Barriers to firm growth in open economies

Facundo Piguillem; Loris Rubini

doi:10.1515/bejm-2018-0003

Article

Barriers to firm growth in open economies

Facundo Piguillem and Loris Rubini

Published/Copyright: August 25, 2018

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From the journal The B.E. Journal of Macroeconomics Volume 19 Issue 1

Abstract

Studies measuring barriers to firm growth assume economies are closed, ignoring information on firm exports. We argue that this information is key to interpreting data and improving the accuracy of model predictions. To do this, we develop a dynamic model with export and domestic barriers. We show theoretically that the closed economy model underestimates barriers and amplifies counterfactuals. By calibrating the model to a set of European countries, we find that the quantitative difference is significant: for example, the closed economy model fails to see that Italian firms are very efficient exporters but poor innovators, and instead concludes that they are mediocre innovators. In terms of predictions, the closed economy model delivers an elasticity of welfare to innovation costs between 31 and 64 percent larger than the open economy model.

Keywords: firm dynamics; innovation and trade; pareto size distribution of firms

JEL Classification: F1; L11; O3

Acknowledgement

We thank Tim Kehoe, Kim Ruhl, Victor Rios Rull, Fernando Alvarez, Ariel Burstein, Richard Rogerson, Marina Azzimonti, Klaus Desmet, Juan Carlos Hallak, Gueorgui Kambourov (the editor) and two anonymous referees for helpful comments. We benefited from talks at ASU, San Andres, WVU, Di Tella, ITAM, the conferences REDG 2012, SED 2012 and Mid West Trade 2012. Loris Rubini gratefully acknowledges financial support from the Spanish Ministry of Science and Innovation (ECO2008-01300 and ECO2011-27014). All errors are our own.

Appendices

A The endogenous distribution of firms

Define Z=[z1,z2]

μ^(t+dt,Z)=∫Zμ^(t,z−z˙dt)e−δdtdz

Taking limits as z1→z2→z

μ^(t+dt,z)=μ^(t,z−z˙dt)e−δdt

For small dt, the following holds:

μ^(t+dt,z)≈μ^(t,z)+μ^1(t,z)dtμ^(t,z−z˙dt)≈μ^(t,z)−μ^2(t,z)z˙dte−δdt≈(1−δdt)

Thus,

μ^(t,z)+μ^1(t,z)dt=μ^(t,z)−μ^2(t,z)z˙dt−δdt(μ^(t,z)+μ^2(t,z)z˙dt)

Note that in steady state μ^1(t,z)=0. Putting all together,

μ^(t,z)=μ^(t,z)−μ^2(t,z)z˙dt−δdt(μ^(t,z)+μ^2(t,z)z˙dt)

Ignoring all terms with dt elevated to a power larger than 1,

μ^(t,z)=μ^(t,z)−μ^2(t,z)z˙dt−δμ^(t,z)dt

Cancelling terms and dividing by dt,

δμ^(t,z)=−μ^2(t,z)z˙

Define the steady state distribution as μ(z)=μ^(t,z) for all t. For non exporters, the distribution is

δμ(z)=−μ′(z)gd(z)z

To solve, use the border condition μ(1) = M. For exporters

δμ(z)=−μ′(z)gxz

To solve, use the border condition μ(z_x) = μ_d(z_x), where μ_d(z_x) is the measure of non exporters that reach the export threshold.

The solution to these distributions works as follows. Start with the exporter distribution. The differential equation can be written as

(28)−μ′(z)μ(z)=δg(z)z

where g(z) = g_x for exporters and g_d(z) for non exporters. For exporters, integrating on both sides,

log⁡(μ(z))=log⁡(z−δ/gx)+Cx

where C_x is the constant of integration, and is determined using the border condition. Taking exponentials yields the distribution of exporters.

For non exporters, we can only integrate both sides of (28) given our guess for the growth rates. The equation becomes

−μ′(z)μ(z)=δ(a/z+b+cz+dz2)

Integrating on both sides,

log⁡(μ(z))=−δ(alog⁡(z)+bz+cz22+dz33)+Cd

where C_d is the constant of integration and is determined using the border condition μ(1) = 1 (the distribution is normalized by the measure of entrants M). Taking exponentials yields the distribution of non exporters.

