Questions: Factor Demands and Substitution Elasticity

Question 1

A government raises the minimum wage by 20%. A manufacturing firm uses labor and automated machinery as inputs. If the elasticity of substitution between labor and capital is high (σ ≈ 2), what is the most likely outcome?

• A. Employment falls sharply as the firm substitutes toward capital, since high σ means inputs are easily swapped
• B. Employment is largely unchanged because high σ means the firm is already using the optimal mix
• C. Employment rises as higher wages attract more productive workers, increasing output
• D. Employment falls slightly because high σ means the firm is locked into its current input mix

Answer: (A)

High substitution elasticity means the firm can readily swap between inputs when relative prices change. When wages rise, the price of labor increases relative to capital, so a firm with high σ responds by substituting toward capital — automating tasks, reducing headcount. The higher σ is, the more dramatic this shift. The common misconception is option D, which gets the direction backwards: high σ implies flexibility and large substitution responses, not rigidity.

Question 2

A Leontief production function describes inputs that must be used in strictly fixed proportions (like a pilot and a plane). What is the elasticity of substitution for this technology?

• A. σ = 1, because cost-minimizing firms always adjust input ratios proportionally to price ratios
• B. σ = ∞, because the firm can always hire more pilots without needing more planes
• C. σ = 0, because no matter how wages or rental costs change, the firm cannot alter its capital-labor ratio
• D. σ > 1, because complementary inputs are more substitutable than independent inputs

Answer: (C)

The Leontief technology requires inputs in a fixed ratio — adding more of one input without the other yields no additional output. Therefore, even if wages rise dramatically, the firm cannot substitute capital for labor; it must continue using inputs in the same proportions. This corresponds to σ = 0. The Cobb-Douglas function, by contrast, has σ = 1 (constant unit elasticity of substitution), and the CES production function generalizes to any σ ≥ 0.

Question 3

A Cobb-Douglas production function always has an elasticity of substitution equal to one, meaning a 10% increase in the wage-rental ratio causes exactly a 10% increase in the capital-labor ratio.

• T. True
• F. False

Answer: True

This is the defining property of the Cobb-Douglas technology. The cost shares of labor and capital are constant (equal to the output elasticities α and 1−α), and the capital-labor ratio responds proportionally to changes in relative factor prices. This makes Cobb-Douglas a useful benchmark: σ = 1 everywhere, regardless of the level of output or input prices.

Question 4

If the elasticity of substitution between labor and capital is near zero, a large increase in the minimum wage will cause a large reduction in employment.

• T. True
• F. False

Answer: False

Near-zero substitution elasticity means the firm has almost no ability to swap between inputs — it must use them in nearly fixed proportions. When wages rise, a firm with σ ≈ 0 cannot easily replace workers with machines, so the employment effect is small. Large employment effects from wage increases require high substitution elasticity (σ >> 0), where firms can readily automate. This is why empirical estimates of σ are central to predicting minimum wage employment impacts.

Question 5

A tax is imposed on capital income. Using the concept of substitution elasticity, explain how the ability to substitute between capital and labor determines who ultimately bears the burden of this tax.

Model answer: If substitution elasticity is high, capital can effectively 'flee' the tax by shifting production toward labor-intensive methods. As capital becomes more expensive, firms substitute toward labor, reducing demand for capital and shifting the burden partly onto labor (via lower wages) and onto consumers. With high σ, capital bears less of the tax because it can escape via substitution. With low σ, capital cannot substitute away, so it bears more of the burden. In the extreme Leontief case (σ = 0), capital and labor are used in fixed proportions, so the tax burden stays on capital with minimal shifting.

The incidence of any factor tax depends on how mobile and substitutable the taxed factor is. High substitution elasticity gives capital 'mobility' across production methods — even without moving geographically, firms reduce their capital intensity in response to higher capital costs. This reduces the effective tax burden on capital and spreads it to other factors. Low elasticity traps capital in its current use and forces it to absorb the full tax.