Private labels and exports: trading variety for volume

Abstract

This paper explores the role of private label trade intermediation in shaping the range and diversity of exports and imports. Whereas direct sales maintain a firm’s unique product characteristics, or ‘brand equity’, trade through an intermediary often takes the form of ‘private label’ sales, under which multiple firms’ output is pooled and re-sold under a new private label brand created by the intermediary. This paper shows that these private label arrangements result in greater total export and import volumes and lower average prices for consumers, but fewer independent varieties available to consumers in equilibrium. Normative implications are mixed: consumers trade variety for volume, independent exporters face greater competition from the new private label products, and intermediary firms can capture more of the gains from trade. We explore the implications of competition at the intermediary level and trade costs for the equilibrium pattern of private label and direct exporting and importing activities.

Appendix

1.1 Direct exports only

In this Appendix we solve for equilibrium in the basic case when firms can export directly or not at all. In this scenario, a $\lambda$-type firm solves the following profit-maximization problem:

$$\begin{aligned} \max _{q}\left( \left( \lambda \alpha -\eta \frac{Q}{L}-\frac{\gamma q}{L} -c-c^{DE}\right) q-F^{DE}\right) , \end{aligned}$$

where $Q=Q^{W}+Q^{H,DE}$ is the aggregate output sold in the Foreign market. The profit-maximizing output of a $\lambda$-type firm is then:

$$\begin{aligned} q^{DE}(\lambda )=\frac{L}{2\gamma }\left( \lambda \alpha -\eta \frac{Q}{L} -c-c^{DE}\right) , \end{aligned}$$

(3)

with associated profit:

$$\begin{aligned} \pi ^{DE}\left( \lambda \right) =\frac{L}{4\gamma }\left( \lambda \alpha -\eta \frac{Q}{L}-c-c^{DE}\right) ^{2}-F^{DE}. \end{aligned}$$

(4)

Then the threshold $\lambda ^{DE}$ implicitly equals to:

$$\begin{aligned} \lambda ^{DE}=\frac{1}{\alpha }\left( 2\sqrt{\frac{\gamma F^{DE}}{L}}+\eta \frac{Q}{L}+c+c^{DE}\right) . \end{aligned}$$

(5)

Aggregating over the set of exporting firms, $\lambda \in [\lambda ^{DE},1]$, per capita output sold in the Foreign market equals:

$$\begin{aligned} \frac{Q}{L}=\frac{Q^{W}}{L}+\frac{1}{L}\int _{\lambda ^{DE} }^{1}q^{DE}(\lambda )d\lambda . \end{aligned}$$

(6)

Using Eqs. (5) and (6) we can derive the equilibrium level of $\lambda ^{DE}$:

$$\begin{aligned} \lambda ^{DE}=1-\frac{1}{\alpha \eta }\left( \sqrt{D}-2\sqrt{\frac{\gamma F^{DE}}{L}}\eta -2\alpha ^{3}\gamma \right) , \end{aligned}$$

(7)

where we use the term D simply as a placeholder for the (somewhat messy) expression:

$$\begin{aligned} D\equiv 4\eta ^{2}\gamma \left( \frac{F^{DE}}{L}+\frac{\alpha ^{3}}{\eta } \left( \frac{\alpha ^{3}\gamma }{\eta }+\alpha -\frac{\eta Q^{W}}{L} -c-c^{DE}\right) \right) . \end{aligned}$$

In words, the expression in (7) tells us that when the only way to reach Foreign consumers is via direct exporting, more exporters will undertake the direct export channel when the (fixed and variable) trade costs are lower, consumers value product diversity more, the Foreign market is larger, and there is less competition from the rest of the world.

1.2 Private labels

In this Appendix we first solve for equilibrium in the case of private label trade intermediation and then derive the effects of private label intermediation on Home exporting firms.

First, we find the contract offered by the IR in equilibrium. The inverse demand for private label k product is:

$$\begin{aligned} p_{k}=\lambda _{k}\alpha -\eta \frac{Q}{L}-\frac{\gamma q_{k}}{L}, \end{aligned}$$

where $Q=Q^{W}+Q^{H,DE}+Q_{k}^{H,PL}$. The profit of a Home firm that accepts the private label contract is thus:

$$\begin{aligned} \pi ^{PL}=\left( \lambda _{k}\alpha -\eta \frac{Q}{L}-\frac{\gamma q_{k}}{L} -c-\Delta \right) q_{k}-f. \end{aligned}$$

Solving, the profit-maximizing output equals:

$$\begin{aligned} q_{k}=\frac{L}{2\gamma }\left( \lambda _{k}\alpha -\eta \frac{Q}{L}-c-\Delta \right) , \end{aligned}$$

so that the profit of a Home firm exporting under a private label contract is:

$$\begin{aligned} \pi ^{PL}=\frac{L}{4\gamma }\left( \lambda _{k}\alpha -\eta \frac{Q}{L} -c-\Delta \right) ^{2}-f. \end{aligned}$$

(8)

It is straightforward to see that the profit of a direct exporter is the same as in (4) with the only difference that now total demand Q includes output of the private label product channeled through IR, $Q_{k}^{H,PL}$.

