Local competition, entry, and agglomeration

Abstract

This paper analyzes competition between two spatially differentiated multi-product retailers who encounter entry from a low-cost discounter. We assess how entry affects the pricing of the incumbent stores and the role played by the location of the entrant. Our primary objective is to identify how traditional retailers respond to new forms of low-cost retailing. Results show that post entry, the prices for some products are higher than the pre entry. However, which product prices increase depends on the incumbent’s location. Contrary to conventional wisdom, we find that the store closer to the entrant is better off compared to the incumbent located further away. We empirically demonstrate the main workings of our theory using sales data from several grocery stores that saw entry by discount stores in their trading areas.

Appendix

This appendix contains proofs of Propositions 1, 2, & 3 as well as of Lemmas 1 & 2.

1.1 Proof of Proposition 1

The proof follows the argument given in Lal and Matutes (1989). That argument applied to the present context is provided in a Supplemental Appendix.

1.2 Proof of Lemma 1

(i) Since Big Basket convenience shoppers visit store C to buy product 3, the assumption that store C has the lowest price for product 2 implies that these consumers will buy it there while buying product 3. Therefore, they will visit exactly one more store (either A or B) for product 1. (ii) Since no consumer will visit a number of stores greater than the number of products demanded, it is sufficient to show that Small Basket convenience shoppers will not visit two stores. Define a two-store shopping plan as a pair (s ₁,s ₂), ${s_i} \in \left\{ {A,B,C} \right\}$, ${s_1} \ne {s_2}$, which prescribes the consumer to buy product i from store s _i . Since C does not offer product 1, we have four possible two-store shopping plans: (A,B), (B,A), (A,C) or (B,C). First, we show that shopping plans (A,B) and (B,A) cannot be an equilibrium shopping plan for Small Basket convenience shoppers because they are dominated by either (A,C) or (B,C). Denote c _x (s ₁, s ₂) as the cost incurred by a consumer located at x carrying out the shopping plan inclusive of price. Since p _1A ,p _1B >K, then for any x,

$$ \begin{array}{*{20}{c}} {\min \left\{ {{c_x}\left( {A,B} \right),{c_x}\left( {B,A} \right)} \right\} = \min \left\{ {{p_{{1A}}},{p_{{1B}}}} \right\} + \min \left\{ {{p_{{2A}}},{p_{{2B}}}} \right\} + t + s} \\ { > \min \left\{ {{p_{{1A}}},{p_{{1B}}}} \right\} + {p_{{2C}}} + t + s} \\ { = \min \left\{ {{c_x}\left( {A,C} \right),{c_x}\left( {B,C} \right)} \right\}.} \\ \end{array} $$

Next, we exclude (A,C). Suppose this plan were part of an equilibrium. Then we must have p _2C + s < p _2A . But then no one will buy product 2 at A. Hence, there is a profitable deviation for A to lower its price to ${p_{{2A}}} = {p_{{2C}}} + s = K + s$ and receive positive profits. It remains to show that (B,C) cannot arise in equilibrium. Suppose prices p _1A , p _2A , p _1B , p _2B yielded (B,C) as the optimal shopping plan for some Small Basket convenience shoppers. We show that such pricing behavior yields a contradiction. The hypothesis implies that p _1B < p _1A . And since p _2C < p _2A , no value buyers will patronize A. A Small Basket convenience buyer located at x will buy the bundle from A if it is cheaper then buying the bundle from B

$$ {p_{{1A}}} + {p_{{2A}}} + xt + s < {p_{{1B}}} + {p_{{2B}}} + \left( {1 - x} \right)t + s $$

and cheaper than (B,C)

$$ {p_{{1A}}} + {p_{{2A}}} + xt + s < {p_{{1B}}} + {p_{{2C}}} + t + 2s. $$

Let x ₂ and x _2A be defined, respectively, as satisfying the above two conditions with equality, then

$$ {x_2} = \tfrac{1}{{2t}}\left[ {\left( {{p_{{1B}}} + {p_{{2B}}}} \right) - \left( {{p_{{1A}}} + {p_{{2A}}}} \right) + t} \right]\,{\text{and}}\,{x_{{2A}}} = \tfrac{1}{t}\left[ {\left( {{p_{{1B}}} + {p_{{2C}}}} \right) - \left( {{p_{{1A}}} + {p_{{2A}}}} \right) + t + s} \right]. $$

