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

𝑈
=
{
Σ
𝑖
(
𝑥
𝑖
)
𝜎
−
1
𝜎
}
𝜎
𝜎
−
1

The Greek letter sigma (
𝜎
) 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
Links to an external site.
), a CES (Constant Elasticity of Substitution) utility function is used. A typical expression of the CES preferences is

𝑈
=
{
Σ
𝑖
(
𝑥
𝑖
)
𝜎
−
1
𝜎
}
𝜎
𝜎
−
1

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

BlackTom AI · Accepted Answer

Elasticity of substitution (σ) measures how strongly relative quantities respond to relative price changes: a higher σ means consumers more readily substitute between varieties, so the varieties behave more like homogeneous goods.

Relevant concepts: CES preferences U = (Σ x_i^{(σ-1)/σ})^{σ/(σ-1)} imply a constant elasticity of substitution σ between any two varieties. For two goods (x1, x2) the first-order condition and marginal utilities give the relationship between consumption ratios and prices that defines σ.

Step-by-step reasoning and intermediate steps:
1. For CES utility the marginal utility ratio satisfies MU1/MU2 = (x1/x2)^{-1/σ} (up to a common positive factor). Setting MU1/MU2 = p1/p2 (consumer optimality) yields
2. p1/p2 = (x1/x2)^{-1/σ}  =>  x1/x2 = (p1/p2)^{-σ}.
3. Take logs: ln(x1/x2) = −σ ln(p1/p2). Differentiate to get the elasticity of the quantity ratio with respect to the price ratio:
   d ln(x1/x2) / d ln(p1/p2) = −σ.
This shows σ is exactly the (absolute) elasticity of substitution: a larger σ means a given relative price change produces a larger change in relative quantities demanded.

Why the correct answer (“more homogeneous”) is correct: If σ is large, a small price increase for one variety causes consumers to switch heavily to others, so varieties are close substitutes and effectively homogeneous in consumers’ eyes. In the limit σ → ∞ the CES aggregator approaches perfect substitutes (linear utility), i.e., complete homogeneity.

Why other answers would be incorrect: Saying a higher σ makes varieties “more differentiated” or “less substitutable” contradicts the definition and the derived relation x1/x2 = (p1/p2)^{-σ}. Those statements imply smaller responsiveness to price differences, which corresponds to a lower σ (σ → 0 gives perfect complements/Leontief).

Conclusion: therefore a higher σ implies the product varieties are more homogeneous, matching the provided correct answer.

类似问题

The elasticity of substitution (sigma) between good 1 and good 2 is defined as where d indicates 'change'. Compute sigma using the following data. Answer the number with one decimal (e.g., 1.24 1.2 and 1.25 1.3) in the box below. Before After p1 10 11 p2 10 10 x1 100 60 x2 100 140

Among the following choices, the elasticity of substituion must be highest between ...

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