Openness, Managerial Incentives and Heterogeneous Firms

Zhihong Yu1

GEP, University of Nottingham

June, 2009

Abstract This paper examines the effects of trade openness on the optimal incentives firms provide to

their managers (and hence firm-level productivity) in the new heterogeneous-firm trade framework.

Export market selection and firm heterogeneity are shown to be the key determinants of firm

owners’ optimal choices of the incentive contracts, which affect the cost-reduction efforts of their

managers. I show that openness to trade generally has a positive impact on the equilibrium power of

the incentive contract. This is due to the carrot-and-stick effect resulting from the Melitz-type inter-

firm reallocation towards ex post lowest-cost exporters, which unambiguously increases the

variation of firm profits and thus the value of cost-reduction. Interestingly, whilst falling variable

trade cost unambiguously increases managerial incentives, a reduction in fixed trade costs has

ambiguous effects on managerial incentives, and might even generate within-firm productivity

losses due to the adverse inter-firm reallocation effects. Hence, the model establishes a causal link

between the trade-induced Melitz-type reallocation effect and the within-firm productivity changes,

both of which have been shown to be important sources of aggregate productivity gains by recent

empirical evidence.

JEL Classification: F12, F13 Keywords: Openness; Trade costs; Managerial Incentives; Heterogeneous Firms, Productivity

1 We thank Spiros Bougheas, Carl Davidson, Rod Falvey, Arijit Mukherji , Marc Melitz, Hitoshi Sato, and participants at the seminar of Michigan State University , Midwest International Economics Group Meeting Fall 2007 at Michigan University at Ann Arbor, and Annual Conference of Royal Economic Society 2008 at Warwick University for helpful comments on the earlier version of this paper. Correspondence: Zhihong Yu, C4, Sir Clive Granger Building, Nottingham University, University Park, Nottingham, NG72RD, United Kingdom. Email: zhihong.yu@nottingham.ac.uk

1

1. Introduction

The relationship between openness and productivity might be one of the most fundamental

issues in international economic studies. Although it is often argued by policy makers and

business leaders that openness could enhance productivity, there has been little consensus on

the precise mechanisms underlying this relation. Recently, a rich body of micro-level empirical

trade study has shed some new light on this issue by identifying a robust link between firm

heterogeneity and international trade. One new empirical regularity emerging from the data is

that firms exhibit substantial differences in their productivity levels even within very narrowly

defined industries,2 and such heterogeneity is important to international trade. Specifically,

exporting firms are more productive, larger and more skill intensive than non-exporters, and

such performance gaps are mainly due to their self-selection into the export market, rather than

learning-by-exporting. (See, e.g., the pioneering work by Bernard and Jensen 1999 and Clerides

et al. 1998). On the other hand, the literature has also provided strong evidence on firm level

productivity gains due to increasing exposure to trade (see, e.g., Pavnick 2002; Bernard, Jensen

and Schott, 2006). In particular, after analyzing the US plants’ responses to falling trade costs,

Bernard, Jensen and Schott (2006) conclude: “We find that greater exposure to international

trade via declining trade costs promotes productivity gains at three levels: across industries

within manufacturing, across plants within industries, and within plants … we provide the first

comprehensive evidence of a relationship between trade liberalization and productivity

growth within plants in a developed economy…” (p. 918 and p. 934, emphasis added).

Whilst the intra-industry across plant/firm reallocation effects of trade− driven by firms’

self selection into the export market − has been explored extensively in the heterogeneous firm

trade models pioneered by Melitz (2003),3 yet, it is much less understood why increasing

openness could boost within-firm productivity growth, which is perhaps an equally important

channel as the reallocation effect via which trade increases aggregate productivity. One

seemingly obvious explanation from the industrial organisation literature is the “innovation

upon competition” effect: if increasing product market competition unambiguously stimulates

firms’ incentives to find cost-reducing innovation, then falling trade barriers that lead to fiercer

competition from foreign rivals should raise home firms’ productive efficiency. However, it is

2 See e.g. Bernard, Eaton, Jensen and Kortum (2003) who show the dramatic differences in labour productivity across US plants within the same four digit industry. 3 Melitz (2003) shows that opening up to trade leads to aggregate productivity gains by reallocating market shares towards more productive firms that export and expand, away from the least productive firms that remain purely domestic and shrink or exit. Bernard, Redding and Schott (2006) incorporate heterogeneous firms into Heckcher-Ohlin framework and show opening up to trade leads to both across industry and within industry reallocation of resources. See also Bernard et al. (2003) ,Yeaple (2004) , Baldwin and Forslid (2004) and Jørgensen and Schröder (2008) as important contributions to this literature.

2

well-known from the theoretical IO literature that the effects of competition on firm’s internal

(in)efficiency is ambiguous (see, e.g., Holmstrom, 1982; Nalebuff and Stiglitz, 1983, Hart,

1983; Scharfstein, 1988; Schmidt, 1997; Raith, 2003), and none of these models explicitly

examines the effects of trade openness. Further, as the current wave of globalization is

increasingly driven by multilateral reduction in trade costs, the import competition argument

may not capture the full effects of openness when an economy is increasingly opened up to

both import competition and export opportunities.4

The purpose of this paper is to provide a new mechanism for understanding the link between

trade openness and firm productivity, in terms of the interaction between firm heterogeneity

and managerial incentive contracts. Motivated by recent findings that managerial incentives and

behavior could be an important determinant of firm productivity, I incorporate the managerial

incentive mechanism into the new heterogeneous firm trade framework. 5 One salient feature of

the model is that the Melitz-type inter-firm reallocation effect becomes the key determinant of

the trade-induced changes in managerial incentives (and hence firm productivity). This is due to

the “carrot and stick” effect: while trade openness induces more competition and therefore

penalizes high-cost firms by reducing their profits (the stick), it could reward some lowest-cost

firms with the opportunity to increase their sales in the foreign markets and therefore earn

greater profits (the carrot). One important consequence is that the variation of firm profits

becomes larger, and the firm owners face higher stakes in a more open economy, even if the

variance of the underlying cost uncertainty remains unchanged. As a result, the value of the

managers’ cost-reduction efforts become greater, and firm owners are willing to transfer more

rents to their managers, giving them stronger incentives to work harder and boost firm

productivity. However, it is shown that the carrot and stick effect disappears when openness

induces all firms to export, or may even work in the “wrong” direction and possibly generate

productivity losses when increasing openness is driven by falling fixed trade costs. Hence, a

causal link between the effects of trade on inter-firm reallocation and within-firm productivity

is established, which crucially depends on the assumption of firm heterogeneity and export

market selection.

Based on Melitz (2003), Raith (2003) and Schmidt (1997), I build a monopolistic

competition model where risk-neutral firm owners (principals) provide incentive contracts to

the wealth-constraint managers (agents) to motivate them exert efforts on cost-reduction

activities. Due to the moral hazard problem, the contract is contingent on the ex post marginal

4 Another often-cited hypothesis is learning-by-exporting, but the empirical evidence on this is notoriously weak and ambiguous. (see, e.g., Clerides et. al., 1998; Bernard and Jensen, 1999). 5 Bandiera, Barankay and Rasul (2007) show from their firm-level field experiment that stronger managerial incentives raises labour productivity. Also, see Bloom and Reenen (2007) who provide evidence for a positive association between management practice and firm productivity.

3

cost of the firm that is ex ante uncertain. If the managers are paid positive rents under the

optimal incentive scheme due to the wealth-constraint, managerial effort and firm productivity

can then be derived as functions of the market fundamentals (entry costs, product

differentiation) and, most importantly, the degree of openness of the economy. Opening up to

trade unambiguously reduces firms’ domestic profits as it introduces foreign competition, but

also provides firms with new opportunities of earning extra export profits as foreign markets

are opened up. In a Melitz-type open economy where only ex post low-cost firms self-select

into the export market, the well-known Melitz-type reallocation effect takes place: the profits of

ex post lowest (high)-cost firms that become exporters (non-exporters) see an increase

(decrease) in their total profits. This has an important second order effect: the profit differential

between ex post low and high cost levels becomes unambiguously greater, and thus the value of

managerial effort increases as cost-reduction now generates more profit gains. As a result, firm

owners provide stronger incentives to their managers to stimulate them work harder, which

reduces managerial slack and generates within-firm productivity gains. By contrast, it is shown

that in a Krugman-type open economy where all firms export, trade openness does not induce

such Melitz-type reallocation in favor of ex post low-cost firms, and thus the carrot-and-stick

effect is not at work, leaving the value of cost reduction and managerial incentives unaffected.

Such contrast highlights the vital role of firm heterogeneity and export market selection in

generating the positive effects of trade on managerial incentive and firm productivity in our

model.

Another interesting contrast arises when we compare the effects of falling variable trade

costs with that of a reduction in fixed trade costs. Whilst falling variable trade costs (such as

reciprocal reduction in tariff) do unambiguously increase managerial incentives due to the

carrot-and-stick effect, the effect of falling fixed trade costs (such as reductions in regulatory

trade barriers) on managerial incentives is actually ambiguous because of the ambiguous inter-

firm reallocation effect: a decrease in the fixed trade costs actually lowers the profits for ex post

lowest-cost firms, but increases the profits of ex post medium-cost firms. Interestingly, when

falling fixed trade costs only induces a small increase in the new entry of the medium-cost

firms into the export market, an adverse inter-firm reallocation effect occurs, which reallocates

profits to ex post high-cost firms, leading to a decrease in the value of cost reduction. As a

result, in this case firms provide weaker incentives to their managers, generating within-firm

productivity losses.

Note that compared to the previous literature, the Melitz-type inter-firm reallocation driven

changes in managerial incentives is a relatively new channel via which trade may affect firm

level productivity. In Horn, Lang, Lundgren (1995)’s pioneering work on international trade

and managerial incentives, trade boosts managerial incentives through very different

4

mechanisms. In their oligopolistic competition model, opening up to free trade increases total

outputs for all firms, which raises the demand for labour, and thus rising labour cost. Since a

firm’s incentive to invest in cost-reductions increases with its level of output as well as the

labour costs , managerial incentives increase when the economy moves from autarky to free

trade. More recently, Trindade (2005) abstracts from the principal-agent mechanism, and

focuses on the leisure-income trade-off decisions by firm owners. He reveals that opening up to

free trade stimulates owner-managers to work harder by reducing the real price of consumption

basket and , therefore, increasing the real reward of their efforts. Note that both papers assume

away the existence of trade costs in the open economy, so export market selection and inter-

firm reallocation play little role in the trade-induced firm level productivity gains in their

models (since opening up to costless trade leads all firms to export). In a related study,

Davidson, Matusz and Shevchenko (2008) do take into account of the importance of export

selection, but stress the role of search-generated unemployment and labour heterogeneity. They

build a model with perfectly competitive product market and imperfect labor market with

heterogeneous workers, and reveal that openness could lead to firm productivity gains/losses

due to changing skill mix within firms that differ in their technology choices.

