Quantitative skills and calculations

Economics uses numbers to measure what theory describes: how strongly buyers react to a price change, how fast an economy grows, how much a tax costs society. Every calculation in the course is short, and none needs more than a four-function calculator. Marks are lost on method, units and wording far more often than on difficult arithmetic.

Calculations appear in Paper 2 (part (b) and sometimes parts (c)–(f)), at both SL and HL, and throughout part (a) of HL Paper 3. Paper 1 has no calculator and no calculations. Each calculation is grouped by unit, with what it measures in one sentence, the formula, the units, a fully worked example and the usual traps. HL-only calculations sit under headings marked “(HL only)”.

How calculations are marked

Calculation items are marked with an analytic markscheme: each mark is for a specific step or a specific answer. A typical 2-mark item gives 1 mark for valid working (the formula with the right numbers substituted) and 1 mark for the correct answer with its unit. A 3-mark item usually has a further step, such as finding a percentage change before using it.

Markschemes typically follow your own figures through: if an earlier answer is wrong but the later method is right, the later step can still earn its marks. This only works when the working is written down. A bare number that is wrong earns nothing; a wrong number reached by a visible, correct method usually earns the method mark.

Four habits prevent most lost marks. Give units on every answer ($ billion, thousand units, %, or “no units” for an elasticity). Keep the sign: −0.8 and 0.8 are different answers for a YED. Round only at the end, to two decimal places unless the question says otherwise. Read the scale of each axis and table header, since “$ billion” and “thousand tonnes” change the answer by a factor of a thousand or more.

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All the calculations at a glance

The table lists every calculation in the course by unit. SL students meet the SL rows in Paper 2; HL students meet every row, mostly in Paper 3.

Skills every calculation uses

Percentage change

Most calculations in economics are built on percentage change: elasticities, growth rates, inflation rates and exchange rate changes are all percentage changes. It measures how large a change is relative to the starting value, so a $1 rise matters more on a $2 good than on a $200 one.

Percentages and percentage points

When the variable is already a percentage, such as an unemployment rate or an interest rate, two different changes can be described. The difference between the two rates is measured in percentage points; the relative change is a percentage. An unemployment rate that rises from 4% to 6% has risen by 2 percentage points, which is a 50% increase in the rate.

Reading diagrams and tables

Many HL calculations are areas on a diagram. Surpluses and welfare losses are triangles; tax revenue, subsidy costs and spending are rectangles. Every area in the course is one of these two shapes, or a trapezium that splits into a rectangle and a triangle.

On a market diagram the width is a quantity and the height a price, so the area is a sum of money. If quantity is in thousands of units and price in dollars, an area of 75 means $75 000. Read the axis titles before calculating anything.

Unit 2: Microeconomics

Price elasticity of demand (PED)

Price elasticity of demand measures how responsive the quantity demanded of a good is to a change in its price. It has no units, because it divides one percentage by another.

PED is negative for almost every good, because price and quantity demanded move in opposite directions. It is usually quoted as an absolute value (the minus sign ignored), and its size is what matters: between 0 and 1 demand is price inelastic, above 1 price elastic, and exactly 1 unitary elastic.

Questions also give the PED and ask for one of the other values. Rearranging the formula gives the percentage change in quantity demanded: multiply PED by the percentage change in price.

Income elasticity of demand (YED)

Income elasticity of demand measures how responsive the quantity demanded of a good is to a change in consumers’ income. Unlike PED, its sign matters: positive for a normal good, negative for an inferior good. Above +1 the good is income elastic (often a luxury); between 0 and +1 it is income inelastic (a necessity).

Price elasticity of supply (PES)

Price elasticity of supply measures how responsive the quantity supplied is to a change in the good’s price. It is positive, because quantity supplied rises with price. Above 1 supply is price elastic; below 1 it is price inelastic.

The same rearrangement finds any missing value in an elasticity. If YED for restaurant meals is +1.5 and income rises by 4%, quantity demanded rises by 1.5 × 4 = 6%. If PED is −0.5 and a city wants quantity demanded to fall by 10%, the price must rise by −10 ÷ −0.5 = 20%. If PES is 0.75 and quantity supplied rose by 9%, the price must have risen by 9 ÷ 0.75 = 12%.

