Dual transfer pricing with internal and external trade

Abstract

This paper examines a transfer pricing problem between two divisions of a decentralized firm. The selling division is privately informed about its own costs and produces a good that is sold both externally in an intermediate market and internally within the firm. Unlike most previous work, we focus on dual transfer pricing systems that allow the selling division to be credited for an amount that differs from the amount charged to the buying division. We identify conditions under which efficient decentralized trade and external price setting incentives can be provided with a properly chosen set of dual transfer prices that do not rely on direct communication. Instead, the optimal dual transfer prices will depend only on public information about the market price charged by the upstream division in the external market, which indirectly communicates information about production costs to the downstream division. For a variety of well-known demand functions, the optimal transfer prices will be linear functions of the market price. Our main results hold when the upstream division faces multiple internal buyers or faces a binding capacity constraint.

Appendix: Proofs

Proof of Lemma 1

As outlined in the text, the transfer prices in (11) and (12) achieve first best. Now, we prove necessity. Since first best requires $t_{d}(p_{u})=c(\theta _{c})$ for all $\theta _{c}$, Assumption 1 ensures uniqueness of $t_{d}(p_{u})=p_{u}+Q_{u}(p_{u})/Q_{u}^{ \prime }(p_{u})$.

Now we show the uniqueness of $t_{u}(p_{u})$ up to a constant, A. The upstream profit from internal trade, ${\varPi }_{u}^{int}(t_{d})=(t_{u}-c(\theta _{c}))\cdot Q_{d}(t_{d})$, must be locally maximized where $t_{d}=c(\theta _{c})$ for all values of $\theta _{c}$, or the upstream division would have an incentive to distort $p_{u}$ away from $p_{u}^{*}$.

Consider an alternative upstream transfer pricing function with an added term, $D(t_{d}(p_{u}))$ [since each possible value of $c(\theta _c)$ yields a unique value of $p_{u}^{*}$, we can always write $t_{u}$ as a function of $t_{d}$].

$$\begin{aligned} t_{u}(p_{u})=t_{d}(p_{u})+\dfrac{\int _{t_{d}}^{\infty }{Q_{d}(s)\;ds+A+D} \left( t_{d}(p_{u})\right) }{Q_{d}\left( t_{d}(p_{u})\right) }. \end{aligned}$$

(38)

The associated upstream profit from internal trade is given by:

$$\begin{aligned} {\varPi }_{u}^{int}(t_{d})=(t_{d}-c(\theta _{c}))\cdot Q_{d}(t_{d})+A+\int _{t_{d}}^{\infty }Q_{d}(s)\;ds+D(t_{d}). \end{aligned}$$

(39)

If $D(t_{d})$ is not constant, we will show that there is some $c(\theta _{c})$ for which ${\varPi }_{u}^{int}(t_{d})$ is not maximized at $t_{d}=c(\theta _{c})$.

Assume that $D(t_{d})$ is not constant. Then there are some values $a\ne b$ within the range $(\underline{c},\overline{c})$ such that $D(b)>D(a)$. W.l.o.g., assume that $b>a$. Denoting the slope of the secant line between D(a) and D(b) as $S=(D(b)-D(a))/(b-a)>0$, we can divide the interval [a, b] into n equal subintervals. At least one of these subintervals $[a^{\prime },b^{\prime }]$ must satisfy $D(b^{\prime })-D(a^{\prime })\ge [D(b)-D(a)]/n$.

Now consider the case where the realization of $\theta _{c}$ is such that $c(\theta _{c})=a^{\prime },$

$$\begin{aligned} {\varPi }_{u}^{int}(b^{\prime })-{\varPi }_{u}^{int}(a^{\prime })&=\left( b^{\prime }-c(\theta _{c})\right) \cdot Q_{d}(b^{\prime })-\int _{c(\theta _{c})}^{b^{\prime }}Q_{d}(s)ds+D(b^{\prime })-D(c(\theta _{c})) \\&>(b^{\prime }-c(\theta _{c}))\cdot \left[ Q_{d}(b^{\prime })-Q_{d}\left( c(\theta _{c})\right) +\frac{D(b)-D(a)}{n\cdot (b^{\prime }-c(\theta _{c})) }\right] \\&=(b^{\prime }-c(\theta _{c}))\cdot \left[ Q_{d}(b^{\prime })-Q_{d}\left( c(\theta _c)\right) +S\right] , \end{aligned}$$

(40)

where the last equality holds because $n(b^{\prime }-c(\theta _{c}))=(b-a)$. Since $Q_{d}$ is continuous over the compact interval [a, b], it is uniformly continuous by the Heine–Cantor theorem. Since $Q_{d}(b^{\prime })-Q_{d}(c(\theta _{c}))<0$ goes to zero uniformly as n becomes large and S does not (because it is constant), it will always be possible to pick a large enough n (and a corresponding $a^{\prime }$ and $c(\theta _{c})=a^{\prime }$) such that the quantity in the brackets will be strictly positive.

