Demand analysis studies how the quantity demanded of a commodity depends on its own price, the prices of related goods (substitutes / complements), consumer income, tastes and other factors. It is at the heart of price theory, marketing and policy analysis.
The basic demand function takes the form:
where \(Q\) is quantity demanded, \(P\) the price of the good, \(P_i\) prices of related goods, \(Y\) consumer income, \(T\) tastes / preferences.
The price elasticity of demand measures the responsiveness of quantity demanded to a change in the good's own price:
\[ E_p \;=\; \dfrac{\%\,\text{change in quantity demanded}}{\%\,\text{change in price}} \;=\; \dfrac{\Delta Q / Q}{\Delta P / P} \;=\; \dfrac{P}{Q}\cdot \dfrac{dQ}{dP}. \]Since \(dQ/dP\) is usually negative, \(E_p\) is negative; we often quote \(|E_p|\).
| \(|E_p|\) | Demand type | Example |
|---|---|---|
| 0 | Perfectly inelastic | Life-saving drug |
| 0 < |E| < 1 | Inelastic (essential) | Salt, electricity |
| = 1 | Unitary elastic | Some clothing |
| 1 < |E| < ∞ | Elastic (luxury) | Foreign travel, restaurant meals |
| ∞ | Perfectly elastic | Identical commodities in pure competition |
Point elasticity at a specific (P, Q):
For a linear demand \(Q = a - b P\): \(E_p = -bP/Q\).
Arc elasticity between two observations:
If price falls from ₹50 to ₹40 (20 % drop) and quantity demanded rises from 100 to 130 (30 % rise):
\(E_p = 30/(-20) = -1.5\). |E| = 1.5 → elastic.
Linear demand \(Q = 100 - 2P\). At \(P = 20\): \(Q = 60\). Then \(E_p = -2 \cdot 20/60 = -0.667\) → inelastic at this price.
At \(P = 40\): \(Q = 20\); \(E_p = -2 \cdot 40/20 = -4\) → highly elastic. Note elasticity varies along a linear demand curve.
Demand for a good often depends on multiple variables. The partial elasticities measure responsiveness to each variable holding the others constant.
When household income rises 10 %, demand for restaurant meals rises 18 %. \(E_Y = 18/10 = 1.8\) → luxury good.
Price of petrol rises 15 %; demand for cars falls 6 %. \(E_{\text{car, petrol}} = -6/15 = -0.4\) → complementary goods.
Each data type addresses different aspects of demand. Time-series suffers from identification problem (price and quantity may both depend on unobserved factors); cross-sectional gives a snapshot but cannot show price response.
W. W. Leontief's method (1929) fits a constant-elasticity demand curve to time-series data using a log-linear relationship:
\[ \log Q \;=\; a + b \log P. \]Here \(b\) is the constant price elasticity of demand: \(E_p = b\).
Apply ordinary least squares to \((\log P_t, \log Q_t)\) pairs from time-series data:
Leontief's method is simple but assumes constant elasticity across the entire range of data — a strong assumption.
For 5 years: \(\log P\) values 1.0, 1.1, 1.2, 1.3, 1.4; \(\log Q\) values 2.5, 2.4, 2.3, 2.2, 2.1.
Sums: \(\sum \log P = 6.0\), \(\sum \log Q = 11.5\), \(\sum (\log P)(\log Q) = 2.5+2.64+2.76+2.86+2.94 = 13.7\), \(\sum (\log P)^2 = 1+1.21+1.44+1.69+1.96 = 7.3\).
\(\hat b = \dfrac{5 \cdot 13.7 - 6 \cdot 11.5}{5 \cdot 7.3 - 6^2} = \dfrac{68.5 - 69}{36.5 - 36} = \dfrac{-0.5}{0.5} = -1\). So price elasticity \(\approx -1\) (unitary elastic).
A. C. Pigou proposed two practical approaches based on either time-series data or family-budget data.
If demand depends on price and income, Pigou suggests a multiple-regression form:
Estimated by least squares; \(b\) is the price elasticity and \(c\) is the income elasticity.
To isolate price effect, Pigou suggests deflating prices and quantities by population, GDP, etc., to control for trend.