Consumer and producer surplus (HL only)

Consumer surplus is the difference between what consumers are willing to pay and what they actually pay; producer surplus is the difference between the price producers receive and the lowest price at which they would supply. On a diagram with straight-line curves, each is a triangle.

Price ceilings and price floors (HL only)

A price ceiling set below the equilibrium price creates excess demand; a price floor set above it creates excess supply. The quantity actually traded is always the smaller of quantity demanded and quantity supplied. The table gives the demand and supply schedules for the same market.

Indirect taxes (HL only)

A specific tax adds a fixed amount per unit to producers’ costs, so the supply curve shifts up by the tax. After the tax, consumers pay the new market price and producers keep that price minus the tax. The diagram shows a $1.50 tax on the same market.

Tax incidence depends on elasticities: consumers pay the larger share when demand is less elastic than supply, as here. An ad valorem tax is a percentage of the price, so the tax per unit is the rate × the pre-tax price.

Subsidies (HL only)

A per-unit subsidy lowers producers’ costs, so supply shifts down by the subsidy. Consumers pay the new, lower market price; producers receive that price plus the subsidy.

Welfare loss from externalities (HL only)

Where an externality exists, the market produces where private costs equal private benefits, not where marginal social cost equals marginal social benefit. The welfare loss is the triangle between the social curves, from the socially optimal quantity to the market quantity.

Costs, revenues and profit (HL only)

A firm maximises profit where marginal cost equals marginal revenue. The definitions all follow from total revenue (TR) and total cost (TC).

The table gives the prices and total costs of a firm with market power (output in units a day, money in $ thousand).

Since AR is above AC, the firm earns abnormal profit. Where AR equals AC it earns normal profit, and where AR is below AC it makes a loss.

Concentration ratios and payoff matrices (HL only)

A concentration ratio is the combined market share of the largest firms in an industry. A four-firm ratio (CR4) adds the shares of the four biggest firms. If the four largest supermarket chains hold 31%, 24%, 12% and 9% of sales, CR4 = 31 + 24 + 12 + 9 = 76%, a high figure typical of oligopoly.

A payoff matrix shows each firm’s profit for every combination of strategies. In the matrix, two firms each choose a high or a low price; each cell shows (Firm A’s profit, Firm B’s profit) in $ million.

Unit 3: Macroeconomics

Nominal GDP and GNI

Gross domestic product measures the total value of final goods and services produced within a country in a year. The expenditure approach adds the four components of spending on domestic output.

Gross national income measures the income earned by a country’s residents, wherever their factors of production are located. It adds net income from abroad: income earned abroad by residents minus income paid to non-residents.

The table gives national income data for Country A in 2024 (all figures in $ billion, current prices).

Real GDP, real GNI and per capita values

Nominal values are measured at current prices, so they rise with inflation even when output does not. Real GDP removes the effect of price changes using a price deflator, an index of the price level with a value of 100 in the base year.

The same formula turns nominal GNI into real GNI. Dividing a real value by the population gives a per capita value, a better guide to the average standard of living.

The rate of economic growth

Economic growth is measured as the percentage change in real GDP over a period, usually a year. Real values are used so that inflation is not counted as growth.

For Country A, growth in 2024 = (942.22 − 933.33) ÷ 933.33 × 100 = 0.95%. The same formula applied to nominal GDP gives 8.16%, which overstates growth because it includes the rise in prices. If a question gives a real GDP index or real GDP directly, no deflating is needed.

The unemployment rate

The unemployment rate is the percentage of the labour force that is unemployed. The labour force is everyone in work plus everyone actively looking for work; people who are neither (students, retired people, those not seeking work) are left out.

The inflation rate from a consumer price index

Inflation is a sustained rise in the general price level, measured by the consumer price index. A CPI prices a fixed basket of goods, with quantities that reflect what a typical household buys, so goods bought in larger quantities carry more weight. The index is the cost of the basket in a year relative to the base year; the inflation rate is the percentage change in the index.