This paper might also provide an analytical mechanism to understand the insight from

Costantini and Melitz (2007) that the non-technical aspects of trade liberalisation, such as the

“timing of announcement”, could also have important implications for firm level adjustments to

increasing openness. In our model, since a firm’s managerial incentive depends on the

principal’s expectation of the trade costs, and has to be set before a firm’s export and

production decisions are made, an “announcement” of trade liberalization could lead to greater

managerial incentives, and thus productivity gains, even it is (unexpectedly) not implemented

later in practice. This is in contrast to the original Melitz model where the inter-firm

reallocation of firms’ outputs, and the resulting aggregate productivity gains, can only be

realized until the trade costs are de facto reduced (firms cannot increase their exports to the

foreign markets unless the trade cost are practically lowered).

The remainder of the paper is divided into four sections. In the next section, we introduce

the model in the closed economy. Section 3 discusses the open economy model and the impacts

of trade, followed by an analysis on the effects of falling trade costs in section 4. Some

concluding remarks are made in Section 5. For expositional convenience, most of the details of

the proofs are shown in the Appendix.

2. The Closed Economy

The model builds upon the workhorse Krugman’s (1979, 1980) model of intra-industry trade

in the presence of monopolistic competition and increasing return to scales, yet incorporating

across firm heterogeneity as in Melitz (2003) and the within firm principal-agent problem as in

Schmidt (1997) and Raith (2003). In particular, we assume in a single industry each firm i

consists of a principal (owner) who makes entry, pricing and contract decisions, and an agent

(manager) who can exert effort to influence the firm productivity (marginal cost).

We start with considering the case of the closed economy, where the timing/sequencing of

firm behaviors is set as follows. During the (entry) stage I, prospective firm owners decide

whether to enter the industry or not. Upon entry, she has to pay a sunk cost of setting up a plant

and then hire a manager to organize the production. This is followed by the (contract) stage II,

where each owner offers the manager an incentive contract to motivate him exert greater effort.

Finally, at the (production) stage III, the firm level marginal production costs, which depend on

the managerial efforts, will be realized and publically observable. Based on their costs, firms

make decisions on their prices and outputs to maximize profits. Products are sold and payments

to the owners and the managers are received. Next we show the details of each stage and solve

the equilibrium by backward induction.

2.1 Demand and production

The preferences of a representative consumer are assumed to be the well-known C.E.S

(Constant Elasticity of Substitution) form over a continuum of varieties indexed by i:

ρρ ⎟⎠⎞⎜

⎝⎛= ∫ Θ∈i i diqU

1

, where 10 << ρ . The measure of the set Θ represents the mass of the

varieties available to consumers. The elasticity of substitution between varieties is assumed to

be constant given by 1>ρ

UQ ≡

11−

=σ . Consumer behavior can be modeled as if they were

consuming an aggregate product with aggregate price

σσ −

Θ∈

− ⎟⎠⎞⎜

⎝⎛= ∫

11

1

i i dipP (1)

This yields a constant elasticity of demand function for each variety:

σ

σ

−

−

⋅=⎟⎠⎞

⎜⎝⎛= i

ii pA

PpQq (2)

5

where 1

EAP σ−≡ is the demand shifter common for each individual variety i, and E is the

aggregate spending.

Production requires a single factor (labor) that is inelastically supplied at L. Technology of

production is characterized by a cost function with constant marginal cost and fixed entry cost:

, where is the total costs, and are constant marginal cost and fixed

entry cost in labor units, respectively, and w is the common wage rate hereafter normalized to

one. It is well-known that under the Dixit-Stiglitz monopolistic competition, the pricing rule by

a profit maximizing firm will be a constant mark-up over its marginal cost:

(i i i EC a q f= + )w iC ia Ef

ρi

iap = . Firms’

market profits and revenues are then given by:

Eiiiiii faBCqpa −⋅=−= −σπ 1)( , (3) σσ −⋅⋅== 1)( iiiii aBqpar

where

( ) ( )1 11 1 1B A Eσ σρ ρ ρ ρ− −≡ − = − ⋅Pσ−

a

(4)

B is a transformed demand shifter that is exogenous from the point of view of individual firms.

It can also be regarded as an inverse measure of the “degree of market competition”: lower B

corresponds to a lower aggregate price (P), leading to a lower firm-level profit, reflecting a

greater degree of product market competition.

2.2 The optimal incentive contract

In the previous section, we showed a firm’s price and production decisions at stage III,

taking its marginal cost ( ) as given. Now we turn to the details of stage II, in which firm-

level marginal costs are determined endogenously. We assume that at the beginning of this

stage, each owner hires an identical manager, whose main task is to exert effort e to organize

the production, and thus determine the costs. Furthermore, the outcome of is uncertain

because cost is also affected by a random influence

i

i

ie

iα ：

Assumption 1 ( )

ii

i

aeαγ

= , where [0, )ie ∈ ∞ , '( ) 0eγ > , (0) 0γ = , lim ( )e

eγ→∞

= ∞ and iα is

an i.i.d. random variable with cumulative distribution )(αF with support [0 . , )∞

6

This assumption implies that has c.d.f. ia [ ]( ; ) ( )iG a e F a eγ= i with support . The

(unconditional) expected marginal cost is

[0, )∞

( )( )i

a E aeα

γ≡ = , where ( )Eα α≡ is the expected

value of α . Increasing managerial effort ( ) therefore reduces firm-level cost on average (ie a

decreasing in ), and improves firm productivity ie ( )1 ii

i i

ea

γϕα

= = in terms of first order

stochastic dominance. In the following analysis , for expositional convenience we adopt

( )e eγ = 。

Both the owner and the manager are risk neutral. The owner’s payoff function is

iii W−=Π π ， where denotes the wage payment to the manager. The manager‘s utility

function is

iW

( )M iU W M ei= − ,where ( )iM e

) 0

is the manager’s disutility, with

, and M M'( ) 0,iM e M ''( ) 0e> >i (0) '(0= = . We assume ei is unobservable, so the

manager may get slack, not exerting his maximum effort. To solve this moral hazard problem,

the owner offers the manager a compensation scheme contingent on the realized cost .

Specifically, we assume the following (linear) payment scheme:

ia

( / )i i i iW s b aα= + (5)

where is fixed salary, is the “piece rate” representing the power of the incentive contract,

and the total bonus

is ib

( /i )iab α increases with ex post firm productivity (1 ./ )ia 6 The manager’s

expected compensation is therefore

( )i i iW E W s b e≡ = + i i (6)

The owner’s net expected payoff ( )i iE Wπ iΠ = − , where iπ represents the expected market

profits. Using (3) , we obtain ( firm index i is suppressed hereafter):

0( ) ( ; ) ( ) - Ea d G a e B V e fπ π

∞= =∫ (7)

where , and 1 1

0( ) ( ; )V e a dG a e eσ σ∞ − −≡ =∫ Ω 1

0( )dFσα α

∞ −∫Ω = . 7 Clearly, '( ) 0eπ > i.e. the

(expected) market profit increases with managerial effort. The owner’s problem is to make the

7

6 As it is well-known that the optimal form of incentive contract is very complicated even in very simple settings, in this paper the linearity of the incentive contract is assumed just for the sake of simplicity and tractability of the model. Such linear contract has been adopted by various previous studies, see e.g. Raith (2003) and Grossman and Helpman (2004).

8

{optimal contractual choice },s b that maximizes her net expected payoff:

{ } { }, , , ,

max ( ) max ( ) Es b e s b eE W BV e s beπΠ = − = − − − f (8)

Subject to:

{ }arg max ( ) arg max ( )e e

e W M e s be M e⎡ ⎤∈ − = + −⎣ ⎦(IC) (9)

(PC) ( ) ( ) rW M e s be M e U− = + − ≥ (10)

(WC) (11)

The

0≥W

incentive-compatible condition (IC) gives the optimal managerial effort for given

compensation scheme { }sb, . This condition can be replaced with the following first order

condition:

(IC’) (12)

, the manager will exert greater effort (

'( )b M e=

Clearly e), the larger the piece rate (b). Hence, the total

expected bonus (incentive pay) paid to the manager ( ) ( ) '( )D e b e e eM e≡ = also increases

with e . Further, the participation constraint (PC) gives th tion acceptable

by the manager: he will accept any contract with pair

e minimum compensa

{ }sb, that yields an expected utility of at

least his outside option which is denoted as 0rU ≥ . Mo er, the wealth constraint (WC) says

that the manager’s payment is non-negative state of nature. In the absence of (WC), it is

well known that, since the manager is risk neutral, it is optimal for the owner to “sell the store”

to the manager for a lump sum payment, and makes the manager residual claimant on profits.

This possibility is ruled out by assuming the manager has no wealth.

We impose the following assumption to ensure that the owner’s

reov

for any

optimization problem is

Assumption 2

globally concave:

''( ) ''( ) ''( )min ,'( ) '( ) '( )

M e D e V eM e D e V e

⎧ ⎫>⎨ ⎬

⎩ ⎭

7 It is noteworthy that under the present setup all entrants produce after entry, so the present model abstracts from the “rationalization effects” of trade on aggregate productivity via the exit of unproductive firms. In Melitz (2003), due to the existence of fixed production costs, some high-cost firms do not produce and immediately exit after entry. In our model , the reason for shutting down such “firm exit” effects is to sharply focus on what is novel here, namely the productivity effects of trade via changes in within-firm productivity, instead of the exit of unproductive firm. Allowing firm exit after entry will add complications to the model without altering the main insights and qualitative results of our model.

