All models are wrong, but some are useful. — George E. P. Box

And, some model would have particularly peculiar assumptions that makes them easy and handy.

Motivation

The neoclassical growth model was designed to explain several facts observed in developed economies over long periods (the “Kaldor facts” from data from the U.S. (1900-2000)):

  1. Output per capita grows at roughly constant rate
  2. Capital-output, investment-output, and consumption-output ratios are roughly constant
  3. Interest rates are roughly constant
  4. Factor shares (capital vs labor income) are roughly constant

Uzawa’s Theorem

Consider $(K_t, L_t, C_t, Y_t)_t$ governed by: $$ \begin{align*} &\dot K_t = Y_t - \delta K_t - C_t && \text{ accounting formula}\cr & Y(t) = \tilde F(K(t), L(t), \tilde X(t)) && \text{CRS in $(K, L)$ production func.} \end{align*} $$ Assume constant growth rate respectively in production, capital, labor and consumption: $$ \frac{\dot Y}Y = g_Y,\frac{\dot K}K = g_K, \frac{\dot C}C = g_C, \frac{\dot L}L = n. $$ Then, the above conditions (CRS + constant growth) implies

  1. Balanced growth: $g_Y =g_c = g_K = g$

  2. Labor augmenting technical change: exists CRS $F(K, XL)$ and $\dot X/X = g - n$ such that $$ F(K, XL) = \tilde F(K, L, X). $$

Proof sketch

  • From $\dot{K} = g_K K$ and accounting: $(g_K + \delta)K = Y - C$
  • This gives: $(g_K + \delta)K(0) = e^{(g_Y - g_K)t}Y(0) - e^{(g_C - g_K)t}C(0)$
  • Differentiating w.r.t. $t$ and requiring this holds for all $t$ forces $g_Y = g_C = g_K$ This prove 1.
  • For 2, write $e^{-g(t-\tau)}Y(t) = \tilde{F}[e^{-g(t-\tau)}K(t), e^{-n(t-\tau)}L(t), \tilde{X}(\tau)]$
  • Using CRS and requiring this holds for all $\tau$ yields the labor-augmenting form with $X(t) = e^{(g-n)t}$

Note: here if we further assume (i) constant ratio $K_t/X_tL_t \equiv c$ and (ii) impose firm’s zero-profit condition, we will get that price ($w$ or $r$) is also constant.

What Preference Supports Constant Growth?

CRS + constant growth + constant factor ratio => constant interest rate and wage. Now further assume that labor is inelastically supplied, consider a representative households’ problem: $$ \begin{align*} & \max_{C, K} \int e^{-\rho t}U(C_t), \text{d}t\cr & s.t.\ \dot K_t = wL_t + rK_t - \delta K_t - C_t, \forall t. \end{align*} $$ The intertemporal optimal condition (the “Euler” condition) can be derived from the Hamiltonian optimal:

Let $H(C, K, \lambda) = U(C) - \lambda \dot K$ and sub in the $\dot K = (\cdots)$: $$ \begin{cases} \dot \lambda = \rho \lambda - H_K(C, K, \lambda)&\Rightarrow \dot \lambda = (\rho - r + \delta) \lambda\cr H_C = 0 & \Rightarrow U’(C) = \lambda \end{cases}\quad \Rightarrow \frac{U’’(c) \dot c}{U’(c)} = \rho - r + \delta. $$ One preference function that satisfies $\frac{U’’(c) \dot c}{U’(c)} = const$ with $\dot c/c = g$ is the Constant Relative Risk Aversion (CRRA) utility, or isoelastic utility: $$ U(C) = \frac1{1 - \sigma}C^{1 - \sigma}. $$

Scaling back to a steady-state neoclassical growth model

The growing representative household’s problem, say, fix $L_t = 1$: $$ \begin{align*} & \max_{C, K} \int e^{-\rho t}U(C_t), \text{d}t\cr & s.t.\ \dot K_t = wL_t + rK_t - \delta K_t - C_t, \forall t. \end{align*} $$ Normalize $c_t = C_t/X_t L_t = C_t/X_t$, similarly $k_t = K_t/X_t$. Take $$ f(k_t) := F(K_t, X_tL_t)\circ K_t(k_t) = F(k_tX_t, X_tL_t) $$ Take $$ \tilde \rho = \rho - (1 - \delta)g_x. $$ Then an isomorphic neoclassical growth model appears with steady state $(k^{ss}, c^{ss})$ corresponding to the OG model. $$ \begin{align*} & \max_{C, K} \int e^{-\tilde \rho t}U(c_t), \text{d}t\cr & s.t.\ \dot k_t = f(k_t) - \delta k_t - c_t, \forall t. \end{align*} $$