If a firm uses n inputs (n>2), what inequality does the...

If a firm uses $n$ inputs $(n>2)$, what inequality does the theory of revealed cost minimization imply about changes in factor prices $\left(\Delta w_{i}\right)$ and the changes in factor demands $\left(\Delta x_{i}\right)$ for a given level of output?

Key Concepts

Revealed Cost Minimization

Revealed cost minimization refers to the behavioral implication that firms choose input combinations that minimize costs given the input prices and desired output level. When firms’ observed choices are consistent with cost minimization, any changes in factor prices and the corresponding adjustments in input demands will satisfy specific inequalities that reflect optimality conditions, allowing us to validate whether the behavior is consistent with cost minimizing behavior.

Cost Function Duality

Duality in production theory pertains to the relationship between a firm's production function and its cost function. Instead of focusing directly on input combinations, analysis via the cost function summarizes the minimum cost of producing a given output. This duality provides a powerful framework where properties of the cost function, such as homogeneity and concavity with respect to input prices, translate into specific restrictions on how input demands respond to changes in prices.

Shephard’s Lemma

Shephard’s Lemma is a key result that connects the cost function to input demand functions. It states that the partial derivative of the cost function with respect to an input’s price yields the conditional demand for that input, assuming cost minimizing behavior. This link is critical in revealing how optimal input choices adjust with changes in factor prices and underpins the derivation of inequalities that relate these price adjustments to changes in input usage.

Concavity of the Cost Function

The concavity of the cost function in input prices is central to ensuring that the cost minimization problem has well?behaved solutions. Specifically, the cost function's concavity implies that its Hessian matrix (the matrix of second order partial derivatives) is negative semi-definite. This property guarantees that the combined variations in input prices and input quantities—when examined via the optimality conditions—satisfy an inequality (commonly expressed as the sum of the product of changes in factor prices and changes in factor demands being less than or equal to zero) reflecting the substitution and adjustment behavior inherent in cost minimization.

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