Hence, Small Basket convenience shoppers with x<x ₂ prefer the bundle at A to the bundle at B and with x<x _2A prefer the bundle at A to (B,C). Similarly, Small Basket convenience shoppers at x prefer the bundle at B over (B,C) if ${p_{{1B}}} + {p_{{2B}}} + \left( {1 - x} \right)t + s < {p_{{1B}}} + {p_{{2C}}} + t + 2s,$ or equivalently, if $x > {x_{{2B}}} = \tfrac{1}{t}\left[ {{p_{{2B}}} - {p_{{2C}}} - s} \right]$. We observe that ${x_2} = \left( {{x_{{2A}}} + {x_{{2B}}}} \right)/2$, which implies that x ₂ lies between x _2A and x _2B . Thus, there are two possibilities to consider. First, x _2B < x ₂ < x _2A implies that all Small Basket convenience shoppers prefer buying the bundle at either A or at B to shopping around (i.e. to (B,C), which contradicts the hypothesis that (B,C) is optimal for some Small Basket consumers. Hence, x _2A < x ₂ < x _2B . Big Basket convenience shoppers located a x will shop at A (for product 1) and C (for products 2 and 3) if it is cheaper than shopping at B (for product 1) if

$$ {p_{{1A}}} < {p_{{1B}}} + \left( {1 - x} \right)t, $$

or equivalently if $x < {x_1} = \left( {{p_{{1B}}} - {p_{{1A}}} + t} \right)/t$. Thus profits to each store can be written as

$$ \begin{array}{*{20}{c}} {{\pi_A} = \alpha \left( {1 - \beta } \right)\left( {{p_{{1A}}} + {p_{{2A}}} - 2K} \right){x_{{2A}}} + \alpha \beta \left( {{p_{{1A}}} - K} \right){x_1},\,{\text{and}}} \\ {{\pi_B} = \alpha \left( {1 - \beta } \right)\left( {{p_{{1B}}} + {p_{{2B}}} - 2K} \right)\left( {1 - {x_{{2B}}}} \right) + \left[ {\left( {1 - \alpha } \right) + \alpha \beta \left( {1 - {x_1}} \right) + \alpha \left( {1 - \beta } \right)\left( {{x_{{2B}}} - {x_{{2A}}}} \right)} \right]\left( {{p_{{1B}}} - K} \right).} \\ \end{array} $$

Prices must be mutually optimal for the two firms. Therefore, interior optima must satisfy the first order conditions for profit maximization, which imply the following prices:

$$ \begin{gathered} \begin{array}{*{20}{c}} {{p_{{1A}}} = K + \frac{{1 + \alpha }}{{3\alpha }}t - \frac{{1 - \beta }}{3}s,} \hfill & {{p_{{2A}}} = K + \frac{s}{2},} \hfill \\ {{p_{{1B}}} = K + \frac{{2 - \alpha }}{{3\alpha }}t - \frac{{1 - \beta }}{3}s,} \hfill & {{\text{and}}\,{p_{{2B}}} = K + \frac{{t + s}}{2}.} \hfill \\ \end{array} \hfill \\ \begin{array}{*{20}{c}} {} \hfill & {} \hfill \\ \end{array} \hfill \\ \end{gathered} $$

It follows that

$$ {x_{{2B}}} - {x_{{2A}}} = \frac{{\left( { - 2 - \alpha } \right)t - \alpha \left( {5 + \beta } \right)s}}{{6\alpha \beta t}} < 0, $$

or equivalently x _2B < x _2A , which is a contradiction. Note that if $H > \tfrac{1}{4}\left( {5K + 3L} \right) + t + \tfrac{3}{2}s$, then the following corner solution is mutually optimal for both firms:

$$ p_{{1A}} = \frac{1} {2}{\left( {K + L + t} \right)},\;p_{{2A}} = K + \frac{s} {2}, $$

$$ p_{{1B}} = L,\;p_{{2B}} = K + \frac{{t + s}} {2}. $$

This leads to ${x_{{2B}}} - {x_{{2A}}} = \left( {K - L - 2s} \right)/t$, which is less than 0 since L>K. Hence, x _2B <x _2A , again a contradiction. Q.E.D.

