Public transfers of climate-mitigation technologies:
The crowding-out eect on relocation
This version: January 31, 2017
Technology transfers can come from government decisions or be a
business choice. Indeed, when rms relocate, they generate technology transfer. These two types of decision seek divergent goals. The purpose of this
paper is to investigate the relationship between the rms' incentives to relocate and the countries' incentives to transfer climate-mitigation technologies.
We consider two countries (home and foreign) implementing a carbon tax.
The government in the home economy decides to transfer or not its technology to the foreign economy, and rms located in the home economy may
decide to relocate their production in the foreign economy. We consider two
types of rms. At home, there are only relatively clean rms, while in the
foreign economy relatively clean and dirty rms coexist. We show that the
governments' transfer of technology decreases the incentives to relocate.
Abstract.
Keywords.
Technology transfer; Carbon tax; Relocation; Trade; Imperfect
JEL codes.
L13, Q53, Q58.
competition.
1. Introduction
A promising avenue to combat climate change proposed during COP 16 consists in enhancing the transfer of green technologies from developed countries to developing countries. Indeed, green technologies are concentrated in developed countries ([Dechezleprêtre et al., 2011]),
while developing countries now produce the majority of the world's CO2 emissions. Being potentially benecial for developing countries, the transfer of green technologies may
have adverse eects on developed economies. Indeed, sharing innovations may enable
rms located in developing countries to reduce more rapidly their emissions, but it may
also increase their competitiveness. Technology transfers can come from government
decisions or be a business choice. Indeed, when rms relocate, transfers of technology
occur. These two types of decision seek divergent goals; while rms seek to maximize
their prots, governments are expected to maximize the welfare of their country. The
purpose of this paper is to study the interactions between public and private transfers of
clean technologies.
Technology may mainly be purchased in the market in which innovators sell their
licenses. Developing countries' rms claim that these licenses may be too expensive and
1
2
that they cannot aord for them. However, governments in developed countries have
several instruments at their disposal to transfer technologies. For instance, they may decide to relax the intellectual property rights (IPP) on green innovations ([Maskus, 2010]).
By imposing compulsory licensing, a government allows a foreign rm to use a patented
process without the consent of the patent owner. Nevertheless, compensations may be
required. Firms may claim that such policies look like expropriation. The developed
countries can subsidize the southern rms to purchase licenses. This paper focuses on
this way for governments to transfer climate-mitigation technologies.
The technology diusion can also be done through rms' relocation. Indeed, rms
settle in a foreign country bringing their knowledge and technologies. Relocation directly
aects the technology used in the host country, but it may also enhance the diusion of
technology through knowledge spillovers. Indeed, it is easier to imitate rms, which are
located in developing countries. By hiring employees abroad, the rms train new employees and may generate horizontal spillovers and by cooperating with local suppliers they
may also generate vertical technology transfers. However, relocation can be particularly
detrimental for a country since they induce a destruction of human and physical capitals,
leading to a loss of specic knowledge and skills. During the past 25 years, employment in
manufacturing as a share of total employment has fallen dramatically in the world's most
advanced economies. Hence, relocation may increase unemployment rate. Apart from
these economic consequences, relocation may also be detrimental for the environment.
Indeed, rms, which relocate in countries implementing more lenient environmental regulations, may contribute to increase emissions by using dirtier technology or by producing
more.
Various tools may be used to prevent rms from relocating. First, subsidies can be
given to rms in order to reduce the production costs. Another solution for the government could be to purchase goods to rms that are exposed to a high risk of relocation.
Nationalizing those rms is also an alternative, since it may prevent them easily from
closing their domestic plants, but is politically sensitive. This paper demonstrates that
government transfers of technologies may prevent rms from relocating.
We develop a simple partial equilibrium model to fathom the economics of the international diusion of climate mitigation technologies in a world with risk of relocation. The
model describes the interactions between two regions, home and foreign, implementing
a carbon tax. In each region, rms produce the same homogeneous polluting good. We
consider two types of rms: relatively clean and dirty rms. The cleanliness of a rm is
given by its emissions intensity, that is the number of emissions by unit produced. We
assume that the home economy is more advanced than the foreign economy, in the sense
that in this region, all rms use a relatively clean production process, while in the foreign
economy, relatively clean rms and dirty rms coexist. Moreover, in the home economy,
the environmental willingness to pay and the production costs are higher. These assumptions reect the facts that the environmental awareness increases with the development,
and that labour is usually more expensive in advanced economies.
