autonomous and induced investment; the accelerator

1. Micro‑economics foundations (AS 1.1‑2.5)

1.1 Demand and supply

  • Demand curve – downward sloping; shows the quantity of a good that consumers are willing to buy at each price.
  • Supply curve – upward sloping; shows the quantity firms are willing to produce at each price.
  • Market equilibrium – the point where the two curves intersect (price = Pe, quantity = Qe).
  • Shifts – changes in determinants (income, tastes, input prices, technology, expectations, number of buyers/sellers) move the whole curve, not just a movement along it.

1.2 Elasticities

ElasticityFormulaInterpretation
Price elasticity of demand (PED) \(\displaystyle \varepsilon_{d}=\frac{\%\Delta Q_{d}}{\%\Delta P}\) ‑ |ε| > 1 = elastic; |ε| < 1 = inelastic; |ε| = 1 = unit‑elastic.
Price elasticity of supply (PES) \(\displaystyle \varepsilon_{s}=\frac{\%\Delta Q_{s}}{\%\Delta P}\) Positive; larger values indicate a more responsive supply.
Income elasticity of demand (YED) \(\displaystyle \varepsilon_{y}=\frac{\%\Delta Q_{d}}{\%\Delta Y}\) >0 = normal good; <0 = inferior good; >1 = luxury.
Cross‑price elasticity of demand (XED) \(\displaystyle \varepsilon_{xy}=\frac{\%\Delta Q_{d}^{x}}{\%\Delta P^{y}}\) >0 = substitutes; <0 = complements.

1.3 Consumer and producer surplus

  • Consumer surplus (CS) – the difference between what consumers are willing to pay (area under the demand curve) and what they actually pay (price).
  • Producer surplus (PS) – the difference between the price received and the minimum price at which firms are willing to supply (area above the supply curve).
  • Both are measures of welfare; a change in price or a policy shift can increase or decrease CS and PS.

1.4 Example – Coffee market

Demand: \(Q_{d}=120-2P\); Supply: \(Q_{s}=20+3P\).
Equilibrium: \(120-2P=20+3P\Rightarrow P_{e}=20\), \(Q_{e}=80\).
If the government imposes a £5 per‑cup tax on producers, supply shifts up to \(Q_{s}=20+3(P-5)\). The new equilibrium price to consumers rises to £23, reducing CS and increasing PS by the tax revenue.

2. Government micro‑intervention (AS 3.1‑3.3)

2.1 Why intervene? (Market failure)

  • Public goods – non‑rival and non‑excludable (e.g., street lighting).
  • Externalities – costs or benefits that affect third parties (e.g., pollution, education).
  • Merit and demerit goods – goods that are under‑ or over‑consumed relative to the socially optimal level.
  • Information asymmetry – one side of a transaction has better information (e.g., used‑car market).

2.2 Main policy tools

ToolHow it worksTypical welfare effect
Taxes (excise, Pigouvian) Raise the price of a demerit good or internalise a negative externality. Reduces over‑consumption; generates revenue.
Subsidies Lower the price of a merit good or internalise a positive externality. Encourages under‑consumed goods; may create fiscal cost.
Price controls (ceilings/floors) Set a maximum (ceiling) or minimum (floor) price. Can cause shortages (ceilings) or surpluses (floors) if set away from equilibrium.
Quotas / import licences Limit the quantity of a good that can be imported or produced. Protect domestic industry; may raise domestic price.
Direct provision Government produces the good itself (e.g., NHS). Ensures universal access; cost borne by public finances.
Information & education campaigns Change consumer preferences or awareness. Often low‑cost; impact depends on behavioural response.

3. The circular flow of income

3.1 Closed‑economy model (households ↔ firms)

  • Factor market – households sell labour, land, capital, entrepreneurship; receive factor‑income (wages, rent, interest, profit).
  • Product market – firms sell goods and services; households spend factor‑income on these outputs.
  • Money continuously circulates from factor‑income to consumption expenditure and back.
Diagram 1 – Closed‑economy circular flow (households, firms, factor market, product market).

3.2 Open‑economy model (adding government and foreign sector)

  • Government – collects taxes (T) and makes transfers; injects spending (G) into the product market.
  • Foreign sector – exports (X) are injections; imports (M) are leakages.

