[PPT] - Economies of Scope and Trade Niklas Herzig Bielefeld University PowerPoint Presentation

SLIDE 1

Economies of Scope and Trade

Niklas Herzig

Bielefeld University

June 2016

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 1 / 53

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Overview

1

Literature overview

2

Eckel and Neary (2010)

3

Nocke and Yeaple (2014)

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 2 / 53

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Literature overview

Product symmetry Product asymmetry Products symmetric on both the demand and supply side Products asymmetric on the demand side Allanson and Montagna (IJIO 2005) Bernard, Redding and Schott (AER 2010, QJE 2011) Nocke and Yeaple (IER 2014) Products asymmetric on the supply (cost) side Arkolakis, Ganapati and Muendler (2015) Mayer, Melitz and Ottaviano (AER 2014) Cannibalization Ju (RIE 2003) Eckel and Neary (RES 2010) Feenstra and Ma (2008) Baldwin and Gu (2009) Eckel, Iacovone, Javorcik and Neary (JIE 2015) Dhingra (AER 2013) Qiu and Zhou (JIE 2013) Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 3 / 53

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Overview - Preview of results Scope adjustment to trade liberalization

classes of scope reactions: economy-wide reduction

Baldwin and Gu (2009) Eckel and Neary (2010) Mayer, Melitz and Ottaviano (2014)

ambiguous reaction

Bernard, Redding and Schott (2011)

heterogeneous reaction throughout the firm distribution

Dhingra (2013) Qiu and Zhou (2013) Nocke and Yeaple (2014)

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 4 / 53

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Eckel and Neary (2010)

Eckel, Carsten and Peter Neary (2010). Multi-Product Firms and Flexible Manufacturing in the Global Economy, Review of Economic Studies 77(1),

pp. 188-217.

Preferences and Demand two-tier utility function: U[u(z)] = 1 u(z)dz (1) with u(z) = a N q(i)di − 1 2b

(1 − e)

N q(i)2di + e N q(i)di 2 q(i): consumption of (horizontally diff.) product variety i, i ∈ [0, N] and N: measure of diff. varieties produced in each industry z, z ∈ [0, 1]

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 5 / 53

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Eckel and Neary (2010)

utility maximization problem: max

q(i)

U[u(z)] subject to 1 N p(i)q(i)didz ≤ I p(i): price of variety i and I: individual income FOC: inverse individual demand function: λp(i) = a − b

(1 − e)q(i) + e

N q(i)di

(2)

λ: Lagrange multiplier (consumer’s marginal utility of income) L (homogeneous) consumers in each of k identical countries, integrated goods markets and free trade (single variety price worldwide)

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 6 / 53

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Eckel and Neary (2010)

market demand for variety i: x(i) = kLq(i) inverse world market demand function: p(i) = a′ − b′[(1 − e)x(i) + eY ] (3) a′ ≡ a/λ, b′ ≡ b/λkL and Y ≡ N

0 x(i)di: industry output

Production and Supply “flexible manufacturing”technology (core competence): cj(i): marginal cost of firm j to produce variety i (independent of output, but different across products: c′

j > 0 and cj(0) = c0 j ; e.g. linear:

cj(i) = c0

j + γi)

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 7 / 53

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Eckel and Neary (2010)

Figure 1

i a′ − b′e(X + Y ) cj(0) cj(i) δ 2b′(1 − e)X

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 8 / 53

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Eckel and Neary (2010)

single-stage Cournot game profit maximization problem: max

xj(i)

πj = δj [pj(i) − cj(i)] xj(i)di − F δj: mass of products produced (scope) and F: fixed cost FOC: (i) scale ∂πj ∂xj(i) = pj(i) − cj(i) − b′ [(1 − e)xj(i) + eXj] = 0

proof

(4) Xj ≡ δj

0 xj(i)di: firm’s aggregate output

xj(i) = a′ − cj(i) − b′e (Xj + Y ) 2b′(1 − e) (5)

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 9 / 53

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Eckel and Neary (2010)

