The elasticity of substitution (sigma) between good 1 and good 2 is defined as

𝜎
=
𝑑
(
𝑥
1
𝑥
2
)
(
𝑥
1
𝑥
2
)
𝑑
(
𝑝
2
𝑝
1
)
(
𝑝
2
𝑝
1
)

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

Question

The elasticity of substitution (sigma) between good 1 and good 2 is defined as

𝜎
=
𝑑
(
𝑥
1
𝑥
2
)
(
𝑥
1
𝑥
2
)
𝑑
(
𝑝
2
𝑝
1
)
(
𝑝
2
𝑝
1
)

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

BlackTom AI · Accepted Answer

Elasticity of substitution σ is the proportional change in the goods ratio divided by the proportional change in the price ratio:
σ = [Δ(x1/x2) / (x1/x2)] ÷ [Δ(p2/p1) / (p2/p1)].

1) Relevant concepts
- x1/x2 is the quantity ratio; p2/p1 is the price ratio.  
- Δ denotes change (after − before).  
- The formula uses proportional (relative) changes, not absolute levels.

2) Compute the quantity ratios and their changes
- Before: x1/x2 = 100/100 = 1.  
- After: x1/x2 = 60/140 = 3/7 ≈ 0.428571.  
- Change: Δ(x1/x2) = 0.428571 − 1 = −0.5714286 = −4/7.  
- Proportional change in quantities = Δ(x1/x2) / (x1/x2 before) = (−0.5714286) / 1 = −0.5714286.

3) Compute the price ratios and their changes
- Before: p2/p1 = 10/10 = 1.  
- After: p2/p1 = 10/11 ≈ 0.9090909.  
- Change: Δ(p2/p1) = 0.9090909 − 1 = −0.09090909 = −1/11.  
- Proportional change in prices = (−0.09090909) / 1 = −0.09090909.

4) Plug into the formula and simplify
σ = (−0.5714286) / (−0.09090909) = (4/7) / (1/11) = (4/7) × 11 = 44/7 ≈ 6.285714.

5) Rounding and interpretation
- Using the stated rounding rule (one decimal, ties up), 6.285714 rounds to 6.3.  
- The negatives cancel, yielding a positive σ, which indicates the goods are substitutable; σ ≈ 6.3 implies a high degree of substitutability.

6) Why other numeric choices would be incorrect
- Any alternative like 6.2 or 6.4 would result from incorrect arithmetic or improper rounding (6.2857 rounds to 6.3). Exact fraction 44/7 confirms the computed value.

Final answer: 6.3 (matches 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

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