Trade, Resource Use and Pollution: A Synthesis

Abstract

We develop a two-sector dynamic model of trade and the environment. Both resource-good and manufacturing sectors generate environmental burdens that can adversely affect the environment in a country over time. The resource-good sector is environmentally sensitive, with its productivity suffering in line with declines in the environmental stock. Countries can be categorized into two types: one in which the resource-good sector is more environmentally harmful than the manufacturing sector, and another in which the opposite is true. At every point in time, labor allocations and specialization patterns under free trade are dependent on country size, technology, household preferences, and environmental stocks. Trading countries’ types have implications for the dynamics of environmental stocks and the corresponding environmental and welfare consequences of trade. If two trading countries are of the same type, one country exports its dirtier goods and the other must export its cleaner goods; consequently, at least one country gains from trade. If the two countries are of different types, they can export their respective dirtier goods and may both lose from trade.

Appendix

1.1 A.1 Proof of Lemma 1

Proof of Part (i)

Assume to the contrary that the sign of $\left( l_{f}A\left( S\right) -l_{m}a\right)$ changes as S varies within $\Gamma$. From the continuity of $A\left( \cdot \right)$, there exists $S_{c}\in \Gamma$ such that $l_{f}A\left( S_{c}\right) =l_{m}a$. Since $S_{c}\in \Gamma$, there exists $\beta _{c}\in \left[ 0,1\right]$ such that $S_{\infty }\left( \beta _{c}\right) =S_{c}$, from which it follows, according to the definition of $S_{\infty }\left( \cdot \right)$, that

$$\begin{aligned} G\left( S_{c}\right) =\left( l_{f}A\left( S_{c}\right) \beta _{c}+l_{m}a\left( 1-\beta _{c}\right) \right) L=l_{m}aL. \end{aligned}$$

This means that ${\dot{S}}=0$ holds for any $\beta \in \left[ 0,1\right]$ as long as $S=S_{c}$, which together with the stability condition and the continuity of $S_{\infty }\left( \beta \right)$ further implies that $S_{\infty }\left( \beta \right) =S_{c}$ for all $\beta \in \left[ 0,1\right]$.⁴⁰ Hence, $\Gamma =\left\{ S_{c}\right\}$, and thus $l_{f}A\left( S\right) -l_{m}a=0$ holds for all $S\in \Gamma$. This leads to a contradiction.

Proof of Part (ii)

The continuity of $S_{\infty }\left( \beta \right)$ implies that $\Gamma$ can be written as a closed interval:

$$\begin{aligned} \Gamma =\left[ \gamma _{\min },\gamma _{\max }\right] , \end{aligned}$$

where $\gamma _{\min }$ and $\gamma _{\max }$ are the left and right endpoints of the interval. It then suffices to show that S cannot escape $\Gamma$ through the two endpoints during dynamic transitions: ${\dot{S}}\ge 0$ for any $\beta \in \left[ 0,1\right]$ if $S=\gamma _{\min }$, and ${\dot{S}}\le 0$ for any $\beta \in \left[ 0,1\right]$ if $S=\gamma _{\max }$. In what follows, we prove only the former: ${\dot{S}}\ge 0$ for any $\beta \in \left[ 0,1\right]$ if $S=\gamma _{\min }$. The latter can be proven similarly, and thus the proof is omitted here.

According to part (i), we proceed by considering two cases.

Case 1: $l_{f}A\left( S\right) =l_{m}a$ for all $S\in \Gamma$. It then follows that

$$\begin{aligned} {\dot{S}}=G\left( S\right) -l_{m}aL, \end{aligned}$$

which gives $S_{\infty }\left( \beta \right) =\gamma _{\min }=\gamma _{\max }$ for any $\beta \in \left[ 0,1\right]$. This implies that once the economy reaches the steady state, it remains there regardless of the labor allocation.

Case 2: $l_{f}A\left( S\right) \ne l_{m}a$ for all $S\in \Gamma$. Assume to the contrary that there exists $\beta _{1}\in \left[ 0,1\right]$ such that ${\dot{S}}<0$ if $S=\gamma _{\min }$. Let $S_{1}=S_{\infty }\left( \beta _{1}\right)$. Then ${\dot{S}}=0$ and $\partial {\dot{S}}/\partial S<0$ holds for $\beta =\beta _{1}$ and $S=S_{1}$. Hence, for sufficiently small $\delta >0$, ${\dot{S}}>0$ holds for $\beta =\beta _{1}$ and $S=S_{1}-\delta$. Recalling that we have assumed that ${\dot{S}}<0$ holds for $\beta =\beta _{1}$ and $S=\gamma _{\min }$, there exists $S_{2}\in \left( \gamma _{\min },S_{1}\right)$ such that ${\dot{S}}=0$ holds if $\beta =\beta _{1}$ and $S=S_{2}$. That is,

$$\begin{aligned} G\left( S_{2}\right) =\left( l_{f}A\left( S_{2}\right) \beta _{1}+l_{m}a\left( 1-\beta _{1}\right) \right) L. \end{aligned}$$