The table shows a simple basket of three items, with the quantities a typical household buys and their prices in 2023 and 2024.

Between two years that are not the base year, the inflation rate is still the percentage change in the index. If the index rises from 111.17 to 115.52, inflation is (115.52 − 111.17) ÷ 111.17 × 100 = 3.91%, not 115.52 − 111.17 = 4.35 percentage points.

The real interest rate

The real interest rate measures the return on saving, or the cost of borrowing, after allowing for inflation. It is approximately the nominal interest rate minus the inflation rate.

With a nominal rate of 5.25% and inflation of 3.4%, the real rate is 5.25 − 3.4 = 1.85%. With a nominal rate of 4% and inflation of 7.5%, it is −3.5%: savers lose purchasing power and borrowers gain. The answer is in percent and can be negative.

Weighted price indices (HL only)

Official price indices weight each category of spending by its share of the household budget, so a 10% rise in housing costs matters more than a 10% rise in the price of books. Given a price index for each category and its weight, the overall index is the weighted average of the category indices.

The table gives the weight of each category in household spending and its price index for 2024 (base year 2023 = 100).

Government debt as a percentage of GDP (HL only)

Debt is a stock and GDP a yearly flow, so the ratio compares what a government owes with the size of the economy that supports it. If Country A’s government owes $742 billion and nominal GDP is $1 060 billion, debt = 742 ÷ 1 060 × 100 = 70.0% of GDP. Use nominal GDP, since debt is also measured at current prices. A budget deficit adds to the debt each year; the ratio falls when nominal GDP grows faster than the debt.

Constructing a Lorenz curve (HL only)

A Lorenz curve plots the cumulative percentage of income against the cumulative percentage of households, ranked from poorest to richest. Quintile data give each fifth’s share of income, so the plotted points are running totals.

The Gini coefficient is the area between the line of perfect equality and the Lorenz curve (A) divided by the whole area under the line of perfect equality (A + B). For this curve, the area under the Lorenz curve, added up as five trapezia, is 0.304 of the square, so the Gini = (0.5 − 0.304) ÷ 0.5 = 0.39.

Indirect tax paid (HL only)

A percentage tax such as a value added tax is usually charged on the price before tax. If a pair of shoes costs $50 before a 20% VAT, the tax is 0.2 × $50 = $10 and the shopper pays $60. When the price given already includes the tax, the tax is the rate divided by (100 + the rate) of the price paid.

Comparing the tax paid with each household’s income shows whether an indirect tax is regressive. A household on $20 000 that spends $15 000 on goods with 20% VAT pays $3 000, or 15% of its income; a household on $100 000 that spends $40 000 pays $8 000, or 8%.

Total tax and average tax rates (HL only)

In a progressive income tax, each rate applies only to the income inside its band. The marginal tax rate is the rate on the last dollar earned; the average tax rate is total tax divided by total income.

The table shows a country’s income tax schedule.

The Keynesian multiplier (HL only)

An increase in an injection (investment, government spending or exports) raises national income by more than the injection, because each round of spending becomes someone’s income and part of it is spent again. The multiplier (k) depends on how much of each extra dollar of income is spent on domestic output, the marginal propensity to consume (MPC), or on how much leaks out in saving, taxes and imports.

Money creation (HL only)

When a bank receives a new deposit it keeps a fraction as reserves (the minimum reserve requirement) and lends the rest, which is deposited again and partly lent again. The maximum total of deposits created from a new deposit is the deposit divided by the reserve ratio. With a new deposit of $1 000 and a 10% reserve requirement, deposits can rise to 1 000 ÷ 0.1 = $10 000, of which $9 000 is new lending. In practice the figure is lower, because banks hold extra reserves and people hold some cash.

Unit 4: The global economy

Prices in different currencies

An exchange rate is the price of one currency in terms of another, so it converts a price from one currency to another. Multiply when converting into the currency that appears as “units per one”; divide when converting the other way.

Changes in the value of a currency

The percentage appreciation or depreciation of a currency is the percentage change in its price in terms of the other currency. Express the exchange rate with the currency in question as the “one” unit first.

