Solow-Swan Model: Show that $\overline{Q_t} = A_t f(\overline{Z}) e^{gt}$ along the balanced growth path

Question

Suppose a Solow-Swan model with $Y_t = F(K,AL)$ as the production function. Suppose the growth rate of $A$ and $L$ are $g > 0$ and $n > 0$ respectively. It is also given that $\dot{K} = sY - \delta K$ where $\delta$ is the rate of depreciation and $s$ is the savings rate. If $Z = \frac{K}{AL}$ and $Q_t = \frac{Y_t}{L_t}$, show that $\overline{Q_t} = A_t f(\overline{Z}) e^{gt}$ where $\overline{Q}$ is the balanced growth path value of $Q$, $\overline{Z}$ is the steady state value of $Z$ and $f(Z) := F(Z,1)$.

Attempt:

$Q_t = A_t \frac{F(K_t,A_tL_t)}{A_tL_t} = A_t f(Z_t) = A_t f(Z_t)$. At the steady state, $\frac{\dot{Z}}{Z} = \frac{s f(Z)}{Z} - (n + \delta + g) = 0 \implies f(\overline{Z}) = \frac{n+\delta+g}{s}\overline{Z}$.

Along the balanced growth path, let's assume $\frac{\dot{Y}}{Y} = g_Y$.

How do I proceed from here?

Answer 1

To answer the question it is necessary to remember the behaviour of the variables in the steady state, so of course it is necessary to have solved previously the model for the steady state.$^1$

In your question we have the Solow model with labour augmenting technical progress, that is the production function is

$Y_t=f(K_t,A_t L_t)$

where $A_t$ represents technical progress and $A_t L_t$ is the so-called effective labour.

Solving the model for the steady state, we obtain that the variables per effective labour are constant, that is $Z_t\equiv \frac {K_t}{A_tL_t}= \overline {Z}$ and $y_t \equiv \frac {Y_t}{A_tL_t}$ are constant, where $\overline {Z}$ is the constant steady state value of $Z_t$.

Therefore we have, in steady state:

$y_t\equiv \frac {Y_t}{A_tL_t}= f(\overline {Z})\;\;\;\;\quad (1)$

from which

$ Q_t\equiv \frac {Y_t}{L_t}= A_tf(\overline {Z})\qquad (2)$

But, by assumption in the model, we have $^2$:

$A_t= A_0 e^{gt}\qquad \;\;\;\;\;\;\;\;\;\;\;(3)$

where $g$ is the rate of growth of technology $A_t$ and $A_0$ is the initial value of $A_t$.

Therefore, substituting in $(2)$, we obtain:

$ Q_t\equiv \frac {Y_t}{L_t}= A_0 f(\overline {Z})e^{gt}.\qquad (4)$ $$\;$$

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$^1$ Of course, the solution of the model is too long to be reported here, you can look at a good text of macroeconomics, as Romer, Advanced macroeconomics or at some standard references on economic growth, as the books of Acemoglu or Barro & Sala i Martin.

$^2$ In the model there is the assumption that $A_t$ grows at constant rate $g$. Remember, very important, that the exponential form with $e^{at}$, as in $(3)$, is the mathematical expression to denote a growth at constant rate $a$ of a variable. If you calculate the growth rate in that case you can realize it.