In Paul Krugman's articile on the trade model with a monopolistically competitive market (https://www.sciencedirect.com/science/article/pii/0022199679900175), a CES (Constant Elasticity of Substitution) utility function is used. A typical expression of the CES preferences is

The Greek letter sigma () indicates the elasticity of substitution. A higher value of sigma means that the variety of the product is ...

Question

In Paul Krugman's articile on the trade model with a monopolistically competitive market (https://www.sciencedirect.com/science/article/pii/0022199679900175), a CES (Constant Elasticity of Substitution) utility function is used. A typical expression of the CES preferences is

The Greek letter sigma () indicates the elasticity of substitution. A higher value of sigma means that the variety of the product is ...

BlackTom AI · Accepted Answer

The elasticity of substitution σ measures how easily consumers switch consumption between different varieties: a larger σ means varieties are closer substitutes, i.e., more homogeneous.

1) Relevant concepts
- CES utility: U = (Σ_i x_i^{(σ-1)/σ})^{σ/(σ-1)}. The parameter σ is the constant elasticity of substitution between any two varieties.
- First-order conditions for utility maximization with budget constraint link marginal utilities to prices and deliver demand responses to price changes that reveal substitutability.

2) Step-by-step reasoning and derivation
Let α = (σ−1)/σ so U = (Σ_i x_i^α)^{1/α}. Marginal utility of variety i:
MU_i = ∂U/∂x_i = (1/α)(Σ_k x_k^α)^{1/α − 1} · α x_i^{α−1} = (Σ_k x_k^α)^{1/α − 1} x_i^{α−1}.
Thus MU_i / MU_j = x_i^{α−1} / x_j^{α−1}.

Compute exponent α−1 = (σ−1)/σ − 1 = −1/σ. So
MU_i / MU_j = (x_i / x_j)^{−1/σ}.

From utility maximization with prices p_i, MU_i = λ p_i, so MU_i/MU_j = p_i/p_j. Combine:
(p_i/p_j) = (x_i/x_j)^{−1/σ} → (x_i/x_j) = (p_i/p_j)^{σ}.

3) Interpretation and limit cases
- The exponent σ on relative prices means the percentage change in quantity ratios in response to a price ratio change equals σ. A larger σ implies a large change in quantities for a small price change — varieties are easily substituted (close substitutes).
- Limits: σ → ∞ → goods are perfect substitutes (homogeneous). σ → 1 yields the Cobb–Douglas-type case (more distinct love of variety), implying less substitutability.

4) Why the correct answer is correct
Because higher σ increases substitutability: consumers treat varieties as more interchangeable, so the set of varieties behaves more like homogeneous products.

5) Why other options are incorrect
- “More differentiated” is the opposite: higher σ reduces differentiation.
- “More/less expensive” are irrelevant: σ governs substitutability, not price level.

6) Conclusion
Therefore a higher σ means the variety of the product is more homogeneous, confirming the provided answer.

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