We assume that rms in the home economy may decide to relocate to the other region
at a x and homogeneous cost. By relocating its production, a home rm benets from
lower production cost and weaker environmental regulation. We assume that there is an
Public transfers of climate-mitigation technologies
3
innovator in the home economy selling the cleaner technology at a at rate. Cleaner rms
located in both economy have already bought the technology. The home government may
decide to purchase licenses to this innovator and give them freely to foreign dirty rms.
By doing so, it subsidies the purchase of licenses in the foreign economy. By deciding to
transfer its technology, the home government makes the cleaner technology freely available for rms located in the foreign economy. We assume that adoption is costless, so
that if a transfer occurs all rms produce with the same technology.
Bilateral trade is assumed and this paper demonstrates that the transfer decreases
the incentives of relatively-clean rms to relocate since it eliminates their comparative
advantage. In the long run, the transfer decreases the number of relocated rms which
leads to a less concentrated market in the home economy. In such a case the reduction of
global emissions is a sucient condition for the transfer since it increases the consumer
surplus and prots at home.
The paper is structured as follows. We begin by reviewing the related literature in
Section 2. Section 3 presents the modeling assumptions. Section 4 analyzes the crowdingout eect in the short-run. Section 4 focuses on the long-run market structures and the
decision to transfer technology. Section 5 discusses the robustness of the results. Section
6 derives policy implications and concludes the paper.
2. Relation to the literature
This paper contributes to several strands of the literature.
First, this paper contributes to the literature which focuses on the incentives of developed economies to transfer clean technologies to developing countries. The pioneering
work by [Stranlund, 1996] highlights the eect of emissions on the incentive to transfer
since he assumes no trade. Papers that are more recent focus on the eects of trade on
the incentives to transfer green technologies. [Stephan and Muller-Furstenberger, 2015]
analyze the incentive to transfer energy-saving technologies when there is trade on energy, while [Helm and Pichler, 2015] assume trade in a global carbon market. Finally
[Glachant et al., 2016] assume that there is trade in the polluting good market. These
studies highlight the fact that transferring clean technology does not necessarily improve
the environment, and that the transfer decision is highly aected by the terms of trade.
The present paper is complementary and shows that promoting the diusion of clean
technologies may prevent rms from relocating. This result may give further incentive
to developed countries to transfer their clean technologies. The paper also shows that in
the long-run the emissions decrease is a sucient condition to the public transfer while
in [Glachant et al., 2016], it is a necessary condition.
This paper is also related to papers studying how to reduce the risk of relocation in
a context of environmental regulation. For instance, [Martin et al., 2014], in a context
of pollution of permits, determine the number of free allowances that are sucient to
prevent rms from relocating. [Hoel, 1997] assumes that countries implement a pollution
tax to maximize the welfare. He also assumes no historical location and highlights the
trade-o a government faces when setting its environmental regulation. On the one hand,
a government wants to attract industry, but on the other hand, it wants to locate the
4
pollution abroad (no trans-boundary pollution). More recently, [Ikefuji et al., 2016] consider a global pollution and studies how the commitment to environmental regulations
aects the rm's relocation choice. He considers a Cournot-duopoly. The characteristics
of pollution highly aects the results. In the present paper, we also focus on global pollution, and we study how the public transfer aects the relocation choice.
3. The set-up
The model describes two countries k = {H, F } where H and F respectively denote the
home and the foreign country. In each country there are respectively NH and NF rms
producing an homogenous polluting good. In each country, consumers purchase goods.
The prices are given by the inverse demand function: pH = aH − QH and pF = aF − QF
where aH and aF are respectively the market size in the home and in the foreign country.
Production creates emissions and we assume that abatement technologies are not
available. There are two types of rms, clean ones emitting µc units of emissions by unit
produced, and dirty ones emitting µd units of emissions by unit produced, where µd > µc .
