Question

Consider a Solow model with the production function: \(Y=K^\alpha L^{1-\alpha}\), where \(Y\), \(K\), and \(L\) are output, capital, and labour, respectively. \(\alpha\) is a positive constant. The savings rate, depreciation rate, and labour growth rates are \(20\%\), \(7\%\), and \(3\%\), respectively. If \(\alpha=0.5\), then the steady-state capital-labour ratio is (in integer).

Hint:

In the Solow model, steady state occurs when \(sy=(n+\delta)k\). For Cobb-Douglas production, first convert the production function into per-worker form.

Answer 1

Correct Answer: 4

Step 1: Write output per worker.
Dividing $Y=K^\alpha L^{1-\alpha}$ by $L$ gives $y=k^\alpha$ with $k=\dfrac{K}{L}$.

Step 2: Use the steady state rule.
At steady state actual investment equals break even investment,
\[ s\,k^\alpha=(n+\delta)k \]

Step 3: Put in the values.
With $s=0.20$, $\delta=0.07$, $n=0.03$ and $\alpha=0.5$, note $n+\delta=0.10$, so
\[ 0.20\,k^{0.5}=0.10\,k \]

Step 4: Solve for k.
\[ \frac{0.20}{0.10}=\frac{k}{k^{0.5}}=k^{0.5}\;\Rightarrow\;2=k^{0.5} \]
Squaring gives $k=4$.
\[ \boxed{4} \]