We focus on the case in which some firms choose not to export at all, so that $\underline{\lambda }>0.$ In this case, using (2) we have that the output of a Home firm exporting under the private label contract is:

$$\begin{aligned} q_{k}=\sqrt{\frac{Lf}{\gamma }}. \end{aligned}$$

To find ${\overline{\lambda }}$ we use conditions (1) and (2) to get $\pi ^{DE}({\overline{\lambda }}) =0$. We then have:

$$\begin{aligned} {\overline{\lambda }}=\frac{1}{\alpha }\left( 2\sqrt{\frac{\gamma F^{DE}}{L}} +\eta \frac{Q}{L}+c+c^{DE}\right) . \end{aligned}$$

(9)

Then using condition (2) we can find $\lambda _{k}$:

$$\begin{aligned} \lambda _{k}=\frac{1}{\alpha }\left( 2\sqrt{\frac{\gamma f}{L}}+\eta \frac{Q }{L}+c+\Delta \right) . \end{aligned}$$

Finally, using $\lambda _{k}=\dfrac{\underline{\lambda }+{\overline{\lambda }} }{2}$, we derive the value of the measure of Home firms exporting under the private label:

$$\begin{aligned} K=2\frac{1}{\alpha }\left( 2\sqrt{\frac{\gamma F^{DE}}{L}}-2\sqrt{\frac{ \gamma f}{L}}+c^{DE}-\Delta \right) \end{aligned}$$

(10)

As one would expect, a higher per unit fee $\Delta$ or a higher fixed fee f charged by the IR decrease the measure of firms that accept a private label contract.

We are now ready to characterize the contract that maximizes the retailer’s profit:

$$\begin{aligned} \Pi =Kq_{k}\left( \Delta -c_{R}\right) +Kf-F_{R} \end{aligned}$$

Using (10), we have that:

$$\begin{aligned} \Pi =2\frac{1}{\alpha }\left( 2\sqrt{\frac{\gamma F^{DE}}{L}}-2\sqrt{\frac{ \gamma f}{L}}+c^{DE}-\Delta \right) \left( \left( \Delta -c_{R}\right) \sqrt{ \frac{Lf}{\gamma }}+f\right) -F_{R}. \end{aligned}$$

Maximizing the IR’s profit gives us the equilibrium contact:

$$\begin{aligned} \Delta &= c_{r} \\ \sqrt{\frac{\gamma f}{L}} &= \frac{2}{3}\sqrt{\frac{\gamma F^{DE}}{L}}+\frac{ 1}{3}c^{DE}-\frac{1}{3}c_{r}. \end{aligned}$$

Next, we can solve for the thresholds ${\overline{\lambda }},\underline{\lambda }$ and the average value of brand equity of the private label product $\lambda _{k}.$ First, the per capita output sold in the target Foreign market is equal to

$$\begin{aligned} \frac{Q}{L}= & {} \frac{Q^{W}}{L}+\frac{1}{L}\left( \int _{{\overline{\lambda }} }^{1}q^{DE}(\lambda )d\lambda +Kq_{k}\right) \nonumber \\= & {} \frac{Q^{W}}{L}+\frac{\left( 1-\left( \underline{\lambda }+K\right) \right) }{4\gamma \alpha }\left( \frac{4}{\alpha }\sqrt{\frac{\gamma F^{DE}}{ L}}+1-\left( \underline{\lambda }+K\right) \right) \nonumber \\&+\frac{2}{\alpha \gamma } \left( \frac{2}{3}\sqrt{\frac{\gamma F^{DE}}{L}}+\frac{1}{3}c^{DE}-\frac{1}{3 }c_{r}\right) ^{2}. \end{aligned}$$

(11)

We can now use (11), (9), and $\underline{\lambda }={\overline{\lambda }}-K,$ to solve for the lower threshold $\underline{\lambda }$:

$$\begin{aligned} \underline{\lambda }=1-K-\frac{1}{\alpha \eta }\left( \sqrt{D-2K^{2}\alpha ^{4}\eta ^{2}}-2\eta \sqrt{\frac{\gamma F^{DE}}{L}}-2\alpha ^{3}\gamma \right) . \end{aligned}$$

(12)

The upper threshold and average then follow immediately.

