Bilateral Information Sharing and Pricing Incentives in a Retail Channel

Abstract

This chapter evaluates the impact of sharing information on wholesale and retail pricing incentives as well as on the distribution of economic rents. We consider a model in which the manufacturer distributes its product to one or more retailers. Each firm receives a private signal as an estimate of stochastic consumer demand. We show that, in the absence of information sharing, the retailer is able to use the wholesale price to infer the manufacturer’s private signal. This creates a pricing distortion which benefits the retailer. Downward sharing of the manufacturer’s private signal eliminates this distortion. In contrast, when the retailer shares its private signal upstream, the manufacturer is able to set price closer to retailer’s value, thus capturing downstream consumer surplus. In general, the manufacturer benefits from more information sharing at the loss of downstream retailers and consumers. Hence, information sharing arrangements in equilibrium require side payments and/or sufficient cost savings (e.g., reduced inventory costs).

Appendix

This appendix provides the proofs of all propositions and corollaries stated in the main text.

Proof of Proposition 1

A comparison of the expected values of ( 16.11) and ( 16.12) with the expected values of ( 16.16) and ( 16.17) gives the results of part (i). To deduce part (ii), we compute realized profits to each firm in the two regimes.

NS: The realized quantity sold is

$$\displaystyle\begin{array}{rcl} q^{NS}& =& a - br^{NS} + u {}\\ & =& \frac{a} {4} \frac{3\sigma + 2s} {2\sigma + s} - \frac{\sigma (3\sigma + 2s)} {4(\sigma +s)(2\sigma + s)}x_{M} - \frac{\sigma } {2(2\sigma + s)}x_{R} + u. {}\\ \end{array}$$

Computing profits at the retailer

$$\displaystyle\begin{array}{rcl} \varPi _{R}^{NS}& =& (r^{NS} - w^{NS})q^{NS} {}\\ & =& \frac{\varDelta } {2b}{\biggl [a\frac{3\sigma + s} {2\sigma } + \frac{\sigma +2s} {2(\sigma +s)}x_{M} + x_{R}\biggr ]} \times q^{NS}, {}\\ \end{array}$$

and at the manufacturer

$$\displaystyle\begin{array}{rcl} \varPi _{M}^{NS}& =& w^{NS}q^{NS} {}\\ & =& \frac{\varDelta } {2b}{\biggl [a \frac{\sigma } {2\sigma + s} + \frac{\sigma ^{2}} {(2\sigma + s)(\sigma +s)}x_{M}\biggr ]} \times q^{NS}, {}\\ \end{array}$$

where $\varDelta \equiv \sigma /(2\sigma + s)$. Taking expectations over (u, x _M , x _R ) gives

$$\displaystyle{ E(\varPi _{R}^{NS}) = \frac{\varDelta ^{2}} {16b}{\biggl [a^{2}{\biggl (\frac{3\sigma + 2s} {\sigma } \biggr )}^{\!\!\!2} + \frac{9\sigma ^{2} + 20\sigma s + 8s^{2}} {\sigma +s} \biggr ]} }$$

(16.28)

and

$$\displaystyle{ E(\varPi _{M}^{NS}) = \frac{\varDelta ^{2}} {8b}{\biggl [a^{2}\frac{3\sigma + 2s} {\sigma } + \frac{\sigma (3\sigma + 2s)} {\sigma +s} \biggr ]}, }$$

(16.29)

which makes use the following facts:

$$\displaystyle{ \begin{array}{rll} E(x_{M}x_{R})& = E(u^{2} +\varepsilon _{M}u +\varepsilon _{R}u +\varepsilon _{M}\varepsilon _{R}) =\sigma \\ E(x_{i})^{2} & = E(u^{2} + 2\varepsilon _{i}u +\varepsilon _{i}) =\sigma +s, \\ E(x_{i}u)& = E(u^{2} + u\varepsilon _{i}) =\sigma, \end{array} }$$

(16.30)

for i = M, R.

