Shephard's Lemma and the Expenditure Function | 谢泼德引理与支出函数

📚 Shephard’s Lemma and the Expenditure Function | 谢泼德引理与支出函数

In microeconomic theory, the expenditure function plays a central role in the dual approach to consumer choice. It tells us the minimum amount of money a consumer needs to achieve a given level of utility, given market prices. Shephard’s lemma, a key result in duality theory, links the expenditure function directly to the compensated (Hicksian) demand function.

在微观经济理论中，支出函数在消费者选择的偶性方法中占据核心地位。它告诉我们，在给定市场价格下，消费者达到某一效用水平所需的最低货币支出。谢泼德引理是偶性理论中的关键结果，它将支出函数直接与补偿（希克斯）需求函数联系起来。

1. The Expenditure Function | 支出函数的定义

Let p = (p₁, p₂, …, pₙ) be the vector of prices and u be a target utility level. The expenditure function e(p, u) is defined as the minimum expenditure required to reach utility level u:

设 p = (p₁, p₂, …, pₙ) 为价格向量，u 为目标效用水平。支出函数 e(p, u) 定义为达到效用水平 u 所需的最小支出：

e(p, u) = minₓ { p·x : U(x) ≥ u }

Here x is a consumption bundle and U(x) is the utility function. The expenditure function inherits a number of important properties from the underlying consumer problem.

这里 x 是消费束，U(x) 是效用函数。支出函数从潜在的消费者问题中继承了许多重要性质。

2. Basic Properties of e(p, u) | 支出函数的基本性质

The expenditure function has four standard properties:

支出函数具有四个标准性质：

Nondecreasing in prices: If pᵢ′ ≥ pᵢ, then e(p′, u) ≥ e(p, u).
价格非递减： 若 pᵢ′ ≥ pᵢ，则 e(p′, u) ≥ e(p, u)。
Homogeneous of degree 1 in p: e(tp, u) = t e(p, u) for t > 0.
关于价格一次齐次： 对于 t > 0，e(tp, u) = t e(p, u)。
Concave in p: e(tp + (1−t)p′, u) ≥ t e(p, u) + (1−t) e(p′, u).
关于价格凹性： e(tp + (1−t)p′, u) ≥ t e(p, u) + (1−t) e(p′, u)。
Continuous in p and u (provided the utility function is well-behaved).
关于 p 和 u 连续（在效用函数性质良好时成立）。

3. The Hicksian Demand Function | 希克斯需求函数

The Hicksian (compensated) demand function h(p, u) is the bundle that solves the expenditure-minimisation problem. It is the cheapest way to achieve utility u at prices p.

希克斯（补偿）需求函数 h(p, u) 是解决支出最小化问题的消费束，它是在价格 p 下达到效用 u 的最便宜方式。

For a well-behaved utility function, the expenditure function and the Hicksian demand function are intimately connected through Shephard’s lemma.

对于性质良好的效用函数，支出函数与希克斯需求函数通过谢泼德引理紧密相连。

4. Shephard’s Lemma | 谢泼德引理

Shephard’s lemma states that the partial derivative of the expenditure function with respect to a price equals the Hicksian demand for that good:

谢泼德引理指出，支出函数对某一价格的偏导数等于该商品的希克斯需求：

∂e(p, u) / ∂pᵢ = hᵢ(p, u)

This result holds whenever the expenditure function is differentiable at the relevant point. It is a direct application of the envelope theorem to the expenditure-minimisation problem.

当支出函数在相关点可微时，该结论成立。它是包络定理直接应用于支出最小化问题的结果。

5. Intuition Behind the Lemma | 引理背后的直觉

Suppose the price of good i rises by one unit. To maintain the same utility level, the consumer must receive extra income. How much extra income is needed? Roughly speaking, the consumer still buys hᵢ units of good i, so each unit of price increase costs an additional hᵢ dollars. This is why the derivative of minimum expenditure equals the compensated quantity demanded.

假设商品 i 的价格上涨一个单位。为了维持相同的效用水平，消费者需要额外收入。需要多少额外收入？大致而言，消费者仍然购买 hᵢ 单位的商品 i，因此每单位价格上涨需要额外花费 hᵢ 美元。这就是为什么最小支出的导数等于补偿需求数量。

If the consumer could adjust their bundle, the marginal adjustment effect is zero because h(p, u) already minimises expenditure at the original prices. This is the envelope argument.

如果消费者可以调整其消费束，边际调整效应为零，因为 h(p, u) 已经在原价格下使支出最小化。这就是包络论证。

6. Derivation Using the Envelope Theorem | 利用包络定理推导

Let L = p·x + λ(u − U(x)) be the Lagrangian for the expenditure-minimisation problem. The first-order conditions are:

设 L = p·x + λ(u − U(x)) 为支出最小化问题的拉格朗日函数。一阶条件为：

∂L/∂xᵢ = pᵢ − λ ∂U/∂xᵢ = 0

The minimum value function is e(p, u) = L(x*, λ*, p, u). By the envelope theorem, the derivative of the value function with respect to pᵢ equals the partial derivative of the Lagrangian with respect to pᵢ, holding x* and λ* fixed:

最小值函数为 e(p, u) = L(x*, λ*, p, u)。根据包络定理，值函数对 pᵢ 的导数等于拉格朗日函数对 pᵢ 的偏导数，保持 x* 和 λ* 不变：

∂e/∂pᵢ = ∂L/∂pᵢ = xᵢ* = hᵢ(p, u)

7. Example: Cobb-Douglas Utility | 例子：柯布-道格拉斯效用

Consider a two-good Cobb-Douglas utility function U(x₁, x₂) = x₁^a x₂^(1−a), with 0 < a < 1. Solving the expenditure-minimisation problem gives the Hicksian demands:

考虑两商品柯布-道格拉斯效用函数 U(x₁, x₂) = x₁^a x₂^(1−a)，其中 0 < a < 1。求解支出最小化问题得到希克斯需求：

h₁ = (a/(p₁))^(1−?) …

Actually, the standard solution is:

实际上，标准解为：

h₁(p, u) = a(p₂/(p₁(1−a)))^(1−a) u, h₂(p, u) = (1−a)(p₁/(p₂a))^a u

The expenditure function is e(p, u) = (p₁/a)^a (p₂/(1−a))^(1−a) u. Differentiating e with respect to p₁ gives exactly h₁, confirming Shephard’s lemma.