National‑income identity:

\[ Y = C + I + G + (X - M) \]
Diagram 2 – Open‑economy circular flow (adds government and foreign sector).

4. National‑income aggregates (Cambridge 9708 4.1‑4.2)

AggregateDefinition (market‑price basis)Relation to other aggregates
GDP (Gross Domestic Product) Market value of all final goods and services produced within a country’s borders in a given period. GDP = GNP – Net factor income from abroad (NFI).
GNI (Gross National Income) GDP + NFI – the total income earned by residents, whether domestically or abroad. GNI = GDP + NFI.
NNI (Net National Income) GNI – Depreciation (consumption of fixed capital). NNI = GNI – Depreciation.

Illustrative UK 2023 figures: GDP ≈ £2.7 trn, NFI ≈ ‑£30 bn, depreciation ≈ £120 bn → NNI ≈ £2.55 trn.

5. Aggregate demand and aggregate supply (syllabus 4.3)

  • Aggregate demand (AD): \(AD = C + I + G + (X-M)\). Downward‑sloping because a higher price level reduces real spending via wealth, interest‑rate and exchange‑rate effects.
  • Short‑run aggregate supply (SRAS): upward‑sloping; some input prices are sticky, so firms increase output when the price level rises.
  • Long‑run aggregate supply (LRAS): vertical at potential output (YP); determined by factor endowments, technology and institutional factors.

5.1 Typical shifts

ShockAD shiftSRAS shiftLRAS shift
Expansionary fiscal policy (↑G) Right
Oil price rise Left
Technological progress Right (lower costs) Right (higher YP)

6. Investment: autonomous and induced (syllabus 5.1‑5.2)

6.1 Definition

Investment (I) = expenditure on new capital goods (machinery, factories, housing) that adds to the productive capacity of the economy.

6.2 Autonomous investment (Ia)

  • Does not vary with the current level of income.
  • Determined by long‑run expectations and external factors:
    • Business confidence about future profitability.
    • Interest rates and the cost of borrowing.
    • Technological change and innovation.
    • Government policy (tax incentives, subsidies, public‑investment programmes).
  • In the short‑run Keynesian model it is treated as a constant.

6.3 Induced investment (Ii)

Varies with national income. The simplest functional form is linear:

\[ I_i = \alpha \, \Delta Y \]
  • \(\alpha\) = marginal propensity to invest (MPI); the proportion of any change in income that is spent on new capital.
  • Typical empirical range for advanced economies: 0.05 – 0.15.

6.4 Total investment function

\[ I = I_a + \alpha \, \Delta Y \]

This expression underpins the multiplier‑accelerator model.

7. The accelerator principle (syllabus 5.3)

7.1 Concept

Firms aim to keep a constant capital‑output ratio \(v\) (capital required per unit of output). When output changes, firms adjust their desired capital stock, generating induced investment.

7.2 Simple accelerator model

Desired capital stock:

\[ K^{d}=vY \]

Change in desired capital stock:

\[ \Delta K^{d}=v\Delta Y \]

Since investment is the change in the capital stock, the accelerator equation is:

\[ I = I_a + v\Delta Y \]
  • \(v\) is the accelerator coefficient (units of capital per unit of output).
  • A larger \(v\) → a stronger response of investment to output changes.
Diagram 3 – Accelerator feedback loop: higher output → higher desired capital → induced investment → further output increase.

7.3 Numerical illustration

  • Autonomous investment \(I_a = £20\) bn
  • Accelerator coefficient \(v = 0.2\) (0.2 units of capital per £1 of output)
  • Initial output \(Y_0 = £500\) bn

If output rises by £10 bn (\(\Delta Y = 10\) bn), induced investment = \(v\Delta Y = 0.2 \times 10 = £2\) bn, so total investment = £22 bn.

8. Interaction of the multiplier and the accelerator (syllabus 5.4)

8.1 Keynesian multiplier

\[ k = \frac{1}{1 - c(1-t)} \]
  • \(c\) = marginal propensity to consume (MPC).
  • \(t\) = tax rate (as a proportion of income).