Proof of Proposition 1

Part (i) Follows from Lemma 1.

Part (ii) If headquarters does not know upstream elasticity, $\eta _{u}(p_{u},\theta _{u})$, then the downstream transfer price cannot be set such that $c(\theta _{c})$ can be inferred from $p_{u}^{*}$. That is, it is not possible to set the downstream price such that $t_{d}(p_{u}^{*}(c(\theta _{c}),\theta _{u}))=c(\theta _{c})=p_{u}^{*}(c(\theta _{c}),\theta _{u})\cdot [1-\eta _{u}(p_{u}^{*}(c(\theta _{c}),\theta _{u}),\theta _{u})]$ for all $(\theta _{u},\theta _{c})$.

If headquarters does not know downstream elasticity, the upstream external price will not be set at the first-best level. Specifically, condition (10) shows that the upstream division will choose $\hat{p_{u}}=p_{u}^{*}$ when

$$\begin{aligned} (t_{u}(p_{u}^{*})-c(\theta _{c}))\cdot \dfrac{\partial Q_{d}(t_{d},\theta )/\partial t_{d}}{Q_{d}(t_{d},\theta )}\cdot \dfrac{ \partial t_{d}(p_{u}^{*})}{\partial p_{u}}+\dfrac{\partial t_{u}(p_{u}^{*})}{\partial p_{u}}=0. \end{aligned}$$

(41)

For a given realization of $c(\theta _{c})$, all of the terms in the expression above are known to headquarters except for the demand elasticity term, $\dfrac{\partial Q_{d}(t_{d},\theta )/\partial t_{d}}{ Q_{d}(t_{d},\theta )}$, which is random and unobserved by headquarters. Equation (41) will hold for all values of $\theta$ if and only if $\partial t_{u}(p_{u}^{*})/\partial p_{u}=0$ and $\partial t_{d}(p_{u}^{*})/\partial p_{u}=0$, but part (iii) of Proposition 1 shows that $\partial t_{d}(p_{u}^{*})/\partial p_{u}=0$ is not part of the optimal solution. Therefore the upstream price will not be set to the first-best level.

Part (iii) To show that $t_{d}(p_{u})$ must be a function of $p_{u}$, we consider the linear transfer prices, $t_{i}(p_{u})=\alpha _{i}\cdot p_{u}+\delta _{i}$ for $i=u,d$, and show that the first-order conditions for optimality are not met when $\alpha _{u}=\alpha _{d}=0$ and that it is not optimal to set $\alpha _{u}\ne 0$ when $\alpha _{d}=0$. Therefore it cannot be optimal to set a downstream transfer price that does not depend on $p_{u}$. Note that using linear transfer prices is without loss of generality, since we are simply showing that the downstream transfer price cannot be constant.

The firm’s problem is:

$$\begin{aligned} \underset{\alpha _{u},\alpha _{d},\delta _{u},\delta _{d}}{\max }&E\left[\left(\hat{p} _{u}(\alpha _{u},\alpha _{d},\delta _{u},\delta _{d},\theta )-c(\theta _{c})\right)Q_{u}\left(\hat{p}_{u}(\alpha _{u},\alpha _{d},\delta _{u},\delta _{d},\theta ),\theta _{u}\right)\right. \\& \left.+\,R(Q_{d}(\alpha _{d},\delta _{d},\theta _{d}),\theta _{d})-c(\theta _{c})Q_{d}(\alpha _{d},\delta _{d},\theta _{d})\right], \end{aligned}$$

(42)

subject to:

$$\begin{aligned} Q_{d}(\cdot ) \in \underset{q_{d}}{\arg \max }\left\{R(q_{d},\theta _{d})-(\alpha _{d}p_{u}+\delta _{d})\text { }q_{d}\right\} \end{aligned}$$