On the other hand, since $S_{2}\in \Gamma$ and $S_{2}\ne S_{1}$, there exists $\beta _{2}\in \left[ 0,1\right]$ such that $\beta _{2}\ne \beta _{1}$ and $S_{2}=S_{\infty }\left( \beta _{2}\right)$:

$$\begin{aligned} G\left( S_{2}\right) =\left( l_{f}A\left( S_{2}\right) \beta _{2}+l_{m}a\left( 1-\beta _{2}\right) \right) L. \end{aligned}$$

Recalling that in this case $l_{f}A\left( S\right) \ne l_{m}a$ for all $S\in \Gamma$, we have $l_{f}A\left( S_{2}\right) \ne l_{m}a$, which together with the two equations above yields $\beta _{1}=\beta _{2}$, leading to a contradiction.

1.2 A.2 Proof of Proposition 1

According to part (i) of Lemma 1, one of the following holds: $l_{f}A\left( S\right) >l_{m}a$ for all $S\in \Gamma$, or $l_{f}A\left( S\right) <l_{m}a$ for all $S\in \Gamma$, or $l_{f}A\left( S\right) =l_{m}a$ for all $S\in \Gamma$. Countries satisfying the first are of BT type; countries satisfying the second are of CT type. The third knife-edge case is ignored.

1.3 A.3 Proof of Proposition 2

Proof of Part (i)

Recall that $S_{\infty }^{\prime }\left( \beta \right) <0$ in BT-type countries and $S_{\infty }^{\prime }\left( \beta \right) >0$ in CT-type countries. It then follows from (13) that $\text {SMRT}<\text {MRT}$ in BT-type countries and $\text {SMRT}>\text {MRT}$ in CT-type countries.

Proof of Part (ii)

Note that a straight line connecting $\left( aL,0\right)$ with another point on the long-run PPF is actually the short-run PPF that corresponds to the environmental stock required for sustainable production at that point. Since the slope (absolute value) of this short-run PPF is equal to the marginal rate of transformation (from manufacturing to resource goods), the result follows immediately from part (i).

Proof of Part (iii)

Use (13) and $\beta =1-X_{m}/aL$ to obtain

$$\begin{aligned} T_{\infty }^{\prime \prime }=\frac{d}{dX_{m}}\left( \frac{dT_{\infty }\left( X_{m}\right) }{dX_{m}}\right) =\frac{d\beta }{dX_{m}}\frac{d}{d\beta }\left( \frac{dT_{\infty }\left( X_{m}\right) }{dX_{m}}\right) =\frac{\Delta }{a^{2}L}, \end{aligned}$$

where $\Delta \equiv \beta A^{\prime }S_{\infty }^{\prime \prime }+\beta A^{\prime \prime }S_{\infty }^{\prime 2}+2A^{\prime }S_{\infty }^{\prime }$. It follows from (9) that

$$\begin{aligned} \beta A^{\prime }S_{\infty }^{\prime \prime }=\beta A^{\prime }L\frac{\varPhi _{1}L}{\varPhi _{2}^{2}}\left( 2l_{f}A^{\prime }+\frac{\varPhi _{1}}{\varPhi _{2}}\left( l_{f}A^{\prime \prime }\beta L-G^{\prime \prime }\right) \right) , \end{aligned}$$

where $\varPhi _{1}\equiv l_{f}A-l_{m}a$, $\varPhi _{2}\equiv G^{\prime }-l_{f}A^{\prime }\beta L$. Similarly, we have $\beta A^{\prime \prime }S_{\infty }^{\prime 2}=\left( \varPhi _{1}L/\varPhi _{2}^{2}\right) \varPhi _{1}A^{\prime \prime }\beta L$ and $2A^{\prime }S_{\infty }^{\prime }=\left( \varPhi _{1}L/\varPhi _{2}^{2}\right) 2\varPhi _{2}A^{\prime }$. These together yield

$$\begin{aligned} \Delta =\frac{\varPhi _{1}L}{\varPhi _{2}^{2}}A^{\prime 2}\left( 2\frac{G^{\prime }}{A^{\prime }}-\frac{\varPhi _{1}}{\varPhi _{2}}\beta L\frac{d}{dS}\left( \frac{G^{\prime }}{A^{\prime }}\right) \right) . \end{aligned}$$

It then follows that

$$\begin{aligned} T_{\infty }^{\prime \prime }=\frac{\varPhi _{1}A^{\prime 2}}{a^{2}\varPhi _{2}^{2}}\left( 2\frac{G^{\prime }}{A^{\prime }}-\frac{\varPhi _{1}}{\varPhi _{2}}\beta L\frac{d}{dS}\left( \frac{G^{\prime }}{A^{\prime }}\right) \right) . \end{aligned}$$

Noting that $\beta \rightarrow 0$ as $X_{m}\rightarrow aL$, we have

$$\begin{aligned} \lim _{X_{m}\rightarrow aL}T_{\infty }^{\prime \prime }=\frac{2\varPhi _{1}A^{\prime }G^{\prime }}{a^{2}\varPhi _{2}^{2}}. \end{aligned}$$

Stability requires that $G^{\prime }-l_{f}A^{\prime }\beta L<0$ hold for all $\beta \in \left[ 0,1\right]$, which implies $\lim _{X_{m}\rightarrow aL}G^{\prime }<0$ (since $\beta \rightarrow 0$ as $X_{m}\rightarrow aL$ and $G^{\prime }-l_{f}A^{\prime }\beta L\rightarrow G^{\prime }$ as $\beta \rightarrow 0$). This means that $T_{\infty }^{\prime \prime }$ has the opposite sign to $\varPhi _{1}$ as $X_{m}\rightarrow aL$, which is negative in BT-type countries but positive in CT-type countries.

