Why does cutting taxes by $100 have a smaller effect on GDP than increasing expenditures by $100

The tax impact on GDP is always indirect unlike the government spending. This means tax rebates given to a person DOES NOT count as expenditure for the giver (the Government) but only a fraction of it will counts in the multiplier chain as the beneficiary starts spending it.

Explanation:

The GDP is the aggregate market value of all the final goods and services produced in a country.

Using the expenditure approach, It can can be calculated by the sum of consumption, investment, government expenditures and the net export

Y = C+I+G + (X-M)

Multiplier Effect: This is he concept that a change in any of aggregate spending will produce more than a proportional change in the GDP.

For example, to cause a change of $600 in the real real GDP only a change of less than $600 is required in any of the autonomous spending.

Expenditure Multiplier is the amount by which the GDP will change if autonomous expenditure changes by a given amount.

It is calculated as follows: 1 / (1-MPC).

Marginal Propensity to consume (MPC) is the portion of additional income that is spent. If the MPC is 0.75, then the expenditure multiplier will be = 1 / (1-0.75) = 4.

Using our question, assuming a MPC of 0.75, an increase of $100 in government expenditure will increase GDP by

100 * multiplier = $100 * 4 = $400

Tax multiplier: Also, a change in tax rate also causes a change in the GDP. A change in tax by a given amount will result in larger change in the real GDP but in an opposite direction. It is calculated as

Tax multiplier = - MPC / (1-MPC)

With a MPC of 0.75, the tax multiplier = - (0.75 / (1-0.75) = - 3.

The magnitude of the change is 1 less than that of the expenditure multiplier. Also, the negative sign implies that there is an inverse relationship between tax change and real GDP

Using our question, the increase in the GDP as a result of tax cut of $100 is:

Increase in GDP = 3 * 100 = $300

Observe that the impact of the tax on GDP is less than that of government spending. The tax impact is always indirect unlike the government spending. This means tax rebates given to a person DOES NOT count as expenditure for the giver (the Government) but only counts for the beneficiary as he starts spending.

So the tax cut of $100 does count as expenditure for the government, as it receives no value. It is not an exchange. And only a proportion of the $100 rebate will start counting as expenditure when the beneficiary start to spend it