Dynamic Externalities: Theory and Empirical Analysis

Abstract

This chapter focuses on dynamic externalities that are a source of competitive advantages, and reviews the related empirical studies. Research on technical-knowledge spillover, the core of dynamic externalities, is being widely conducted, mainly in the field of industrial organization theory. However, there are few empirical studies that have considered technical-knowledge spillover from the viewpoint of industrial agglomeration. Some studies estimate the extent and type of dynamic externalities, and find evidence consistent with dynamic externalities. Despite the different data sources used, methodologies are similar. This chapter reviews the main methodologies of dynamic externalities, and discusses empirical analysis issues of previous studies.

This chapter is based on Otsuka (2004) “Dynamic externalities of industrial agglomeration: A survey,” published in Okayama Economic Review (Vol. 35, No. 4, pp. 27–50, in Japanese).

Appendix: Solow Residual Measurement Bias

Two points are described in this appendix, as problems related to the measurement of the Solow residual discussed in this chapter: (1) the presence of measurement bias that occurs when assuming perfect competition and constant returns to scale, and (2) differences using value-added versus gross output as the output element.

• 1. The basic model

For simplicity, the Hicks neutral-type production function shall be assumed.

$$ V= AF\left(K,L\right), $$

(A.1)

where, V is the value added as the output element, and K and L are the capital input and the labor input. A represents the production technologies, and technological progress can be ascertained through the production function shift term.

Conducting a logarithmic total differential on (A.1), and dividing both sides by V, the formula becomes as follows.

$$ \frac{dV}{V}=\frac{dA}{A}+A\frac{\partial F}{\partial K}\frac{K}{V}\frac{dK}{K}+A\frac{\partial F}{\partial L}\frac{L}{V}\frac{dL}{L}. $$

(A.2)

Here, when taking x = log X for the arbitrary variable X, (A.2) can be rewritten as follows.

$$ dv= da+A\frac{\partial F}{\partial K}\frac{K}{V} dk+A\frac{\partial F}{\partial L}\frac{L}{V} dl. $$

(A.3)

Assuming perfect competition, (A.4) is established from the first-order condition for profit maximization.

$$ A\frac{\partial F}{\partial K}=\frac{P_K}{P},\kern0.30em A\frac{\partial F}{\partial L}=\frac{P_L}{P}, $$

(A.4)

where P is the output price, while P _K and P _L represent the capital and labor prices, respectively.

Further, assuming the condition of constant returns to scale, the following formula is established from Euler’s theorem.

$$ A\frac{\partial F}{\partial K}\frac{K}{V}+A\frac{\partial F}{\partial L}\frac{L}{V}=1. $$

(A.5)

From this relation between (A.4) and (A.5),

$$ PQ={P}_KK+{P}_LL $$

(A.6)

is established as “exhaustion of the total product theorem.” Therefore, when rewriting (A.3) from (A.4) and (A.5), the following formula is established.

$$ dv= da+\left(1-{s}_L^v\right) dk+{s}_L^v dl $$

(A.7)

where, ${s}_L^v$ is labor share.

$$ {s}_L^v\equiv \frac{P_LL}{PV}. $$

Therefore, the value-added base Solow residual (SR ^v) becomes as follows,

$$ {SR}^v\equiv dv-\left[\left(1-{s}_L^v\right) dk+{s}_L^v dl\right]= da. $$

(A.8)

The Solow residual is consistent with the rate of technological progress.

• 2. Markup and economies of scale

Next, it is assumed that producers have “monopoly power” and do not set price equal to marginal cost. Then, from the first-order condition for profit maximization by producers with monopoly power, the formula becomes as follows.

$$ A\frac{\partial F}{\partial K}=\frac{P_K}{P}\mu, \kern0.30em A\frac{\partial F}{\partial L}=\frac{P_L}{P}\mu, $$

(A.9)

where, μ is the price markup and is defined as the ratio of price P and the marginal cost MC.

$$ \mu \equiv \frac{P}{MC}=\frac{\eta -1}{\eta }. $$

η is the price elasticity of demand.

Further, assuming increasing returns to scale,

$$ A\frac{\partial F}{\partial K}\frac{K}{V}+A\frac{\partial F}{\partial L}\frac{L}{V}=\gamma $$

(A.10)

is established. γ is the parameter indicating homogeneity.

From this relation between (A.9) and (A.10), the formula becomes as follows.

$$ PQ=\frac{\mu }{\gamma}\left[{P}_KK+{P}_LL\right]. $$

(A.11)

Therefore, if μ = γ is not established, exhaustion of the total product theorem is also not established.

On rewriting (A.3) from (A.9) and (A.10), the formula becomes as follows.

$$ dv= da+\left(\gamma -\mu {s}_L^v\right) dk+\mu {s}_L^v dl. $$

(A.12)

In other words,

$$ {\displaystyle \begin{array}{c}{SR}^v\equiv dv-\left[\left(1-{s}_L^v\right) dk+{s}_L^v dl\right]\\ {}= da+{s}_L^v\left(\mu -1\right)\left( dl- dk\right)+\left(\gamma -1\right) dk.\end{array}} $$

(A.13)

In the event of the existence of imperfect competition and economies of scale, μ > 1 and γ > 1. Under these conditions, the value-added based Solow residual is not consistent with technological progress, and measurement bias occurs according to the degree of imperfect competition and economies of scale. Therefore, measurement bias according to imperfect competition and economies of scale also occurs for the effects of industrial agglomerations that are measured as an element of the Solow residual.

• 3. Gross output and value-added as the output element

Finally, the Solow residual is different when gross output is used as the output element compared to when value-added is used.

With gross output as Y and intermediate input goods as M, the respective share for labor and intermediate input goods on a total output basis are as follows.

$$ {s}_L\equiv \frac{P_LL}{PY},{s}_M\equiv \frac{P_MM}{PY}. $$

According to Basu and Fernald (1997, 2002), the value-added growth rate dv is expressed as follows.

$$ dv=\frac{dV}{V}=\frac{PdY-{P}_M dM}{PY-{P}_MM}=\frac{dy-{s}_M dm}{1-{s}_M}. $$

(A.14)

Then, introducing (A.7) into (A.14) and arranging it based on (A.8),

$$ dy=\left(1-{s}_M\right)\left[{SR}^v+\left\{\left(1-{s}_L^v\right) dk+{s}_L^v dl\right\}\right]+{s}_M dm $$

(A.15)

is obtained.

Here, the gross-output based Solow residual (SR) is defined as follows.

$$ SR\equiv dy-\left[\left(1-{s}_L-{s}_M\right) dk+{s}_L dl+{s}_M dm\right]. $$

(A.16)

Therefore, arranging (A.15),

$$ {SR}^v=\frac{1}{1-{s}_M} SR $$

(A.17)

is established. This indicates that the Solow residual based on value-added is not independent of the variations in the share of intermediate input s _M .

• 4. Summary

As was shown above, as perfect competition and constant returns to scale are assumed when measuring the Solow residual, if producers face imperfect competition and possess production technologies providing increasing returns to scale, measurement bias occurs for the Solow residual. Further, when using value-added to measure the Solow residual, the value-added based Solow residual is affected by variations in the share of intermediate input. Therefore, it is judged highly likely that systematic bias occurs in the dynamic externalities evaluated using the value-added based production function because of the influence of the share of intermediate goods.