8.2 Combined multiplier‑accelerator process

Assume an initial autonomous injection \(\Delta A\) (e.g., a rise in G). The first round raises income by \(k\Delta A\). This extra income induces investment \(\alpha k\Delta A\), which becomes a new autonomous injection for the next round, and so on.

RoundAutonomous injectionChange in incomeInduced investment
1\(\Delta A\)\(k\Delta A\)\(\alpha k\Delta A\)
2\(\alpha k\Delta A\)\(k\alpha k\Delta A\)\(\alpha k\alpha k\Delta A\)
3\(\alpha k\alpha k\Delta A\)\(k(\alpha k)^2\Delta A\)\(\alpha k(\alpha k)^2\Delta A\)

The total change in income is the sum of an infinite geometric series:

\[ \Delta Y_{\text{total}} = \frac{k\Delta A}{1 - \alpha k}\qquad\text{provided }\alpha k < 1 \]

A product \(\alpha k\) close to 1 gives a large overall impact.

8.3 Example – UK fiscal stimulus 2024

  • \(c = 0.75\), \(t = 0.20\) → \(k = \dfrac{1}{1-0.75\times0.8}=3.33\).
  • \(\alpha = 0.10\) (MPI) → \(\alpha k = 0.333\).
  • Autonomous increase in government spending \(\Delta A = £5\) bn.
  • Total predicted rise in output: \[ \Delta Y = \frac{3.33 \times 5}{1-0.333}\approx £25\text{ bn} \]

9. Macro‑policy mix (syllabus 5.5‑5.6, AS 5.1‑5.4, A‑Level 10.1‑10.3)

9.1 Fiscal policy

  • Components: Government spending (G), taxation (T), and borrowing (ΔB).
  • Expansionary move: ↑G or ↓T (or a combination) → raises autonomous injection \(\Delta A\).
  • Contractionary move: ↓G or ↑T → reduces \(\Delta A\).
  • Fiscal policy directly influences the multiplier and, via the accelerator, the induced‑investment channel.

9.2 Monetary policy

  • Instruments: policy interest rate, reserve‑requirement ratio, open‑market operations.
  • Lower interest rates:
    • Reduce the cost of borrowing → raise autonomous investment \(I_a\).
    • May increase the marginal propensity to invest \(\alpha\) (stronger accelerator).
    • Can also raise consumption (higher \(c\)) → larger multiplier.
  • Higher rates have the opposite effects.

9.3 Supply‑side (structural) policy

  • Improving education and training, investing in infrastructure, deregulation and tax reforms.
  • Shift LRAS right (higher potential output) and can lower the accelerator coefficient \(v\) by making capital more productive.

9.4 Exchange‑rate policy (A‑Level 10.2)

  • Fixed exchange rate – government intervenes to maintain a set rate; can affect net exports (X‑M) and thus AD.
  • Floating exchange rate – market‑determined; changes in interest rates cause capital flows that appreciate or depreciate the currency, influencing export competitiveness.
  • Depreciation → ↑X, ↓M → rightward AD shift; appreciation does the opposite.

9.5 Policy conflicts

  • Stabilisation vs. growth – a large fiscal stimulus may boost output (good for growth) but risk demand‑pull inflation.
  • Fiscal‑monetary coordination – if fiscal expansion is matched by monetary tightening, the net effect on AD may be muted.
  • Supply‑side vs. demand‑side – supply measures improve LRAS without creating inflationary pressure, whereas demand measures shift AD.

10. Labour market (A‑Level 8.3)

10.1 Demand for labour

  • Derived from the marginal product of labour (MPL); firms hire up to the point where MPL = real wage (w).
  • Shift factors: technology, product demand, capital intensity, labour‑saving innovation.

10.2 Supply of labour

  • Depends on population, participation rates, education, and wage expectations.
  • Higher real wages encourage more people to work (upward‑sloping supply curve).

10.3 Wage determination & market structures

  • Perfectly competitive labour market – wage equals MPL; equilibrium employment.
  • Monopsony – a single large employer faces an upward‑sloping labour supply; sets wage below MPL, causing under‑employment.
  • Trade unions – can push wages above equilibrium, potentially creating a labour surplus (unemployment).

