🚀 Long-Run Economic Growth Models

While business cycle theory focuses on short-run deviations from potential output, long-run growth theory explains the exponential expansion of potential GDP and living standards over decades and centuries.

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1. ⚙️ The Solow-Swan Neoclassical Growth Model (1956)

The Solow model assumes an aggregate production function with constant returns to scale:

$$Y=F(K,AL)$$

where $K$ is capital, $L$ is labor (growing at population growth rate $n=\dot{L}/L$), and $A$ is labor-augmenting technological knowledge (growing at rate $g=\dot{A}/A$).

1.1 Intensive Form (Per Effective Worker)

Let $k\equiv \frac{K}{AL}$ and $y\equiv \frac{Y}{AL}=f(k)$. For Cobb-Douglas $Y=K^{\alpha }(AL{)}^{1-\alpha }⟹y=k^{\alpha }$.

Gross investment per effective worker is $i=sf(k)$, where $s$ is the exogenous savings rate. Depreciation and dilution rate is $(n+g+\delta )k$.

1.2 The Fundamental Differential Equation of Capital Accumulation

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

  Output & Investment per effective worker
    ▲
    │                                  y = f(k)
    │                                 /
    │                           Break-even (n+g+δ)k
    │                          /     /
    │                         /     /  s·f(k)
    │                        /     /  /
 y* │───────────────────────•─────/──/
    │                      /     /  /
    │                     /     /  /
    │                    /     •  /
    │                   /     /  /
    └──────────────────┴─────┴──┴────────► Capital per effective worker (k)
                            k*

1.3 Steady-State Equilibrium ($\dot{k}=0$)

$$sf(k^{\ast })=(n+g+\delta )k^{\ast }$$

For Cobb-Douglas $f(k)=k^{\alpha }$:

$$s(k^{\ast }{)}^{\alpha }=(n+g+\delta )k^{\ast }⟹(k^{\ast }{)}^{1-\alpha }=\frac{s}{n+g+\delta }⟹k^{\ast }={\left(\frac{s}{n+g+\delta }\right)}^{\frac{1}{1-\alpha }}$$$$y^{\ast }=(k^{\ast }{)}^{\alpha }={\left(\frac{s}{n+g+\delta }\right)}^{\frac{\alpha }{1-\alpha }}$$

1.4 Steady-State Growth Rates

• Capital per effective worker ($k$): $0\mathrm{\%}$
• Output per effective worker ($y$): $0\mathrm{\%}$
• Output per worker ($Y/L=Ay$): $g$ (Rate of technological progress)
• Total National Output ($Y$): $n+g$

Solow Paradox & Key Policy Insight

In the neoclassical model, changes in the savings rate $s$ produce level effects (permanently higher $y^{\ast }$), but no long-run growth effect. Sustained per-capita GDP growth is driven solely by exogenous technological progress $g$.

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2. 👑 The Golden Rule Level of Capital Accumulation

The Golden Rule capital stock $k_{gold}^{\ast }$ maximizes steady-state consumption per effective worker $c^{\ast }=(1-s)f(k^{\ast })=f(k^{\ast })-(n+g+\delta )k^{\ast }$:

$$\frac{d{c}^{\ast }}{d{k}^{\ast }}=f^{\prime }(k_{gold}^{\ast })-(n+g+\delta )=0⟹MPK=n+g+\delta$$

For Cobb-Douglas $f(k)=k^{\alpha }$:

$$\alpha (k_{gold}^{\ast }{)}^{\alpha -1}=n+g+\delta ⟹s_{gold}=\alpha$$

The optimal savings rate maximizing long-run consumption equals the capital share of income $\alpha$.

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3. 💡 Endogenous Growth Theory (Romer & Lucas)

To explain technological progress $g$ endogenously rather than treating it as an exogenous "manna from heaven", endogenous growth models introduce:

1. $AK$ Model: Constant marginal returns to aggregate capital ($\alpha =1⟹Y=AK$), eliminating diminishing returns via learning-by-doing and knowledge spillovers.
2. R&D-Based Models (Paul Romer 1990): Non-rivalrous ideas and patented intermediate varieties generated by monopolistically competitive R&D labs.
3. Human Capital (Robert Lucas 1988): Accumulation of education and skill spillovers.

$$\text{Long-run growth rate in AK model: }\gamma =\frac{\dot{Y}}{Y}=sA-\delta$$

A higher savings/investment rate $s$ directly accelerates the long-run growth rate $\gamma$ indefinitely!

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4. 🎯 Olympiad-Level Worked Master Problem

Master Problem: Solow-Swan Quantitative Steady State

Problem: An economy has production function $Y=K^{1/3}(AL{)}^{2/3}$. Savings rate is $s=32\mathrm{\%}$, depreciation rate is $\delta =5\mathrm{\%}$, population growth is $n=2\mathrm{\%}$, and technological progress is $g=1\mathrm{\%}$.

1. Compute the steady-state capital per effective worker $k^{\ast }$ and output per effective worker $y^{\ast }$.
2. Compute the Golden Rule capital stock $k_{gold}^{\ast }$ and Golden Rule savings rate $s_{gold}$.
3. Determine whether the economy is dynamically efficient or inefficient.

Step-by-Step Rigorous Solution:

1. Calculate Steady-State $k^{\ast }$ and $y^{\ast }$:

   $$\alpha =1/3,1-\alpha =2/3$$$$n+g+\delta =0.02+0.01+0.05=0.08$$$$k^{\ast }={\left(\frac{s}{n+g+\delta }\right)}^{\frac{1}{1-\alpha }}={\left(\frac{0.32}{0.08}\right)}^{\frac{1}{2/3}}=(4{)}^{3/2}=(\sqrt{4}{)}^3=2^3=8$$$$y^{\ast }=(k^{\ast }{)}^{1/3}=8^{1/3}=2$$

   Steady-state consumption: $c^{\ast }=(1-s)y^{\ast }=(1-0.32)(2)=0.68\times 2=1.36$.

2. Compute Golden Rule Values:

   $$s_{gold}=\alpha =1/3\approx 33.33\mathrm{\%}$$$$k_{gold}^{\ast }={\left(\frac{1/3}{0.08}\right)}^{3/2}={\left(\frac{0.3333}{0.08}\right)}^{3/2}=(4.167{)}^{1.5}\approx 8.51$$$$y_{gold}^{\ast }=(8.51{)}^{1/3}\approx 2.04$$$$c_{gold}^{\ast }=(1-1/3)(2.04)=\frac{2}{3}\times 2.04=1.361$$

3. Dynamic Efficiency Assessment:

   • Current savings rate $s=32\mathrm{\%}<s_{gold}=33.33\mathrm{\%}⟹k^{\ast }=8<k_{gold}^{\ast }=8.51$.
   • The economy is dynamically efficient (below the Golden Rule capital stock). Increasing savings would temporarily lower consumption today but permanently raise future steady-state consumption.