Chapter 8 & 9: The Economy in the Very Long Run - The Solow Growth Model

Only 38% of the African population has access to electricity.
Internet penetration rate is less than 10%.
Only a quarter of Africa’s road network is paved.
Poor infrastructure adds 30% to 40% to the costs of goods traded among African countries.
This adversely affects private sector development and foreign direct investment (FDI).
A World Bank study found that poor infrastructure reduces national economic growth by two percentage points every year.
It also cuts business productivity by as much as 40%.
This makes Africa the region with the lowest productivity levels in the world, despite its mineral and natural resources.

a. Supply:
- The production function is given by $Y = F(K, L)$ , where:
  - $Y$ is output.
  - $K$ is capital.
  - $L$ is labor.
- Constant returns to scale: $zY = F(zK, zL)$ for any positive $z$ .
- Set $z = 1/L$ , so $Y/L = F(K/L, 1)$ .
- Output per worker is a function of the amount of capital per worker.
- In lowercase notation: $y = Y/L$ , $k = K/L$ , so $y = f(k)$ .
- Slope: $MPK = \Delta y/ \Delta k$ (Marginal Product of Capital).
- Diminishing marginal product of capital: As $k$ increases, $MPK$ decreases.
b. Demand for Goods:
- $c + i$ (per-worker version)
- $y = c + i$
- $i = y – c = \text{saving}$
- $s$ is the saving rate, where 0 < s < 1.
- $i = sy = sf(k)$

Two forces influence $k$ : investment and depreciation.
- Investment: new capital.
- Depreciation: wearing out of old capital.
- A constant fraction $\delta$ of the capital stock wears out each year.
- Depreciation per worker = $\delta k$ :
$\Delta k = i – \text{depreciation} = sf(k) – \delta k$
$k^*$ : steady-state level of capital.
When $k = k^*$ , $\Delta k = 0$ , so $sf(k) = \delta k$
The steady state represents the long-run equilibrium of the economy.
Example: If $Y = K^{0.5}L^{0.5}$ , $s = 0.3$ , $\delta = 0.1$ , then $k^* = ?$
- Divide by $L$ : $y = k^{0.5}$
- $0.3k^{0.5} = 0.1k$
- $k^* = 9$

What amount of capital accumulation is optimal?
- Optimal saving rate and optimal steady state.
Maximize economic well-being → Maximize consumption → golden rule level of capital $k^*_{gold}$
$c = y – i = y – \delta k$ ( $i = \delta k$ in the steady state)
Golden Rule: $MPK = \delta$

a. In the steady state, $K$ and $Y$ must also be growing at $n$ .
- $k = K/L$ , $\Delta k = 0$ , $L$ grows at rate $n$
- $y = Y/L$ , $\Delta y = 0$ , $L$ grows at rate $n$ → Explain sustained growth in total output $Y$
b. $n$ rises, $k^*$ falls, output per worker $y$ falls → Explain why some countries are poor.
c. Golden rule: $c = y – i = y - (\delta+n)k$
- $MPK = \delta + n$

$Y = F(K, L*E)$ , where:
- $E$ : the efficiency of labor
- $L*E$ : the number of effective workers
Assume technological progress causes $E$ to grow at a constant rate $g$
$L * E$ grows at rate $n+g$ .

Capital and output per effective worker:
- $k = K/(L*E)$
- $y = Y/(L*E)$
- $y = f(k)$
Per effective worker production function: $\Delta k = i – \delta k – (n+g)k$
Break-even investment: $i = \delta k + (n+g)k$