Motivation

Paradox: Standard productivity-based theories (see Supply Side Structural Change in Macroeconomics) predict that sectors with higher productivity growth should have declining employment shares. This is counter-intuitive, and false in reality:

Japan (1960-1990): Rapid manufacturing productivity growth → increased manufacturing GDP share

A lot of empirical evidence (common knowledge…) shows, countries with faster manufacturing TFP growth did not experience faster manufacturing employment decline. And cross-country data shows no negative correlation between manufacturing productivity and manufacturing employment. [Source pending]

So another solution is Trade — introduces a comparative advantage effect that can reverse the weird productivity-based prediction.

Model

Countries: Home and Foreign (*). Each country has 1 unit of labor. Assume there are three sectors, which:

  • Agriculture (A): Tradeable numeraire, endowment $y$ (no production)
  • Manufacturing (M): Tradeable, productivity $X_M$ (Home), $X_M^{\star}$ (Foreign)
  • Services (S): Non-tradeable, productivity $X_S$ (Home), $X_S^{\star}$ (Foreign)

Country’s preference $$ U = (C_A - \gamma_A)^\alpha \left[\beta_M(C_M - \gamma)^\theta + \beta_S C_S^\theta\right]^{(1-\alpha)/\theta} $$ Technology: Linear in labor $$ Y_M = X_M L_M, \quad Y_S = X_S L_S $$ Budget constraint: $$ C_A + p_M C_M + p_S C_S \leq y + w. $$ Labor feasibility: $$ L_M + L_S = 1, \quad L_M^{\star} + L_S^{\star} = 1. $$

Equilibrium Conditions

Firm optimality: $$ p_S = \frac{w}{X_S}, \quad p_M = \frac{w}{X_M} $$ Free trade in M and A: $$ p_M = p_M^{\star} \implies \frac{w}{X_M} = \frac{w^{\star}}{X_M^{\star}}. $$ Market clearing:

  • Global: $C_M + C_M^{\star} = X_M L_M + X_M^{\star} L_M^{\star}$
  • Local services: $C_S = X_S L_S$, $C_S^{\star} = X_S^{\star} L_S^{\star}$

After some algebra:

At equilibrium, employment in manufacturing satisfies

$$ L_M = \frac{\frac{\alpha}{2}\left(1 - \frac{X_M^{\star}}{X_M}\right) + \frac{\gamma}{X_M} + \left(\frac{\beta_M}{\beta_S}\right)^\sigma \left(\frac{X_S}{X_M}\right)^{1-\sigma}}{1 + \left(\frac{\beta_M}{\beta_S}\right)^\sigma \left(\frac{X_S}{X_M}\right)^{1-\sigma}}\tag{$\star$}. $$

What’s left is just to stare at $(\star)$

Demand-driven case ($\gamma > 0$, $\sigma = 1$):

Global productivity growth in M: $\frac{\Delta X_M}{X_M} = \frac{\Delta X_M^{\star}}{X_M^{\star}} > 0$ $$\implies \Delta L_M < 0, \quad \Delta L_M^{\star} < 0$$

National productivity growth in M: $\frac{\Delta X_M}{X_M} > 0$, $\frac{\Delta X_M^{\star}}{X_M^{\star}} = 0$ $$\text{sign}[\Delta L_M] = \text{sign}\left[\frac{\alpha}{2} - \frac{\gamma}{X_M^{\star}}\right], \quad \Delta L_M^{\star} < 0$$

Trade effect: Comparative advantage can cause $\Delta L_M > 0$ despite productivity gains.

Supply-driven case ($\gamma = 0$, $\sigma < 1$):

Similar ambiguity arises from competition between:

  1. Relative supply effect: Higher $X_M$ reduces demand for M labor
  2. Trade effect: Higher $X_M$ increases comparative advantage in M

Concluding, in other words

  1. Productivity gains in manufacturing cause global decline in manufacturing employment
  2. At the national level, trade effects can dominate, leading to increased manufacturing employment in countries with higher productivity growth
  3. Resolves the Japan puzzle: High manufacturing productivity → comparative advantage → increased manufacturing share