In a country, output is produced with labor and physical capital. The production function in per-worker terms is y =𝑘^(1/2). The depreciation rate is 2%. The investment rate (γ) is determined as follows:


γ = 0.20 if y < 10


γ = 0.40 if y > 10


Draw a diagram showing the steady state(s) of this model. Calculate the values of any steady state levels of k and y. Also, indicate on the diagram and describe briefly in words how the levels of y and k behave outside of the steady state. Comment briefly on the stability of the steady state(s). Draw a Solow growth graph with all the values afterwards.

The graph above illustrates the Solow growth model for the given production function, investment rates, and depreciation. Key features:

1. **Production Function (Blue Curve)**: This represents \( y = k^{1/2} \), showing how output per worker increases with capital per worker.
2. **Investment Curves (Green and Orange)**:
   - The **green dashed line** represents investment when \( \gamma = 0.20 \), relevant when \( y < 10 \).
   - The **orange dashed line** represents investment when \( \gamma = 0.40 \), applicable when \( y > 10 \).
3. **Depreciation Line (Red)**: This is the linear function \( \delta k \), showing the constant depreciation rate of 2%.

The two steady states are marked:
- **First steady state** at \( k = 100 \), where output per worker \( y = 10 \), and investment with \( \gamma = 0.20 \) intersects depreciation. This point is **unstable**.
- **Second steady state** at \( k = 400 \), where output per worker \( y = 20 \), and investment with \( \gamma = 0.40 \) intersects depreciation. This point is **stable**.

### Summary of Behavior:
- Below \( k = 100 \), capital accumulates, pushing the economy toward the unstable steady state.
- Between \( k = 100 \) and \( k = 400 \), capital accumulates due to the higher investment rate, moving the economy to the stable steady state.
- Above \( k = 400 \), capital depreciates, bringing the economy back to the stable point.

Would you like any further details or explanations?

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**5 Related Questions**:
1. How does a change in the depreciation rate affect the steady states?
2. What happens if the investment rate does not increase after \( y = 10 \)?
3. How would population growth impact the steady-state values?
4. Can we derive a long-term growth rate in this model?
5. What happens to output when technological progress is added?

**Tip**: In the Solow model, a higher investment rate leads to a higher steady-state capital level, but diminishing returns to capital still hold.

In a country, output is produced with labor and physical capital. The production function in per-worker terms is y =𝑘^(1/2). The depreciation rate is 2%. The investment rate (γ) is determined as follows: γ = 0.20 if y < 10; γ = 0.40 if y > 10. Draw a diagram showing the steady state(s) of this model. Calculate the values of any steady state levels of k and y. Also, indicate on the diagram and describe briefly in words how the levels of y and k behave outside of the steady state. Comment briefly on the stability of the steady state(s). Draw a Solow growth graph with all the values afterwards.

Explore the Solow growth model with a per-worker production function y = k^(1/2), a 2% depreciation rate, and variable investment rates. This problem includes steady state analysis, a graph of the production function, and stability conditions of the steady states. Suitable for undergraduate economics students.

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