This assumption says that the manager’s disutility function is sufficiently steep , which

guarantees that problem (8) has a unique internal solution.8

To understand the source of potential contractual friction, next we derive the first best effort

level that could be achieved as if effort were contractible: Fe

{ } '( )arg max ( ) ( ) '( )

FF

e F

M ee e M e BV e

π= − ⇒ = (13)

Note that assumption 2 ensures the unique solution to the first best problem. The manager is

paid if and only if he chooses rFF UeMW += )( Fee = , which just gives him the reserve

utility. In the absence of the (WC), the owner can always achieve the first best allocation even

if e is unobservable, by choosing ( )F Fb b e '( )FM e≡ = , and '( ) ( )F r F F Fs U M e e M e= − +

such that the manager’s expected utility just equals his outside option:

( , , ) ( )F F F Fb e M e− rW s U= . However, the wealth constraint (WC) requires that the manager

cannot be asked to pay a fine in case of a very bad outcome of the productivity. Under the worst

possible situation where firm productivity is zero ( 01=

a), the manager receives no bonus

( 0=⎟⎠⎞

⎜⎝⎛

ab ) , and gets paid his fixed salary ( ) only. So the (WC) implies that must be

non-negative:

Fα

Fs Fs

0Fs ≥ ⇔ ( ) '( ) ( )F F F F rI e M e e M e U≡ − ≤ (14)

Note that , so '( ) 0FI e > (14) is satisfied only if is sufficiently small or is sufficiently

large. To make things more interesting, we impose the following assumption to rule out this

possibility:

Fe rU

Assumption 3 '( ) ( )F F F rM e e M e U− >

Assumption 3 says that the first best effort is sufficiently large, so that under the first best

compensation scheme { },F Fs b the lowest possible compensation paid to the manager becomes

negative ( ) , violating (WC). Hence, under this assumption the manager’s wealth 0Fs <

9

8 For example , when ( ) keM eθ

θ= , θ >2, assumption 2 is equivalent to 1θ σ> − , requiring that the

slope of the disutility function is sufficiently large.

constraint leads to contractual friction that prevents the first best effort being implemented.9

The optimal contractual choice is thus the second best allocation, given as follows.

Lemma 1a Under assumption 1-3, for given B, the second best managerial effort

{ }* max ,s Le e= e , where

se : '( )'( )

s

s

D eBV e

= , (OI: Optimal Incentive) (15)

Le : '( ) ( )L L L rM e e M e U− = (16)

The optimal contractual choice is given by:

(17) * '(b M e= *)

(18) *s = 0

The total (expected) managerial compensation is given by ( *) '( *)W D e eM e= ≡ . Furthermore,

the second best effort is lower than the first best effort, i.e., .

* Fe e<

Proof: See appendix. ■

It is important to note that (WC) is always binding under the second best solution, otherwise the

first best effort can always be implemented that violates our previous assumption. The level of

the second best managerial effort ( ) depends on whether (PC) is binding or not. Note that

the (OI) condition implies

*e

se is increasing in B.10 Hence, when B is sufficiently large, s Le e>

so that (PC) is not binding( * se = e ). In this case, as was in Schmidt (1997), the owner offers a

contract that yields positive (expected) rent for the manager in order to maximize her net payoff

due to the wealth constraint.11 On the other hand, when the market is sufficiently competitive

(B small), is low so that (PC) is binding (se s Le e< ). Choosing * se e= is not feasible since the

manager whose expected utility derived from the contract falls below his outside option will

reject such a compensation scheme. The owner therefore has to raise the piece rate that

increases the manager’s expected payment until he just accepts the contract. Note that in this

9 Although the rest of the paper focuses on the case of second best allocation under assumption 2 , the main qualitative results remain the same when the assumption is relaxed and the first best effort can be achieved. Proof available from the author upon request. 10 Note that assumption 2 ensures

[ ]

'

2'( ) ''( ) '( ) '( ) ''( ) 0'( ) '( )

B D e D e V e D e V ee V e V e

⎛ ⎞∂ −= =⎜ ⎟∂ ⎝ ⎠

> .

10

11 The manager’s rent is ( ) ( ) '( ) ( ) 0M r s s r s s s rU U W e M e U M e e M e U− = − − = − − > , because [ ]'( ) ( )

0M e e M e

e∂ −

>∂

, and s Le e> .

case the optimal effort is determined only by the shape of the manager’s disutility function Le

( )M e and outside option ( ), but independent of the market condition (B). rU

2.3 Equilibrium

Finally we consider stage I, where the owner makes entry decisions on whether to pay the

sunk entry cost to start the firm or not, so the market condition variable (B) becomes

endogenous at this stage. We assume that there is an unbounded mass of prospective owners, so

free entry into the industry drives owners’ expected net pay off to zero:

Ef

11

( ) ( *) ( *) 0EE W BV e D e fπΠ = − − − == (ZEP: Zero expected payoff) (19)

LeWhen (PC) is binding ( se < ), * Le e= , so equation (19) implies that in equilibrium

( )( )L

L

D e fE

V e+B = , which pins down the equilibrium firm level variables such as quantity, profit

and revenue as given in equation (2)-(3). The more interesting case is when (PC) is not binding

( Lese > ) , so * se e= , in which case the industry equilibrium can be determined by combining

the ZEP condition (19) and the OI condition (15):

'( )'( )

s

s

D eBV e

= (OI)

( ) ( )s s EBV e D e f− = (ZEP)

The equilibrium se and B are thus determined by the two different relationships between these

two key endogenous variables. First, as can be seen in figure 1a, the (OI) condition defines an

increasing relationship between se and B , whereas the (ZEP) condition defines a U shape

relationship. The intuition can be described as follows. Along the ZEP curve, each pair of

se and B yields the same (zero) net expected payoff for prospective owners. When se is low

and B is high, so that the pair { },se B is located at the upper-left of point E, an increase in se

tends to increase the net payoff. i.e. ⇔>∂Π∂ 0)(

seE '( ) '( )s sBV e D e> . This is because the

marginal increase in profits from cost reduction is high (due to high demand shifter B) and the

increase in compensation is low (due to low se and therefore low marginal disutility of the

manager). Hence, when se increases B has to decline so as to keep the net payoff equal zero,

yielding a downward slope. However, when the pair { },se B passes point E so that se is high,

an increase in se tends to decrease owner’s payoff i.e., ( ) 0

s

Ee

∂ Π< ⇔

∂'( ) '( )s sBV e D e< . This is

because the corresponding marginal increase in the manager’s compensation will be so high,

due to the manager’s steep disutility function, that it exceeds the marginal increase in market

profit. This requires an increase in B in order to keep the owner’s net payoff unchanged,

implying an upward slope. The intersection of the (OI) and (ZEP) schedules is always at the

ZEP curve’s bottom point E, which defines a unique pair of at the equilibrium. ( , )se B

More formally, substituting (15) into (19) we can obtain the equilibrium : se

'( )( ) ( ) ( )'( )

s

12

s s s Es

D e V e fV e

≡ − =J e D e

(0)

(20)

It can be shown that , '( ) 0J e ≥ 0J = and lim ( )e

J e→∞

= ∞ , therefore (20) has a unique

internal solution for se 12. The following proposition summarizes the above results:

PROPOSITION 1 In the closed economy, under assumption 1-3 , there exists a unique

industry equilibrium in the second best managerial effort : when (PC) is binding , *e * Le e=

as per (16) , and )(

)(*)(L

L

eVeDeB +

= Ef * se e=; when (PC) is not binding , as per (20) , and

))

s

s

('('

*)(eVeD

eB = as per (15) . The corresponding optimal contractual choice { }*,s b* are

given by (17)-(18).

Since determines the equilibrium distribution of firm level marginal costs

, all the economy-wide aggregate variables of the closed economy , such as

the number of firms/varieties (N) and the aggregate price (P), can all be written as functions of

only

*e

*) =( ; *)G a e F

*e

(ae

13.

(1V e

)( *) ( *)

ENB e

−=

ρ , (21) ( ) 1

0 0( )dF( ; *a adG *)e eα α α

∞ ∞−

∫ ∫= = , (22)

12

' ''( )

)'( ) '( )'( ) ( ) '( ) ( )

'( '( ) '( )D e D e D e'( )J e V

)

e V D e V eV e V e V e

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= + − =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

e >0 , using footnote 11.

13 Rewrite the aggregate price as a function of N and e :

( ( )1 11

1 1 1σ1 110

( ; )iiP p N a a eσσ σσdi dG ρ

∞−− −−= =∫ ∫− − =

1 111 σ1 ( )N V eσ ρ −− −

1 1( )

Substitute this expression to [4], we obtain 1 1(1 ) (1 )E P Eσ σρ ρ ρ−B V e N− −⋅ −≡ ⋅ ⋅ − = − . Rearrange this equation we obtain the equilibrium N and P shown below.

11

1

( *)(1 )

B ePE

σ

σρ ρ

−

−

⎛ ⎞= ⎜ −⎝ ⎠

⎟ , (23) 1

11 1(1 )( *)

EU EP B e

σσ

σ σρ ρ −− −⎛ ⎞−

= = ⎜ ⎟⎝ ⎠

. (24)

Further inspections on (21)-(24) reveals that all these variables are monotonic functions of e* :

higher managerial effort (e*) lowers the number of varieties, reduces average costs of the

industry, but leads to higher aggregate price and therefore lower consumer welfare. 14 It looks

somewhat surprising that average cost and aggregate price move opposite directions when the

managerial effort increases. Intuitively, when managers work harder, firm productivity

increases, which deters new entrants who faces more productive competitors. As a result, the

number of surviving firms/varieties (N) decreases, which tends to raise aggregate price (P).

Overall, the variety effect dominates the productivity effect, so aggregate price increases and

consumers are worse off, even if firms become more productive. Note that this result runs

contrast to Melitz (2003) where the productivity effect- in terms of the exit of least productive

firms from the industry- dominates, so there is always a positive correlation between industry

productivity and consumer welfare.15 What we have shown here, however, is that consumers

may not always get better off when the industry becomes more productive, especially when

increases in within-firm productivity might lead to falling number of varieties that has an

adverse effect on consumer utility.

3. The Open Economy

We now examine the effects of trade openness in a world that is composed of m+1 symmetric

countries.16 This implies that wage rate will be equalized across countries, which is again

14 Note that . This is because when (PC) is not binding (0*)(' >eB see =* ), using footnote (13),

0')(')('

*)(' >⎟⎟⎠

⎞⎜⎜⎝

⎛=

s

s

eVeD

eB ; when (PC) is binding ( Lee =* ) , we can

show [ ) >− EL f ] 0()('')('' =⎟⎟

⎠

⎞

L

LE eJeVeVf

)()(*)(' ⎜⎜

⎝

⎛ +=

L

L

eVeDeB , using (20) and . sL ee >

15 Consider, for example, the effect of an increase in the fixed cost of entry. In the original Melitz model , this will lead to lower aggregate productivity as more unproductive firms can survive after entry, and thus a decrease in welfare. But in our model, according to equation (20) and (15), increasing Ef leads to increasing

se and B, implying an increase in firm productivity but a decrease in welfare. Also see Raith (2003) obtaining similar results using a circle model.