1.4 A.4 Proof of Proposition 3

We derive the conditions for the respective world specialization patterns to arise, taking the environmental stocks S and $S^{*}$ as given. As byproducts, we obtain the labor allocations $\beta$ and $\beta ^{*}$ and the world relative price P. Note that some results in Table 2 further use the spending ratio definition $q\equiv b/\left( 1-b\right)$.

Pattern (f,d)

If pattern (f,d) arises, Home produces only the resource good, $X_{f}=A\left( S\right) L$, and Foreign produces both goods, $X_{f}^{*}=A^{*}\left( S^{*}\right) \beta ^{*}L^{*}$ and $X_{m}^{*}=a^{*}\left( 1-\beta ^{*}\right) L^{*}$. The world relative price of the manufactured good with respect to the resource good satisfies

$$\begin{aligned} P=\frac{A^{*}\left( S^{*}\right) }{a^{*}}, \end{aligned}$$

(29)

which depends only on Foreign’s condition. In pattern (f,d), the resource good is produced in both countries, and thus the wage in Home is $w=A\left( S\right)$ and that in Foreign is $w^{*}=A^{*}\left( S^{*}\right)$. Since the manufactured good is supplied only by Foreign, the world market-clearing condition requires, given the same preference in the two countries, $\left( 1-b\right) \left( wL+w^{*}L^{*}\right) =PX_{m}^{*}$. This gives Foreign’s labor allocation:

$$\begin{aligned} \beta ^{*}=b-\left( 1-b\right) \frac{z}{v}, \end{aligned}$$

(30)

where, as defined in (19) and (20),

$$\begin{aligned} v\equiv \frac{A^{*}\left( S^{*}\right) a}{A\left( S\right) a^{*}},\quad z\equiv \frac{aL}{a^{*}L^{*}}. \end{aligned}$$

Since Foreign produces both goods, we have $\beta ^{*}>0$, which together with (30) yields $v>\left( 1-b\right) z/b$. Recalling that a necessary condition for Foreign to export the manufactured good is $v\le 1$, the necessary condition for pattern (f,d) to arise is therefore

$$\begin{aligned} \frac{1-b}{b}z<v\le 1. \end{aligned}$$

(31)

A sufficient condition can be obtained by excluding $v=1$:

$$\begin{aligned} \frac{1-b}{b}z<v<1. \end{aligned}$$

(32)

Pattern (f,m)

In pattern (f,m), Home produces only the resource good, $X_{f}=A\left( S\right) L$, and Foreign produces only the manufactured good, $X_{m}^{*}=a^{*}L^{*}$. The wage in Home is $w=A\left( S\right)$, whereas that in Foreign is $w^{*}=a^{*}P$. Since both countries completely specialize, the world relative price of the manufactured good, P, is determined such that world supply equals world demand, $b\left( wL+w^{*}L^{*}\right) =X_{f}$, which gives

$$\begin{aligned} P=\frac{\left( 1-b\right) A\left( S\right) L}{ba^{*}L^{*}}. \end{aligned}$$

(33)

For the two countries to completely specialize and Home to export the resource good, the necessary condition is $A^{*}\left( S^{*}\right) /a^{*}\le P\le A\left( S\right) /a$, which yields

$$\begin{aligned} v\le \frac{1-b}{b}z\le 1. \end{aligned}$$

(34)

A sufficient condition can be obtained by excluding $v=1$:

$$\begin{aligned} v\le \frac{1-b}{b}z\le 1 \text{ and } v\ne 1. \end{aligned}$$

(35)

Pattern (d,m)

In pattern (d,m), Home produces both goods, $X_{f}=A\left( S\right) \beta L$ and $X_{m}=a(1-\beta )L$, and Foreign produces only the manufactured good, $X_{m}^{*}=a^{*}L^{*}$. The world relative price of the manufactured good depends only on Home’s condition:

$$\begin{aligned} P=\frac{A\left( S\right) }{a}. \end{aligned}$$

(36)

The wage in Home is $w=A\left( S\right)$, whereas that in Foreign is $w^{*}=a^{*}P=A\left( S\right) a^{*}/a$. Since the resource good is supplied only by Home, the market-clearing condition requires $b\left( wL+w^{*}L^{*}\right) =X_{f}$. This gives Home’s labor allocation:

$$\begin{aligned} \beta =b\left( 1+\frac{1}{z}\right) . \end{aligned}$$

(37)

