Economic Growth Theory

Production Function

$Y_t = F(K_t, A_t L_t)$ — labor-augmenting (Harrod-neutral) technological progress.

Assumptions:

• $F_K > 0$, $F_{KK} < 0$ (positive but diminishing marginal product)
• Constant Returns to Scale (CRS): $F(\lambda K, \lambda AL) = \lambda F(K, AL)$, $\forall \lambda > 0$
• Inada Conditions: $F_K(0, AL) = \infty$, $F_K(\infty, AL) = 0$

Capital Accumulation & Saving

Constant saving rate $s \in (0,1)$: $S_t = sY_t$. In a closed economy, $I_t = S_t$.

Capital accumulation with depreciation rate $\delta$:

Exogenous growth: $A_{t+1} = (1+g)A_t$, $L_{t+1} = (1+n)L_t$.

Intensive Form

Detrend the model by writing in efficiency units:

$y_t \equiv \frac{Y_t}{A_t L_t} = f(k_t)$ where $k_t \equiv \frac{K_t}{A_t L_t}$

Using CRS: $y = F(\frac{K}{AL}, 1) \equiv f(k)$.

Discrete Time

From $K_{t+1} = sF(K_t, A_t L_t) + (1-\delta)K_t$, divide by $A_{t+1}L_{t+1}$:

Continuous Time

$\dot{K}(t) = sF(K, AL) - \delta K$

Using $k = \frac{K}{AL}$ and the growth rates $\dot{A}/A = g$, $\dot{L}/L = n$:

The term $\delta + g + n$ is the effective depreciation rate — capital per effective worker depreciates through physical wear, technological obsolescence, and dilution by labor force growth.

Steady State

Steady state is where $k_{t+1} = k_t$ (discrete) and $\dot{k} = 0$ (continuous):

At steady state, investment exactly covers effective depreciation. Capital per effective worker and output per effective worker are constant.

Comparative Dynamics: Saving Rate

How does $k^*$ change with $s$? Implicitly differentiate $sf(k(s)) = (\delta+g+n)k(s)$:

The denominator is negative at steady state (the $sf(k)$ curve crosses $(\delta+g+n)k$ from above), so an increase in $s$ raises $k^*$.

Saving Rate and Growth

• Saving rate has no effect on long-run growth rate of output per worker — that is $g$.
• Saving rate determines the long-run level of output per worker.
• An increase in $s$ leads to higher growth in the short run, not forever. The boost fades as the economy converges to a new, higher steady state.

Consumption Maximization

$c^* = f(k^*) - (\delta+g+n)k^*$

First-order condition:

Marginal product of capital equals the effective depreciation rate.

Optimal Saving Rate

At the golden rule, $sf(k^*) = (\delta+g+n)k^*$ and $f'(k^*) = \delta+g+n$, implying:

For Cobb-Douglas, the optimal saving rate equals the capital share $\alpha$.

Convergence Rate

$\psi'(k^*) = sf'(k^*) - (\delta+g+n) = \frac{sf'(k^*)k^*}{f(k^*)}(\delta+g+n) - (\delta+g+n)$

where $\epsilon \equiv \frac{f'(k^*)k^*}{f(k^*)}$ is the elasticity of output with respect to capital (the capital share in Cobb-Douglas).

Balanced Growth

With labor-augmenting technology, the model exhibits a balanced growth path (BGP) where:

• $Y/AN$ and $K/AN$ are constant in steady state.
• Output $Y$ grows at rate $g + n$.
• Output per worker $Y/N$ grows at rate $g$.
• Capital per worker $K/N$ grows at rate $g$.

The standard Solow model without technological progress predicts that capital accumulation alone cannot sustain long-run growth. Sustained growth requires $g > 0$.

Social Planner's Problem

Maximize: $\displaystyle \sum_{t=0}^{\infty} \beta^t u(c_t)$, subject to:

$c_t + k_{t+1} = f(k_t) + (1-\delta)k_t$

Lagrangian with multiplier $\lambda_t$:

$\mathcal{L} = \sum_{t=0}^{\infty} \beta^t u(c_t) + \lambda_t [f(k_t) + (1-\delta)k_t - c_t - k_{t+1}]$

Key Optimality Condition: Euler Equation

FOCs yield the consumption Euler equation:

With CRRA utility $u(c) = \frac{c^{1-\sigma} - 1}{1-\sigma}$:

Consumption grows when the return to capital exceeds the rate of time preference.

Phase Diagram & Saddle Path

The system has two differential equations:

• $\dot{k} = f(k) - c - \delta k$ (resource constraint)
• $\dot{c} = \frac{c}{\sigma}[f'(k) - \delta - \rho]$ (Euler, where $\rho = -\ln\beta$)

The steady state is saddle-path stable: for any initial $k_0$, there is a unique $c_0$ that puts the economy on the convergent trajectory. All other paths violate the transversality condition.

Setup

Each generation $t$:

• Young: Supplies 1 unit of labor, earns wage $w_t$, consumes $c_{1t}$, saves $s_t$.
• Old: Consumes $c_{2,t+1} = (1 + r_{t+1})s_t$ from savings plus interest.

Maximizes: $u(c_{1t}) + \beta u(c_{2,t+1})$ subject to the intertemporal budget constraint.

Dynamic Inefficiency

A key insight of the OLG model: competitive equilibria may be dynamically inefficient — the economy could over-accumulate capital beyond the golden rule level. This cannot happen in the RCK model with infinitely-lived agents because the transversality condition rules it out.

Growth Rate Decomposition

Start with $Y = A K^\alpha N^{1-\alpha}$. Taking logs and differencing:

Growth in output per worker:

Output per worker growth = TFP growth + capital deepening (weighted by $\alpha \approx 1/3$).

Levels Decomposition

Comparing two countries $i, j$ at a point in time:

$\displaystyle \ln\frac{Y_i}{N_i} - \ln\frac{Y_j}{N_j} = (\ln A_i - \ln A_j) + \alpha\!\left(\ln\frac{K_i}{N_i} - \ln\frac{K_j}{N_j}\right)$

Most of the income gap between rich and poor countries is explained by TFP differences, not capital differences.

TFP as a Residual

Measured TFP is the "Solow residual" — a measure of our ignorance:

$\widehat{\ln A} = \ln Y - \alpha\ln K - (1-\alpha)\ln N$

It captures everything not explained by measured inputs: technology, institutions, human capital, political stability, etc.

Why Does TFP Differ?

TFP is not just literal technology — it's anything that affects production efficiency:

• Political stability and rule of law
• Property rights protection
• Contract enforceability
• Financial market development
• Corruption levels

Cross-country regressions show strong correlations between institutional quality and GDP per capita. Causality is harder to establish, but natural experiments (e.g., North vs South Korea) suggest institutions matter enormously.