Shephard’s Lemma: What It Says, Why It’s True, and How It Links Expenditure to Hicksian Demand

Shephard’s Lemma: What It Says and Why It’s True

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Shephard’s Lemma is a core topic in microeconomics tutoring and appears frequently in graduate consumer theory, duality, and optimization assignments. Students often struggle because the lemma connects three different objects: the expenditure function, Hicksian demand, and cost minimization. This page breaks the idea down clearly so you can master it for problem sets and exams.

Formally, if \(E(p, u)\) is the minimum expenditure needed to achieve utility \(u\) at prices \(p\), then:

\(\frac{\partial E(p, u)}{\partial p_i} = h_i(p, u)\)

Why does this work? Because the expenditure function is defined as the value of a cost-minimization problem. When you differentiate the value of an optimization problem with respect to a parameter, the envelope theorem tells you that the derivative equals the optimal choice of the variable associated with that parameter. Here, the parameter is the price, and the optimal choice is Hicksian demand.

1. Start with the cost-minimization problem. Minimize \(p \cdot x\) subject to \(u(x) \ge u_0\).
2. Write the Lagrangian. \(L = p \cdot x + \lambda (u_0 – u(x))\).
3. Derive the FOCs. These give the Hicksian demand functions \(h(p, u_0)\).
4. Plug the optimal bundle back into the objective. This gives the expenditure function \(E(p, u_0)\).
5. Apply the envelope theorem. The derivative of the minimized value with respect to a price equals the optimal quantity of that good.
6. Conclude the lemma. \(\partial E / \partial p_i = h_i(p, u_0)\).

Suppose a consumer has Cobb–Douglas utility \(u(x_1, x_2) = x_1^{0.5} x_2^{0.5}\). The Hicksian demands are:

\(h_1 = \frac{u \cdot p_2}{p_1 + p_2}, \quad h_2 = \frac{u \cdot p_1}{p_1 + p_2}\)

The expenditure function is:

\(E(p_1, p_2, u) = u (p_1 p_2)^{0.5}\)

Differentiate with respect to \(p_1\):

\(\frac{\partial E}{\partial p_1} = \frac{u}{2} \sqrt{\frac{p_2}{p_1}} = h_1\)

This verifies Shephard’s Lemma.

• Confusing Hicksian demand with Marshallian demand.
• Forgetting that the expenditure function comes from cost minimization, not utility maximization.
• Trying to differentiate before solving the minimization problem.
• Mixing up the roles of prices and utility in the expenditure function.

Shephard’s Lemma is essential for duality, Slutsky decomposition, welfare analysis, and deriving compensated elasticities. It appears in nearly every graduate microeconomics exam and is foundational for understanding how consumers adjust to price changes while holding utility constant.

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