支出函数为 e(p, u) = (p₁/a)^a (p₂/(1−a))^(1−a) u。对 e 关于 p₁ 求导恰好得到 h₁，验证了谢泼德引理。

8. Relation to the Indirect Utility Function | 与间接效用函数的关系

The expenditure function is the inverse of the indirect utility function in a well-defined sense. Let v(p, y) be the maximum utility obtainable with income y. Then:

支出函数在明确的意义上是间接效用函数的逆。设 v(p, y) 为收入 y 下可获得的最大效用。则：

e(p, u) = y ⇔ v(p, y) = u

This duality implies that e(p, v(p, y)) = y and v(p, e(p, u)) = u. Shephard’s lemma can also be derived from Roy’s identity using this inverse relationship.

这种偶性意味着 e(p, v(p, y)) = y 且 v(p, e(p, u)) = u。利用这一逆关系，也可以从罗伊恒等式推导出谢泼德引理。

9. Shephard’s Lemma vs. Roy’s Identity | 谢泼德引理与罗伊恒等式

Roy’s identity gives the Marshallian (ordinary) demand from the indirect utility function:

罗伊恒等式从间接效用函数给出马歇尔（普通）需求：

xᵢ(p, y) = − (∂v/∂pᵢ) / (∂v/∂y)

Shephard’s lemma gives the Hicksian demand from the expenditure function. The two demands are related by the identity xᵢ(p, y) = hᵢ(p, v(p, y)). Thus Shephard’s lemma is the expenditure-side analogue of Roy’s identity.

谢泼德引理从支出函数给出希克斯需求。两种需求通过恒等式 xᵢ(p, y) = hᵢ(p, v(p, y)) 相联系。因此，谢泼德引理是罗伊恒等式在支出方面的对应。

10. Elasticity Relations | 弹性关系

Shephard’s lemma can be used to derive the Slutsky equation, which decomposes the effect of a price change on Marshallian demand into substitution and income effects:

谢泼德引理可用于推导斯勒茨基方程，该方程将价格变化对马歇尔需求的影响分解为替代效应和收入效应：

∂xᵢ/∂pⱼ = ∂hᵢ/∂pⱼ − xⱼ (∂xᵢ/∂y)

The substitution term ∂hᵢ/∂pⱼ is obtained by differentiating the Hicksian demand, which is itself the gradient of the expenditure function. Symmetry of the Slutsky matrix follows from Young’s theorem applied to e(p, u).

替代项 ∂hᵢ/∂pⱼ 通过对希克斯需求求导得到，而希克斯需求本身又是支出函数的梯度。斯勒茨基矩阵的对称性来自对 e(p, u) 应用杨定理。

11. Applications in Applied Economics | 在应用经济学中的应用

Shephard’s lemma is widely used in cost-of-living index construction. The change in the expenditure function due to price changes measures the true cost-of-living index. Since the derivative gives Hicksian demands, researchers can recover compensated demand elasticities from observed expenditure data.

谢泼德引理广泛用于生活成本指数的构建。由价格变化引起的支出函数变化度量了真实生活成本指数。由于导数给出希克斯需求，研究者可以从观测到的支出数据中恢复补偿需求弹性。

In production theory, the same lemma applies to cost functions: the derivative of the cost function with respect to an input price gives the conditional factor demand. This makes Shephard’s lemma a cornerstone of empirical industrial organisation and trade policy analysis.

在生产理论中，同一引理适用于成本函数：成本函数对投入品价格的导数给出条件要素需求。这使得谢泼德引理成为实证产业组织和贸易政策分析的基石。

12. Common Pitfalls and Exam Tips | 常见错误与考试提示

Students often confuse Hicksian and Marshallian demands. Remember: Shephard’s lemma differentiates the expenditure function, so it yields h(p, u), which holds utility constant. Roy’s identity differentiates the indirect utility function and yields x(p, y), which holds income constant.

学生常混淆希克斯需求与马歇尔需求。记住：谢泼德引理是对支出函数求导，因此得到 h(p, u)，即保持效用不变；罗伊恒等式是对间接效用函数求导，得到 x(p, y)，即保持收入不变。

Only differentiate the expenditure function with respect to prices, not with respect to u.
检查可微性条件；在非光滑点（如角点解）需使用次梯度。
Check differentiability conditions; at non-smooth points (corner solutions) use subgradients.
清楚地区分补偿需求与普通需求。
Clearly distinguish compensated demand from ordinary demand.

When working through exam problems, first write down the expenditure-minimisation problem, then solve for h(p, u), integrate or differentiate to verify Shephard’s lemma. This systematic approach avoids sign errors.

在解决考试问题时，首先写出支出最小化问题，然后求解 h(p, u)，求导或积分以验证谢泼德引理。这种系统方法可以避免符号错误。

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Shephard’s Lemma and the Expenditure Function | 谢泼德引理与支出函数