Equilibrium Income and Output in Three Sector Economy

The three sector economy consists of three economic units:households, business firm, and the government. The national output of the economy thus comprises of the monetary value of final consumption (C), final investment goods (I), and final government purchases (G).
Firms and government employ factors of production to produce goods and services. This creates a factor market, where factor payments in the form of wage, rent, interest, and profit are paid to the suppliers of production factors. The summation of such factor payment is termed as national income (NI) at factor cost.

In the three sector economy, households and firms pay taxes to the government, which becomes a source of income for the government. Households also make saving which gives rise to financial markets. These are the sources of credit to the business firms for making investment. The government makes purchases with the help of fund collected through taxes and loan from financial market.

Government Purchases (G)

Since government purchases are assumed to be autonomous, it will be parallel to the output/income line. Symbolically, it is represented as
G= G_a
Where, G_a is autonomously determined government purchases of goods and services.

Consumption Expenditures

Consumption expenditures of household sectors depend upon their disposable income left after payment of taxes to the government. Generally, it has a direct relationship with the size of disposable income, which can be expressed as
C= C_a + λY_D
Where, C= Household consumption expenditure
C_a= Autonomous consumption
λ= Marginal propensity to consume
Y_D= Disposable income

Disposable Income

Disposable income is the income earned by the household sector after the payment of personal taxes net of governmental transfer payments. It is expressed as
Y_D= Y – T_n
Where, T_n = Net tax (tax adjusted for governmental transfer payments to individuals)
Thus, net tax can be expressed as
T_n= T_a – R_a
Where, T_a= Gross autonomous tax
R_a= Autonomous transfer payments
Substituting the value of net tax to the equation of disposable income,
Y_D= Y – (T_a – R_a)
Or, Y_D= Y – T_a + R_a
Thus,
Y_D= Y – T_a [since, R_a= 0 by assumption]
As stated earlier, aggregate expenditure (AE) in the three economic model is the sum of household’s consumption expenditure (C), business investment expenditure (I), and government purchase expenditure (G), AE can be expressed as
AE = C + I + G

Investment Expenditures

The expenditures made by the business firms on capital goods such as fixed structures, machinery, equipment, and inventories for the production of goods and services. Although, investment can be both autonomous and induced, it is assumed to be autonomous in the current scenario. Symbolically, it is expressed as
I= I_a

The Three-Sector AE Line

The aggregate expenditure of the three sectors of the economy has been diagrammatically presented below
Figure: Expenditure line in three sector model

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Government affects the consumption, production, and investment made by households and firms through taxes and other spending. Transfer payments and subsidies upon private sectors affect their income and saving as well.
Therefore, different fiscal policies can be helpful to describe how equilibrium output/income are determined. In order to determine economy’s equilibrium income/output, three major fiscal policy models can be specified.

1.First Fiscal Model and Equilibrium Level of Income/Output

The model assumes that government taxes (T) are autonomous. Government follows a lump sum tax policy which means individuals and firms should pay a fixed amount of tax regardless of their level of income.
The autonomous tax component is represented as
T= T_a
The first fiscal model assumes there are no transfer payments or subsidies provided by the government. This can be expressed as
R= 0
Where R refers to government transfer payments.

Equilibrium Level of Output/Income: Equations

Equilibrium level of output/income is attained when planned aggregate expenditure is equal to actual output/income. Symbolically,
Y = AE
Substituting AE with its components, we get
Y= C + I + G
Or Y= C_a + λY_D + I_a + G_a
Or Y = C_a + λ(Y – T_n) + G_a
Or, Y – λY= C_a – λT_a + I_a + G_a [since, transfer payments= 0; net tax T_n = T= T_a]
Or Y (1 – λ) = C_a – λT_a + I_a + G_a
Then, the final equilibrium income is

Thus, the equilibrium level of income is the sum total of three autonomous expenditures minus autonomous tax multiplied by the multiplier 1/ (1 – λ) = 1/ (1 – MPC)

2.Second Fiscal Model and Equilibrium Level of Income/Output

The second fiscal model differs from the first model in that it takes into consideration the transfer payments made by the government to individuals. Although the second model assumes that government makes transfer payments, they are assumed to be autonomous, which means they are not affected by the individual’s level of income.
Thus, the assumption of government transfer payments in the second fiscal model can be expressed as
R= R₀
Where, R= Transfer payments
R₀= Fixed or given value of R
The assumptions made in the first fiscal model are followed in the second fiscal model as well. This means that the second fiscal model also assumes government purchases expenditure is autonomous (G= G₀), taxes are autonomous (T=T₀), and business investment is autonomous (I= I₀). The consumption of household depends on their level of disposable income.

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Equilibrium Level of Income/Output: Equations

Considering the assumptions of the model, the equilibrium level of income can be derived with aggregate expenditure and actual output/income. Symbolically,https://69e12cce12149ece2d5c76f08b27a8e6.safeframe.googlesyndication.com/safeframe/1-0-37/html/container.htmlY= AE
Substituting AE with the components,
Y= C + I + G
Or, Y= C_a + λY_D + I_a + G_a
Or, Y= C_a + λ (Y – T_n) + I_a + G_a [T_n is the net tax]
Or, Y= C_a + λ [Y – (T – R) + I_a + G_a] [T is the gross tax]
Or, Y= C_a + λ [Y – (T_a – R_a) + I_a + G_a] [Taxes and Transfer payments are autonomous]
Or, Y= C_a + λY – λT_a + λR_a + I_a + G_a
Or, Y – λY= C_a – λT_a + λR_a + I_a + G_a
Or, Y (1-λ) = C_a – λT_a + λR_a + G_a
So, the final equilibrium income is

3.Third Fiscal Model and Equilibrium level of Income and Output

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In this model, the tax is divided into two parts: T=Ta+t.Y
where, T=Tax, t.Y= Induced Tax
The transfer payments is autonomous R=Ra
The assumptions made in the first fiscal model are followed in the Third fiscal model as well. This means that the Third fiscal model also assumes government purchases expenditure is autonomous (G= Ga), and business investment is autonomous (I= Ia). The consumption of household depends on their level of disposable income.
Equilibrium Level of Input and Output : Equations
AS=AE
Y=C+I+G
Y= a+b(Y-Ta-t.Y+Ra)+Ia+Ga
Y=a+b.Y-b.Ta-b.t.Y+b.Ra+Ia+Ga
Y-b.Y+b.t.Y=a-b.Ta+b.Ra+Ia+Ga
Y(1-b+b.t)=a-b.Ta+b.Ra+Ia+Ga
Y=a-b.Ta+b.Ra+Ia+Ga/(1-b+b.t)
Adopted from: www.businesstopia.com

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