In this section, we focus on the home country. The result of the foreign country can be obtained analogously.
5.1 A Partial Equilibrium Analysis
Similar to Fan et al. (2013), we fix the wage in this section. We first analyze the impact on patterns of specialization. Then we analyze the impact on cuto↵ productivity level. The average sectoral productivity level increases with the cuto↵ productivity level.
Taking the derivative of A and A with respect to ⌧ , we can observe that depending on , trade liberalization has di↵erent impacts on patterns of specialization. The result is summarized by Figure (2). It is caused by the interaction between the home market e↵ect and Ricardo’s comparative advantage.
When the di↵erence in size is significant, the decrease in ⌧ induces pro-ducers to leave sectors of the large country where that country has a weak comparative advantage. Therefore, if > ⌧2✓, the range of sectors in which production occurs in only the home country is reduced, and the range of sectors in which only the foreign country produces increases. By contrast, when < ⌧ 2✓, the range of sectors in which production occurs in only the home country increases, and the range of sectors in which only the foreign country produces is reduced. When the di↵erence in size is not so signif-icant, that is, when 2 ⌧ 2✓, ⌧2✓ , the impact is similar to the ordinary Ricardian model, which increases the range of sectors in which only home
Labor reallocation accompanies the change of specialization patterns. It also depends on the relative size and is summarized by Figures (4) to (6).
The solid line shows the allocation of labor before trade liberalization. The dotted line shows the result after trade liberalization. As will be explained latter, the dash-dot lines in Figure (4) are used to analyze the impact of trade liberalization. The relationship is not necessarily linear. We draw a linear relationship to illustrate the result clearly without a↵ecting the generality of the qualitative result. In general, because the home market e↵ect is weakened when trade costs fall, when is higher, more sectors experience a decrease in labor demand while fewer sectors experience an increase in demand.
We briefly describe how the figures are derived. Without loss of gener-ality, the figures are drawn under the assumption that i is the same for all i. Therefore, the maximum labor demand, i⇣
L + w 1L⇤⌘
, does not vary with i. This level of demand is reached when the production occurs in only the home country. It is represented by the horizontal line in those figures.
The line with negative slope shows the labor demand when both countries produce. As we mentioned, in this scenario, labor demand starts decreasing with Ai; when Ai is high enough, labor demand reaches zero.
To show how labor demand changes with ⌧ , by taking the derivative of the right-hand side of equation (37) with respect to ⌧ , it can be shown that the derivative is positive if Ai is greater than
(40) Aˆl= ⌧✓+ 21
1 + ⌧✓ 21
!1+✓1 w 1,
where the subscript l means the threshold is related to the labor reallocation.
The derivative is negative if Ai< ˆAl. It is zero when Ai = ˆAl.
. It is summarized by Figure (3). If 2 ⌧ 2✓, ⌧2✓ , ˆAl 2 A, A . Hence, when trade is liberalized, as is shown by Figure (5), the line is rotated clock-wise, as ˆAl is the pivot point. The rotated line intersects with the upper bound of labor demand and the horizontal axis at A0 and A0, respectively.
Those are the new thresholds used to determine the patterns of specializa-tion. In Figure (4), if we also draw the graph of equation (37) when it is defined over Ai A or Ai A0, we obtain the dash-dot lines. The dash-dot line extending from the solid line represents the scenario before trade liber-alization. The other represents the scenario afterwards. As shown by Figure (3), when > ⌧2✓, ˆAl < A; therefore two dash-dot lines must intersect at Aˆl. As shown by Figure (4), when trade is liberalized, the solid line and its extended dash-dot line rotate clockwise, as ˆAl is the pivot point. The new thresholds used to determine patterns of specialization are also obtained.
Hence, for sectors with Ai2 A0, A , their labor demand decreases. Similar logic can be applied to derive Figure (6).
Now we analyze the impact of trade liberalization on ˆ'id. The average sectoral productivity strictly increases in it. By substituting Li and L⇤i obtained in the previous section into equation (29), we obtain
ˆ the definition of A, after some calculation it can be shown that the values of the first and second expressions are equalized. A similar calculation shows
Ai = A.
As we have mentioned, because there are no entrants in sectors with Ai A, in this scenario ˆ'id would be the lowest productivity level needed to maintain zero gross profit at home if there are entrants. Because the home country does not produce, in this case the change of ˆ'id should not be viewed as an impact on productivity distribution. It is the change of
ˆ
'⇤id that reflects this impact because production takes place wholly in the foreign country .5 The change in ˆ'⇤id can be derived when we analyze the impact on productivity distribution in the foreign country. We still analyze how ˆ'idchanges although there are no production activities because welfare is a function of it as we show in appendix.
