Competitive Delivered Pricing by Mail-Order and Internet Retailers

Abstract

Mail-order and internet sellers must decide how customers pay shipping charges. Typically, these sellers choose between two pricing policies: either “uniform pricing,” where the firm delivers to any customer at a fixed delivery charge (that may be volume dependent), or “mill pricing,” where the firm bills the customer a distance-related shipping charge. This paper studies price competition between a mail-order (or internet) seller and local retailers, and the mail-order firm’s choice of pricing policy. The price policy choice is studied when retailers do not change price in reaction to the mail-order firm’s policy choice, and when they do. In the second case, a two-stage non-cooperative game is used and it is found that for low customer willingness to pay, mill pricing is favored but as willingness to pay rises, uniform pricing becomes more attractive. These results are generalized showing that larger markets, higher transportation rates, higher unit production cost, and greater competition between retailers all increase profit under mill pricing relative to uniform pricing (and vice versa). On the other hand, cost asymmetries that favor the mail-order firm will tend to induce uniform rather than mill pricing. Some empirical data on retail and mail-order sales that confirm these results are presented.

Appendix

Proof of Proposition 3.1

Here we assume that ${p_{rU}} = {p_{rM}} = 0$ and seek the optimal price choice by the mail-order firm under mill pricing and uniform pricing. Without loss of generality I assume t=1. Ignoring the Max, Min, and []⁺ constraints mail-order profit can be written as:

$$ \begin{array}{*{20}{c}} {{\Pi _U} = 2{{\left[ {a - {p_U}} \right]}^ + }\int\limits_{Max\left[ {.5,Min\left[ {{{\left[ {{p_U} - {p_{rU}}/t} \right]}^ + },0} \right]} \right]}^{.5} {\left( {{p_U} - \left( {.5 - x} \right)} \right)dx = \left( {a - {p_U}} \right)\left( { - .1125 + {p_U} - 1.5p_U^2} \right)} } \hfill \\ {{\Pi _M} = 2\int\limits_{Min\left( {.5,Max\left( {0,.25 + {p_M} - {p_{rM}}/2t} \right)} \right.}^{.5} {{p_m}{{\left[ {a - {p_M} - \left( {.5 - x} \right)} \right]}^ + }dx} = {p_M}\left( { - .03125 + .25a - {p_M}\left( {.125 + .5a - .375{p_M}} \right)} \right)} \hfill \\ \end{array} $$

(A.1)

This simplification assumes an interior solution. Differentiating and solving for the zero of the resulting polynomials yields:

$$ \begin{array}{*{20}{c}} {{p_U} = \frac{1}{9}\left( {2 + 3a - 3\sqrt {{\frac{7}{{36}} - \frac{2}{3}a + {a^2}}} } \right)} \hfill \\{{p_M} = \frac{4}{9}\left( {\frac{1}{4} + a - \sqrt {{\frac{{13}}{{64}} - \frac{5}{8}a + {a^2}}} } \right)} \hfill \\\end{array} $$

(A.2)

I substitute these functions into the appropriate profit functions, and then equate the profit functions. This yields a quadratic function of a, which when solved yields solutions: a = 0.1482, and a = 0.3384. The first solution is a spurious one in that it violates the region of integration in (5.1), and thus the hypothesis of (A.1) is violated. Eq. (A.1) coincides with (5.1) at a = 0.3384 and is the dividing point for optimality of the pricing policies. Above this solution, uniform pricing gives higher profit than mill (and vice versa). By substituting uniform prices into the uniform profit function and setting the equation equal to zero yields zero profit when $a = .166\overline {66}$. QED

Proof of Lemma 4.1

As the mail-order and the retailer’s profit functions are symmetric, the reaction functions for the mail-order and retailers are the same (with prices reversed). Solving for a symmetric equilibrium is done by solving the fixed point problem by composing the mail-order reaction function with the retailer reaction function. If a Nash equilibrium exists, each firm’s price must lie in the interval [0,a].¹⁰ In general, if a firm chooses p ϵ [0,a], then it will earn zero profit at the boundary and non-negative profit at any interior price. The derivative of the mail-order and retailer profit with respect to their price choices using profit functions Equation (2.2) is:

