ONLINE APPENDIX TO: FINANCING CONSTRAINTS, FIRM DYNAMICS, AND INTERNATIONAL TRADE∗ Till Gross†

St´ephane Verani‡

This online appendix contains additional derivations and extensions.

1 1.1

Derivations Optimal revenue function for intermediate good producers

The results of this paper are robust to a wide range of production technologies. However, in the computational implementation we assume production functions with constant returns to scale and a constant elasticity of substitution. We will follow this approach here, as it allows us to derive some properties analytically. ∗ This version: May 3, 2013. The views expressed in this paper do not reflect the views of the Board of Governors of the Federal Reserve System or its staff. † till [email protected]. Department of Economics, Carleton University. ‡ [email protected], corresponding author. Federal Reserve Board. Mail Stop 145. Constitution Ave NW, Washington, DC 20551. Tel: 1-(202)-912-7972. Fax: 1-(202)-475-6363

1

The production function of the final goods producer is

Z Y =

y(ω)

σ−1 σ

σ ! σ−1

Z dω +

y(ωf )

σ−1 σ

dωf

,

(1)

Ωf

Ω

where σ > 1 is the elasticity of substitution between input varieties. The first-order condition for the maximization of equation (1) with respect to variety ω ˆ ∈ Ω yields the inverse demand functions for the intermediate goods producers:1

p(ω) = y(ω)−1/σ Y 1/σ .

(2)

Consider a Cobb-Douglas production function for the intermediate goods producer, such that G(k, n) = k ηk nηn . The cost minimization then implies that the quantity sold by a domestic firm is

qD = Rηk +ηn (1 + ηn /ηk )−ηk [w(1 + ηk /ηn )]−ηn = Rν x.

(3)

where ν = ηk + ηn is the returns to scale parameter of the production function G(k, n) and x = (1+ηn /ηk )−ηk [w(1+(1+ηk /ηn )]−ηn reflects the impact of wages and the shares of capital and labor in production. When a firm maximizes its cash-flow Fi (R) − R(1 + r), the optimal amount of resources used by a non-exporting firm is then 1  1/σ (1−1/σ)  1−ν(1−1/σ) Y x ν(1 − 1/σ) eD = π R . 1+r

1

(4)

Due to constant returns to scale of the production function the final goods producer’s profits will be zero.

2

From the underlying first-order condition one can deduce that ν(1 − 1/σ) < 1 must hold, or otherwise there would not be a finite optimum. This means that if the production G(k, n) exhibits increasing returns to scale, the reduction in marginal cost by expanding capacity must not out-pace the reduction in marginal revenue. Analogously, we can derive the expressions for goods sold domestically and abroad by an exporter

(R − IE )ν x = (R − IE )ν B σ/(σ−1) , and qE = ∗ 1−σ 1 + (Y /Y )(1 + IT ) qE∗ =

(R − IE )ν x(Y ∗ /Y )(1 + IT )−σ = (R − IE )ν B ∗ σ/(σ−1) , 1 + (Y ∗ /Y )(1 + IT )1−σ

(5) (6)

where

 B= ∗

B =



x ∗ 1 + (Y /Y )(1 + IT )1−σ

1−1/σ

x(Y ∗ /Y )(1 + IT )−σ 1 + (Y ∗ /Y )(1 + IT )1−σ

1−1/σ

, and

(7)

.

(8)

The quantity sold domestically depends positively on the domestic demand Y and negatively on the foreign demand parameter Y ∗ . The higher the trade cost, the more goods an exporter sells at home. The reverse applies to goods sold abroad. A firm operating at full scale when it is profitable to trade requires period resources "

1/σ ∗ 1/σ ∗ B )ν(1 − 1/σ) eE = π (Y B + Y R 1+r

3

1 # 1−ν(1−1/σ)

+ IE .

