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The Keynesian Model: Output - Principles of Micro - RIT - Dr. Jeffrey Burnette

The Keynesian Model: Output

The Keynesian Model or Short Run Model

Assumptions

• Each market is analyzed on an aggregate scale (One interest rate, One type of labor,One type of output)

Differences from the Classical Model

• The Keynesian model does not assume that markets clear in the short-run. In fact,it believes that prices are sticky downward. That is, prices may not decrease to bringthe market to equilibrium. This is why it focuses on the short-run.

• In the short-run spending depends upon income.

• Whereas the classical model focused on equilibrium in the labor market determining thelevel of output, the Keynesian model will focus on the domestic spending of consumers.

The Keynesian Model w/out Government

• Households have 2 choices for spending their income, Purchase goods and services(Consumption) or Save.

• Firms have 2 types of expenditures, purchase goods and services (Consumption) orpurchase equipment and structures from borrowed funds (Planned Investment).

• In the short-run, planned investment is taken to be independent of income.

• There is a funds market where savings is made available to firms that wish to borrow.

Disposable Income = Income - Taxes

Since there is no government, taxes are equal to zero. Therefore, real income is equal toreal disposable income. Y = YD

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Consumption Function - Algebraic equation that describes the relationship between con-sumption and real (disposable) income. There is a positively sloped relationship betweenreal consumption and spending and real (disposable) income.

C = f(Y )

= a + bY

Where the intercept “a” is called autonomous consumption and the slope “b” is calledthe marginal propensity to consume.

Autonomous Consumption - the part of consumption spending that is independent ofincome. This is the intercept of the consumption function.

Marginal Propensity to Consume (MPC) - The amount by which consumption spend-ing rises when (disposable) income rises by one dollar. This is the slope of the consumptionfunction

MPC =∆C

∆Y(1)

In the above graph, autonomous consumption is 1000 So, when there is no income thetotal level of consumption is $1000. This $1000 must have come out of savings. The marginalpropensity to consume is 0.80. In other words, if given an additional dollar of income therewould be an increase in consumption of $0.80. Likewise, if income decreases by $1 thenconsumption would decrease by $0.80.

The Savings Function- Algebraic equation that describes the relationship between sav-ings and real (disposable) income. There is a positively sloped relationship between realconsumption and spending and real (disposable) income.

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Marginal Propensity to Save (MPS) - The amount by which savings rises when (dis-posable) income rises by one dollar. This is the slope of the savings function.

MPS =∆S

∆Y(2)

Deriving the Savings Function

Since savings by definition is the amount left over income after consumption has takenplace, we can represent savings as a function of income.

Y ≡ C + S

or

S ≡ Y − C (3)

We now have savings written solely as a function of income, because we know thatconsumption is a function of income. If the consumption function is known deriving thesavings function is just a matter of algebra.

S = Y − (a + bY )

= Y − a− bY

= −a + (1− b)Y

Here the intercept is the same magnitude as autonomous consumption but it has adifferent sign. This is because when we have no income and still spend on consumption

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we must have gotten it from savings. The slope of the savings function, the MPS, is (1 - b)or (1 - MPC). This is due to the fact that any additional dollar of real (disposable) incomemust be spent either on consumption or savings.

MPC + MPS = 1 (4)

Example Given the consumption function of C = 1000 + 0.8Y we would like to know thesavings function.

We substitute the consumption function into equation (3). This gives us

S = Y − (1000 + 0.8Y )

= Y − 1000− 0.8Y

= −1000 + 0.2Y

So, we find that when income is equal to $0, consumption is $1000 and savings is -$1000.In addition, the MPC was 0.8 and the MPS is 0.2. So if given an additional dollar of incomewe will spend $0.80 and save $0.20, which of course sums to $1.

Solving for Equilibrium

The Keynesian Model defines equilibrium like the Classical Model. The total amount peoplewant to purchase should be equal to the total amount firms want to produce.

