On the Microeconomics of Specialization: the Role of Non-Convexity

Abstract

This paper investigates the microeconomics of specialization and its effects on firm productivity. We define economies of specialization as the productivity gains obtained under greater specialization. The paper shows how scale effects and non-convex technology affect economies of specialization. Using a nonparametric approach, we present an empirical analysis applied to Korean farms. The results indicate that non-convexity is prevalent especially on large farms. We find that non-convexity generates large productivity benefits from specialization on larger farms (but not on smaller farms), providing a strong incentive for large farms to specialize. We evaluate the linkages between non-convexity, firm size and management.

Appendix

Proof of Proposition 1: Eq. (2) can be alternatively written as $EP\left(z,{z}^1,\dots, {z}^K,T\right)={\sum}_{k=1}^KD\left({z}^k,T\right)\hbox{--} K\ D\left(z/K,T\right)+K\ D\left(z/K,T\right)\hbox{--} D\left(z,T\right)$. Given (4) and (5), this gives the decomposition in (3).

For K > 1, note that $T\left\{\begin{array}{c}\supset \\ {}=\\ {}\subset \end{array}\right\}\ K\ T$ under $\left\{\begin{array}{c}IRS\\ {}CRS\\ {}DRS\end{array}\right\}$. It follows from (1) that $D\left(z,T\right)=\underset{\beta }{ \sup }\ \left\{\beta :\left(z+\beta\ g\right)\in T\right\}$ $\left\{\begin{array}{c}\ge \\ {}=\\ {}\le \end{array}\right\}\underset{\beta }{ \sup }\ \left\{\beta :\left(z+\beta\ g\right)\in K\ T\right\}$under $\left\{\begin{array}{c}IRS\\ {}CRS\\ {}DRS\end{array}\right\}$. Letting b = β/K, we have $\underset{\beta }{ \sup }\ \left\{\beta :\left(z+\beta\ g\right)\in K\ T\right\}=K\underset{b}{ \sup }\ \left\{b:\left(z/K+b\ g\right)\in T\right\}=K\ D\left(z/K,T\right)$. Combining these results gives the inequalities in (4).

From (1), we have $D\left({z}^k,T\right)=\underset{\beta_k}{ \sup}\left\{{\beta}_k:\left({z}^k+{\beta}_k\ g\right)\in T\right\}$ and ${\sum}_{k=1}^K\left(1/K\right)\ D\left({z}^k,T\right)=\underset{\beta }{ \sup }\ \left\{{\sum}_{k=1}^K{\beta}_k/K:\left({z}^k+{\upbeta}_k\ g\right)\in T,k=1,\dots, K\right\}$. Assume that the set T is convex. Then, (z ^k + β _k  g) ∈ T for all k implies that ${\sum}_{k=1}^K\left[{z}^k/K+\left({\beta}_k/K\right)g\right]\in T$. Letting $\alpha ={\sum}_{k=1}^K{\beta}_k/K$, it follows that $\underset{\beta }{ \sup }\ \left\{{\sum}_{k=1}^K{\beta}_k/K:\left({z}^k+{\upbeta}_k\ g\right)\in T,k=1,\dots, K\right\}\le \underset{\alpha }{ \sup }\ \left\{\alpha :\left({\sum}_{k=1}^K{z}_k/K+\alpha\ g\right)\in T\right\}=D\left({\sum}_{k=1}^K{z}_k/K,T\right)$. When $z={\sum}_{k=1}^K{z}^k$, this yields ${\sum}_{k=1}^K\left(1/K\right)\ D\left({z}^k,T\right)\le D\left(z/K,T\right)$, which gives the first inequality in (5).