$$ \frac{{\partial {\Pi _M}}}{{\partial {p_M}}}\left\{ {\begin{array}{*{20}{c}} {\left( {\frac{{{p_M}}}{2}} \right)\left( {a - {p_M} - \left( {.25 + \left( {{p_M} - {p_{rM}}} \right)/2} \right)} \right)} \hfill & {if\;{p_M} + .25 + \left( {{p_M} - {p_{rM}}} \right)/2 \leqslant a} \hfill \\ {\frac{1}{2}{{\left( {a - {p_M}} \right)}^2}{p_M}} \hfill & {if\;{p_M} \leqslant a,{p_M} + .25 + \left( {{p_M} - {p_{rM}}} \right)/2 > a\,and\,\left( {{p_M} - {p_{rM}}} \right)/2 \leqslant .25} \hfill \\ 0 \hfill & {if\left( {{p_M} = {p_{rM}}} \right)/2 > .25,or\;{p_M} > a} \hfill \\ \end{array} } \right. $$

(A.3)

Three different cases can be described as follows: the first case is where both firms serve and all customers are served, the second when the mail-order is a local monopolist, and the third when the mail-order serves no customers. When the mail-order firm is a monopolist, it can be easily shown that its monopoly price is a/3. Solving for the mail-order’s reaction function (confirming that second-order conditions do hold):

$$ {{\text{p}}_{\text{M}}}\left( {{{\text{p}}_{{\text{rM}}}}} \right) = \left\{ {\frac{1}{9}\left( {1 + 4a + 2{{\text{p}}_{{\text{rM}}}} - 4\sqrt {{{a^2} - \frac{{5a}}{8}\left( {1 + 2{{\text{p}}_{{\text{rM}}}}} \right) + \frac{{13}}{{16}}{{\left( {\frac{1}{2} + {{\text{p}}_{{\text{rM}}}}} \right)}^2}}} } \right)} \right.if\left( {\frac{a}{3} - {{\text{p}}_{{\text{rM}}}}} \right)/2 \leqslant .25\frac{a}{3}\;if\left( {\frac{a}{3} - {{\text{p}}_{{\text{rM}}}}} \right)/2 > .25. $$

(A.4)

Second-order conditions hold for the reaction function. As profit is zero at the boundary points, this reaction function maps an opponent’s price into the interior of [0,a]. Solve for the fixed point, for now ignoring the case where local prices can exceed a and thus generate negative demand. This is equivalent to setting: ${{\hbox{p}}_{\rm{M}}}\left( {{{\hbox{p}}_{\rm{rM}}}\left( {\hbox{x}} \right)} \right) = {\hbox{x}}$, or explicitly:

$$ {\hbox{p}} = \frac{1}{9}\left( {1 + 4{\hbox{a}} + 2{\hbox{p}} - 4\sqrt {{{{\hbox{a}}^2} - \frac{{5a}}{8}\left( {1 + 2{\hbox{p}}} \right) + \frac{{13}}{{16}}{{\left( {\frac{1}{2} + {\hbox{p}}} \right)}^2}}} } \right), $$

(A.5)

yielding the solution conditioned on a: ${p^* }(a) = \frac{1}{8}\left( {\left( {3 + 4{\hbox{a}} - \sqrt {{13 - 8{\hbox{a}}\left( {1 + 2{\hbox{a}}} \right)}} } \right.} \right)$.

This relationship only holds when this price causes the different firm’s market boundaries to intersect, and all customers are served with positive demand. That corresponds to condition $\frac{1}{8}\left( {\left( {3 + 4{\hbox{a}} - \sqrt {{13 - 8{\hbox{a}}\left( {1 + 2{\hbox{a}}} \right)}} } \right.} \right) + .25 > a$, which assures that prices at the boundary points are less than a, yielding the reaction function:

$$ {p^ * }\left( a \right) = p_{rM}^ * = p_{rM}^ * = \left\{ {\frac{1}{8}\left( {\left( {3 + 4a - \sqrt {{13 - 8a\left( {1 + 2a} \right)}} } \right.} \right)\;if\;\frac{1}{8}\left( {\left( {3 + 4a - \sqrt {{13 - 8a\left( {1 + 2a} \right)}} } \right.} \right) + .25 > a\;\frac{a}{3}} \right.\;otherwise $$

(A.6)

Solving $\frac{1}{8}\left( {\left( {3 + 4{{a}} - \sqrt {{13 - 8{{a}}\left( {1 + 2{{a}}} \right)}} } \right.} \right) + .25 = a$ yields the solution a = 3/8. So (A.4) can be written as

$$ {p^* }(a) = p_{rM}^* = p_M^* = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{8}\left( {\left( {3 + 4a - \sqrt {{13 - 8a\left( {1 + 2a} \right)}} } \right)if\,a \geqslant 3/8} \right)} \\{\frac{a}{3}if\,a\, < 3/8} \\\end{array}, } \right. $$

(A.7)

which are the Nash prices on the Mill subgame: neither of the two firms wishes to revise its price at these market prices. I can conclude that at the Nash equilibrium outcome: when a < 3/8, retailers and mail-order firm act as local monopolists: the market areas are strictly separated, with some customers not served. But when a ≥ 3/8, the mail-order and retailers compete for customers, and all customers are served.