B Productivity

The goal is to derive the reduced form for aggregate output

Qj=ZjNpj

where Qj=[∫1∞qj(z)σ−1σμj(dz)+∫zx∗∞qj∗(z)σ−1σμ∗(dz)]σσ−1 and N_pj is labor used for production. Let n_dj(z) denote labor for production of units sold domestically and nj,∗(z) for exports. With some algebra, we find

ndj(z)=(σ−1)πdjwj(1+τlj)znj,∗(z)=(σ−1)πxj−πdjwj(1+τlj)z

Labor used in production is

Npj=(σ−1)[πdjwj(1+τlj)∫1∞zμj(dz)+πxj−πdjwj(1+τlj)∫zxj∞zμj(dz)]

From trade balance,

(29)σ(πxj−πdj)∫zxj∞zμ(dz)=σπdj(wj(1+τlj))1−σX∗

where X∗=(1+τx∗)1−σ(w∗(1+τl∗))1−σ∫zx∗∞zμ∗(dz). The left hand side of equation (29) is exports and the right hand side is imports. X^* is supply of foreign goods, which we take as given following the small open economy assumption. We can rewrite total production labor as

Npj=(σ−1)πdj(wj(1+τlj))[∫1∞zμj(dz)+(wj(1+τlj))σ−1X∗]

Since πdj=QjPjσ(wj(1+τlj))1−σσ−σ(σ−1)σ−1.

Npj=((σ−1)σ)σQjPjσ(wj(1+τlj))σ[∫1∞zμj(dz)+(wj(1+τlj))σ−1X∗]Npj=((σ−1)σ)σQjPjσ(wj(1+τlj))σZ~j

where Z~j=∫1∞zμj(dz)+(wj(1+τlj))σ−1X∗.

Next consider the price P_j. By definition,

Pj1−σ=∫1∞pj(z)1−σμj(dz)+(1+τx∗)1−σ∫zx∗∞p∗(z)1−σμj(dz)=(σσ−1(wj(1+τlj)))1−σ(∫1∞zμj(dz)+(wj(1+τlj))σ−1X∗)=(σσ−1(wj(1+τlj)))1−σZ~j

Thus,

Npj=((σ−1)σ)σQjPjσ(wj(1+τlj))σP1−σ(σσ−1(wj(1+τlj)))σ−1=(σ−1)σQjPjwj(1+τlj)=(σ−1)σQjwj(1+τlj)σσ−1(wj(1+τlj))Z~j11−σ=QjZ~j11−σ

Rearranging,

Qj=ZjNpj

where

Zj=[∫1∞zμj(dz)+(wj(1+τlj))σ−1X∗]1σ−1

C Proof of Proposition 2

Proof.

Using equation (15) evaluated at z = 1 and using the free-entry condition,

wjκe=πjj+κI,jgdj(1)22

Exporter profits are

πxj=πjj+(Djwjw∗)2πd∗

where D_j = 1 + τ_xj. Thus,

πxj=wjκe−wjκIjgdj(1)22+(Djwjw∗)1−σπd∗

Rearrange the above equation to introduce the growth rate g_x into it. To do so, multiply both sides by 2[(ρ+δ)2κIjwj]−1 to obtain

2πxj(ρ+δ)2κIjwj=2κeκIj(ρ+δ)−gdj(1)2(ρ+δ)2+2πd∗κIjwj(ρ+δ)2(Djwjw∗)1−σ

Using equation (12) on the left hand side of the above equation,

(30)1−(1−gxj(ρ+δ))2=2κeκIj(ρ+δ)−gdj(1)2(ρ+δ)2+2πd∗wjκIj(ρ+δ)2(Djwjw∗)1−σ

To introduce κjc, notice that in the closed economy the following must hold:

1−(1−κe(ρ+δ)κjc)2=1−(1−gxj(ρ+δ))2

Introducing the last expression in equation (30) and simplifying,

2κeρ+δ[1κjc−1κIj]=(κe(ρ+δ)κjc)2−gdj(1)2(ρ+δ)2+2πd∗κIjwj(ρ+δ)2(Djwjw∗)1−σ

From free entry in the closed economy, gxj=κeκjc, so

(31)2κe(ρ+δ)[1κjc−1κIj]=gxj2−gdj(1)2+2πd∗wjκIj(Djwjw∗)1−σ

Replacing D_j = 1 + τ_xj in (31), we obtain (22). Since profits are always positive and by Proposition 1, g_xj > g_dj(1), the right hand side of the above equation is positive. Thus, it must be the case that 1κjc>1κIj, or κjc<κIj. □

D Proof of Corollary 1

Taking the difference of (20) for the two countries and reorganizing:

κd2−κd1=α(κI2−κI1)+χ2ρ[gx12−gx12+gd2(1)2−gd1(1)2]+χ2ρ[2π2κI,1w1(D1w1w2)1−σ−2π1κI,2w2(D2w2w1)1−σ]

where 0<α=κd2κd1κI2κI1<1 and χ=κd2κd1κe>0.