Figure 2

x(i) p(i) c(i) a′ − b′eY p(i) = a′ − b′[(1 − e)x(i) + eY ] a′ − b′e(X + Y )

MR(i) = a′ − b′[2(1 − e)x(i) + e(X + Y )]

x(i)

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 10 / 53

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Eckel and Neary (2010)

pj(i) = 1 2

a′ + cj(i) − b′e(Y − Xj)
(6)

(ii) scope ∂πj ∂δj = [pj(δj) − cj(δj)] xj(δj) = 0 (7) product range: output of the marginal variety (δj) zero: xj(δj) = 0 cj(δj) = a′ − b′e (Xj + Y ) (8) pj(δj) = a′ − b′eY labour productivity (LP) of multi-product firms: labour as the only factor of production and economy-wide and perfectly competitive labour market unit cost of producing each variety: c(i) = wγ(i)

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 11 / 53

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Eckel and Neary (2010)

total labour input: l = δ γ(i)x(i)di d ln LP d ln θ = δ

0 h(i) d x(i) d ln θ di

δ

0 h(i)x(i)di

− d ln l d ln θ (9) θ: any exogenous variable and h(i): weight of variety i x(i) = w [γ(δ) − γ(i)] 2b′(1 − e) l = wβ(δ) 2b′(1 − e) with β(δ) ≡ δ γ(i) [γ(δ) − γ(i)] di d ln LP d ln θ = ∂ ln LP ∂ ln δ d ln δ d ln θ

proof Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 12 / 53

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Eckel and Neary (2010)

choice of weights h(i): ∂ ln LP ∂ ln δ

h(i)=γ(i)

= δ

0 γ(i) ∂x(i) ∂ ln δ di

δ

0 γ(i)x(i)di

− ∂ ln l ∂ ln δ = 0

proof

Proposition 1: With given technology, any shock which raises the product range δ (a) leaves LP unchanged when output changes are marginal cost-weighted, (b) reduces LP when output is a simple aggregate

proof and

(c) reduces LP but by less when output changes are price-weighted

proof . Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 13 / 53

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Eckel and Neary (2010)

Industry Equilibrium symmetric Cournot oligopoly with an exogenously given number of firms m in each of k countries industry output: Y = kmX FOC for scope (rewrite (8)): wγ(δ) = a′ − e(1 + km)b′X ⇒ scope: δ(X) FOC for scale (integrate over (5)): X =

a′ − wµ′

γ

δ

△1 b′ with △1≡ 2(1−e)+eδ(1+km) > 0

proof ⇒ scale: X(δ)

with µ′

γ ≡ 1 δ

δ

0 γ(i)di

d ln X d ln δ = a′ − wγ(δ) − e(1 + km)b′X a′ − wµ′

γ

proof Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 14 / 53

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Eckel and Neary (2010)

Figure 3

X δ

s❝♦♣❡✿ δ(X) s❝❛❧❡✿ X(δ) Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 15 / 53

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Eckel and Neary (2010)

Effects of Globalization globalization: increase in the number of countries k participating in the global economy two channels: market-size effect (L ↑) competition effect (m ↑) Proposition 2: The market-size effect of an increase in k is an equi-proportionate increase in the output of each variety and of total

utput, but no change in firm scope.

Proposition 3: The competition effect of an increase in k is a uniform absolute fall in the output of each variety, coupled with falls in both total firm output and firm scope, but a rise in industry output.

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 16 / 53

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Eckel and Neary (2010)

Figure 4

i x(i)

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 17 / 53

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Eckel and Neary (2010)

full effect:

n firm output:

d ln X d ln k = 1 − eδkm △1

proof

(10) where △1 −eδkm =△0 (≡ 2(1 − e) + eδ) > 0

n variety output:

d ln x(i) d ln k = 1− ekmα(δ) △1 [γ(δ) − γ(i)] = △0 △1 +

1 − △0

△1 µ′

γ − γ(i)

γ(δ) − γ(i)

proof

(11)

γPE: labour requirement of the threshold variety whose output is

unchanged

γPE = △0

△1 γ(δ) +

1 − △0

△1

µ′

γ

proof Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 18 / 53

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Eckel and Neary (2010)

Proposition 4: The total effect of an increase in k is a rise in total output coupled with a fall in scope. Relatively high-cost varieties are discontinued

r produced in lower volumes, whereas more is produced of all varieties

with average costs or lower. → “leaner and meaner“-response of multi-product firms to globalization Corollary 1: Firm productivity is unaffected by the market-size effect, but rises with the competition effect of an increase in k.

