7. In a country the production function is y = 𝑘!/#. The fraction of output invested, γ, is 0.25. The

depreciation rate δ, is 0.05.

a. What are the steady-state levels of capital per worker, k, and output per worker, y?

b. In year 1, the level of capital per worker is 16. In a table such as the following one, show how

capital and output change over time (the beginning is filled in as a demonstration). Continue this

table up to year 8.

Year Capital

K

Output

Y=𝑘!/#

Investment

γ y

Depreciation

δk

Change in

Capital stock

γ y- δk

1 16 4 1 0.8 0.2

2 16.2

3

c. Calculate the growth rate of output between years 1 and 2.

d. Calculate the growth rate of output between years 7 and 8.

e. Comparing your answers from parts c and d, what can you conclude about the speed of output

growth as a country approaches its steady state?

We are given the following production function and parameters:

- Production function: \( y = k^{1/2} \), where \( y \) is the output per worker and \( k \) is the capital per worker.
- Fraction of output invested: \( \gamma = 0.25 \)
- Depreciation rate: \( \delta = 0.05 \)

We are tasked with finding the steady-state levels of capital per worker and output per worker and also constructing a table to track the capital and output over time.

### **Part A: Steady-State Levels of Capital and Output per Worker**

In the Solow model, the steady state occurs when the change in capital per worker (\( \Delta k \)) is zero. This happens when the investment per worker equals the depreciation of capital per worker.

The equation for capital accumulation is:

\[
\Delta k = \gamma y - \delta k
\]

At steady state, \( \Delta k = 0 \), which gives:

\[
\gamma y = \delta k
\]

Since the production function is \( y = k^{1/2} \), substitute this into the equation:

\[
\gamma k^{1/2} = \delta k
\]

Now, solve for \( k \) at the steady state:

\[
\gamma k^{1/2} = \delta k
\]
\[
k^{1/2} = \frac{\delta}{\gamma} k
\]
\[
k = \left( \frac{\gamma^2}{\delta^2} \right)^{1/3}


In a country, the production function is y = k^(1/2). The fraction of output invested, γ, is 0.25. The depreciation rate δ, is 0.05. a. What are the steady-state levels of capital per worker, k, and output per worker, y? b. In year 1, the level of capital per worker is 16. In a table, show how capital and output change over time up to year 8. c. Calculate the growth rate of output between years 1 and 2. d. Calculate the growth rate of output between years 7 and 8. e. Compare the speed of output growth as a country approaches its steady state.

Explore the Solow growth model with a detailed analysis of the production function y = k^(1/2), capital accumulation, and the steady-state level of capital and output per worker. Learn how capital changes over time and how growth rates evolve as a country approaches its steady-state equilibrium. Suitable for undergraduate or graduate-level students.

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