Index Numbers and Inflation Calculations

When people in Pakistan say "everything has become expensive", economists answer with a single number: the inflation rate. Behind that number sits an index number, a clever way of summarising price changes across hundreds of goods into one figure. This article shows how index numbers are built, how inflation is calculated from them, and how to use them to convert salaries and interest rates into real terms.

What Is an Index Number?

An index number measures the change in a variable relative to a base period, which is assigned the value 100. If an index stands at 150, the variable is 50% higher than in the base period.

Price Relatives

For a single item, the price relative is

$$ I = \frac{p_1}{p_0} \times 100 $$

where \( p_0 \) is the base-period price and \( p_1 \) the current price. If flour rose from PKR 60 to PKR 100 per kg, its price relative is \( (100/60) \times 100 \approx 166.7 \): a 66.7% increase.

A single item tells us little about the cost of living. We need to combine many items, and since a family spends far more on some items than others, they must be weighted.

Weighted Aggregate Indices

The three classic formulas differ in which quantities they use as weights.

Laspeyres index (base-period quantities):

$$ L = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100 $$

It answers: "How much more does the original basket cost today?"

Paasche index (current-period quantities):

$$ P = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100 $$

It answers: "How much more does today's basket cost than it would have at base prices?"

Fisher's ideal index is the geometric mean of the two:

$$ F = \sqrt{L \times P} $$

Worked Example 1: A Household Basket

A Rawalpindi household's monthly consumption of three items is shown below (prices in PKR, quantities per month).

Step 1: Base-period expenditure \( \sum p_0 q_0 \)

$$ 60(20) + 120(30) + 150(40) = 1{,}200 + 3{,}600 + 6{,}000 = 10{,}800 $$

Step 2: Original basket at current prices \( \sum p_1 q_0 \)

$$ 100(20) + 200(30) + 270(40) = 2{,}000 + 6{,}000 + 10{,}800 = 18{,}800 $$

Step 3: Laspeyres index

$$ L = \frac{18{,}800}{10{,}800} \times 100 \approx 174.07 $$

Step 4: Current basket at current and base prices

$$ \sum p_1 q_1 = 100(18) + 200(28) + 270(35) = 1{,}800 + 5{,}600 + 9{,}450 = 16{,}850 $$

$$ \sum p_0 q_1 = 60(18) + 120(28) + 150(35) = 1{,}080 + 3{,}360 + 5{,}250 = 9{,}690 $$

Step 5: Paasche index

$$ P = \frac{16{,}850}{9{,}690} \times 100 \approx 173.89 $$

Step 6: Fisher index

$$ F = \sqrt{174.07 \times 173.89} \approx 173.98 $$

Interpretation: the cost of this basket has risen by about 74%. The Paasche index is slightly lower because the household cut back on items whose prices rose, and Paasche weights reflect those smaller current quantities. Laspeyres, which assumes no substitution, tends to overstate the rise in the cost of living; Paasche tends to understate it. Fisher's index lies between the two.

Sanity check: each item's price relative is 166.7 (flour), 166.7 (milk) and 180 (petrol). Any sensible weighted average must lie between 166.7 and 180, and 174 does. Petrol carries the largest expenditure weight, which pulls the index towards 180.

From Index Numbers to Inflation

National statistics offices, such as the Pakistan Bureau of Statistics, publish a Consumer Price Index (CPI), essentially a large Laspeyres-type index of a representative basket of goods and services. The inflation rate between two periods is the percentage change in the CPI:

$$ \pi = \frac{\text{CPI}_{\text{new}} - \text{CPI}_{\text{old}}}{\text{CPI}_{\text{old}}} \times 100\% $$

Note that inflation is a percentage change, not a difference in index points. A rise from 250 to 262.5 is 12.5 points but \( 12.5/250 = 5\% \) inflation.

Worked Example 2: Real Wages and Real Interest

Part (a): Has a salary increase beaten inflation?

Bilal's salary rose from PKR 80,000 to PKR 95,000 a month. Over the same period, the CPI rose from 200 to 240.

Step 1: Nominal increase.

$$ \frac{95{,}000 - 80{,}000}{80{,}000} = 18.75\% $$

Step 2: Inflation.

$$ \frac{240 - 200}{200} = 20\% $$

Step 3: Deflate the new salary to old-period rupees.

$$ \text{Real salary} = 95{,}000 \times \frac{200}{240} \approx \text{PKR } 79{,}166.67 $$

Step 4: Real change.

$$ \frac{79{,}166.67 - 80{,}000}{80{,}000} \approx -1.04\% $$

Despite an 18.75% raise, Bilal's purchasing power has fallen by about 1%. The quick check \( 1.1875 / 1.20 - 1 \approx -0.0104 \) confirms this.

Part (b): Real return on a deposit.

A savings certificate pays 15% a year while inflation is 5%. The Fisher equation gives the exact real rate:

$$ 1 + r_{\text{real}} = \frac{1 + i}{1 + \pi} = \frac{1.15}{1.05} \approx 1.0952 $$

So the real return is about 9.52%. The popular shortcut \( i - \pi = 10\% \) is close when rates are low but becomes less accurate at the high inflation rates Pakistan has experienced in some years.

Practical Issues With Index Numbers

1. Choice of base year. Bases are updated periodically because spending patterns change; an old base can give misleading weights.
2. Splicing. When a new base is introduced, the old series is rescaled so that the two can be joined. If the old index was 180 in the year that becomes the new base (100), divide every old value by 1.8.
3. Quality change. A 2026 smartphone is not the same product as a 2016 one, which complicates price comparisons.
4. Substitution bias. Consumers switch to cheaper goods, which a fixed-basket Laspeyres index ignores.
5. Different indices for different purposes. Wholesale or producer price indices track business input costs; sensitive price indicators track essential items weekly.

Did You Know?

• Étienne Laspeyres and Hermann Paasche were 19th-century German economists who proposed their indices in 1871 and 1874 respectively.
• Irving Fisher called his formula "ideal" because it passes several tests, including the time-reversal test, that the other two fail.
• If prices rise at 10% a year, they roughly double in about 7 years, by the Rule of 72.

Key Takeaways

• An index number compares a value with a base period set to 100.
• Laspeyres uses base quantities, Paasche current quantities, and Fisher is their geometric mean.
• Inflation is the percentage change in a price index, not the change in index points.
• Deflate nominal values by multiplying by \( \text{CPI}_{\text{base}} / \text{CPI}_{\text{current}} \).
• Use the Fisher equation \( (1+i)/(1+\pi) - 1 \) for the exact real interest rate.

Need help with index number, CPI or inflation questions for your economics or business statistics course? Amathify's subject experts and PhD scholars provide step-by-step solutions via WhatsApp or the Ask a Question form.