DS: The realized quantity sold is

$$\displaystyle\begin{array}{rcl} q^{DS}& =& a - br^{DS} + u {}\\ & =& \frac{a} {4} - \frac{\varDelta } {2}x_{R} -\frac{\varDelta (4\sigma + 3s)} {4(\sigma +s)} x_{R} + u. {}\\ \end{array}$$

Computing profits at the retailer

$$\displaystyle\begin{array}{rcl} \varPi _{R}^{DS}& =& (r^{DS} - w^{DS})q^{DS} {}\\ & =& \frac{1} {4b}{\biggl [a + \frac{\varDelta s} {\sigma +s}x_{M} + 2\varDelta x_{R}\biggr ]} \times q^{DS}, {}\\ \end{array}$$

and at the manufacturer

$$\displaystyle\begin{array}{rcl} \varPi _{M}^{DS}& =& w^{DS}q^{DS} ={\biggl [ \frac{a} {2b} + \frac{\sigma } {2b(\sigma +s)}x_{M}\biggr ]} \times q^{DS}. {}\\ \end{array}$$

Taking expectations over (u, x _M , x _R ) gives

$$\displaystyle{ E(\varPi _{R}^{DS}) = \frac{1} {16b}{\biggl [a^{2} + \frac{\sigma \varDelta (2\sigma + 5s)} {\sigma +s} \biggr ]} }$$

(16.31)

and

$$\displaystyle{ E(\varPi _{M}^{DS}) = \frac{1} {8b}{\biggl [a^{2} + \frac{\sigma ^{2}} {\sigma +s}\biggr ]}. }$$

(16.32)

Using ( 16.28) and ( 16.31), the difference E(Π _R ^DS −Π _R ^NS) is given by the expression in ( 16.18). For the manufacturer, use ( 16.29) and ( 16.32) to compute

$$E(\Pi _M^{NS} - \Pi _M^{DS}) = \frac{1}{{8b}}[{a^2}\underbrace {\left( {1 - \frac{{\sigma (3\sigma + 2s)}}{{{{(2\sigma + s)}^2}}}} \right)}_ - + \underbrace {\left( {\frac{{{\sigma ^2}}}{{\sigma + s}} - \frac{{{\sigma ^3}(3\sigma + 2s)}}{{{{(2\sigma + s)}^2}(\sigma + s)}}} \right)}_ - ] < 0.$$

Q.E.D.

Proof of Corollary 1

The change in expected consumer surplus can be computed using ( 16.19) for k = NS, DS. This leads to

$$\displaystyle\begin{array}{rcl} E(CS^{NS} - CS^{DS})& =& \frac{b} {2}[E(r^{NS})^{2} - E(r^{DS})^{2}] - a[E(r^{NS}) - E(r^{DS})] {}\\ & & -[E(ur^{NS})^{2} - E(ur^{DS})^{2}] {}\\ & =& \frac{1} {32b(2\sigma + s)^{2}} {}\\ & & \times {\biggl [a^{2}(5\sigma + 3s)(\sigma +s) + (7\sigma ^{2} + 8\sigma s + 3s^{2}) \frac{\sigma ^{2}} {\sigma +s}\biggr ]}, {}\\ \end{array}$$

which makes use of the facts in ( 16.30) to compute the expectations in the brackets above and some amount of algebra. The expression above is obviously positive. Q.E.D.

Proof of Proposition 2

The DS case is analyzed in the proof of Proposition 1. We derive here the results for FS. Note that part (i) of the proposition follows from the expressions ( 16.25) and ( 16.26). Part (ii) requires the computation of profits in FS regime.

FS: The realized quantity sold is

$$\displaystyle{ q^{FS} = a - br^{FS} + u = \frac{a} {4} -\frac{3\varDelta } {4}(x_{R} + x_{R}) + u. }$$

Computing profits at the retailer

$$\displaystyle{ \varPi _{R}^{FS} = (r^{FS} - w^{FS})q^{FS} = \frac{1} {4b}\left [a +\varDelta (x_{M} + x_{R})\right ] \times q^{FS}, }$$

and at the manufacturer

$$\displaystyle{ \varPi _{M}^{FS} = w^{FS}q^{FS} = \frac{1} {2b}\left [a +\varDelta (x_{M} + x_{R})\right ] \times q^{DS}. }$$

Taking expectations over (u, x _M , x _R ) gives

$$\displaystyle{ E(\varPi _{R}^{FS}) = \frac{1} {16b}[a^{2} +\varDelta ^{2}(4\sigma + 2s)] }$$

(16.33)

and

$$\displaystyle{ E(\varPi _{M}^{FS}) = \frac{1} {8b}[a^{2} +\varDelta ^{2}(7\sigma + 2s)]. }$$

(16.34)