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 19 / 53

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Eckel and Neary (2010)

Globalization and Product Variety number of varieties per firm δ ↓ + number of firms m ↑ → total variety effect? N = kmδ: total number of varieties produced in a symmetric equilibrium market-size effect: unaffected competition effect: conflicting effects (m ↑ and δ ↓) d ln N d ln k = 1 + d ln δ d ln k = 1 − eδkm △1 α(δ) δαδ Proposition 5: In partial equilibrium, an increase in the number of countries cannot lower the total number of varieties if the function relating costs to varieties has constant curvature, but it may do so if the technology is sufficiently flexible.

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 20 / 53

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Nocke and Yeaple (2014)

Nocke, Volker and Stephen Yeaple (2014). Globalization and Multiproduct Firms, International Economic Review 55(4), pp. 993-1018. Closed Economy discrete-time, infinite horizon model with a single (differentiated goods) sector and a single factor of production (labor) mass L of identical consumers (workers) with a per-period CES utility function: Us =

Ω

xs(ω)

σ−1 σ dω

σ

σ−1

(12) with xs(ω): consumption of product ω ∈ Ω in period s and σ > 1: elasticity of substitution in each period: single unit labor supply of each worker (ws ≡ 1) aggregate demand: Xs(ω) = Asps(ω)−σ (13)

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 21 / 53

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Nocke and Yeaple (2014)

with ps(ω): price of product ω in period s and As ≡ L

Ω ps(ω)−(σ−1)dω

market entry: irrecoverable setup cost F e (fraction F/F e ∈ (0, 1] used to build firm-specific capital equipment, remaining fraction 1 − F/F e spent

n intangibles)

upon entry: random draw of its (time-invariant) type ( θ, K) from G with support (0, 1/(σ − 1)) × [1, ∞)

θ: organizational efficiency

K: organizational capital size of product portfolio (scope): N products (for each product: irrecoverable one-time development cost f )

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 22 / 53

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Nocke and Yeaple (2014)

marginal cost of product ω: c(ω; kω; θ) =

zk−

θ ω

if kω ≥ 1 ∞

therwise

with z > 0: cost parameter and kω: amount of organizational capital allocated to product ω (

ω∈I kω ≤ K with I: set of products managed by

the firm) at the end of each period: probability of dying: 1 − β ∈ (0, 1) sequence in each period:

1 market entry decision 2 scope decision and organizational capital allocation decision 3 pricing decision Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 23 / 53

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Nocke and Yeaple (2014)

monotonic transformation: θ ≡ θ(σ − 1) (organizational efficiency) ֒ → firm type (θ, K) with distribution function G on support Θ ≡ (0, 1) × [1, ∞) pricing decision: p(ω; kω; θ) =

σ

σ − 1

c(ω; kω; θ)

scope decision: Lemma 1: A firm of type (θ, K) chooses to manage no more than K products, i.e., N(θ, K) ≤ K. Moreover, it allocates kω = K/N(θ, K) units

f organizational capital to each one of its N(θ, K) products.

proof

marginal cost of a firm of type (θ, K): c(θ, K) = z

K

N(θ, K) −

θ σ−1

(14)

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 24 / 53

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Nocke and Yeaple (2014)

profit-maximizing price: p(θ, K) =

σ

σ − 1

z
K

N(θ, K) −

θ σ−1

(15) per-period profits: π(θ, K) = N(θ, K)[p(θ, K) − c(θ, K)]Ap(θ, K)−σ = N(θ, K)(1 − β)f ζ