1.3 Proof of Lemma 2

Suppose product 1 is cheaper at A: p _1A  < p _1B . Then all Big Basket consumers will shop at retailer C for good 3. While there, these consumers also buy product 2 at C. Furthermore, all Big Basket consumers buy product 1 from A: Big Basket value buyers since it has the lowest price; convenience shoppers since they, already at retailer C, do not benefit from incurring costs traveling to B. Now consider Small Basket consumers. Small Basket convenience shoppers shop at A and C for products 1 and 2, respectively. Small Basket Convenience consumers shop either at A or at B and buy both products at one of these stores. (Recall that convenience shoppers visit no more than one store for two products, by Lemma 3.) A Small Basket convenience shopper located at x will shop at A if

$$ x < {x_S} \equiv \frac{{\left( {{p_{{1B}}} + {p_{{2B}}}} \right) - \left( {{p_{{1A}}} + {p_{{2A}}}} \right)}}{{2t}}. $$

(As noted previously, these consumers make their retailer choice based solely on the price of the bundle of the items.) The consumer behavior described above generates the following profit functions for each retailer:

$$ {\pi_A} = \left( {1 - \alpha } \right)\left( {{p_{{1A}}} - K} \right) + \alpha \beta \left( {{p_{{1A}}} - K} \right) + \alpha \left( {1 - \beta } \right)\left( {{p_{{1A}}} + {p_{{2A}}} - 2K} \right){x_S} $$

$$ {\pi_B} = \alpha \left( {1 - \beta } \right)\left( {{p_{{1B}}} + {p_{{2B}}} - 2K} \right)\left( {1 - {x_S}} \right). $$

We now show that there exists a profitable deviation whenever p _1A < p _1B . First suppose p _1A > K. Then B can always lower its price on product 1 and raise its price on product 2 so that p _2A + p _2B does not change, but K < p _2A < p _1A . In such a deviation, retailer B retains the same Small Basket convenience shoppers and steals all value buyers buying product 2. Hence, this is a profitable deviation for B. Next, suppose p _1A  = K <  p _1B . Then clearly A can raise its price on product 1 slightly and lower its price on product 2 while preserving the sum p _1A +p _2A . This is profitable since it retains the same customers, but extracts positive revenues from its value buyers. Hence, it is not possible to have an equilibrium with p _1A  < p _1B . Finally, we consider the case when p _1A  = p _1B and show there always exists a profitable deviation. Note that this implies the following profits for the two retailers:

$$ {\pi_A} = \frac{{\left( {1 - \alpha } \right)}}{2}\left( {{p_{{1A}}} - K} \right) + \alpha \beta \left( {{p_{{1A}}} - K} \right){x_B} + \alpha \left( {1 - \beta } \right)\left( {{p_{{1A}}} + {p_{{2A}}} - 2K} \right){x_S} $$

$$ {\pi_B} = \frac{{\left( {1 - \alpha } \right)}}{2}\left( {{p_{{1B}}} - K} \right) + \alpha \beta \left( {{p_{{1B}}} - K} \right)\left( {1 - {x_B}} \right) + \alpha \left( {1 - \beta } \right)\left( {{p_{{1B}}} + {p_{{2B}}} - 2K} \right)\left( {1 - {x_S}} \right), $$

where ${x_i} \in \left( {0,1} \right)$ is the fraction of consumers with basket size i=B,S shopping at A. If p _1A  = p _1B  = K, then either retailer, say A, can earn more profits by slightly raising its price on product 1 since it would loose no revenue from the value segment and improve revenues from the convenience segment. If p _1A  = p _1B  = K then either retailer, say A, can be profitable by lowering p _1A by a small amount (some $\varepsilon > 0$, sufficiently small) while raising p _2A so that p _1A +p _2A remains constant since it experiences a discrete gain in value buyers and no loss from the convenience shoppers. Q.E.D.

1.4 Proof of Proposition 2

(i) The first order conditions of (1) with respect to (p _1A , p _2A ) and (2) with respect to (p _1B , p _2B ) imply interior maxima, as presented in part (i). To validate that these prices are, in fact, an equilibrium, we verify that that the outcome is

• (a) consistent with our assumptions on consumer behavior

• (b) not subject to profitable deviations by either retailer, for all α and β sufficiently large.

