First Fiscal Model and Equilibrium Level of Income/Output

First Fiscal Model and Equilibrium Level of Income/Output

The model assumes that government taxes (T) are autonomous, that is independent of the income level. 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= Ta

Government also makes payments to individuals in the form of transfer payment as social security benefits. However, 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.

Government Purchases (G)

In order to stabilize the economy, government uses fiscal policy tools such as government purchases of gross domestic product (GDP). Government purchases include capital goods as well as consumption goods comprising of governmental expenses on national security defense, federal expenses, etc.

The relation between aggregate income or production and government purchases has been explained in the diagram below:

The figure shows two types of government purchases. EG is the autonomous government purchase, which is unaffected by production or income. EG’ on the other hand, is the induced government purchase which is affected by production or income.

Since government purchases are assumed to be autonomous, it will be parallel to the output/income line. Symbolically, it is represented as

G= Ga

Where, Ga 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= Ca + λYD

Where, C= Household consumption expenditure

                Ca= Autonomous consumption

                λ= Marginal propensity to consume

                YD= 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

YD= Y - Tn

Where, Tn = Net tax (tax adjusted for governmental transfer payments to individuals)

Thus, net tax can be expressed as

Tn= Ta - Ra

Where, Ta= Gross autonomous tax

                Ra= Autonomous transfer payments

Substituting the value of net tax to the equation of disposable income,

YD= Y – (Ta – Ra)

Or, YD= Y – Ta + Ra


YD= Y – Ta [since, Ra= 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= Ia

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

The line C + I + G is positively sloped indicating that greater levels of income and output lead to greater aggregate expenditure made by the three macroeconomic sectors. The other lines C and C + I represent how the expenditures increase at each level, beginning with consumption, then adding investment, and government expenditures.

Here, investment expenditure and government expenditures are assumed to be autonomous, and thus, they are parallel as seen in the figure. The C + I line is parallel to the line C because investment is considered autonomous and constant. The lines C + I + G and C + I are also parallel with respect to autonomous government purchases. If any of the expenses are seen as induced, the slopes become steeper.

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= Ca + λYD + Ia + Ga

Or Y = Ca + λ(Y – Tn) + Ga

Or, Y – λY= Ca – λTa + Ia + Ga [since, transfer payments= 0; net tax Tn = T= Ta]

Or Y (1 – λ) = Ca – λTa + Ia + Ga

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)

Equilibrium Level of Output/Income with Saving and Investment Equality: The Leakages-Injections Approach

Equilibrium of output/income can also be determined with the equality of saving and investment, which is termed as the leakages-injections approach.

Injections are the non-consumption expenditures on the total number of goods and services produced in an economy. In the three sector model, government purchases (G) and investment expenditures (I) are the injections which are injected into the circular flow of consumption of the economy.

On the other hand, leakages are the withdrawals like saving (S) and taxes (T) made from the circular flow of production, consumption and income.

The three sector economy assumes equilibrium at a point where a balance exists between the sum of saving(S) and taxes (T), and the sum of government expenditures (G) and investment expenditures (I). The equilibrium between injections and leakages is the same as equilibrium between aggregate expenditure (AE) and aggregate productivity (Y= GDP).

We know, AE= C + I + G

And, the income generated through aggregate production is used by the household sectors for consumption (C), saving (S), and taxes (T). Symbolically,

Y= C + S + T

Substituting the equations into Y = AE,

C + S + T= C + I + G

Or, S + T = I + G

The equation above gives an equilibrium condition where leakages are equal to injections, when income equals aggregate expenditure is given.

 Equilibrium Income in the First Fiscal Model: Leakages-Injections Approach

Under the first fiscal model, injections-leakages equality is presented as

S + Ta = Ia + Ga [investment, tax, and govt. purchases are autonomous]

Or, YD – C + Ta = Ia + Ga [since saving= disposable income – consumption expenditure]

Or, YD – (Ca + λYD) + Ta = Ia + Ga [consumption depends on disposable income]

Or, Y – Ta – Ca – λ (Y – Ta) = Ia + Ga [disposable income = income – taxes]

Or, Y – Ca – λY + λTa = Ia + Ga

Or, YλY = Ca – λTa + Ia + Ga

Or, Y (1 – λ) = Ca – λTa + Ia + Ga

Finally, equilibrium income


Equilibrium Income: Graphical Illustration

The graphs below illustrate the determination of equilibrium income/output in the three sector under the first fiscal model.

The first graph shows the equilibrium income level through aggregate expenditure and aggregate output approach. The upward sloping expenditure lines in the three sector economy shows that total expenditure increases with the increase in income and output. The aggregate expenditure line is obtained by sequentially adding the expenses in the household sector, business investments and government purchases. Equilibrium occurs at point E where the three sector aggregate expenditure line intersects the 450 line. The equilibrium level of output and income is Ye.

The second graph illustrates the determination of income/output equilibrium by the method of leakages-injections method or the saving-investment method. The vertical axis shows leakages-injections while the horizontal axis shows aggregate income/output. Line S refers to saving and I is the investment line. Assuming governmental purchases to be autonomous, and adding it to autonomous investment, I + G is obtained.

The first leakage is the savings made by households which depends on their income level. In addition to this, taxes are the governmental leakages. Assuming taxes to be autonomous, the line S + T is obtained, which is parallel to the saving line S and equal to the marginal propensity to save (MPS).

Equilibrium through this method is similar to the two sector model, with only a variation in the number of leakages and injections. In the figure, S + T intersects with I + G at point E1 and gives the equilibrium level of income at point OYe.

Major derivations from the leakages-injections equilibrium

  • The equilibrium level of aggregate income/output is dependent on the overall height of the lines rather than on the mix of injections and leakages that make up each lines.

For instance, if the total autonomous investment and government purchases is $ 50 billion, it doesn’t matter if the investment is $ 40 billion, and government purchases is $ 10 billion or vice versa because the level of equilibrium would be the same at $ 50 billon.

  • Fiscal policy effect can be seen with shifts in the line I + G line and S + T line.
  • The vertical gap between the line S + T and line I + G is the unplanned inventory. If leakages equal injections, inventories do not change. If injections exceed leakages, inventories decline, and if leakages exceed injections, inventories increase.