K

N(θ, K) θ (16) with ζ ≡

A σ(1−β)f

σ−1

σz

σ−1, A =

(1−β)L M[

Θ N(θ,K)p(θ,K)−(σ−1)dG(θ,K)] and M:

mass of entrants in each period firm’s market value: v(θ, K) = π(θ, K) 1 − β = N(θ, K)f ζ

K

N(θ, K) θ (17)

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 25 / 53

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Nocke and Yeaple (2014)

scope decision: max

N≥0 Nf ζ

K N θ − Nf (18) Proposition 1: In equilibrium, a firm of type (θ, K) chooses to manage N(θ, K) =

K

if θ ∈ (0, θ] K[(1 − θ)ζ]

1 θ

if θ ∈ [θ, 1) (19) products, where θ ≡ (ζ − 1)/ζ ∈ (0, 1).

proof

market entry decision:

Θ

ve(θ, K)dG(θ, K) − F e = 0 (20) with ve(θ, K) ≡ π(θ,K)

1−β − N(θ, K)f

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 26 / 53

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Nocke and Yeaple (2014)

labor market equilibrium: L = M 1 − β

Θ

N(θ, K)

Ap(θ, K)−σc(θ, K)
dG(θ, K)

+M

f
Θ

N(θ, K)dG(θ, K) + F e

L = σM
f
Θ

N(θ, K)dG(θ, K) + F e

(21)

equilibrium given by N(·, ·), p(·, ·), M, ζ satisfying (15), (16), (18), (19) und (20) Open economy decision adjustment: for each product: sales location decision (domestically

r domestically and abroad)

with exports: one-time irrecoverable cost f x and iceberg-type trade cost τ > 1

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 27 / 53

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Nocke and Yeaple (2014)

pricing decision: domestic price: p(ω; kω; θ) = (σ/(σ − 1))c(ω; kω; θ) price abroad: p∗(ω; kω; θ) = τp(ω; kω; θ) Lemma 2: A firm of type (θ, K) chooses to manage no more than K products, i.e., N(θ, K) ≤ K. Generically, it exports all of its products, denoted δx(θ, K) = 1, or none, δx(θ, K) = 0. In either case, the firm allocates the same amount kω = K/N(θ, K) of its organizational capital to each one of its N(θ, K) products.

proof

marginal cost: c(θ, K) = z

K

N(θ, K) −

θ σ−1

(22) price for the domestic and foreign market: p(θ, K) =

σ

σ − 1

c(θ, K) and p∗(θ, K) = τ
σ

σ − 1

c(θ, K)

(23)

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 28 / 53

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Nocke and Yeaple (2014)

scope, capital allocation and export status decision: max

N∈[0,K],δx∈{0,1} N

f ζ(1 + δxρ)

K N θ − (f + δxf x)

(24)

with ρ ≡ τ −(σ−1): measure of trade freeness assumption: ln(1 + f x/f ) ln(1 + ρ) > ζ > 1 (25) Proposition 2: In the equilibrium of the open economy, the export decision of a firm of type (θ, K) is given by δx(θ, K) =

if θ ∈ (0, θx)

1 if θ ∈ (θx, 1) (26) with θx ≡ 1 −

ln(1+ρ) ln(1+f x/f ) ∈ (θ, 1).

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 29 / 53

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Nocke and Yeaple (2014)

Proposition 2 (cont.): The firm’s equilibrium number of products is N(θ, K) =        K if θ ∈ (0, θ] K((1 − θ)ζ)

1 θ

if θ ∈ [θ, θx) K

1+ρ

1+f x/f

(1 − θ)ζ

1

θ

if θ ∈ (θx, 1).

proof

(27) per-period profit: π(θ, K) = N(θ, K)(1 − β)f ζ[1 + δx(θ, K)ρ]

K

N(θ, K) θ (28) market entry decision:

Θ

ve(θ, K)dG(θ, K) − F e = 0 (29) with ve(θ, K) = π(θ,K)

