All prices in an economy double while a consumer's target utility level remains unchanged. What happens to the value of the expenditure function e(p, u)?
AIt stays the same — the consumer can achieve the same utility with the same bundle
BIt more than doubles — the consumer must substitute toward goods that got relatively less expensive
CIt exactly doubles — because the expenditure function is homogeneous of degree 1 in prices
DIt depends on the specific utility function
The expenditure function is homogeneous of degree 1 in prices: e(λp, u) = λ · e(p, u) for any scalar λ > 0. When all prices double (λ = 2), minimum expenditure exactly doubles. Intuitively, the optimal bundle does not change when all prices double proportionally (relative prices are unchanged), so the cost of that bundle simply doubles. Option D is the tempting wrong answer — while the specific value of e depends on the utility function, the homogeneity property holds for all well-behaved utility functions.
Question 2 Multiple Choice
Which of the following correctly describes the expenditure function e(p, u)?
AThe total amount a consumer actually spends given income m and prices p
BThe maximum utility achievable with income m at prices p
CThe minimum income needed to reach utility level u at prices p
DThe marginal cost of increasing utility by one unit at prices p
The expenditure function answers the question: 'What is the least I must spend to achieve exactly utility u, given prices p?' It is defined as e(p, u) = min_x {p·x : u(x) ≥ u}. Option A confuses the function (a mapping from prices and utility to a number) with actual realized expenditure. Option B describes the indirect utility function — the dual object. Option D is not a standard concept in this context.
Question 3 True / False
The expenditure function e(p, u) and the indirect utility function v(p, m) contain exactly the same information about consumer preferences.
TTrue
FFalse
Answer: True
These two functions are inverses of each other: e(p, v(p, m)) = m and v(p, e(p, u)) = u. The primal problem (maximize utility given income) and the dual problem (minimize expenditure given utility) yield the same underlying preference structure, just represented from opposite perspectives. Any welfare question you can answer with one representation you can answer with the other — the choice between them is purely a matter of mathematical convenience.
Question 4 True / False
Because achieving higher utility generally requires more spending, the expenditure function is convex in u.
TTrue
FFalse
Answer: False
The expenditure function is non-decreasing in u (higher utility targets require at least as much spending), but the relevant convexity property is concavity in prices, not convexity in utility. The concavity in prices reflects the consumer's ability to substitute toward relatively cheaper goods when a price rises, which means expenditure does not rise as fast as a proportional price increase would suggest. The confusion between 'non-decreasing' and 'convex' is a common error.
Question 5 Short Answer
How does the expenditure function differ from simply 'the amount a consumer spends,' and what is its duality relationship with the indirect utility function?
Think about your answer, then reveal below.
Model answer: The expenditure function e(p, u) is not a number but a function that maps any combination of prices and a target utility level to the minimum cost of achieving that utility. Actual expenditure is a single number for a specific situation. The duality relationship is that e and the indirect utility function v are inverses: e(p, v(p, m)) = m (spending the minimum to reach the utility you'd get from income m costs exactly m) and v(p, e(p, u)) = u (using the minimum expenditure to achieve u gives you exactly utility u).
This duality means the expenditure-minimization and utility-maximization problems are two sides of the same coin. Any result derived from one can be translated to the other. In practice, the expenditure function is easier to use for welfare analysis and for cleanly separating income and substitution effects via Hicksian demand.