Consider the version of the Solow model where the population grows at an exogenous rate
𝑛
¯
=
2.1
%
,
𝐴
¯
=
1.5
,
𝑠
¯
=
.3
,
𝑑
¯
=
.045
, and
𝛼
=
1
3
. Based on this information you know that, in the long run, the growth rate of
𝑌
𝑡
(total output) is ________ percent.

Round your answer to the nearest tenth of a percent.

Question

Consider the version of the Solow model where the population grows at an exogenous rate 
𝑛
¯
=
2.1
%
, 
𝐴
¯
=
1.5
, 
𝑠
¯
=
.3
, 
𝑑
¯
=
.045
, and 
𝛼
=
1
3
. Based on this information you know that, in the long run, the growth rate of 
𝑌
𝑡
 (total output) is ________ percent.

Round your answer to the nearest tenth of a percent.

BlackTom AI · Accepted Answer

In the Solow model without technological progress, aggregate output in the steady state grows at the rate of population growth n.

1) Relevant knowledge points and concepts
- Cobb–Douglas production: Y = K^α (A L)^(1−α). Here A is a level of technology (given as Ā = 1.5) but no growth rate for A is provided, so Adot/A = 0. L grows at rate n = 2.1% (population growth). 
- Define capital per effective worker k ≡ K/(A L). The Solow k̇ equation is k̇ = s f(k) − (n + δ) k. In steady state k̇ = 0, so k* is constant.
- In balanced growth with constant k*, aggregate K and Y grow at the same rate as A times L; if A is constant, that rate equals n.

2) Step-by-step reasoning and intermediate steps
- Start from k ≡ K/(A L). Take time derivatives (or log-differentiate):
  ln K = ln k + ln A + ln L
  ⇒ K̇/K = k̇/k + Ȧ/A + L̇/L.
- At the steady state of the Solow model, k̇ = 0. Given Ȧ/A = 0 (A constant) and L̇/L = n, we get
  K̇/K = 0 + 0 + n = n.
- Use the production function to get Ẏ/Y. Log-differentiate Y = K^α (A L)^(1−α):
  Ẏ/Y = α K̇/K + (1 − α)(Ȧ/A + L̇/L).
- Substitute K̇/K = n and Ȧ/A = 0, L̇/L = n:
  Ẏ/Y = α n + (1 − α) n = n (α + 1 − α) = n.

3) Numerical substitution and rounding
- n = 2.1% (given), so long-run growth rate of total output Yt = 2.1%. Rounded to the nearest tenth: 2.1%.

4) Why the correct answer is correct
- Because with no technological progress (A constant), the only source of sustained growth in aggregate output in the Solow framework is population growth. Capital and output grow at the population rate in steady state while capital per effective worker and output per effective worker are constant.

5) Why other plausible ideas are incorrect
- The saving rate s, depreciation δ, and capital share α determine the steady-state levels (k*, y*), not the long-run growth rate when technology is constant. Ā = 1.5 is a level parameter, not a growth rate; it does not add to long-run growth unless A grows over time.

Conclusion: The long-run growth rate of Yt is 2.1%, which matches the provided answer.

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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