It remains to show that when a ≥ 3/8, the symmetric price pair in (A.7) is the unique Nash solution on the Mill subgame. This is most easily demonstrated by viewing the vector field diagrams in Fig. 6 showing the mail-order and retailer profit gradients with respect to their own price decisions. Panels with a=.375,.6,.8, 1.0, 2.0, 10.0 show that there is a unique symmetric local Nash equilibrium. As a Nash equilibrium is also a local Nash equilibrium, (A.7) indicates that my solution is the unique Nash equilibrium. If multiple Nash equilibria existed there would be multiple stationary points which are the points with converging vectors. Other values of a ≥ .375 give identical results. Finally, I note that there is no incentive for one of the firms to undercut its rival’s profit by pricing at:${{{p}}_{{M}}} = {{{p}}_{{rM}}} - .5 - \varepsilon$ for some small ε > 0. This contrasts with the Bertrand analysis usually found, for example in D’Aspremont et al. (1979). That follows as p*(a) in (A.7) is monotone increasing and converges asymptotically to 0.5: thus undercutting equilibrium prices results in losses! QED

Proof of Lemma 4.2

Assuming p _U ≤ a, the Uniform pricing mail-order firm’s profit is

$$ {\Pi_U}\left( {{p_U},{p_{rU}}} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{\left( {{{a}} - {{{p}}_{{U}}}} \right)}}{2}\left( {{{{p}}_{{U}}} - \frac{1}{4}} \right)if\,{{{p}}_{{U}}} - {p_{rU}} \leqslant 0} \\{\left( {{{a}} - {{p}}} \right)\left( {\left( {2 + {p_{rU}}} \right)\left( {\frac{{16{{p}} + 2{p_{rU}} - 1}}{{16}}} \right) - \frac{{3{{{p}}^2}}}{2}} \right)if\,0 \leqslant {{{p}}_{{U}}} - {p_{rU}} \leqslant .5} \\{0\quad \quad \quad if\,{{{p}}_{{U}}} - {p_{rU}} \geqslant .5} \\\end{array} } \right. $$

(A.8)

In the first case, the mail-order firm serves the market alone as a monopolist, in the second, it shares the market with the retailer, and in the third it does not serve any customers.

Mail-order profit is negative for p _U  = 0, and zero when p _U = a, so if a positive profit exists then the maximum must be in the interior of the interval [0,a], and satisfy first-order conditions. But of course, profit may be negative at all points in the interval. A sufficient condition that assures that the profit in the interval has a single positive interior maximum is that the profit function has a one singularity on [0,a] and the second-order condition is negative. To find the mail-order reaction function to the retailer’s price, I compute the first derivatives for mail-order profit, assuming that p _U  ≤ a:

$$ \frac{{d{\Pi_U}\left( {{p_U},{P_{rU}}} \right)}}{{d{p_U}}} = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{8} + \frac{a}{2} - {p_U}\quad if\,0 \geqslant {P_{rU}} - {P_{rU}}} \\{\frac{{9p_u^2}}{2} + \frac{{\left( {1 + 2{p_{rU}}} \right)\left( {1 + 8a + 2{p_{rU}}} \right)}}{8} - {p_U}\left( {2 + 3a + 4{p_{rU}}} \right)\quad if\,0 \leqslant {p_U} - {p_{rU}} \leqslant .5} \\{0\quad if\,{p_U} - {p_{rU}} \geqslant .5} \\\end{array} } \right.. $$

(A.9)

The mail-order firm will be a monopolist if its monopoly uniform price is less than the retailer’s Mill price. The mail-order monopoly price can be easily derived by exploiting the first order condition, above, and is ${p_{uniform - monopoly}} = 0.125 + 0.5\,a$. But what if p _rU is less than this amount? In that case, the mail-order firm can choose to share the market or to serve no customers.

