Separate a price level, its growth rate, and a causal account of persistent inflation.
By the end: Calculate price changes and evaluate demand, supply, and monetary explanations without confusing identities with causes.
Inflation is the rate of change of a price index over a stated interval. A simple fixed-basket index compares the cost of the same quantities today with their cost in a base period. If a basket cost 100 and now costs 110, the increase is 10%. An official consumer index uses many products, expenditure weights, and methods for changing quality and consumption patterns; our three-date example is deliberately simpler.
Different households buy different baskets. If energy has a 10% initial expenditure weight and its price rises 50% while everything else is unchanged, this fixed basket rises 5%. With a 40% energy weight it rises 20%. Neither is a mistake. A representative index is not a promise that every household experiences the same change, and asset prices such as existing share prices are not automatically consumer-price components.
A percentage change compares a difference with the amount at the beginning of the interval. If a basket's cost rises from 80 to 88, the increase is 8. Divide 8 by 80 to obtain 0.10, or 10%. The new cost is 110% of the old cost, but inflation is 10%, not 110%. To make the initial index equal 100, divide every basket cost by 80 and multiply by 100. The two index levels are then 100 and 110. Index points are scaled numbers, not currency units that someone can withdraw from an account.
Construct an additional paper basket with four bread units at 2 each and two energy units at 6 each. Its initial cost is 8 for bread plus 12 for energy, totaling 20. Later, bread costs 2.50 and energy costs 9. Keep the quantities fixed. Bread now costs 10 and energy costs 18, so the same basket costs 28. The increase is 8 on an initial cost of 20, giving 40% inflation for this basket. An index with initial level 100 therefore reaches 140. The quantities matter as much as the individual price changes.
Weights express each item's initial share of the basket's spending. Bread initially takes 8 of 20, or 40%, while energy takes 60%. Bread's price rises 25% and energy's rises 50%. Forty percent of the 25% rise contributes 10 percentage points to basket inflation. Sixty percent of the 50% rise contributes 30 percentage points. Together they give 40%. Taking the simple average of 25% and 50% would give 37.5% and would incorrectly treat initial spending on the two categories as equal. These weights belong to this fixed basket and this starting date.
A price index tries to compare prices for comparable consumption, not every change in a household's expenditure. Suppose the household reacts by buying only one energy unit while keeping four bread units. Its later spending is 10 plus 9, totaling 19, below the original 20. That does not establish falling prices. The household consumed less energy. The fixed original basket still costs 28. An official index may update quantities or allow for substitution using stated methods, but it must distinguish price comparison from changes in what people buy. Lower expenditure can reflect reduced consumption rather than cheaper living.
Product changes create another comparison problem. A price of 12 for a package cannot be compared with an earlier price of 10 without knowing the package size and contents. If the old package held ten identical units, each cost 1. If the new package holds eight, each costs 1.50. The package price rose 20%, but the price per comparable unit rose 50%. Quality changes are harder because different characteristics may matter to different buyers. Official measurement methods address such comparisons; our fixed baskets avoid them by assuming unchanged goods. This assumption makes arithmetic transparent, not universally adequate.
A price index moving from 100 to 110 and then 112.2 has inflation of 10% followed by 2%. That is disinflation, a fall in the inflation rate. Prices are still rising and are 12.2% above the initial level. Deflation means a falling price level over the interval. If an index jumps to 110 and then stays there, its next annual inflation rate is zero, though the earlier loss of purchasing power remains.
One-time relative-price shocks can raise measured inflation temporarily. Persistence depends on how costs, margins, wages, demand, and expectations subsequently adjust. A continuing series of supply disruptions differs from a single shock whose price effect stops growing. Claims about persistence require evidence on the breadth, timing, and propagation of changes, not just one headline number.
A level and its growth rate answer different questions. An index of 150 says the basket costs one and a half times its cost at a base level of 100. It says nothing about the most recent year's change unless we know the preceding level. If that level was 145, the increase is 5 divided by 145, about 3.45%. If it was 160, the decline is 10 divided by 160, or 6.25%. The same current level can accompany inflation or deflation. A chart of levels and a chart of inflation rates therefore need different labels.
In an additional paper case, monthly index levels are 100 in January, 108 in February, and 108 in every month through the following February. The initial monthly rise is 8%; subsequent monthly changes are zero. Yet the following January remains 8% above the first January. In the following February, the comparison is 108 against 108, so annual inflation is zero. The earlier price increase has dropped out of the twelve-month comparison. This is a base effect, a change in the reported rate caused by which earlier observation is used. It need not indicate a new fall in prices.