13

16 One natural way to examine the impacts of openness is to consider the effect of an increase in market size of the economy as characterized in the previous section. However, from equation (20) the equilibrium managerial effort is independent of the size of the market, implying that opening up to costless trade does not

normalized to one. The sequencing of firm behaviors is similar to the case of the closed

economy, but now there is a new stage (export decision) inserted between the contract stage

and production stage, during which the owner decides whether to enter the export market or not,

depending on the realised marginal cost (see figure 2). It is important to note that the optimal

contractual choice of incentives were made by the principals before the export status was

decided. Hence, openness affects firms’ managerial incentives mainly via its impacts on the

principals’ expected payoff functions given the new opportunities of exporting and threats of

foreign competition. In a partial equilibrium framework with a single monopolist firm, one

would expect trade openness brings additional selling opportunities in the foreign markets, and

therefore raises the firm’s expected profit and output, which should provide the principal with

stronger incentives to raise managerial effort and thus boost firm productivity. However, this

line of reasoning is perhaps too simple to capture the general equilibrium feedback effects of

openness in a world with a large number of firms competing in the same industry as considered

here. Under free entry, opening up to trade not only brings extra opportunities of selling abroad,

but also introduces more competition from both foreign rivals and new domestic entrants. As a

result, firm level profit/output could either increase or decrease, leaving the total effects of

openness on the value of cost reduction and managerial effort much more complicated and

intriguing. This begs the following important questions: What are the conditions for trade

openness to generate within-firm productivity gains via the managerial incentive mechanism?

Can firm heterogeneity play a crucial role? What is the link between the trade-induced inter-

firm reallocation and within-firm productivity changes? To answer these questions, in the

following sections we start with analyzing the effects of trade openness in the original

Krugman-style framework where all firms export in the open economy, and then examine the

question in the Melitz-style framework where firms self-select into the export markets in terms

of their heterogeneous marginal costs. Comparing and contrasting these two cases help to

reveal the vital role of firm heterogeneity and export market selection in the impacts of trade on

managerial incentives (and hence firm productivity).

3.1 Krugman-style framework: all firms export

14

1

We first consider the open economy version of the model where exporting only incurs ice-

berg variable cost τ >

but no fixed costs. This set up is very close to the original Krugman

affect firm productivity. This leads us to consider the open economy version of the model with costly trade as described below.

intra-industry trade model , insofar as all firms export in equilibrium under CES demand,

despite their heterogeneous costs.17

Specifically, for firms with any given cost , they can always earn positive export profits

(note our symmetric country assumption) :

a1( ) ( ) 0x a mB a σπ τ −= > . Let 1( )d a Ba σπ −=

represent the domestic profit, then the firm level total market profit is given by:

. Hence, the firm’s expected market profit

is

1 1( ) ( ) ( ) (1 )d X Ea a a f B m σ σπ π π τ − −= + − = + Ef−a

1

0( ) ( ; ) (1 ) ( ) Efa dG a e B m V eσπ π τ

∞ −= = +∫ − . As was in the case of the closed economy, the

firm owner makes contractual choice that optimizes her net expected payoff ( )E WπΠ = − ,

with the same managerial incentive structure (W) as per (5) , and subject to all the constraints

as per (9)-(11). Reasoning analogous to the analysis of the closed economy, optimal managerial

effort depends on whether (PC) is binding or not. When (PC) is not binding, the second best

effort se is given by

1(1 ) '( ) '( )s sB m V e D eστ −+ = (OI) (25)

And the equilibrium is characterized by the free entry condition that yields zero net expected

payoff :

1(1 ) ( ) ( )s s EB m V e D e fστ −+ − = (ZEP) (26)

Combining the above two equations we obtain the equilibrium condition:

'( )( ) ( ) ( )'( )

ss s s

s

D eJ e V e D e fV e

≡ − E= (27)

The impact of trade openness can be assessed by comparing the trade equilibrium with the

autarky equilibrium as shown in the last section. Let superscript A denote the autarky variables,

e.g., Ase and AB denote the equilibrium effort and the degree of market competition in autarky

as per (20) and (15), respectively. Comparing (27) and (20), it is straightforward that As se e= ,

which says that managerial effort in the open economy is identical to that in autarky. More

interestingly, since (27) implies that se is independent of τ , it also implies that falling ice-berg

trade cost has no effect on optimal effort in the open economy. The same result applies to the

15

17 Note that the principal-agent framework requires the existence of cost uncertainty (otherwise the moral hazard does not exist ), and thus firms cannot be absolutely homogeneous in their costs. But firms are homogeneous in export status since all of them export.

case when (PC) is binding. 18On the other hand, trade openness does have an impact on the

(inverse) market competition measure B, which is lower in trade than autarky,19 generating

welfare gains (bigger U) in the open economy as per (24). Intuitively, openness increases the

number of sellers in each market, leading to more competition and higher consumer welfare

due to love of variety, which replicates the classic result of the original Krugman model.

Why in this case trade openness does not lead to increasing managerial incentives, even if

firms face more selling opportunities abroad? Note that the optimal level of managerial effort

increases with the value of cost reduction - defined as the profit differences between high and

low cost levels drawn from ,( ; )G a e 20 which is determined by the slope of the firm’s profit

schedule as a function of marginal cost (a). It turns out that, when the economy is opened up,

an increase in firms’ (expected) profits due to the new exporting opportunity is completely

offset by the a fall in the domestic profits due to increasing foreign competition, so that the net

effect of moving from autarky to trade, or falling ice-berg cost, on firms’ profit schedule is

zero.21 Hence, the firm level profit schedule as a function of marginal cost is unaffected, so the

value of cost reduction also remains unchanged, leaving no incentives for firm owners to

increase managerial efforts. 22

However, the “all firms export” scenario clearly does not fit well with the large body of

evidence on firm-level export mentioned in the introduction. This leads us to investigate the

problem further in the Melitz-type framework in the next section, where cost heterogeneity

plays a crucial role in firms’ self-selection into the export market .

16

18 As was shown in (16), the lower limit of the second best effort(eL) is only determined by the manager’s disutility function and outside options, but independent of the profit schedules and market conditions. So is also unaffected by trade openness and trade costs.

Le

19 When (PC) is not binding , (25) implies that ( )1

'( ) '( )'( )'( ) 1

AAs s

Ass

D e D eB BV eV e m στ −

= <+

= ; when (PC) is

binding , free entry implies that ( )1

( ) ( )( )( ) 1

AAL E L E

ALL

D e f D e fB BV eV e m στ −

+ += <

+= .

20 It is common in the Schumpeterian literature that the incentives to innovate depends on the differences, rather than the absolute size of pre- and post innovation rents. See e.g.,Raith (2003) and Trindade (2005). 21 Note that 1 '( ) '( )(1 )

'( ) '( )

As s

As s

D e D eB mV e V e

στ −+ = = , so we have

and 1 1 1( ) ( ) (1 ) ( )A Ax da a B m a B aσ σ σπ π τ π− − −+ = + = = a 1(1 ) / 0B m στ τ−⎡ ⎤∂ + ∂⎣ ⎦ = .

22 Note that this result contrasts with Horn, Lang and Lundgren (1995) in an oligopolistic competition model, where the effect of increasing export profit/output always dominates that of the shrinking domestic profit/output, leading to increasing managerial effort. We show that their result may not apply universally: in the original Krugman-style model with endogenous firm productivity, the opposite effects of trade openness on firm profit would just cancel each other out, leaving no within-firm productivity gains from trade.

3.2 Melitz-style framework : export market selection with heterogeneous costs

17

0>f

As was in Melitz (2003),we assume that exporting incurs not only per unit trade costs, but

also significant fixed costs per export market that are invariant to the export volume

(see, e.g., Robert and Tybout, 1997). This fixed trade cost assumption is crucial for firms’ self-

selection into the export markets. In contrast to the last section where all firms can earn positive

export profits regardless of their heterogeneous costs, now only a fraction of firms with the

lowest marginal costs can overcome the fixed costs of exporting and break into the export

market.

X

3.2.1 Equilibrium

In the presence of fixed trade costs, the potential export profit for firms with given cost a is:

( )[ ]XX faBma −= −στπ 1)( (28)

Let denote the upper bound cost for exporters, then this cut off must satisfy Xa

0)( =XX aπ ⇔ 111

−−= φσBaX (29)

where τφσ 1

1−

≡ Xf is a measure of trade costs. Firms with marginal costs will export to

all foreign markets, whilst those with will remain purely domestic. So a firm’s total

market profit depends on its export status:

Xa a<

Xa a≥

Ed fBaa −= −σπ 1)( , non-exporter (30) Xa a≥

=)(aπ

( ) EXXd fmfmBaaa −−+=+ −− σσ τππ 11 1)()( , exporters 0 Xa a< <

Hence, the total expected market profit by a prospective entrant can be written as : 23

0 0

( ) ( ; ) ( ) ( ; ) ( ) ( , , , )Xa

d Xa dG a e a dG a e BV e Y B e f fπ π π τ∞

= + = +∫ ∫ X E−

(31)

where

23 Using (28) and (29) , we obtain

1 1

0 0 0( ) ( ; ) 1 ( ) 1 ( )X X Xa a ea

X XX X X

a eaa dG a e m f dF ae m f dFa

σ σ

π αα

− −⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞= − = −⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∫ ∫ ∫

)(1),,,(0

1

ααατ

ασ

dFmffeBY X XXX ∫

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛=

−

, 1

11( )X Xa e Bσ eα γ φ−−≡ = .

As was in the case of the closed economy, the owner’s problem is to design a contract that

maximizes her expected net payoff ( )E π WΠ = − , given the managerial compensation scheme

as per (5), subject to the constraints as per (9)-(11). To ensure that the owner’s problem is

globally concave and has a unique solution, we make the following assumption:

Assumption 4 [ ] ' '( , , , ) / '( ) '( ) '( )min ,

'( ) '( )e XY e B f V e D e M e

e V eτ

V e

⎡ ⎤∂ ⎛ ⎞ ⎛<

⎞⎢ ⎥⎜ ⎟ ⎜∂ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

Under assumption 1-4, reasoning analogous to the case of the closed economy, the optimal

managerial effort and contract choice is given by:

Lemma 1b In the open economy, for given B the first best effort is given by Fe

'( ) ( , , , ) '( )FF e F X FBV e Y B e f M eτ+ = (32)

The second best managerial effort { }* max ,s Lee = e , where

se : '( ) ( , , , ) '( )ss e s X sBV e Y B e f D eτ+ = , (OI: Optimal Incentive) (33)

Le : '( ) ( )L L L rM e e M e U− = (34)

The optimal incentive contract is given by:

* '(b M e= *) , . (35) *s = 0

The total (expected) managerial compensation is given by ( *) * '( *)W D e e M e= ≡ .