The middle equation above has a first order equation that can be solved:

$$ {p_U}\left( {{p_{rU}}} \right) = \frac{{\left( {4 + 6a + 8{p_{rU}} - \sqrt {{36{a^2} + 7{{\left( {1 + {p_{rU}}} \right)}^2} - 24a\left( {1 + 2{p_{rU}}} \right)}} } \right)}}{{18}}. $$

(A.10)

Second-order derivatives confirm this is a maximum. This expression is the mail-order firm’s profit maximizing price reaction to the retailer’s price as long as the implied solution is above the monopoly price, but below a. This can be guaranteed as long as: $0.125 + 0.5\,a \geqslant {p_{rU}} \geqslant \frac{{6a - 1}}{2}$. When ${p_{rU}} < \frac{{6a - 1}}{2}$, the solution to the mail-order optimization with respect to p _U is p _U (p _rU ) = a which implies that the mail-order has no sales and thus zero profit. Thus, the mail-order reaction function is:

$$ {p_U}\left( {{p_{rU}}} \right) = \left\{ {\begin{array}{*{20}{c}} {p * = 0.125 + 0.5\,a\;\quad if\quad {p_{rU}} \geqslant p * } \\{p** = \frac{{\left( {4 + 6a + 8{p_{rU}} - \sqrt {{36{a^2} + 7{{\left( {1 + {p_{rU}}} \right)}^2} - 24a\left( {1 + 2{p_{rU}}} \right)}} } \right)}}{{18}}if\quad p * \geqslant {p_{rU}} \geqslant \frac{{6a - 1}}{2}} \\{a\quad \quad \quad if\frac{{6a - 1}}{2} \geqslant {p_{rU}} \geqslant 0} \\\end{array} } \right.. $$

(A.11)

Similarly, I compute the retailer’s profit, derivative of profit with respect to its Mill price, and its price reaction as a function of the mail-order’s Uniform price choice. I assume that p _U and p _rU are both less than a.

$$ {\prod _{rU}}\left( {{p_U},{p_{rU}}} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{1}{2}{{\left( {a - {p_M}} \right)}^2}{p_M}\,if\;{p_U} - {p_{rU}} \leqslant 0} \\ {\frac{1}{2}\left( {2a - {p_U} - {p_{rU}}} \right)\left( {{p_U} - {p_{rU}}} \right){p_{rU}}\;if\;0 \leqslant {p_U} - {p_{rU}} \leqslant .5} \\ {0\quad \quad \quad \quad if\,{p_U} - {p_{rU}} \geqslant .5} \\ \end{array} } \right.. $$

(A.12)

The top term represents the retailer as a monopolist, the middle term is valid when the retailer and the mail-order share the market, and the bottom when the mail-order serves the market alone. I assumed that p _U  < a, so the middle function is always strictly positive in the interval [0, p _U ], and zero at the endpoints. It is important to note that the second derivative of the middle term is −2a + 3p _rU , thus the middle term is concave when ${p_{rU}} \leqslant \frac{{2a}}{3}$, and convex when ${p_{rU}} \geqslant \frac{{2a}}{3}$. A non-negative function` on a finite interval which is zero at the end points and initially concave and then convex implies that this function must be unimodal in p _rU , and thus, I will need to analyze one stationary point, which will be the maximizer of retailer profits. The derivative is needed to find the stationary point:

$$ \frac{{d{\Pi_{rU}}\left( {{p_U},{P_{rU}}} \right)}}{{d{p_{rU}}}} = \left\{ {\begin{array}{*{20}{c}} 0 \hfill & {if\,0 \geqslant {p_{rU}} - {p_U}} \hfill \\{{p_U}\left( {a - \frac{{{P_U}}}{2}} \right) - {p_{rU}}\left( {2a - \frac{{3{p_{rU}}}}{2}} \right)} \hfill & {if\,0 \leqslant {p_U} - {p_{rU}} \leqslant .5} \hfill \\{\frac{1}{2}{{\left( {a - {p_{rU}}} \right)}^2}{p_{rU}}} \hfill & {if\,{p_U} - {p_{rU}} \geqslant .5} \hfill \\\end{array} } \right., $$

(A.13)

and the reaction function becomes:

$$ {p_{rU}}\left( {{p_U}} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{a}{3}\,if\,{p_U} - {p_{rU}} \geqslant .5} \\{\frac{{\left( {2a - \sqrt {{4{a^2} - 6a{p_U} + 3p_u^2}} } \right)}}{3}\,if\,.5 \geqslant {p_U} - {p_{rU}}.} \\\end{array} } \right. $$

