Principles of Economics/Multipliers

The Concept
The multiplier effect refers to the idea that an initial spending rise can lead to even greater increase in national income. In other words, an initial change in aggregate demand can cause a further change in aggregate output for the economy. This is because the newly circulated money changes hands several times, each time shrinking in amount due to savings yet adding more to economic activity. Eventually, this cycle involves infinitesimally small amounts of money, so the cycle actually does come to an end, based on the basic geometric series rule:


 * $$ (r)^1 + (r)^2 + (r)^3 + ... = \frac{ 1 }{ 1 - r } - 1 = \frac{ r }{ 1 - r } $$


 * For example: A company spends $1 million to build a factory. This money goes to builders, who in total now have $1 million. They may save 25% of that and use the rest to buy goods. The producers of those goods obtain 75% of the original $1 million and save 25% of that, then spend some more. The multiplier here is 4, because an initial $1 million injected into the economy eventually becomes used the equivalent of four times.

The Multipliers
Note: In the following examples the multiplier is the right-hand-side equation without the first component.


 * y is original output (GDP)
 * $$b_C$$ is marginal propensity of consumption (MPC)
 * $$b_T$$ is original income tax rate
 * $$b_M$$ is marginal propensity to import
 * $$\Delta y$$ is change in output (equivalent to GDP)
 * $$\Delta a_T$$ is change in lump-sum tax rate
 * $$\Delta b_T$$ is change in income tax rate
 * $$\Delta G$$ is change in government spending
 * $$\Delta T$$ is change in aggregate taxes
 * $$\Delta I$$ is change in investment
 * $$\Delta X$$ is change in exports

Standard Lump-sum Tax Equation
$$\Delta y = \Delta a_T * \frac{- b_C}{1 - b_C(1 - b_T) + b_M}$$

Note: only $$\Delta a_T$$ is here because if this is a change in lump-sum tax rate then $$\Delta b_T$$ is implied to be 0.

Standard Income Tax Equation
$$\Delta y = \Delta b_T * \frac{- b_C * y}{1 - b_C(1 - b_T) + b_M}$$

Note: only $$\Delta b_T$$ is here because if this is a change in income tax rate then $$\Delta a_T$$ is implied to be 0.

Standard Government Spending Equation
$$\Delta y = \Delta G * \frac{1}{(1 - b_C)(1 - b_T) + b_M}$$

Standard Investment Equation
$$\Delta y = \Delta I * \frac{1}{1 - b_C(1 - b_T) + b_M}$$

Standard Exports Equation
$$\Delta y = \Delta X * \frac{1}{1 - b_C(1 - b_T) + b_M}$$

Balanced-Budget Government Spending Equation
$$\Delta y = \Delta G * 1$$

$$\Delta y = \Delta T * 1$$