3.1.1. Assumptions

The model describes two countries $j=\{N,S\}$ where N and S denote the North and the South respectively. In country j, there are $m_{j}$ firms producing a homogeneous polluting good, and consumers purchasing the goods. The prices are given by the inverse demand function: $p_{j}=a_{j}-Q_{j}$ where $a_{j}$ is the market size in country j, and $Q_{j}$ the quantity consumed in country j.

Production generates emissions that create a global damage and we assume that abatement technologies are not available. The production technology is characterized by an emission intensity parameter μ. We consider two technologies: a relatively clean technology, and a dirty technology. The relatively clean technology creates $\mu ^{c}$ units of emissions per unit produced, and the dirty one creates $\mu ^{d}>\mu ^{c}$ units of emissions per unit produced. We assume that in the North all the firms use the relatively clean technology, while in the South both technologies are used. We denote by $m_{S}^{d}$ the number of dirty southern firms, and by $m_{S}^{c}$ the number of relatively clean southern firms. The number of firms located in the South is then $m_{S}=m_{S}^{d}+m_{S}^{c}$. For simplicity, we refer to relatively clean firms as clean firms, even if they also pollute.

Let us assume that there is an innovator in the North which sells the clean technology. The clean technology price is denoted by K and is assumed to be exogenous. Firms using the clean technology have already purchased the technology and, therefore, this cost represents a sunk cost to the firm. Nevertheless, dirty firms have the possibility to buy clean technology on the patent market. We consider that there is no cost of adoption.

Let us assume single-plant firms that are located in a given country but still serve both markets (North and South). In other words, they produce at home and export to foreign countries. This assumption will be relaxed in section 4.2. The production of a northern firm i sold in the North and in the South is denoted by $r_{NN_{i}}$ and $r_{NS_{i}}$ respectively. Let us also denote by $r_{SN_{i}}^{d}$ and $r_{SS_{i}}^{d}$ ($r_{SN_{i}}^{c}$ and $r_{SS_{i}}^{c}$) the production of a dirty southern (clean southern) firm i, sold respectively in northern and southern markets. The market clearing condition implies that $Q_{N}=\sum _{i=1}^{m_{N}}r_{NN_{i}}+\sum _{i=1}^{m_{S}^{c}} r_{SN_{i}}^{c}+\sum _{i=1}^{m_{S}^{d}}r_{SN_{i}}^{d}$ and $Q_{S}=\sum _{i=1}^{m_{N}}r_{NS_{i}}+\sum _{i=1}^{m_{S}^{c}} r_{SS_{i}}^{c}+\sum _{i=1}^{m_{S}^{d}}r_{SS_{i}}^{d}$. Transport is costly and let t be the constant unit transportation cost.

We consider that each country implements an emissions tax. Indeed, many developing countries have implemented or plan to implement environmental regulation. For instance, carbon taxes were launched in Chile and Colombia. Moreover, according to the World Bank and Ecofys (2018) in Argentina and South-Africa, an Emissions Trading Scheme or carbon pricing are scheduled and these instruments are under consideration in Brazil, Côte d'Ivoire, Thailand and Vietnam. Let us denote by $c_{j}$ the marginal production cost in country j and by $\tau _{j}>0$ the carbon tax implemented in country j. In the South, clean and dirty firms have the same marginal production cost $c_{S}$, hence the marginal production cost is country-specific. We assume that the marginal production cost and the emissions tax are higher in the North. Let us use the following notations: $\Delta \tau \equiv \tau _{N}-\tau _{S}>0$, $\Delta c\equiv c_{N}-c_{S}>0$ and $\Delta \mu \equiv \mu ^{d}-\mu ^{c}>0$.

3.1.2. Production and prices

Let us now determine the equilibrium production levels. Northern firms bought the technology in the past, and the costs of buying the clean technology no longer appear in their profit. Each northern firm solves the following problem:

\[ \max_{r_{NN_{i}},r_{NS_{i}}}\pi _{N_{i}} \left(m_{N},m_{S}^{c},m_{S}^{d}\right) =(p_{N}-c_{N}-\tau _{N} \mu^{c})r_{NN_{i}}+(p_{S}-c_{N}-\tau _{N}\mu^{c}-t)r_{NS_{i}}.\]

In the South, dirty and clean firms coexist. They respectively solve the following problems.

\begin{align*} & \max_{r_{SN_{{i}}}^{d},r_{SS_{{i}}}^{d},} \pi_{S_{{i}}}^{d}\left( m_{S}^{d},m_{S}^{c},m_{N}\right) =(p_{N}-c_{S}-\tau_{S} \mu ^{d}-t)r_{SN_{{i}}}^{d}+(p_{S}-c_{S}-\tau _{S}\mu^{d})r_{SS_{{i}}}^{d} \\ & \max_{r_{SN_{{i}}}^{c},r_{SS_{{i}}}^{c}}\pi_{S_{{i}}}^{c} \left(m_{S}^{c},m_{S}^{d},m_{N}\right) =(p_{N}-c_{S}-\tau_{S} \mu ^{c}-t)r_{SN_{{i}}}^{c}+(p_{S}-c_{S}-\tau _{S}\mu^{c})r_{SS_{{i}}}^{c}. \end{align*}

