quantity theory of money (MV = PT)

Quantity Theory of Money (QTM) – Cambridge AS & A Level (Topic 9.4)

The Quantity Theory of Money links the amount of money circulating in an economy to the overall price level and the volume of transactions. It provides the basis for many macro‑economic arguments about inflation, monetary policy and the long‑run neutrality of money.

1. The Equation of Exchange

\[ MV = PT \]

• M – Money supply (nominal value of money held by the public).
• V – Velocity of money – the average number of times a unit of money is spent in a given period.
• P – General price level (e.g., CPI index).
• T – Transaction volume – real output measured as the total value of transactions (often approximated by real GDP, Y).

Re‑arranged forms (useful for analysis)

\[ P = \frac{MV}{T}\qquad\text{or}\qquad M = \frac{PT}{V} \] These expressions show how a change in any one variable must be offset by a change in at least one of the others.

2. Functions of Money and Their Relevance to the QTM (AO1)

Function	Explanation	Link to QTM
Medium of exchange	Facilitates transactions without barter.	Ensures that V can be measured – if money is not accepted, velocity falls.
Unit of account	Provides a common measure for valuing goods, services and debts.	Allows the price level P to be expressed as a single index.
Store of value	Enables wealth to be saved and retrieved in the future, provided inflation is low.	If inflation expectations rise, households may spend faster, increasing V.

3. Money‑Supply Aggregates (AO1)

Aggregate	Typical UK components	Relevance for QTM
M0 (monetary base)	Banknotes, coins, and commercial banks’ reserves with the central bank.	Basis for the money multiplier; not used directly in the QTM.
M1 (narrow money)	M0 + demand deposits (current accounts).	Useful for very short‑run cash‑based analysis.
M2 (broad money)	M1 + savings deposits, time deposits, money‑market funds.	Most commonly used in Cambridge questions because it best approximates the “money” that participates in everyday transactions.
M3 (very broad)	M2 + large time deposits, institutional money‑market funds, other liquid assets.	Occasionally mentioned in advanced discussions of policy transmission.

4. The Money Multiplier and the Role of Commercial Banks (AO2)

Commercial banks create money by accepting deposits and extending loans. The relationship between the monetary base (B) and the broader money supply (M) is expressed by the money multiplier (m):

\[ m \;=\; \frac{M}{B} \;=\; \frac{1}{c+r} \]

• c = cash‑holding ratio (cash held by the public as a proportion of deposits).
• r = reserve‑requirement ratio (required reserves as a proportion of deposits).

How changes affect the multiplier

• Higher r → lower m (banks can lend less).
• Higher c → lower m (more cash is held outside the banking system).
• Both ratios move in opposite directions to the multiplier; the effect is symmetric because they appear in the denominator.

Numerical example

If r = 0.10 and c = 0.05, then

\[ m = \frac{1}{0.10+0.05}=6.67 \]

A £1 increase in the monetary base can therefore increase the broad money supply by £6.67, assuming no excess reserves and a stable cash‑holding behaviour.

Diagram (Figure 1) – a straight line through the origin with the monetary base (B) on the horizontal axis and the resulting money supply (M) on the vertical axis; the slope of the line equals the multiplier m. (Insert a simple line diagram when presenting the notes.)

5. Monetary‑Policy Tools (AO2)

Tool	Mechanism	Typical impact on M, V and P
Open‑market operations (OMO)	Buying (expansionary) or selling (contractionary) government securities.	Directly changes the monetary base → shifts M via the multiplier; V usually unchanged in the medium‑run.
Reserve‑requirement ratio	Alters the proportion of deposits banks must hold as reserves.	Changes the multiplier (m) → alters M; V and T are assumed constant in the long run.
Discount (policy) rate	Interest rate charged to commercial banks for borrowing from the central bank.	Influences banks’ willingness to lend, affecting M and potentially V in the short run.
Quantitative easing (QE)	Large‑scale purchase of longer‑term securities when conventional tools are exhausted.	Expands the monetary base; the effect on P depends on the stability of V and T.

6. Core Assumptions of the Quantity Theory (AO2)

1. Velocity (V) is stable in the medium‑to‑long run (or changes predictably).
2. Transaction volume (T) is determined by real factors—technology, resources, labour – and is independent of the money supply in the long run.
3. Money neutrality – changes in M affect only nominal variables (P) and not real output (Y) in the long run.