(43)

$$\begin{aligned} \hat{p}_{u}(\cdot ) \in \underset{p_{u}}{\arg \max }\left\{ \left( p_{u}-c(\theta _{c})\right) \text { }Q_{u}(p_{u},\theta _{u})+(\alpha _{u}p_{u}+\delta _{u}-c(\theta _{c}))\cdot Q_{d}(\alpha _{d},\delta _{d},\theta _{d})\right\} . \end{aligned}$$

(44)

The associated first-order conditions are:

$$\begin{aligned} \alpha _{d}:\;&E\left[ \left( Q_{u}(\cdot )+(p_{u}-c(\theta _{c}))Q_{u}^{\prime }(\cdot )\right) \dfrac{\partial \hat{p}_{u}}{\partial \alpha _{d}}+(R_{q}(\cdot )-c(\theta _{c}))\left( \dfrac{\partial Q_{d}}{\partial \alpha _{d}}+\dfrac{ \partial Q_{d}}{\partial \hat{p}_{u}}\dfrac{\partial \hat{p}_{u}}{\partial \alpha _{d}}\right) \right] =0 \end{aligned}$$

(45)

$$\begin{aligned} \delta _{d}:\;&E\left[ (Q_{u}(\cdot )+(p_{u}-c(\theta _{c}))Q_{u}^{\prime }(\cdot ))\dfrac{\partial \hat{p}_{u}}{\partial \delta _{d}}+(R_{q}(\cdot )-c(\theta _{c}))\left( \dfrac{\partial Q_{d}}{\partial \delta _{d}}+\dfrac{ \partial Q_{d}}{\partial \hat{p}_{u}}\dfrac{\partial \hat{p}_{u}}{\partial \delta _{d}}\right) \right] =0 \end{aligned}$$

(46)

$$\begin{aligned} \alpha _{u}:\;&E\left[ (Q_{u}(\cdot )+(p_{u}-c(\theta _{c}))Q_{u}^{\prime }(\cdot ))\dfrac{\partial \hat{p}_{u}}{\partial \alpha _{u}}+(R_{q}(\cdot )-c(\theta _{c}))\dfrac{\partial Q_{d}}{\partial \hat{p}_{u}}\dfrac{\partial \hat{p}_{u}}{\partial \alpha _{u}}\right] =0 \end{aligned}$$

(47)

$$\begin{aligned} \delta _{u}:\;&E\left[ (Q_{u}(\cdot )+(p_{u}-c(\theta _{c}))Q_{u}^{\prime }(\cdot ))\dfrac{\partial \hat{p}_{u}}{\partial \delta _{u}}+(R_{q}(\cdot )-c(\theta _{c}))\dfrac{\partial Q_{d}}{\partial \hat{p}_{u}}\dfrac{\partial \hat{p}_{u}}{\partial \delta _{u}}\right] =0. \end{aligned}$$

(48)

Let $\alpha _{u}=\alpha _{d}=0$. Then the transfer prices are constant, and $\hat{p}_{u}=p_{u}^{*}$, which implies that $Q_{u}(\cdot )+(p_{u}-c(\theta _{c}))\cdot Q_{u}^{\prime }(\cdot )=0$. Furthermore, $\alpha _{d}=0$ implies $\partial Q_{d}/\partial \hat{p}_{u}=0$. Then (47) and (48) are satisfied, and (45) and (46) reduce to:

$$\begin{aligned} \alpha _{d}&:\;\;E\left[ (R_{q}(\cdot )-c(\theta _{c}))\cdot \dfrac{ \partial Q_{d}}{\partial \alpha _{d}}\right] =0 \end{aligned}$$

(49)

$$\begin{aligned} \delta _{d}&:\;\;E\left[ (R_{q}(\cdot )-c(\theta _{c}))\cdot \dfrac{ \partial Q_{d}}{\partial \delta _{d}}\right] =0. \end{aligned}$$

(50)