The relationship between ˆ'id and i before trade liberalization is shown by the solid lines in Figures (7) to (9). The dotted lines show how the relationship is changed by the trade liberalization. i0 and i0 correspond to A0 and A0, respectively. The impact is a function of . Without loss of generality, those figures are drawn under the assumption that ai increases when i increases while a⇤i is the same for all i. Under these assumptions, ˆ'id decreases with i and it starts increasing in sectors where the home country does not produce. The relationship is not necessarily linear. We draw a linear relationship to illustrate the result clearly without a↵ecting the generality of the qualitative result.
We use Figure (10) as an example to demonstrate how we derive the result of trade liberalization. The figure shows the case when 2 1, ⌧2✓ . The solid lines are the graph before trade liberalization. The dotted lines
5The change of ˆ'⇤ie may be even more appropriate to reflect the impact because all products come from the foreign country via exports. However, it can be immediately confirmed that ˆ'⇤ie= ˆ'⇤idby substituting Li= 0 into equations (30) and (31). Therefore, any impact on ˆ'⇤ie is reflected in the change of ˆ'⇤id.
show the result of trade liberalization. Line AA is the graph of the first expression of (41). The graph also shows the value of the expression when i > i. Line BB is the graph of the second expression. The value when i < i or i > i is also shown. Line CC is the graph of the last expression. We also draw the graph of this expression when i < i. If we take the derivative of the second expression of (41) with respect to ⌧ and after some rearrangement, then we obtain @ ˆ'id @⌧ > 0 if
Ai > ˆA' =
✓⌧✓+ ⌧ ✓ 2
◆1+✓1 w 1,
where the subscript ' means that the threshold is associated with produc-tivity. The derivative is negative when Ai < ˆA'. The derivative is zero when Ai = ˆA'. Simple calculation shows that the relative position of ˆA', A, and A on the real line depends on the value of . Figure (11) summarizes this information. When 2 1, ⌧2✓ , ˆA' 2 A, A . Therefore, after trade liberalization, line CC rotates clockwise with the pivot point lying between i and i. Because the value of the last expression increases when ⌧ decreases, line CC moves up. The new thresholds, i0 and i0 determining the patterns of specialization are also obtained. The impact of trade liberalization when
> ⌧2✓ or when < 1 can be derived analogously.
As these figures illustrate, because of the home market e↵ect the impact of trade liberalization is a function of . When > 1, sectors where the large country has a weak—but not too weak—comparative advantage experience a decrease in ˆ'id. However, it is impossible to observe a decrease in ˆ'idwhen
< 1. This result is in contrast to the one-sector models in Melitz (2003) and Melitz and Ottaviano (2008) where trade liberalization always increases
is because high-productivity firms are more capable of overcoming the fixed exporting costs not because of higher markup since elasticity is exogenous.
In our model, it results from the decrease of the markup; therefore, trade liberalization does not necessarily reduce markup.
5.2 Why Productivity Cuto↵ Decreases with Trade Liberal-ization
Now we analyze why, in some sectors, trade liberalization decreases ˆ'idand therefore it also decreases the average sectoral productivity. Taking the derivative of equation (29) with respect to ⌧ , we have
@ ˆ'id
The terms in the brackets capture all e↵ects that influence ˆ'id. This occurs through two channels. The first term captures the anticompetitive e↵ect.
The increase of ⌧ reduces the competitiveness of the foreign producers when they sell in the home country. Therefore, it reduces the competition level and increases markup. This reduces ˆ'id. The second channel comes from the reallocation of labor across sectors. It is captured by the last two terms. As the measure of entrants in a sector strictly increases with the labor allocated to the sector, the competition level, markup, and therefore ˆ'id, change. The impact of this channel can be positive, negative, or zero depending on how the labor is reallocated across sectors.
It is now clear why ˆ'iddoes not change in those sectors in which produc-tion occurs in only the home country before and after trade liberalizaproduc-tion.
Because there are no foreign competitors, there is no e↵ect operating through
the first channel. When w is fixed, Li does not change and L⇤i remains zero;
thus, the impact of the second channel is also missing. On the contrary, for sectors in which only the foreign country produces before and after trade liberalization, ˆ'id increases. Because trade liberalization increases the com-petitiveness of foreign producers and Li remains equal to zero and L⇤i does not change because the wage is fixed, trade liberalization increases the cuto↵
productivity level mainly through the first channel.