1−β − N(θ, K)(f + δx(θ, K)f x)

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 30 / 53

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Nocke and Yeaple (2014)

labor market equilibrium: L = AM 1 − β

σ

σ − 1 −σ

Θ

(1 + δx(θ, K)ρ)N(θ, K)c(θ, K)−(σ−1)dG(θ, K) +M

Θ

(f + δx(θ, K)f x)N(θ, K)dG(θ, K) + F e

= σM
Θ

(f + δx(θ; K)f x)N(θ, K)dG(θ, K) + F e

(30)

equilibrium given by N(·, ·), p(·, ·), p∗(·, ·), δx(·, ·), M, ζ satisfying (21), (22), (25), (26), (27), (28), (29) Effects of globalization reduction in the iceberg-type trade cost τ (increase in the trade freeness parameter ρ)

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 31 / 53

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Nocke and Yeaple (2014)

Lemma 3: Consider an increase in trade freeness from ρ to ρ

′ > ρ. This

lowers the effective market size facing non-exporters, i.e., ζ

′ < ζ, and raises

the effective market size facing exporters, i.e., ζ

′(1 + ρ ′) > ζ(1 + ρ). proof

Proposition 3: Consider an increase in trade freeness from ρ to ρ

′ > ρ.

This induces the thresholds for exporting and for maximal diversification to fall: θx′ < θx and θ

′ < θ.

Proposition 4: Consider an increase in trade freeness from ρ to ρ

′ > ρ.

This causes firms that initially sold only domestically to drop products, i.e., N(θ, K)

′ ≤ N(θ, K) for all θ ∈ (0, θx), with a strict inequality if

θ ∈ (θ

′, θx), and all continuing exporters to increase the number of

products they manage, i.e., N(θ, K)

′ > N(θ, K) for all θ ∈ (θx, 1). proof Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 32 / 53

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Nocke and Yeaple (2014)

Corollary 1: Consider an increase in trade freeness from ρ to ρ

′ > ρ. For

firms that initially sold only domestically, this results in higher TFP, i.e., c(θ, K)

′ ≤ c(θ, K) for all θ ∈ (0, θx), with a strict inequality if θ ∈ (θ ′, θx).

For continuing exporters, this results in lower TFP, i.e., c(θ, K)

′ > c(θ, K)

for all θ ∈ (θx, 1).

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 33 / 53

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Appendix

ften used terms:

α(δ) = δ

γ(δ) − µ′

γ

β(δ) = δ
γ(δ)µ′

γ − µ′′ γ

= α(δ)µ′

δ − δσ2 γ

αδ = δγδ βδ = µ′

γαδ

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 34 / 53

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Appendix

∂πj ∂xj(i) = pj(i) − cj(i) + δj ∂pj(i∗) ∂xj(i) xj(i∗)di∗ = 0 i = i∗ : ∂pj(i∗) ∂xj(i) = −b′ and i = i∗ : ∂pj(i∗) ∂xj(i) = −b′e ∂πj ∂xj(i) = pj(i) − cj(i) − b′ [(1 − e)xj(i) + eXj] = 0, Xj ≡ δj xj(i∗)di∗ (4)

back Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 35 / 53

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Appendix

d ln LP d ln θ = ∂ ln LP ∂ ln θ + ∂ ln LP ∂ ln δ d ln δ d ln θ l = ψ(θ)β(δ) and ∂ ln l ∂ ln θ = ∂ ln l ∂θ ∂θ ∂ ln θ = ψ′ ψ θ x = ψ(θ) [γ(δ) − γ(i)] and ∂x ∂ ln θ = ∂x ∂θ ∂θ ∂ ln θ = [γ(δ) − γ(i)] ψ′θ δ

0 h(i) ∂x ∂ ln θdi

δ

0 h(i)x(i)di

= δ

0 h(i)ψ′(θ)θ [γ(δ) − γ(i)] di

δ

0 h(i)ψ(θ) [γ(δ) − γ(i)] di

= ψ′ ψ θ d ln LP d ln θ = ∂ ln LP ∂ ln δ d ln δ d ln θ ∂ ln LP ∂ ln θ = 0

back

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 36 / 53

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Appendix

∂ ln LP ∂ ln δ

h(i)=γ(i)