By calculating the first-order conditions and solving the system of equations, we obtain the production:

\begin{align*} r_{NN}(m_{N},m_{S}^{c},m_{S}^{d})& =\frac{a_{N}-c_{N}-\mu ^{c}\tau _{N}-(\Delta c+\mu ^{c}\tau _{N}-t)m_{S}+\left(\mu ^{c}m_{S}^{c}+\mu ^{d}m_{S}^{d}\right) \tau _{S}}{m_{N}+m_{S}+1} \\ r_{NS}(m_{N},m_{S}^{c},m_{S}^{d})& =\frac{a_{S}-c_{N}-\mu ^{c}\tau _{N}-\left( \Delta c+\mu ^{c}\tau _{N}+t\right) m_{S}+\left( \mu ^{c}m_{S}^{c}+\mu ^{d}m_{S}^{d}\right) \tau _{S}-t}{m_{N}+m_{S}+1} \\ r_{SS}^{d}(m_{S}^{d},m_{S}^{c},m_{N})& =\frac{a_{S}-c_{S}-\mu ^{d}\tau _{S}+(\Delta c+\mu ^{c}\tau _{N}-\mu ^{d}\tau _{S}+t)m_{N}-\Delta \mu m_{S}^{c}\tau _{S}}{m_{N}+m_{S}^{d}+m_{S}^{c}+1} \\ r_{SN}^{d}(m_{S}^{d},m_{S}^{c},m_{N})& =\frac{a_{N}-c_{S}-\mu ^{d}\tau _{S}+\left( \Delta c+\mu ^{c}\tau _{N}-\mu ^{d}\tau _{S}-t\right) m_{N}-\Delta \mu m_{S}^{c}\tau _{S}-t}{m_{N}+m_{S}+1} \\ r_{SS}^{c}(m_{S}^{c},m_{S}^{d},m_{N})& =\frac{a_{S}-c_{S}-\mu ^{c}\tau _{S}+(\Delta c+\mu ^{c}\Delta \tau +t)m_{N}+\Delta \mu m_{S}^{d}\tau _{S}}{m_{N}+m_{S}+1} \\ r_{SN}^{c}(m_{S}^{c},m_{S}^{d},m_{N})& =\frac{a_{N}-c_{S}-\mu ^{c}\tau _{S}+\left( \Delta c+\mu ^{c}\Delta \tau -t\right) m_{N}+\Delta \mu m_{S}^{d}\tau _{S}-t}{m_{N}+m_{S}+1}. \end{align*}

In each market, a clean southern firm produces more than a dirty southern firm. Moreover, a clean southern firm produces more than a clean northern firm in the southern market since it benefits from a low tax, a low production cost and does not pay for the transportation cost. Obviously, an increase either in the northern tax or in the northern production cost increases the production of southern firms, while an increase either in the southern tax or in the southern production cost increases the production of northern firms. Note that as a firm increases its production, its profit increases as well. From the functional forms chosen, the profits are equal to $\pi _{N}=r_{NN}^{2}+r_{NS}^{2}$; $\pi _{S}^{c}={r_{SS}^{c}}^{2}+{ r_{SN}^{c}}^{2}$; $\pi _{S}^{d}={r_{SS}^{d}}^{2}+{r_{SN}^{d}}^{2}$. At the equilibrium, prices are equal to:

\begin{align*} p_{N}& =\frac{a_{N}+m_{N}(c_{N}+\mu ^{c}\tau _{N})+m_{S} \left(t+c_{S}\right) +\left( \mu ^{c}m_{S}^{c}+\mu ^{d}m_{S}^{d}\right) \tau _{S}}{m_{N}+m_{S}+1} \\ p_{S}& =\frac{a_{S}+c_{S}m_{S}+m_{N}\left( c_{N}+\mu ^{c}\tau _{N}+t\right) +\left( \mu ^{c}m_{S}^{c}+\mu ^{d}m_{S}^{d}\right) \tau _{S}}{m_{N}+m_{S}+1}. \end{align*}

In each region, the price increases with the transportation cost, the taxes, the production costs and the market size.

3.1.3. Relocation

Northern firms may relocate to the South, that is, instead of serving both markets from the North, they may decide to relocate their production in the South and serve both markets from the South. In such a case, the firm will continue to produce in only one country and export its production to the other country. Let us assume a constant and symmetric cost of relocation $C^{R}$. Let $X=\mu ^{c}\Delta \tau +\Delta c>0$ be the unit gain from relocating (without taking into account the relocation cost).

A clean single-plant firm located in the North relocates its production if, and only if, the profit made in the South net of relocation costs is higher than the current profit in the northern country. Moreover, the single-plant firm anticipates that relocation modifies market structures. A northern single-plant firm relocates its production if, and only if, the profit realized in the South minus the relocation cost is higher than the current profit in the northern country. Stated differently, if:

(1)\begin{equation} \pi _{S}^{c}\left( m_{S}^{c}+1,m_{S}^{d},m_{N}-1\right) -C^{R}>\pi_{N} \left( m_{N},m_{S}^{c},m_{S}^{d}\right).\tag{1}\end{equation}

Northern firms have the incentive to relocate their production in the South as long as their profit net of the relocation cost is larger than the northern profit. Hence, at equilibrium, the number of firms that relocate their production is such that clean firms obtain the same profit regardless of their location.

Let us now study how an improvement in the technology used in the South affects the incentives to relocate. We study in a complementary way how technological diffusion affects global emissions. Technology improvement can take different forms and can come from different channels. In this section, we study first the effect of a decrease in the dirty southern firms' emission intensity and second the effect of an increase in the number of clean southern firms.

