Shephard27s lemma

What Is Shephard's Lemma?

Shephard's lemma is a fundamental principle in microeconomic theory that relates a firm's or consumer's optimal behavior to their underlying cost function or expenditure function. Specifically, it states that the partial derivative of a cost function with respect to the price of an input gives the conditional demand for that input. Similarly, in consumer theory, the partial derivative of an expenditure function with respect to the price of a good yields the Hicksian (or compensated) demand function for that good. Shephard's lemma is a powerful tool for analyzing how changes in input prices affect the quantities of inputs demanded by firms or the quantities of goods demanded by consumers when a specific output level or utility level is maintained.

History and Origin

Shephard's lemma is named after American economist Ronald Shephard, who formally proved it in his 1953 book, "Theory of Cost and Production Functions."⁷ However, similar concepts and results had been explored earlier by economists like John Hicks (1939) and Paul Samuelson (1947). The equivalent result in the context of consumer theory was notably derived by Lionel W. McKenzie in 1957. The lemma is a cornerstone of duality theory in economics, which establishes deep connections between primal problems (like profit maximization or utility maximization) and their dual counterparts (like cost minimization or expenditure minimization).⁶

Key Takeaways

• Shephard's lemma provides a direct link between a cost (or expenditure) function and the conditional demand for inputs (or goods).
• It is derived from the principle of cost or expenditure minimization, implying that firms and consumers are making optimal choices.
• The lemma is a powerful tool in microeconomic analysis, particularly for understanding producer behavior and consumer responses to price changes while maintaining output or utility.
• It forms a crucial part of duality theory, allowing economists to infer underlying production or utility structures from observed cost or expenditure data.

Formula and Calculation

Shephard's lemma is expressed mathematically through partial differentiation.

For a firm's cost function, (C(w, q)), where (w) is a vector of input prices and (q) is the output quantity, the demand for input (i), denoted as (x_i), is given by:

Here, (x_i(w, q)) represents the conditional input demand for input (i).

For a consumer's expenditure function, (e(p, u)), where (p) is a vector of good prices and (u) is a given level of utility, the Hicksian (compensated) demand for good (i), denoted as (h_i), is given by:

Here, (h_i(p, u)) represents the Hicksian demand function for good (i).

Interpreting Shephard's Lemma

Shephard's lemma offers a direct and intuitive interpretation: the rate at which the minimum cost (or expenditure) changes as the price of a specific input (or good) increases by a small amount is precisely equal to the quantity of that input (or good) being used (or demanded) at the cost-minimizing (or expenditure-minimizing) point.⁵ This means that if a firm is minimizing its costs for a given output, increasing the price of, say, labor by one dollar will increase total costs by the amount of labor currently employed. Similarly, for a consumer, if they are minimizing the expense to achieve a certain level of utility, an increase in the price of a good will raise their minimum expenditure by the quantity of that good they are consuming. This relationship holds because at the optimum, any slight increase in an input price must be fully reflected in the cost if the firm (or consumer) is already choosing the cheapest way to produce (or consume).

Hypothetical Example

Consider a hypothetical firm, "Green Widgets Inc.," that produces widgets using two inputs: labor (L) and capital (K). The prices for these inputs are wage ((w)) and rental rate ((r)), respectively. Suppose Green Widgets Inc.'s cost function for producing (q) widgets is given by:

This function represents the minimum cost to produce (q) units of output given the input prices.

To find the firm's conditional demand for labor ((L)) and capital ((K)) at a specific output level (q), we apply Shephard's lemma:

1. Demand for Labor (L): Take the partial derivative of the cost function with respect to the wage rate ((w)):

   $$L(w, r, q) = \frac{\partial C(w, r, q)}{\partial w} = \frac{\partial}{\partial w} (2 \cdot q \cdot w^{0.5} \cdot r^{0.5}) = 2 \cdot q \cdot 0.5 \cdot w^{-0.5} \cdot r^{0.5} = q \cdot \sqrt{\frac{r}{w}}$$

   So, the conditional demand for labor is (L = q \cdot \sqrt{r/w}).

2. Demand for Capital (K): Take the partial derivative of the cost function with respect to the rental rate ((r)):

   $$K(w, r, q) = \frac{\partial C(w, r, q)}{\partial r} = \frac{\partial}{\partial r} (2 \cdot q \cdot w^{0.5} \cdot r^{0.5}) = 2 \cdot q \cdot 0.5 \cdot w^{0.5} \cdot r^{-0.5} = q \cdot \sqrt{\frac{w}{r}}$$

   Thus, the conditional demand for capital is (K = q \cdot \sqrt{w/r}).

If Green Widgets Inc. wants to produce 100 widgets ((q=100)), and the wage rate is $25 ((w=25)) and the rental rate is $16 ((r=16)):

• Labor demanded: (L = 100 \cdot \sqrt{16/25} = 100 \cdot (4/5) = 80) units.
• Capital demanded: (K = 100 \cdot \sqrt{25/16} = 100 \cdot (5/4) = 125) units.