Using ( 16.33) and ( 16.34), the difference E(Π _R ^FS −Π _R ^DS) is given by the expression in ( 16.24). For the manufacturer, use ( 16.33) and ( 16.34) to compute

$$\displaystyle{ E(\varPi _{M}^{FS} -\varPi _{ M}^{DS}) = \frac{\varDelta ^{2}} {8b}{\biggl [\frac{3\sigma ^{2} + 5\sigma s + s^{2}} {\sigma +s} \biggr ]} > 0.\quad \text{Q.E.D.} }$$

Proof of Corollary 2

Following the computation in Corollary 1 and using the corresponding derivations for the FS regime we have

$$\displaystyle{ E(CS^{DS} - CS^{FS}) = \frac{3s\sigma ^{2}} {32b(\sigma +s)(2\sigma + s)} > 0, }$$

which makes use of the fact that E(r ^FS) = E(r ^DS). Q.E.D.

Proof of Proposition 3

NS: We start with the no sharing NS regime and specify decision rules for the three firms as follows

$$\displaystyle{ w = f_{M}^{NS}(x_{ M}) =\mu _{0} +\mu _{1}x_{M} }$$

(16.35)

$$\displaystyle{ r_{i} = f_{i}^{NS}(w,x_{ i}) =\rho _{0i} +\rho _{1i}w +\rho _{2i}x_{i},\quad i = 1,2. }$$

(16.36)

M sets a price w according to f _M ^NS and retailers respond by simultaneously maximizing

$$\displaystyle\begin{array}{rcl} E(\varPi _{i}\vert w,x_{i})& =& (r_{i} - w)E(q_{i}\vert w,x_{i}) \\ & =& (r_{i} - w){\biggl [ \frac{a} {b + d} - \frac{br_{i}} {b^{2} - d^{2}} + \frac{dE(r_{j}\vert w,x_{i})} {b^{2} - d^{2}} + E(u\vert w,x_{i})\biggr ]}.{}\end{array}$$

(16.37)

for i = 1, 2 and j ≠ i. Retailer i’s conditional expectations of its rivals’ price is facilitated by ( 16.36):

$$\displaystyle{ E(r_{j}\vert w,x_{i}) =\rho _{0j} +\rho _{1j}w +\rho _{2j}E(x_{j}\vert w,x_{i}),\quad i = 1,2\text{ and }j\neq i. }$$

Again, perfect inference of x _M is possible from observing wholesale price w and inverting ( 16.35). Specifically, retailer i’s conditional expectations in ( 16.37) are

$$\displaystyle\begin{array}{rcl} & & E(x_{j}\vert w,x_{i}) = E(u\vert w,x_{i}) = \frac{\sigma (x_{i} + x_{M})} {2\sigma + s} = \frac{\sigma (x_{i} + (w -\mu _{0})/\mu _{1})} {2\sigma + s}, {}\\ \end{array}$$

which were similarly derived in Sect. 16.2.

The manufacturer sets wholesale price w to maximize expected profits:

$$\displaystyle\begin{array}{rcl} E(\varPi _{M}\vert x_{M})& =& wE(q_{1} + q_{2}\vert x_{M}) \\ & =& w{\biggl \{ \frac{2a} {b + d} - \frac{(b - d)} {b^{2} - d^{2}}[E(r_{1}\vert x_{M}) + E(r_{2}\vert x_{M})] + 2E(u\vert x_{M})\biggr \}},{}\end{array}$$

(16.38)

where M’s conditional expectations are computed using ( 16.36) and ( 16.8) so that

$$\displaystyle{E(x_{i}\vert x_{M}) = E(u\vert x_{M}) = \frac{\sigma } {\sigma +s}x_{M}.}$$

As before, maximizations of ( 16.37) for each i = 1, 2 and ( 16.38) yields three first order conditions which may be used to match coefficients with ( 16.35) and ( 16.36), respectively. This leads to a linear system of eight equations and eight unknown characterizing the equilibrium decision rules. However, symmetry across retailers reduces the solution by three variables leading to the following solution:

$$\displaystyle\begin{array}{rcl} w^{NS}& =& \varGamma {\biggl [a + (b + d) \frac{\sigma } {\sigma +s}x_{M}\biggr ]} {}\\ r_{i}^{NS}& =& \mu _{ 0}^{NS} +\mu _{ 1}^{NS}w +\mu _{ 2}^{NS}x_{ i}, {}\\ \end{array}$$

