassical theory of savings and investment

1. In Keynes, since consumption is a function of disposable income, and saving is income not spent, saving is also primarily a function of disposable income. S is a passive residual, determined by disposable income and the marginal propensity to consume. Keynes did not believe it was legitimate to hold income constant when analyzing aggregate saving, as in neoclassical theory. He also disagreed with the neoclassical belief that saving is primarily a function of the rate of interest.

Loanable Funds Market

Interest Rate

S, I

S (Savings)

I (Investment)

i*

S = I0

Keynes’s critique of the neoclassical theory of savings and investment

2. Historical experience of the Great Depression:

interest rates very low, no investment;

wages low, no labor demand; how long is the long run?.

3. S = I is the macroeconomic equilibrium condition in both Keynes and neoclassical, but in Keynes I => S through changes in Y and in neoclassical S => I through changes in i. In addition, in Keynes the two may be equal at a whole range of potential levels of output and income, only one of which is full employment, while in neoclassical the two may be equal only at full employment.

4. Keynes did not believe it was legitimate to hold the state of investor expectations constant in analyzing aggregate investment, as in neoclassical theory. He also disagreed with the neoclassical view that investment is primarily a function of the rate of interest. Expected profitability of investors and lending institutions both required for investment to take place.

5. Keynes distinguished between risk, which is calculable, and uncertainty, which is not conducive to statistical probability. He believed most important determinants of investment described by uncertainty, not risk. In neoclassical theory, uncertainty in this sense is not recognized. Also, even under risk, the confidence of whether one will ‘beat the odds’ is subject to unpredictable variation. Mass psychology subject to waves of optimism and pessimism.

6. Business and political climate will influence investment decisions, as will many other factors, not all of which appear immediately relevant, at least on the surface.

7. In a modern capitalist economy with high-tech financial institutions and advanced instruments of credit, a ‘pool’ of savings is not necessary to finance investment. Banks are private, profit-maximizing institutions and will not pass up the chance to make profits if they believe a loan will be profitable. They will always make a loan and worry about reserve requirements at the end of the day (often borrowing themselves to meet their requirements).

8. In Keynes, the rate of interest is not determined by savings and investment, but by the supply and demand for money. This is Keynes’s liquidity preference theory (more on this later).

9. Separation of ownership and management means those who own do not necessarily know the business well, and those who manage may have different interests and incentives than if they also owned. Makes investment more unstable.

10. Speed of asset revaluation increasingly faster and faster. Assets are revalued within the space of seconds, and ability to react immediately, without having to wait to see if a change is a temporary deviation, creates instability. Self-fulfilling prophecies become a characteristic of the system (for example, people think an asset’s value is going to go down, so they sell and because people sell, the value goes down).

fiscal policy for full employment: eliminating a recessionary gapKeynes’s demonstration of the possibility of

the economy being in macroequilibrium, with S = I, below full employment provides a theoretical justification for more interventionist policies by the government.

Fiscal policy: the attempt to affect macroeconomic variables (such as C, I, Y) through government spending and tax policies.

Increasing GIf Yf = $2000 billion and Ye is $1600, how

much does the government have to increase spending to push the economy to Yf?

Increasing GIf Yf = $2000 billion and Ye is $1600, how much

does the government have to increase spending to push the economy to Yf? Not $400.

Increasing GIf Yf = $2000 billion and Ye is $1600, how much

does the government have to increase spending to push the economy to Yf? Not $400. If G increased by $400 then:

Y = C + I + GC = a + bY

Increasing GIf Yf = $2000 billion and Ye is $1600, how much

does the government have to increase spending to push the economy to Yf? Not $400. If G increased by $400 then:

Y = C + I + GC = a + bY

Ye = 1/(1 - .75) * 100 + 300 + 400= 4 (800)= $3200

Way past Yf—impossible, so inflation will occur. What happened?

closing the recessionary gapThe government spending of 400, like all other

autonomous expenditures, had a multiplier effect, in this case of 4, and so increased total output and income not by 400 but by 1600.

How much do we need to increase G by to just get the economy to full employment?

By the size of the gap, or the amount we need to increase total spending (Yf – Ye) divided by 4.

gap = Yf – AS@Yf = Sf – I = (Yf – Ye)/mult. = 100

Keynesian Model

Expenditure

Y

45°

C

0

a

AS = C + I

S = - a + (1 – b) Y

I

YfY*Y1

I + G

a + I

- a

I

I + G

C + I + G

a + I + G

full employmentNotice that at Yf, S = I + G; (I + G can be thought of

as private and public investment)Notice the aggregate spending function with

government (still no foreign trade), or the C + I + G line:

has a y-intercept of (a + I + G);has a slope = mpc; andintersects the 45 degree line at Yf

Keynesian Model

Expenditure

Y

45°

C

0

a

AS = C + I

S = - a + (1 – b) Y

I

YfY*Y1

I + G

a + I

- a

I

I + G

C + I + G

a + I + G

Multiplier with TaxesLet's add taxes and government spending into

the multiplier formula!First, we begin with:Y = C + I + G (1) Then we take our consumption function:C = a + bYd (2) Only now we have to account for the fact that Y

and Yd are not equalYd = Y - T (3)

Multiplier with TaxesYd = Y - T (3)because disposable income is aggregate income

less taxes. Since taxes can be determined by the tax rate

times aggregate income:T = tY (4)Then:Yd = Y - tY (5)

Multiplier with TaxesSubstituting equation (5) into the

consumption function:C = a + b(Y - tY) (6)And substituting equation (6) into equation

(1):Y = a + b(Y - tY) + I + G (7)

Y = a + b(Y - tY) + I + G (7)

We then solve for Y: 

Y = a + bY - btY + I + G (8)

Y - bY + btY = a + I + G (9) Y(1 - b + bt) = a + I + G (10)

Y(1 - b + bt) = a + I + G (10)

Y = 1 * (a + I + G) (11) 1- b + bt

 So the multiplier with taxes is:1/(1 - b + bt) (12) And the multiplier times total injections (a + I + G)

will give us the equilibrium level of output and income.

Multiplier with taxes – numerical exampleGiven:C = $100 + .8YdI = $50G = $350t = .25Find: Ye, value of mult., T, Yd, C, & SDoes S = I? Why or Why Not?

Multiplier with taxes – numerical exampleYe = 1/(1-b+b[t]) (a + I + G )

= 1/.4 (100 + 50 + 350) = 2.5 (500)= 1250

Multiplier with taxes – numerical exampleValue of the multiplier = 1/(1-b+b[t]) = 1/(1-.8+.8[.25]) = 1/.4 = 2.5

[since (.8 x .25) =.2]

Multiplier with taxes – numerical example T = .25 (1250) = 312.50 Yd = Y – T = 1250 – 312.50 = 937.50 C = a + bYd = 100 + .8(937.50)

= 100 + 750 = 850 S = -a + (1-b)Yd = -100 + (1-.8)937.50

= -100 + 187.50 = 87.50

Multiplier with taxes – numerical exampleDoes S = I? Why or why not?

Multiplier with taxes – numerical exampleDoes S = I? Why or why not?No, total injections = total withdrawals

I + G = S + T = 50 + 350 = 87.50 + 312.50 = 400 = 400

Multiplier with taxes – numerical exampleDoes S = I? Why or why not?

I + G = S + T = 50 + 350 = 87.50 + 312.50 = 400 = 400and the: public deficit = private sector surplusG – T = S – I = 350 – 312.50 = 87.50 – 50

= 37.50 = 37.50