Economy in the Very Long Run: The Solow Growth Model

Chapter 8 & 9: The Economy in the Very Long Run

The Solow Growth Model

Infrastructure Development in Africa

  • 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 annually.

  • It also cuts business productivity by as much as 40%.

  • This makes Africa the region with the lowest productivity levels globally, despite its resources.

I. The Accumulation of Capital

  • Assume labor force and technology are fixed.

1. The Supply and Demand for Goods
a. Supply: The Production Function

  • Y = F(K, L)

  • Where:

    • Y = Output

    • K = Capital

    • L = Labor

  • Constant returns to scale: zY = F(zK, zL) for any positive z

  • Set z = 1/L

  • Y/L = F(K/L, 1)

  • Output per worker is a function of the amount of capital per worker.

  • In lowercase:

    • y = Y/L

    • k = K/L

  • Therefore, y = f(k)

  • Slope: MPK = \frac{\Delta y}{\Delta k}

  • 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 = saving

  • s is the saving rate, 0 < s < 1

  • i = sy = sf(k)

2. Growth in the Capital Stock and the Steady State

  • 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

3. Capital Stock Dynamics

  • \Delta k = i – depreciation

  • \Delta k = sf(k) – \delta k

  • k^*: steady-state level of capital

  • When k = k^*, \Delta k = 0, 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 what is k^*?

  • Divide by L: y = k^{0.5}

  • 0.3k^{0.5} = 0.1k

  • k^* = 9

4. Important Lessons

  • The saving rate is a key determinant of the steady-state capital.

  • Higher s, k^* rises, and y rises, but only temporarily.

  • In the new steady state, k and y are constant.

  • y = Y/L (L is fixed, so Y is fixed.)

  • The level of Y increases, not the growth rate of it.

  • G and C impact saving, therefore economic growth.

II. The Golden Rule Level of Capital

  • What amount of capital accumulation is optimal?

  • Optimal saving rate and optimal steady state.

  • Maximize economic well-being, which means 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

III. Population Growth

  • Assume labor force grows at a constant rate n.

1. The Steady State with Population Growth

  • The growth in the number of workers causes capital per worker to fall.

  • \Delta k = i – \delta k – nk = i – (\delta+n)k

  • \Delta k = 0 in the steady state, so i = (\delta+n) k

  • (\delta+n)k: break-even investment, to keep k constant

2. The Effects of Population Growth
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

  • This explains sustained growth in total output Y.

b.

  • If n rises, k^* falls, and output per worker y falls.

  • This explains why some countries are poor.

c.

  • Golden rule: c = y – i = y - (\delta+n)k

  • MPK = \delta + n

IV. Technological Progress

1. Labor Augmenting Technology (the efficiency of labor)

  • 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.

2. The Steady State with Technological Progress

  • Capital and output per effective worker

  • k = \frac{K}{L*E}

  • y = \frac{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

3. The Effects of Technological Progress
a. Can explain persistently rising living standards (Y/L).

  • In the steady state, y is constant.

  • S y = \frac{Y}{L*E}

  • Y grows at rate n+g.

  • Output per (actual) worker \frac{Y}{L} grows at rate g.

  • \frac{Y}{L} and \frac{K}{L} have grown at 2% per year.

b. Golden rule:

  • c = y – i = f(k) – (\delta+n+g)k

  • Golden rule: MPK = \delta+n+g

  • MPK – \delta = n+g

  • n+g: the growth rate of Y, total output

  • This equation can be used to find out if we have the golden rule level of capital.

Gross National Savings as a Percent of GDP 2023

  • Ireland: 61.2%

  • Singapore: 54.6%

  • UAE: 47.8%

  • China: 44%

  • South Korea: 35.4%

  • Russia: 29.8%

  • India: 28.3%

  • Germany: 26.2%

  • Australia: 26.1%

  • Japan: 24.7%

  • France: 21.8%

  • Canada: 20.1%

  • US: 18.2%

  • Brazil: 16.8%

  • UK: 16.4%

  • Greece: 5.9%