Solving (43) gives the downstream division’s first-order condition: $R_{q}(q_{d},\theta )=\alpha _{d}\hat{p}_{u}+\delta _{d}$. Using the implicit function theorem, we get $\dfrac{\partial Q_{d}}{\partial \alpha _{d}}=\dfrac{\hat{p}_{u}}{R_{qq}}$ and $\dfrac{\partial Q_{d}}{ \partial \delta _{d}}=\dfrac{1}{R_{qq}}$. Since $\alpha _{d}=0$ implies $R_{q}=\delta _{d}$, we can rewrite:

$$\begin{aligned} \alpha _{d}&:\;\;E\left[ (\delta _{d}-c(\theta _{c}))\cdot \dfrac{\hat{p} _{u}}{R_{qq}}\right] =0 \Rightarrow \delta _{d}=\frac{E\left[ c(\theta _{c})\cdot \hat{p }_{u}\cdot R_{qq}^{-1}\right] }{E\left[ \hat{p}_{u}\cdot R_{qq}^{-1}\right] } \end{aligned}$$

(51)

$$\begin{aligned} \delta _{d}&:\;\;E\left[ (\delta _{d}-c(\theta _{c}))\cdot \dfrac{1}{R_{qq}} \right] =0 \Rightarrow \delta _{d}=E\left[ c(\theta _{c})\right] , \end{aligned}$$

(52)

where we use that $R(\cdot )$ (by assumption) and $Q_{d}(\cdot )$ (since $t_{d}$ does not depend on $\hat{p}_{u}$) are independent of $c(\theta _{c})$. Since both equations cannot be fulfilled simultaneously for all $\theta$, one or both transfer prices must depend on $p_u$.

If only the upstream transfer price depends on $p_{u}$, then Condition 10 reduces to:

$$\begin{aligned} \dfrac{\partial t_{u}(p_{u})}{\partial p_{u}}\cdot Q_{d}\left( t_{d},\theta _{d}\right) =0. \end{aligned}$$

(53)

Since the upstream division will only set the external price efficiently if the equality above holds and since an inefficient external price provides no benefit to the firm when $t_{d}$ does not depend on $p_{u},\,\frac{\partial t_{u}(p_{u})}{\partial p_{u}}=0$ will always be optimal if $t_{d}$ does not depend on $p_{u}$. Since one or both transfer prices must depend on $p_{u}$, it will always be optimal for $t_{d}$ to depend on $p_{u}$.

Proof of Proposition 2

By Lemma 1, $t_{d}(p_{u})=p_{u}+Q_{u}(p_{u})/Q_{u}^{\prime }(p_{u})$. Therefore $t_{d}(p_{u})$ is linear in $p_{u}$ if and only if $Q_{u}/Q_{u}^{\prime }$ is linear.

The upstream transfer price, $t_{u}(t_{d}(p_{u}))$, only depends on $p_{u}$ through the downstream transfer price, $t_{d}(p_{u})$. Thus, if $t_{d}(p_{u})$ is linear in $p_{u}$, then $t_{u}(t_{d}(p_{u}))$ will be linear in $p_{u}$ if and only if it is linear in $t_{d}$, that is, if $t_{u}(t_{d}(p_{u}))=\alpha \cdot t_{d}+\beta$. For first best to hold, the upstream transfer price must be set such that the upstream division’s profit from internal trade, ${\varPi }_{u}^{int}=(t_{u}(t_{d})-c(\theta _{c}))\cdot Q_{d}(t_{d})$, is maximized when $t_{d}=c(\theta _{c})$ for all $c(\theta _{c})$ (as outlined in the text). Thus $t_{u}(t_{d}(p_{u}))$ must be such that the derivative of ${\varPi }_{u}^{int}$ with respect to $t_{d}$ is equal to zero when $t_{d}=c(\theta _{c})$. Writing this first-order condition as:

$$\begin{aligned} t_{u}^{\prime }(t_{d})\cdot Q_{d}(t_{d})+(t_{u}(t_{d})-c(\theta _{c}))\cdot Q_{d}^{\prime }(t_{d})=0, \end{aligned}$$

(54)

substituting $t_{u}(t_{d})=\alpha \cdot t_{d}+\beta$ and $t_{d}=c(\theta _{c})$ into (54), and solving gives:

$$\begin{aligned} \frac{Q_{d}(t_{d})}{Q_{d}^{\prime }(t_{d})}=\frac{(1-\alpha )\cdot t_{d}-\beta }{\alpha }. \end{aligned}$$

(55)

Thus a linear upstream transfer price will be optimal only if $Q_{d}/Q_{d}^{\prime }$ is linear.