10.4 Policy tools

ToolObjectiveTypical effect on AD/LRAS
Minimum wage Increase real wages for low‑paid workers May raise consumption (AD) but can increase labour costs → leftward SRAS.
Training programmes Raise skill levels, productivity Shift LRAS right (more efficient labour).
Unemployment benefits Provide income support Increase consumption (AD) but may reduce job‑search effort.

11. International economics (AS 6.1‑6.5, A‑Level 11.1‑11.6)

11.1 Why trade?

  • Comparative advantage – countries specialise in goods with lower opportunity cost.
  • Result: higher world output, gains from trade for all participants.

11.2 Protectionist instruments and welfare effects

InstrumentDiagram effectWelfare impact
Tariff Domestic price rises above world price; quantity falls. Consumer surplus ↓, producer surplus ↑, government revenue ↑ → net loss (dead‑weight loss).
Quota Quantity limited; price rises to the quota price. Similar dead‑weight loss; rents accrue to licence holders.
Subsidy to exporters Domestic price falls; exports increase. Producer surplus ↑, government outlay ↑ → net loss.

11.3 Balance of payments (BoP)

  • Current account = (X‑M) + net primary income + net secondary income.
  • Capital & financial account records net capital flows.
  • BoP identity: Current account + Capital & financial account + Errors & omissions = 0.

11.4 Exchange‑rate effects

  • Marshall‑Lerner condition: a depreciation improves the trade balance if \(|\varepsilon_{x}| + |\varepsilon_{m}| > 1\) (sum of price elasticities of export and import demand).
  • J‑curve: after a depreciation, the trade balance may initially worsen before improving as volumes adjust.

11.5 Development and sustainability

  • Key indicators: GDP per capita, Human Development Index (HDI), Gini coefficient (inequality), poverty rates.
  • Inclusive growth – growth that is broad‑based and reduces inequality.
  • Sustainable growth – growth that can be maintained without depleting natural resources or causing environmental damage.

12. Key points to remember

  • Micro‑foundations: demand‑supply curves, elasticities, consumer & producer surplus.
  • Government can intervene to correct market failure using taxes, subsidies, price controls, quotas, direct provision and information campaigns.
  • Closed‑economy circular flow links households and firms; open‑economy adds government and foreign sector (taxes, transfers, exports, imports).
  • GDP, GNI and NNI differ in scope (domestic vs. national) and whether depreciation is deducted.
  • AD = C + I + G + (X‑M); SRAS slopes upward, LRAS is vertical at potential output.
  • Investment = autonomous component (\(I_a\)) + induced component (\(\alpha\Delta Y\)).
  • Accelerator: \(I = I_a + v\Delta Y\); \(v\) measures the required capital per unit of output change.
  • Multiplier‑accelerator total impact: \(\displaystyle \Delta Y_{\text{total}} = \frac{k\Delta A}{1-\alpha k}\). Larger \(\alpha k\) → larger overall effect.
  • Fiscal policy changes \(\Delta A\); monetary policy influences \(I_a\) and \(\alpha\); supply‑side measures shift LRAS; exchange‑rate policy affects net exports.
  • Labour market equilibrium is where real wage equals marginal product of labour; monopsony and unions can cause wage‑employment distortions.
  • International trade generates gains from comparative advantage; protectionist tools create dead‑weight losses. Exchange‑rate movements affect the trade balance via the Marshall‑Lerner condition and the J‑curve.
  • Policy makers must balance stabilisation (inflation, unemployment), growth (shifting LRAS), and distributional objectives (equity, sustainability).

13. Suggested diagram set

  • (a) Closed‑economy circular flow.
  • (b) Open‑economy circular flow.
  • (c) AD/AS framework with typical shocks.
  • (d) Accelerator feedback loop.
  • (e) Demand‑supply diagram with price elasticity illustration.
  • (f) Consumer & producer surplus diagram.
  • (g) Labour‑market diagram (MPL = wage).
  • (h) Trade‑policy diagram (tariff impact on CS, PS, government revenue).
  • (i) Balance‑of‑payments flow chart.
  • (j) Exchange‑rate depreciation and the J‑curve.

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