Furthermore, (32) and (33) imply that and Fe se increase in B , and * Fe e< .

Proof : See Appendix. ■

Again, free entry condition leads to zero expected payoff in equilibrium:

( ) ( *) ( , *, , ) ( *) 0E XE W f BV e Y B e f D e fπ τΠ = − − = + − − =E [ZEP] (36)

Hence, analogous to the case of the closed economy, equation (33) and (36) define one upward

sloping curve and one U-shaped relationship between B and se . It is shown in the appendix

that these two schedules identify a unique pair of { },se B . We can thus draw the following

conclusions on the equilibrium managerial effort and contract choice in the open economy:

18

PROPOSITION 2 In the open economy where only low cost firms self-select into the export

markets, under assumption 1-4, there exists a unique industry equilibrium in the second best

managerial efforts . When (PC) is binding, *e * Le e= as per (34) ; when (PC) is not binding ,

* se e= that is determined by (33) and (36) . The market condition (B) is determined by (36)

for forgiven e*. The optimal contract choice { }*, *s b are given by (35).

Proof : see appendix. ■

The equilibrium pair { }*,e B thus determines the export cost cut off ( as per (29), and the

ex post fraction of firms that export

)Xa

( ) ( )ex X Xp G a F α= = . Following the line of reasoning

similar to the one used in the closed economy, the number of firms (N) and varieties ( ) are

also uniquely determined in equilibrium:

vN

( )1

(1 )( *) 1 ex

ENBV e mp σ

ρτ −

−=

+, 1

1(1 )(1 )( *) 1

exv ex

ex

mpEN N mpBV e mp σ

ρτ −

⎛ ⎞+−= + = ⎜ ⎟+⎝ ⎠

(37)

Regarding the other economy-wide variables, the average firm productivity , the aggregate

price and welfare can also be written as functions of and B only, just the same expressions

as those given in the closed economy as per

*e

(22) to (24).

3.2.2 Impact of trade openness

With the equilibrium firm level and economy-wide variables determined, we are now in the

position to examine the impact of a transition from the closed economy (variables with

superscript A) to the open economy. Our main interest is what happens to the pair { }*,e B .

Firstly, when (PC) is binding in both autarky and the open economy, it follows immediately

from lemma 1a and lemma 1b that the optimal effort does not change ( ). Further, it can

be shown that

ALL ee =

,A AvvB B N N< >

. 24 Hence, opening up to trade makes markets becomes more

competitive as there are more sellers in each economy, generating welfare gains due to

24 When (PC) is binding, from [36] we obtain ( ) ( , , , ) ( )

( ) ( )AL E L X L E

L L

D e f mY B e f D e fB BV e V e

τ+ + += < = , and

19

1 1

1 1(1 ) (1 )( ) 1 ( ) ( , , , ) 1

(1 ) (1 ) ( ) ( )

ex exv

L ex L E L X ex

AA

L E L

m p m pE ENBV e m p D e f Y B e f m p

E E ND e f B V e

σ σ

ρ ρτ τ

ρ ρ

− −

⎛ ⎞ ⎛ ⎞+ +− −= =⎜ ⎟ ⎜ ⎟+ + − +⎝ ⎠ ⎝ ⎠

− −> = =

+

τ

increasing number of total varieties available for consumers, rather than increasing managerial

effort and within-firm productivity25. These effects are identical to those in the Krugman-style

framework as described in section 3.1.

The most interesting case is when (PC) is not binding in both autarky and trade

( ) , in which case the impact of trade can be analyzed as follows. As was

shown in Figure 1b,inspection of the (OI) condition in the open economy as per

AL

AsLs eeee << ,

(33) relative to

the one in the closed economy as per (15) reveals that the OI curve shifts downwards. 26

Reasoning analogously, the ZEP schedule also shifts downwards in the open economy. This

implies that at the open economy equilibrium (E’) , B must be lower than that at the closed

economy (E). Furthermore, substituting (33) into (36) and rearranging, we obtain:

( ) ( , , , ) ( )As s X E sJ e H B e f f J eτ+ = = ⇒ ( ) ( ) ( , , , )A

s s sJ e J e H B e fτ− = X (38)

where( , , , ) ( )

( , , , ) ( , , , )'( )

se s Xs X s X

s

Y B e f V eH B e f Y B e f

V esτ

τ τ⋅

≡ − . It is shown in the appendix that

( , , , )s XH B e fτ <0, which implies that As se e> .27 Hence, moving from autarky to trade leads to

greater managerial effort ( se ) and higher power of the incentive contract ( '( )sb M e= ), as well

as increasing competitiveness of the market, reflected in the lower firm level demand shifter (B)

faced by each firm. Again , welfare also increases , as U decreases in B.

However, note that here the source of the welfare gain is very different from that in the

previous case where (PC) is binding in both autarky and trade. Here when (PC) is not binding,

it is the rising within-firm productivity, rather than an increase in the number of varieties, that

makes consumers better off. Inspection of the equilibrium number of domestic firms ( and

total varieties ( ) as per

)N

vNvN (37) relative to those in autarky as per (21) reveals that is not

necessarily greater than AN : openness allows consumers to access foreign varieties, but may

also lead to a loss of domestic varieties , so the overall effects of openness on the total number

of varieties become ambiguous. It is the positive welfare gains from increasing within-firm

25 From (23) U decreases with B, so since AUU > ABB < .

26 Note that (33) can be rewritten as ( , , , )'( )

'( ) '( )se s Xs

s s

Y B e fD eBV e V e

τ= − . Comparing with (15), it

follows immediately that for given se , B must be lower on (33) than (15), since ( , , , )s XY B e f esτ∂ ∂ >0. 27 Note that '( ) 0J e >

20

productivity, which dominates the potential welfare loss from falling domestic varieties, that

leads to improving consumer utility.28

Furthermore, it is straightforward that if (PC) is not binding in the open economy, but

binding in the closed economy, optimal effort also increases after trade openness , and we can

rule out the possibility that (PC) is binding in the open economy but not binding in the closed

economy. 29 Hence, we obtain the first main result of our paper:

PROPOSITION 3 In the open economy where only low cost firms self-select into the export

markets , under assumption 1-4, As se e> , and A

LL ee = ABB < . Hence,

(i) when (PC) is not binding ( see =* ), moving from autarky to costly trade leads to an

increase in the power of the incentive contract, within-firm productivity and

consumer welfare, although the total number of varieties might decrease.

(ii) when (PC) is binding ( Lee =* ), moving from autarky to costly trade does not affect

the power of the incentive contract and within-firm productivity, but the total

number of varieties increases , generating consumer welfare gains.

3.2.3 Carrot and Stick Effect

Proposition 3 reveals that, trade openness could indeed lead to an increase in managerial

incentive within the Melitz-style framework, which runs contrast to those results from the

Krugman-style framework as shown in section 3.1. So what is the main mechanism driving

such differences? Why is that the selection of low-cost firms into the export market become the

vital condition for the trade-induced within-firm productivity gains?

Recall that the optimal managerial incentive increases with the value of the cost-reducing

managerial effort, which is determined by the profit differences between low and high levels of

firms’ ex post marginal costs. In the Melitz-type framework, when (PC) is not binding, moving

from autarky to costly trade will unambiguously raise the value of managerial effort, because

the market profits of lowest cost exporting firms will increase, whilst those of high cost non-

exporters shall decrease, leading to increasing profit differences between these two types of

firms. We label this as the “carrot and stick” effect, which is shown more clearly in figure 3. 28 This result mirrors the “anti-variety” effect as was pointed out in the original Melitz(2003) model , and elaborated in Baldwin and Forslid (2004) . Also see Falvey, Greenaway and Yu (2004). 29 Note that (PC) is not binding in the open economy implies that A

s Le e e> = L , meaning that effort is greater in trade than autarky. Further, (PC) is not binding in the closed economy but binding in the open economy implies that A A

s Le e e> =

21

L , and , leading to . This is impossible, since we have just shown

.

Ls ee < Ass ee <

Ass ee >

The black line represents the profit in the open economy (π ) as a function of 1a σ− as per (30) ,

and the dashed line represents the autarky profit schedule Aπ as per (3). Note that the existence

of the export cut off cost induces a kink on Xa ( )aπ , and to the right (left) of the kink the

slope of the π is greater (smaller) than that of Aπ .30 Consequently, those firms to the right of

the intersection of these two lines that draw very low costs such as , will find their profits

unambiguously greater in the open economy than those in autarky (the carrot effect) , whilst

those drawing very high costs such as

La

Ha will receive absolutely lower profits in the open

economy (the stick effect). As a result, the profit difference between and La Ha in the open

economy, denoted as ( ,La a )HπΔ , must be greater than that in the autarky ( ),

meaning a greater value of cost-reduction that leads to higher managerial effort.

),( HLA aaπΔ

It is noteworthy that firm heterogeneity and export market selection plays a crucial role in

generating the carrot and stick effect. Recall that in the original Krugman framework where all

firms export, the extra export profit is just offset by the decrease in domestic profit for all firms,

leaving the profit schedule and the value of cost reduction unchanged. Here in the Melitz

framework where only a fraction of firms export, those ex post high-cost non-exporters (such as

firms with Ha ) earn zero export profit and therefore have to suffer a profit loss since their

domestic profits fall, i.e., . However, it is shown in the appendix that those ex

post lowest cost exporters (such as firms with ) will find their extra export profit more than

compensate their losses in the domestic market, i.e., , leading to an

increase in total profit. This implies that the variation of firm profits becomes greater, although

the distribution of the underlying cost uncertainty (

( ) (Ad H Haπ π< )a

a

La

( ) ( ) ( )AX L L d La aπ π π> −

( )F α ) remains unchanged. As a result, the

principals have more at stake in the open economy than autarky: they will be penalized more

severely if receiving a bad cost draw, but could also be rewarded by much more upon a very

good draw. This motivates the principals to design a higher power incentive contract to increase

the agent’s effort, so as to increase their chances of seizing the carrot and avoiding the stick.

Note that the above results do not depend on the specific setup of the model (e.g. CES

demand and monopolistic competition): as long as trade costs are high so that only lowest-cost

firms can afford to export, trade openness will unambiguously increase the value of cost-

reduction by reducing (increasing) the profits of high-cost non-exporters (lowest-cost exporters).

