Steady State / Balanced Growth Model

Introduction

• In the absence of technical progress the economy is expected to grow at a constant rate
• $y,K,L\to v$
• $sy$ is the net investment
• $(n+d)k$ is the gross investment
• capital deepening
• capital widening

Beauty of the Model

Positive effect: net investment

Negative effect: gross investment (depreciation and boom in population)

In the long run, both of them are getting used up.

The economy will eventually converge back to $k^{\ast }$

• it is not in the economy’s fate to remain at $k^{\ast }$ If there is technical progress, the economy will eventually shift

Assumptions

• We assume, there is no change in technological progress / productivity/ $A$.
• We assume that the labor force growing over time at an exogenous rate $$\frac{\Delta L}{L}=n$$
• We assume that $I=sy$
• We assume that constant fraction $d$ of the existing capital stock, depreciates each period.
  • So the total depreciation is $dk$.

Explanation

The steady state balanced growth is defined to be a situation where $y,k,L$ are growing at the same rate over a period of time.

$$\frac{\Delta Y}{Y}=\frac{\Delta K}{K}=\frac{\Delta L}{L}=n$$

In equilibrium(or Steady State - Balanced Growth Model) $\boxed{sy=(n+d)k}$ where $sy$ is the investment-per-worker and $(n+d)k$ is the gross-investment-per-worker, which is needed to keep the $k$ unchanged for the labor-force growing at $n$.¹

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The steady-state value of $k$ and $y$ are depicted as $k^{\ast }$ and $y^{\ast }$.

To the left of $k^{\ast }$, $sy>(n+d)k$ so, $\Delta k>0$ and thus, $k$ would increase towards $k^{\ast }$. (capital deepening)

On the right of $k^{\ast }$, our $sy<(n+d)k$ so, $\Delta k<0$ and thus, $k$ would decrease towards $k^{\ast }$. (capital widening)

Thus, the $k$ converges to $k^{\ast }$ from either directions.

• In the long run, the steady-state (balanced growth) without productivity change $Y,K,L$ are growing at the same rate $n$.

• In the steady-state, the positive impact of investment on $k$, just balances the negative effects of Depreciation and the growth of labor force.

Technical Progress and Growth

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With $A$ remaining unchanged, the system has a tendency to converge at $k^{\ast }$. If there is rise in the productivity from $A\to A^{\prime }$ even if $s$ unchanged, the economy will move to a new steady state, with higher $k$ and $y$.

Limitations

1. The technical progress captured by $A$ has been taken to be determined exogenously. (we don’t know²)
2. This model
   • ignores entrepreneurship
   • ignores the strength of the institutions
3. Too much of sticking to³ INADA conditions.

Footnotes

1. Labor needs capital $k$ to work, but $k$ depreciates at $d$ over time and thus we need to ensure that the existing labor has enough capital to work with, thus we increase capital by $d$ over time. But labor is also growing at a rate of $n$ and so for the new labor, we need to increase the existing capital by $n$. Thus the total (gross) investment is $(n+d)k$ ↩

2. This was explained by Endogenous Growth Model ↩

3. Stickiness ↩