3.3.1. An increase in the share of clean firms among the firms in the South

Keeping the number of southern firms $m_{S}=m_{S}^{d}+m_{S}^{c}$ constant, let us first analyze the effect of an increase in $m_{S}^{c}$ on the individual production of the different types of firms. An increase in the share of clean firms in the South decreases the production of all types of firms. Indeed, it transforms one dirty southern firm into a clean southern firm and then reduces its production cost, inducing a reduction in production for all the other firms. An increase in the share of clean firms in the South has two effects on global production. On the one hand, each firm produces individually less, and on the other hand, the share of clean firms increases, while the share of dirty firms decreases. The second effect prevails on the first one and the overall production increases. The effect of an increase in $m_{S}^{c}$ (keeping the number of southern firms constant) on total emissions is given by:

\begin{align*} \left.\frac{\partial E}{\partial m_{S}^{c}}\right\vert _{m_{S}=m_{S}^{d}+m_{S}^{c}}&= \frac{\Delta \mu }{m_{N}+m_{S}+1}[2(\mu ^{d}+\mu ^{c}) \tau_{S}-2m_{N}(\mu ^{c}\tau _{N}-\mu ^{d}\tau _{S}+\Delta c) \\ &\quad +2(m_{S}^{c}-m_{S}^{d})\Delta \mu \tau_{S}+t+2c_{S}-a_{N}-a_{S}]. \end{align*}

An increase in the share of southern clean firms decreases the individual production and consequently emissions. The newly clean firms produce more since they are more efficient, but they pollute less per unit produced. As a result, an increase in the share of clean firms in the South has an ambiguous effect on emissions. It increases global emissions when the market sizes are low, the transportation cost is high, and when the competitive gap between the two regions is relatively low (low $c_{N}$ and $\tau _{N}$ and high $c_{S}$). Put differently, the conditions under which emissions increase with a rise in the share of clean firms among firms in the South are close to those under which emissions increase with a decrease in $\mu ^{d}$. Let us focus on the effects of an increase in the share of clean firms in the southern firms on the incentives to relocate. By calculating equation (1) and studying how an increase in $m_{S}^{c}$ affects it keeping $m_{S}=m_{S}^{d}+m_{S}^{c}$ constant, we deduce the following proposition (proof in online appendix B).

If the number of clean southern firms increases, while the number of southern firms stays constant, all firms decrease their production. Hence, the firm's profit decreases regardless of its decision (relocating or staying). Nevertheless, since the relocated single-plant firm increases its overall production when it relocates, the effect of the increase in the share of clean firms among the southern firms is all the more so as the single-plant firm has relocated. Hence, an increase in the number of clean southern firms decreases the northern firms' incentives to relocate.

3.3.2. An increase in the number of firms located in the South

Keeping the number of clean firms constant, i.e., $m^{c}=m_{N}+m_{S}^{c}$, let us first analyze the effect of an increase in $m_{S}^{c}$ on the individual production of the different types of firms,

\begin{align*} & \left.\frac{\partial r_{SS}^{d}}{\partial m_{S}^{c}}\right\vert _{m^{c}=m_{N}+m_{S}^{c}}= \left.\frac{\partial r_{SS}^{c}}{\partial m_{S}^{c}}\right\vert _{m^{c}=m_{N}+m_{S}^{c}}= \left.\frac{\partial r_{NS}}{\partial m_{S}^{c}}\right\vert _{m^{c}=m_{N}+m_{S}^{c}}= -\frac{X+t}{m_{S}^{d}+m^{c}+1}<0 \\ & \left.\frac{\partial r_{SN}^{d}}{\partial m_{S}^{c}}\right\vert _{m^{c}=m_{N}+m_{S}^{c}}= \left.\frac{\partial r_{SN}^{c}}{\partial m_{S}^{c}}\right\vert _{m^{c}=m_{N}+m_{S}^{c}}= \left.\frac{\partial r_{NN}}{\partial m_{S}^{c}}\right\vert _{m^{c}=m_{N}+m_{S}^{c}}= -\frac{X-t}{m_{S}^{d}+m^{c}+1}. \end{align*}

An increase in $m_{S}^{c}$ decreases the individual production sold on the southern market. However, the effect on the individual production sold on the northern market depends on the value of transportation cost. If the transportation cost is relatively low as compared to the unit gain from relocating, relocation strengthens competition and then induces a decrease in the firms' individual production. However, if the transportation cost is relatively high as compared to the unit gain from relocating, relocation softens competition and firms' individual production increases. Nevertheless, note that the total production of each firm (for sales at home and exports) decreases with the replacement of a clean firm in the North by a clean firm in the South. Indeed, by summing the production for sales at home and exports, we get:

\begin{align*} \left.\frac{\partial \left( r_{SS}^{d}+r_{SN}^{d}\right) }{\partial m_{S}^{c}}\right\vert _{m^{c}=m_{N}+m_{S}^{c}} &= \left.\frac{\partial \left(r_{SS}^{c}+r_{SN}^{c}\right) }{\partial m_{S}^{c}}\right\vert _{m^{c}=m_{N}+m_{S}^{c}}= \left.\frac{\partial \left( r_{NS}+r_{NN}\right) }{\partial m_{S}^{c}}\right\vert _{m^{c}=m_{N}+m_{S}^{c}} \\ &= -\frac{2X}{m_{S}^{d}+m^{c}+1}<0. \end{align*}

The effect of an increase in $m_{S}^{c}$ on global emissions, keeping the number of clean firms constant, is equal to ${\partial E}/{\partial m_{S}^{c}}\vert _{m^{c}=m_{N}+m_{S}^{c}}={2(\mu ^{c}-m_{S}^{d}\Delta \mu )X}/{(m_{S}^{d}+m^{c}+1)}$. Two effects are at stake. First, the newly-established clean firm in the South produces more than when it was located in the North. Second, the production and the emissions of the others firms decrease. The first effect prevails when the technological gap is relatively low, and when there are only a few dirty firms (low $m_{S}^{d}$).