Allowing increasing elasticity of product substitution due to trade openness will only reinforce

22

30 Note that , and in the appendix we can prove that . ABB < ABmB >+ − )1( 1 στ

the effects of increasing value of cost-reduction, and, therefore, strengthen the positive effects

of trade on managerial incentives.31

Readers familiar with the Melitz model may find the “carrot and stick effect” is closely

related to the well-known trade-induced intra-industry reallocation effects highlighted in Melitz

(2003). What is new in our model, is that such inter-firm reallocation effect could also become

the key driving force behind the within-firm productivity effects of trade openness. In other

words, by incorporating the principal-agent mechanism, our model reveals that there could exist

a causal link between the Melitz-type reallocation effect and the within-firm productivity gains,

with the unidirectional causality running from the former to the later. This is because while

openness reallocates the market shares towards low cost firms, the principals may not only

react passively by adjusting their output margins, but also foresee such threats and opportunities

and adapt their incentive contract margins proactively to induce agents to work harder, even

though this is achieved with higher agency costs.

4. Trade Liberalisation

In the previous section we have examined the equilibrium outcomes of moving from autarky

to trade. However, very few of the economies of the world today are under an autarky regime, it

is therefore more important to understand how could an incremental decline in the costs of

international trade affect within-firm managerial incentives and productivity, once an economy

is already opened up. In particular, we are interested in whether the “carrot and stick” effect is

still at work when the economy is increasingly exposed to trade? We will show in the

following analysis that a decrease in different types of trade costs – variable trade cost τ

versus fixed trade costs - could lead to very different outcomes on firm productivity, depending

on whether the “carrot and stick” mechanism is operating in the “right” direction as was

described in the last section.

4.1 Falling variable trade costs τ

We first analyze the effects of trade liberalization characterized by falling per unit trade cost

τ , driven either by improving technology in communication/transportation or multi-lateral

agreements to reduce tariff . Consider a decrease of such cost from τ to 'τ τ< . When (PC) is

binding, reasoning analogous to the case of transition from autarky to trade, the within-firm

23

31 For given cost reduction , firm profit would increase by more when products become closer substitutes, since demand would be more elastic to price cuts.

managerial incentive and productivity are unaffected by trade costs. The more interesting case

is when (PC) is not binding: differentiating the open economy equilibrium conditions as per (33)

and (36) with respect to τ , we obtain the following result:

PROPOSITION 4 In the open economy where only low-cost firms self-select into the export

market, the equilibrium level of se is decreasing in τ , and the equilibrium level of B is

increasing τ .Hence, when (PC) is not binding, falling variable trade costs leads to an increase

in the power of incentive contract, within-firm productivity, average managerial compensation ,

and consumer welfare.

Proof: see appendix. ■

The mechanism driving this result is very similar to that described in the last section: a

decrease in τ unambiguously increases the profit difference between lowest-cost exporters and

high-cost non-exporters, leading to greater value of cost reduction and managerial effort. In

other words, the carrot and stick effect just described in the last section is working at the “right”

direction. More formally, we can show that (i) ( ) 0d aπ τ∂ ∂ > : whenτ decreases, the ex post

high cost firms that do not export will suffer from a profit loss due to increasing competition in

the domestic market. (ii) [ ]( ) ( )d Xa aπ π∂ + τ∂ <0: the ex post low cost firms that export will

see a net increase in their total profits, since their profit loss in the domestic market

( 0>∂∂π τd ) will be more than compensated by the increase in the export profits

( 0<∂∂π τX ), the result is a net increase in the total profit )(aπ when τ falls. This can be

seen more clearly in figure 4a: at the lower level of per unit trade cost 'τ , the kink of the profit

schedule shifts to the left ( ) . To the right (left) of the new kink, the slope of the profit

function becomes steeper (flatter).

'Xa a> X

32 As a result, the profit difference ( πΔ ) between and La

Ha has to be unambiguously greater. This suggests that the cost-reducing managerial effort

becomes more “valuable” to the owners, leading them to choose a higher power incentive

contract to induce greater effort by the managers, in order to decreases (increase) the firm’s

probability of realizing a high (low) cost and become an ex post non-exporter (exporter).33

The above results are consistent with the empirical findings from Bernard, Jensen and Schott

(2007) that reductions in transport costs simulate the within-firm productivity of U.S. firms.

32 See appendix for proof. 33 Recall that the probability that a firm draw a cost below ,a )();( aeFeaG = ,is increasing in managerial

effort e .

24

Our model suggests that the principal-agent mechanism and the Melitz-type reallocation effect

might provide a plausible explanation for such firm level productivity gains from falling trade

costs. Another empirical implication of Proposition 4 is that a positive correlation between

product market competition and firm productivity – as was suggested by many empirical IO

works - could be driven by a third factor, namely falling international trade costs, rather than a

direct causality between the two. As was described above, in our model falling trade costs

simultaneously leads to increasing firm productivity (greater effort e) resulting from the carrot-

and-stick effect, and increasing product market competition (falling demand shifter B) due to

lower selling prices of foreign competitors, without a direct causality running from the latter to

the former. Hence, from a theoretical point of view our model has identified one potential

source of the endogeneity problem that plagues some previous empirical works on the relation

between competition and efficiency, i.e. the level of variable trade costs that could vary both

over time and across industries.

25

ort outputs accordingly.

Finally, our model might also provide an analytical mechanism to understand the insight

from Costantini and Melitz (2007) that the non-technical aspects of trade liberalisation, such as

the “timing of announcement”, could also have important implications for firm level

adjustments to openness. 34 In particular, in our model the timing of the within-firm

productivity improvement, relative to that of trade liberalization, could be very different from

that of the trade-induced inter-firm reallocation. Whilst the latter has to occur after trade

liberalization is practically implemented (exporters cannot expand their foreign sales until the

trade cost is really lowered), the former could take place before the trade costs is reduced in

practice, because the managerial incentives/efforts are determined by the principal’s

expectation on the level of the trade costs, and set ahead of a firm’s export and production

decision (see figure 2). An important policy implication is that the within-firm increase in

managerial incentives and productivity might be realized instantaneously following an

“announcement” of trade liberalization, even it is (unexpectedly) not implemented later in

practice. This is in contrast to the original Melitz model where the reallocation driven aggregate

productivity gains can only be realized when the trade costs are de facto reduced, and firms

adjust their exp

4.2 Falling fixed trade costs Xf

As was rightly pointed out by Baldwin and Forslid (2004), among most industrialized

countries tariff are already at a low level, and the main barriers to trade are mainly related to

34 The complex dynamic effects of trade liberalization are analyzed in Costantini and Melitz(2007) using simulation methods.

standard and regulation that impose substantial fixed costs of trade on potential exporters. One

relatively unexplored question in the literature is how would a fixed cost trade liberalisation

affect within-firm productivity? Again, when (PC) is not binding we can analyse such effects

by differentiating equation (33) and (36) with respect to Xf :

Proposition 5 In the open economy where only low-cost firms self-select into the export

market, the equilibrium level of B is increasing Xf , and the equilibrium level of se is

increasing in Xf if and only if 1exX

ex X

pfp f

∂−

∂< . Hence, when (PC) is not binding, falling fixed

trade costs unambiguously increase consumer welfare, but has ambiguous effects on

managerial incentives and firm productivity: it may lead to weaker incentives and firm-level

productivity losses, if the ex post fraction of firms that export is relatively inelastic to changes

in fixed trade cost.

Proof. See appendix ■

To see the intuition behind this result, we need to examine whether the “carrot and stick”

effect is still at work when Xf falls. Differentiating firm profit as per (30)with respect to Xf ,

we find that falling Xf reduces firms’ domestic profits , as well as the total profits of ex post

exporters if their marginal cost are sufficiently low:35

1( ) 0d

X X

a B af f

σπ −∂ ∂= >

∂ ∂ (39a)

[ ]

11

1 1( ) ( ) >0 ( )d X

X X

a a Ba a mf f

σσπ π

τ−

− −∂ + ⎡ ⎤∂⇔ < ≡ +⎢ ⎥∂ ∂⎣ ⎦

(39b)

More clearly, as was shown in figure 4b, when falls, the profits of ex post lowest-cost firms

decrease, so do the ex post highest-cost firms. Only those medium-cost firms see an increase in

their profits. Falling fixed trade costs, therefore, does not necessarily reallocate market profits

towards the ex post lowest-cost firms, and thus the profit difference between low-cost-exporters

and high-cost-non-exporters may or may not increase (

Xf

( 'X )fπΔ may be smaller than ( X )fπΔ ).

As a result, falling generally has an ambiguous effect on the value of cost reduction, and

hence managerial incentives and firm productivity. In other words, it is possible that falling

Xf

26

35 Note that ( )1 1( ) 1X X

a B m af f

σ σ mπ τ − −∂ ∂= + −

∂ ∂, and see appendix for the proof that . Xa a<

regulatory trade barriers generates weaker incentives and within-firm productivity losses. This

could happen when the entry of new exporters induced by falling Xf is relatively small,

reflected in a small elasticity of the fraction of firms that export ( exp ) with respect to Xf .

Intuitively, the ex post medium-cost firms that could increase their profits are mostly new

export entrants. So a small entry of new exporters means only a small fraction of medium-cost

firms that export can increase their profits. Thus, there is an adverse inter-firm reallocation

effect: most of the ex-post exporters with relatively low cost substantially reduce their profits,

leading to a lower value of cost reduction, and thus weaker incentives. However, despite the

possibility of productivity losses, consumers are always better off. This is because falling fixed

trade costs increases the number of imported varieties available in each economy, and this

positive “variety effect” turns out to dominate the ambiguous productivity effect , which

generates welfare gains.

Note that the ambiguous productivity effect of falling Xf runs contrast to the original

Melitz (2003) model. In his model, although the most productive firms lose their market shares

when the fixed trade costs decrease, the positive contribution of the exit of the least productive

firms and the expansion of new exporters to aggregate industry productivity dominates, leading

to unambiguously positive productivity gains. Our model shows that, in the presence of the

within-firm managerial incentive mechanism, the productivity effect of falling fixed trade costs

could actually be more intricate. It is possible that the potential within-firm productivity loss

due to the adverse reallocation effect of falling Xf overturns the positive productivity gains as

described in the original Melitz model, leading to overall productivity losses from a decrease in

the regulatory barriers to international trade.