Since Home produces both goods, we have $\beta <1$, and therefore using (37), $\left( 1-b\right) z/b>1$. Recalling that a necessary condition for Home to export the resource good is $v\le 1$, the necessary condition for pattern (d,m) is therefore

$$\begin{aligned} v\le 1<\frac{1-b}{b}z. \end{aligned}$$

(38)

A sufficient condition follows by excluding $v=1$:

$$\begin{aligned} v<1<\frac{1-b}{b}z. \end{aligned}$$

(39)

Pattern (d,f)

In pattern (d,f), we have $X_{f}=A\left( S\right) \beta L$, $X_{m}=a\left( 1-\beta \right) L$, and $X_{f}^{*}=A^{*}\left( S^{*}\right) L^{*}$. The world relative price of the manufactured good is given by (36). The wages are $w=A\left( S\right)$ and $w^{*}=A^{*}\left( S^{*}\right)$. The market-clearing condition requires $\left( 1-b\right) \left( wL+w^{*}L^{*}\right) =PX_{m}$, which gives Home’s labor allocation:

$$\begin{aligned} \beta =b-\left( 1-b\right) \frac{v}{z}. \end{aligned}$$

(40)

It follows from $\beta >0$ that $v<bz/\left( 1-b\right)$, which together with $v\ge 1$ gives the necessary condition for pattern (d,f):

$$\begin{aligned} 1\le v<\frac{b}{1-b}z. \end{aligned}$$

(41)

Again, excluding $v=1$ yields a sufficient condition:

$$\begin{aligned} 1<v<\frac{b}{1-b}z. \end{aligned}$$

(42)

Pattern (m,f)

In pattern (m,f), we have $X_{m}=aL$ and $X_{f}^{*}=A^{*}\left( S^{*}\right) L^{*}$. The wages are $w=aP$ and $w^{*}=A^{*}\left( S^{*}\right)$. The world relative price of the manufactured good is determined by the market-clearing condition $b\left( wL+w^{*}L^{*}\right) =X_{f}$, which gives

$$\begin{aligned} P=\frac{\left( 1-b\right) A^{*}\left( S^{*}\right) L^{*}}{baL}. \end{aligned}$$

(43)

For the two countries to completely specialize and Foreign to export the resource good, the necessary condition is $A\left( S\right) /a\le P\le A^{*}\left( S^{*}\right) /a^{*}$, which yields

$$\begin{aligned} 1\le \frac{b}{1-b}z\le v. \end{aligned}$$

(44)

A sufficient condition can be obtained by excluding $v=1$:

$$\begin{aligned} 1\le \frac{b}{1-b}z\le v \text{ and } v\ne 1. \end{aligned}$$

(45)

Pattern (m,d)

In pattern (m,d), we have $X_{m}=aL$, $X_{f}^{*}=A^{*}\left( S^{*}\right) \beta ^{*}L^{*}$, and $X_{m}^{*}=a^{*}\left( 1-\beta ^{*}\right) L^{*}$. The world relative price of the manufactured good is given by (29). The wages are $w=aP$ and $w^{*}=A^{*}\left( S^{*}\right)$. The market-clearing condition requires $b\left( wL+w^{*}L^{*}\right) =X_{f}^{*}$, which gives Foreign’s labor allocation:

$$\begin{aligned} \beta ^{*}=b\left( 1+z\right) . \end{aligned}$$

(46)

It follows from $\beta ^{*}<1$ that $bz/\left( 1-b\right) <1$, which together with $v\ge 1$ gives the necessary condition for pattern (m,d):

$$\begin{aligned} \frac{b}{1-b}z<1\le v. \end{aligned}$$

(47)

A sufficient condition follows by excluding $v=1$:

$$\begin{aligned} \frac{b}{1-b}z<1<v. \end{aligned}$$

(48)

Pattern (d,d)

In pattern (d,d), both countries produce both goods. A necessary condition for pattern (d,d) to arise is that $v=1$. The wages are $w=A\left( S\right)$ and $w^{*}=A^{*}\left( S^{*}\right)$. The market-clearing condition requires $b\left( wL+w^{*}L^{*}\right) =A\left( S\right) \beta L+A^{*}\left( S^{*}\right) \beta ^{*}L^{*}$, which gives (22), namely,

$$\begin{aligned} \beta z+\beta ^{*}=b\left( z+1\right) . \end{aligned}$$

With (22) as the only constraint on the two variables, $\beta$ and $\beta ^{*}$ are indeterminate (and thus other patterns may arise).

1.5 A.5 Proof of Proposition 4

Let $\beta _{T}$ and $\beta _{T}^{*}$ denote the labor allocations in Home and Foreign at the trade steady state. Supposing that the trade pattern is Home exporting the resource good and Foreign exporting the manufactured good at the trade steady state, we have

$$\begin{aligned} \beta _{T}>b,\quad \beta _{T}^{*}<b. \end{aligned}$$

Consider two countries of the same type. If both countries are the BT type,

$$\begin{aligned} S_{\infty }\left( \beta _{T}\right) <S_{\infty }\left( b\right) ,\quad S_{\infty }^{*}\left( \beta _{T}^{*}\right) >S_{\infty }^{*}\left( b\right) . \end{aligned}$$

If both countries are the CT type, then

$$\begin{aligned} S_{\infty }\left( \beta _{T}\right) >S_{\infty }\left( b\right) ,\quad S_{\infty }^{*}\left( \beta _{T}^{*}\right) <S_{\infty }^{*}\left( b\right) . \end{aligned}$$

In either case, the environmental stock in one country increases and that in the other decreases at the trade steady state (vis-a-vis those at the autarkic steady state).

