This is a problem from Theory of Income II problem sets. Courtesy to Elle Edmonds who worked out this problem with great clarity. All the contributions are hers and mistakes are mine.

Inequality in the Neoclassical Growth Model

For standard NGM: given CRS technology $F(K_t, X_t L_t)$ with capital accumulation $\dot{K}_t = I_t - \delta K_t$ and TFP growth $\dot{X}_t/X_t = g > 0$.

Agents:

  • Measure $\pi_c$ of capitalists: endowed with $k_0$, no labor, rent capital competitively
  • Measure $\pi_w$ of workers: no initial wealth, each supplies $1/\pi_w$ units of labor inelastically

Preferences: Both types maximize $\int_0^\infty e^{-\rho t} \ln C_{it}, dt$, $ i \in {c, w}$.

Markets: Complete asset markets (workers and capitalists trade freely).

Questions:

  • Define a competitive equilibrium.
  • Show there exists $k_0^\star$ such that $k_0 = k_0^\star$ implies the economy is immediately on a balanced growth path.
  • On the BGP, characterize the ratio of aggregate wage income ($w_t L_t$) to aggregate rental income ($r_t K_t$). Does it change over time?
  • Is the assumption of complete markets (asset trade between types) essential for part (c)?
  • On the BGP, compare the interest rate $r_t - \delta$ to the growth rate of wages $\dot{w}_t/w_t$. Interpret in light of the claim that “$r > g$ implies rising inequality.”

Basic Setup: within the framework of NGM:

  • Production: $C_t + I_t = F(K_t, X_t L_t)$, CRS and concave
  • Capital: $\dot{K}_t = I_t - \delta K_t$
  • Technology on labor: $X_t$ exogeneous and grows at rate $g$.
  • Agents: $\pi_c$% capitalists + $\pi_w$% workers. Capitalists only supplies capital whereas workers only supply labor.
  • Preferences structure for both type of agent: $\int_0^\infty e^{-\rho t} \ln {C}_{it} , \text d t,i \in \lbrace c,w\rbrace$,

Competitive Equilibrium Conditions

Capitalist’s Problem: $$ \begin{align*} &\max \int_0^\infty e^{-\rho t} \ln C_t^c , dt \cr &\text{s.t.} \quad C_t^c + I_t^c + S_t^c = r_t^k K_t^c + r_t^a A_t^c \cr &\quad\quad \dot{A}_t^c = S_t^c \quad \text{(bond accumulation)} \cr &\quad\quad \dot{K}_t^c = I_t^c - \delta K_t^c \quad \text{(capital accumulation)} \end{align*} $$

Worker’s Problem: $$ \begin{align*} &\max \int_0^\infty e^{-\rho t} \ln C_t^w , dt \cr &\text{s.t.} \quad C_t^w + S_t^w = \frac{w_t}{\pi_w} + r_t^a A_t^w\cr & \quad\quad \dot A_t^w = S_t^w \end{align*} $$

Firm’s Problem: $$ \begin{align*} &\max_{L_t^d, K_t^d} F(K_t^d, X_t L_t^d) - w_t L_t^d - r_t^k K_t^d \cr &\text{FOCs:} \quad F_L = \frac{w_t}{X_t}, \quad F_K = r_t^k \end{align*} $$

Market Clearing: $$ \begin{align*} \text{Labor:} && L_t^d &= \pi_w L_t^s = 1 \cr \text{Capital:} && K_t^d &= \pi_c K_t^c = K_t \cr \text{Bonds:} && \pi_w A_t^w + \pi_c A_t^c &= 0 \cr \text{Goods:} && C_t + I_t &= F(K_t, X_t L_t) \end{align*} $$ where $C_t = \pi_c C_t^c + \pi_w C_t^w$ and $I_t = \pi_c I_t^c$.

Existence of BGP

On BGP, all variables grow at constant rates; define $k_t \equiv K_t/X_t$ (it should be constant in BGP).

  • From the capitalist’s problem: $$ \frac{\dot{C}_t^c}{C_t^c} = r_t^k - \rho - \delta = g \tag{1} $$

    Note: The capitalist maximizes: $\int_0^\infty e^{-\rho t} \ln C_t^c , dt$ subject to their wealth accumulation. With complete asset markets (ie bond trading is free and everyone faces the same interest rate $r^a_i$) we can consolidate capitalist’s budget into a single wealth constraint.

    Let $W_t^c = K_t^c + A_t^c$ be total wealth. The return on wealth is $r_t^a = r_t^k - \delta$ (no-arbitrage between bonds and capital). So for capitalists: $$ \dot{W}_t^c = r_t^a W_t^c - C_t^c. $$ The Euler Equation will implies $(1)$. (You can derive it from differentiating the HJB function or working on the Hamiltonian optimality condition)

  • Firm FOC /w CRS implies $r_t^k = F_K(K_t, X_t) = F_K(k_t, 1) \equiv f’(k_t)$

  • On BPG: $f’\left(\frac{K_0^*}{X_0}\right) = g + \rho + \delta$ (need to be constant)

Trick for analyzing factor growth rate in BGP

Start with the capital accumulation equation, $\dot K_t = F(K_t, X_t L_t) - C_t - \delta K_t$, divide by $K_t$ on both sides $$ \frac{\dot K_t}{K_t} = \frac{F(K_t, X_tL_t)}{K_t} - \frac{C_t}{K_t} - \delta $$ For the LHS to be constant (which is what BGP requires), the RHS must be constant. Note that also, /w CRS $F$: $$ \frac{F(K_t, X_tL_t)}{K_t} = F(1, \frac{X_tL_t}{K_t}) $$ So basically squint your eyes and you can tell which equals which. Eventually, since $F$ is CRS and concave, $f’ > 0$ and decreasing in $k$ $\Rightarrow$ unique $k_0^*$ exists.

Ratio of Wage to Rental Income on BGP

$$ \begin{align*} \text{Wage income:} && w_t \cdot 1 &= w_t \cr \text{Rental income:} && r_t^k K_t & \cr \text{Ratio:} && \frac{w_t L_t}{r_t^k K_t} &= \frac{X_t F_L(\cdot)}{F_K(\cdot) K_t} \end{align*} $$

On BGP with $k_t = k^\star$ constant: $$ \displaystyle\frac{w_t}{r^k K_t} = \frac{w_0/X_0}{r^k k^\star} = \text{constant} $$ Since $w_t$ and $K_t$ both grow at rate $g$: ratio is constant on BGP.

Asset Trading is Necessary for BGP

Without asset markets ($A_t^c = A_t^w = 0$): $$ \begin{align*} \text{Workers:} && C_t^w &= w_t/\pi_w \quad \text{(hand-to-mouth)} \cr \text{Capitalists:} && &\text{must self-finance all investment} \end{align*} $$

With complete markets: both agents face $r^a = r^k - \delta$, ensuring both Euler equations yield growth rate $g$.

Interest Rate vs Wage Growth on BGP

$$ \begin{align*} \text{Interest rate:} && r^k &= g + \rho + \delta \quad \text{(constant)} \cr \text{Wage growth:} && \dot{w}_t/w_t &= g \cr \text{Net return:} && r^A &= r^k - \delta = g + \rho \cr \text{Comparison:} && r^A - g &= \rho > 0 \end{align*} $$

So $r > g$ on BGP, but this does not imply increasing inequality because:

  • Both wages and capital income grow at rate $g$
  • The ratio of labor to capital income is constant
  • $r > g$ reflects time preference $\rho$, required for transversality