5. Concluding Remarks

This paper has studied how trade openness affects managerial incentives and firm-level

productivity by incorporating the principal-agent mechanism into the new heterogeneous firm

trade model. When trade liberalization is driven by reciprocal reduction in variable trade costs,

the variation of firm profits increases ,and thus the principal has more at stake , due to the trade

induced carrot-and-stick effect resulting from the Melitz-type inter-firm reallocation. As a

consequence, the value of cost reduction increases, which motivates the principals to provide

stronger incentives to their managers to reduce costs. Hence, for reasons independent of

learning-by-exporting forces, export selection and increasing export opportunity leads to firm-

level productivity gains, However, when trade liberalisation is driven by falling fixed trade cost,

27

28

an adverse inter-firm reallocation effect may occur, and the carrot-and-stick effect could even

work in the “wrong” direction, leading to ambiguous effects on firm-level productivity . Thus, a

new determinant of the trade-induced within-firm productivity gains, i.e. the direction of the

inter-firm reallocation effect, is revealed.

The qualitative results of the paper seem quite general, which do not rely on the

monopolistic competition model used here. The central assumption of firm heterogeneity and

export selection can be easily verified in a model with an alternative demand system or an

oligopolistic competition market structure. Furthermore, our modeling of the optimal

managerial incentive contract leads to a reduced-form profit function that would resemble one

with a continuous “innovation” choice by the firm with a general trade-off between high fixed

costs (managerial compensation in our model) and a greater degree of cost reduction.36 The

advantage of our model is that it explains where such trade-off could come from, and how and

why it could be affected by openness.

Our model parsimoniously captures the significantly complex role of managerial incentive

structures in shaping firm behaviors. To achieve this parsimony, we abstracted in our model

from many features of managerial characteristics in the real world. For example, we omitted

the heterogeneity of managers’ abilities and their attitude towards risks, and we ignored the

dynamics of managerial behaviors. Future research could be directed towards extending this

line of research to a comprehensive framework to explain the complex interactions between

international trade and managerial behaviors. We hope our model has brought trade theory

closer to the evidence, and serves as an important first step towards that end.

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36 Such trade-off can also come from, for example, cost-reducing R&D investments with some degree of uncertainty in outcome leading to firm heterogeneity. I thank Marc Melitz for pointing this out.

29

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31

AB

[36]

OI [33] [15]

0

B

e

EA

E

ZEP [19]

se

Figure 1b. Effects of moving from autarky to open economy

B

0

E

(19) ZEP

B

e

(15) OI

Figure 1a. Equilibrium in the closed economy

Ase

32

( )aπ

π

Stick

Carrot

( )A aπ

1a σ−

( )1Ha σ−

( )1Xa σ−

( )1La σ− 1a σ−

Figure 3. Value of cost-reduction and trade openness

( , )AL Ha aπΔ

( , )L Ha aπΔ

Ef−

1 0 2 3

Firm owners pay sunk entry cost to enter the industry

Optimal incentive contract designed

Managers choose effort (e)

Marginal cost (a) realized

Figure 2. Timing of firm behavior in the open economy

t 4 5

Production and sales of the product undertaken, profits realized.

Export mentry decision made

arket

Stage I : industry entry

Stage II : contract and cost

Stage III: export market entry

Stage IV: production and sales

π

33

( ; ')aπ τ

π

( ; ')Xa fπ

Stick

Carrot

( ; )aπ τ

1a σ−

( )Ha 1 σ−

( )1'Xa σ− ( )1

Xa σ− ( )L

1a σ− 1a σ−

Figure 4a. Value of cost-reduction and falling per unit trade costs

( )π τΔ

( ')π τΔ

Ef−

( ; )Xa fπ

1a σ−

( )1Ha σ−

( )1'Xa σ− ( )1Xa σ− ( )1a σ−

( )1La σ− 1a σ−

( 'X )fπΔ

Ef−

Figure 4b. Value of cost-reduction and falling fixed trade costs

π

( )fπΔ X

Appendix

A. PROOF OF LEMMA 1a

1. Substituting (12) to [8], we can rewrite the owner’s problem as: { },

max ( ) '( ) Ee sBV e eM e s f+ − − . Clearly,

for given , the optimal choice of s that maximizes the owner’s net payoff should be zero. So the owner’s

second best problem becomes:

e

{ }max ( ) ( ) EeBV e D e f− − (A.1)

Subject to

(PC) ( ) ( ) ( ) '( ) ( ) rW M e D e M e eM e M e U− = − = − ≥ (A.2)

Note that , since and ( ) ( )D e M e> '( ) '( ) 0D e M e− ≥ (0) (0) 0D M− = . The first order condition for

solution to (A.1) is '( )'( )

D eV e

B , and assumption 2 ensures that the second order condition is satisfied.

Combined with constraint (A.2), we obtain the second best solution as shown in the Lemma.

=

2. To show { }* max ,F > = s Le e e e , note that assumption 3 ensures that F Le e> . So we need to show

that sF ee > . Using (13) and (15), we obtain )(')('

)(')('

s

s

F

F

eVeD

eVeM

= . Hence, ⇒ '( ) '( )D e M e>

'( )'( )

'( )'( )

M e DV e

eV e

⇒ F se e>< .■

B. PROOF OF LEMMA 1b

Using (31), the first best effort is given by 

{ } { }arg max ( ) ( ) arg max ( ) ( , , , ) ( )F X Ee e

e e M e BV e Y B e f M eπ τ= − = + − f− (A.3)

First order condition of problem (A.3) leads to the solution of as per (32). Assumption 4 ensures that the

second order condition always holds.

Fe

1. For the second best effort , reasoning analogous to the proof of lemma 1a, the optimal choice of s must be

zero , and the (IC) condition implies '( )b M e= . So '( ) ( )W s be M e e D e= + = = , and the owner’s

second best problem becomes:

{ }max ( ) ( , , , ) ( )Xe EBV e Y B e f D e fτ− − − (A.4)

Subject to the (PC) condition as per (A.2). The first order condition of (A.4) is

'( ) ( , , , ) '( )e s XBV e Y B e f D eτ+ = (A.5)

A-1

Assumption 4 ensures that the second order condition always holds. Combined with the (PC) constraint

we obtain the results in (33) and (34).

2. Equation (33) can be rewritten as

'( ) ( , , , )'( )sS e s X

S

D e Y e B fB

V eτ−

= (A.6)

It is obvious that the r.h.s. of [A.6] decreases in B, and it also follows immediately from assumption 4 that the

r.h.s. of [A.6] is increasing in se as '( ) ( , , , )

0'( )sS e s X

SS

D e Y e B fe

V eτ−⎛ ⎞

∂⎜⎝ ⎠

∂ >⎟ . Now suppose B is

decreasing in se . If this is true, then when se increases , the r.h.s. of [A.6] increases, but the l.h.s. of [A.6]

will decrease, thus [A.6] cannot hold, leading to contradiction. Therefore B must be increasing in se

F se e

.

Reasoning analogously, from equation (32) and assumption 4 , we can show that e increases in B.

Furthermore, from (32) and (33), and using the condition , it is straightforward that .

Combined with assumption 3, we have

F

'( ) '( )D e M e> >

{ }* ma> = x ,F s Le ee e . ■

C. PROOF OF PROPOSITION 2

The uniqueness of the equilibrium se in the open economy can be showed as follows. Equation (33) implies

that B can be expressed as an implicit function of se . So, we can rewrite the ZEP condition as :

( )( ) ( ) ( ) ( ), , , ( )s s s s s X sQ e B e V e Y B e e f D e fτ≡ + − E= [A.7]

Taking differentiation of both hand sides of [A.7] with respect to se yields:

( ) ( )( ) '( ) ( ) , , , '( ) , , , ( ) '( ) '( )s

ss s B s X s e s X s s s

s

Q e B e V e Y B e f B e Y B e f B e V e D ee

τ τ∂= + + + −

∂ [A.8]

Using equation (33) we obtain:

( )( ) '( ) ( ) , , , 0ss s B s X

s

Q e B e V e Y B e fe

τ∂⎡= +⎣∂

⎤ >⎦ [A.9]

Since is monotonically increasing in e , there exists a unique internal solution ( )Q e se s.t. ( )s Ef=Q e . ■

D. PROOF OF ( , , , ) 0s XH B e fτ <

Rewrite [33] as ( )'( ) , , ,

'( )ss e s X

s

D e Y B e fB

V eτ−

= and substitute it into (36), the left hand side of (35) can be

written as ( ) ( ), , ,'( ) ( ) ( ) , , , ( ) ( ) ( , , , )'( ) '( )

se s XsS s s X s s s

s s

Y B e fD e V e D e Y B e f V e J e H e B fV e V e

ττ τ− + − = + X

A-2

Let ≡)( XZ α )(10

1

αααα

σ

dFX X∫⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛

−

, so we can write )(),,,( XXXs ZmffeBY ατ = . Using

Leibiniz integral rule , ( ) 2 1

0'( ) 1 ( )X

X XZ dασ σ Fα σ α α− −= − ∫ α . So

( ) ( )

( )[ ]

11 1

0

1

, , , '( ) 1 ( )

= 1 ( , , , ) ( )

X

s

Xe s X X X X X

s X X X

Y B e f mf Z e e mf dF

e Y B e f mf F

σα ατ α α σ

ασ τ

−− −

−

⎛ ⎞= = − ⎜ ⎟⎝ ⎠

− +

∫ α

α

Hence , using ( ) 1( ) 1'( )

s

s

V e eV e

σ −= − , we obtain

( ) ( ), , ,( , , , ) , , , ( ) ( ) 0

'( )se s X

s X s X s X Xs

Y B e fH e B f Y B e f V e mf F

V eτ

τ τ≡ − = − α < . [A.10] ■

E. PROOF OF PROPOSITION 4

1. prove 0Bτ∂

>∂

. Differentiating (33) and (36) with respect to τ , we obtain :

( , , ) ( , , ) ( , , )'( ) '( ) '( )s s s ss s s

s

Y e B e Y e B Y e BB V e BV e D eB e

τ τ ττ τ

⎡ ⎤∂ ∂ ∂ −∂∂ ⎡ ⎤+ + + − =⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦ τ [A.11]

( )

2 2

2

( , , ) ( , , ) ( , , )'( ) ''( ) ''( )s s s ss s s

s ss

Y e B e Y e B Y e BB V e BV e D ee B ee

2τ τ ττ τ

⎡ ⎤⎡ ⎤∂ ∂ ∂ −∂∂+ + + − =⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂ ∂∂⎢ ⎥⎣ ⎦ ⎣ ⎦ τ∂

[A.12]

Using (33), from [A.11] we obtain

( )sB Y V e Y

Bττ∂ −∂⎛ ⎞ ⎡= ⎜ ⎟ ⎢∂ ∂∂ ⎝ ⎠ ⎣

∂ ⎤+ ⎥⎦ [A.13]

since 0Yτ∂

<∂

, and 0YB∂

>∂

, we have Bτ∂∂

>0.

