ECON7870 Chap.6 Indirect Utility and Expenditure Duality
Indirect Utility and Expenditure Duality
Two optimisation problems
Utility maximisation holds prices and income fixed, yielding Marshallian demand and indirect utility. Expenditure minimisation holds prices and a utility target fixed, yielding Hicksian demand and minimum expenditure.
The two value functions invert one another in the relevant argument.
Use duality as an error check
Minimum expenditure at the utility achieved by an income should return that income. Likewise, giving the minimum required expenditure back to the primal problem should reproduce the target.
This round trip checks quantities, value and units together.
Keep values and quantities distinct
Indirect utility and expenditure are value functions: one returns achieved utility and the other minimum cost. Marshallian and Hicksian demand return quantities. Envelope results connect them, but the maintained conditions differ.
Label the inputs beside each expression so a price derivative of expenditure is not confused with a price response at fixed income.
Check economic properties
Indirect utility rises with income and falls when prices rise. Minimum expenditure rises with the utility target, cannot fall when a price rises and scales proportionally when every price scales together.
These direction and homogeneity checks often detect a missing exponent or price term before the original optimisation is repeated.
Close the round trip
Price the compensated quantities and confirm they reproduce the expenditure function. Insert them into utility and confirm the target. Next use that expenditure as income in the primal problem and recover the same bundle.
Satisfying several independent identities makes the derivation credible and turns duality into an audit method rather than an extra set of formulas.
Trace the maintained variable
Write the fixed item beside every derivative or comparison. A Marshallian price response keeps nominal income fixed and therefore combines substitution with purchasing-power change.
A Hicksian response keeps utility fixed by changing expenditure and isolates substitution. Roy's identity and Shephard's lemma recover different demands because their value functions embody those different controls. After using either result, test units and reconstruct the associated value: price compensated quantities to recover minimum expenditure, or insert ordinary demands into utility to recover indirect utility.
This labelled route prevents a correct derivative from receiving the wrong economic interpretation.
What this chapter covers
- 01
Indirect utility
- 02
Expenditure minimisation
- 03
Hicksian demand
- 04
Primal and dual problems
- 05
Expenditure properties
- 06
Shephard's lemma
- 07
Money-metric welfare
Worked example · free
Price a utility target
- 3Use MRS and prices to obtain the cost-minimising ratio.
- 3Impose the utility target and solve quantities.
- 2Compute expenditure and verify the primal round trip.
Key terms
- Indirect Utility
- The maximum attainable utility after optimising at stated prices and income.
- Expenditure Function
- The minimum spending required to reach a utility target at stated prices.
- Marshallian Demand
- Demand from utility maximisation with nominal income fixed.
- Hicksian Demand
- Demand from expenditure minimisation with utility fixed.
- Shephard's Lemma
- The result that a price derivative of the expenditure function gives compensated demand for that good.
Indirect Utility and Expenditure Duality FAQ
Why are indirect utility and expenditure called dual?
They describe the same preferences from opposite constraints. One asks how much utility a budget can buy; the other asks how much budget a utility target costs. Feeding the optimal value from one into the other returns the original constraint under standard conditions.
What does Shephard's lemma provide?
Differentiate the expenditure function with respect to a good's price while holding utility and other prices fixed. The derivative is Hicksian demand for that good. The result links a value function to compensated quantities and supplies a direct route to substitution analysis.
How can I check a derived expenditure function?
It should be non-decreasing in prices, homogeneous of degree one in all prices and concave in prices. Doubling every price should double required expenditure for the same utility. A violation usually signals a missing square root, exponent or price factor.
Exam move
Solve a primal and dual problem side by side using the same preferences. Circle which variable is fixed and which is solved in each. Then substitute the achieved utility into expenditure and the required expenditure into indirect utility to verify the round trip.
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