This example shows how Shephard's lemma allows us to derive the optimal input prices directly from the cost function.

Practical Applications

Shephard's lemma has broad practical applications in economics and finance:

• Empirical Estimation: Economists use the lemma to estimate production function and demand functions. By observing firms' cost data and input prices, researchers can infer the underlying technology and how firms respond to price changes.⁴
• Policy Analysis: It helps policymakers understand the impact of taxes, subsidies, and regulations on firms' production costs and input choices. For instance, a carbon tax on energy inputs can be analyzed using Shephard's lemma to predict its effect on energy demand and overall production costs.
• Environmental Economics: In green finance and environmental policy, Shephard's lemma can be integrated into models to analyze how policies, such as those that increase the "cost" of pollution, influence a firm's demand for cleaner technologies and impact green innovation.³
• Consumer Welfare: In consumer theory, it allows for the derivation of Hicksian demand curves, which are essential for measuring the welfare effects of price changes, such as those caused by excise taxes or trade policies, by holding utility constant.
• Cost Minimization: Firms seeking profit maximization implicitly engage in cost minimization. Shephard's lemma provides a direct method to determine the optimal mix of inputs required to produce a specific output quantity at the lowest possible cost, given prevailing input prices.

Limitations and Criticisms

While Shephard's lemma is a powerful tool, its applicability depends on certain underlying assumptions about the cost or expenditure function. These include:

• Differentiability: The cost or expenditure function must be differentiable with respect to input prices. This implies smooth changes in input demand as prices change, which may not always hold true in real-world scenarios, particularly with discrete input choices or technologies.²
• Convexity: The underlying preferences (for consumers) or technology (for firms) must exhibit convexity. For firms, this means the input requirement set is a convex set, implying that averaging two input bundles that can produce a given output can also produce that output. For consumers, it means preferences are convex, ensuring a unique expenditure-minimizing bundle. If these assumptions are violated, the lemma may not provide accurate results.¹
• Cost Minimization: The lemma assumes that the firm (or consumer) is actively minimizing costs (or expenditures). If decision-makers are not perfectly rational or face other constraints (e.g., imperfect information, managerial slack), the observed input demands may deviate from those predicted by Shephard's lemma.
• Fixed Output/Utility: The lemma applies when output (for firms) or utility (for consumers) is held constant. It does not directly account for situations where output levels change in response to price shifts or other market dynamics.

Despite these limitations, Shephard's lemma remains a cornerstone of economic theory, especially in theoretical and empirical analyses where these assumptions are reasonable approximations.

Shephard's Lemma vs. Roy's Identity

Shephard's lemma and Roy's Identity are both fundamental results in microeconomic theory that relate optimal choice functions to value functions, but they do so in different contexts and yield different demand functions.

The primary distinction lies in what is held constant. Shephard's lemma keeps the output level (for firms) or utility level (for consumers) fixed, illustrating how demand for an input or good changes when prices change while maintaining that specific level. Roy's Identity, conversely, keeps income (or budget) fixed and describes how Marshallian demand changes in response to price variations. Both are derived from duality in economic optimization problems but serve distinct analytical purposes.

FAQs

What is the core idea behind Shephard's Lemma?

The core idea of Shephard's lemma is that if a firm or consumer is minimizing costs or expenditures, then the amount of an input or good they demand is precisely equal to how much their minimum cost or expenditure would increase if the price of that input or good were to slightly rise. It directly links the rate of change of a cost or expenditure function to the quantity of a specific input or good.

How is Shephard's Lemma used in production theory?

In production function, Shephard's lemma is used to derive a firm's conditional input prices by taking the partial derivative of the firm's cost function with respect to each input's price. This allows economists to understand how a firm's demand for labor, capital, or other inputs changes as their prices fluctuate, assuming the firm maintains a specific output level and minimizes costs.

Can Shephard's Lemma be applied to consumers?

Yes, Shephard's lemma applies to consumers in the context of consumer theory. When applied to a consumer's expenditure function, it states that the partial derivative of the expenditure function with respect to the price of a good yields the Hicksian (or compensated) demand function for that good. This demand function shows the quantity of a good a consumer would demand to achieve a specific level of utility at minimum cost.

What assumptions are necessary for Shephard's Lemma to hold?

For Shephard's lemma to hold, the underlying cost or expenditure function must be continuous, differentiable, and convex in input prices. It also assumes that the firm or consumer is engaging in cost or expenditure minimization—that is, they are making rational, optimal choices given their objectives and constraints.

How does Shephard's Lemma relate to marginal cost?

While Shephard's lemma directly links the partial derivative of the total cost function with respect to an input price to the conditional input demand, marginal cost refers to the change in total cost resulting from producing one additional unit of output. These are distinct but related concepts within the broader framework of cost theory. Shephard's lemma focuses on how input prices affect input demand at a given output level, while marginal cost focuses on how total cost changes with output.