(9)

1.2

Workers’ decision rules

We assume that the instantaneous utility function for workers is

u(cw , 1 − h) = log(cw ) + λ log(1 − h) ,

(10)

where λ > 0 is the elasticity of leisure. The optimal decision rules for savings and labor are:

d0 = h=

w[(1 + r)βˆ − 1] ˆ + (1 + r)βd (1 + r) − (1 − γw ) ˆ + r) − (1 − γw )(1 + λ) (1 + r) + λ(1 − γw )β(1

(1 + λ)[(1 + r) − (1 − γw )] ˆ − γw )) λ(1 + r)(1 − β(1 − d. w(1 + λ)

(11) (12)

It follows that workers’ deposits increase with age as long as the interest rate plus the ˆ principal is greater than the inverse of the workers’ discount factor (i.e., (1 + r) > 1/β). Otherwise workers’ debt level increases up to a point where d0 = d = −w/(r + γw ), which is always less than the maximum amount of debt a worker can service in perpetuity working full time (dmin = w/r). Deposits depend positively on the interest rate (∂d0 /∂r > 0), and on wages. Labor supply depends positively on the interest rate for younger workers with few deposits and negatively when d > w(1 + r − (1 − γw ))−2 . It always depends positively on

4

wages.2 It follows that aggregate deposits and labor supply are

D= H= −

1.3

w(1 − γw )((1 + r)βˆ − 1) ˆ (1 − (1 − γw )(1 + r)β)((1 + r) − (1 − γw )) ˆ + r) − (1 − γw )(1 + λ) (1 + r) + λ(1 − γw )β(1 (1 + λ)[(1 + r) − (1 − γw )] ˆ − γw ))(1 − γw )((1 + r)βˆ − 1 λ(1 + r)(1 − β(1 ˆ (1 + λ)(1 − (1 − γw )(1 + r)β)((1 + r) − (1 − γw ))

(13) (14) .

Final good market clearing

To show that Y = Cw + Ce + K, start from the zero profit condition for final goods producers and invoke the market clearing condition for intermediate goods:

Z Y =

Z pydω + X

Z pf yf dωf =

Z pqdω + X

pf qf dωf

(15)

where we omit the argument ω in p(ω) for notational simplicity. Given the balanced trade condition, the market clearing for exported goods and the definition of revenues, it follows that Y

=

R

pqdω + X

=

R

pqdω +

=π

R

R

R

pf qf dωf

p∗ q ∗ dω ∗

F (R)dµ .

Using the definition of entrepreneurial consumption, and the intermediaries’ balanced 2

Labor supply decreases linearly in deposits, which implies that workers may choose negative working hours if they accumulate enough deposits. Given our calibration, this does not occur.

5

budget condition yields

Y

=π

R

F (R)dµ

= Ce + π

R

τ dµ

= Ce + (1 + r)

R

Rdµ + (1 + r)ΓI0 − rZ − Γb S .

Capital market clearing allows to substitute for rZ:

Y

= Ce + (1 + r) = Ce +

R

R

Rdµ + (1 + r)ΓI0 − rZ − Γb S

Rdµ + ΓI0 − Γb S + rD .

Plugging in for the use of resource advancements R = k + wn, and using the definition of total capital expenditures, K, allows us to write:

Y

= Ce +

R

Rdµ + ΓI0 − Γb S + rD

= Ce +

R

kdµ + ΓE IE + ΓI0 − Γb S +

= Ce + K +

R

R

nwdµ + rD

nwdµ + Dr .

Finally, the labor market clearing condition, and the aggregate budget constraint for workers Cw + D = wH + D(1 + r), complete the proof:

Y

= Ce + K +

R

nwdµ + Dr

= Ce + K + wH + Dr = Ce + Cw + K .

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1.4

Stochastic process for firm equity

Let A be any subset of B(S). The stochastic process for a perpetually regenerating firm can be written as Xt+1 = Tω (Xt , t ), (t )t≥0 ∼ φω ∈ P(Z), X0 = V0 ∈ S

(16)

where Tω : S ×Z → S is a collection of measurable functions indexed by ω ∈ Ω the parameter space, (t )∞ t=1 is a sequence of independent random shocks with (joint) distribution φω , and S and Z are the state space and the probability space respectively. It follows for any x ∈ {x : Vr < x < VD and V L (x) < Vr }

P (x, A) =

    (1 − γ)(1 − π)α(V L (x)) + γ if A = {V0 }         (1 − γ)(1 − π)(1 − α(V L (x)) if A = {Vr }    (1 − γ)π         0

