Keynesian Model of Income Determination in A Two Sector Economy.

Introduction

            This model assumes that the aggregate supply curve is perfectly elastic up to the full employment level of output after which it becomes perfectly inelastic. Hence price level, until the full employment level, will be determined solely by the height of the supply curve. Hence, the price variable gets less attention while entire focus is on the determination of equilibrium level of income, which is determined solely by the aggregate demand.

Aggregate Demand in a Two Sector Economy

            The following are the postulations for the above analysis.

1. The prices are constant or invariable

2. Given the price level, the firms are willing to sell any amount of the output at that price level

3. The short run aggregate supply curve is perfectly elastic or flat

4. Investment is assumed to be autonomous and thus independent of the income level

5. There exist only two sectors in the economy, the households and the firms

Aggregate demand is the total amount of goods demanded in an economy. The aggregate demand function can be expressed as

                                    AD = C + I

Where, C = aggregate demand for consumers goods

             I = aggregate demand for investment goods

Determination of equilibrium income or output in a Two Sector Economy

In the most basic terms, an economy can be said to be in equilibrium when the production plans of the firms and the expenditure plans of the households are realized.

Below are the postulations of the analysis

1. There exists only two sectors of the economy, there is no government sector and foreign sector

2. All the factors of production are owned by the households who sell the factor services to earn an income. With a part of this income, they purchase goods and services and save the rest

3. As there is no government in the economy there are no taxes and subsidies and no government expenditure

4. As there are no foreign sectors in the economy there are no exports and imports and external inflows and outflows

5. As far as the firms are concerned there are no undistributed profits

6. All the prices are constant and does not change

7. The technology and the supply of capital are given

8. According to Keynesian theory, there are two approaches, they are Aggregate Demand - Aggregate Supply Approach and Saving Investment Approach

Let us see few illustrations which explain the two sector models

Illustration 12

The fundamental equations in a two sector economy are given as consumption function C = 400 + 0.75 Y, investment Ī = 400. You are required to ascertain the following.

1. Equilibrium level of income

2. Equilibrium level of consumption

Solution

In a two sector economy, the equilibrium level of income is

                                                Y         =              1      ( Ca + Ī )
                                                                          1 – b

In the equation, Ca = 400, Ī = 400, b = 0.75

By substituting the above we get

                                                Y         =               1       (400 +400)
                                                                        1 – 0.75

                                                Y         =          1 / 0.25 (800)

                                                Y         =          4 x 800

(a) Thus, the equilibrium level of income is 3,200

In a two sector economy, the consumption function is

                                                C         =          Ca + b Y

In the above equation, Ca = 400, Ī = 400 and b = 0.75

By substituting the above values we obtain,

                                                C         =          400 + 0.75Y

                                                C         =          400 + 0.75 (3,200)

                                                C         =          400 + 2400

(b) Thus the equilibrium of level of consumption is 2,800

Illustration 13

The fundamental equations in a two sector economy are given as: Consumption C = 300 + 0.8Y and the investment function Ī = 400.

1. Derive the saving function

2. Find the equilibrium level of productivity the equating the saving leakages to the investment injections

Solution

The saving function is given by

S          =          Y – C

                                                            S          =          Y – 300 + 0.8Y

                                                            S          =          – 300 + 1Y – 0.8Y

                                                            S          =          – 300 + 0.2Y

(a) Hence, the saving function is given by S            =          - 300 + 0.2Y

The equilibrium level of productivity can be determined by equating the saving leakages to the investment injections.

Thus,
                                                            - 300 + 0.2Y   =          400

                                                            - 300 – 400     =          - 0.2Y

                                                            - 700               =          - 0.2Y

Or                                                        0.2Y                =          700

(b) Thus, the equilibrium level of productivity is 3,500

Illustration 14

For a two sector economy we have the following equation for consumption function

C = 120 + 0.75Y, determine the following

1. If investment in a year is $70 millions what will be the equilibrium level of income or productivity

2. If full employment level of income is $920 millions what investment is required to be undertaken to ensure equilibrium at full employment

Solution

                                                          Y         =          C + I

                                                          Y         =          120 + 0.75Y + 70

                                               Y – 0.75Y       =          120 + 70

                                             Y (1 – 0.75)     =          190

                                              0.25Y              =          190

                                                          Y         =          190 / 0.25

(a) Thus, if the investment in a year is $70 millions, then the equilibrium level of 
      income or Productivity (Y) will be 760

To ensure full employment equilibrium investment should be equal to the saving gap at full employment income. With the given full employment income equal to $920 millions,

                                                            SF = YF – CF
Thus,
YF – (120 + 0.75YF)

                                                            920 – (120 + 0.75 (920))

                                                            800 – 0.75 (920)

                                                            SF        =          800 – 690

(b) Thus, investment required for full employment equilibrium is S110 millions

Illustration 15

If in a two sector economy Consumption C = 900 + 0.8Y and Investment I = 1,080 then

1. Determine the equilibrium level of income and consumption
2. Derive the saving function and determine the saving at the equilibrium level
3. Determine the equilibrium level of income by equating planned investment

Solution

The equilibrium condition is given as Y = C + I

Thus,
                                                            Y         =          900 + 0.8Y + 1,080

                                                            Y         =          1,980 + 0.8Y

                                                Y – 0.8Y         =          1,980

                                                0.2Y               =          1,980

                                                            Y         =          1,980 / 0.2

(a)        Thus, the equilibrium level of income (Y) is 9,900

The consumption function C = 900 + 0.8Y

When Y = 9,900

C         =          900 + 0.8(9,900)

C         =          900 + 7,920
           

1. Thus, the equilibrium level of consumption C is 8,820

The saving function is given by S = Y – C

                                                            S          =          Y – (900 + 0.8Y)

                                                            S          =          Y – 0.8Y – 900

                                                            S          =          0.2Y – 900

2.       Thus the saving function is given by S = 0.2Y – 900

At equilibrium level,
S          =          0.2 (9,900) – 900

S          =          1980 – 900

3.       The saving function at Equilibrium will be S = 1080

The planned saving is given by S        =          - 900 + 0.2Y

In equilibrium, planned saving equals planned investment

Thus,
                                                            - 900 + 0.2Y    =          1080

                                                                        0.2Y    =          1080 + 900

                                                                        0.2Y    =          1980

                                                                        Y         =          1980 / 0.2

4. Thus, the equilibrium level of income (Y) is 9,900