The question is asking for the expression for the savings rate $s$ that maximizes steady-state consumption per worker in the Solow model, given the production function $f(k_t) = A k_t^\alpha$ and the consumption function:

$c^* = (1 - s) A^{1 - \alpha} \left( \frac{s}{\delta} \right)^{\frac{\alpha}{1 - \alpha}}$

To determine the savings rate $s$ that maximizes steady-state consumption $c^*$, we will use calculus. Here's a step-by-step explanation:

1. Setup the Problem

The steady-state consumption function is given by:

$c^* = (1 - s) A^{1 - \alpha} \left( \frac{s}{\delta} \right)^{\frac{\alpha}{1 - \alpha}}$

• $A$ is the productivity parameter.
• $s$ is the savings rate.
• $\alpha$ is the output elasticity of capital (a parameter between 0 and 1).
• $\delta$ is the depreciation rate.

Our goal is to maximize $c^*$ with respect to $s$.

2. Differentiate $c^*$ with Respect to $s$

To find the value of $s$ that maximizes steady-state consumption, we need to differentiate $c^*$ with respect to $s$ and set the derivative equal to zero.

The expression for $c^*$ can be written as:

$c^*(s) = (1 - s) \cdot f(s)$

where:

$f(s) = A^{1 - \alpha} \left( \frac{s}{\delta} \right)^{\frac{\alpha}{1 - \alpha}}$

The derivative of $c^*(s)$ using the product rule is:

$\frac{d}{ds} c^*(s) = \frac{d}{ds} \left[ (1 - s) f(s) \right]$

By the product rule:

$\frac{d}{ds} c^*(s) = \frac{d}{ds} (1 - s) \cdot f(s) + (1 - s) \cdot \frac{d}{ds} f(s)$

3. Maximizing Consumption

To maximize $c^*$, set the derivative equal to zero. This yields the first-order condition:

$\frac{d}{ds} c^*(s) = 0$

Solving this equation leads to the result that the savings rate $s$ that maximizes steady-state consumption is:

$s = \alpha$

4. Conclusion

The correct answer is $s = \alpha$, which is option (1).

Reasoning:

The Solow model's fundamental result is that the savings rate that maximizes steady-state consumption occurs where the marginal product of capital equals the depreciation rate. This optimal savings rate is given by $s = \alpha$, where $\alpha$ is the elasticity of output with respect to capital in the production function.

Would you like a more detailed walkthrough of the derivative or additional clarification?

Here are 5 related questions for deeper understanding:

1. How does the savings rate $s$ affect long-term capital accumulation in the Solow model?
2. What is the significance of the production function $f(k_t) = A k_t^\alpha$ in determining economic growth?
3. How would changes in the depreciation rate $\delta$ impact the steady-state consumption?
4. Can the Solow model explain differences in income across countries based solely on the savings rate?
5. What role does technological progress play in the Solow model, and how does it affect the steady-state?

Tip:

In optimization problems, always remember to check the second-order condition to ensure you are finding a maximum and not a minimum!