The cleaner technology is sold by an innovator, who is located in the home country. The
innovator sells the technology at a at rate X on a competitive market. Moreover, we
assume that in the home country, all the rms have already bought the cleaner technology, while in the foreign country, clean and dirty rms coexist µF = {µc , µd }. Let us
denote NH , the number of clean rms in the home economy, NFd , the number of dirty
foreign rms, and NFc the number of clean foreign rms. The cleaner rms in the foreign
country are rms which have purchased the technology. The number of rms located in
the foreign economy is then: NF = NFd + NFc . Note that for simplicity, we refer to clean
rms, the cleaner rms, even if they also pollute.
Domestic and foreign rms may sell either at home or abroad. Said dierently, bilateral trade is assumed. The production of a home rm sold in the home and in the foreign
economy is respectively denoted by rHH and rHF . Let us also denote rF Hd and rFd F (rFc H
and rFc F ), the production of a foreign dirty (foreign clean) rms, sold respectively in the
home and in the foreign market. Transport is costly, and symmetric, and a unit transportation cost t is assumed.1
Let us denote ck the production cost in region k and τk > 0 the carbon tax implemented in the country k. We do not consider cases where governments subsidize
production. In the foreign country, clean and dirty rms have the same production costs
cF . Thus the production costs can be interpreted as labor costs. We assume that the
production cost is higher in the home economy (cH > cF ) and that the emissions tax is
also higher (τH > τF ).
The home government may transfer the technology to the rms in the South by subsidizing the purchase of the technology. If the home government transfers its technology, all
rms in the foreign economy use the clean technology µc . To enter into details the home
1 This
assumption diers from [Glachant et al., 2016] in which no transportation costs is considered
and a global market for polluting good is assumed.
Public transfers of climate-mitigation technologies
5
government directly purchases the technology to the innovator and gives it to the dirty
foreign rms. The home government transfers its technology if it increases its welfare.
Emissions generate a global damage, assumed to be linear, and the welfare is dened as
the sum of the consumer surplus, the sum of the domestic prots including the prots of
the innovator and the tax revenue minus the environmental damage.
The rms in the home country may relocate in the foreign country. Let us assume
a constant and unique cost of relocation C R . A clean rm located at home relocates its
production if, and only if, the prots realized in the foreign country is higher than the
current prots in the domestic country minus the relocation cost. Moreover, the rm
takes into account that relocation modies market structures.
4. The crowding-out effect
Firms' production.
(4.1)
max
rHHi ,rHFi
The rms in the home country solve the following problem.
πHi (NH , NFd , NFc ) = (pH − cH − τH µc ) rHHi + (pF − cH − τH µc − t) rHFi
In the foreign country, dirty and clean rms coexist. They respectively solve the
following problems.
(4.2)
max
d
d
rF
H ,rF F
i
(4.3)
πFd i (NFd , NFc , NH ) = (pH − cF − τF µd − t) rFd Hi + (pF − cF − τF µd ) rFd Fi
i
max πFc i (NFc , NFd , NH ) = (pH − cF − τF µc − t) rFc Hi + (pF − cF − τF µc ) rFc Fi
c
c
rF
H ,rF F
i
i
By calculating the rst-order conditions and solving the system of equations, we obtain
the following productions, which are detailed in Appendix.
An increase in µd increases the productions of clean rms (rHH , rHF , rFc H and rFc F )
since it increases their comparative advantage. This increase is particularly signicant
when the carbon tax is high. Note that the prices of the polluting good in both markets
increase with the emission intensity, the production costs and the market size.
If the home government transfers its technology, all rms in the
foreign economy then use the clean technology µc . The transfer costs NFd X to the government which is a lump-sum transfer from the government to the domestic innovator.
By studying how the transfer aects the outcome, we deduce the following lemma:
Public Transfer.
Lemma
1. The public transfer
- increases the consumer surplus, and decreases the prot at home,
- decreases productions and emissions from clean rms located in both economies
- increases the production of foreign dirty rms, and has an ambiguous eect on their
emissions.
