Expenditure Function

The expenditure function tells you the smallest amount of money a consumer needs to spend to reach a target utility level, given market prices.

Definition

Let (u(x)) be a utility function over a bundle (x), and let (p) be a price vector. For a target utility level (\bar u), the expenditure function is:

[ e(p,\bar u) = \min_{x \ge 0} p \cdot x \quad \text{s.t.} \quad u(x) \ge \bar u. ]

The bundle that solves this problem is the Hicksian (compensated) demand (h(p,\bar u)).

What it measures (economic intuition)

(e(p,\bar u)) is a cost: “How expensive is it to achieve (\bar u) when prices are (p)?”
Holding (\bar u) fixed makes this a natural tool for welfare comparisons, because it keeps utility constant while prices change.

Key properties (useful for theory)

Under standard assumptions, the expenditure function is:

Nondecreasing in prices: if some prices rise (and others do not fall), it is (weakly) more expensive to reach the same utility.
Homogeneous of degree 1 in prices: (e(\lambda p,\bar u)=\lambda e(p,\bar u)) for (\lambda>0).
Concave in prices: it behaves like a cost function when you vary prices.

Connection to cost-of-living and welfare

Because it prices a fixed utility target, (e(p,\bar u)) is closely related to cost-of-living comparisons. A common object is the cost-of-living index holding base-period utility (u_0) constant:

[ \text{COLI}(p_1,p_0;u_0)=\frac{e(p_1,u_0)}{e(p_0,u_0)}. ]

The expenditure function is also used to define:

Compensating variation (CV): how much money would keep you at your original utility after prices change.
Equivalent variation (EV): how much money you would pay (or need) to be as well off as under the new prices.

Worked example (Cobb-Douglas)

If (u(x_1,x_2)=x_1^{\alpha}x_2^{1-\alpha}) with (0<\alpha<1), the expenditure function is:

[ e(p,\bar u)=\bar u,\frac{p_1^{\alpha}p_2^{1-\alpha}}{\alpha^{\alpha}(1-\alpha)^{1-\alpha}}. ]

This expression makes two ideas clear:

higher prices raise the cost of reaching the same (\bar u),
the (\alpha) terms capture how expenditure shares depend on preferences.

Knowledge Check

### What does the expenditure function \(e(p,\bar u)\) measure? - [x] The minimum spending needed to reach utility \(\bar u\) at prices \(p\) - [ ] The maximum utility achievable with income \(\bar u\) at prices \(p\) - [ ] The revenue a firm earns at prices \(p\) - [ ] The quantity demanded when income rises by \(\bar u\) > **Explanation:** By definition, \(e(p,\bar u)\) solves an expenditure-minimization problem subject to achieving at least \(\bar u\). ### The Hicksian (compensated) demand \(h(p,\bar u)\) is: - [x] The cost-minimizing bundle that reaches \(\bar u\) at prices \(p\) - [ ] The utility-maximizing bundle given income \(m\) - [ ] A demand curve that ignores substitution - [ ] A reduced-form forecasting equation > **Explanation:** Hicksian demand is the argmin of \(p\cdot x\) subject to \(u(x)\ge \bar u\). ### How does the expenditure function scale when all prices are multiplied by \(\lambda>0\)? - [x] It scales proportionally: \(e(\lambda p,\bar u)=\lambda e(p,\bar u)\) - [ ] It is unchanged: \(e(\lambda p,\bar u)=e(p,\bar u)\) - [ ] It scales by \(\lambda^2\) - [ ] It becomes undefined > **Explanation:** The expenditure function is homogeneous of degree 1 in prices because it is a cost/minimum-expenditure object. ### In a cost-of-living comparison using \(e(p,\cdot)\), what is held constant? - [x] Utility (the target \(\bar u\) or base-period utility) - [ ] Nominal income - [ ] The set of goods available - [ ] The inflation rate by definition > **Explanation:** The whole point is to ask how much spending is needed to reach the same utility level under different price vectors.