(A.14)

Second-order conditions confirm that the reaction function is appropriate. To find equilibrium prices, the fixed point equation:

$$ {p_U}\left( {{p_{rU}}(x)} \right) = x $$

(A.15)

is solved. The resulting equation yields a quartic equation that has a (messy) closed-form solution that Mathematica 6.03 can easily compute. Of the four roots, the appropriate one yields the solution $x(a) = {p_U}(a)$ that has increasing profit for all non-negative a. ¹¹ With this solution to the fixed point equation (A.15), retailer’s prices can be found with (A.14). Using the profit function (A.8) with these solutions results in the equilibrium profit function for the Uniform pricing mail-order company. Figure 7 plots the Uniform pricing mail-order’s profit as a function of a.

Although in equilibrium it must be true that the retailers earn strictly positive profit (as these firms use Mill pricing, and thus always have a local monopoly of sorts), the mail-order firm earns zero profit if the market is too small. At some positive value of a, equilibrium profit for the mail-order firm using Uniform pricing will be zero for all values of a less than or equal to this number. To find this value of a, the solution to the fixed point equation is solved for the specific value of a satisfying: ${p_U}\left( {{p_{rU}}(a)} \right) = a$. The solution is quickly found by numerical methods. This equation states that at the appropriate value of a, the Uniform pricing firm sets an equilibrium price of a, so that it serves no customers, and thus has zero profit. By numerical analysis, the solution is a=.1875. Since I can solve the fixed point equation ${p_U}\left( {{p_{rU}}(x)} \right) = x$ for all a>.1875, a Nash equilibrium with both firms serving customers is shown to exist. For values of a ≤ .1875, only the retailers serve customers using monopoly prices.

Uniqueness of the price equilibrium for any value of a is demonstrated by viewing the vector field diagram in Fig. 8 which shows the vector of the mail-order and retailers’ profit derivative with respect to its own price choice on the domain $\left( {{p_u},{p_{rU}}} \right) \in \left[ {0,a} \right] \times \left[ {0,a} \right]$. Each firm can guarantee itself at least zero profit by setting its own price to a. As the fixed point equation was constructed from the respective firm’s “reaction functions,” any solution to the fixed point equation must correspond to a Nash equilibrium. Figure 8 shows that there is only a single Nash equilibrium for each value of a>.1875. If multiple Nash equilibria existed, there would be multiple points with converging vectors but there is only one. Thus, the Nash equilibrium is unique. QED

Proof of Proposition 5.1

First, note that if Mill (Uniform) pricing is chosen in Nash equilibrium, then the payoff to the mail-order must equal its payoff on the Mill (Uniform) subgame. Thus, any Mill (Uniform) pricing Nash equilibrium must be essentially unique. Figure 9(a) shows the Nash equilibrium payoff to the mail-order firm on the Mill subgame, and the maximum payoff that the mail-order firm can guarantee itself on the Uniform subgame no matter the retailers’ price. In other words, the latter is the mail-order’s smallest possible payoff if it plays a Uniform strategy pricing to maximize its minimum payoff. Figure 9(b) shows the Nash equilibrium payoff to the mail-order firm on the Uniform subgame and the mail-order’s smallest possible payoff if it plays a Mill pricing strategy to maximize its minimum payoff. Denote the unique Nash equilibrium (conditioned on a) in the two subgames by $\left( {p_M^*,p_{rM}^*} \right)$ and $\left( {p_U^*,p_{rU}^*} \right)$, respectively. Viewing Fig. 9(a), below a < .547, the most the mail-order firm can guarantee itself on the Uniform subgame is less than equilibrium payoff on the Mill subgame. By Lemma 4.2, for all such a, a Nash equilibrium with Mill pricing exists on the entire game, but when a>.547 choosing Uniform prices always yields a higher return than playing Mill prices. Observing Fig. 9(b), for all a ≥ .382, the most the mail-order firm can guarantee itself on the Mill subgame is less than the equilibrium payoff on the Uniform subgame, and thus, the mail-order will never choose to play the Mill subgame. Therefore, a Nash equilibrium with Uniform pricing with positive demand exists for the entire game. Within the interval [.3821,.547] there are two essentially unique equilibria: the mail-order playing Mill or Uniform pricing are both Nash, outside this interval there is just one essentially unique equilibrium. In this interval the mail-order firm’s payoffs are different as they reflect the payoffs of play on different subgames as is shown in Fig. 7. QED