Consider now two countries of different types. If Home is of BT type (thus Foreign of CT type),

$$\begin{aligned} S_{\infty }\left( \beta _{T}\right)<S_{\infty }\left( b\right) ,\quad S_{\infty }^{*}\left( \beta _{T}^{*}\right) <S_{\infty }^{*}\left( b\right) . \end{aligned}$$

If Home is of CT type (thus Foreign of BT type), then the opposite holds:

$$\begin{aligned} S_{\infty }\left( \beta _{T}\right)>S_{\infty }\left( b\right) ,\quad S_{\infty }^{*}\left( \beta _{T}^{*}\right) >S_{\infty }^{*}\left( b\right) . \end{aligned}$$

In either case, the environmental stocks in the two countries change in the same direction at the trade steady state (vis-a-vis those at the autarkic steady state).

From similar arguments, the same conclusion holds if the trade pattern is reversed.

1.6 A.6 Proof of Proposition 5

We derive welfare impacts of trade on each country by comparing the trade steady-state utility level ($V_{T}$ for Home and $V_{T}^{*}$ for Foreign) with autarky ($V_{A}$ for Home and $V_{A}^{*}$ for Foreign). It is helpful to proceed in two cases categorized by the trade pattern.

Home Exports the Resource Good

Suppose that Home exports the resource good at the trade steady state and thus Foreign exports the manufactured good. We consider the welfare impacts of trade first on Foreign and then on Home.

Foreign either remains diversified ($\beta _{T}^{*}>0$) or specializes in manufacturing ($\beta _{T}^{*}=0$) at the trade steady state. We show that in either case Foreign gains from trade.

If Foreign remains diversified, the world relative price at the trade steady state can be expressed by, according to Table 2,

$$\begin{aligned} P_{T}=\frac{A^{*}\left( S_{\infty }^{*}\left( \beta _{T}^{*}\right) \right) }{a^{*}}. \end{aligned}$$

Since Foreign exports the manufactured good ($\beta _{T}^{*}<b$) and is of BT type, we have

$$\begin{aligned} P_{T}>P_{A}^{*}=\frac{A^{*}\left( S_{\infty }^{*}\left( b\right) \right) }{a^{*}}. \end{aligned}$$

(49)

Similar to (16), Foreign’s utility level at the autarkic steady state can be expressed by

$$\begin{aligned} V_{A}^{*}&=B+\ln L^{*}+b\ln A^{*}\left( S_{\infty }^{*}\left( b\right) \right) +\left( 1-b\right) \ln a^{*}\nonumber \\&=B+\ln L^{*}+\ln a^{*}+b\ln P_{A}^{*}, \end{aligned}$$

(50)

which together with (26) yields

$$\begin{aligned} V_{T}^{*}-V_{A}^{*}=b\ln \frac{P_{T}}{P_{A}^{*}}. \end{aligned}$$

(51)

It follows from (49) that $V_{T}^{*}>V_{A}^{*}$ holds (i.e., Foreign gains from trade).

If Foreign specializes in manufacturing, it holds that

$$\begin{aligned} P_{T}\ge \frac{A^{*}\left( S_{\infty }^{*}\left( 0\right) \right) }{a^{*}}, \end{aligned}$$

(52)

from which we also obtain (49). Since (51) still holds in this case, we have, again, that $V_{T}^{*}>V_{A}^{*}$ (i.e., Foreign gains from trade).

Home either remains diversified ($\beta _{T}<1$) or specializes in the resource good ($\beta _{T}=1$) at the trade steady state. We show that Home necessarily gains when remaining diversified and may gain or lose when specialized.

If Home remains diversified, we have, according to Table 2,

$$\begin{aligned} P_{T}=\frac{A\left( S_{\infty }\left( \beta _{T}\right) \right) }{a}. \end{aligned}$$

(53)

Since Home exports the resource good ($\beta _{T}>b$) and is of BT type,

$$\begin{aligned} P_{T}<\frac{A\left( S_{\infty }\left( b\right) \right) }{a}=P_{A}. \end{aligned}$$

(54)

It follows from (16) and (25) that

$$\begin{aligned} V_{T}-V_{A}=b\ln \frac{P_{T}}{P_{A}}, \end{aligned}$$

(55)

which together with (54) yields $V_{T}<V_{A}$ (i.e., Home loses from trade).