Let us focus on the effects of an increase in the share of clean firms in the southern firms on the incentives to relocate. By calculating equation (1) and studying how an increase in $m_{S}^{c}$ affects it, keeping $m^{c}=m_{N}+m_{S}^{c}$ constant, we deduce the following proposition (proof in online appendix C).

If the number of clean southern firms increases, while the number of clean firms remains constant, the firm's profits made in the southern market decrease, but the profits made in the northern market decrease if the transportation cost is low. Hence, it affects the clean firm's profit in the same manner regardless of its decision (relocating or staying). Nevertheless, since the newly relocated firm increases its overall production when it relocates, the effect of the substitution of a northern clean firm by a southern one is all the more so as the firm has relocated. Hence, an increase in the number of clean southern firms decreases the northern firms' incentives to relocate.

4.1.1. Southern firms' decisions

We analyze here the possibility for southern firms to purchase and adopt the clean technology. We assume no adaptation cost and adoption cost is simplified to a patent price K. The timing is the following. In stage 1, southern firms decide whether to adopt the technology. In stage 2, northern firms decide whether to relocate. Finally, in stage 3, firms produce and sell the good on the two markets for products.

Stage 3. The third stage is similar to the one defined in section 3.1, except that at the last stage $m_{N}=m_{N}^{0}-{l}$, $m_{S}^{d}={ m_{S}^{d}}^{0}-k$ and $m_{S}^{c}={l}+k$ where ${l}$ is the number of relocated firms if there is adoption, and k is the number of dirty southern firms adopting the technology.

Stage 2. At equilibrium a northern single-plant firm is indifferent between relocating and remaining located in the North. Hence, at equilibrium, the number of relocated firms l is given by the following equality:Footnote ⁷

(2)\begin{equation} \pi _{S}^{c}\left( l+k,{m_{S}^{d}}^{0}-k,m_{N}^{0}-l\right) -C^{R}= \pi_{N}\left( m_{N}^{0}-l,l+k,{m_{S}^{d}}^{0}-k\right). \tag{2}\end{equation}

By solving (2), we define ${l}(k)$ given by:

(3)\begin{align} l(k)& =\frac{a_{S}\left( X+t\right) +a_{N}\left( X-t\right) }{2\left( { {X}^{2}}+{{t}^{2}}\right) }-\frac{\left( \mu ^{c}\left( \tau _{N}+ \tau_{S}\right) +t+c_{N}+c_{S}\right) X}{2\left( {{X}^{2}}+{{t}^{2}}\right) } \nonumber \\ &\quad +\frac{\left( {m_{S}^{d}}^{0}-k\right) \Delta \mu \tau _{S}X}{{{X} ^{2}}+{{t}^{2}}}-\frac{C^{R}\left( m_{N}^{0}+{m_{S}^{d}}^{0}+1\right) }{ 4\left( {{X}^{2}}+{{t}^{2}}\right) }+\frac{m_{N}^{0}-{m_{S}^{d}}^{0}}{2}. \tag{3}\end{align}

The number of firms that relocate depends negatively on the relocation cost and positively on the southern market size. The more costly relocation is, the fewer the firms that relocate. Moreover, the higher the southern market size is, the more profitable for northern firms relocation is and the more firms relocate. However, the effect of northern market size on the number of relocated firms is ambiguous and depends on the transportation cost. If the transportation cost is relatively low as compared to the unit gain from relocating, the number of relocated firms increases with the northern market size. Indeed, since it is cheap to transport goods, relocation decreases the marginal cost to produce and sell on the northern market. The greater the northern market size is, the more northern firms relocate. However, if the transportation cost is relatively high as compared to the unit gain from relocating, the relocated firm will be less efficient in the northern market. Thus, the greater the northern market size is, the fewer the northern firms that relocate.

By deriving $l(k)$ with respect to k, we obtain: ${\partial l(k)}/{ \partial k}=-{\Delta \mu \tau _SX }/{({{X}^{2}}+{{t}^{2}})}<0$. From this equation, the following corollary is deduced.

This corollary is a direct implication of proposition 2. Adoption increases the share of clean firms on the southern market, which decreases the gains from relocation. Note that adoption highly reduces relocation when the transportation costs are low. If so, clean southern firms are highly competitive in the northern market, and northern firms have more incentive to supply this market locally.

Stage 1. Southern firms purchase the technology from the innovator anticipating the possible relocation of northern firms. A dirty southern firm adopts the cleaner technology if, and only if, the profit made when it is clean minus the adoption cost is higher than the profit it gets when it is dirty. At equilibrium, the number of adoptions k is such that the profit of a southern firm is the same with the two technologies:

(4)\begin{equation} \pi _{S}^{c}\left( {l}(k)+k,{m_{S}^{d}}^{0}-k,m_{N}-{l}(k)\right) -K=\pi_{S}^{d}\left( {m_{S}^{d}}^{0}-k,{l}(k)+k,m_{N}-{l}(k)\right). \tag{4}\end{equation}

By replacing ${l}(k)$ and solving (4) with respect to k, we are able to define the number of firms that adopts the clean technology:

(5)\begin{align} k^{\ast }& =\frac{C^{R}\left( m_{N}^{0}+{m_{S}^{d}}^{0}+1\right) X}{ 4\Delta \mu {{t}^{2}}\tau _{S}}+\frac{X\left( \left( m_{N}^{0}+{ m_{S}^{d}}^{0}\right) {t}^{2}+\left( m_{N}^{0}+{m_{S}^{d}}^{0}+1\right) { X}^{2}\right) }{2\Delta \mu {t}^{2}\tau _{S}} \nonumber \\ &\quad -\frac{K\left( m_{N}^{0}+{m_{S}^{d}}^{0}+1\right) \left( {X}^{2}+{t} ^{2}\right) }{\left( 2{\Delta \mu }{t}\tau _{S}\right) ^{2}}-\frac{ \left( m_{N}^{0}+{m_{S}^{d}}^{0}+1\right) {X}^{2}}{2{t}^{2}} \nonumber \\ &\quad +\frac{a_{N}\left( X+t\right) -a_{S}\left( X-t\right) }{2\Delta \mu t\tau _{S}}-\frac{\mu ^{d}\left( \tau _{S}+\tau _{S}\right) +t+2c_{S}}{2\Delta \mu \tau _{S}}-\frac{m_{N}^{0}-{m_{S}^{d}}^{0}}{2}. \tag{5}\end{align}

By studying (5), the following lemma is deduced.

