Consider a continuous$-$time version $of$ the Solow model with the following

duction function:

$Y=L^{}{K}^{}{x}^{1--}$

where $Lis$ labor, $Kis$ capital and $xis$ land. Assume that $+<mn\; id="EditorElement\_177468">1</mn>$ and the

supply $of$ land and labor are fixed:

$x^{}=0$ and $L^{}=0$

$(a)$ What are the four ingredients for any growth model? Write down the three

ingredients $to$ the continuous$-$time Solow model that are not explicitly

specified above.

$(b)$ Rewrite the production function $in$ per capita terms where $y=\frac{Y}{L},$

$k=\frac{K}{L}$ and $x=\frac{x}{L},$ and express the growth rate $of$ the capital$-$labor

ratio $as$ a function $of$ the capital$-$labor ratio and the land$-$labor ratio.

$(c)$ Show that there exists a unique steady state capital$-$labor ratio $in$ this

model.

$(d)$ Draw $\frac{k^{}}{k}as$ a function $ofk$ and use the graph $to$ argue that this steady

state $is$ stable.

$(e)$ Solve for the steady state capital$-$labor ratio and the steady state output

per worker.

Now assume instead that the labor force $L$ grows $at$ a constant rate $ofn.$

$(f)$ What $is$ the definition $of$ a balanced growth path?

$(g)$ Explain why the growth rates $of$ output, capital, consumption and invest$-$

ment must $be$ equal along the balanced growth path.

$(h)$ Use the production function $to$ derive a relationship between the growth

rate $of$ output and the growth rate $of$ labor.

$(i)$ What $is$ the growth rate $of$ output along the balanced growth path? What

$is$ the growth rate $of$ output per worker along the balanced growth path?