§10 Saving, Capital Accumulation, and Output

Interactions between Output and Capital

• Assume $N$ is constant (no population growth) and recall from previous lecture

  $$\frac{Y_t}{N} = f \left( \frac{K_t}{N} \right) \quad \text{with } f' > 0, f'' < 0$$

• Assume the economy is closed and there is no public deficit, so

  $$I_t = S_t$$

• Moreover, assume private saving is proportional to income $S = sY$, so that

  $$I_t = sY_t$$

Interactions between Output and Capital

• The evolution of the capital stock is

  $$K_{t+1} = (1 - \delta)K_t + I_t$$

• In per worker terms is

  $$\frac{K_{t+1}}{N} = (1 - \delta)\frac{K_t}{N} + s\frac{Y_t}{N}$$

• Or

  $$\frac{K_{t+1}}{N} - \frac{K_t}{N} = s\frac{Y_t}{N} - \delta\frac{K_t}{N}$$

• The change in the capital stock per worker is equal to the saving per worker minus depreciation.

• Using the production function to substitute $Y/N$, we get a difference equation for $K/N$:

  $$\frac{K_{t+1}}{N} - \frac{K_t}{N} = sf \left( \frac{K_t}{N} \right) - \delta\frac{K_t}{N}$$

Getting a Sense of Magnitudes

• Assume $Y = \sqrt{K}\sqrt{N}$ so that

  $$\frac{K_{t+1}}{N} - \frac{K_t}{N} = s\sqrt{\frac{K_t}{N}} - \delta\frac{K_t}{N}$$

• Solving for the steady state (i.e. $K/N$ constant)

  $$\frac{K^*}{N} = \left(\frac{s}{\delta}\right)^2$$

  $$\frac{Y^*}{N} = \sqrt{\frac{K^*}{N}} = \frac{s}{\delta}$$

• In the long run, output per worker doubles when the saving rate doubles.

  $$\begin{aligned}\frac{C^*}{N} &= \frac{Y^*}{N} - \delta \frac{K^*}{N} \\&= \frac{s(1 - s)}{\alpha}\end{aligned}$$

What if there is population growth?

• The evolution of the capital stock doesn’t change

  $$K_{t+1} = (1 - \delta)K_t + sY_t$$

• In per worker terms is a little trickier…

  $$\begin{aligned} \frac{K_{t+1}}{N_{t+1}} &= (1 - \delta) \frac{K_t}{N_{t+1}} + s \frac{Y_t}{N_{t+1}} \\ &= (1 - \delta) \frac{N_t}{N_{t+1}} \frac{K_t}{N_t} + s \frac{N_t}{N_{t+1}} \frac{Y_t}{N_t} \\ &\approx (1 - \delta)(1 - g_N) \frac{K_t}{N_t} + s(1 - g_N) \frac{Y_t}{N_t} \\ &\approx (1 - \delta - g_N) \frac{K_t}{N_t} + s \frac{Y_t}{N_t} \end{aligned}$$

• or

  $$\begin{aligned}\frac{K_{t+1}}{N_{t+1}} - \frac{K_t}{N_t} &= s\frac{Y_t}{N_t} - (\delta + g_N)\frac{K_t}{N_t} \\&= sf\left(\frac{K_t}{N_t}\right) - (\delta + g_N)\frac{K_t}{N_t}\end{aligned}$$

— Apr 18, 2025

§10 Saving, Capital Accumulation, and Output by Lu Meng is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. Permissions beyond the scope of this license may be available at About.