Because of the definition of exporters’s profits

κd2−κd1=α(κI2−κI1)+χ2ρ[gx12−gx12+gd2(1)2−gd1(1)2]+χ2ρ[2(πx1−π1)κI,1w1−2(πx2−π2)κI,2w2]

κd2−κd1=α(κI2−κI1)+χ2ρ[gx12−gx12+gd2(1)2−gd1(1)2]+ρχ2[hx1−hd1−(hx2−hd2)]

Using the optimal innovation policy of the exporter

hx1−hx2=(1−gx2ρ)2−(1−gx2ρ)2=(gx2ρ)2−(gx1ρ)2+2ρ(gx1−gx2)

Therefore:

κd2−κd1=α(κI2−κI1)+χ[gx1−gx2]+χ2ρ[gd2(1)2−gd1(1)2]+ρχ2[hd2−hd1]

Similarly,

κd2−κd1=α(κI2−κI1)+χ[gx1−gx2]+χ2ρ[gd2(1)2−g1,22−gd1(1)2+g,112]+χ[g1,2−g1,1]

where_{ȃ g1,i=(ρ+δ)(1−1−hdi)} for all i.

κd2−κd1=α(κI2−κI1)+χ[gx1−gx2+g1,2−g1,1]+χ2ρ[gd2(1)2−g1,22−gd1(1)2+g1,12]

where gdi(1)≥gi=ρ(1−1−hdi)

E Proof of Proposition 5

When κ_x = 0, given the closed form solution for the variables in equilibrium derived in the main section of the paper, and using trade balance, wages are

wσ−1=∫zdμ(z)X∗πx−πdπd=∫zdμ(z)X∗w1−σπ~πd⇒w2(σ−1)=κeδκIκe−κe2π~X∗1πdw=(κeδκIκe−κe2π~X∗1πd)12(σ−1)

To solve this, we need to know the value of π_d. Notice that

πxw=κe(1−κe2κI)=πd+w1−σπ~w

where π~=(1+τx)1−σπ∗w∗1−σ.

Introducing in this expression the value for w defines the following implicit function

πd2σ−12(σ−1)((δκI−κe)X∗π~)12(σ−1)+πdσ2(σ−1)((δκI−κe)X∗π~)σ2(σ−1)π~=κe(1−κe2κI)

Next we build towards showing that ∂Zx∂κI>∂Zc∂κI. We cannot show it generally, but we can find a sufficient condition for this to happen. This condition is that σ < 3/2.

We first show that ∂πxπd∂κI<0, which, by Proposition 4, implies that ∂Zx∂κI>∂Zc∂κI. To show ∂πxπd∂κI<0 we proceed by contradiction. Thus, we show that if ∂πxπd∂κI≥0, then it must be the case that ∂πd∂κI≥0. But we also show that under our sufficient condition this cannot happen. We start by showing this last result, and then the main proposition.

Lemma 1

If σ < 3/2

∂πd/∂κI<0

Proof.

Using the implicit function theorem, we show that if σ < 3/2 then ∂πd/∂κI<0. Notice, this is a sufficient condition, but it will help us prove that ∂πx/πd/∂κI>0.

Define

π^d=πd12(σ−1)

Then the equation that defines π^d is

(32)F=π^d2σ−1(X∗π~)12(σ−1)(δκI−κe)12(σ−1)+π^dσ(X∗π~)σ2(σ−1)(δκI−κe)σ2(σ−1)π~−κe+κe22κI=0

The implicit function theorem says

∂π^d∂κI=−∂F∂κI∂F∂π^d

It is easy to check that ∂F∂π^d>0. So we need to check that ∂F∂κI>0.

