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Business Economics · Determination of National Income

Investment Multiplier and Government Sector: Formulas and Numericals

Updated 1 October 2026 · Fact-checked

The investment multiplier shows how much equilibrium income rises for each extra rupee of investment: k = 1 ÷ (1 − MPC) = 1 ÷ MPS. With taxes and imports, divide 1 by (1 − MPC(1 − t) + MPM). Find the total leakage rate, take its reciprocal, then multiply by the change in spending.

Understand Investment Multiplier and Government Sector

Income is not changed only by the first round of spending. When investment rises by ₹100, that money becomes income for workers and suppliers. They spend part of it and save the rest. The part spent becomes someone else's income, and the chain continues. The final rise in income is a multiple of the original rise in investment. This is the multiplier.

The size of the multiplier depends on how much of each extra rupee is spent again. The share spent is the marginal propensity to consume (MPC). The share saved is the marginal propensity to save (MPS), and MPC + MPS = 1. With MPC = 0.8, the rounds add up to 100 + 80 + 64 + ... = 100 ÷ (1 − 0.8) = ₹500. So k = 5.

Anything that removes money from the spending chain is a leakage. In a simple economy the only leakage is saving. In a larger economy, taxes and imports also leak out. More leakage means a smaller multiplier. A higher MPC means a larger one.

The government sector adds two tools. Government spending (G) is an injection, like investment, so its multiplier is the same 1 ÷ (1 − MPC) when taxes are lump sum. Taxes (T) reduce disposable income, so they cut consumption by only MPC times the tax. The tax multiplier is therefore smaller in size than the spending multiplier and is negative.

If the government raises G and T by the same amount, income rises by exactly that amount. This is the balanced budget multiplier, equal to 1. Foreign trade works the same way: exports are an injection, imports are a leakage, and a higher marginal propensity to import lowers the multiplier.

Key formulas to remember

Investment multiplier (two sector)
k = ΔY ÷ ΔI = 1 ÷ (1 − MPC) = 1 ÷ MPS
Valid when saving is the only leakage. MPC + MPS = 1.
Equilibrium income (two sector)
C = a + bY, so Y = (a + I) ÷ (1 − b)
a is autonomous consumption, b is MPC.
Equilibrium income with lump-sum tax
C = a + b(Y − T), so Y = (a − bT + I + G) ÷ (1 − b)
Consumption depends on disposable income Y − T.
Government expenditure multiplier
ΔY ÷ ΔG = 1 ÷ (1 − b)
For lump-sum taxes. Same size as the investment multiplier.
Tax multiplier
ΔY ÷ ΔT = −b ÷ (1 − b)
Negative: a tax rise lowers income. Smaller in size than the G multiplier.
Balanced budget multiplier
(1 ÷ (1 − b)) + (−b ÷ (1 − b)) = 1
Equal rise in G and T raises income by the same amount.
Multiplier with proportional tax
k = 1 ÷ (1 − b(1 − t))
t is the income tax rate.
Multiplier with tax and imports
k = 1 ÷ (1 − b(1 − t) + m)
m is the marginal propensity to import. Denominator equals total leakage rate.
Four-sector equilibrium
Y = C + I + G + X − M
Exports X are an injection, imports M a leakage.

How to solve Investment Multiplier and Government Sector questions

Use this method for any multiplier or equilibrium income question.

  1. 1Identify the sectors given: households only, plus government, plus foreign trade.
  2. 2Write down MPC (b), tax type (lump sum T or rate t) and import rate m, if any. If MPS is given, get MPC = 1 − MPS.
  3. 3If asked for a multiplier, build the denominator: 1 − b(1 − t) + m. Drop any term not given.
  4. 4If asked for equilibrium income, write Y = C + I + G + X − M. Put the consumption function in with disposable income, then solve for Y.
  5. 5If asked for a change in income, compute ΔY = multiplier × change in the injection. Use the tax multiplier −b ÷ (1 − b) for a lump-sum tax change.
  6. 6Check the sign: spending rise gives higher income, tax rise gives lower income.
  7. 7Match your answer to the options and check units (₹ crore or a pure number).

Quickest way: Leakage reciprocal shortcut

When to use it: For any multiplier MCQ with numbers given. Takes under 30 seconds.

  1. Add all leakage rates: MPS (or 1 − b), then adjust for tax as b·t, and add m. Total leakage = 1 − b(1 − t) + m.
  2. Take the reciprocal. That is the multiplier.
  3. Multiply by the change in injection to get ΔY.
  4. For a lump-sum tax change, multiply ΔT by −b ÷ (1 − b), which is the multiplier times b with a minus sign.
  5. Use options to eliminate: a multiplier with taxes or imports must be smaller than 1 ÷ MPS.
  6. If an equilibrium question needs long algebra, test the options by plugging Y into C + I + G. Skip it if it takes more than about a minute.

Common mistakes in Investment Multiplier and Government Sector

  • Using 1 ÷ MPC instead of 1 ÷ (1 − MPC).

    Students remember 'reciprocal' but not of what.