AE = Y

C + I = C + S

I = S

One major difference here is that since we are dealing with the short run, we do notanalyze the funds market. Firms have decided the amount of planned investment basedupon the current interest rate and both are held constant.

In our case we will assume that planned investment is equal to zero.

Solving for equilibrium can be done in one of two ways; Using the consumption functionor using the savings function. In both cases we will achieve the same result due to the factthat C + I = Y simplifies down to

I = S.

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The Consumption Route

To solve for equilibrium using the consumption function we will use C + I = Y. Here wehaven’t broken income into its two components from the equilibrium condition.

Substituting our consumption function, the level of planned investment into our equilib-rium condition and solving we have

C + I = Y1000 + 0.8Y + 0 = Y

-0.8Y -0.8Y1000 = 0.2Y

Y∗ = 5000

We can draw a line on the consumption function graph such that every point along thex-axis, in this case income, is equal to the same point on the y-axis, in this case aggregateexpenditure. Since every point along this line requires that income is equal to aggregateexpenditure this line represents all possible equilibrium points. Every equilibrium point willbe on this line. Due to the fact that this line bisects our graph we call it the 45◦ line.

In our example I = 0 so, aggregate expenditure is equal consumption. We will be inequilibrium if C = Y. So, where C crosses the 45◦ line is our equilibrium level of output.This happens when income is equal to 5000. When income is 5000, then consumption is1000 + 0.8(5000) = 5000. So, consumption is equal to income and we are in equilibrium ifplanned investment is zero.

The Savings Route

To solve for equilibrium using the savings function we will use the simplified equilibriumcondition, I = S.

Substituting our savings function, the level of planned investment into our equilibriumcondition and solving we have

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I = S

0 = −1000 + 0.2Y

1000 = 0.2Y

Y ∗ = 5000

Here again we find that the equilibrium level of output is 5000. If we substitute 5000 intothe savings function we should find that Savings is equal to zero. S=-1000 + 0.2(5000)=0.

In the above graph, since investment is equal to zero we find that we have equilibriumwhen the savings function crosses the x-axis.

Example 2

In this example we will stick with the same consumption function, but planned investmentis now equal to 100.

C= 1000 + 0.8YI = 100



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C + I = Y1000 + 0.8Y + 100 = Y

-0.8Y -0.8Y1100 = 0.2Y

Y∗ = 5500

In our example I = 100 so, aggregate expenditure is equal consumption plus 100. Wewill be in equilibrium ifC + I = Y. So, where C + I crosses the 45◦ line is the equilibrium level of income forthis situation. This happens when income is equal to 5000. When income is 5500, thenconsumption is 1000 + 0.8(5500) = 5400 and Investment is equal to 100. So, consumptionplus planned investment is equal 5500, and 5500 is the equilibrium level of output. Youshould notice that aggregate expenditure has shifted up by 100 for each level of income.This is because in the short-run investment is independent of income.

The Savings Route


I = S

100 = −1000 + 0.2Y

1100 = 0.2Y

Y ∗ = 5500

Here again we find that the equilibrium level of output is 5500. If we substitute 5500 intothe savings function we should find that Savings is equal to 100. S=-1000 + 0.2(5500)=100,which is equal to the level of investment.

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In the above graph, since investment is equal to 100 we find that we have equilibriumwhen the savings function crosses the investment line. This happens when income is equalto 5500.

The Investment Multiplier

Notice that when we increased planned investment from 0 to 100 the equilibrium level ofoutput increased from 5000 to 5500. This is no accident. A change in planned investmentwill result in a proportional change in the equilibrium level of output. In this case an increaseof 100 in investment brought about an increase of 500 to output. Therefore, the investmentmultiplier is 5.

Im =1

MPS(5)

In our case the multiplier is Im = 10.2

= 5. Once the multiplier has been determined we canuse it to calculate the new level of equilibrium instead of solving things via the equilibriumcondition.