7. Deriving the Direct Relationship Between Money Supply and Prices (AO1)

Holding V and T constant, a proportional change in M leads to an identical proportional change in P:

\[ \% \Delta P = \% \Delta M \qquad (\Delta V = 0,\; \Delta T = 0) \]

This is the classic “inflation‑money‑supply” link used by monetarists.

8. Why Velocity May Vary (AO2)

• Technological change – e‑payments, digital wallets reduce the need for cash.
• Expectations of future inflation – higher expected inflation can accelerate spending (higher V).
• Financial‑market conditions – credit crunches or liquidity traps lower V.
• Institutional factors – tax policies, payment‑system regulations, and the prevalence of cashless transactions.

9. Short‑Run vs. Long‑Run: The Role of Money in the Real Economy (AO2)

• Short‑run (Keynesian) view: A change in M can affect interest rates, aggregate demand and therefore real output (Y) and employment. The Phillips‑type trade‑off between inflation and output is relevant.
• Long‑run (Monetarist) view: Once prices and expectations adjust, V and T return to their “natural” levels and money becomes neutral – only P changes.

Diagram (Figure 2) – a short‑run aggregate‑demand (AD) curve shifting left/right with a change in M, showing a temporary impact on output (Y) and price level (P). (Insert when presenting.)

10. Evaluation of the Quantity Theory (AO3)

• Velocity instability: Empirical evidence (e.g., during financial crises) shows large swings in V, weakening the proportionality assumption.
• Measurement of T: Real output is usually measured by real GDP, not the total number of transactions. Using GDP can under‑state the true transaction volume, especially in economies with large informal sectors.
• Choice of money aggregate: M2 is preferred for QTM analysis because it captures the bulk of money used in transactions while excluding assets that are not readily spendable. Using M0 or M3 can distort the predicted impact on P.
• Short‑run non‑neutrality: In the presence of price‑stickiness, a rise in M can lower interest rates, boost investment and raise T, offsetting part of the price‑level rise.
• Policy relevance: Modern central banks target inflation (through interest‑rate policy) rather than the money supply because of the above uncertainties. Nevertheless, the QTM remains a useful benchmark for long‑run inflation expectations.
• Liquidity traps: When interest rates are near zero, increases in M may not translate into higher V, limiting the effectiveness of monetary expansion.

11. Numerical Example (AO1 & AO2)

Assume an economy with the following data (values in £ billions unless otherwise stated):

• M = 500 (broad money, M2)
• V = 4 transactions per year (assumed stable)
• T = 2 000 (real GDP in real terms)

Using the QTM:

\[ P = \frac{MV}{T}= \frac{500 \times 4}{2\,000}=1.0 \]

The central bank expands the monetary base by 10 % via open‑market purchases, raising M to 550 (all else equal).

\[ P' = \frac{550 \times 4}{2\,000}=1.10 \]

Result: the price level rises by 10 % – a direct illustration of the proportional relationship.

Evaluation point: The calculation holds V and T constant. In reality, a 10 % increase in M could lower interest rates, stimulate investment, raise real GDP (T) and possibly reduce V (if households postpone spending). The net effect on P would therefore be smaller than 10 %.

12. Suggested Diagrams for Revision (AO2)

13. Summary (Key Take‑aways)

• The Quantity Theory of Money is expressed by MV = PT, linking money, velocity, price level and transaction volume.
• Under the assumptions of constant V and T, a proportional change in the money supply leads to an identical proportional change in the price level.
• M2 is the aggregate most commonly used for QTM analysis because it best approximates the money that actually circulates in transactions.
• The money multiplier shows how the monetary base is amplified by commercial‑bank activity; changes in reserve‑requirement or cash‑holding ratios shift the multiplier.
• Monetary‑policy tools (OMO, reserve ratio, discount rate, QE) influence the monetary base and, indirectly, the price level, but their effectiveness depends on the stability of V and the responsiveness of T.
• In practice, velocity is not perfectly stable, real output can respond to monetary changes in the short run, and modern policy focuses on interest‑rate targeting rather than direct money‑supply targets.