where

$$\displaystyle\begin{array}{rcl} \varGamma & =& \frac{4bd(\sigma +s)^{2} + 2\sigma (\sigma +s)(2b^{2} + bd + d^{2}) +\sigma ^{2}(4b^{2} - d^{2})} {2[4b^{2}(2\sigma + s) - d^{2}\sigma ^{2}]}; {}\\ \mu _{0}^{NS}& =& \frac{b - d} {2b - d}{\biggl [a - \frac{2ab(\sigma +s)} {2b(2\sigma + s) - d\sigma }\biggr ]}; {}\\ \mu _{1}^{NS}& =& \frac{\sigma (b^{2} - d^{2})} {2b(2\sigma + s) - d\sigma };\quad \text{and} {}\\ \mu _{2}^{NS}& =& \frac{b} {2b - d}{\biggl [1 - \frac{2(b - d)(\sigma +s)} {\varGamma 2b(2\sigma + s) - d\sigma }\biggr ]}. {}\\ \end{array}$$

Note that Γ has the following properties:

$$\displaystyle{ \varGamma < \frac{1} {2}\quad \text{for all }d < b\quad \text{and}\quad \lim _{b\downarrow d}\varGamma = \frac{1} {2}. }$$

(16.39)

DS: In this regime both retailers have knowledge of the manufacturer’s realized signal x _M , but the manufacturer has no additional information relative to NS. This information structure implies the following linear forms of decision rules:

$$\displaystyle\begin{array}{rcl} w& =& f_{M}^{DS}(x_{ M}) =\mu _{0} +\mu _{1}x_{M} {}\\ r_{i}& =& f_{i}^{DS}(w,x_{ i},x_{M}) =\rho _{0i} +\rho _{1i}w +\rho _{2i}x_{i} +\rho _{3i}x_{M};\quad i = 1,2. {}\\ \end{array}$$

Following the same process as in the NS regime without the inference of x _M leads to the following equilibrium decision rules:

$$\displaystyle\begin{array}{rcl} w^{DS}& =& \frac{1} {2}{\biggl [a + (b + d) \frac{\sigma } {\sigma +s}x_{M}\biggr ]}{}\end{array}$$

(16.40)

$$\displaystyle\begin{array}{rcl} r_{i}^{DS}& =& \mu _{ 0}^{DS} +\mu _{ 1}^{DS}w +\mu _{ 2}^{DS}x_{ i} +\mu _{ 3}^{DS}x_{ M},{}\end{array}$$

(16.41)

where

$$\displaystyle\begin{array}{rcl} & \mu _{0}^{DS} = \frac{a(b-d)} {2b-d};\quad \mu _{1}^{DS} = \frac{\sigma (b^{2}-d^{2})} {2b(2\sigma +s)-d\sigma };\quad \mu _{2}^{DS} = \frac{b} {2b-d};& {}\\ & \text{and}\quad \mu _{3}^{DS} = \frac{2b\sigma (b^{2}-d^{2})} {(2b-d)[2b(2\sigma +s)+d\sigma ]}. & {}\\ \end{array}$$

The difference in expected wholesale prices: $E(w^{NS} - w^{DS}) = (\varGamma -\tfrac{1} {2})a$, which from (39) is shown negative for d < b and equal zero in the limit d ↑ b.

Finally, the difference in retail prices

$$\displaystyle{E(r^{NS} - r^{DS}) = (\rho _{ 0}^{NS} -\rho _{ 0}^{DS}) + a(\varGamma \rho _{ 2}^{NS} -\tfrac{1} {2}\rho _{2}^{DS})}$$

is seen negative for d < b by noting that ρ ₀ ^NS < ρ ₀ ^DS for d < b, ρ ₂ ^NS = ρ ₂ ^DS for d ≤ b and reusing ( 16.39). Again, ( 16.39) implies this difference is zero in the limit d ↑ b. Q.E.D.

Proof of Proposition 4

To prove this we derive the equilibrium outcome in the FS regime and compare it to the outcome in the DS regime derived in the proof of Proposition 3. In the FS case, we assume all signals x ₁, x ₂, x _M are known by all firms. Hence, the assumed linear form for the decision rules imply

$$\displaystyle\begin{array}{rcl} w& = & f_{M}^{FS}(x_{ 1},x_{2},x_{M}) =\mu _{0} +\mu _{1}x_{1} +\mu _{2}x_{2} +\mu _{3}x_{M} \\ r_{i}& = f_{i}^{FS}(w,x_{i},x_{j},x_{M})& \\ & = & \rho _{0i} +\rho _{1i}x_{i} +\rho _{2i}x_{j} +\rho _{3i}w +\rho _{4i}x_{M};\quad i = 1,2;\ j\neq i.{}\end{array}$$

