The concept and formula of the multiplier

The multiplier illustrates how an initial change in spending can lead to a larger overall increase in gross domestic product (GDP), also referred to as national income. This occurs because additional spending generates income, part of which is then spent again, creating further income in a chain reaction.

The multiplier effect

An injection into the economy, such as increased investment or government spending, raises incomes for recipients. These individuals spend a portion of their extra income, generating more income for others. The process continues through multiple rounds, with the total rise in national income exceeding the initial injection. The effect diminishes over time due to withdrawals (leakages) like saving, taxation, or imports, which remove money from circulation.

Formula for the multiplier

$\text{Multiplier}=\frac{\text{change in income}}{\text{change in injection}}=\frac{\mathrm{\Delta }Y}{\mathrm{\Delta }J}$

Where:

• ΔY = Change in national income
• ΔJ = Change in an injection (e.g., investment, government spending, or exports)

The multiplier in different economic models

The size of the multiplier varies depending on the economic model's complexity, particularly the types of injections and withdrawals included. More withdrawals generally result in a smaller multiplier.

Multiplier in a closed economy (two sectors)

This model involves households and firms only, with one injection (investment) and one withdrawal (saving). The equilibrium condition is: consumption (C) + investment (I) = national income (Y), and I = S.

$\text{Multiplier}=\frac{1}{\text{mps}}$

Where:

• mps = Marginal propensity to save

An alternative form is:

$\text{Multiplier}=\frac{1}{1-\text{mpc}}$

Where:

• mpc = Marginal propensity to consume

Multiplier in a closed economy with government sector

This model adds the government sector, creating an additional injection (government spending) and withdrawal (taxation). The equilibrium condition becomes: C + I + G = Y, and I + G = S + T.

$\text{Multiplier}=\frac{1}{\text{mps}+\text{mrt}}$

Where:

• mps = Marginal propensity to save
• mrt = Marginal rate of taxation

Multiplier in an open economy with government sector

This model includes all four sectors: households, firms, government, and foreign trade. It adds exports as an injection and imports as a withdrawal. The equilibrium condition is: C + I + G + (X - M) = Y, and I + G + X = S + T + M. The inclusion of three withdrawals (saving, taxation, imports) reduces the multiplier's size.

$\text{Multiplier}=\frac{1}{\text{mps}+\text{mrt}+\text{mpm}}$

Where:

• mps = Marginal propensity to save
• mrt = Marginal rate of taxation
• mpm = Marginal propensity to import

Average and marginal propensities and rates

Propensities measure how changes in income affect consumption, saving, taxation, and imports. Average propensities relate to total income, while marginal propensities focus on changes in income.

Propensities to save

Saving represents disposable income not spent on consumption.

Average propensity to save (aps) - The proportion of total income saved:

$\text{aps}=\frac{\text{S}}{\text{Y}}$

Where:

• S = Saving
• Y = Total income

Marginal propensity to save (mps) - The proportion of additional income saved:

$\text{mps}=\frac{\mathrm{\Delta }\text{S}}{\mathrm{\Delta }\text{Y}}$

At low income levels, dissaving may occur (spending exceeds income). As income rises, both aps and mps typically increase.

Propensities to consume

Consumption is primarily influenced by disposable income levels.

Average propensity to consume (apc) - The proportion of total income spent on consumption:

$\text{apc}=\frac{\text{C}}{\text{Y}}$

Where:

• C = Consumption
• Y = Total income

Marginal propensity to consume (mpc) - The proportion of additional income spent on consumption:

$\text{mpc}=\frac{\mathrm{\Delta }\text{C}}{\mathrm{\Delta }\text{Y}}$

The relationship between these measures is: apc = 1 - aps. As income increases, the proportion spent (apc) tends to decrease.

Rates of taxation

Average rate of taxation (art) - The proportion of total income paid in taxes:

$\text{art}=\frac{\text{T}}{\text{Y}}$

Where:

• T = Taxation
• Y = Total income

Marginal rate of taxation (mrt) - The proportion of additional income paid in taxes:

$\text{mrt}=\frac{\mathrm{\Delta }\text{T}}{\mathrm{\Delta }\text{Y}}$

Propensities to import

Average propensity to import (apm) - The proportion of total income spent on imports:

$\text{apm}=\frac{\text{M}}{\text{Y}}$

Where:

• M = Imports
• Y = Total income

Marginal propensity to import (mpm) - The proportion of additional income spent on imports:

$\text{mpm}=\frac{\mathrm{\Delta }\text{M}}{\mathrm{\Delta }\text{Y}}$

These vary widely across countries, depending on factors like domestic production capacity.

National income determination

National income (GDP) is determined at the point where aggregate demand equals aggregate supply, or equivalently, where aggregate expenditure matches total output. This balance ensures economic equilibrium.

Components of aggregate expenditure

Aggregate expenditure consists of consumption (C), investment (I), government spending (G), and net exports (X - M):

$\text{Aggregate expenditure}=\text{C}+\text{I}+\text{G}+(\text{X}-\text{M})$

How equilibrium is achieved

• If aggregate expenditure exceeds current output, firms increase production, raising GDP.
• If aggregate expenditure falls below current output, firms cut production, lowering GDP.
• This process is represented in the Keynesian 45° diagram, where the aggregate expenditure curve intersects the 45° line (representing output equals expenditure).
• Note: An aggregate demand curve shows total spending against price levels, while an aggregate expenditure curve shows total spending against income levels.

Effects of changes in aggregate demand on national income

Changes in aggregate demand, through injections or withdrawals, lead to multiplied effects on national income. The total change equals the initial shift multiplied by the multiplier.

How changes impact national income

An increase in spending (e.g., higher investment) raises national income by the injection amount times the multiplier. The increase happens gradually as recipients spend part of their new income, continuing until withdrawals match the injection.

The paradox of thrift

A rise in saving reduces aggregate demand, causing GDP to fall. This is known as the paradox of thrift, where higher saving lowers overall income, reducing the total amount that can be saved in the economy.

If households increase saving, consumption falls, leading to lower production and incomes. Although individuals aim to save more, the reduced economic activity means total savings may actually decrease.

Worked example - Calculating the change in national income

In a closed economy with a government sector, the marginal propensity to save is 0.1 and the marginal rate of taxation is 0.15. If government spending increases by £300 million, calculate the multiplier and the total rise in national income.

Step 1: Identify the values

• mps = 0.1
• mrt = 0.15
• Change in injection (government spending) = £300 million

Step 2: Calculate the multiplier

$\text{Multiplier}=\frac{1}{\text{mps}+\text{mrt}}$

$\text{Multiplier}=\frac{1}{0.1+0.15}=\frac{1}{0.25}=4$

Step 3: Calculate the change in national income

$\text{Change in national income}=\text{change in injection}\times \text{multiplier}$

$\text{Change in national income}=\pounds 300\text{ million}\times 4=\pounds \mathrm{1,200}\text{ million}$