Proof. Proof in Appendix 9
6
The transfer of technology increases the production of foreign dirty rms (rFd H and
rFd F ) and decreases the production of clean rms in both regions. Indeed, clean rms
loose their comparative advantage in both markets, while dirty rms become more competitive. Thus, the transfer decreases the prot of home rms. However, it increases
the consumer surplus since it increases the overall production sold in both regions. The
transfer decreases the emissions from clean rms in both regions, while it has an ambiguous eect on the foreign dirty rms' emissions. Indeed, they produce more, but emit less
by unit produced. As a result, the public transfer has an ambiguous eect on the environment. Moreover, it decreases the tax revenue since the domestic production decreases.
The transfer decreases the overall emissions if:
aH +aF −t+2 µc NH τH −2 µd (NH + 1) + µd − µc NFc + µc τF −2 (cF (NH + 1) − cH NH ) ≥ 0
The transfer decreases the overall emissions when NFc is low, τH is large, and τF is low.
Indeed, a lax environmental regulation in the foreign economy implies that dirty rms
produce and emit a large amount without transfer. Thus, by transferring its technology,
the home government is able to signicantly decrease their emissions. Using the same
argument, when the number of clean rms in the foreign economy is low, dirty rms
produce large quantity since the market is then concentrated, and since they only compete
with few clean rms.
The government incentives to transfer are given by the dierence between the home
welfare without and with transfer minus the transfer cost NFd X . In Section 5, we focus
on the conditions to transfer in the long-run.
A home rm relocates its production if and only if, the prots
realized in the foreign country is higher than the current prots in the domestic country
minus the relocation cost. Said dierently, if:
Firms' relocation.
πFc H (NFc + 1, NFd , NH − 1) + πFc F (NFc + 1, NFd , NH − 1) − C R >
(4.4)
πHF (NH , NFd , NFc ) + πHH (NH , NFd , NFc )
We use the properties of the linear demand, such that if the demand function is
P (Q) = a − Q, then πj = rj2 . Thus, π1 − π2 is equal to (r1 − r2 ) (r1 + r2 ). The dierences
between foreign and domestic quantities respectively sold in the domestic and the foreign
markets are equal to:
NH + NFd + NFc (µc (τH − τF ) + cH − cF − t)
+
− 1) −
=
NH + NFd + NFc + 1
c
d
c
N
+
N
+
N
(µ (τH − τF ) + cH − cF + t)
H
F
F
rFc F (NFc + 1, NFd , NH − 1) − rHF (NH , NFd , NFc ) =
NH + NFd + NFc + 1
rFc H (NFc
1, NFd , NH
rHH (NH , NFd , NFc )
By assumption µc (τH − τF ) + cH − cF + t is positive, while µc (τH − τF ) + cH − cF − t
can be positive or negative. By studying how a decrease in µd aects (5.1), we deduce
the following proposition.
Proposition
cate.
1. The transfer of technology decreases the home rms' incentive to relo-
Public transfers of climate-mitigation technologies
7
To give the intuition, consider for a moment that there is no trade. Since the markets
for products in the two countries are independent, the transfer does not aect home
prots. However, the prot of foreign clean rms decreases with the transfer. Indeed,
if a transfer occurs, clean rms do not longer have a comparative advantage when they
relocate. As a result, the transfer decreases the incentive to relocate.
Turning back to the case in which trade is bilateral, by relocating, a home rm
benets from low labor costs, lenient environmental regulation, but has to pay for the
transportation cost to sell the good to the home market. Hence, relocation increases its
prot related to the foreign market, especially if its comparative advantage is high (high
µd ). On the opposite, relocation has an ambiguous eect on the prot related to the
home market. Nevertheless, if relocation increases (decreases) the home rm's prot in
the domestic market, public transfer decreases this gain (increases this loss). As a result,
the transfer decreases the incentive to relocate.
5. Long-term market structures and public transfers
We now use a three-stage game to endogenize the rms' location:
Stage 1. The home country decides whether it transfers its technology to the foreign economy.
Stage 2. Firms decide whether they relocate.
Stage 3. Firms produce and sell the good in the local market for products.
We solve this problem backwards. The third stage is similar to the one dened in section
4. We focus then on the second stage.
Home rms decide whether they relocate their production or not and they
act sequentially. Firms have the incentives to relocate their production in the foreign
economy as long as their prot net of the relocation cost is larger than the home prot.
