# Tax multiplier – Derivation, Formula, and Graphical Representation

## Introduction

The tax multiplier briefly explains a measurement of the change in GDP i.e Gross Domestic Product with respect to the change in government taxes. The tax multiplier could be complex as well as simple corresponding to the tax changes whether they impact only the consumption part of the overall components of the GDP.

## Tax multiplier

The tax multiplier is applied by the governments, investors and economists (economic utility definition) to figure out the changes in the fiscal policy of taxation impact the aggregate production. Taxes are an issue at both macro and micro levels. We would study more about the simple tax which considers only consumption expenditure affected by the change in production resulting from the effect of changes in government taxes. It provides us with valuable information.

However, the taxes affect the consumer’s disposable income as well which impacts both consumption and the savings rate. As a result, the complex multiplier explains changes in government taxes resulting in changes in aggregate production impact the aggregate expenditures as a whole. Although it includes more variables but provides a better reflection of the complex nature of economics.

#### GDP ( Gross Domestic Product )

Gross domestic product refers to the value of all goods and services produced within the domestic boundary or country in a given time frame. Therefore, the tax multiplier calculator is a measure of this effect.

#### Marginal propensity to consume

Both simple and complex method evaluates MPC. It refers to the percentage of one unit increase in income that is spent by the consumer. For example, if Suresh receives a bonus check of \$ 100 and spends it all then the MPC will be equal to 1 ( \$100 spent / \$100 increase in income). However, if \$60 is saved, then MPC = 0.4 ( \$40 spent / \$100 increase in income )

#### Marginal propensity to save

It explains the vice versa situation of marginal propensity to consume. Since it’s the inverse relationship we can simplify it by stating ( 1-MPC) i.e it explains the percentage of one unit of an income that is saved by the consumer.

This must also be noted that MPS + MPC = 1 by studying the inverse relationship explained above in detail for both the terminologies.

## Tax multiplier example

Supposedly the US government decreases the tax rate by 5.6% with a tax volume costing \$428 million. As a result, the consumer expenditure will increase for the same worth since they will have higher disposable income since the tax paid amount will be less.

Now, the consumers will also save more since the tax burden on them has been decreased by the diminishing government tax rate. Therefore, with a decrease in the tax rate, the disposable income, consumption as well as need to increase production to meet demands all occurs. As a result, in future, the GDP would increase by the number of Tax Multiplier examples to equate to a decrease in tax rates.

However, if the government tries to build a vice versa situation by increasing the tax rate by 5.6 % with the worth \$428 million so the consumer spending will decrease by the same amount. Furthermore, disposable income, consumption, savings and production will decrease. At last, the GDP will also decrease by tax multiplier example to compensate for the increase in the government tax rate.

## Tax multiplier Derivation

The first step in deriving any multiplier is finding out their equilibrium with income i.e Y . As a result, starting with the following set of equations:

C = m ( DI ) + b

I = Io
G = Go
T = To
X = M = 0

Where, C = consumption , b = autonomous consumption , DI = disposable income , m = marginal propensity to consume , M = imports , X = exports , I = investment expenditure, T = tax, G = government spending  ( the “o” subscripts explain that these variables are autonomus expenditures)

Now from macro economics we learned,

Y = [ m ( Y – To) + b] + Io + Go

∆Y =  (-mTo + b + Io + Go) / (1 – m)        ………………(1)

Considering changes in taxes, T = To + DT

Y = [m(Y – [To + DT]) + b] + Io + Go + (0 – 0)

∆Y  = (-m[To + DT] + b + Io + Go)/(1 – m)    …………….(2)

Now, subtract (1) from (2) , we get

DY =  [(-m [To + DT] + b + Io + Go)/(1 – m)] – [(-m To + b + Io + Go)/(1 – m)]
=  -m DT/(1 – m)
= -m DT/(1 – m)
DY/DT = -m/(1 – m)

This must be noted that the multiplier makes just an unrealistic assumption i.e the tax multiplier derivation we get while working with it does not changes the basic results but puts a restriction on the variability of details of conclusions that might otherwise lie.

## Tax multiplier formula

We studied that increase in tax rate decreases the income( circular flow of income and expenditure) for the consumers i.e they hold a contractionary effect. Explaining in symbols, a change in tax i.e increase (∆T) results in a change or decrease in income (∆Y). So the ratio of ∆Y/∆T is termed a tax multiplier. Therefore, the amount of decline in national income due to an increase in tax rate depends on consumption or MPC. Denoted by KT, the formula would be:

∆Y/∆T = -MPC / ( 1 – MPC)

∆Y / ∆T = – MPC / MPS

Therefore, the tax multiplier calculator is absolute and negative or one less than govt. expenditure multiplier. Lets suppose MPC = ¾ so KT would be (-¾) / ( 1 – ¾) = -3 . Now, these being proportions explain that 20 crores of increase in tax results in 60 crores of decline in income.

Taking complex tax multiplier into consideration if the MPC = ¾ and KG = 4 (government expenditure) . This explains that with 20 crore increase in government expenditure will lead to an 80 crore of decrease in income.

Thus, KT is negative and is denoted by KT, K or KG.

#### Problems based on Tax Multiplier

Question 1: In a nation personal spending increases by 500 lakh due to an increase in disposable income by 650 lakh. Now, the government wants to increase their GDP by 250 lakh for the present year. Therefore, suggest the tax policy to the level of GDP that the nation wants to achieve.

• Marginal propensity to consume can be calculated as –

MPC = change in consumption/change in disposable income

= 500 / 650

= 0.77

• Therefore, Tax multiplier of the nation will be as follows –

Multiplier = – MPC / ( 1 – MPC)

= – 0.77 / ( 1- 0.77)

= – 3.33

• So, for the desired GDP level, the decline in tax receipt will be as follows –

–  ΔY / decrease in tax receipt = Tax multiplier

Decrease in tax receipt = -250 / (3.33)

= 75 lakh

As a result, the government has to decrease the taxes by 75 lakhs to achieve the required level of the target level of GDP.

