Sourcing product quality for foreign market entry

Abstract

This study employs a differentiated Cournot model with both horizontal and vertical product differentiation. It scrutinizes the implications of frictions over manufacturing high quality (upstream market power) on the endogenous foreign market entry mode and product quality choice. Both high-quality exports and subsidiary sales require high-quality inputs supplied by a monopoly upstream firm, and thus are costly. Relative product quality, product substitutability, and product-specific trade costs are the key for the variation in input/output prices, sales and the markup between an exporter and a multinational, and have significant trade policy implications for high-quality exports and subsidiary sales.

Appendix

1.1 A.1 Proof of Proposition 1

Comparing a multinational’s profits from subsidiary sales of different quality, given by Eqs. (3) and (5) shows that

$$\begin{aligned} \lim \limits _{{t\rightarrow 0}}\left[ \pi _h^{FDI}({\overline{u}},\cdot )-\pi _h^{FDI}({\underline{u}},\cdot )\right] =\frac{(2 {\overline{u}}+(4-3 \sigma ) {\underline{u}}) (({\overline{u}}-{\underline{u}})-({\underline{u}}-(\sigma {\underline{u}}/2)))}{2 (4-\sigma ^2)^2}, \end{aligned}$$

which is clearly negative for $({\overline{u}}/{\underline{u}})<(4-\sigma )/2 \iff ({\overline{u}}-{\underline{u}})-({\underline{u}}-(\sigma {\underline{u}}/2))<0$. Moreover, $\partial [\pi _h^{FDI}({\overline{u}},\cdot )-\pi _h^{FDI}({\underline{u}},\cdot )]/\partial t < 0$ for any $t<{\overline{t}}={\overline{u}}-(\sigma {\underline{u}}/2)$, and at $t=({\overline{u}}-{\underline{u}})-({\underline{u}}-(\sigma {\underline{u}}/2))<{\overline{t}}$, $[\pi _h^{FDI}({\overline{u}},\cdot )-\pi _h^{FDI}({\underline{u}},\cdot )]=0$. Thus, for $({\overline{u}}/{\underline{u}})>(4-\sigma )/2$, $[\pi _h^{FDI}({\overline{u}},\cdot )-\pi _h^{FDI}({\underline{u}},\cdot )]>0$ so long as $t<({\overline{u}}-{\underline{u}})-({\underline{u}}-(\sigma {\underline{u}}/2))$, whereas for $({\overline{u}}/{\underline{u}})<(4-\sigma )/2$, $[\pi _h^{FDI}({\overline{u}},\cdot )-\pi _h^{FDI}({\underline{u}},\cdot )]<0$ for any $t<{\overline{t}}$ completing the proof.

1.2 A.2 Proof of Proposition 3

Comparing an exporter’s profits from its sales of different quality, given by Eqs. (6) and (8) shows that

$$\begin{aligned} \lim \limits _{{\tau \rightarrow 0}}\left[ \pi _h^{T}({\overline{u}},\cdot )-\pi _h^{T}({\underline{u}},\cdot )\right] =\frac{(2 {\overline{u}}+(4-3 \sigma ) {\underline{u}}) (({\overline{u}}-{\underline{u}})-({\underline{u}}-(\sigma {\underline{u}}/2)))}{2 (4-\sigma ^2)^2}, \end{aligned}$$

which is clearly negative for $({\overline{u}}/{\underline{u}})<(4-\sigma )/2 \iff ({\overline{u}}-{\underline{u}})-({\underline{u}}-(\sigma {\underline{u}}/2))<0$. When $\tau$ approaches to its prohibitive level ${\overline{\tau }}={\underline{u}}-(\sigma {\underline{u}})/2$, this difference gets

$$\begin{aligned} \lim \limits _{{\tau \rightarrow {\overline{\tau }}}}\left[ \pi _h^{T}({\overline{u}},\cdot )-\pi _h^{T}({\underline{u}},\cdot )\right] =\left( \frac{{\overline{u}}-{\underline{u}}}{4-\sigma ^2}\right) ^2 \end{aligned}$$

