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.