= δ

0 γ(i) ∂x(i) ∂ ln δ di

δ

0 γ(i)x(i)di

− ∂ ln l ∂ ln δ = 1 l δ γ(i)∂x(i) ∂ ln δ di − ∂ ln l ∂ ln δ = 1 l δ ∂γ(i)x(i) ∂ ln δ di − ∂ ln l ∂ ln δ = δ ∂ ln l ∂l ∂γ(i)x(i) ∂ ln δ di − ∂ ln l ∂ ln δ = ∂ ln l ∂ ln δ δ ∂γ(i)x(i) ∂l di − ∂ ln l ∂ ln δ = 0

back Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 37 / 53

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Appendix

∂ ln LP ∂ ln δ

h(i)=1

= ∂ ln X ∂ ln δ − ∂ ln l ∂ ln δ = ∂ ln α(δ) ∂ ln δ − ∂ ln β(δ) ∂ ln δ (since X = δ w [γ(δ) − γ(i)] 2b′(1 − e) di = w 2b′(1 − e)δ

γ(δ) − µ′

γ

=

wα(δ) 2b′(1 − e)) = δαδ α(δ) − δβδ β(δ) = − δ2αδσ2

γ

α(δ)β(δ) < 0

back Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 38 / 53

SLIDE 39

Appendix

∂ ln LP ∂ ln δ

h(i)=p(i)

= δ

0 p(i) ∂x(i) ∂ ln δ di

δ

0 p(i)x(i)di

− ∂ ln l ∂ ln δ p(i)x(i) = 1 2

a′ + wγ(i) − b′e(Y − X)
w

2b′(1 − e) [γ(δ) − γ(i)] = 1 2

wγ(i) + wγ(δ) + 2b′eX
w

2b′(1 − e) [γ(δ) − γ(i)] = w 1 2 (γ(i) + γ(δ)) + e α(δ) 2(1 − e)

w

2b′(1 − e) [γ(δ) − γ(i)]

back Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 39 / 53

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Appendix

X = δ a′ − wγ(i) − b′e(X + Y ) 2b′(1 − e) di = 1 2b′(1 − e) δ (a′ − wγ(i) − b′e(1 + km)X)di X

1 + b′eδ(1 + km)

2b′(1 − e)

=

1 2b′(1 − e) δ (a′ − wγ(i))di X 2b′(1 − e) + b′eδ(1 + km) 2b′(1 − e)

=

δ 2b′(1 − e)

a′ − wµ′

γ

X =

δ b′(2(1 − e) + eδ(1 + km))

a′ − wµ′

γ

back

Niklas Herzig (Bielefeld University) Economies of Scope and Trade June 2016 40 / 53

SLIDE 41

Appendix

ln X = ln

a′ − wµ′

γ

δ
− ln
(2(1 − e) + eδ(1 + km)) b′

µ′

γ = 1

δ δ γ(i)di d dδµ′

γ = γ(δ)δ −

δ

0 γ(i)di

δ2 = γ(δ) − µ′

γ

δ d ln X d ln δ = δ

a′ − wµ′

γ

δ
a′ − wµ′

γ

+ δ
−w γ(δ) − µ′

γ

δ

−

δ (2(1 − e) + eδ(1 + km))b′

eb′(1 + km)
d ln X

d ln δ = a′ − wγ(δ) a′ − wµ′

γ

− eb′(1 + km)X a′ − wµ′

γ

= a′ − wγ(δ) − eb′(1 + km)X a′ − wµ′

γ

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SLIDE 42

Appendix - Industry Equilibrium Comparative Statics

△1

e(1 + km)