which is positive. Note that $\partial [\pi _h^{T}({\overline{u}},\cdot )-\pi _h^{T}({\underline{u}},\cdot )]/\partial \tau > 0$ at $\tau < {\underline{u}}-(\sigma {\underline{u}})/2-({\overline{u}}-{\underline{u}})/3$, and that at $\tau ={\underline{u}}-(\sigma {\underline{u}})/2-({\overline{u}}-{\underline{u}})$, $[\pi _h^{T}({\overline{u}},\cdot )-\pi _h^{T}({\underline{u}},\cdot )]=0$. It is now straightforward to show that, when $({\overline{u}}/{\underline{u}})<(4-\sigma )/2$, $[\pi _h^{T}({\overline{u}},\cdot )-\pi _h^{T}({\underline{u}},\cdot )]<0$ for $\tau < {\underline{u}}-(\sigma {\underline{u}})/2-({\overline{u}}-{\underline{u}})$, and $[\pi _h^{T}({\overline{u}},\cdot )-\pi _h^{T}({\underline{u}},\cdot )]>0$ for ${\underline{u}}-(\sigma {\underline{u}})/2-({\overline{u}}-{\underline{u}})< \tau < {\overline{\tau }}$. As for $({\overline{u}}/{\underline{u}})>(4-\sigma )/2$, $[\pi _h^{T}({\overline{u}},\cdot )-\pi _h^{T}({\underline{u}},\cdot )]>0$ at both $\tau =0$ and $\tau ={\overline{\tau }}$, and $\partial ^{2} [\pi _h^{T}({\overline{u}},\cdot )-\pi _h^{T}({\underline{u}},\cdot )]/\partial \tau ^{2} < 0$, and thus $[\pi _h^{T}({\overline{u}},\cdot )-\pi _h^{T}({\underline{u}},\cdot )]>0$ for any $\tau <{\overline{\tau }}$, completing the proof.

1.3 A.3 Proof of Proposition 5

The fixed investment cost thresholds are such that $\pi _h^{FDI} ({\underline{u}}, \cdot )-\pi _h^{T} ({\overline{u}}, \cdot )=0$ at $G=G_1$, and $\pi _h^{FDI} ({\overline{u}}, \cdot )-\pi _h^{T} ({\overline{u}}, \cdot )=0$ at $G=G_2$, where

$$\begin{aligned} G_1&=\frac{(\tau -(({\overline{u}}-{\underline{u}})-({\underline{u}}-(\sigma {\underline{u}}/2))))(({\overline{u}}+(1-\sigma ){\underline{u}}+({\underline{u}}-(\sigma {\underline{u}}/2))-\tau )}{(4-\sigma ^2)^2}\\ G_2&=\frac{(\tau -t)(2{\overline{u}}-\sigma {\underline{u}}-(\tau +t))}{(4-\sigma ^2)^2}. \end{aligned}$$

$\pi _h^{FDI} ({\underline{u}}, \cdot )-\pi _h^{T} ({\overline{u}}, \cdot )>0$ when $G<G_1$, and $\pi _h^{FDI} ({\overline{u}}, \cdot )-\pi _h^{T} ({\overline{u}}, \cdot )>0$ when $G<G_2$, where $G_1\ge 0$ if (and only if) $\tau \ge (({\overline{u}}-{\underline{u}})-({\underline{u}}-(\sigma {\underline{u}}/2)))$, and $G_2\ge 0$ if (and only if) $\tau \ge t$, completing the proof. As for the changes in the fixed investment cost thresholds:

$$\begin{aligned} \frac{\partial G_1}{\partial \tau }&= \frac{2 {\overline{u}}-\sigma {\underline{u}}-2 \tau }{\left( 4-\sigma ^2\right) ^2}>0, \quad \forall \tau< {\overline{\tau }}; \quad \frac{\partial G_1}{\partial t} = 0 \\ \frac{\partial G_2}{\partial \tau }&= \frac{2 {\overline{u}}-\sigma {\underline{u}}-2 \tau }{\left( 4-\sigma ^2\right) ^2}>0, \quad \forall \tau< {\overline{\tau }}; \quad \frac{\partial G_2}{\partial t} = -\frac{2 {\overline{u}}-\sigma {\underline{u}}-2 t }{\left( 4-\sigma ^2\right) ^2}<0, \quad \forall t<{\overline{t}} \end{aligned}$$

Note that the prohibitive cost thresholds are such that ${\overline{\tau }}={\underline{u}}-\sigma {\underline{u}}/2$ and ${\overline{t}}={\overline{u}}-\sigma {\underline{u}}/2$.