(16.42)

Expected profit to retailer R _i given x ₁, x ₂, x _M and w is

$$\displaystyle\begin{array}{rcl} & & E(\varPi _{i}\vert w,x_{1},x_{2},x_{M}) \\ & & \qquad = E\biggl \{(r_{i} - w)\biggl [ \frac{a} {b + d} - \frac{br_{i}} {b^{2} - d^{2}} \\ & & \qquad \qquad \ \ + \frac{d} {b^{2} - d^{2}}E(r_{j}\vert w,x_{1},x_{2},x_{M}) + E(u\vert w,x_{1},x_{2},x_{M})\biggr ]\biggr \}{}\end{array}$$

(16.43)

and to the manufacturer

$$\displaystyle\begin{array}{rcl} & & E(\left.\varPi _{M}\right \vert x_{1},x_{2},x_{M}) \\ & & \qquad = E{\biggl \{w{\biggl [ \frac{a} {b + d} - \frac{br_{i}} {b^{2} - d^{2}} + \frac{dr_{j}} {b^{2} - d^{2}} + E(u\vert w,x_{1},x_{2},x_{M})\biggr ]}\biggr \}}.{}\end{array}$$

(16.44)

Following DeGroot (1970) we have that

$$\displaystyle{E(u\vert x_{1},x_{2},x_{M}) = \frac{\sigma } {3\sigma + s}(x_{1} + x_{2} + x_{M}).}$$

Furthermore, because retailers i = 1, 2 in the FS regime are in all aspects symmetric, we can exploit the fact that their decision rules will be identical in a symmetric equilibrium. Using these facts and the first order condition for maximizing R _i ’s expected profit ( 16.43), we can solve for the equilibrium coefficients of ( 16.42):

$$\displaystyle{ r^{FS} ={\biggl [ \frac{a} {2} + \frac{b + d} {2} \frac{\sigma } {3\sigma + s}(x_{1} + x_{2} + x_{M})\biggr ]}\frac{3b - 2d} {2b - d}, }$$

(16.45)

which has the expectation: $E(r^{FS}) = (a/2)[(3b - 2d)/(2b - d)]$. Using the retailers’ reaction, ( 16.42), the first order condition from the maximization of ( 16.44) permits the solution

$$\displaystyle{ w^{FS} = \frac{a} {2} + \frac{b + d} {2} \frac{\sigma } {3\sigma + s}(x_{1} + x_{2} + x_{M}), }$$

(16.46)

which has the expectation $E(w^{FS}) = a/2$. Part (i) of the proposition follows from taking expectations of w ^DS and r ^DS, given in Eqs. ( 16.40) and ( 16.41) derived in the proof of Proposition 3.

To establish part (ii) of the proposition, note that the covariances can be computed directly using ( 16.40), ( 16.41), ( 16.45), and ( 16.46) and making use of the facts in ( 16.30):

$$\displaystyle\begin{array}{rcl} \mathop{\mathrm{cov}}(w^{DS},r^{DS})& =& \frac{b + d} {2} \frac{\sigma ^{2}} {\sigma +s}{\biggl \{\!\! \frac{b^{2} - d^{2}} {2b(2\sigma + s) - d\sigma }\!{\biggl [\sigma + \frac{2b} {2b - d}(\sigma +s)\biggr ]}\! + \frac{b(b + d)} {2(2b - d)}\!\biggr \}} {}\\ \mathop{\mathrm{cov}}(r^{DS},u)& =& \frac{\sigma ^{2}} {2b - d}{\biggl \{\frac{(b^{2} - d^{2})(4b - d)} {2b(2\sigma + s) - d\sigma } + \frac{b(b + d)} {2(\sigma +s)} \biggr \}} {}\\ \mathop{\mathrm{cov}}(w^{FS},r^{FS})& =& \frac{3\sigma (b + d)^{2}} {4} \frac{3b - 2d} {2d - d} {}\\ \mathop{\mathrm{cov}}(r^{FS},u)& =& \frac{3\sigma ^{2}(b + d)} {2(3\sigma + s)} \frac{3b - 2d} {2d - d}. {}\\ \end{array}$$

Evaluating the limits of these expressions for d ↑ b implies the orderings stated in the proposition. This guarantees a d < b such that the ordering holds in the left neighborhood of b. Q.E.D.