∆Y = Im ∗∆I (6)

Y ∗New = Y ∗

Old + ∆Y (7)

In our case the change in the equilibrium level of income is equal to 10.2∗ 100 = 5*100

= 500. We know that the old level of equilibrium was 5000 and that when investment isincreased by 100 then equilibrium output increases by 500. So, the new level of output inequilibrium is 5000 + 500 = 5500. Exactly what we calculated earlier.

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The Keynesian Model w/ Government

Adding government to the Keynesian model is very similar to the addition of government inthe classical model.

• The addition of government means that now there are household income divided threeways; Consumption and Saving as before, and Taxes paid to the government.

• There is no change in the types of expenditures by firms. While it is true that gov-ernment does tax firms we will make the simplifying assumption that all governmentincome is derived via a tax on households.

• The funds market now has three players, households, firms and the government.

• In the short-run, planned investment is taken to be independent of income.

• In the short-run, government expenditure is taken to be independent of income.

• There is a funds market where savings is made available to government and firms thatwish to borrow.

Using the above assumptions we can now define aggregate expenditure and aggregateoutput.

Aggregate Expenditure (AE) = Consumption + Planned Investment + Government Expendi-ture = C + I + G

Aggregate Output = Aggregate Income = Y = Consumption + Savings + Taxes = C + S + T

Disposable Income = YD = Y - T

So, we will have equilibrium in the classical model if

AE = Y

C + I + G = C + S + T

I + G = S + T

The Keynesian model differs due to its focus on the consumption function’s dependenceon income in the short-run. The problem with our equilibrium conditions above is thatconsumption and savings are functions of disposable income and not total income and we areinterested in the equilibrium level of total income. So, we will have to do a little substitution.This is best demonstrated by using our example. We will add government in parts. Firstwe will focus on the impact of government expenditure, then we will examine the impact oftaxes.

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Example 2 Con’t

We now modify our example but keep the same consumption function, planned invest-ment is 100 and we will assume government expenditure to be 100. In this example sincetaxes are equal to zero disposable income is equal to gross income.

C= 1000 + 0.8YD

I = 100G = 100T = 0



C + I + G = YC + I + G = YD - T

1000 + 0.8YD + 100 + 100 = YD - T-0.8Y -0.8YD

1200 = 0.2YD

YD∗ = 6000

In our example I = 100, and G = 100 so, aggregate expenditure is equal consumptionplus 200. We will be in equilibrium if C + I + G = Y. Again, since T = 0, YD = Y.The addition of government is very similar to adding planned investment. The aggregateexpenditure shifts up by another 100 above the initial level when G was equal to zero.

The Savings Route


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I + G = S + T

100 + 100 = −1000 + 0.2Y D + 0

1200 = 0.2Y D

Y D∗ = 6000

Here again we find that the equilibrium level of output is 6000. If we substitute 6000 intothe savings function we should find that Savings is equal to 100. S=-1000 + 0.2(6000)=200.

The Government Multiplier

Notice that when we added government and it went from 0 to 100 the equilibrium levelof output increased from 5500 to 6000. This is no accident. A change in governmentexpenditure will result in a proportional change to the equilibrium level of output. In thiscase an increase of 100 in government expenditure brought about an increase of 500 tooutput. Here the government multiplier is 5.

Gm =1

MPS(8)

In our case the multiplier is Gm = 10.2

= 5. Once the multiplier has been determinedwe can use it to calculate the new level of equilibrium instead of solving things via theequilibrium condition.

∆Y = Gm ∗∆G (9)

In our case the change in the equilibrium level of income is equal to 10.2∗ 100 = 5*100

= 500. We know that the old level of equilibrium was 5500 and that when governmentexpenditure is increased by 100 then equilibrium output increases by 500. So, the new levelof output in equilibrium is 5500 + 500 = 6000. Exactly what we calculated earlier.