2(1−e)δγδ α(δ)

d ln X d ln δ

=
△1

e(1 + km)

d ln L

−ekm δ 1

d ln m +

△0 e

d ln k −

δµ′

γ

γ(δ) 2(1 − e) α(δ) d ln w d ln X = d ln L − eδkm △1 d ln m + △0 △1 d ln k − 2(1 − e)δµ′

γ

△1 α(δ) d ln w d ln δ = −eδkmα(δ) △1 δαδ (d ln m + d ln k) − 2(1 − e)δµ′

δ+ △1 α(δ)

△1 δαδ d ln w

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SLIDE 43

Appendix - Industry Equilibrium Comparative Statics

d ln x(i) = d ln L − ekmα(δ) △1 [γ(δ) − γ(i)]d ln m + △0 △1 +

1 − △0

△1

µ′

γ − γ(i)

γ(δ) − γ(i)

d ln k − 2(1 − e)γ(i) − eδ(1 + km)
µ′

γ − γ(i)

△1 [γ(δ) − γ(i)]

d ln w

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SLIDE 44

Appendix - Industry Equilibrium Comparative Statics

= △0 △1 +

1 − △0

△1 µ′

γ − γ(i)

γ(δ) − γ(i) (γ(i) − γ(δ)) △0 △1 =

1 − △0

△1 µ′

γ − γ(i)

γPE = γ(i)

= △0 △1 γ(δ) +

1 − △0

△1

µ′

γ

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SLIDE 45

Appendix

Having incurred development cost for N = #I products: max

{kω}ω∈I

1 1 − β

ω∈I+≡{ω∈I|kω≥1}

(p(ω, kω, θ) − c(ω, kω, θ)) x(ω, kω, θ) = A (1 − β)σ σ − 1 σz σ−1

ω∈I+≡{ω∈I|kω≥1}

(kω)θ subject to

ω∈I kω ≤ K

bjective function increasing and concave in the kω’s → full capital

endowment exhaustion and choice of either kω = k ≥ 1 or kω = 0 (latter not a solution!) → N(θ, K) ≤ K and kω = K/N(θ, K)

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SLIDE 46

Appendix

max

N≥0

Nf ζ K N θ − Nf FOC: f ζ K N θ + Nf ζθ K N θ−1 − K N2

− f

!

= 0 f ζ K N θ (1 − θ) = f (1 − θ)f ζK θ N(θ, K)−θ = f

N(θ, K) = K [(1 − θ)ζ]

1 θ

N(θ, K) > 0 for all (θ, K) ∈ Θ and

N(θ, K) ≤ K N(θ, K) = min{K, N(θ, K)}

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SLIDE 47

Appendix (cont.)

N(θ, K) strictly decreasing in θ:

∂ N(θ, K) ∂θ 1 K = − ζ θ2 [(1 − θ)ζ]

1−θ θ (θ + (1 − θ) ln((1 − θ)ζ))

≡Ψ(θ)

∂ N(θ, K) ∂θ < 0 ⇔ Ψ(θ) > 0 Ψ(0) = ln(ζ) > 0 as ζ > 1 by assumption Ψ

′(θ) = − ln((1 − θ)ζ) and Ψ ′′(θ) =

1 1 − θ > 0 unique minimum: Ψ

′(θm) = 0 ⇔ θm = 1 − 1

ζ = ζ−1 ζ

with Ψ(θm) = θm > 0

N(θ, K) = K

⇔ θ = ζ − 1 ζ

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SLIDE 48

Appendix

Having incurred development cost and given exporting decision: max

{kω}ω∈I

ζf

ω∈I+≡{ω∈I|kω≥1}

[1 + χ(ω)ρ](kω)θ subject to

ω∈I kω ≤ K (χ(ω) = 1 if ω is exported, χ(ω) = 0 otherwise)

bjective function increasing and concave in kω’s → full capital

endowment exhaustion and choice of kω ∈ {0, kx} with kx ≥ 1 if χ(ω) = 1 and kω ∈ {0, kd} with kd ≥ 1 if χ(ω) = 0 (allocation of zero

rganizational capital not optimal!) → N(θ, K) ≤ K

Lagrangian: L = f ζN[(1 − δ)(kd)θ + δ(1 + ρ)(kx)θ] − λN

(1 − δ)kd + δkx − K

N

with δ as the share of exported products and λ as the Lagrange multiplier
n the firm’s organizational capital constraint → Lagrangian linear in δ →