(17)

if A = {V H (x)} otherwise

For any x ∈ {x : Vr < x < VD and Vr ≤ V L (x) ≤ VD and V H (x) ≤ VD }

P (x, A) =

    γ if A = {V0 }         (1 − γ)(1 − π) if A = {V L (x)}    (1 − γ)π         0

H

if A = {V (x)} otherwise

7

(18)

For any x ∈ {x : Vr < x < VD and Vr ≤ V L (x) ≤ VD and V H (x) ≥ VD }

P (x, A) =

    γ         (1 − γ)(1 − π)    

if A = {V0 } if A = {V L (x)}

(1 − γ)πδ(V H (x)) if A = {VE }        (1 − γ)π(1 − δ(V H (x))) if A = {VD }         0 otherwise

(19)

For any x ∈ {x : VE < x < Ve and V L (x) ≤ VE and VE ≤ V H (x) < Ve }

P (x, A) =

    γ         (1 − γ)(1 − π)δ(V L (x))    

if A = {V0 } if A = {VE }

(1 − γ)(1 − π)(1 − δ(V L (x))) if A = {VD }        (1 − γ)π if A = {V H (x)}         0 otherwise

(20)

For any x ∈ {x : VE < x < Ve and VE ≤ V L (x) ≤ Ve and V H (x) ≥ Ve }

P (x, A) =

    γ if A = {V0 }         (1 − γ)(1 − π) if A = {V L (x)}    (1 − γ)π         0

if A = {Ve } otherwise

8

(21)

And for x = {Ve }

P (x, A) =

    γ    

if A = {V0 }

(1 − γ) if A = {Ve }        0 otherwise

(22)

For each A ∈ B(S), P (·, A) is a non-negative function on B(S), and for each x ∈ S, P (x, ·) is a probability measure on B(S). Therefore, for any initial distribution ψ, the stochastic process X defined on S ∞ is a time-homogeneous Markov chain.

2

Extensions

2.1

Unit-costs of trade

While iceberg costs are pervasive in the international trade literature, the existence of unitcosts of trade cannot be ruled out. For instance, Hummels and Skiba (2004) and Irarrazabal, Moxnes, and Opromolla (2011) find evidence of unit-costs of trade. In this section, we derive the implications of the presence of such unit-costs of trade on exporter dynamics. In particular, when the production function exhibits decreasing returns to scale, the ratio of exports to total sales increases as available resources increase. This implies that older exporters export proportionally more than younger ones on average, and that the typical exporter export more intensively over time. Let IT be the iceberg trade costs and IˆT the per-unit trade costs. An exporting firm’s

9

maximum revenues for given resources R are

FE (R) = maxq,q∗ ,J Y 1/σ q 1−1/σ + Y ∗ 1/σ q ∗ 1−1/σ ˆ s.t. q + q ∗ (1 + IT ) ≤ G(J)

(23)

JpJ + q ∗ IˆT + IE ≤ R .

For ease of exposition, let J be the combined optimal inputs of n and k and pJ be the costs of ˆ these inputs. With a constant elasticity of substitution, G(J) has the same scale-properties as G(k, n) in the main text. We have also substituted in for prices, so that Y 1/σ q 1−1/σ = p(q)q and similarly for exports. Let λ denote the Lagrange multiplier for the production constraint and µ for the resource constraint. The first-order conditions for q, q ∗ , and J with IT = 0 are then

(1 − 1/σ)q −1/σ Y 1/σ = λ (1 − 1/σ)q ∗ −1/σ Y ∗ 1/σ = λ + µIˆT ˆ 0 (J) = µpJ . λG

(24) (25) (26)

First, combining equations (24) and (25) yields the ratio of domestic to foreign sales:

q∗ Y∗  µ −σ . = 1 + IˆT q Y λ

(27)

From equation (26), we see that the fraction µ/λ depends on the returns to scale of the production function: ˆ 0 (J) µ G = . λ pJ 10

(28)

