Consumer Theory and Duality Explained: Utility, Expenditure and the True Cost of a Price Rise

Understanding consumer theory and duality — simply put

Consumer theory describes how a household chooses what to buy when its income is limited and every good has a price. The household ranks bundles of goods by how much it likes them, and picks the best bundle it can afford. Duality is the observation that this choice can be read backwards: the bundle that gives the most satisfaction for a given budget is also the cheapest bundle that delivers that level of satisfaction. Two problems, one answer. That second reading turns well-being into money, which is what lets economists say how much a price rise actually costs someone.

How does it actually help?

Take a UK household facing a jump in energy prices. It spends a share of its budget on energy and the rest on everything else. A standard way to represent its preferences is the Cobb-Douglas utility function:

The household spends m in total, pays p_E per unit of energy and p_O per unit of everything else, and cannot spend more than it has: p_E x_E + p_O x_O ≤ m. Maximizing u under that budget gives a simple rule: spend the fraction a of income on energy, so x_E = a·m / p_E. That rule is the household's Marshallian demand — demand as a function of prices and income.

Duality asks the reverse question: what is the least money that reaches a given utility level u at today's prices? The answer is the expenditure function, e(p, u). If energy prices rise, e(p, u) tells you exactly how much extra income would leave the household as well off as before. That number is the compensating variation, the theoretically correct cost of the price rise.

Why does this matter?

Every cost-of-living calculation rests on this distinction. A consumer price index prices a fixed basket: it asks what last year's bundle costs at this year's prices. Households do not keep buying last year's bundle when one price jumps; they turn the heating down and shift spending elsewhere. The expenditure function accounts for that, so the gap between the two is the well-known substitution bias in fixed-basket price indices. Governments that index benefits and pensions, central banks that target inflation, and regulators that assess the welfare cost of a tax or a tariff all rely on these tools.

What makes the model reliable?

The results hold when preferences are well behaved: the household can rank any two bundles, its rankings are consistent, more is preferred to less, and it prefers balanced bundles to extreme ones. Under those conditions demand is well defined, utility maximization and expenditure minimization give the same bundle, and the demand curves the model produces slope downward once income effects are removed. The assumptions, and what breaks when each one fails, are set out in full below.

The takeaway

Utility maximization tells you what a household buys; expenditure minimization tells you what its well-being costs. Put together, they measure the real burden of a price change in pounds or euros, not in an abstract unit of satisfaction. The rest of this guide works through a real price shock, then sets out the full mechanics.

A real example: the 2022 UK energy price cap rise

On 1 April 2022 Ofgem, the UK energy regulator, raised the price cap for a typical dual-fuel household on a default tariff from £1,277 to £1,971 a year, an increase of about 54% affecting roughly 22 million customers. In the year to March 2022 the Office for National Statistics' Family Spending survey put average household spending at £528.80 a week, of which £25.70 went on electricity, gas and other fuels.

Those two figures pin down the model. Annual spending is m = £528.80 × 52 ≈ £27,498. The energy budget share is a = 25.70 / 528.80 ≈ 0.0486, so the household spends about £1,336 a year on energy — close to Ofgem's typical bill, which is a useful check that the numbers hang together. The price shock is a ratio of r = 1,971 / 1,277 ≈ 1.543 on energy, with other prices held fixed to isolate its effect.

The fixed-basket figure asks what the old energy bill costs at the new price: a·m·(r − 1) = £1,336 × 0.543 ≈ £726. That is what a basket-based price index implicitly assumes the household needs.

The compensating variation (CV) comes from the expenditure function. For Cobb-Douglas preferences, the money needed to reach the original utility rises in proportion to r^a, so CV = m·(r^a − 1) ≈ £586. It is smaller than £726 because a household given that money would not rebuy its old bundle; it would use less energy and more of everything else, and reach the same well-being more cheaply.

The equivalent variation (EV) asks the mirror question: how much income would the household have to lose, at the old prices, to end up as badly off as the price rise leaves it? EV = m·(1 − r^(−a)) ≈ £574. CV and EV differ because of the income effect; for a good that takes under 5% of the budget the difference is small.

Reading the result

The £140 gap between the fixed-basket and expenditure-function measures is substitution bias, about half a percentage point of total spending. It is the reason statistical agencies update basket weights regularly and why some use chained or superlative indices, which approximate the expenditure function better than a fixed basket.