Question 2: For a nation the personal consumption declines by 200 million affected by the decline in the disposable income of 450 million. So, the government wishes to decline the tax receipts by 100 million to lighten the load on disposable income. Find out the GDP level which will be increased because of this step taken by the government.

• Marginal propensity to consume = Change in consumption / change in disposable                       income

= – 200 / ( – 450 )

= 0.44

•  Multiplier = – MPC / ( 1 – MPC )

= – 0.44 / ( 1 – 0.44 )

= – 0.80

• Now, upliftment in the level of GDP due to tax decrease will be computed as follows:

Increase in GDP = – ΔT X  Multiplier

= – 100 X ( – 0.8)

= 80 million

As a result, the GDP gets uplifted by 80 million due to the 100 million cut in the tax receipts.

#### Explanation to Calculate

The tax multiplier formula could be derived from the following ladder in order:

• The first step includes finding out the marginal propensity to consume, which is depicted by the ratio of spending or consumption to the changes in extra income ( disposable ) for the whole entire nation.
• The Second step includes finding out the multiplier represented by the negative marginal propensity to consume divided by one minus the marginal propensity to consume again or instead of one minus MPC, MPS could also be written.
• At last, the ratio of change in income divided by change in tax receipts has termed a multiplier.

## Tax Multiplier Graph

Graphically we can show the relationship between consumption leading to change in aggregate demand, income and taxes. The tax multiplier graph is shown in the following figure. The consumption line and aggregate demand before the tax changes are denoted by C1 and C1 + I + G, respectively. Therefore, the equilibrium created would exist at the OY1 level of income.

Now, let’s increase the taxes which decreases the consumption and shifts the line to C2 resulting in a downward shift of the aggregate demand as well as C2 + I + G. Therefore, the equilibrium declines to the OY2 level of income. As a result, it explains the effect of an increase in taxes on income is contractionary.

Here the difference between government expenditure (KG) and tax multiplier graph (KT) should be clear which can be explained below:

This explains that KT is negative showing an inverse relationship and one less than KG. The government expenditure multiplier and tax multiplier are termed fiscal multipliers as they are related to the changes in activities of government relating to changes in taxation plans and expenditures.

## Government Spending Multiplier vs Tax Multiplier

Keynes explains the investment multiplier which is static and simple depending upon initially with investment and consumption only termed a two-sector model. However, after Keynes, to make these multipliers more practical, many economists added different variables to create others represented as complex multipliers.

The dynamic multipliers include the government spending multiplier, balanced budget multiplier, tax multiplier and foreign trade multiplier. Among these, we will discuss the government spending multiplier vs tax multiplier. Adding government spending and taxes to Keynes’s model, it becomes a three-sector model.

#### Government Expenditure Multiplier

This type of Keynesian multiplier is an expenditure multiplier that measures the proportion of change in income due to autonomous consumption expenditure by the government. Therefore, KG is denoted as

∆Y / ∆G = 1 / ( 1- c )

Here, the income is taken on the horizontal axis and governmental spending expenditure on the vertical axis i.e C + I + G while graphical representation. Adding govt expense to the initial two-sector model, the consumer equilibrium shifts towards the right and the increase in income is more than their expense.

For example, if c= ⅔ then KG = 1 / ( 1 – ⅔) = 3 which explains that the government spending multiplier is more than the unit.

#### Tax multiplier

It explains that when the tax is changed by the government so the relation between the national income and the disposable income changes. As understood before, when government increases taxes.

The marginal or additional propensity to consume of the individual decreases since their extra or disposable income has been reduced. This falls the national income as well due to less spending by the people. However, a reduction in taxes raises the national income. This has been explained in detail in simple as well as complex ways above.

#### Limitations of the tax multiplier

The concept studied till now, even government spending multiplier vs tax multiplier there are limitations attached to these multipliers which are as follows:

• It excludes the counting of transfer payments and only accounts for the government spending on services and goods. Moreover, the transfer payments would offset the negative aspect of the multiplier.
• Assumption includes uniform MPC for one who pays taxes or sells services and goods to the government.
• When the matter of taxes is undertaken, they affect either the consumption or investment depending on the kind of taxpayer. Which explains whether tax is charged on fixed income groups or the business community.

## Use of tax multiplier

From an economic viewpoint, the tax multiplier is an important concept since taxes form an indispensable part of the economic system working at both micro and macro economic levels. So it becomes necessary to understand the way government takes decisions regarding the requirements of change in tax policy.

However, the multiplier does not count into the metric of policy making in tax but surely impacts the decisions regarding the GDP of the country. As explained when the taxes rise, the disposable income or extra income of the consumer decreases which impacts the consumption and is computed by the multiplier.

However if the tax affected all the components of the GDP then the complex multiplier formula would have been as follows:

Tax multiplier = – MPC / [ 1 – ( MPC X ( 1 – MPT) + MPI + MPG + MPM )]

Where, MPT = Marginal Propensity to Tax

MPG = Marginal Propensity of Government Expenditures

MPC = Marginal Propensity to Consume

MPI = Marginal Propensity to Invest

MPM = Marginal Propensity to Import

## Conclusion

The tax multiplier, therefore, concludes the change in the income or the GDP level due to a change in the taxation of the country. It must be noted that there exists a negative relation between tax receipts and GDP level where an increase in one leads to a decrease in the other and vice versa.

Moreover, the tax multiplier formula with examples has been discussed to provide a detailed explanation of the concept. To aid the government and economists with interpretations of the tax policy, the multiplier is therefore being used.