Southern firms adopt technology to reduce the unit production cost and prevent some firms from settling in the southern market. They have more incentive to adopt technology when the gains on the southern market are high. This is the case when the southern market size is large and the transportation costs are relatively high since it means that the southern market is profitable and not significantly exposed to northern firms' competition. This is also the case when the size of the northern market and the relocation costs are high, since it means that only a few northern firms will relocate.

We can deduce from the previous results that reducing patent prices (for instance, by allowing for pricing differentiation of patents or relaxing intellectual property rights) decreases relocation. Such policies would increase adoption, reduce relocation and may increase emissions. Note that reducing patent prices has two effects on the innovator located in the North. On the one hand, it increases the number of southern firms that are willing to purchase the technology and consequently increases the innovator's profit, but on the other hand, it decreases the revenue of each patent sold. The effect on the innovator's profit depends on the price elasticity of demand for clean technology.

4.1.2. Northern government's decision

The northern government may transfer the technology to the dirty southern firms by subsidizing the purchase of the technology. We assume that the government directly purchases the technology from the innovator and grants it to the dirty southern firms. Moreover, the government decides to purchase licenses for all dirty southern firms or none of them and is thus unable to discriminate between firms. The goal of the northern government is to increase its welfare. Before introducing the welfare function, let us first introduce some additional assumptions. We consider that emissions generate global damage which is assumed to be linear. Let us denote by $\delta _{N}$ the marginal damage in the North.Footnote ⁸ The northern welfare is defined as the sum of the consumer surplus, the sum of the northern profits, the regulator's revenue minus the environmental damage. The northern welfare is $W_{N}=SC_{N}+\Pi _{N}+\Pi _{I}+RR_{N}-\delta _{N}(E_{N}+E_{S})$ where $\Pi _{I}$ is the profit of the innovator and $RR_{N}$ the revenue of the northern regulator. The latter is equal to the tax revenue minus the subsidies for the southern firms. Subsidizing southern firms ($m_{S}^{d}K$) is a lump-sum transfer from the government to the northern innovator. Put differently, subsidizing southern firms is a neutral operation for the northern welfare. Indeed, the patents given to southern firms induce a profit for the northern innovator, but patents are paid for by the northern government.

The decision to improve southern technology is taken by the northern government, while previously the southern firms were deciding whether to adopt it or not. The game is characterized by the following timing: at stage 1, the North decides whether it subsidizes the purchase of licenses for southern firms. At stage 2, firms decide whether they relocate. Finally, at stage 3, firms produce and sell the good on the two markets for products. We solve this problem backwards and as previously we focus on the first two stages, since the third stage is similar to the one defined in section 3.1.

Stage 2. Following the same method as in section 4.1.1, the number of relocations l can be defined. Similarly to adoption, the subsidies affect the market structure. The subscript referring to the case in which the northern government subsidizes the purchase of patents for dirty southern firms is denoted by G (for government), and the case in which the northern government does not subsidize the purchase of patents for dirty southern firms is now denoted by WG. The number of relocated firms with subsidies is denoted by $l^{G}={l}({ m_{S}^{d}}^{0})$, while the number of relocated firms without subsidies is denoted by $l^{WG}={l}(0)$. Since ${l}(k)$ decreases with k, we immediately deduce $l^{G}<{l}(k^{\ast })<l^{WG}$. The number of firms that relocate is the lowest when the northern government decides to subsidize the technology adoption for all dirty southern firms. We deduce the following corollary.

This corollary is a direct implication of proposition 1. Indeed, subsidizing the purchase of patents may be understood as an improvement in the dirty technology, which reduces the incentive to relocate.

Stage 1. The northern government decides whether it subsidizes the purchase of patents abroad anticipating firms' relocation. If the northern government subsidizes, the market structure will be as follows: $m_{S}^{d}=0$, $m_{S}^{c}={m_{S}^{c}}^{0}+{m_{S}^{d}}^{0}+l^{G}$ and $m_{N}={m_{N}}^{0}-l^{G}$. In contrast, if the northern government does not subsidize, $m_{S}^{d}={m_{S}^{d}}^{0}$, $m_{S}^{c}={m_{S}^{c}}^{0}+l^{WG}$ and $m_{N}={ m_{N}}^{0}-l^{WG}$.

By studying the effect of the subsidies on the different components of the northern welfare, the following lemma is deduced (proof in online appendix D).

In the northern market, subsidies decrease the quantity sold by clean firms from both countries, and increase the quantity sold by dirty firms. Nevertheless, the northern consumer surplus increases since subsidies decrease the number of inefficient firms, while keeping the total number of firms constant. The subsidy may increase the industry's profit in the North. Indeed, it decreases the individual profit but increases the number of firms located in the North. The crowding-out effect lowers the positive effect of the subsidy on the northern consumer surplus. Indeed, the subsidy decreases the marginal production cost, which decreases the price but also impedes relocation which lowers this price decrease when transportation costs are relatively high compared to the unit gain from relocating.

