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Complex Multiplier

 Complex Multiplier

Illustration 32

Presume the consumption function is C = 100 + 0.8Yd and investment is I = 200 millions. The government expenditure at $ 180 millions where as the tax function is a proportional income tax function where T = 0.10Y.

  1. Find the equilibrium level of income in an economy
  1. Find the revenue from taxes at the equilibrium level of income. Is the government budget balanced?
  1. Suppose there is an increase in investment from 200 to 240 million dollars, what is the equilibrium level income
  1. What is the revenue from taxes at the new equilibrium level of income? Is there a balanced government budget?

Solution

  1. The equilibrium condition is given as Y = C + I + G
  2. Here,
                            C         =          100 + 0.8Yd

                            C         =          100 + 0.8(Y – T)

                            C         =          100 + 0.8(Y – 0.10Y)

                            C         =          100 + 0.8(0.9Y)

                            C         =          100 + 0.72Y
    Thus,
                            Y         =          100 + 0.72Y + 200 + 180

                Y – 0.72 Y      =          100 + 200 + 180

                            0.28Y  =          480

                            Y         =          480 / 0.28

    The equilibrium level of income is $1,714.28 millions.

  3. The revenue from taxes at the equilibrium level of income.
  4. The tax function is a proportional income tax function where T = 0.10Y

    Thus, T = 0.10(1,714.28)        =          171.42 million dollars

    Hence the revenue from taxes at the equilibrium level of income is at $171.42 millions where as the government expenditure is at $180 millions. Therefore, there is a budget deficit of $8.58 millions.

  5. When there is an increase in investment from $200 millions to $240 millions,
  6.                         Y         =          100 + 0.72Y + 240 + 180

                Y – 0.72Y       =          100 + 240 + 180

                            0.28Y  =          520

                            Y         =          520 / 0.28

    The equilibrium level of income is 1,857.14 million dollars.

  7. The revenue from taxes at the new equilibrium level of income is

                            T          =          0.10 (1,857.14)
               
    =          185.714 million dollars

    Hence, the revenue from taxes at the equilibrium level of income is $ $185.714 millions whereas the government expenditure is at 180 million dollars. Therefore, there is a budget surplus of $5.714 millions. Due to the higher income level, there are larger tax revenues leading to a budget surplus.

Illustration 33

In an economy, C = 100 + 0.8Yd, I = $200 millions government expenditures is at $100 millions whereas T = $40 millions.

  1. Find the equilibrium level of income in an economy.
  2. Find the equilibrium level of consumption and saving at the equilibrium level of income.
  3. Depict the injections leakages equality at the equilibrium level.

Solution

1. The equilibrium condition in an economy is given as Y = C + I + G

Here,                           C         =          100 + 0.8Yd

                                    C         =          100 + 0.8(Y – T)

                                    C         =          100 + 0.8(Y – 40)

                                    C         =          100 + 0.8Y – 32
Thus,
                                    Y         =          100 + 0.8Y – 32 + 200 + 100

                        Y – 0.8Y         =          368

                        0.2Y                =          368

                                    Y         =          368 / 0.2

The equilibrium level of income is $1,840 millions.

2. When the equilibrium level of income is $1,840 millions,

Equilibrium level of consumption:

                                    C         =          100 + 0.8Yd

                                    C         =          100 + 0.8(Y – T)

                                    C         =          100 + 0.8(Y – 20)

                                    C         =          100 + 0.8(1,840) – 32

                                    C         =          100 + 1472 – 32

                                    C         =          1,540

The equilibrium level of consumption is $1,540 millions.

Equilibrium level of saving:

                                    S          =          Y – C – T

                                    S          =          1840 – 1540 – 40 = 260

The equilibrium level of saving is $260 millions.

3. Injections    =          I + G

    Leakages     =          S + T

                                    Injections        =          200 + 100        =          300

                                    Leakages         =          260 + 40          =          300

This depicts the injections leakages equality at the equilibrium level.

Illustration 34

For a closed economy, the following data is given

                        Consumption, C                      =          100 + 0.8Yd

                        Investment, I                           =          140

                        Government Expenditure, G  =          400

                        Transfer Payments, TR           =          200

                        Rate of income tax, t              =          0.2

  1. Write the reduced form of equation for equilibrium income
  2. Find out equilibrium income
  3. What is the value of tax multiplier?

Solution

1. Reduced form of the equation for equilibrium income is

                                    Y         =                 1         (a + bTR + I + G)
                                                            1 – b + bt

2. Substituting the values of parameters in the above equation for the equilibrium income, we have

                                    Y         =                 1      (100 + 0.8*200 + 140 + 400)
                                                            1 – 0.8 + 0.8*0.2

                                                =                1    (100 + 160 + 140 + 400)
                                                            1 – 0.64
                                   
                                                =            1   (800)
                                                            0.36

                                                =          2,222

Thus, the equilibrium income is 2,222

                        Tax multiplier            =                 1      
                                                                        1 – b + bt

=                    1              
              1 – 0.8 + 0.8*0.2

=               1                 =            1       
            1 – 0.64                       0.36

=          2.77

Illustration 35

Presume we have an economy characterised by the following functions:

                        C         =          200 + 0.8Yd

                        Ī           =          200
           
Ḡ         =          200

Ť          =          200

Values are in million dollars.

  1. Find the equilibrium level of income.
  2. How much increase in income will take place if government expenditure on goods and services increases by $120 millions?
  3. Find the tax multiplier and balanced budget multiplier
  4. Find the equilibrium level of national income if T = Ť + tY = 200 + 0.25Y

Solution

1. Equilibrium Y = C + Ī + Ḡ

                        Y         =          a + bYd + Ī + Ḡ

                        Y         =          a + b(Y – Ť) + Ī + Ḡ

Substituting the values of a, b, Ť, Ī and Ḡ we have,

                        Y         =          200 + 0.8(Y – 200) + 200 + 200
                                   
=          200 + 0.8Y – 160 + 200 + 200

            Y – 0.8 Y        =          440

            0.2Y                =          440

                        Y         =          440 / 0.2
                       
Y         =          2,200

2. Government expenditure multiplier Gm     =            1       
                                                                                    1 – b

                                                                        =              1                  =          5
                                                                                    1 – 0.8

Increase in national income if government expenditure increases by $120 millions.

                                    Δ Y      =          Gm * DG

                                                =          5 * 120            =          600

3. Tax Multiplier,

                                    Tm       =          - b       
                                                            1 – b

                                                =          - 0.8                 =        - 0.8                 =          - 4
                                                            1 – 0.8                           0.2

                                                =         
Balanced Budget Multiplier

                                    Gm + Tm

                                    5 + (- 4)           =          1

4. The equilibrium level of national income when,

                                    T          =          200 + 0.25Y    =          Ť + tY

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

                                                =                 1                     (200 - 0.8*200 + 200 + 200)
                                                            1 – 0.8 + 0.8*0.25
                                               
                                                =                   1             (440)   
                                                            1 – 0.8 + 0.2
                                               
                                                =            1     (440)
                                                            0.4

                                                =          1,100

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