1.4 A.4 Proof of Proposition 6

The fixed investment cost thresholds are such that $\pi _h^{FDI} ({\underline{u}}, \cdot )-\pi _h^{T} ({\overline{u}}, \cdot )=0$ at $G=G_3$, and $\pi _h^{FDI} ({\underline{u}}, \cdot )-\pi _h^{T} ({\underline{u}}, \cdot )=0$ at $G=G_4$, where

$$\begin{aligned} G_3&=\frac{(\tau +(({\underline{u}}-(\sigma {\underline{u}}/2))-({\overline{u}}-{\underline{u}})))(({\overline{u}}+(1-\sigma ){\underline{u}}+({\underline{u}}-(\sigma {\underline{u}}/2))-\tau )}{(4-\sigma ^2)^2}\\ G_4&=\frac{4\tau ((2-\sigma ){\underline{u}}-\tau )}{(4-\sigma ^2)^2}. \end{aligned}$$

$\pi _h^{FDI} ({\underline{u}}, \cdot )-\pi _h^{T} ({\overline{u}}, \cdot )>0$ when $G<G_3$, and $\pi _h^{FDI} ({\underline{u}}, \cdot )-\pi _h^{T} ({\underline{u}}, \cdot )>0$ when $G<G_4$, completing the proof. As for the changes in the fixed investment cost thresholds:

$$\begin{aligned} \frac{\partial G_3}{\partial \tau }&= \frac{2 {\overline{u}}-\sigma {\underline{u}}-2 \tau }{\left( 4-\sigma ^2\right) ^2}>0, \quad \forall \tau< {\overline{\tau }}; \quad \frac{\partial G_3}{\partial t} = 0 \\ \frac{\partial G_4}{\partial \tau }&= \frac{4 (2-\sigma ) {\underline{u}}-8 \tau }{\left( 4-\sigma ^2\right) ^2}>0, \quad \forall \tau < {\overline{\tau }}; \quad \frac{\partial G_4}{\partial t} = 0 \end{aligned}$$

Note that the prohibitive cost thresholds are such that ${\overline{\tau }}={\underline{u}}-\sigma {\underline{u}}/2$ and ${\overline{t}}={\overline{u}}-\sigma {\underline{u}}/2$.

1.5 A.5 Proof of Proposition 7

Comparing a multinational’s profits from subsidiary sales of different quality (in the case that high quality products are manufactured through vertical integration), given by Eqs. (3) and (9) shows that

$$\begin{aligned} \pi _h^{v,FDI}({\overline{u}},\cdot )-\pi _h^{FDI}({\underline{u}},\cdot )=\frac{4( ({\overline{u}}-{\underline{u}})-t)(({\overline{u}}+(1-\sigma ){\underline{u}}-t)}{(4-\sigma ^2)^2}, \end{aligned}$$

which is clearly positive if $t<({\overline{u}}-{\underline{u}})$, and negative if otherwise, given input trade costs are below their prohibitive levels such that $t<{\overline{t}}={\overline{u}}-\sigma {\underline{u}}/2$, completing the proof of (i). Replacing t by $c+t$ in profits given by Eq. (9) leads to a difference in profits given above, in which t should be replaced by $c+t$ (and assuming away prohibitive costs should follow $c+t<{\overline{u}}-\sigma {\underline{u}}/2$). Thus, in such a case, $\pi _h^{v,FDI}({\overline{u}},\cdot )>\pi _h^{FDI}({\underline{u}},\cdot )$ if $c<({\overline{u}}-{\underline{u}})-t$, completing the proof of (ii). Similarly, including fixed costs specific to vertical integration in profits given by Eq. (9) leads to a fixed costs threshold (e.g., ${\overline{C}}$), such that $\pi _h^{v,FDI}({\overline{u}},\cdot )-\pi _h^{FDI}({\underline{u}},\cdot )=0$ at ${\overline{C}}$ which is given by the displayed equation above. Consequently, $\pi _h^{v,FDI}({\overline{u}},\cdot )-\pi _h^{FDI}({\underline{u}},\cdot )>0$ for fixed costs smaller than ${\overline{C}}$, completing the proof of (iii).

1.6 A.6 Proof of Proposition 8

Comparing an exporter’s profits from its sales of different quality (in the case that high quality products are manufactured through vertical integration), given by Eqs. (6) and (10) shows that

$$\begin{aligned} \pi _h^{v,T}({\overline{u}},\cdot )-\pi _h^{T}({\underline{u}},\cdot )=\frac{4 ({\overline{u}}-{\underline{u}})({\overline{u}}+(1-\sigma ){\underline{u}}-2\tau )}{(4-\sigma ^2)^2}, \end{aligned}$$