Adding Taxes

In the above case with government, YD = Y. So we didn’t need to substitute to get theequilibrium level of income. Adding taxes is different from adding government or investmentspending in that investment and government spending are direct injections into the economy.Taxes are more of an indirect injection in that changing taxes changes the consumptionfunction. If the government lowers taxes only a percentage of this change in taxes will bespent due to the marginal propensity to consume.

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C = a + bYD

= a + b(Y-T)= a + bY - bT= a - bT + bY

Officially, the consumption is a function of disposable income and not gross income,however we can rewrite it as a function of gross income to find equilibrium.

Example 2 Con’t

Again we modify our example we keep the same consumption function, planned invest-ment is 100 and government expenditure is 100, but now taxes are assumed to be 100.

C= 1000 + 0.8YD

I = 100G = 100T = 100

Following the steps for substituting taxes into the consumption function we get

C = 1000 + 0.8YD

= 1000 + 0.8(Y-100)= 1000 + 0.8Y - 80= 1000 - 80 + 0.8Y= 920 + 0.8Y


Since we just substituted taxes into the consumption function we don’t need to modifythe right hand side of the equilibrium condition.

C + I + G = Y920 + 0.8Y + 100 + 100 = Y

-0.8Y -0.8Y1120 = 0.2Y

Y∗ = 5600

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In our example I = 100, and G = 100 so, aggregate expenditure is equal consumptionplus 200. We will be in equilibrium if C + I + G = Y. Since T = 100, YD = Y - 100, andtherefore the equilibrium level of disposable income is YD = 5500.

The Savings Route


I + G = S + T

100 + 100 = −1000 + 0.2Y D + 100

1100 = 0.2Y D

Y D∗ = 5500

Y ∗ = 5600

The Tax Multiplier

Notice that when we added taxes and they went from 0 to 100 the equilibrium level ofoutput decreased from 6000 to 5600. This is no accident. A change in taxes will result ina proportional change in the equilibrium level of output. In this case an increase of 100 totaxes brought about a decrease of 400 in output. Here the government multiplier is -4.

Tm =−MPC

MPS(10)

In our case the multiplier is Tm = −0.80.2

= -4. Once the multiplier has been determinedwe can use it to calculate the new level of equilibrium instead of solving things via theequilibrium condition.

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∆Y = Tm ∗∆T (11)

In our case the change in the equilibrium level of income is equal to −0.80.2∗100 = -4*100 =

-400. We know that the old level of equilibrium was 6000 and that when taxes are increasedby 100 then equilibrium output decreases by 400. So, the new level of output in equilibriumis 6000 - 400 = 5600. Exactly what we calculated earlier.

There is a relationship between the government spending and tax multipliers. Like theMPC and MPS they sum to 1.

Gm + Tm =1

MPS+−MPC

MPS

=1−MPC

MPS

=MPS

MPS

Gm + Tm = 1 (12)

Okun’s Law - There is a correlation between output and the unemployment rate. A 1percentage point change in the unemployment rate is associated with a 2 percent change ofoutput in the opposite direction.

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Sample Problem -

1. In the Keynesian cross assume the following information:

C = 1200 + 0.6 YD I = 300 G = 200 T = 100

(a) What is the equilibrium level of income, consumption, savings and disposableincome?

(b) Graph consumption and aggregate expenditure as a function of Y (not YD) anddemonstrate the equilibrium values of consumption and income you calculated in(a).

(c) Assume that the level of output is 3200. Explain how the economy adjusts toequilibrium. Specifically mention inventory levels. Demonstrate this on yourgraph in (a).

(d) The President’s Council of Economic Advisors anticipate private investment todecrease by $150. By how much would the government need to change taxes toprevent the looming recession?

(e) What would happen to this economy if society becomes more thrifty and savesmore at each level of income? Why do you suppose this result is called the paradoxof thrift?

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