Next, we analyze how the availability of the private label export channel affects the exporting firms. We already know that total Home exports rise and there are fewer Home direct exporters. Next, we find the effect on total number of Home exporters. Using (7) and (12) we can derive the following:

$$\begin{aligned} \underline{\lambda }-\lambda ^{DE}=\frac{K}{\alpha \eta }\left( \sqrt{\frac{D }{K^{2}}}-\sqrt{\frac{D}{K^{2}}-2\alpha ^{4}\eta ^{2}}-\alpha \eta \right) . \end{aligned}$$

We see that $\underline{\lambda }>\lambda ^{DE}$ if and only if

$$\begin{aligned} \frac{D}{K^{2}}<\frac{\left( \alpha \eta \right) ^{2}}{4}\left( 1+2\alpha ^{2}\right) ^{2}, \end{aligned}$$

or, substituting for K and D, the mass of exporting firms decreases iff:

$$\begin{aligned} \frac{16\left( \frac{\gamma F^{DE}}{L}+\frac{\alpha ^{3}\gamma }{\eta } \left( \frac{\alpha ^{3}\gamma }{\eta }+\alpha -\left( \eta \frac{Q^{W}}{L} +c+c^{DE}\right) \right) \right) }{\left( \frac{2}{3}\sqrt{\frac{\gamma F^{DE}}{L}}+\frac{1}{3}\left( c^{DE}-c_{r}\right) \right) ^{2}}<\left( 1+2\alpha ^{2}\right) ^{2}, \end{aligned}$$

(13)

which is generally ambiguous as it depends on parametric assumptions. Condition (13) will be satisfied when the international retailer’s cost advantage, $(c^{DE}-c_{r})$ is sufficiently high, substitutability between varieties, $\eta$, is sufficiently high, and/or the rest of the world’s exports per foreign consumer, $\frac{ Q^{W}}{L}$ are sufficiently high.

1.3 Variation in variable trade cost

In this Appendix we show that in the case of a uniform increase in the variable trade costs for both direct and intermediated trade, the negative extensive margin effect dominates and the aggregate Home exports fall while the output of a direct exporter rises.

First, using (11) and (12), and then differentiating Q with respect to t we get that the effect on total exports is negative:

$$\begin{aligned} \frac{dQ}{dt}=-L\frac{2\sqrt{\frac{\gamma F^{DE}}{L}}+(1-{\overline{\lambda }} )\alpha }{\sqrt{D-2K^{2}\alpha ^{4}\eta ^{2}}}<0. \end{aligned}$$

Next using (3) and differentiating with respect to t we can also show that the effect on the quantity of a direct exporter is positive

$$\begin{aligned} \frac{dq^{DE}}{dt}=-\frac{\eta }{2\gamma }\frac{dQ}{dt}-\frac{L}{2\gamma }= \frac{\alpha ^{3}L}{\sqrt{D-2K^{2}\alpha ^{4}\eta ^{2}}}>0. \end{aligned}$$

1.4 Variation in fixed trade cost

In this Appendix we show that the effects of an increase in fixed cost of direct exporting on aggregate Home exports and individual output of a direct exporter are of opposite sign and the direction of each effect depends on the parameter’s values.

Using (11) and differentiating with respect to $F^{DE}$ we have that the effect on aggregate exports is

$$\begin{aligned} \frac{dQ}{dF^{DE}}=2L\left( \frac{\sqrt{\frac{\gamma }{LF^{DE}}}}{\sqrt{ D-2K^{2}\alpha ^{4}\eta ^{2}}}\left( \frac{2}{3}K\alpha ^{3}-\sqrt{\frac{ \gamma F^{DE}}{L}}\right) \right) . \end{aligned}$$

Hence the sign of $\dfrac{dQ}{dF^{DE}}\gtrless 0$ depends on the sign of the following term

$$\begin{aligned} \left( \frac{8}{9}\alpha ^{2}-1\right) \sqrt{\frac{\gamma F^{DE}}{L}}+\frac{4 }{9}\alpha ^{2}\left( c^{DE}-c_{r}\right) \gtrless 0 \end{aligned}$$

If $\alpha , \gamma F^{DE}/L$ or $\left( c^{DE}-c_{r}\right)$ are high then we see higher aggregate exports in response to an increase in the fixed cost of direct exports. And since

$$\begin{aligned} \frac{dq^{DE}(\lambda )}{dF^{DE}}=-\frac{\eta }{2\gamma }\frac{dQ}{dF^{DE}} \end{aligned}$$

we have the opposite effect on the individual quantity of each direct exporter.

1.5 Reducing monopoly power

In this Appendix we solve for the effects on outputs and the measure of Home exporters in the case of an exogenous reduction in the monopoly power of the retailer, i.e. a decrease in fixed retailing fee f.

