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Development Economics & Growth Theory

22 min Economics

Development economics studies why some countries are rich and others poor, and what policies foster long-run prosperity. The Solow–Swan growth model provides the workhorse framework: output per worker depends on capital per worker and technology, and the economy converges to a steady state where investment exactly offsets depreciation and labor force growth. Beyond Solow, modern growth theory emphasizes total factor productivity, institutions, human capital, and the role of geography in determining living standards. Poverty measurement, the Human Development Index, and inequality metrics like the Gini coefficient translate development theory into policy-relevant numbers.

Solow Growth Model

Aggregate Production Function

$$Y = F(K, L) = K^\alpha L^{1-\alpha}, \quad 0 < \alpha < 1$$

In per-worker terms, letting $k = K/L$ and $y = Y/L$:

$$y = k^\alpha$$

Capital Accumulation & Steady State

Capital per worker evolves as:

$$\dot{k} = sf(k) - (n + \delta)k$$

$s$ = savings rate, $n$ = population growth rate, $\delta$ = depreciation rate. At the steady state $\dot{k}=0$:

$$sk^{*\alpha} = (n+\delta)k^* \implies k^* = \left(\frac{s}{n+\delta}\right)^{\frac{1}{1-\alpha}}$$

Golden Rule

The savings rate $s^{\text{gold}}$ that maximizes steady-state consumption per worker satisfies:

$$f'(k^*_{\text{gold}}) = n + \delta \implies \alpha k^{*\alpha - 1} = n + \delta$$

Growth Accounting (Solow Residual)

$$\frac{\Delta Y}{Y} = \frac{\Delta A}{A} + \alpha\frac{\Delta K}{K} + (1-\alpha)\frac{\Delta L}{L}$$

$\Delta A/A$ = Total Factor Productivity (TFP) growth — often called the Solow residual — captures technological change and efficiency gains.

Human Development Index (HDI)

$$HDI = \left(I_{\text{health}} \times I_{\text{education}} \times I_{\text{income}}\right)^{1/3}$$

Each sub-index is $I = (x - x_{\min})/(x_{\max} - x_{\min})$, ranging 0–1.

Gini Coefficient

Measures income inequality on a 0–1 scale. Geometrically:

$$G = \frac{A}{A+B}$$

where $A$ is the area between the Lorenz curve and the line of perfect equality, and $A+B$ is the total area below the line. $G=0$ means perfect equality; $G=1$ means one person holds all income.

Example 1 — Steady-State Capital

$\alpha = 0.3$, $s = 0.2$, $n = 0.01$, $\delta = 0.09$. Find $k^*$.

  1. $k^* = (0.2/0.10)^{1/0.7} = 2^{1.429} \approx 2.69$ units of capital per worker.
Example 2 — Growth Accounting

Output grows $4\%$, capital grows $3\%$, labor grows $1\%$, $\alpha = 0.4$. Find TFP growth.

  1. $\Delta A/A = 4\% - 0.4(3\%) - 0.6(1\%) = 4\% - 1.2\% - 0.6\% = 2.2\%$.
Example 3 — Gini

In a two-person economy, one person earns $\$30$k and the other earns $\$90$k. Compute the Gini.

  1. Income shares: 25\% and 75\%.
  2. Lorenz curve: $(0,0) \to (0.5, 0.25) \to (1, 1)$. Area under Lorenz $= 0.5(0.25)+0.5(0.25+1) \cdot 0.5 = 0.125+0.3125 = 0.4375$. Area under equality line $= 0.5$.
  3. $G = (0.5-0.4375)/0.5 = 0.125$. (Low inequality since the gap is modest.)

Practice Problems

1. $s=0.3$, $n=0.02$, $\delta=0.08$, $\alpha=0.5$. Find $k^*$ and $y^*$.
2. In the Solow model, a rise in $s$ raises $k^*$. Does it raise the long-run growth rate?
3. TFP growth: output $+5\%$, capital $+4\%$, labor $+2\%$, $\alpha=0.3$. Find $\Delta A/A$.
4. What is the golden rule savings rate if $\alpha=0.4$, $n=0.01$, $\delta=0.04$?
5. Country A has $k=10$, $y=k^{0.5}$, $s=0.2$, $\delta=0.1$, $n=0.05$. Is capital above or below steady state?
6. An HDI health index: life expectancy $75$ years, min $20$, max $85$. Compute $I_{\text{health}}$.
7. Why does the Solow model predict conditional convergence?
8. Distinguish Solow's exogenous growth from Romer's endogenous growth theory.
9. A country with $G=0.60$ vs one with $G=0.25$ — which has more equal distribution?
10. How does the poverty trap relate to the S-shaped production function in development models?
Show Answer Key

1. $k^* = (0.3/0.10)^2 = 9$; $y^* = 9^{0.5} = 3$.

2. No. A higher $s$ raises the level of $k^*$ and $y^*$ but not the long-run growth rate (which equals $n+g$). Only TFP growth raises the long-run growth rate.

3. $\Delta A/A = 5 - 0.3(4) - 0.7(2) = 5 - 1.2 - 1.4 = 2.4\%$.

4. Golden rule: $f'(k^*)= n+\delta \Rightarrow 0.4k^{-0.6}=0.05 \Rightarrow k^*=(8)^{1/0.6}\approx 27.9$; $s^{\text{gold}} = (n+\delta)k^*/y^* = 0.05 \cdot k^{*1-0.4}/1 = \alpha = 0.40$ (for Cobb-Douglas, golden rule savings rate $= \alpha$).

5. Steady state: $0.2 k^{0.5} = 0.15k \Rightarrow k^* = (0.2/0.15)^2 \approx 1.78$. $k=10 > k^*$, so capital is above steady state; it will fall.

6. $I = (75-20)/(85-20) = 55/65 \approx 0.846$.

7. Poor countries (low $k$) have higher marginal product of capital, so they grow faster and catch up to rich countries — but only conditionally (same $s$, $n$, $\delta$, technology).

8. Solow: technology is exogenous (falls from the sky). Romer: technology is created by intentional R\&D and knowledge spillovers, making growth sustainable and policy-responsive.

9. $G=0.25$ is more equal.

10. If the production function has an S-shape, there may be a low-capital equilibrium from which a poor country cannot escape through normal saving — it needs a "big push" (external investment or aid) to jump to the high-capital equilibrium.

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