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The Keynesian Model Of Income Determination In A Three Sector Economy

 Determination of Equilibrium Income or Output in a Three Sector Economy

Introduction

            The action of government relating to its expenditures, transfers and taxes is called the fiscal policy. Here we focus on three fiscal policy models which are in increasing order of complexity, with the emphasis being on the government expenditure, taxation and the income level.

Determination of Equilibrium Income or Output In a Three Sector Economy

Though the government is involved in a variety of activates three of them are of greater relevance to us in the present context. Hence we will focus on these activities of the government, which are discussed below:

  1. Government Expenditure

    This includes goods purchased by the central, state and the local government and also the payments made to the government employees.

  1. Transfers

    These are those government payments which do not involve any direct services by the recipient for instance welfare payments, unemployment insurance and others.

  1. Taxes

    These include taxes on property, income and goods. Taxes can be classified into two categories, direct taxes and indirect taxes. Direct taxes are levied directly and include personal income and corporate income tax. They are paid as a part of the price of the goods.

    We simplify our analysis by making a few postulations, which are as follows.

  1. The government purchases factor services from the household sector and goods and services from the firms.

  2. Transfer payment includes subsidies to the firms and pensions to the household sector.

  3. The government levels only direct taxes on the household sector. We here introduce the notion of an income leakage and an injection. In a two sector model, a part of the current income stream leaked out as saving whereas injections in the form of investment were injected into the system.

In a three sector model taxes, like saving, are income leakages whereas government expenditures like investment are injections.

Let us see few illustrations relating to a three sector economy.

Illustration 21

In a two sector economy, the basic equations are as follows:

The Consumption function is C = 200 + 0.8Yd and investment is I = 300 millions. The equilibrium level of income is 2500 millions. Presume the government is added to this two sector model, which then becomes a three sector economy. The government expenditure is at 100 millions

  1. Determine the equilibrium level of income in the three sector economy
  2. What is the multiplier affect of the government expenditure? Is it of the same magnitude as the multiplier effect of a change in the autonomous investment?
  3. Presume that there is a balanced budget in that the entire government expenditure is financed from a lump sum tax. Find the new equilibrium level of income in the three sector economy.

Solution

The equilibrium condition in the three sector economy is given as

                                    Y         =          C + I + G
Thus,
                                    Y         =          200 + 0.8Y + 300 + 100

                                    Y         =          600 + 0.8Y

                        Y – 0.8Y         =          600

                                    0.2Y    =          600

                                    Y         =          600 / 0.2

The equilibrium level of income in the three sector economy is 3,000 millions, which is an increase by 500 millions over the two sector economy.

Government Expenditure Multiplier

                                    GM       =          Δ Y      =             1    
                                                            Δ G                    1 – b

                                                =          1 / 1 – 0.80

                                                =          5

Investment Multiplier, m        =         Δ Y      =               1       
                                                          Δ I                     1 – b

Where b is the marginal propensity to consume,

Thus the magnitude of the multiplier effect is the same as that of a change in government expenditure.

                                    G         =          T          =          100 millions
Thus,
                                    C         =          200 + 0.8 (Y -100)

                                    C         =          200 – 80 + 0.8Y

                                    C         =          120 + 0.8Y

But,                             Y         =          C + I + G

                                    Y         =          120 + 0.8Y + 300 + 100

                        Y – 0.8Y         =          120 + 400

                                    0.2Y    =          520

                                   Y         =          520 / 0.2

The new equilibrium level of income in the three sector economy, when there exists a balanced budget is 2,600 millions.

Illustration 22

In an economy, the full employment output occurs at 2000 millions. The marginal propensity to consume is 0.8 and the equilibrium level of output is currently at 1600 millions. Suppose the government aspires to achieve the full employment output, find the change in

  1. The level of government expenditures
  2. Net lump sum tax
  3. The level of government expenditures and the net lump sum tax when the government aims at bringing the productivity the full employment while keeping the budget balanced

Solution

We have, GM   =

Δ Y      =            1       
Δ G                  1 - b

Where, Δ G     =          Change in government expenditure

            b          =          Marginal propensity to consume

            Δ Y      =          Change in income

            GM       =          Government expenditure multiplier

For instance,

            b          =          0.80

            Δ Y      =          2000 – 1600

            Δ Y      =          400

Thus,
                        400 / Δ G         =          1 / 1- 0.8

                                    Δ G      =          400 (0.2)

Thus, the level of government expenditures required to achieve the full employment output is 80 millions

We have, GF    =

                                    Δ Y      =          - b      
                                    Δ T                  1 – b
Where,
                                                            Δ T      =          Change in tax

                                                            b          =          marginal propensity to consume

                                                            Δ Y      =          Change in income

                                                            GF        =          Government tax multiplier

As the tax multiplier is negative, an increase in tax leads to a decrease in the equilibrium level of income.

For instance,
                                                b          =          0.80

                                                Δ Y      =          2000 – 1600 = 400
Thus,
                                                400      =          - 0.80  
                                                Δ T                  1 – 0.80

                                    -0.8 Δ T           =          400 (0.20)

The net lump sum tax is – 100 millions. There should be a decrease in lump sum tax by 100 millions

The next equation to solve is

                                                Δ Y      =              1               (-b Δ T + Δ Ḡ)
                                                                          1 – b

But,                                         Δ G      =          Δ T
Thus we can write
                                                Δ Y      =              1               (-b Δ Ḡ + Δ Ḡ)
                                                                          1 – b
Or
                                    Δ Y (1-b)         =          Δ Ḡ (-b +1)

Or                                Δ Y (1-b)         =          Δ Ḡ (1-b)

Or                               
                                                Δ Y      =          1 – b    =          1
                                                Δ G      =          1 – b

                                                Δ Y      =          Δ G      =          400

The required increase in the level of government expenditures and the net lump sum tax us 400 millions.

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