∂F∂κI=δ[π^d2σ−1(X∗π~)12(σ−1)(δκI−κe)12(σ−1)+σπ^dσ(X∗π~)σ2(σ−1)(δκI−κe)σ2(σ−1)π~]2(σ−1)(δκI−κe)−κe22κI2>δ[π^d2σ−1(X∗π~)12(σ−1)(δκI−κe)12(σ−1)+π^dσ(X∗π~)σ2(σ−1)(δκI−κe)σ2(σ−1)π~]2(σ−1)(δκI−κe)−κe22κI2=δ[κe(1−κe2κI)]2(σ−1)(δκI−κe)−κe22κI2=δ2(σ−1)(δκI−κe)πxw−gx22

The first term on the third line comes from rearranging the expression F. The second term comes from the expressions derived previously for π_x/w and the equilibrium value for g_x.

Multiplying the equation by κ_I gives

δκI2(σ−1)(δκI−κe)πxw−κIgx22=πxw(δκI2(σ−1)(δκI−κe)−1)>πxw(12(σ−1)−1)>0

⊡

We use the lemma for the proof of the proposition. The proposition says

∂Zx∂κI>∂Zc∂κI

To prove it, we proceed by contradiction. So suppose this is not true. Then if ∂Zx∂κI≤∂Zc∂κI it must be the case that

∂πxπd∂κI≤0

From the definition of π_x,

πxπd=1+w1−σπdπ~⇒∂w1−σπd∂κI≤0

From trade balance,

ExportsImports=πx−πdπdZcX∗w1−σ=1

We know that ∂πx−πdπd∂κI≤0 and ZcX∗∂κI<0. Then we must have ∂w1−σ∂κI>0. Since ∂w1−σπd∂κI<0, this implies that ∂πd∂κI>0, which is a contradiction.

F The firm size distributions in the data

Figure 9:

Firm size distributions.

Figure 10:

Exporters’ distributions.

Figure 11:

Non-exporters’ distributions.

G Fit of the approximation

Recall that our solution for the non exporter growth rate involves a differential equation with no closed form solution. Since we need a closed form to derive the distribution of firms, we approximate the non exporter with the following functional form

gdi(z)=(ai+biz+ciz2+diz3)−1

In this section, we discuss the goodness of this fit. Table 17 shows the values we compute for the variables a, b, c, and d for each country. Figure 12 through Figure 16 show how good this approximation is for the growth rates and the non exporter value function.

Table 17:

Fitted values.

Country	a	b	c	d
France	39.93	44.58	−54.27	14.69
Germany	23.39	66.24	−62.75	15.83
Italy	−20.83	275.59	−296.97	91.45
Spain	36.33	50.34	−56.99	15.17
UK	42.65	31.97	−47.01	13.80

Figure 12:

France.

Figure 13:

Germany.

Figure 14:

Italy.

Figure 15:

Spain.

Figure 16:

UK.

H Alternative values of σ

Changing σ does not affect the computed values of the costs κ_I and κ_x, and only affects the estimates of τ_x. Table 18 shows how these values change.

Table 18:

Estimates of 1 + τ_x under different specifications of σ.

σ	4.00	5.00	6.00
France	1.00	1.00	1.00
Germany	1.00	1.00	1.00
Italy	0.93	0.92	0.92
Spain	1.69	1.42	1.30
UK	0.88	0.92	0.94

Notice that when changing σ both the elasticity of substitution and a parameter that affects firm productivity are changing. This assumption follows Atkeson and Burstein (2010) and greatly simplifies the analysis by making profits linear in productivity.

There is no reason to believe this result would change if σ did not affect firm productivity, since one of the calibration targets is the trade volume. To see this, consider the result in Rubini (2014) that the effect of a larger σ is to amplify the share of imports within the economy, in a setting with innovation where σ does not affect the productivity of a firm. Intuitively, a larger σ increases the elasticity of substitution between domestic and imported goods, driving consumers to purchase more imports that are relatively cheaper. Given that the export share is a target in the calibration, this would require a larger τ to offset this change. There is no reason to expect additional first order effects. It is not clear whether this bias would be larger for Germany or the other countries, having little effect on the relative cost, as in the present case.

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The online version of this article offers supplementary material (DOI: https://doi.org/10.1515/bejm-2018-0003).

Published Online: 2018-08-25

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https://doi.org/10.1515/bejm-2018-0003

Keywords for this article

firm dynamics; innovation and trade; pareto size distribution of firms