Thus, we can dene the equilibrium rms' location. The number of rms that relocates
their production is given by the prots of the last rm that is willing to relocate its
production. Assuming that there is no technology transfer, this rm n is indierent
between staying in the home economy or relocating in the foreign economy:
Stage 2.
πF Hc (n, NFd , NH − n) + πF Fc (n, NFd , NH − n) − C R =
(5.1)
πHF (NH − n + 1, NFd , n − 1) + πHH (NH − n + 1, NFd , n − 1)
By solving (5.1), we dene n given in appendix 10. By deriving n with respect to µd ,
we obtain:
∂n
>0
∂ µd
Corollary
1. The public transfer of technology decreases the number of relocated rms
in the long run (nT < nSQ ).
8
This corollary is a direct implication of Proposition 1. Indeed, since the transfer
decreases the incentives to relocate, the transfer induces in the home economy a less
concentrated market-structure in the long run.
The home government decides whether it transfers its technology to the foreign
country.2 It anticipates that the transfer aects the rm's location.
Stage 1.
In the long run, there are (NH − n) at home, and n clean rms in the foreign market.
By studying the eect of the technology transfer on the outcome, the following proposition
is deduced:
Proposition
2. In the long run, the transfer
- increases the consumer surplus at home,
- increases the gross prot and the emissions at home,
- has an ambiguous eect on the emissions from the foreign dirty rms,
- decreases the emissions from the foreign clean rms.
At the status quo (without transfer), the global emissions are given by:
SQ
SQ
SQ
SQ
E = NH − nSQ µc (rHH
+ rHF
) + NFd µd (rFd H + rFd F ) + nSQ µc (rFc H SQ + rFc F SQ )
In the home market, public transfer decreases the quantity sold by clean rms from
both regions, and increases the quantity sold by dirty since they catch up their technical
gap . The overall eect is such that the consumer surplus increases. Moreover, even if the
home individual prot decreases the gross prot increases since public transfer decreases
in the long run relocation, and thus increases the number of home rms.
In the foreign market, public transfer has an ambiguous eect on the price and on
the quantity sold by clean rms from both regions. Indeed, two eects are at stake.
Public transfer cancels the comparative advantage of clean rms, which decreases their
production. But it also decreases the competition in the foreign market since less local
rms, saving transportation costs, are on this market. Public transfer only decreases the
price and the individual production of clean rms if µc (τH − τF ) + cH − cF − t > 0, that
is if the transportation costs is relatively large.
Public transfer has an ambiguous eect on the emissions of initially-dirty rms since
they produce more but emit less by unit produced. It decreases the emissions of foreign
clean rms, and it increases the emissions of home rms.
6. Discussion and Robustness
We study the robustness of our results.
2 The
welfare is convex in µd , thus the government chooses to transfer or not its entire technology
(corner solutions)
Public transfers of climate-mitigation technologies
9
A crucial assumption of this paper is that domestic rms
close their plants in the home country and settle in the foreign country. Multinational
companies may have plants in various countries. We consider then rms deciding to
build a plant in the foreign country, while keeping its plant at home. The government
cares about the employment level. If a rm decides to open a plant abroad, then this
decreases the production in the home economy and consequently the employment (since
the production level aects the labor demand), since the competition in the foreign market
is ercer. In such a context, public transfers of climate-mitigation technologies decrease
the incentive to open a plant abroad by reducing the comparative advantage of domestic
rms. Even considering multinational companies, our results are still valid.
Multinational companies.
Foreign rms may imitate the climatemitigation technologies used by rms which have relocated their production to their
country or which have purchased the technology to the innovator. Indeed, it is easier
to copy and imitate a rm located in the foreign economy than a rm located in the
home economy. We consider then that the emission intensity of dirty rms decreases
with relocation. The more home rms relocate, the more the foreign dirty rms imitate,
the higher spillovers are. Taking into account imitation and spillovers, the incentives to
relocate are lower since the rms imitate and the comparative advantage is lower. As a
result, in the long-run, the long-run market structure would be even less concentrated
than without transfer.
Imitation and knowledge spillovers.