Glachant et al. (Reference Glachant, Ing and Nicolaï2017) study in a close setting the incentives to transfer clean technology. The main difference between the two papers is that in our paper, firms can relocate, while in Glachant et al. (Reference Glachant, Ing and Nicolaï2017) market structures are exogenous. As in Glachant et al. (Reference Glachant, Ing and Nicolaï2017), the subsidy may increase or decrease global emissions. Indeed, two effects are in opposition: firms in the South pollute less by production unit but since they are more competitive, they produce more. However, in Glachant et al. (Reference Glachant, Ing and Nicolaï2017), the technology transfer increases the welfare if and only if it decreases total emissions, while in this paper the technology transfer may increase welfare even if it increases global emissions. Indeed, the transfer decreases the number of relocated firms and it may increase both the consumer surplus and the northern profits.

4.2.1. Creation of a new subsidiary from scratch

Northern firms may decide to build a new subsidiary in the South, while keeping their plant in the North. Let us denote by $\hat {l}$ the number of northern single-plant firms that relocate and by s the number of northern multi-plant firms that open a subsidiary abroad and whose northern plant is still active.

Stage 3. This stage is similar to the one defined in section 3.1 except that the market structure is different. In the northern market there are $m_{N}=m_{N}^{0}-\hat {l}$, $m_{S}^{d}={m_{S}^{d}}^{0}$ and $m_{S}^{c}=\hat {l}$ firms operating, while in the southern market there are $m_{N}=m_{N}^{0}-\hat {l}-s$, $m_{S}^{d}={m_{S}^{d}}^{0}$ and $m_{S}^{c}=\hat {l }+s$ firms operating. To determine the production levels we simply replace $m_{N}$, $m_{S}^{d}$ and $m_{S}^{c}$ by their corresponding value.

Stage 2. At equilibrium a northern single-plant firm is indifferent between relocating and remaining located in the North. Hence, at equilibrium, the number of relocated firms $\hat {l}$ is given by the following equality:

(6)\begin{align} & \pi _{SN}^{c}\left( \hat{l},{m_{S}^{d}}^{0},m_{N}^{0}-\hat{l}\right) + \pi_{SS}^{c}\left( \hat{l}+s,{m_{S}^{d}}^{0},m_{N}^{0}-\hat{l}-s\right) -C^{R} \nonumber \\ &\quad =\pi _{NN}\left( m_{N}^{0}-\hat{l},\hat{l},{m_{S}^{d}}^{0}\right) + \pi_{NS}\left( m_{N}^{0}-\hat{l}-s,\hat{l}+s,{m_{S}^{d}}^{0}\right). \tag{6}\end{align}

Relocation corresponds to the transformation of a clean firm in the North into a clean firm in the South. Put differently, the total number of firms selling on each market is unchanged by a relocation. However, the creation of a subsidiary in the South corresponds to the transformation of a clean exporter from the North to the South into a clean firm from the South selling only in its domestic market. We deduce from equation (6) the number of single-plant firms that relocate, which is:

\begin{align*} \hat{l}(s) &= \frac{a_{S}\left( X+t\right) +a_{N}\left( X-t\right) }{2\left( {{X}^{2}}+{{t}^{2}}\right) }- \frac{\left(\mu ^{c}\left( \tau_{N}+\tau _{S}\right) +t+c_{N}+c_{S}\right) X}{2\left( {{X}^{2}}+{{t}^{2}}\right) } \\ &\quad +\frac{{m_{S}^{d}}^{0}\Delta \mu \tau _{S}X}{{{X}^{2}}+{{t}^{2}}}- \frac{C^{R}\left( m_{N}^{0}+{m_{S}^{d}}^{0}+1\right) }{4\left( {{X}^{2}}+{{t}^{2}}\right) }+ \frac{m_{N}^{0}-{m_{S}^{d}}^{0}}{2}-\frac{s{{\left(X+t\right) }^{2}}}{2\left( {X}^{2}+{t}^{2}\right)}. \end{align*}

As in equation (3), the number of firms that relocate depends negatively on the relocation cost and positively on the southern market size. As usual, the number of single-plant firms that relocate increases with the northern market size when the transportation cost is relatively low. We immediately obtain the derivative of the number of single-plant firms that relocate relative to the number of created subsidiaries ${\partial \hat {l}(s)}/{\partial s}=-{( X+t) ^{2}}/{2(X^{2}+t^{2})}<0$. From this inequation, the following corollary is deduced.

The firm that creates a subsidiary in the South becomes more competitive than before. This effect induces a reduction in the number of northern single-plant firms that relocate. The production of northern firms in the northern market is not affected by the subsidiary opening. However, the firm located in the North which opens the subsidiary no longer supplies the southern market. As a result, when a firm opens a subsidiary, production from the northern plant decreases. Opening of a subsidiary increases the total number of southern firms and more precisely increases the number of clean southern firms. Hence, as in the previous cases, competition on the southern market is strengthened, which reduces the number of northern firms that relocate.

