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.