Collecting tax revenue may be costly for the government and some distortionary taxes may also exist. Said dierently, there may be a
shadow cost of public funds. A shadow cost of public fund reduces the tax revenue and
increases the cost of transferring the technology. In Section 3, the subsidies given to
foreign dirty rms to buy the cleaner technology were a lump-sum transfer between the
government and the innovator. However, if there is a cost of public fund, the cost of
subsidizing foreign rms is higher than the innovator's prot. However, the existence of
a shadow cost of public funds only aects the decision to transfer without aecting the
rms' incentive to relocate. Our results are then still valid.
Shadow cost of public funds.
We assume that the
market for technology is competitive. However, if the innovator has a market power, then
he set the price of the innovation such that foreign rms are indierent between adopting
the cleaner technology or producing with the dirty technology. If so, it increases the cost
of transferring, since it increases the price of technology. However, such an assumption
does not modify the decisions to relocate.
Imperfect competition in the market for technology.
We have assumed that the decisions for technology adoption were exogenous. If we make the decisions endogenous, the
anticipation of the public transfer reduces the incentive to adopt the technology.
Endogenous choice for technology adoption.
Optimal taxation. If countries do not cooperate in environmental regulation, and
implement the emissions tax that maximize their own welfare, then the eect of public
transfer on the incentives to relocate mainly depend on the eect of public transfer on the
environmental regulation. However, public transfer has an ambiguous eect on environmental regulations. On the one hand, public transfer allows governments to decrease the
emissions taxes, since for a given level of production (and a given price for consumers),
10
the environmental damage is lower. This eect is particularly signicant in the foreign
government, since it also increase the prots. This eect, may increase the incentive to
relocate. On the other hand, for the home rms the competition is ercer, which may lead
to a decrease in the home emissions tax, which would reduce the incentive to relocate.
7. Conclusion
In this paper, the relation between the rms' incentives to relocate and the countries' incentives to transfer climate-mitigation technologies has been investigated. We have shown
that technological transfer may decrease the incentives to relocate. Thus, the transfer of
technologies may be used to prevent rms from relocating, which is currently a hot topic
issue. [Glachant and Dechezleprêtre, 2016], show that climate-friendly technologies spill
over boundaries through market mechanisms and foreign development investments. In
other words, clean technologies are going to be transferred anyway but it is more favorable for developed country to transfer themselves and to keep the domestic industries at
home than loosing on the two dimensions.
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730-741.
Appendix
8. Productions and prices
(NFd + NFc ) (t + cF − cH ) − µc NFd + NFc + 1 τH + µc NFc + µd NFd τF + aH − cH
rHH =
