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Suppose that the production function f(k) = 2k and A =1 so that the growth of the labor-augmenting technology (A/A) = 0. What is the growth rate of (k.)/(k)?

To find the growth rate of \( \frac{\dot{k}}{k} \), let's break down the equation given in the image and use the information provided.

### Given Information

The equation for capital accumulation per effective worker \( \hat{k} \) is:
\[
\dot{\hat{k}} = s f(\hat{k}) - (n + \delta + g) \hat{k}
\]

where:
- \( f(\hat{k}) = \hat{k}^{0.5} \),
- \( s = 0.20 \),
- \( n = 0.02 \),
- \( \delta = 0.10 \),
- \( g = 0.08 \).

However, you've specified a modified production function:
\[
f(k) = 2k
\]

with \( A = 1 \) and a zero growth rate for labor-augmenting technology \( \left( \frac{\dot{A}}{A} = 0 \right) \).

This simplifies the growth rate analysis because we’re dealing with \( k \) and not \( \hat{k} \).

### Steps to Solve for \( \frac{\dot{k}}{k} \)

Given the production function \( f(k) = 2k \), we can substitute this into the capital accumulation equation without considering the adjustments for effective labor (since \( A \) is constant and does not grow). Therefore, the equation for \( \dot{k} \) becomes:

\[
\dot{k} = s f(k) - (n + \delta) k
\]

Substitute \( f(k) = 2k \):
\[
\dot{k} = s \cdot 2k - (n + \delta) k
\]

Expanding:
\[
\dot{k} = (2s - n - \delta) k
\]

Now, divide both sides by \( k \) to find \( \frac{\dot{k}}{k} \):
\[
\frac{\dot{k}}{k} = 2s - n - \delta
\]

### Substitute the Values

Using the given values:
- \( s = 0.20 \),
- \( n = 0.02 \),
- \( \delta = 0.10 \).

\[
\frac{\dot{k}}{k} = 2 \cdot 0.20 - 0.02 - 0.10
\]

Calculate each term:
1. \( 2 \cdot 0.20 = 0.40 \),
2. \( 0.40 - 0.02 = 0.38 \),
3. \( 0.38 - 0.10 = 0.28 \).

### Final Answer
\[
\frac{\dot{k}}{k} = 0.28
\]

The growth rate of \( \frac{\dot{k}}{k} \) is **0.28** or **28%**.

Would you like additional details on this calculation or further analysis?

### Related Questions:
1. How does the growth rate change if the depreciation rate \( \delta \) changes?
2. What impact does the savings rate \( s \) have on \( \frac{\dot{k}}{k} \)?
3. How would the solution differ if \( A \) were growing at a positive rate?
4. What does this growth rate imply for long-term capital accumulation?
5. How would labor growth \( n \) affect the growth rate \( \frac{\dot{k}}{k} \)?

### Tip
Understanding how each parameter affects the growth rate can help in economic modeling, especially in long-term growth predictions.

Suppose that the production function f(k) = 2k and A = 1 so that the growth of the labor-augmenting technology (A/A) = 0. What is the growth rate of (k.)/(k)?

This solution covers the growth rate of capital (k̇ / k) using a Solow model where f(k) = 2k and A = 1. With parameters s = 0.20, n = 0.02, δ = 0.10, the resulting growth rate is 0.28 or 28%. Learn how savings, depreciation, and population growth impact economic growth.

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