Returning to a previous price level requires reversing the increase in currency amounts, not repeating its percentage with a minus sign. After a rise from 100 to 120, the amount to reverse is 20. Relative to the new level 120, that is a fall of one sixth, about 16.67%. A 20% fall would instead remove 24 and leave 96. This asymmetry explains why a positive inflation year followed by an equally large negative percentage does not restore the original level. It also matters when comparing changes in purchasing power with changes in basket prices.
Suppose a worker's annual nominal income rises from 1,000 to 1,100 while the basket index rises from 100 to 120. Assume the basket initially costs 100 currency units, so the income buys ten baskets. Later it buys about 9.17 baskets costing 120 each. Nominal income increased 10%, but purchasing power fell about 8.33%. Compare the income growth factor 1.10 with the price factor 1.20 by division. This case assumes that the basket represents the worker's spending and ignores taxes. Different income growth or expenditure weights would give different results. Inflation alone does not determine every household's change in living standards.
A price fall in one category also differs from broad deflation. Suppose initial spending is 20 on energy and 80 on everything else. Energy prices fall 25% while other prices rise 10%, with quantities fixed. Energy's cost becomes 15 and the rest becomes 88. The total is 103, a 3% basket increase. Energy contributes negative 5 percentage points while other items contribute positive 8. Describing this as falling prices without specifying energy would misstate the overall basket. Measures that exclude selected volatile categories answer different measurement questions; exclusions do not make those purchases irrelevant to household budgets.
Use the previous year's level, not the original base, to compute this year's change.
Year 1 inflation: 10%. Year 2 inflation: 2%. Inflation falls from 10% to 2%. Prices still rise.
| Fixed basket | Energy weight | Energy price change | Other prices | Basket increase |
|---|---|---|---|---|
| Lower energy exposure | 10% | +50% | Unchanged | 5% |
| Higher energy exposure | 40% | +50% | Unchanged | 20% |
The four buttons above choose complete constructed paths at annual dates. They do not set a policy rate, change a money supply, or calculate a household budget. Switching paths replaces all three observations; it does not continue after the previously selected endpoint. The vertical axis starts at 90, so a line's apparent height is not a price measured from zero. Read the printed index levels and the reported annual rates together. All paths start with the same 100 and first reach 110. Their different second years isolate what can happen after that common initial increase.
Compare Continued 10% inflation with One-time level jump. In the former, the next increase is 10% of 110, which is 11, giving 121. In the latter, the next increase is zero, leaving 110. The 11-point gap between their endpoints represents the extra second-year price rise in the continued-inflation path. It does not represent a measured benefit from any particular policy. The widget supplies no production, wage, energy-supply, or demand equations. To explain why a country followed either pattern, we would need observations and a mechanism beyond these index calculations.
Now select Deflation after a shock. Five percent of the year-one level 110 is 5.5. Subtracting gives 104.5. The basket is cheaper than one year earlier but still more expensive than at the start. Assume the basket initially costs 100 currency units. A fixed deposit of 110 then buys one basket at the first-year price of 110 and about 1.053 baskets at the second-year price of 104.5. This purchasing-power improvement does not increase the number in the deposit account. Price changes and ledger balances are distinct measurements. The example assumes the same basket remains relevant at each date.
For guided exploration on paper, keep the first two levels fixed at 100 and 110 and try a final level of 108.9. The second-year decrease is 1.1 divided by 110, giving 1% deflation. Relative to the start, prices remain 8.9% higher. Then try 113.3 instead. The second-year increase is 3.3 divided by 110, giving 3% inflation, while the cumulative rise is 13.3%. These are additional paper paths, not editable settings in the widget. Their purpose is to check whether the denominator changes appropriately when the comparison date changes.
Inflation can change the real burden of a fixed nominal contract without changing its legal principal. Suppose a borrower owes 100 and a lender holds the corresponding claim. If the basket price rises 25%, the purchasing power represented by that 100 falls to 80% of its earlier amount, because 100 divided by 125 is 0.8. The lender's real claim falls while the borrower's real obligation falls. Their nominal entries remain 100. This is not a bank repayment, an asset impairment entry, or automatic deletion of a seller's deposit. Indexed contracts and changing income require different calculations.
Nominal GDP can be written as a chosen money stock times its measured GDP velocity, where velocity is nominal GDP per unit of that stock during the period. It can also be written as the price level times real output under consistent definitions. Here the price measure is the GDP deflator, the price component of domestic output, not necessarily the consumer basket. This quantity identity alone does not fix velocity or output, establish causation, or say which money aggregate matters. Money growth can accompany rising prices, higher output, or increased desire to hold liquid balances.
