Investment and Capital Accumulation
Assume that private saving is proportional to income: \(S=s\cdot Y\)
- \(s\) is saving rate (\(0<s<1\)).
Combine \(I=S\) and \(S=s\cdot Y\), in year \(t\) we have: \(I_{t}=s\cdot Y_{t}\).
Investment is proportional to output: higher output implies higher saving, then higher investment.
The evolution of capital stock is given by: \(K_{t+1}=(1-\delta)K_{t}+I_{t}\).
Substitute \(I_{t}\) by \(I_{t}=s\cdot Y_{t}\) and divide both sides by \(N\): \(\frac{K_{t+1}}{N}=(1-\delta)\frac{K_{t}}{N}+s\frac{Y_{t}}{N}\).
Rearrange it: \(\frac{K_{t+1}}{N}-\frac{K_{t}}{N}=s\frac{Y_{t}}{N}-\delta\frac{K_{t}}{N}\).
Plug \(\frac{Y_{t}}{N}=f(\frac{K_{t}}{N})\) in: \[
\frac{K_{t+1}}{N}-\frac{K_{t}}{N}=s\cdot f(\frac{K_{t}}{N})-\delta\frac{K_{t}}{N}
\]
Steady-state Capital and Output
The state in which output per worker and capital per worker are no longer changing is called steady state of the economy.
Recall the evolution of capital per worker: \[
\frac{K_{t+1}}{N}-\frac{K_{t}}{N}=s\cdot f(\frac{K_{t}}{N})-\delta\frac{K_{t}}{N}
\]
- In steady state, no change in capital per worker, so that the left hand side \(\frac{K_{t+1}}{N}-\frac{K_{t}}{N}=0\). Thus the investment should be equal to depreciation since right hand side \(s\cdot f(\frac{K_{t}}{N})-\delta\frac{K_{t}}{N}\) has to equal to zero.
If we denote the steady-state capital per worker as \(\frac{K^*}{N}\), then: \[
s\cdot f(\frac{K^*}{N})=\delta\frac{K^*}{N}
\]
Given steady-state capital per worker, \(\frac{K^*}{N}\), we can denote steady-state output per worker as \(\frac{Y*}{N}\), so that: \[
\frac{Y^*}{N}=f(\frac{K^*}{N})
\]