The balance of payments

The balance of payments records all transactions between a country’s residents and the rest of the world. The current account is the sum of the trade in goods balance, the trade in services balance, primary income (wages, interest, profits and dividends) and secondary income (current transfers). The capital account and financial account record transfers of capital and flows of investment and reserve assets. Credits (money flowing in) are positive and debits negative, and the accounts must sum to zero, with net errors and omissions as the balancing item.

The table gives the balance of payments data for Country C (US$ billion).

Free trade diagrams (HL only)

Under free trade, the domestic price equals the world price. When the world price is below the domestic equilibrium, domestic consumers buy more than domestic firms supply and the gap is imported. Import expenditure is the world price × the quantity imported.

At the world price of $200, domestic firms supply 20 thousand bicycles and consumers buy 80 thousand, so imports are 80 − 20 = 60 thousand and import expenditure is $200 × 60 000 = $12 million. When the world price is above the domestic equilibrium, the same method gives exports (domestic supply − domestic demand) and export revenue (world price × exports).

Tariffs, quotas and production subsidies (HL only)

A quota of 30 thousand bicycles has the same effect on price and quantities, because the domestic price rises until imports equal 30 thousand (at $300). The difference is area c: instead of tariff revenue for the government, it becomes quota rent for the foreign exporters or licence holders, (300 − 200) × 30 000 = $3 million.

A production subsidy of $100 per bicycle to domestic firms leaves the price at $200 but lets producers receive $300, so domestic supply rises to 40 thousand while consumers still buy 80 thousand. Imports fall to 40 thousand (import expenditure $8 million), the subsidy costs the government $100 × 40 000 = $4 million, producers’ revenue including the subsidy rises to $12 million, and the welfare loss is triangle b only ($1 million), since consumers are unaffected.

Opportunity cost and comparative advantage (HL only)

A country has comparative advantage in a good if it can produce it at a lower opportunity cost than another country. The opportunity cost of one unit of a good is the amount of the other good given up.

The Marshall–Lerner condition (HL only)

A depreciation improves the current account only if the sum of the price elasticities of demand for exports and imports (as absolute values) is greater than 1. If PED for exports is 0.4 and for imports 0.5, the sum is 0.9: below 1, so the current account worsens at first, as the J-curve shows. If, after two years, the elasticities rise to 0.8 and 0.7, the sum is 1.5 and the depreciation improves the current account.

Traps that cost marks in every unit

The same few errors account for most of the marks lost on calculations. The table lists them with the fix for each.

Practice set

Try each item before opening the answer. Every answer shows the working that would earn full marks.

Summary

• Every calculation is marked for method and answer: write the formula, the substitution and the answer with its unit.

• Percentage change divides by the old value; elasticities put the percentage change in quantity on top.

• Macro calculations use real values for growth and living standards, the labour force for unemployment, and weighted baskets for inflation.

• HL market calculations are rectangles (revenue, spending, tax, subsidy cost) and triangles (surplus, welfare loss) on a diagram.

• HL macro calculations include the multiplier with every leakage, tax bands applied band by band, and Lorenz curves from cumulative shares.

• HL trade calculations read imports and prices off the diagram; comparative advantage comes from opportunity cost, not output.

Self-check

• To do:

  I can calculate a percentage change and tell percentages from percentage points.

• To do:

  I can calculate PED, YED and PES, and use them to find a missing change in quantity or total revenue.

• To do:

  I can calculate nominal and real GDP and GNI, per capita values, growth, unemployment, inflation and real interest rates.

• To do:

  I can calculate the BoP balances, a price in another currency and a currency’s percentage change.

• To do:

  I can calculate surplus, tax, subsidy and price control effects and welfare loss from a diagram (HL).

• To do:

  I can calculate MR, MC, AR, AC and profit from a table and find the profit-maximising output (HL).

• To do:

  I can calculate a weighted price index, tax paid and average tax rates, and construct a Lorenz curve (HL).

• To do:

  I can calculate the multiplier, opportunity costs and the effects of tariffs, quotas and subsidies (HL).