If Home specializes in the resource good at the trade steady state, then

$$\begin{aligned} P_{T}\le \frac{A\left( S_{\infty }\left( 1\right) \right) }{a}. \end{aligned}$$

(56)

Noting that $A\left( S_{\infty }\left( 1\right) \right) /a<P_{A}$ (since Home is of BT type), we have that $P_{T}<P_{A}$. It follows from (16) and (25) that

$$\begin{aligned} V_{T}-V_{A}=\ln \frac{A\left( S_{\infty }\left( 1\right) \right) }{A\left( S_{\infty }\left( b\right) \right) }-\left( 1-b\right) \ln \frac{P_{T}}{P_{A}}, \end{aligned}$$

(57)

the sign of which is indefinite given $P_{T}<P_{A}$ and $S_{\infty }\left( 1\right) <S_{\infty }\left( b\right)$. Letting $V_{T}=V_{A}$ in (57) yields the threshold price:

$$\begin{aligned} P_{1}^{\prime \prime }=\frac{A\left( S_{\infty }\left( 1\right) \right) }{a}\left( \frac{A\left( S_{\infty }\left( 1\right) \right) }{A\left( S_{\infty }\left( b\right) \right) }\right) ^{\frac{b}{1-b}}, \end{aligned}$$

(58)

such that $V_{T}>V_{A}$ (i.e., Home gains) if $P_{T}<P_{1}^{\prime \prime }$ and $V_{T}<V_{A}$ (i.e., Home loses) if $P_{T}>P_{1}^{\prime \prime }$.

Foreign Exports the Resource Good

Now suppose that Foreign exports the resource good at the trade steady state. Similar arguments as above apply.

Specifically, Home (exporting the manufactured good) necessarily gains from trade. Foreign (exporting the resource good) loses if it remains diversified at the trade steady state. If Foreign specializes, it follows from (50) and (26) that

$$\begin{aligned} V_{T}^{*}-V_{A}^{*}=\ln \frac{A^{*}\left( S_{\infty }^{*}\left( 1\right) \right) }{A^{*}\left( S_{\infty }^{*}\left( b\right) \right) }-\left( 1-b\right) \ln \frac{P_{T}}{P_{A}^{*}}. \end{aligned}$$

(59)

Letting $V_{T}^{*}=V_{A}^{*}$ gives the threshold price:

$$\begin{aligned} P_{2}^{\prime \prime }=\frac{A^{*}\left( S_{\infty }^{*}\left( 1\right) \right) }{a^{*}}\left( \frac{A^{*}\left( S_{\infty }^{*}\left( 1\right) \right) }{A^{*}\left( S_{\infty }^{*}\left( b\right) \right) }\right) ^{\frac{b}{1-b}}, \end{aligned}$$

(60)

such that $V_{T}^{*}>V_{A}^{*}$ (i.e., Foreign gains) if $P_{T}<P_{2}^{\prime \prime }$, and $V_{T}^{*}<V_{A}^{*}$ (i.e., Foreign loses) if $P_{T}>P_{2}^{\prime \prime }$.

Combining (58) and (60) gives (27).

For a given world specialization pattern at the trade steady state, the welfare impacts of trade on Home and Foreign can be obtained by applying the relevant results derived above.

1.7 A.7 Proof of Proposition 6

We examine the welfare impacts of trade by comparing the trade steady-state utility level ($V_{T}$ for Home and $V_{T}^{*}$ for Foreign) with autarky ($V_{A}$ for Home and $V_{A}^{*}$ for Foreign). We proceed in two cases as follows.

Home Exports the Resource Good

Suppose that Home exports the resource good at the trade steady state and thus Foreign exports the manufactured good. We consider the welfare impacts of trade first on Home and then on Foreign.

Home either remains diversified ($\beta _{T}<1$) or specializes in the resource good ($\beta _{T}=1$) at the trade steady state. We show that in either case, Home gains from trade.

If Home remains diversified, we have (53) and (55). Since Home exports the resource good ($\beta _{T}>b$) and is of CT type, $S_{\infty }\left( \beta _{T}\right) >S_{\infty }\left( b\right)$ holds. This implies, by (53), $P_{T}>P_{A}$, and thus by (55), $V_{T}>V_{A}$ (i.e., Home gains from trade).

If Home specializes in the resource good, we have (56) and (57). Note that

$$\begin{aligned} \ln \frac{A\left( S_{\infty }\left( 1\right) \right) }{A\left( S_{\infty }\left( b\right) \right) }>\left( 1-b\right) \ln \frac{A\left( S_{\infty }\left( 1\right) \right) }{A\left( S_{\infty }\left( b\right) \right) }\ge \left( 1-b\right) \ln \frac{P_{T}}{P_{A}}, \end{aligned}$$

where the first inequality comes from $S_{\infty }\left( 1\right) >S_{\infty }\left( b\right)$ (since Home is of CT type), and the second inequality from (56). This together with (57) also gives $V_{T}>V_{A}$ (i.e., Home gains from trade).

Foreign either remains diversified ($\beta _{T}^{*}>0$) or specializes in manufacturing ($\beta _{T}^{*}=0$) at the trade steady state. We show that Foreign necessarily loses when remaining diversified and may gain or lose when specialized.