We have shown that for both transfer prices to be linear, $Q_{d}/Q_{d}^{\prime }$ and $Q_{u}/Q_{u}^{\prime }$ must be linear. The general solutions to the differential equation $F(x)/F^{\prime }(x)=\gamma x+\delta$ are given by:

$$\begin{aligned} F(x) = K\cdot (\delta +\gamma x)^{1/\gamma }\quad \text { if }\quad \gamma \ne 0, \end{aligned}$$

(56)

$$\begin{aligned} F(x) = K\cdot e^{x/\delta }\quad \text { if } \quad \gamma =0. \end{aligned}$$

(57)

Plugging (56) into Lemma 1 yields the form in Proposition 2. (57) is a limiting form of (56) as $\gamma \rightarrow 0$.

Proof of Corollary 1

Follows directly from Proposition 2.

Proof of Corollary 2

Follows from Proposition 1.

Proof of Proposition 3

Proof follows the same argument as for the unconstrained case in Sect. 3.

Proof of Lemma 2

We derive conditions for which the upstream division’s profit function (32), i.e.,

$$\begin{aligned} {\varPi }_{u}(\theta )=\left\{ \begin{array}{lll} (p_{u}-c(\theta _{c}))\cdot Q_{u}(p_{u},\theta _{u})+(t_{u}(p_{u})-c(\theta _{c}))\cdot Q_{d}(t_{d}(p_{u}),\theta _{d}) &{}\quad \hbox {if} &{}\quad p_{u}\ge p_{u}^{c} \\ p_{u}\cdot (K-Q_{d}(t_{d}(p_{u}),\theta _{d}))+t_{u}(p_{u})\cdot Q_{d}(t_{d}(p_{u}),\theta _{d})-c(\theta _{c})\cdot K &{}\quad \hbox {if} &{}\quad p_{u}\le p_{u}^{c}, \end{array}\right. \end{aligned}$$

(58)

is decreasing in $p_{u}$ for $p_{u}>p_{u}^{c}$ and increasing in $p_{u}$ for $p_{u}<p_{u}^{c}$.

Calculating the derivative,

$$\begin{aligned} \dfrac{\partial {\varPi }_{u}}{\partial p_{u}}=\left\{ \begin{array}{lll} (p_{u}-c(\theta _{c}))\cdot Q_{u}^{\prime }+Q_{u}+(t_{u}-c(\theta _{c}))\cdot \dfrac{\partial Q_{d}}{\partial t_{d}}\dfrac{\partial t_{d}}{\partial p_{u}}+\dfrac{\partial t_{u}}{\partial p_{u}}\cdot Q_{d} &{}\quad \hbox {if} &{}\quad p_{u}\ge p_{u}^{c} \\ K-Q_d+(t_{u}-p_{u})\cdot \dfrac{ \partial Q_{d}}{\partial t_{d}}\dfrac{\partial t_{d}}{\partial p_{u}} +\dfrac{\partial t_{u}}{\partial p_{u}}\cdot Q_{d} &{}\quad \hbox {if} &{}\quad p_{u}\le p_{u}^{c}, \end{array} \right. \end{aligned}$$

(59)

noting that $(p_{u}-c(\theta _{c}))\cdot Q_{u}^{\prime }+Q_{u}\le 0$ if $p_{u}\ge p_{u}^{c}$, and simple algebra yields (34).

Proof of Corollary 3

Follows directly from Proposition 1.

Proof of Proposition 4

Noting from the Envelope Theorem that $E_{\theta _{d}}[\int _{t_{d}(p_{u})}^{\infty }Q_{d}(s,\theta _{d})\;ds]=E_{\theta _{d}}\left[ {R}(\cdot )-t_{d}(p_{u})\cdot Q_{d}(\cdot )\right]$, we can restate the upstream division’s expected profit from the internal transaction as in (14), that is:

$$\begin{aligned} E_{\theta _{d}}\left[ \left( t_{u}(p_{u})-c(\theta _{c})\right) \cdot Q_{d}(t_{d}(p_{u}){,\theta _{d}})\right] =E_{\theta _{d}}\left[ {R} (Q_{d}(t_{d}(p_{u}),{\theta _{d}}))-c(\theta _{c})\cdot Q_{d}(t_{d}(p_{u}),{ \theta _{d}})\right] . \end{aligned}$$

The upstream division will maximize its profits from internal trade by setting $p_{u}$ to make the downstream division’s transfer price equal to $c(\theta _{c})$ (which induces the downstream division to choose the optimal internal trade quantity). Noting that $t_{d}(p_{u}^{*})=c(\theta _{c})$ proves the result.