Consider two countries, labeled 1 and 2. Each has the production function
𝑌
𝑖
=
𝐴
¯
𝑖
𝐾
𝑖
𝛼
𝐿
𝑖
1
−
𝛼
, for i = 1, 2. Assume that the countries currently have the same GDP per person,
𝐾
1
=
𝐾
2
, common saving, population growth, and depreciation rates, and
𝐿
1

Question

Consider two countries, labeled 1 and 2. Each has the production function 
𝑌
𝑖
=
𝐴
¯
𝑖
𝐾
𝑖
𝛼
𝐿
𝑖
1
−
𝛼
, for i = 1, 2. Assume that the countries currently have the same GDP per person, 
𝐾
1
=
𝐾
2
, common saving, population growth, and depreciation rates, and 
𝐿
1
<
𝐿
2
. Based on this information you know that, in the long run, GDP per capita in country 1 (
𝑦
1
) is _______  
𝑦
2
.

Hint: Based on the information above, what do you know about 
𝐴
1
 relative to 
𝐴
2
?

BlackTom AI · Accepted Answer

Output per worker for the Cobb–Douglas function is y_i = Y_i/L_i = Ā_i (K_i/L_i)^α. Use this to compare productivities and steady states.

1) Relevant concepts
- Cobb–Douglas production: Y = A K^α L^{1−α} implies y ≡ Y/L = A (K/L)^α.
- Solow steady state (no technological growth): in steady state k* = [s A/(n+δ)]^{1/(1−α)} and y* = A k*^α = A^{1/(1−α)} [s/(n+δ)]^{α/(1−α)}. Thus long-run output per worker y* is increasing in A; specifically y* ∝ A^{1/(1−α)}.

2) Step-by-step reasoning (current situation → A relationship)
- Given current GDP per person equal: y1 = y2.
- Using y_i = Ā_i (K_i/L_i)^α and K1 = K2, set them equal:
  Ā1 (K1/L1)^α = Ā2 (K2/L2)^α.
  With K1 = K2 this simplifies to Ā1 L1^{−α} = Ā2 L2^{−α}, or
  Ā1/Ā2 = (L1/L2)^α.
- Since L1 < L2, the ratio L1/L2 < 1, and with α ∈ (0,1) we get Ā1/Ā2 < 1, so Ā1 < Ā2. In words: given the same total capital but a smaller labor force, country 1 has higher capital per worker; to have equal current y, its measured A must be smaller than country 2’s A.

3) Long-run (steady-state) comparison
- Steady-state output per worker y*_i = constant × Ā_i^{1/(1−α)} (with identical s, n, δ across countries).
- Because Ā1 < Ā2, raising to the positive power 1/(1−α) preserves the inequality: Ā1^{1/(1−α)} < Ā2^{1/(1−α)}.
- Therefore y*_1 < y*_2 in the long run.

4) Why the correct answer is “less than”
- The algebra shows Ā1 < Ā2 given the initial equal y and K1 = K2 with L1 < L2. Since long-run y* is strictly increasing in Ā, country 1’s steady-state y will be lower than country 2’s.

5) Why the other choices are incorrect
- “Greater than”: would require Ā1 > Ā2; but the equality of current y together with K1 = K2 and smaller L1 implies the opposite (Ā1 < Ā2).
- “Equal to”: would require Ā1 = Ā2 (or other compensating differences), which does not hold given L1 < L2 and K1 = K2 combined with equal current y.

Conclusion: Given the algebra and Solow steady-state formula, long-run GDP per capita in country 1 will be less than that in country 2. The correct comparative statement is: y1 is less than y2.

类似问题

If, in addition, the economy's population grows at rate 𝑛 , i.e., 𝐿 𝑡 + 1 = ( 1 + 𝑛 ) 𝐿 𝑡 , you know that the steady-state capital-labor ratio is ________.

Using the Solow model, if, at time t = 0, the initial capital stock is 𝐾 0 = 500 , investment is 𝐼 0 = 240 , and δ = 0.12 is the depreciation rate, capital accumulation from period 0 to period 1 (i.e., Δ 𝐾 1 ) is:

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