If Foreign remains diversified, (51) holds. Since Foreign exports the manufactured good ($\beta _{T}^{*}<b$) and is of CT type, we have

$$\begin{aligned} P_{T}=\frac{A^{*}\left( S_{\infty }^{*}\left( \beta _{T}^{*}\right) \right) }{a^{*}}<\frac{A^{*}\left( S_{\infty }^{*}\left( b\right) \right) }{a^{*}}=P_{A}^{*}, \end{aligned}$$

(61)

which together with (51) gives $V_{T}^{*}<V_{A}^{*}$ (i.e., Foreign loses from trade).

If Foreign specializes in manufacturing, (51) holds, and

$$\begin{aligned} P_{T}\ge \frac{A^{*}\left( S_{\infty }^{*}\left( 0\right) \right) }{a^{*}}. \end{aligned}$$

(62)

Noting that $A^{*}\left( S_{\infty }^{*}\left( 0\right) \right) /a^{*}<P_{A}^{*}$, the sign of (51) remains ambiguous. We further show that Foreign’s gains from trade depend on Home’s specialization pattern: Foreign gains if Home remains diversified and may gain or lose if Home specializes. The details are as follows. (i) If Home remains diversified at the trade steady state, we have $S_{\infty }\left( \beta _{T}\right) >S_{\infty }\left( b\right)$, and thus by (53), $P_{T}>P_{A}$. Note that the trade pattern between two countries of CT type is self-reinforcing in that the trade pattern remains unchanged during a dynamic transition (as long as there are no exogenous shocks).⁴¹ This self-reinforcing feature ensures that if Home exports the resource good at the trade steady state, it also exports the resource good immediately after trade is opened, which requires

$$\begin{aligned} P_{A}\ge P_{A}^{*}. \end{aligned}$$

It then follows that $P_{T}>P_{A}^{*}$ and thus $V_{T}^{*}>V_{A}^{*}$ (i.e., Foreign gains from trade). (ii) If Home specializes in the resource good at the trade steady state, it requires $P_{T}\le A\left( S_{\infty }\left( 1\right) \right) /a$ as given in (56). On the other hand, we have $A\left( S_{\infty }\left( 1\right) \right) /a>P_{A}\ge P_{A}^{*}$ (the first inequality comes from Home being of CT type). Thus, the sign of (51) remains ambiguous. Letting $V_{T}^{*}=V_{A}^{*}$ in (51) yields the threshold price:

$$\begin{aligned} P_{1}^{\prime \prime \prime }=\frac{A^{*}\left( S_{\infty }^{*}\left( b\right) \right) }{a^{*}}=P_{A}^{*}, \end{aligned}$$

(63)

such that $V_{T}^{*}>V_{A}^{*}$ (i.e., Foreign gains) if $P_{T}>P_{1}^{\prime \prime \prime }$ and $V_{T}^{*}<V_{A}^{*}$ (i.e., Foreign loses) if $P_{T}<P_{1}^{\prime \prime \prime }$.

Foreign Exports the Resource Good

Now suppose that Foreign exports the resource good at the trade steady state and thus Home exports the manufactured good. Similar arguments as above apply.

Specifically, Foreign (exporting the resource good) necessarily gains from trade. Home (exporting the manufactured good) loses if it remains diversified. If Home specializes, we have (55) and

$$\begin{aligned} P_{T}\ge \frac{A\left( S_{\infty }\left( 0\right) \right) }{a}. \end{aligned}$$

Noting that $A\left( S_{\infty }\left( 0\right) \right) /a<P_{A}$ (since Home is of CT type), the sign of (55) is ambiguous. We further show that Home’s gains from trade depend on Foreign’s specialization pattern: Home gains if Foreign stays diversified and may gain or lose if Foreign specializes. The details are as follows. (i) If Foreign stays diversified, we have

$$\begin{aligned} P_{T}=\frac{A^{*}\left( S_{\infty }^{*}\left( \beta _{T}^{*}\right) \right) }{a^{*}}>\frac{A^{*}\left( S_{\infty }^{*}\left( b\right) \right) }{a^{*}}=P_{A}^{*} \end{aligned}$$

since Foreign exports the resource good ($\beta _{T}^{*}>b$) and is of CT type. The self-reinforcing trade pattern (which arises between two countries of CT type) ensures that

$$\begin{aligned} P_{A}^{*}\ge P_{A}, \end{aligned}$$

which gives $P_{T}>P_{A}$, and thus $V_{T}>V_{A}$ (i.e., Home gains from trade). (ii) If Foreign specializes in manufacturing, $P_{T}\ge A^{*}\left( S_{\infty }^{*}\left( 0\right) \right) /a^{*}$ is required as given in (52). On the other hand, we have $A^{*}\left( S_{\infty }^{*}\left( 0\right) \right) /a^{*}<P_{A}^{*}$ (since Foreign is of CT type). However, with these constraints, the sign of (55) remains ambiguous. Letting $V_{T}=V_{A}$ in (55) yields the threshold price:

$$\begin{aligned} P_{2}^{\prime \prime \prime }=\frac{A\left( S_{\infty }\left( b\right) \right) }{a}=P_{A}, \end{aligned}$$

(64)

such that $V_{T}>V_{A}$ (i.e., Home gains) if $P_{T}>P_{2}^{\prime \prime \prime }$ and $V_{T}<V_{A}$ (i.e., Home loses) if $P_{T}<P_{2}^{\prime \prime \prime }$.

