ECON7870 Chap.5 Demand with Cobb-Douglas and CES Utility
Demand with Cobb-Douglas and CES Utility
Move from a bundle to a function
Demand expresses the optimal quantity as prices and income vary. Cobb-Douglas exponents, normalised by their sum, become fixed expenditure shares.
Quantities remain responsive because each spending amount is divided by the relevant price.
Measure substitution separately
CES preferences use a curvature parameter to control how strongly the quantity ratio reacts to the price ratio. Elasticity is unit-free and describes responsiveness rather than the starting mix.
Comparative statics should be read from the formula before values are substituted.
Keep spending and quantity apart
A fixed expenditure share is a monetary statement. If a good's price doubles while income is fixed, its spending share can remain unchanged while its quantity halves. Compute the spending amount first and divide by the matching price second.
Then restore the budget equation to verify that the derived quantities exhaust income under monotonicity.
Read a formula before using numbers
Mark whether each variable enters directly, inversely or not at all. Cobb-Douglas demand is proportional to income and inverse in own price. A variable absent from one demand can still affect utility or another good.
This comparative reading gives a direction prediction and prevents a calculator result from being accepted when it contradicts the formula's structure.
Explain elasticity completely
The sign reports direction, the absolute magnitude reports responsiveness, and the definition names the variable causing the percentage change.
Substitution elasticity concerns a quantity ratio responding to a relative price ratio, not own-price demand alone.
Also state whether the elasticity is constant or local, because a single estimate should not be projected across a large policy change without support.
Check comparative statics in layers
First inspect the formula for sign and proportionality, then calculate the new quantity, and finally translate the result into spending.
Under Cobb-Douglas demand, an own-price rise lowers quantity inversely while leaving that good's expenditure share fixed. Under CES preferences, a relative-price change also alters the quantity ratio according to the substitution elasticity. Keep this response distinct from income elasticity and own-price elasticity.
Reporting the initial and final ratios, the price-ratio change and the maintained income assumption gives a complete interpretation and reveals whether a numerical answer moves in the direction predicted by the demand expression.
What this chapter covers
- 01
Cobb-Douglas demand
- 02
Expenditure shares
- 03
Own-price response
- 04
Income response
- 05
CES preferences
- 06
Elasticity of substitution
- 07
Homogeneity of demand
Worked example · free
Derive quantities from shares
- 2Translate exponents into expenditure shares.
- 2Divide each spending amount by its own price.
- 2Verify the budget and interpret the shares.
Key terms
- Demand Function
- The optimal quantity expressed as a function of prices, income and other maintained parameters.
- Expenditure Share
- The fraction of income spent on a good at the optimum.
- Own-Price Elasticity
- The percentage change in quantity associated with a one per cent change in its own price.
- Income Elasticity
- The percentage change in demand associated with a one per cent change in income.
- Substitution Elasticity
- The responsiveness of the quantity ratio to a change in the relative price ratio.
Demand with Cobb-Douglas and CES Utility FAQ
Why do Cobb-Douglas expenditure shares stay fixed?
The first-order conditions allocate income in proportions determined by the utility exponents. When a price changes, the monetary share remains fixed but the purchased quantity adjusts inversely with that price. Fixed shares therefore do not imply fixed quantities or unchanged utility.
What does a larger substitution elasticity mean?
It means the chosen input or consumption ratio responds more strongly to a change in relative prices. The statement concerns responsiveness, not which good has the larger quantity initially. A high elasticity represents flatter trade-offs and greater scope to reallocate when one good becomes expensive.
How should I interpret a negative elasticity?
The sign gives direction and the absolute value gives responsiveness. An own-price elasticity of -2 means a one per cent price increase is associated with about a two per cent quantity decrease locally. It is more responsive than -1 even though the number is algebraically smaller.
Exam move
Derive one Cobb-Douglas demand from a Lagrangian, then use the share shortcut on new numbers. For every elasticity, write the percentage response and the variable causing it in words. Compare absolute values only after preserving the sign's directional meaning.
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