which is clearly positive for any non-prohibitive trade costs in final goods such that ${\overline{\tau }}<{\overline{\tau }}={\underline{u}}-\sigma {\underline{u}}/2$. In the case that vertical integration warrants some variable costs, $\tau$ in profits given by Eq. (10) should be replaced by $c+\tau$. In the case of additional variable costs (where prohibitive costs would be assumed away such that $c+\tau <{\underline{u}}-\sigma {\underline{u}}/2$), the difference between profits would be

$$\begin{aligned} \pi _h^{v,T}(c_h,{\overline{u}},\cdot )-\pi _h^{T}(c_h,{\underline{u}},\cdot )=\frac{4( ({\overline{u}}-{\underline{u}})-c)({\overline{u}}+(1-\sigma ){\underline{u}}-2\tau -c)}{(4-\sigma ^2)^2}, \end{aligned}$$

which is clearly positive for $c<({\overline{u}}-{\underline{u}})$. If vertical integration warrants some additional fixed costs (e.g., ${\overline{C}}$), then this should be subtracted from profits given by Eq. (10). In such a case, depending on whether or not there is an additional variable cost (due to vertical integration), either of the two expressions above will be the threshold fixed cost below which exporting a high-quality foreign variety will be preferred over exporting a low-quality foreign variety, completing the proof.

1.7 A.7 Proof of Proposition 10

Propositions 7 and 8 and “Appendix A.5 and A.6” have already shown that producing a high-quality foreign variety is optimal for any non-prohibitive trade costs for an exporter, whereas a multinational prefers manufacturing high quality only if $t<({\overline{u}}-{\underline{u}})$. That is, so as to determine the equilibrium outcome, when $t<({\overline{u}}-{\underline{u}})$, $\pi _h^{v,FDI} ({\overline{u}}, \cdot )$ and $\pi _h^{v,T} ({\overline{u}},\cdot )$ has to be compared, whereas when $t>({\overline{u}}-{\underline{u}})$, $\pi _h^{FDI} ({\underline{u}}, \cdot )$ and $\pi _h^{v,T} ({\overline{u}},\cdot )$ has to be compared. This leads to the fixed investment cost thresholds such that $\pi _h^{FDI} ({\underline{u}}, \cdot )-\pi _h^{v,T} ({\overline{u}}, \cdot )=0$ at $G=G_5$, and $\pi _h^{v,FDI} ({\overline{u}}, \cdot )-\pi _h^{v,T} ({\overline{u}}, \cdot )=0$ at $G=G_6$, where

$$\begin{aligned} G_5&=\frac{4(\tau -({\overline{u}}-{\underline{u}}))(({\overline{u}}+(1-\sigma ){\underline{u}}-\tau )}{(4-\sigma ^2)^2}\\ G_6&=\frac{4(\tau -t)(2{\overline{u}}-\sigma {\underline{u}}-(\tau +t))}{(4-\sigma ^2)^2}=4G_2. \end{aligned}$$

$\pi _h^{FDI} ({\underline{u}}, \cdot )-\pi _h^{v,T} ({\overline{u}}, \cdot )>0$ when $G<G_5$, and $\pi _h^{v,FDI} ({\overline{u}}, \cdot )-\pi _h^{v,T} ({\overline{u}}, \cdot )>0$ when $G<G_6$, where $G_5\ge 0$ if (and only if) $\tau \ge ({\overline{u}}-{\underline{u}})$, and $G_6\ge 0$ if (and only if) $\tau \ge t$, completing the proof. As for the changes in the fixed investment cost thresholds:

$$\begin{aligned} \frac{\partial G_5}{\partial \tau }&= \frac{4(2 {\overline{u}}-\sigma {\underline{u}}-2 \tau ) }{\left( 4-\sigma ^2\right) ^2}>0, \quad \forall \tau< {\overline{\tau }}; \quad \frac{\partial G_5}{\partial t} = 0 \\ \frac{\partial G_6}{\partial \tau }&= \frac{4(2 {\overline{u}}-\sigma {\underline{u}}-2 \tau )}{\left( 4-\sigma ^2\right) ^2}>0, \quad \forall \tau< {\overline{\tau }}; \quad \frac{\partial G_6}{\partial t} = -\frac{4(2 {\overline{u}}-\sigma {\underline{u}}-2 t )}{\left( 4-\sigma ^2\right) ^2}<0, \quad \forall t<{\overline{t}} \end{aligned}$$

Note that the prohibitive cost thresholds are such that ${\overline{\tau }}={\underline{u}}-\sigma {\underline{u}}/2$ and ${\overline{t}}={\overline{u}}-\sigma {\underline{u}}/2$.