Motivation

Baumol (1967): Differential productivity growth across sectors drives “uneven growth” and structural transformation. As sectors experience different rates of technological progress, this causes relative prices to change and labor to reallocate.

Question: Can this supply-side mechanism generate structural change while maintaining balanced growth (Kaldor facts)?

Model

Household utility: $$ \begin{align*} & \int_0^\infty \exp(-\rho t) \frac{c(t)^{1-\theta} - 1}{1-\theta} dt\cr &s.t. \ \dot K(t) = r(t)K(t) - w(t)\bar L -\sum_{i \in \lbrace A, M, S\rbrace}p^iC^i(t) \ \cr & \text{where, }c(t) = \left(\sum_{i \in {A,S,M}} \eta^i C^i(t)^{(\sigma-1)/\sigma}\right)^{\sigma/(\sigma-1)}. \end{align*} $$ Note that $\sigma$ represents elasticity of substitution (constant across sectors)

Technology

For each sector $i$, production is given by the following Cobb-Douglas form. Sector-specific productivity growth rates $g^i$ differ across $i \in {A,M,S}$: $$ \begin{align*} & Y^i(t) = X^i(t) K^i(t)^\alpha L^i(t)^{1-\alpha},\cr & {\dot X^i} = g_i{X^i} \end{align*} $$ We also assume inelastic labor $\bar{L}$; manufacturing produces capital.

Market Clearing

For agriculture and service all outputs are consumed. Manufacturing output is split between consumption and investment: $$ \begin{align*} & Y^A = C^A, Y^S=C^S\cr & Y^M = C^M + \dot K + \delta K. \end{align*} $$

Solution

  • Intratemporal Optimality $$ \frac{c^i}{c^j} = \left(\frac{\eta^i}{\eta^j}\right)^\sigma \left(\frac{p^i}{p^j}\right)^{-\sigma}. $$

  • Because all sectors has the same production technology (Cobb-Douglas with identical $\alpha$) and face the same factor prices ($r, w$) in competitive markets, capital-labor ratios equals across sectors. From sectors’ FOCs: $$ K^i(t)/L^i(t) = k(t), \forall i. $$ And relative prices reflect relative productivity: $$ \frac{p^i(t)}{p^j(t)} = \frac{X^j(t)}{X^i(t)}. $$

  • Similarly, from firm’s FOC, labor allocation satisfies: $$ \frac{L^i(t)}{L^j(t)} = \left(\frac{\eta^i}{\eta^j}\right)^\sigma \left(\frac{X^j(t)}{X^i(t)}\right)^{1-\sigma} $$ So, labor growth rate: $$ \frac{\dot{L}^i(t)}{L^i(t)} - \frac{\dot{L}^j(t)}{L^j(t)} = (1-\sigma)(g^j - g^i). $$

The above implies some weird counter-intuitive facts:

When $\sigma < 1$ (inelastic demand):

  • Faster productivity growth → prices fall
  • But expenditure shares increase (inelastic demand)
  • Labor flows away from productive sectors
  • Asymptotically: All labor concentrates in the most stagnant sector

Last and least,

Balanced Growth Path

From $\dot{K} = X^M K^\alpha \bar{L}^{1-\alpha} - \delta K - \sum_i p^i c^i$, the Euler Equation implies $$ \frac{\partial_t(\sum_i p^i c^i)}{\sum_i p^i c^i} - (1-\theta)\frac{\dot{c}}{c} = \alpha X^M K^{\alpha-1}\bar{L}^{1-\alpha} - \delta - \rho $$ This implies $\theta = 1$ (log utility) is necessary and sufficient for the model to replicate Kaldor facts (constant $K/Y$, constant growth) with uneven sectoral growth.

Empirical Evidence

Baumol et al. (1985) documents for US (1947-76):

  • Stagnant sectors: 0.64% productivity growth, 27.6% → 41.2% employment share
  • Progressive sectors: 2.94% productivity growth, employment share relatively stable
  • Confirms $\sigma < 1$: labor flows toward low-productivity sectors

……Fine