δ ∈ {0, 1}

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SLIDE 49

Appendix

δx = 0: (24) simplifies to (18): Nd(θ, K) = min{K, K[(1 − θ)ζ]

1 θ }

δx = 1: max

N∈[0,K]

N

f ζ(1 + ρ)

K N θ − (f + f x)

FOC:
f ζ(1 + ρ)

K N θ − (f + f x)

− θf ζ(1 + ρ)

K N θ

!

= 0 Nx(θ, K) = min{K, K

1 + ρ

1 + f x/f

(1 − θ)ζ

1

θ

}

ptimal choice: δx(θ, K) = 0 if vd(θ, K) > vx(θ, K) and δx(θ, K) = 1 if

vd(θ, K) < vx(θ, K) vd(θ, K) = max{Kf [ζ − 1], Kf [(1 − θ)ζ]1/θ(θ/(1 − θ))}

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SLIDE 50

Appendix (cont.)

vx(θ, K) = Kf

1 + ρ

1 + f x/f

(1 − θ)ζ

1

θ

1 + f x f θ 1 − θ

vx(θ, K) > Kf [(1 − θ)ζ]1/θ(θ/(1 − θ)) ⇔
1 + ρ

1 + f x/f 1

θ

(1 + f x/f ) > 1

r

θ > θx ≡ 1 − ln(1 + ρ) ln(1 + f x/f ) > θ Kf [(1 − θ)ζ]1/θ(θ/(1 − θ)) > Kf [ζ − 1] ⇔ θ > θ vx(θ, K) > vd(θ, K) ⇔ θ > θx

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SLIDE 51

Appendix

Inserting π(θ, K) and N(θ, K) into (29): f ∞

1

K

(ζ − 1)

θ g(θ, K)dθ + θx

θ

θ

1 − θ

[(1 − θ)ζ]

1 θ g(θ, K)dθ

+ 1

θx

θ(1 + f x/f ) 1 − θ 1 + ρ 1 + f x/f

(1 − θ)ζ

1

θ

g(θ, K)dθ

dK − F e = 0

dζ ζ f ∞

1

K θ ζg(θ, K)dθ + θx

θ

1

1 − θ

[(1 − θ)ζ]

1 θ g(θ, K)dθ

+ 1

θx

1 + f x/f 1 − θ 1 + ρ 1 + f x/f

(1 − θ)ζ

1

θ

g(θ, K)dθ

dK

+ dρ (1 + ρ)f ∞

1

K 1

θx

1 + f x/f 1 − θ 1 + ρ 1 + f x/f

(1 − θ)ζ

1

θ

g(θ, K)dθdK = 0

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SLIDE 52

Appendix

dζdρ < 0 and ζ

′ < ζ

suppose: ζ(1 + ρ) decreasing as well → ζ

′(1 + ρ ′) ≤ ζ(1 + ρ), then LHS of

(29) negative after trade liberalization (contradiction!) → ζ

′(1 + ρ ′) >

ζ(1 + ρ)

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SLIDE 53

Appendix

firm (θ, K) with θ ∈ (0, θ

′]: N(θ, K) ′ = N(θ, K) = K if θ ∈ (0, θ ′] and

N(θ, K)

′ < N(θ, K) = K if θ ∈ (θ ′, θ]

firm (θ, K) with θ ∈ (θ, θx′) ∪ (θx, 1): N(θ, K)

′ < N(θ, K) if θ ∈ (θ, θx′)

and N(θ, K)

′ > N(θ, K) if θ ∈ (θx, 1)

firm (θ, K) with θ ∈ (θx′, θx): N(θ, K)

′

N(θ, K) =

ζ

′

ζ 1

θ

1 + ρ

′

1 + f X/f 1

θ

< 1

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