Combining (63) obtained in the first case and (64) in the second case gives (28).

For a specific world specialization pattern, the welfare impacts of trade on the two countries can be obtained by applying the relevant results derived above. Note that, however, there exists no (stable) pattern (d,d) at the trade steady state when both countries are of CT type. To see this, suppose that $P_{A}>P_{A}^{*}$ such that Home exports the resource good and Foreign exports the manufactured good right after trade opening. Due to the self-reinforcing feature of the trade pattern as discussed above, the same trade pattern prevails at the trade steady state, which requires $P_{T}>P_{A}$ if Home remains diversified and $P_{T}<P_{A}^{*}$ if Foreign remains diversified. Therefore, if both Home and Foreign remain diversified at the trade steady state, $P_{T}>P_{A}>P_{A}^{*}>P_{T}$ holds and leads to a contradiction. Moreover, as a result of intended policies or exogenous shocks, it is possible that there exists a trade steady state with pattern (d,d) between two CT-type countries. Such a trade steady state is, however, also unstable because of the self-reinforcing feature mentioned above (see the Online Appendix for a formal proof).

1.8 A.8 Proof of Proposition 7

We consider first what happens if Home is of BT type (and thus Foreign is of CT type since the two countries are of different types), and then we consider what happens if Foreign is of BT type.

Home is of BT Type

As for part (i), Home (the BT-type country) exports the manufactured good ($\beta _{T}<b$) at the trade steady state, and Foreign (the CT-type country) exports the resource good ($\beta _{T}^{*}>b$). Home either remains diversified ($\beta _{T}>0$) or specializes ($\beta _{T}=0$). If Home remains diversified,

$$\begin{aligned} P_{T}=\frac{A\left( S_{\infty }\left( \beta _{T}\right) \right) }{a}>P_{A}, \end{aligned}$$

where the inequality comes from $\beta _{T}<b$ and the fact that Home is of BT type. If Home specializes,

$$\begin{aligned} P_{T}\ge \frac{A\left( S_{\infty }\left( 0\right) \right) }{a}>P_{A}. \end{aligned}$$

In either case, $P_{T}>P_{A}$ holds. Since Home exports the manufactured good, we have that (55) holds, which yields $V_{T}>V_{A}$. That is, Home (the BT-type country that exports the manufactured good) gains from trade in the long run.

Similarly, Foreign either remains diversified ($\beta _{T}^{*}<1$) or specializes in the production of the resource good ($\beta _{T}^{*}=1$). If Foreign remains diversified,

$$\begin{aligned} P_{T}=\frac{A^{*}\left( S_{\infty }^{*}\left( \beta _{T}^{*}\right) \right) }{a^{*}}>P_{A}^{*}, \end{aligned}$$

where the inequality comes from the fact that $\beta _{T}^{*}>b$ and that Foreign is of CT type. If Foreign stays diversified, the difference in Foreign’s steady-state utility levels under trade and autarky can be expressed by (51), which together with $P_{T}>P_{A}^{*}$ yields $V_{T}^{*}>V_{A}^{*}$. If Foreign specializes,

$$\begin{aligned} P_{T}\le \frac{A^{*}\left( S_{\infty }^{*}\left( 1\right) \right) }{a^{*}}. \end{aligned}$$

(65)

The difference between the utility level at the trade steady state and that in autarky can be expressed by (59). Note that

$$\begin{aligned} \ln \frac{A^{*}\left( S_{\infty }^{*}\left( 1\right) \right) }{A^{*}\left( S_{\infty }^{*}\left( b\right) \right) }>\left( 1-b\right) \ln \frac{A^{*}\left( S_{\infty }^{*}\left( 1\right) \right) }{A^{*}\left( S_{\infty }^{*}\left( b\right) \right) }\ge \left( 1-b\right) \ln \frac{P_{T}}{P_{A}^{*}}, \end{aligned}$$

where the first inequality comes from the fact that Foreign is of CT type and thus $S_{\infty }^{*}\left( 1\right) >S_{\infty }^{*}\left( b\right)$; the second inequality comes from (65). This together with (59) yields, again, that $V_{T}^{*}>V_{A}^{*}$. In either case, Foreign (the CT-type country that exports the resource good) gains from trade.

As for part (ii), Home exports the resource good, and Foreign exports the manufactured good. As discussed in the proof of Proposition 5, Home necessarily loses from trade if it remains diversified and may gain or lose if it specializes. It loses if $P_{T}>P^{\prime \prime }$. As discussed in the proof of Proposition 6, Foreign necessarily loses if remaining diversified and may gain or lose if specializing. It loses if $P_{T}<P^{\prime \prime \prime }$.

Foreign is of BT Type

The arguments above remain true when switching Home (and Home-related variables) with Foreign (and Foreign-related variables).