formulae for and calculation of multiplier in a closed and open economy, with and without a government sector

📚 Circular Flow of Income – A Quick Guide

Think of the economy as a big round‑about where households, firms, the government and the rest of the world keep the traffic moving.

• Households provide labour and receive wages, interest and rent.

• Firms produce goods and services and pay households for their labour.

• The government collects taxes and spends on public goods.

• The rest of the world imports goods from us and exports our goods abroad.

💡 Key Concept: The Multiplier

The multiplier tells us how a change in one part of the economy (like a tax cut or a new road) ripples through the whole system.

It’s like a water‑bottle: pour a little water in, and the whole bottle fills up because of the shape of the bottle.

The shape of the “circular flow bottle” depends on whether the economy is closed or open, and whether the government is involved.

📈 Closed Economy (No Government)

SymbolMeaningFormula
\$MPC\$Marginal Propensity to Consume%
MultiplierHow much total income changes per unit change in autonomous spending\$k = \dfrac{1}{1 - MPC}\$

Example: If \$MPC = 0.8\$, then \$k = \dfrac{1}{1-0.8} = 5\$.

A £1 million increase in investment will raise total income by £5 million.

📊 Closed Economy (With Government)

Now we add taxes.

Let \$MPT\$ = Marginal Propensity to Tax (the fraction of extra income that goes to taxes).

SymbolMeaningFormula
\$MPT\$Marginal Propensity to Tax%
Government MultiplierEffect of a change in government spending\$k_g = \dfrac{1}{1 - MPC + MPT}\$

If \$MPC = 0.8\$ and \$MPT = 0.2\$, then \$k_g = \dfrac{1}{1-0.8+0.2} = \dfrac{1}{0.4} = 2.5\$.

A £1 million tax cut increases disposable income by £1 million, but the total income rises only £2.5 million because some of the extra money is taxed away.

🌍 Open Economy (No Government)

Imports act like a drain: part of the money households spend goes abroad.

Let \$MPI\$ = Marginal Propensity to Import.

SymbolMeaningFormula
\$MPI\$Marginal Propensity to Import%
Open‑Economy MultiplierEffect of a change in autonomous spending\$k_{open} = \dfrac{1}{1 - MPC + MPI}\$

Example: \$MPC = 0.75\$, \$MPI = 0.15\$ → \$k_{open} = \dfrac{1}{1-0.75+0.15} = \dfrac{1}{0.4} = 2.5\$.

The presence of imports halves the multiplier compared to a closed economy with the same MPC.

🏛️ Open Economy (With Government)

Combine taxes and imports.

The government multiplier in an open economy is:

\$k_{g,open} = \dfrac{1}{1 - MPC + MPT + MPI}\$

If \$MPC = 0.7\$, \$MPT = 0.1\$, \$MPI = 0.2\$, then

\$k_{g,open} = \dfrac{1}{1-0.7+0.1+0.2} = \dfrac{1}{0.6} \approx 1.67\$.

🛠️ Quick Calculation Checklist

  1. Identify the type of economy: closed/open, with/without government.

  2. Write down the relevant propensities: \$MPC\$, \$MPT\$, \$MPI\$.

  3. Plug them into the appropriate multiplier formula.

  4. Multiply the autonomous spending change by the multiplier to get total income change.

🤔 Quick Quiz

  • What happens to the multiplier if the marginal propensity to import increases?

  • Why does adding a government sector usually reduce the multiplier?

  • Calculate the multiplier for a closed economy where \$MPC = 0.85\$.

Answers are in the back of the book – or ask your teacher! 🎓