Stage 1. Each firm decides to be a multi-plant firm or single-plant. Each northern firm decides to open a subsidiary taking into account that it also has the possibility to relocate. At equilibrium, the northern single-plant firms are indifferent between the two strategies:

\begin{align*} & \pi _{NN}\left( m_{N}^{0}-\hat{l}(s),\hat{l}(s),{m_{S}^{d}}^{0}\right) +\pi _{SS}^{c}\left( \hat{l}(s)+s,{m_{S}^{d}}^{0},m_{N}^{0}-\hat{l}(s)-s\right) -C^{o} \\ &\quad =\pi _{NN}\left( m_{N}^{0}-\hat{l}(s),\hat{l}(s),{m_{S}^{d}}^{0}\right) +\pi _{NS}\left( m_{N}^{0}-\hat{l}(s)-s,\hat{l}(s)+s,{m_{S}^{d}}^{0}\right), \end{align*}

where $C^{o}$ is the cost to create a subsidiary from scratch. The first part of the equation represents the profit made by a multi-plant firm, while the second part is the profit made by a northern single-plant firm. At equilibrium, the number of subsidiaries at equilibrium is given by:

\begin{align*} s &= -\frac{C^{o}\left( m_{N}^{0}+{m_{S}^{d}}^{0}+1\right) \left( {X}^{2}+ {t}^{2}\right) }{{{\left( X-t\right) }^{2}}{{\left( X+t\right) }^{2}}}+ \frac{C^{R}\left( m_{N}^{0}+{m_{S}^{d}}^{0}+1\right) }{2{{\left( X-t\right) }^{2}}}-\frac{a_{N}}{X-t}+\frac{a_{S}}{X+t} \\ &\quad +\frac{t\left( \mu ^{c}\left( \tau _{N}+\tau _{S}\right) +t+c_{N}+c_{S}\right) }{\left( X-t\right) \left( X+t\right) }-\frac{2{ m_{S}^{d}}^{0}\Delta \mu t\tau _{S}}{\left( X-t\right) \left(X+t\right) }. \end{align*}

We immediately deduce the following lemma.

Proof.

\begin{align*} \frac{\partial s}{\partial{m_N}^0} &= \frac{C^R}{2{( X-t) }^{2}}- \frac{C^o({{t}^{2}}+{{X}^{2}}) }{{{( X-t) }^{2}}{{(X+t) }^{2}}}, \\ \frac{\partial s}{\partial{m_S^d}^0} &= \frac{C^R}{2{{( X-t)}^{2}}}- \frac{C^o( {{t}^{2}}+{{X}^{2}}) }{{{( X-t) }^{2}}{{( t+X)}^{2}}}- \frac{2\Delta\mu t\tau_S}{( X-t) (X+t) }. \end{align*}

The creation of a subsidiary and relocation are two substitutable actions, but they are not taken at the same time. Thus, strategic effects can be generated by the creation of a subsidiary. The number of multi-plant firms depends negatively on the costs to build a subsidiary and depends positively on the relocation costs and on the southern market size. The number of created subsidiaries increases with the northern market size if the transportation cost is high. Indeed, in this case, it is expensive to provide the northern market from abroad, and firms are more willing to become multi-plant. A particularly interesting result is that an increase in ${m_{S}^{d}}^{0}$ and ${m_{N}}^{0}$ leads to an increase in the number of subsidiaries if and only if the relocation costs are high. If relocation costs are low, firms anticipate that an increase in the number of firms will lead to a large number of firms that relocate and therefore have fewer incentives to open a subsidiary.

4.2.2. Purchase of dirty southern firms

Let us now consider that northern firms may purchase a southern firm and turn it into a clean one. Let us denote by $\tilde {l}$ the number of northern single-plant firms that relocate and by b the number of northern firms that purchase a southern dirty firm.

Stage 3. The third stage is similar to the one defined in section 3.1 except that the market structure is different. In the northern market there are $m_{N}=m_{N}^{0}-\tilde {l}$, $m_{S}^{d}={m_{S}^{d}}^{0}-b$ and $m_{S}^{c}=\tilde {l}$ firms operating, while in the southern market there are $m_{N}=m_{N}^{0}-\tilde {l}-b$, $m_{S}^{d}={m_{S}^{d}}^{0}-b$ and $m_{S}^{c}=\tilde {l}+b$ firms operating. To determine the production levels we simply replace $m_{N}$, $m_{S}^{d}$ and $m_{S}^{c}$ by their corresponding value.

Stage 2. At equilibrium a northern single-plant firm is indifferent between relocating and remaining located in the North. Hence, at equilibrium, the number of relocated firms $\tilde {l}$ is given by the following equality:

(7)\begin{align} & \pi _{SN}^{c}\left( \hat{l},{m_{S}^{d}}^{0}-b,m_{N}^{0}-\tilde{l}\right) +\pi _{SS}^{c}\left( \tilde{l}+b,{m_{S}^{d}}^{0}-b,m_{N}^{0}-\tilde{l}-b\right) -C^{R} \nonumber \\ &\quad = \pi _{NN}\left( m_{N}^{0}-\tilde{l},\tilde{l},{m_{S}^{d}}^{0}-b\right) +\pi _{NS}\left( m_{N}^{0}-\tilde{l}-b,\tilde{l}+b,{m_{S}^{d}}^{0}-b\right). \tag{7}\end{align}

The purchase of a dirty firm in the South to transform it into a subsidiary has two effects. First, it transforms a clean exporter from the North to the South into a clean southern firm selling only in its domestic market. Second, it also removes from both markets a dirty southern firm. We deduce from equation (7) the number of single-plant firms that relocate, which is given by:Footnote ¹⁰

\begin{align*} \tilde{l}(b) &=\frac{a_{S}\left( X+t\right) +a_{N}\left( X-t\right) }{ 2\left( {{X}^{2}}+{{t}^{2}}\right) }-\frac{\left( \mu ^{c}\left( \tau _{N}+\tau _{S}\right) +t+c_{N}+c_{S}\right) X}{2\left( {{X}^{2}}+{{t}^{2}}\right) } \\ &\quad +\frac{({m_{S}^{d}}^{0}-b)\Delta \mu \tau _{S}X}{{{X}^{2}}+{{t}^{2}}}- \frac{C^{R}\left( m_{N}^{0}+{m_{S}^{d}}^{0}-b+1\right) }{4\left( {{X}^{2}}+{{t}^{2}}\right) }\notag\\ &\quad + \frac{m_{N}^{0}-{m_{S}^{d}}^{0}+b}{2}-\frac{b{{\left( X+t\right) }^{2}}}{2\left( {{X}^{2}}+{t}^{2}\right) }. \end{align*}