2. Prove 0seτ

∂<

∂. Rearrange [A.12], we obtain

( )

2 2

2

2

'( )

''( ) ''( )

ss ss

s ss

Y B YV ee ee

YBV e D ee

τ ττ

B⎧ ⎫⎡ ⎤−∂ ∂ ∂⎪ ⎪− +⎨ ⎬⎢ ⎥∂ ∂ ∂ ∂ ∂∂ ⎪ ⎪⎣ ⎦⎩=

∂∂ + −∂

⎭ [A.14]

A-3

Using (33), the denominator of the r.h.s. of [A.14] can be rewritten as

( )( )

2

2

'( )''( ) ''( )

'( )s s

s ss s

D e Y e YV e D eV e e

⎡ ⎤− ∂ ∂ ∂+ −⎢ ⎥∂⎣ ⎦

, which is negative by assumption 4. Hence, we only need to

prove the numerator of r.h.s. of [A.14] is positive. Substituting [A.13] into the numerator [A.14] to eliminate

Bτ∂∂

, we need to prove :

2 2

'( ) 0( )

ss s

s

YY YV e

Ye e B

∂

V eB

ττ

⎡ ⎤−∂ ∂∂+ +⎢ ⎥∂∂ ∂ ∂ ∂⎡ ⎤ ⎣ ⎦+⎢ ⎥∂⎣ ⎦

>

2 2

'( )

( )

ss s

s

Y YV ee e

Y YV eB

τ

τ

∂ ∂+

B∂ ∂⇔ >

∂ ∂+

∂ ∂

∂ ∂ [A.15]

Based on [A.10], we obtain ( )( )2

1'( ) 1XX X

s

Y Y F m sf ee

αα στ τ τ

−∂∂ ∂⎡ ⎤= + −⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ , and

( )( )2

1'( ) 1XX X

s

Y Y F m

A-3

sf ee B B B

αα σ −∂∂ ∂⎡ ⎤= + −⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦. Substitute them into [A.15] and use

( )( ) 1'( ) ( ) 1s s sV e V e eσ −= − , [A.15] is equivalent to :

( )X X

s

BY YV e

B

α τ α

τ

∂ ∂ ∂ ∂>

∂ ∂+∂ ∂

[A.16]

Using ( )'( )X X XY mf Z α α ττ∂

= ∂∂

∂ and ('( )X X XY mf Z BB

α α )∂= ∂ ∂

∂, [A.16] is equivalent to

( ) 1( ) '( ) 0s X X XV e mf Z Bα α

−⎡ ∂ ∂ >⎣ ⎤⎦ , which always holds , since Z '( ) 0Xα > and 0X Bα∂ ∂ . Hence,

[A.15] always holds and therefore

>

0seτ

∂<

∂.

3. Based on proposition 3, '( )sb M , e= '( )s sW e are increasing in M e= se , and U is decreasing in B

as per (24), and firm level productivity is increasing in se as per (22). Hence, since 0Bτ>

∂∂ and 0se

τ∂

<∂

,

decreasing τ leads to a greater level of firm , b W , 1a − and U .■

F. CARROT-AND-STICK EFFECT OF FALLING τ

In this appendix we first prove ( ) 0aπτ

∂<

∂ for [ ]0, Xa a∈ . Using (30), we obtain

1 1( ) (1 ) ( 1)a Ba m mB στ −⎡ ⎤⎢ ⎥⎣ ⎦

σ σπ τ στ τ

− −∂ ∂= + − −

∂ ∂. Hence ,

( )( )1

11

Bmm

( ) 0a B σ

σ

σ ττ

πτ τ

−

−

−

+∂ ∂

< ⇔ <∂ ∂

. [A.17]

A-4

Using [A.13], we can obtain ( )( )

'( )( )

( ) '( )X X X

ss X X X

mf ZB Y YV eB V e mf Z B

α α τττ α

− ∂ ∂∂ −∂ ∂⎛ ⎞ ⎡ ⎤= + =⎜ ⎟ ⎢ ⎥∂ ∂∂ +⎝ ⎠ ⎣ ⎦ α∂ ∂. Using Leibniz

integral rule , ( ) 2'( ) 1X Xσα σ α −= − ΘZ , where 1

0( )X dF

α σα α−∫Θ

Θ = , and note that

, we can obtain ( )( )s sV e e ( )1 11

0( ) sdF eσ σα α

∞− −= <∫σ−

( )( ) ( )

1

1 1 11

( 1) ( 1)1

s

s s

e mBBme e m

σ σ mB σ

σ σ σσ

σ τ σ ττ ττ

− − −

− − −−

Θ −∂< =

∂ +Θ+ Θ

−

Hence, [A.17] always holds.

Furthermore, since 0Bτ

∂>

∂ ,

1 1( )( ) 0B mB aa

σ στπτ τ

− −⎡ ⎤∂ +∂ ⎣ ⎦= <∂ ∂

implies that ( )1

0B στ

τ

−∂<

∂ .

Therefore, ( )

11 1

11 0X

X

Ba f

σ σ

σ

τ

τ τ

− −

−

⎡ ⎤∂ ⎢ ⎥

∂ ⎣ ⎦=∂ ∂

< , indicating that both X Xa eα = and (ex Xp F )α= are

decreasing in τ . Hence, in figure 4a , the kink of profit function ( ;a ')π τ must be located at the left to that of

( ; )aπ τ , since a . Further, to the right of the kink, the slope of 'X Xa< ( ; ')aπ τ must be greater than that

of ( ;a )π τ , since 1B στ − decreases in τ . ■

G. CARROT-AND-STICK EFFECT OF OPENING UP TO TRADE

In this appendix we prove , and if is sufficiently low.

Firstly, since

( ) ( )Ad aπ π< a a( ) ( ) ( )A

d Xa aπ π π+ > aAB B<

>

, it is straightforward that . This also

implies that for a a , the slope of

1 1( )d Ea B a f Baσ σ− −= − <

( )da a

( )A AE aπ π=

( )

f−

X π π= is smaller than . For a a ,

. From [A.17] , we know that

( )π

a

A a X<

( )1 11B m σ στ − −= +( )π π

(

( )d Xa a π= + ( )a X Emf f− −

)1

0σ−⎡ ⎤

⎣ ⎦ <1B mτ

τ

∂ +

∂, so ( ) ( )11 A11 limB m B

ττ −

→∞+ > m Bστ − ⎤+ =⎦

σ ⎡⎣ . This implies that the slope of ( )aπ is

steeper than for a a . Hence, there exists a such that for a( )A aπ X< ˆ 0a > a< ,

( ) ( ) 0AX Emf f− + >1 1aσ σ− −( )d Xπ π ( ) ( )a a π+ − 1A B m Bτ⎡ ⎤+ −⎣ ⎦a = . ■

H. PROOF OF PROPOSITION 5

Differentiating (33) and (36) with respect to Xf , we obtain :

A-5

'( ) '( ) '( )ss s s

X X s

e

X

B Y YV e BV e D e Yf B f e f

⎡∂∂ ∂ ∂⎡ ⎤+ + + − =⎢⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦

⎤ −∂⎥ ∂

[A.18]

( )

2 2

2'( ) ''( ) ''( )ss s s

X s X s

eB Y YV e BV e D e2

s X

Yf e B f e fe

⎡ ⎤⎡ ⎤ ∂∂ ∂ ∂+ + + − =⎢ ⎥⎢ ⎥∂ ∂ ∂ ∂ ∂∂⎢ ⎥⎣ ⎦ ⎣ ⎦

−∂∂

[A.19]

Using [33], [A.16] implies that

1

'( )sX X

B Y YV ef f B

−∂ −∂ ∂⎡= +⎢∂ ∂ ∂⎣ ⎦⎤⎥ [A.20]

Since 0, 0X

Y Yf B∂ ∂

<∂ ∂

> , [A.20] implies that 0X

Bf∂

>∂

. Thus, falling fixed trade cost increases welfare U ,

because U decreases in B. Further, from [38] and [A.10], we obtain ( ) ( )s E X XJ e f mf F α= + . Hence,

( )'( )

ex X ex Xs

X s

p f p f mef J e

⎡ ⎤+ ∂ ∂∂ ⎣=∂

⎦ , which implies that sign s

X

ef

⎛ ⎞∂⎜ ⎟∂⎝ ⎠

=sign ( )ex X ex Xp f p f⎡ ⎤+ ∂ ∂⎣ ⎦ . Sine

ex Xp f∂ ∂ <0 , i.e., falling fixed trade cost increases the proportion of exporting firms, falling Xf leads to

ambiguous effects on se , and will reduce se if ( )( )

1/

ex X

ex X

p fp f

− ∂ ∂< , i.e. the elasticity of exp with respect to.

Xf is less than unity.■

I. PROOF Xa a<

In this appendix we prove that 1

11 1 ( ) X

X

Ba mf

σστ

−− −⎡ ⎤∂

≡ +⎢ ⎥∂⎣ ⎦a< . This is equivalent to show that

( )1 1(1 )X X

B mf a mσ στ− −

∂<

∂ + [A.21]

Using [A.20], we obtain

1 1 1 1 1

0 0

( )

( ) ( )( )X

X X

X

Yf mFB

Yf e dF e m dFV eB

ασ σ σ σ σ

α

α α τ α α∞− − − − −

∂−∂∂

= =∂∂ ++∂ ∫ ∫ 1 1 1

0

( )

(1 ) ( )X

XmF

e m dFασ σ σ

α

τ α α− − −<

+ ∫ [A.22]

A-6

Using 1

0( ) 1 ( ) 0X X

XZ dσ

α αα αα

−⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

∫ ,F > , we have .This

is equivalent to ( )

1 1

0( ) ( ) 0X

X XdF Fασ σα α α α− − − >∫

1 1

0( ) ( )X

X Xa e dF Fασ σα α α− − >∫ , since X Xa eα = . Hence,

( )1 11

0

( )(1 )( )X

X

X

ma mdF

α σ1 1(1 )

mF

m e σσ σ σ

αττ α α

− −−<

+∫− −+. Using [A.22] , this implies that [A.21] always holds. ■

A-7