Growth

• One good model.
• Production function, $Y=F(A,K,eL)$ where,
• $Y$ is the output amount (real GDP/flow variable),
• $A$ is the technology/knowledge "amount",
• $e$ is the quality/level of education,
• $L$ is the amount of labor (stock variable),
• $K$ is capital (stock variable).
• $F$ has decreasing marginal returns wrt. each variable.
• Capital depreciates at a linear rate, $0\lt\delta\lt 1$. The depreciation period $t$ is then $\delta\cdot K_{t}$.
• Capital and output measured with the same units.
• $\gamma$ savings rate (exogenous/given/a constant by assumption).
• $K_{t+1}=K_{t}+I_{t}-\delta\cdot K_{t}$.
• Investment is savings, $I_{t}=S_{t}=\gamma\cdot Y_{t}$, where $\gamma$ is the fraction of output that is saved (a constant by assumption of the model).

1. Whether an economic variable is exogenous or endogenous is context dependent.
2. Exogenous: the value of the variable is given; the variable is a constant; economic agents do not choose the value of the variable; etc...
3. Endogenous: economic agents chose the value of the variable; the variable or the value of the variable is determined in equilibrium, etc...

• The output and capital stock converge to their steady-state values.
• At the steady-state, the growth rate, $\%\Delta \mathrm{GDP}=\dfrac{\mathrm{GDP}(t+1)-\mathrm{GDP}(t)}{\mathrm{GDP}(t)}$ is zero.
• If population increases at a linear rate, $L_{t+1}=g\cdot L_{t}$ with $g\gt 1$, the per-capita variables converge to their steady-state values.
• If $A$, the technology/knowledge increases continuously, convergence may not occur: growth rate will remain positive.
• Steady-state equations: $\overline K=\overline K+ \overline I-\delta\cdot \overline K$, $\overline Y=F(A, \overline K, eL)$, $\overline I=s\overline Y$.
• Please, see Mathematica File > Module > Week TBA.

The equations that describe the evolution of output and stock of capital (assuming labor is constant) in the Solow Model are:

$Y_{t}=Y=F(A,K_{t},eL)$
$K_{t+1}=(1-\delta)K_{t}+s\cdot Y_{t}$

In a steady state, variables do not change: $Y_{t+1}=Y_{t}$ and $K_{t+1}=K_{t}$—growth rates are zero—and so the value of the steady-state variables must satisfy:

$Y=F(A,K,eL)$
$K=(1-\delta)K+s\cdot Y$

We use the first equation to eliminate the output variable $Y$ from the second equation. This substitution yields:

$\delta K = s\cdot F(A,K,eL)$

If we assume a specific production function form, $F(A,K,eL)=\sqrt{A\cdot K\cdot eL}$, we can solve for steady-state $K$:

$\delta K = s\cdot \sqrt{A\cdot K\cdot eL}$
$K=\left(\dfrac{s}{\delta}\right)^2\cdot A \cdot eL$
$Y=\dfrac{s}{\delta}A\cdot eL$