The number of single-plant firms that relocate depends negatively on the relocation cost and positively on the southern market size. The effects of northern market size and the number of northern and southern firms on relocation are ambiguous and depend on the transportation cost. Finally, we immediately obtain the derivative of the number of single-plant firms that relocate relative to the number of multi-plant firms,

\[ \frac{\partial \tilde{l}(b)}{\partial b}=\frac{C^{R}}{4\left( {{X}^{2}}+{t}^{2}\right) }- \frac{X\left( \Delta \mu \tau _{S}+t\right) }{{X}^{2}+{t}^{2}}.\]

The following proposition is deduced.

As opposed to the previous channels of technology diffusion (technology adoption, public transfer and subsidiary creation), the purchase of a dirty southern firm does not necessarily decrease the number of relocated firms. In fact, when the relocation costs are sufficiently high relative to the sum of the transportation cost and the unit gain from relocating, the purchase of a southern firm increases the incentives to relocate. Contrary to the case whereby firms create subsidiaries, the purchase of a southern firm affects the production sold on the northern market. It softens competition in the northern market thus increasing the profit made by all types of firms in the northern market. In the southern market, the purchase decreases the number of firms but turns a dirty firm into a clean one. The results depend on the value of the relocation costs. Consider for a moment high relocation costs. In such a case, few firms have an interest in relocating. However, if the number of dirty firms purchased increases, competition in the southern market is reduced and more northern firms have incentives to relocate.

Stage 1. Each northern firm decides either to be a multi-plant or to have only one plant in the North. Put differently, each northern firm decides to open a subsidiary taking into account that it will have the possibility to relocate later in the game. It purchases a southern firm if its profit net of the purchasing costs is larger than without the purchase. At equilibrium, northern firms are indifferent between the two strategies, and the number of firms purchased, i.e., b, is such that:

\begin{align*} & \pi _{NN}\left( m_{N}^{0}-\tilde{l}(b),\tilde{l}(b),{m_{S}^{d}}^{0}-b\right) +\pi _{SS}^{c}\left( \tilde{l}(b)+b,{m_{S}^{d}}^{0}-b,m_{N}^{0}-\tilde{l}(b)-b\right) -C^{p} \\ &\quad =\pi _{NN}\left( m_{N}^{0}-\tilde{l}(b),\tilde{l}(b),{m_{S}^{d}}^{0}-b\right) +\pi _{NS}\left( m_{N}^{0}-\tilde{l}(b)-b,\tilde{l}(b)+b,{m_{S}^{d}}^{0}-b\right), \end{align*}

where $C^{p}$ is the cost to purchase a firm. The first part of the equation represents the profit made by a multi-plant firm purchasing a firm, while the second part is the profit made by a northern single-plant firm. Since after the purchase the multi-plant firm still supplies the northern market locally, the profit made in the northern market at equilibrium is the same with and without purchasing a dirty southern firm. Hence, b is:

\begin{align*}&\pi_{SS}^{c}\left( \tilde{l}(b)+b,{m_{S}^{d}}^{0}-b,m_{N}^{0}-\tilde{l} (b)-b\right) -C^{p}\\ &\quad =\pi _{NS}\left( m_{N}^{0}-\tilde{l}(b)-b, \tilde{l}(b)+b,{m_{S}^{d}}^{0}-b\right).\end{align*}

At equilibrium, the number of firms purchased in the South by multi-plant firms is:

\begin{align*} b &= \frac{2\left( {{X}^{2}}-{{t}^{2}}\right) \left( a_{N} \left(t+X\right) -a_{S}\left( X-t\right) -t\left( \mu ^{c}(\tau _{N}+ \tau_{S})-2{m_{S}^{d}}^{0}\Delta \mu \tau _{S}+t+c_{N}+ c_{S}\right)\right) }{2\left( X^{2}-t^{2}\right) \left( t\left( 2\Delta \mu \tau _{S}+t\right) -{X}^{2}\right) -C^{R}{{\left( t+X\right) }^{2}} +2C^{p}\left( {{t}^{2}}+{{X}^{2}}\right) } \\ &\quad +\frac{\left( m_{N}^{0}+{m_{S}^{d}}^{0}+1\right) \left( 2C^{p} \left({{t}^{2}}+{{X}^{2}}\right) -C^{R}{{\left( t+X\right) }^{2}}\right) }{ 2\left( X^{2}-t^{2}\right) \left( t\left( 2\Delta \mu \tau_{S}+t\right) -{X}^{2}\right) -C^{R}{{\left( t+X\right) }^{2}}+2C^{p}\left( {{t}^{2}}+{{X}^{2}}\right) }. \end{align*}

Analyzing b is not straightforward. First, it is fair to assume that the denominator is positive. Indeed, the cost to purchase a firm ($C^{p}$) must be high enough to ensure that b is lower than $m_{N}^{0}$ and ${m_{S}^{d}} ^{0}$. This assumption implies that northern firms do not purchase all southern firms. Second, under this assumption, an increase in the southern market size decreases the number of multi-plant firms. This counter-intuitive result can be explained as follows: northern firms anticipate that a significant southern market size leads to massive relocations, and thus they have fewer incentives to purchase southern firms anticipating that the competition in this market will be fierce. Moreover, an increase in the northern market size only decreases the number of multi-plant firms if the transportation cost is sufficiently high compared to the unit gain from relocating.