The model also splits the fall in energy use into two parts, the Slutsky decomposition. Setting the original energy quantity to 100:

Almost the whole predicted fall is a substitution effect; the income effect is small because energy is a small share of the budget. The predicted fall of about 35% is far larger than what UK households actually did in 2022. Cobb-Douglas preferences force a price elasticity of exactly −1, while short-run energy demand is much less responsive. With less substitution, the compensating variation moves up toward the fixed-basket £726. The two numbers in the first table are therefore best read as the bounds of the true cost: £726 if households cannot substitute at all, about £586 if they substitute as freely as Cobb-Douglas assumes.

The assumptions

Completeness. The household can compare any two bundles: it prefers one, or is indifferent. Without it, there is no "best" bundle to choose, because some bundles cannot be ranked at all.

Transitivity. If A is preferred to B and B to C, then A is preferred to C. Without it, choices can cycle, and no utility function can represent the preferences, so the entire apparatus below has nothing to work on.

Continuity. Small changes in a bundle produce small changes in how it is ranked. This guarantees that preferences can be written as a continuous utility function, which in turn guarantees that a best affordable bundle exists.

Local nonsatiation. Near any bundle there is a slightly better one. This forces the household to spend its whole budget, which is exactly what makes the two problems dual: if a household left money unspent, the cheapest way to reach its utility would cost less than its income, and the identity e(p, v(p, m)) = m would fail.

Convexity (strict quasi-concavity of utility). Averages are preferred to extremes. It makes demand a single bundle rather than a set, and it is what makes the substitution effect of a price rise negative. Without it, demand can jump discontinuously as prices move.

The formula

The derivations below are optional for anyone who only needs the intuition, but they are what graduate courses and research papers use.

Let p = (p₁, …, pₙ) be the price vector, m income, and x = (x₁, …, xₙ) a bundle. The utility maximization problem (UMP) and the expenditure minimization problem (EMP) are:

The solution to the UMP is Marshallian demand x(p, m); the solution to the EMP is Hicksian (compensated) demand h(p, u). Under local nonsatiation the two problems are linked by four identities:

Shephard's lemma recovers Hicksian demand by differentiating the expenditure function, by the envelope theorem applied to the EMP:

Roy's identity does the same for Marshallian demand from the indirect utility function:

Differentiating the identity h_i(p, u) = x_i(p, e(p, u)) with respect to p_j and applying Shephard's lemma gives the Slutsky equation:

The Slutsky matrix S, with entries ∂h_i/∂p_j, is symmetric and negative semi-definite. Its diagonal entries are therefore non-positive: compensated demand never rises with its own price. Uncompensated demand can, in principle, rise with price (a Giffen good) only if the income effect is large and of the wrong sign.

For the two-good Cobb-Douglas case used in the example, with u = x₁^a x₂^(1−a):

Because e(p, u) is proportional to p₁^a, a price ratio r on good 1 multiplies the cost of the original utility level by r^a. That gives the measures used in the worked example:

Because r^a is a convex function of r, CV is always below the fixed-basket figure. The gap is the substitution bias.

The Slutsky equation is often used in elasticity form, where it can be estimated from data:

With many goods the same structure holds. Demand systems such as the Almost Ideal Demand System are built by specifying an expenditure function directly and deriving demands by Shephard's lemma, which guarantees the results are consistent with utility maximization.

When this isn't the right tool

Cobb-Douglas is too rigid for energy. It fixes every own-price elasticity at −1 and every budget share as constant. Estimating a flexible demand system, or at least a CES function with a measured elasticity of substitution, with regression methods on household data gives a more credible compensating variation.

An average household hides the distribution. Lower-income households spend a larger share of their budgets on energy, so their compensating variation as a share of spending is larger. A policy assessment should compute CV by income group, not for the mean.

Rationality is an assumption. Inattention, habits and reference points mean real households do not always optimize. Behavioral economics documents these departures; the duality framework remains the benchmark they are measured against.

One market at a time. The example holds other prices fixed. In 2022 the energy shock pushed up food and other prices as well, which requires a general equilibrium view.

Further reading

• Understanding Inflation — internal
• Supply and Demand Explained — internal
• Producer Theory and Cost Functions Explained — internal
• General Equilibrium and the Welfare Theorems Explained — internal
• OLS Regression Explained — internal
• ONS, Family spending in the UK: April 2021 to March 2022 — external reference
• Ofgem, Price cap to increase by £693 from April — external reference

Spending levels are from the ONS Family Spending survey for April 2021 to March 2022 and the price change from Ofgem's February 2022 price cap announcement. The compensation figures are EconoLens calculations from a Cobb-Douglas model calibrated to those data, not measured outcomes.