Demand rising faster than productive capacity can raise price pressure. Energy shortages or exchange-rate depreciation can raise costs even with weak output. Pricing power, wage bargaining, expectations, fiscal choices, and monetary responses influence how shocks spread. Their relative importance varies by episode and is empirically contested. A credible explanation states the shock, transmission mechanism, counterfactual, and supporting evidence rather than attributing every price change to a single balance-sheet entry.
Separate a change in available goods from a change in nominal spending. In a paper economy producing only bread, 100 units sell for 2 each, giving nominal sales of 200. If a supply disruption leaves only 80 units and nominal spending stays 200, the average sale price must be 2.50. If spending instead falls to 160, the price can stay 2. These are conditional arithmetic cases, not predictions about how buyers respond. Actual buyers may substitute other products or reduce purchases. The example identifies which behavioral assumption turns a smaller quantity into a higher price.
A demand case needs equally explicit conditions. Suppose bread supply stays at 100 units but buyers actually spend 240 rather than 200 on it. The average price is then 2.40 if all 100 units sell and this spending covers those units. If producers can instead supply 120 units at the old price, 240 of spending can buy more bread without a price rise. Capacity means the feasible production given labor, equipment, and inputs over the relevant time. A new deposit may finance either case, but its creation does not specify which capacity constraint or spending response applies.
Trace that deposit before attributing inflation to it. A bank loan of 40 gives a borrower a deposit asset and loan liability of 40. The bank gains a loan asset and deposit liability of 40. If the borrower pays a producer at another bank, deposits move to the producer and reserves settle between banks. The origination created purchasing power, not 40 of net wealth or new bread. If the borrower instead leaves the deposit idle or purchases an existing bond, the immediate transaction is different. A larger money stock therefore needs an account of use before it becomes a causal explanation of consumer prices.
The quantity identity requires the same discipline. In a one-good paper economy, a chosen money stock of 100 and nominal output of 200 imply measured annual velocity of 2. Velocity here means annual nominal output divided by the stock, not a tracked count of each payment. If money becomes 125 while nominal output stays 200, measured velocity becomes 1.6. If velocity instead stays 2, nominal output becomes 250, but that still leaves open the division between prices and quantities. In actual national accounts this relationship uses the GDP price measure, not automatically the consumer basket index above.
Persistence requires an explanation across time. An imported input becoming costlier can raise a firm's costs, but subsequent prices depend on contract renewal, margins, wages, demand, and input substitution. A wage increase may restore purchasing power, compress margins, raise prices, or accompany higher productivity; its size alone does not choose among these outcomes. To evaluate an episode, compare dates and categories, including import costs, quantities produced, wages, and spending. A proposed causal account should identify what would have happened without the claimed shock. The constructed paths verify level and rate arithmetic, while neither their slopes nor a coincident money increase supply that counterfactual.
An index rises from 200 to 220, then to 224.4. Find both inflation rates and the total price rise since the start.
Each period uses its own starting level as the denominator.
| Date | Index | Change from previous date |
|---|---|---|
| Start | 200 | Not applicable |
| Year 1 | 220 | 10% |
| Year 2 | 224.4 | 2% |
The rates are 10% and 2%. Total prices rise 12.2%. The second period is disinflation, not deflation. Adding the rates gives 12%, which misses compounding.
A chosen money stock rises 10%, its measured velocity falls 10%, and real output is unchanged. What does the quantity identity imply for the price level? Does this prove that extra money reduces prices?
Multiply gross factors: 1.10 and 0.90.
| Factor | Multiplier |
|---|---|
| Money stock | 1.10 |
| Velocity | 0.90 |
| Nominal expenditure | 0.99 |
| Real output | 1.00 |
| Price level | 0.99 |
Nominal expenditure becomes 99% of its starting value. With unchanged real output, the price level falls 1%. This is a conditional accounting example, not evidence that money growth causes deflation. The assumed velocity response needs its own explanation and evidence.
These references support the mechanisms and identify institutional context. They are optional; the explanations and worked answers above stand alone. Numerical examples are teaching cases, not live financial data or personal financial advice.
Price baskets, expenditure weights, annual comparisons, and differences in household experience.
Rates, expectations, asset prices, credit, and uncertain transmission lags. A channel description is not a quantitative causal estimate.
Bank lending, deposit creation and destruction, constraints, and asset purchases. UK institutional explanation, not a universal timetable or a current rate announcement.