    Fix: The multiplier is the reciprocal of the leakage, which is MPS = 1 − MPC. With MPC 0.8, k = 5, not 1.25.

  • Ignoring tax and import leakages when they are given.

    The two-sector formula is learnt first and applied automatically.

    Fix: Read the data first. If t or m is given, use 1 ÷ (1 − b(1 − t) + m).

  • Treating the tax multiplier as positive or equal to the G multiplier.

    Both are 'government' multipliers, so they look alike.

    Fix: A tax rise lowers disposable income and consumption, so the sign is negative. Its size is b ÷ (1 − b), which is smaller than 1 ÷ (1 − b).

  • Writing C = a + bY when taxes exist.

    Students forget that consumption depends on disposable income.

    Fix: Use C = a + b(Y − T), or b(1 − t)Y for a proportional tax.

  • Applying (1 − t) to MPC in the wrong place, such as 1 − (b − t).

    Mixing up the order of brackets.

    Fix: Disposable income is (1 − t)Y, so only that part is multiplied by b: b(1 − t).

  • Thinking the balanced budget multiplier is zero.

    Students assume a rise in G and T cancels out fully.

    Fix: G adds ΔG × 1/(1 − b), T subtracts b/(1 − b) × ΔT. For equal changes the net effect is ΔG, so the multiplier is 1.

Worked examples

Example 1

In a two-sector economy, MPC is 0.8. Investment rises by ₹200 crore. By how much does equilibrium income rise? Options: (A) ₹250 crore (B) ₹800 crore (C) ₹1,000 crore (D) ₹1,600 crore

Show the solution
  1. MPC = 0.8, so MPS = 1 − 0.8 = 0.2.
  2. Multiplier k = 1 ÷ 0.2 = 5.
  3. ΔY = k × ΔI = 5 × 200 = ₹1,000 crore.

Answer: (C) ₹1,000 crore

Example 2

In an economy, C = 100 + 0.75(Y − T), I = ₹200, G = ₹150 and lump-sum tax T = ₹100. What is equilibrium income? Options: (A) ₹1,000 (B) ₹1,500 (C) ₹1,900 (D) ₹2,000

Show the solution
  1. Y = C + I + G = 100 + 0.75(Y − 100) + 200 + 150.
  2. Expand: 0.75(Y − 100) = 0.75Y − 75.
  3. Y = 100 − 75 + 200 + 150 + 0.75Y = 375 + 0.75Y.
  4. Y − 0.75Y = 375, so 0.25Y = 375.
  5. Y = 375 ÷ 0.25 = ₹1,500.
  6. Check: C = 100 + 0.75 × 1,400 = 1,150. C + I + G = 1,150 + 200 + 150 = 1,500. Matches.

Answer: (B) ₹1,500

Example 3

MPC is 0.8, the income tax rate is 25% and the marginal propensity to import is 0.1. What is the value of the multiplier? Options: (A) 1.25 (B) 2 (C) 2.5 (D) 5

Show the solution
  1. Use k = 1 ÷ (1 − b(1 − t) + m).
  2. b(1 − t) = 0.8 × 0.75 = 0.6.
  3. Denominator = 1 − 0.6 + 0.1 = 0.5.
  4. k = 1 ÷ 0.5 = 2.
  5. Note: ignoring imports gives 1 ÷ 0.4 = 2.5, and ignoring both gives 5. Both are traps.

Answer: (B) 2

Exam tips

  • Always scan for t and m before choosing a formula. Examiners often add them to test whether you adjust the multiplier.
  • For 'change in income' questions, the multiplier is usually the whole task. Do it mentally: reciprocal, then multiply.
  • Remember the three government results: G multiplier 1 ÷ (1 − b), tax multiplier −b ÷ (1 − b), balanced budget multiplier 1. Many one-line theory MCQs are based on them.
  • Theory MCQs ask which item is a leakage (saving, taxes, imports) and which is an injection (investment, government spending, exports). Learn the two lists.
  • With 0.25 negative marking, skip a long equilibrium algebra question if you cannot set it up quickly. Return to it if time remains.

Practice questions from Determination of National Income

Investment Multiplier and Government Sector: frequently asked questions

What is the investment multiplier formula for CA Foundation?

It is k = 1 ÷ (1 − MPC), which equals 1 ÷ MPS. It tells you how many times the change in investment gets multiplied into the change in income. With MPC 0.75, k = 4.

How do I calculate the multiplier with taxes?

With a proportional tax rate t, use k = 1 ÷ (1 − MPC(1 − t)). If imports also leak, add the import rate m to the denominator. Taxes and imports both lower the multiplier.

Why is the tax multiplier smaller than the government expenditure multiplier?

Government spending enters income directly in the first round. A tax cut raises disposable income, and households spend only the MPC part of it. So the tax multiplier in size is MPC times the spending multiplier.

What are leakages in the multiplier process?

Leakages are parts of income that are not spent on domestic output in the next round. They are saving, taxes and imports. The larger the leakage, the smaller the multiplier.