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}