savings function: autonomous and induced savings

The Circular Flow of Income (Closed & Open Economy)

1. The Five Main Sectors

• Households – supply labour, land, capital and entrepreneurship; receive income (wages, rent, interest, profit).
• Firms – purchase factors of production, produce goods & services and sell them to households, other firms, the government and the Rest of the World.
• Government – collects taxes, makes transfer payments and purchases goods & services (government spending).
• Financial Institutions – intermediate between savers and borrowers; channel savings into investment.
• Rest of the World (ROW) – trades goods and services with the domestic economy (exports and imports).

2. Injections & Leakages

3. Equilibrium Condition

In equilibrium total injections equal total leakages:

For a closed economy (no trade) the condition reduces to:

4. Link to National‑Income Aggregates

4.1 Expenditure Approach

• GDP (Gross Domestic Product) – total market value of final goods and services produced within the domestic borders.
  $$\text{GDP}=C+I+G+X$$
• GNI (Gross National Income) – GDP plus net primary income earned from abroad (e.g., wages, dividends).
  $$\text{GNI}= \text{GDP}+ \text{Net Primary Income from abroad}$$
• NNI (Net National Income) – GNI less depreciation of the capital stock.
  $$\text{NNI}= \text{GNI} - \text{Depreciation}$$

4.2 Income Approach

The income side of the circular flow adds up the factor‑payments received by households:

Because the same total income is either spent (C + I + G + X) or earned (W + R + I + P), the two approaches must give the same figure.

4.3 How the Flow Diagram Relates to the Aggregates

• Arrows from households to firms represent factor payments (income approach).
• Arrows from firms to households represent expenditure on consumption, investment, government purchases and exports (expenditure approach).
• Leakages (S, T, M) and injections (I, G, X) are the channels through which the circular flow is disturbed or restored, linking directly to the components of the GDP equation.

5. The Multiplier Process

5.1 Definition

The multiplier measures the total change in national income (ΔY) that results from an initial change in an autonomous injection (ΔA) such as government spending, investment or exports.

5.2 Propensities

Because every additional unit of disposable income must be either consumed, saved, taxed or spent on imports:

5.3 Formal Multiplier Formulae

• Closed economy (no trade, no taxes) – only consumption and saving matter:
  $$k_{\text{closed}} = \frac{1}{s} = \frac{1}{1-c}$$
• Closed economy with taxes (but no imports):
  $$k_{\text{closed, tax}} = \frac{1}{s + t} = \frac{1}{1 - c}$$ (since c + s + t = 1)
• Open economy (with taxes and imports):
  $$k_{\text{open}} = \frac{1}{s + t + m} = \frac{1}{1 - c}$$ (again because c + s + t + m = 1)

All three expressions show that the multiplier is greater than one because s + t + m < 1.

5.4 Why the Multiplier Is > 1

1. An autonomous injection (e.g., ΔG) raises income by the same amount in the first round.
2. Households receive that extra income; a part is spent again (c × ΔG) creating a second round of income.
3. The process repeats, each round becoming smaller because a fraction (s + t + m) is withdrawn.
4. The infinite series sums to ΔY = k × ΔA, where k > 1.

5.5 Numerical Example (Open Economy)

Assume:

• MPC c = 0.70
• MPS s = 0.15
• MPT t = 0.10
• MPM m = 0.05

Check: c + s + t + m = 0.70 + 0.15 + 0.10 + 0.05 = 1.00.

Open‑economy multiplier:

If the government increases spending by £10 million, the total rise in equilibrium income will be:

Savings Function

1. Autonomous Savings ( S_a )

• Definition: The amount of saving that occurs even when disposable income is zero.
• Typical sources:
  • Precautionary motives – households keep a cash buffer.
  • Institutional requirements – minimum balances in bank accounts.
  • Policy‑driven incentives – tax‑advantaged pensions, ISAs.
• Appears as the constant term in the savings function.

2. Induced (Marginal) Savings

• Definition: Saving that varies directly with disposable income (Y).
• Captured by the Marginal Propensity to Save (MPS), symbol s, where 0 < s < 1.
• Formula: s = ΔS / ΔY.

3. Combined Savings Function

The total saving in the economy is the sum of autonomous and induced components:

where:

• S – total saving.
• S_a – autonomous saving (constant).
• s – marginal propensity to save (MPS).
• Y – disposable income (after taxes and transfers).

4. Relationship with the Consumption Function

Expenditure identity: Y = C + S.
The A‑Level consumption function is:

Because every extra unit of disposable income is either consumed or saved:

Thus the savings and consumption functions are mirror images of each other.

5. Numerical Example

Assume:

• Autonomous saving S_a = £20 million
• MPS s = 0.25
• Disposable income Y = £200 million

Induced saving = sY = 0.25 × 200 = £50 million.

Total saving = S = £20 m + £50 m = £70 million.

6. Table of Key Components

7. Policy Implications

1. Fiscal policy – A change in tax rates alters disposable income (Y) and therefore the induced saving component sY.
2. Monetary policy – Higher interest rates raise the incentive to save, potentially increasing both S_a (precautionary/interest‑bearing saving) and the MPS s.
3. Supply‑side measures – Compulsory pension contributions or “auto‑enrolment” raise autonomous saving, shifting the whole savings function upward.

8. Key Points to Remember

• The A‑Level savings function is linear: S = S_a + sY.
• Autonomous saving (S_a) is independent of income; induced saving (sY) varies with disposable income.
• Because c + s = 1, any increase in the marginal propensity to consume reduces the marginal propensity to save, and vice‑versa.
• The savings function links directly to the multiplier: a higher MPS (s) makes the multiplier smaller (k = 1/(s + t + m)), while a lower s makes the multiplier larger.
• Understanding savings is essential for analysing how changes in income affect investment through the financial sector and how fiscal/monetary policies can influence overall economic equilibrium.