With constant returns to scale, the above ratio of the value of resource over the value of production is a constant, and so q/q ∗ is also a constant – as with iceberg-style trade costs only. On the other hand, µ/λ decreases as resources increase with decreasing returns to scale, and thus the ratio q/q ∗ decreases in firm size, V . This implies that the ratio of exports to total sales q ∗ /(q + q ∗ ) increases as an exporter grows larger. The opposite is true with increasing returns to scale. The intuition behind this result is as follows: With constant returns to scale, resources can be transformed into production at a constant rate. Iceberg trade costs are expressed in terms of production, and per-unit costs are expressed in terms of resources. The relative costs of iceberg vs. per-unit cost are therefore independent of the size of the firm. With decreasing returns to scale, however, the value of production relative to the value of resources becomes larger the more resources are availiable to the firm; and therefore the per-unit costs of trade compared to the production costs are comparatively less important for larger firms. With iceberg costs, both trade and production cost are in terms of the same unit, so firm size does not matter for export intensity. When both types of trade costs are in place, the ratio of domestic to foreign sales is

Y∗  µ −σ q∗ = 1 + IT + IˆT , q Y λ

and so the same results as above apply.

11

(29)

2.2

Capital-skill complementarity

The empirical trade literature documents that exporters are more capital intensive, hire more high-skilled labor, and pay higher wages than non-exporting firms. In this appendix, we show how our model can be extended to account for differences in skill and capital intensities between exporters and non-exporters. Consider a world with two types of workers, high and low-skilled, with wages wh and wl respectively. Let G(k, nh , nl ) be the intermediate firm production function so that:

iν h 1/ρ , 1 < ηh + ηk < ρ , G(k, nh , nl ) = (k ηk nηhh )1/ρ + ηl nl

(30)

where nh and nl are the high and low-skilled labor employed by the firm. Under the assumption that ηh + ηk < ρ, capital is a complement to high-skilled labor, and an imperfect substitute for unskilled labor. Furthermore, the assumption that 1 < ηh + ηk implies that larger firms employ relatively more capital and high-skilled labor. The parameter ηl determines the relative productivity of unskilled compared to skilled labor, while returns to scale are governed by ν. The efficient allocation of firm resources R (net of any fixed export cost) to the factors of production solves max

k,nh ,nl

h iν 1/ρ ηk ηh 1/ρ (k nh ) + ηl nl

s.t. wl nl + wh nh + k ≤ R ,

12

(31)

and the factor demands are (implicitly) given by

(η +ηh −ρ)/(1−ρ)

wh nh (1 + ηk /ηh ) + wl ζnh k

=R

k = nh ηkηwh h

(32) (33)

(η +ηh −ρ)/(1−ρ)

nl = ζnh k

(34)

where  ζ=

ηh wl ηl wh

ρ 

ηk wh ηh

ηk 1/(1−ρ) .

(35)

The assumption that 1 < ηh + ηk < ρ implies that the ratio of unskilled to skilled labor decreases as the amount of resources R at the firm’s disposal increases. To see this, note that taking the total derivative of the demand for skilled labor (32) yields

dnh = dR

 −1 ηk +ηh −1 ηk ηk + ηh − ρ 1−ρ wh + wh + wl ζnh >0, ηh 1−ρ

(36)

which implies h −1 −1 dn d(nl /nh ) ηk + ηh − 1 ηk +η h 1−ρ =ζ nh <0, dR 1−ρ dR

(37)

where the last inequality holds because 1 < ηh + ηk . Under the assumption that high-skilled workers earn higher wages, we obtain the result that larger firms also pay higher wages on average. Lastly, larger firms have a higher capital to labor ratio, as the ratio of skilled labor to capital is constant. We can therefore conclude that in our model with the above specification of the production function, exporters will pay higher wages and employ more capital and skilled labor relative to unskilled labor than non-exporting firms.

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References David Hummels and Alexandre Skiba. Shipping the good apples out? an empirical confirmation of the alchian-allen conjecture. Journal of Political Economy, 112(6):1384–1402, December 2004. Alfonso A. Irarrazabal, Andreas Moxnes, and Luca David Opromolla. The tip of the iceberg: A quantitative framework for estimating trade costs. Working Papers w201125, Banco de Portugal, Economics and Research Department, 2011.

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