Solow Model

macro

The Solow Model (or Solow Growth Model) focuses on long-run economic growth.

1. Recall from Cobb-Douglas Production Function,

$$Y=A{K}^{\alpha }{L}^{1-\alpha }$$

$\alpha$ is the capital intensity. usually, $[0.3,0.5]$

2. Capital Accumulation Equation

$$\dot{K}=I-\delta K$$

• $I$: investment

3. Investment Equation (What’s new here)

$$I=sY$$

• $s\in (0,1)$ is the investment rate, usually $(10\%,50\%)$ in the data.

Note: $s$ is the parameter.

4. National Income Account (In a close economy)

$$Y=C+I+\underset{=0}{\underbrace{G+NX}}$$

so $C=(1-s)Y$

Summary of the solow model

The Central Equation of Solow Model is:

$$\Delta k=sf(k)-\delta k$$

or we could write as: (after I study ECON4004 Advanced Macroeconomics)

$$\dot{K}=sY-\delta K$$

This is a one-dimensional dynamic system

So the growth rate of the capital would be:

$$\frac{\dot{K}}{K}=sA{\left(\frac{L}{K}\right)}^{1-\alpha }-\delta$$

It determines the behaviors of capital over time ⇒ determines behavior of all the other endogenous variables because they all depend on $k$ .

Per Capital Production Function

Consumption Per Person

Or we could draw like this: the difference is how we treat the vertical axis.

When $sy=s{k}^{\alpha }=\delta k$ , which means $\Delta k=0$, that means the investment is just enough to cover the depreciation. The Steady State occur.

We use $k^{\ast }$ to represents the Steady State situation.

Then we can move to the methodcalled Golden Rule

The technological progress is assumed exogenous (i.e. the value of $g$ is given)

That means:

$$\frac{\Delta E}{E}=g$$

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Exercise

Q1: Derive the steady-state level of $K$ and $Y$. (Hint: set $\dot{K}=0$, so we could derive $K^{\ast }$, put $K^{\ast }$ into the production function for $Y^{\ast }$.)

Answer: $\dot{K}=0⟹sA{\left(\frac{L}{K}\right)}^{1-\alpha }=\delta ⟹K^{\ast }=(\frac{sA}{\delta }{)}^{1/1-\alpha }L$

Q2: What happens when $s$ increases?

The Phase Diagram:

Comparative Static: $s\uparrow$.

At $K^{\ast }$, we have an increase in $s$, $\frac{\dot{K}}{K}$ becomes positive in the short-run. So, $K_t$ rises over time and converges to $K^{\ast \ast }$. At the new steady state, $K$ remains constant in the long-run.

Effects of a higher saving rate:

$s\uparrow$:

• a higher growth of $K$ in the short-run.
• a higher level of $K$ in the long-run.

Here, we do not have economic growth in the long-run. When $K$ converges to a steady state, output $Y$ also converges to a steady state.

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Since the capital accumulation itself won’t generate so much increase in output $Y$ in the long-run, To sustain growth in the long-run, we need technological progress (i.e. growth in $A$).

We assume that $\frac{\dot{A}}{A}=g>0$, here $g$ is a parameter.

The Solow Model treats $g$ as exogenous.

From

$$\begin{array}{rl}Y& =A{K}^{\alpha }{L}^{1-\alpha }\\ \ln Y& =\ln A+\alpha \ln K+(1-\alpha )\ln L\\ \frac{\delta \ln Y}{\delta t}& =\frac{1}{Y}\frac{\delta Y}{\delta t}=\frac{\dot{Y}}{Y}\\ \frac{\dot{Y}}{Y}& =\frac{\dot{A}}{A}+\frac{\alpha \dot{K}}{K}\end{array}$$

2. $\dot{K}=I-\delta K=sY-\delta K⟹\frac{\dot{K}}{K}=s\frac{Y}{K}-\delta$

Balanced Growth Path: Each variable grows at a constant rate in the long-run. So steady-state is a special case of balanced growth path, which is 0

On the BGP, $\frac{\dot{K}}{K}$ is constant (in the long-run). so of course $\frac{Y}{K}$ is constant. The conclusion is that $\frac{\dot{Y}}{Y}=\frac{\dot{K}}{K}$ in the long-run.

The train of thought is definition → implication

We combine the equations, then we could get

$$\frac{\dot{Y}}{Y}=\frac{g}{1-\alpha }$$

• If $g=0$, then $\frac{\dot{Y}}{Y}=0$ in the long-run.

Empirical Implications

In the data, $\frac{\dot{Y}}{Y}\approx 2\%$ in the long-run among developed countries

$g=(1-\alpha )2\%$, since $\alpha \in [\frac{1}{3},\frac{1}{2}]$, so $g\in [1\%,1.3\%]$

A $1\%-1.3\%$ technology growth rate over 2 centuries could increase income level from \1000$to$$52k$

The Solow Model tells us that: technological progress is crucial for economic growth in the long-run.

However, the Solow Model does not tell us where technological progress comes from. It assumes $g$ to be a parameter. It only allows us to explore the proximate causes (capital and technology) of growth.

We want to go one step further to explore the fundamental causes of growth.

Two important parameters:

1. $g=\frac{\dot{A}}{A}$ determines the long-run growth rate.
2. $s=\frac{I}{Y}$ determines the long-run level of output.

The question is why do these two parameters differ across countries?

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Ramsey Model