Based on your answer to the previous question and assuming
𝛾
=
0.75
, you know that the growth rate of
𝑦
𝑡
at time
𝑡
=
0
is ______ percent.

Round your answer to the nearest tenth of a percent.

Question

Based on your answer to the previous question and assuming 
𝛾
=
0.75
, you know that the growth rate of 
𝑦
𝑡
 at time 
𝑡
=
0
 is ______ percent.

Round your answer to the nearest tenth of a percent.

BlackTom AI · Accepted Answer

Because y_t = k_t^γ (Cobb–Douglas per-capita production), the instantaneous growth rate of y equals γ times the instantaneous growth rate of k.

1. Relevant knowledge points and concepts
- For a Cobb–Douglas relation y_t = k_t^γ (with constant γ), taking natural logs gives ln y_t = γ ln k_t. Differentiating with respect to time yields the growth-rate identity: g_y(t) ≡ d ln y_t / dt = γ · d ln k_t / dt ≡ γ · g_k(t).
- Growth rates expressed as percentages are the decimal growth rate multiplied by 100.

2. Complete reasoning process step by step
- Start from ln y_t = γ ln k_t.
- Differentiate: (d/dt) ln y_t = γ (d/dt) ln k_t.
- Therefore g_y(t) = γ · g_k(t). Evaluate at t = 0: g_y(0) = γ · g_k(0).
- Use the previous question’s result for g_k(0). The previous answer gave the growth rate of k at t = 0 as 1.6%.

3. Calculations with intermediate steps
- Convert percentages to decimals for multiplication or multiply directly with percent units: g_y(0) = 0.75 × 1.6% = (0.75 × 1.6) %.
- Compute 0.75 × 1.6 = 1.20.
- So g_y(0) = 1.20%.

4. Why the correct answer is correct
- The result follows exactly from the log-differentiation of the Cobb–Douglas form: output growth equals the capital elasticity times capital growth. No approximation beyond standard log differentiation is needed, so the direct multiplication gives the correct instantaneous percent growth.

5. Why other interpretations would be incorrect
- A common mistake is to add growth rates or to use γ + g_k instead of γ·g_k; that contradicts the log-differentiation result. Another error would be to apply discrete compounding incorrectly; for instantaneous (continuous) growth the multiplicative relation γ·g_k is exact.

6. Conclusion and confirmation
- Therefore the growth rate of y_t at time t = 0 is 1.2%, which matches the provided answer.

Based on your answer to the previous question and assuming 𝛾 = 0.75 , you know that the growth rate of 𝑦 𝑡 at time 𝑡 = 0 is ______ percent. Round your answer to the nearest tenth of a percent.数值题

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