When the e↵ects of both channels are not zero, the result is ambiguous.
If the e↵ect of the second channel is negative and it dominates that of the first channel, the cuto↵ productivity level decreases when trade is more liberalized. We will use the home country as an example to show that the necessary condition for the second channel to be negative is dLi < 0. For the foreign country, the same logic can be applied to show that the condition is dL⇤i < 0. It helps to explain why trade liberalization tends to decrease
ˆ
'id when Ai is higher than a threshold level but lower than an upper bound and when > 1.
Because the wage is fixed, total expenditure in sector i is unchanged.
This implies that wdLi + dL⇤i = 0. Therefore, dL⇤i dLi = w. Using this information, it can be shown that if the e↵ect of the second channel is negative, we have
(42) dLi⇣
1 A1+✓i w1+✓⌧ ✓⌘
< 0.
Therefore, when dLi is negative, the e↵ect of the second channel is negative if and only if
(43) Ai < w 1⌧1+✓✓ .
The result is intuitive. As the decrease in Li must accompany the pro-portional increase in L⇤i, the home producers are replaced by the foreign producers. The reallocation decreases the competition level if and only if the efficiency of the home producers is higher than a threshold level. Fur-thermore, the home producers have to be more efficient to satisfy inequality (42) when the wage is higher or when ⌧ is lower, because both imply that foreign producers are more competitive in the home market. On the con-trary if dL > 0, the e↵ect of the second channel is negative if and only if
(44) Ai > w 1⌧1+✓✓ .
In this case, the reallocation decreases the competition level if and only if the labor requirement of the home producers relative to that of the foreign producers is large enough.
It can be shown that ˆAl is smaller than the right-hand side of the in-equalities (43) and (44). When Ai< ˆAl, the inequality (44) will never hold.
Therefore, if dLi > 0 (this happens when Ai < ˆAl), it must be true that the e↵ect of the second channel is positive. On the contrary, if dLi < 0 (this happens when Ai > ˆAl) and inequality (43) is satisfied, the e↵ect of the second channel is negative.
As dLi < 0 is necessary, it is clear why ˆ'id decreases when is large enough and when Ai belongs to a certain interval, that is, when Ai is high enough but not too high. When two countries are symmetric in size, trade liberalization tends to reduce labor demand in sectors with weak compara-tive advantage. However, when two countries are asymmetric, the reduction is more significant when is higher because the home market e↵ect is weak-ened. Conversely, it is impossible to observe a reduction in labor demand
when is sufficiently small. Furthermore, the reduction of labor demand, if it occurs, is small when the comparative disadvantage is very strong. This is because most of the production activities of those sectors have been concen-trated in the foreign country. If the disadvantage is sufficiently strong, all production occurs in the foreign country before trade liberalization. Then the fall in trade costs has no impact on Li.
5.3 A General Equilibrium Analysis
The change of w a↵ects Li and L⇤i. It also a↵ects the competitiveness of home relative to foreign producers. Therefore, ˆ'id must change with it.
Without endogenizing w, the results obtained in the partial equilibrium analysis might not be robust. Furthermore, because labor demand from the sectors in which production occurs in only one country changes with w, the cuto↵ productivity level of those sectors also changes. This e↵ect is not captured by the partial equilibrium analysis.
Taking the derivative of the left-hand side of equation (39) with respect to ⌧ while holding w constant, we have
Z
i2[i,i]
@li
@⌧ wdi.
The sign of the derivative is ambiguous and depends on the distributions of A and , the size of , and the level of ⌧ . Therefore, the endogenous level of w might change nonmonotonically with ⌧ . Because of the ambiguity, we rely on numerical examples to show the impact under general equilibrium. We also analyze how the change in the cuto↵ productivity level a↵ects welfare.
In the appendix, we show that after a monotonic transformation, the utility
level of the representative consumer in the home country can be written as
(45) ln eV =
Z
i i
✓ ln
"
Li ai +L⇤i
a⇤i
✓Aiw
⌧
◆✓# di.
The utility level of the representative consumer in the foreign country can be obtained analogously. In the next section, we provide numerical examples to show the impact of trade liberalization on cuto↵ productivity level, on patterns of specialization, and on welfare. The result is in contrast to what we obtained in the partial equilibrium analysis.