(a) The conditions $p_{{1B}}^{*} \leqslant L$ and $p_{{1B}}^{*} < p_{{1A}}^{*}$ were assumed in our formulation of profit functions (1) and (2). The first is implied by Assumption 1 if $\alpha > \tfrac{1}{{1 + \beta /4}}$. The second holds if and only if $\alpha > \tfrac{1}{{1 + \beta }}$. Since ${\alpha_{{AE}}} > \tfrac{1}{{1 + \beta /4}} > \tfrac{1}{{1 + \beta }}$, both conditions hold for α > α _AE . Finally, Assumption 3 ensures positive demand for product 2 at retailer A from Small basket convenience shoppers since $p_{{2A}}^{*} \leqslant K + s$. Hence, prices $\left( {p_{{1A}}^{*},p_{{2A}}^{*}} \right)$, $\left( {p_{{1B}}^{*},p_{{2B}}^{*}} \right)$ are consistent with our assumptions on consumer behavior. (b) To check for profitable deviations, first note that the profits at these prices can be expressed as in (4) and (5). Local deviations cannot be profitable by the maximization conditions on (1) and (2). We now check “jump” deviations. First consider a deviation by B with some ${\tilde{p}_{{1B}}} > p_{{1A}}^{*}$. In this case, retailer B looses all value buyers as well as all Big Basket Convenience consumers (x _B  > 1). This leaves B with profits

$$ {\tilde{\pi }_B} = \alpha \left( {1 - \beta } \right)\left( {{{\tilde{p}}_{{1B}}} + {{\tilde{p}}_{{2B}}} - 2K} \right)\left( {1 - {{\tilde{x}}_S}} \right), $$

where, ${\tilde{x}_S} = \left[ {\left( {{{\tilde{p}}_{{1B}}} + {{\tilde{p}}_{{2B}}}} \right) - \left( {p_{{1A}}^{*} + p_{{2A}}^{*}} \right) + t} \right]/\left( {2t} \right) < \tfrac{1}{2}$. This deviation is not profitable, since

$$ \begin{gathered} {\max_{{{{\tilde{p}}_{{1B}}},{{\tilde{p}}_{{2B}}}}}}{{\tilde{\pi }}_B} = \alpha \left( {1 - \beta } \right)\left( {t + 2K} \right) \\ = \alpha \left( {1 - \beta } \right)\left( {p_{{1B}}^{*} + p_{{2B}}^{*} - 2K} \right)\left( {1 - {{\bar{x}}_S}} \right) < \pi_B^{*}\;, \\ \end{gathered} $$

where the inequality can be seen by comparing this expression with (2) at $\left( {p_{{1B}}^{*},p_{{2B}}^{*}} \right)$. Now suppose retailer A deviates by undercutting B for product 1. That is, consider the deviation by A with ${\tilde{p}_{{1A}}} = p_{{1B}}^{*} - \varepsilon$, for small $\varepsilon > 0$. Any such deviation is most profitable if A sets ${\tilde{p}_{{2B}}}$ in order to maintain the bundled price ${\tilde{p}_{{1A}}} + {\tilde{p}_{{2A}}} = t + 2K$. This yields profits of in the limit, with $\varepsilon \downarrow 0$.

$$ \begin{array}{*{20}{c}} {{{\tilde{\pi }}_A} = \alpha \beta \left( {{{\tilde{p}}_{{1A}}} - K} \right) + \alpha \left( {1 - \beta } \right)\left( {{p_{{1A}}} + {p_{{2A}}} - 2K} \right){{\tilde{x}}_S} + \left( {1 - \alpha } \right)\left( {{{\tilde{p}}_{{1A}}} - K} \right)} \\ { = \left[ {\left( {\frac{{2\left( {1 - \alpha } \right)}}{{3\alpha \beta }} + \frac{1}{3}} \right)\left( {\alpha \beta - \alpha + 1} \right) + \frac{{\alpha \left( {1 - \beta } \right)}}{2}} \right]t\;.} \\ \end{array} $$

Using the above and comparing with the profits in (4), we can express the direct gain from deviation:

$$ \pi_A^{*} - {\tilde{\pi }_A} = \frac{{{\alpha^2}\left( {{\beta^2} + 5{\beta^2} - 5} \right) + 5\alpha \left( {2 - \beta } \right) - 5}}{{9\alpha \beta }}t $$

(A.8)

For any $\beta \in \left( {0,1} \right)$, (A.8) is positive for α >  α _AE (β). (ii) follows from direct substitution of equilibrium prices into (3). Q.E.D.

1.5 Proof of Proposition 3

Consider the difference in retailer’s equilibrium profits, as expressed in (4) and (5):

$$ \pi_A^{*} - \pi_B^{*} = \frac{{{\alpha^2}\left( {\beta - 1} \right) + 2\alpha - 1}}{{3\alpha \beta }}t. $$

This difference is positive for $\tfrac{1}{{1 + \sqrt {\beta } }} < \alpha < \tfrac{1}{{1 - \sqrt {\beta } }}$. These conditions, however, are implied by the assumed conditions in Proposition 2. Q.E.D.