From a household-budget survey, divide families into income groups. For each group compute average income \(Y\) and average expenditure on the commodity \(M\):
This gives the income elasticity of demand for the commodity. By computing this for various goods we discover whether they are necessities, luxuries or inferior.
Pigou's method requires only cross-sectional family-budget data — much cheaper to collect than time-series.
From a household survey, average income across 4 income classes is ₹1, 2, 3, 4 lakh; expenditure on milk is ₹6 000, 9 000, 11 000, 12 500.
Fit \(M = a + bY\): \(\sum Y = 10,\; \sum M = 38500,\; \bar Y = 2.5,\; \bar M = 9625\). With \(\sum YM = 1(6000)+2(9000)+3(11000)+4(12500) = 107000\) and \(\sum Y^2 = 30\):
\[ b = \dfrac{\sum YM - n\bar Y\bar M}{\sum Y^2 - n\bar Y^2} = \dfrac{107000 - 4 \cdot 2.5 \cdot 9625}{30 - 4(2.5)^2} = \dfrac{107000 - 96250}{30 - 25} = \dfrac{10750}{5} = 2150 \]i.e. expenditure on milk rises by ₹2 150 per additional lakh of income.
At mid-point \(\bar Y = 2.5,\; \bar M = 9625\): \(E_Y = (2.5/9625) \cdot 2150 = 0.56\) → income-inelastic (necessity).
Time-series data on quantity, price, and income shows multiple-regression results: \(\log Q = 2.5 - 0.8 \log P + 1.2 \log Y\). Price elasticity = −0.8; income elasticity = +1.2 (luxury). Both significant.
The Engel curve shows how expenditure on a particular good varies with household income, holding prices constant. Plotting expenditure (Y-axis) against income (X-axis) gives the Engel curve.
"The poorer a family, the greater is the proportion of total expenditure that must be spent on food."
Stated more formally: as household income rises, the share of income spent on food decreases; the share spent on luxury goods increases.
Implication: \(E_Y\) for food < 1 (necessity), and for luxury goods \(E_Y\) > 1.
Survey shows households with monthly income ₹20 000 spend 50 % on food, while those with ₹2 lakh spend only 20 %. Income elasticity of food is < 1 — consistent with Engel's law.
Vilfredo Pareto (1897) observed empirically that the number of persons \(N\) whose income exceeds a threshold \(X\) follows a power law:
\[ N \;=\; \dfrac{A}{X^\alpha}, \quad A > 0, \alpha > 0, \]or equivalently \(\log N = \log A - \alpha \log X\). Higher \(\alpha\) means more equal distribution.
Pareto's coefficient \(\alpha\) (typically between 1.5 and 2.5) describes how rapidly the number of high earners falls off. The famous "80-20" rule (about 80 % of wealth held by 20 % of people) corresponds to \(\alpha \approx 1.16\).
Plot \(\log N\) vs \(\log X\); the slope of the fitted straight line is \(-\alpha\). Use least squares on log-log data.
From income-tax data, the number of taxpayers with income above various thresholds is tabulated. Fitting log-log gives slope \(-2.0\), so \(\alpha = 2.0\). \(A\) determined from any single (X, N) pair.
Curves of concentration — most famously the Lorenz curve — visualise the inequality in distribution of income, wealth or any other quantity.
Plot the cumulative percentage of population (X-axis) against the cumulative percentage of total income (Y-axis), with the population sorted from poorest to richest.
The Gini coefficient is twice the area between the Lorenz curve and the 45° line; ranges from 0 (perfect equality) to 1 (perfect inequality where one person owns everything).
where \(L(p)\) is the Lorenz function.
5 income classes (sorted poorest to richest), each containing 20 % of population. Income shares: 4 %, 8 %, 14 %, 24 %, 50 %.
Cumulative population %: 20, 40, 60, 80, 100.
Cumulative income %: 4, 12, 26, 50, 100.
Plot (20,4), (40,12), (60,26), (80,50), (100,100). The bow from the 45° line measures inequality.
Using the trapezoidal rule, the area under the Lorenz curve (as a fraction) is 0.284, so Gini \(= 1 - 2(0.284) = 0.43\) — significant inequality.