First, we note that as previously the profit of private label exporters must be equal to zero:

$$\begin{aligned} \pi ^{PL}=q_{k}^{2}\frac{\gamma }{L}-f=0. \end{aligned}$$

Hence, each private label exporter’s output equals

$$\begin{aligned} q_{k}=\sqrt{\frac{fL}{\gamma }}. \end{aligned}$$

Note that as the monopoly power of the retailer is reduced, the fixed fee f decreases, and each private label exporter will export less—that is, $q_{k}$ decreases.

Next, we determine the measure of private label exporters and their total exports. Since profits have to be equal at the threshold, and private label exporters make zero profit, i.e. $\pi ^{DE}({\overline{\lambda }}) =\pi ^{PL}=0$, it follows that

$$\begin{aligned} {\overline{\lambda }}=\frac{1}{\alpha }\left( 2\sqrt{\frac{\gamma F^{DE}}{L}} +\eta \frac{Q}{L}+c+c^{DE}\right) , \end{aligned}$$

and

$$\begin{aligned} \lambda _{k}=\frac{1}{\alpha }\left( 2\sqrt{\frac{\gamma f}{L}}+\eta \frac{Q }{L}+c+c^{R}\right) . \end{aligned}$$

The measure of Home firms exporting under the private label then equals:

$$\begin{aligned} K={\overline{\lambda }}-\underline{\lambda }=2\left( {\overline{\lambda }} -\lambda _{k}\right) =2\frac{1}{\alpha }\left( 2\sqrt{\frac{\gamma F^{DE}}{L} }-2\sqrt{\frac{\gamma f}{L}}+c^{DE}-c^{R}\right) . \end{aligned}$$

As f decreases, K increases and there will be more private label exporters compared to the monopoly case.

The net effects on the total volume of intermediated exports takes some work. Given the mass of private label exporters, total private label exports amount to:

$$\begin{aligned} Q^{H,PL}=Kq_{k}=2\frac{1}{\alpha }\left( 2\sqrt{\frac{\gamma F^{DE}}{L}}-2 \sqrt{\frac{\gamma f}{L}}+c^{DE}-c^{R}\right) \sqrt{\frac{fL}{\gamma }} \end{aligned}$$

and this quantity varies with the fixed retailing fee according to:

$$\begin{aligned} \frac{dQ^{H,PL}}{df}=\frac{1}{\alpha \sqrt{f}}\left( \left( 2\sqrt{\frac{ \gamma F^{DE}}{L}}+c^{DE}-c^{R}\right) \sqrt{\frac{L}{\gamma }}-4\sqrt{f} \right) \end{aligned}$$

In order to sign this derivative it proves convenient to define the following critical value of the fee:²⁷

$$\begin{aligned} {\widetilde{f}}=\frac{L}{\gamma }\left( \frac{2\sqrt{\frac{\gamma F^{DE}}{L}} +c^{DE}-c^{R}}{4}\right) ^{2} \end{aligned}$$

Regarding the effect on the total quantity of private label exports, we have that for $f>{\widetilde{f}}$ the quantity increases as f falls, and then, once $f<{\widetilde{f}}$, starts to decrease as the fee falls even further.

The resulting equilibrium is determined by two conditions:

$$\begin{aligned} \pi ^{DE}\left( \lambda \right) =\frac{L}{4\gamma }\left( {\overline{\lambda }} \alpha -\eta \frac{Q}{L}-c-c^{DE}\right) ^{2}-F^{DE}=0 \end{aligned}$$

and

$$\begin{aligned} \frac{Q}{L}= & {} \frac{Q^{W}}{L}+\frac{Q^{H,DE}}{L}+\frac{Q^{H,PL}}{L}\\= & {} \frac{Q^{W}}{L}+\frac{\left( 1-{\overline{\lambda }}\right) }{4\gamma \alpha }\left( \frac{4}{\alpha }\sqrt{\frac{\gamma F^{DE}}{L}}+1-{\overline{\lambda }} \right) + \\&2\frac{1}{\alpha }\left( 2\sqrt{\frac{\gamma F^{DE}}{L}}-2 \sqrt{\frac{\gamma f}{L}}+c^{DE}-c^{R}\right) \sqrt{\frac{fL}{\gamma }} \end{aligned}$$

We want to understand how these equilibrium values change when f falls. Consider first the case where $f>{\widetilde{f}}$. For this range of market power we have that $\frac{Q}{L}$ and ${\overline{\lambda }}$ increase as f falls. To see this, suppose that $\frac{Q}{L}$ decreases. Then ${\overline{\lambda }}$ decreases as well, which results in higher $Q^{H,DE}$. As $Q^{H,PL}$ also increases, $\frac{Q}{L}$ would rise, which is a contradiction. Now consider the case where $f<{\widetilde{f}}$. In this range, $Q^{H,PL}$ decreases as f falls and ${\overline{\lambda }}$ decreases as well. The argument is the same as before, only with opposite signs.