Cross-industry growth differences with asymmetric industries and endogenous market structure

Chien-Yu Huang; Lei Ji

doi:10.1515/bejm-2017-0045

Article

Cross-industry growth differences with asymmetric industries and endogenous market structure

Chien-Yu Huang and Lei Ji

Published/Copyright: February 22, 2019

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From the journal The B.E. Journal of Macroeconomics Volume 19 Issue 2

Abstract

We develop a two-industry R&D growth model with a free-entry endogenous market structure to evaluate the impact of industrial fundamentals on cross-industry differences of TFP growth and R&D intensity. Endogenous market structure in our model allows the firm’s market size to respond to the firm’s entry and exit which complements the models with an exogenous market structure in the previous literature. We find that surprisingly, an industry with a relatively high R&D productivity or appropriability exhibits “relatively” low in-house innovation growth and R&D intensity during transition. Moreoever, we examine the effects of R&D subsidies and patent breadth policies on industry differences by implementing both asymmetric and symmetric policy rules. We find that only asymmetric R&D subsidies have impacts on TFP growth and R&D intensity differences.

Keywords: cross-industry TFP growth differences; endogenous growth; endogenous market structure

JEL Classification: L10; O40

Acknowledgement

We gratefully acknowledge the research support from China National Philosoply and Social Science Fund (16BJL089).

Appendix

Proof of Lemma 1.

The firm’s current-value Hamiltonian is

CVHij=Gij(PGij−Ai)−ϕi−Rij+qij(αiRij)

where i denotes the industry, and q_ij is the co-state variable. By taking the first-order derivative subject to PGij, the solutions for the prices are mark-ups over variable cost:

(47)PG1≡PG1j=A1λ

(48)PG2≡PG2j=A2λ

The Hamiltonian is linear in R&D expenditure, and so the solution for investment expenditure R_1j is bang-bang:

Rij{=∞if 1/αi>qij>0if 1/αi=qij=0if 1/αi<qij

We rule out the first possibility of Rij=∞ because it is inconsistent with the market equilibrium. We also rule out the other corner solution, 1/α<qij, because it implies no economic growth, and we am interested here in the case where perpetual growth occurs. We thus have the interior solution

(49)1αi=qij

The left side of eq. (49) is the same for all j, so that all firms in industry i choose the same level of R&D, which we denote as R_i.

The Maximum Principle gives the necessary condition for the evolution of the co-state variable q₁, which we can rearrange as

(50)rij=∂Fij∂Zij1qij+qij˙qij

This equation defines the rate of return to R&D (i.e. on quality innovation), with r_ij as the percentage marginal revenue from R&D plus the capital gain (the percentage change in the shadow price). Because 1/αi=qij, we also have q˙ij/qij=0. As with intermediate goods prices, the expressions for the rates of return differ across the two industries. The rate of return for industry 1 in (19) is obtained by substituting (7), (15), (47), and (49) into (50). Following the same steps as in industry 1, we obtain the rate of return to R&D in industry 2. □

Proof of Lemma 3.

Combining (34), (21) and (26) leads to

r=ρ+g=Γg1+(1−Γ)g2+ρ=Γ(r1δ1−α1θ1Z1Z2)+(1−Γ)(r2δ2−α2θ2Z2Z1)+ρ.

Imposing r = r₁ = r₂ leads to

r=−Γα1θ1Z1Z2−(1−Γ)α2θ2Z2Z1+ρ1−Γ/δ1−(1−Γ)/δ2.

Substituting the above equation into Z1˙Z1−Z2˙Z2 leads to

(Z1Z1)˙/(Z2Z2)=Z1˙Z1−Z2˙Z2=α1[(P1−A1)G1−ϕ1]Z1−α2[(P2−A2)G2−ϕ2]Z2=α1[r1/α1δ1−ϕ1Z1]−α2[r2/α2δ2−ϕ2Z2]=(1δ1−1δ2)r−α1θ1Z1Z2+α2θ2Z2Z1=(1δ1−1δ2)−Γα1θ1Z1Z2−(1−Γ)α2θ2Z2Z1+ρ1−Γ/δ1−(1−Γ)/δ2−α1θ1Z1Z2+α2θ2Z2Z1=δ1(1−δ2)δ1δ2−δ2Γ−δ1(1−Γ)α1θ1Z1Z2+−δ2(1+δ1)α2θ2Z2Z1δ1δ2−δ2Γ−δ1(1−Γ)+δ1−δ2δ1δ2−δ2Γ−δ1(1−Γ)ρ

By the no-arbitrage condition r₁ = r₂, the dynamics of the relative number of firms N1/N2 can be expressed as (38), through which we can also obtain the dynamics of the firm size ratio as (39). Combining r₁ = r₂ leads to (40). □

Proof of Lemma 4.

We obtain the quality ratio (45) on the BGP by noting that g1=g2=g∗, r1=r2≡r, and, from the Euler equation, r=g∗+ρ. From those relations, we obtain a quadratic form with two roots:

(Z1Z2)⋆=(δ11−δ1−δ21−δ2)ρ+[(δ21−δ2−δ11−δ1)ρ]2+4δ11−δ1δ21−δ2α1ϕ1α2ϕ22δ11−δ1α1ϕ1>0,(Z1Z2)⋆=(δ11−δ1−δ21−δ2)ρ−[(δ21−δ2−δ11−δ1)ρ]2+4δ11−δ1δ21−δ2α1ϕ1α2ϕ22δ11−δ1α1ϕ1<0.

We discard the negative solution which is economically implausible. We derive the number of firms Ni⋆ by combining (19), (26), (29) and (35). □

Proof of the model results under the general setting of fixed operating costs (i.e.ϕi=Ziβi/Zkβk, whereβi=1+βk). Assume δ1=δ2, and recall that r₁ = r₂

(51)Z1˙Z1−Z2˙Z2=[r1/δ−α1ϕ1Z1]−[r2/δ−α2ϕ2Z2]=−α1ϕ1Z1+α2ϕ2Z2

A simplified case proposed by the referee is to set ϕ1=Z1β+1/Z2β, and ϕ2=Z2β+1/Z1β, Substituting them into the above expression yields,

Z1˙Z1−Z2˙Z2=−α1Z1β+1/Z2βZ1+α2Z2β+1/Z1βZ2=−α1Z1β/Z2β+α2Z2β/Z1β.

On the steady state (i.e. Z1˙Z1=Z2˙Z2), we further arrange the above expression yields

(52)α1Z1β/Z2β=α2Z2β/Z1β(Z1Z2)2β=α2α1Z1Z2=(α2α1)12β

Consider a more general case proposed by the referee which sets ϕ1=Z1β1+1/Z2β1, and ϕ2=Z2β2+1/Z1β2. Substituting them into the expression (51) yields,

Z1˙Z1−Z2˙Z2=−α1Z1β1+1/Z2β1Z1+α2Z2β2+1/Z1β2Z2=−α1Z1β1/Z2β1+α2Z2β2/Z1β2

On the steady state (i.e. Z1˙Z1=Z2˙Z2), we further arrange the above expression yields

(53)α1Z1β1/Z2β1=α2Z2β2/Z1β2(Z1Z2)β1+β2=α2α1Z1Z2=(α2α1)1β1+β2

From expressions (52) and (53), we obtain the similar qualitative results of the impacts of α₁ and L₁ in proposition 2 (except δ since we assume δ1=δ2) as well as the R&D subsidy rate s₁ in Section 4.1,

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Published Online: 2019-02-22

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https://doi.org/10.1515/bejm-2017-0045

Keywords for this article

cross-industry TFP growth differences; endogenous growth; endogenous market structure