NH + NFd + NFc + 1
− NFd + NFc + 1 (t + µc τH ) + µc NFc + µd NFd τF + (cF − cH ) (NFd + NFc ) + aF − cH
rHF =
NH + NFd + NFc + 1
c
c
d
d
c
N
t
+
µ
N
τ
−
µ
(N
+
1)
+
µ
−
µ
NF τF + (cH − cF ) NH + aF − cF
H
H
H
H
rFd F =
d
c
NH + NF + NF + 1
c
− (NH + 1) t − µ NH τH + µd (NH + 1) + µd − µc NFc τF + (cH − cF ) NH + aH − cF
rF Hd =
NH + NFd + NFc + 1
NH t + µc NH τH + µd − µc NFd − µc (NH + 1) τF + (cH − cF ) NH + aF − cF
c
rF F =
NH + NFd + NFc + 1
d
c
d
c
c
−
(N
+
1)
t
+
µ
N
τ
+
µ
−
µ
N
−
µ
(N
+
1)
τF + (cH − cF ) NH + aH − cF
H
H
H
H
F
rFc H =
d
c
NH + NF + NF + 1
The prices are given by:
aH + cH NH + NFc + NFd (t + cF ) + µc NH τH + µc NFc + µd NFd τF
pH =
NH + NFc + NFd + 1
aF + cF NFc + NFd + NH (t + cH ) + µc NH τH + µc NFc + µd NFd τF
pF =
NH + NFc + NFd + 1
9. Proof of Lemma 1
∂ pH
∂ µd
NFd τF
∂ pF
NH +NFc +NFd +1 ∂ µd
∂ πH
2
2
rHH
+ rHF
=
∂ µd
;
=
πH =
;
NFd τF
NH +NFc +NFd +1
2 rHH ∂∂rHH
+ 2 rHF ∂∂rµHF
d
µd
=
Public transfers of climate-mitigation technologies
∂ rHH
∂ µd
∂ rF H c
∂ µd
∂ rF H d
∂ µd
NFd τF
NFd τF
∂ rHF
>
0
=
>0
c
d
d
∂µ
NH +NF +NF +1
NH +NFc +NFd +1
c
N d τF
∂ rF
NFd τF
F
= NH +NFc +N
d +1 > 0
d = N +N c +N d +1 > 0
∂
µ
H
F
F
F
F
d
(NH +NFc +1) τF
(NH +NFc +1) τF
∂ rF
F
= − NH +N c +N d +1 < 0 ∂ µd = − NH +N c +N d +1
F
F
F
F
=
c (N c +1,N −1)
∂ rF
H
H
F
∂ µd
,
,
,
,
,
=
c (N c +1,N −1)
∂ rF
H
F
F
∂ µd
=
∂ rHF (NFc ,NH )
∂ µd
=
13
<0
∂ rHF (NFc ,NH )
∂ µd
=
NFd τF
NH +NFd +NFc +1
∂ (5.1)
2 NFd τF (rHH (NFc , NH ) + rHF (NFc , NH ) − rFc F (NFc + 1, NH − 1) − rFc H (NFc + 1, NH − 1))
=
∂ µd
NH + NFd + NFc + 1
4 NFd NH + NFd + NFc τF (µc (τH − τF ) + cH − cF )
=
2
NH + NFd + NFc + 1
10. Proof of Corollary 1
(cF − cH ) t + 2 NFd µc τH − µd τF + 2 µc τF + 2 cF − aH − aF + t (t + aF − aH )
n=
2 t2 + µc2 (τH − τF )2 + (cH − cF ) (2 µc (τH − τF ) + cH − cF )
µc (τH − τF ) t + 2 NFd µc − µd τF − cH + cF + 2 µc τF + 2 cF − aH − aF
NH − NFd
+
−
2
2 t2 + µc2 (τH − τF )2 + (cH − cF ) (2 µc (τH − τF ) + cH − cF )
NFd τF (µc (τH − τF ) + cH − cF )
∂n
=
>0
∂ µd
t2 + µc2 (τH − τF )2 + (cH − cF ) (2 µc (τH − τF ) + cH − cF )
11. Proof of Proposition 2
By replacing NH by NH − n and NFc by n, we obtain:
SQ
T
T
rHH
− rHH
= rFc H SQ − rFc H T = pSQ
H − pH
µd − µc NFd τF t (t + µc (τH − τF ) + cH − cF )
>0
=
NH + NFd + 1 t2 + µc2 (τH − τF )2 + (cH − cF ) (2 µc (τH − τF ) + cH − cF )
SQ
T
T
− rHF
= rFc F SQ − rFc F T = pSQ
rHF
F − pF
µd − µc NFd τF t (µc (τH − τF ) + cH − cF − t)
=−
NH + NFd + 1 t2 + µc2 (τH − τF )2 + (cH − cF ) (2 µc (τH − τF ) + cH − cF )
The public transfer decreases in the home markets, the clean rms' individual production
and the price. On the opposite, it has an ambiguous eect on the foreign price and on
14
the clean rms' individual production sold on the foreign market
SQ
rH
−
T
rH
=
2 µd − µc NFd τF t2
NH + NFd + 1
∂ rF Hd
=
∂ µd
∂n
∂ µd
∂n
∂ rFd F
∂ µd
=
−
∂ µd
t2 + µc2 (τH − τF )2 + (cH − cF ) (2 µc (τH − τF ) + cH − cF )
(t − µc (τH − τF ) − cH + cF ) − (NH + 1) τF
NH + NFd + 1
(t + µc (τH − τF ) + cH − cF ) + (NH + 1) τF
∂n
2 ∂ µd
∂ rF Hd ∂ rFd F
+
=
−
∂ µd
∂ µd
<0
<0
NH + NFd + 1
(µc (τH − τF ) + cH − cF ) + 2 (NH + 1) τF
NH + NFd + 1
<0
where rH = rHH + rHF , the public transfer decreases the individual prot at home and
increases the overall production of initially dirty rms.