Dynamics and Stability in Retail Competition

Marcelo J. Villena and Axel A. Araneda∗

August 15, 2018

Faculty of Engineering & Sciences, Universidad Adolfo IbáñezAvda. Diagonal las Torres 2640, Peñalolén, 7941169, Santiago, Chile.

Retail competition today can be described by three main features: i) oligopolistic com-petition, ii) multi-store settings, and iii) the presence of large economies of scale. In thesemarkets, firms usually apply a centralized decisions making process in order to take fulladvantage of economies of scales, e.g. retail distribution centers. In this paper, we modeland analyze the stability and chaos of retail competition considering all these issues. Inparticular, a dynamic multi-market Cournot-Nash equilibrium with global economies anddiseconomies of scale model is developed. We confirm the non-intuitive hypothesis thatretail multi-store competition is more unstable that traditional small business that coverthe same demand. The main sources of stability are the scale parameter and the numberof markets.

Keywords: Multi-market Oligopoly, Cournot-Nash competition, Economies of Scale,Stability, Bifurcations, and Chaos.

1 IntroductionIn an oligopolistic setting under a Cournot scheme [1], the strategy of each economicplayer depends on its own quantity decision, and on its rival’s reaction. Puu was thefirst to explicitly show the complex dynamics of the oligopolistic setting under simpleassumptions (isoelastic demand function and constant marginal cost) for two and threeplayers [2–4]. This kind of analysis has grown significantly during the last decade in both,the mathematics and complex systems literature, as well as in the economically-orientedjournals.

Indeed, since the Puu’s approach, several games has been developed for the studyof the market stability, focusing on: different demand or price function [5, 6], numberof players [7, 8], behavioral assumptions (naive [9–11], versus adaptive [12, 13], boundedrationality [6, 14, 15] or heterogeneous expectations [16–19]). In terms of the cost functiondefinition, several developments has been proposed as well, as non-linear cost function[6, 13, 15, 19, 20], capacity constraints |[13, 21–23] and some spillover effects [24–27].

∗Email: axel.araneda@uai.cl, Tel.: +56 2 23311416

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Most works in this line of research have concentrated in single markets with linearproduction structures (i.e. assuming constant returns to scale). Nevertheless, oligopolis-tic competition today seems to present multi-market phenomena and, in some cases, theyshowcase important economies of scale, especially in the retail industry. Indeed, super-market chains and retailers of food, gasoline, supplies and services all compete for marketshare through multi-store formats over geographically separated markets. This localizedcompetition is presented in different levels: city, region, or country. In this context,companies segment their strategies, tailoring their selected outcome for different types ofconsumers and competitors, which vary by geographical location. On the other hand, onthe supply side, multi-market retailers usually try to take full advantages of their size, inother words, their economies of scale. For instance, through the development of distri-butions centers that attend most of the stores in an specific territory. Thus, as the coststructure of multi-market retails depends on the total volume of the produced goods, theindividual cost structure of each store is usually coupled with the whole business. It isimportant to point out that, this system of production, implies a centralized decisionsmaking process, which become in practice an extremely difficult task. Summing up, retailcompetition today can be described by three main features: i) oligopolistic competition,ii) multi-store setting, iii) the presence of economies of scale.

Applications of the Cournot scheme into the multi-market problem has been proposedbefore by economists, for example, in the case of international trade. Some of theseworks modeled the presence of economies of scale, for the domestic and foreign markets,considering the size (quantity produced) and other properties of firms [28–35]. Thus, forinstance in a work of Krugman [30], a multi-market Cournot model with economies ofscale was used to explain the successful performance of Japan as an exporting country atthe beginning of the 1980s.

In theoretical terms, the multi-market oligopoly framework was revisited and general-ized in the seminal paper by Bulow, Geanakoplos and Klemperer [36]. One of their mainremarks is that the presence of a multi-store firm in a market may affect the position inthe others for the presence of demand and/or supply spillovers. In the same line, Bern-heim and Whinston, [37], show that with scale economies, the multi-market contact mayproduce “spheres of influence” [38], that occurs when each of the multi-market competingfirms may be more efficient in some subset of these markets and less efficient in others(symmetric advantage) or when one firm is more efficient in all markets (absolute ad-vantage). Despite these multi-market analysis, this literature has focused mainly on thedemand side of the problem, not the supply side. Specifically, they refer to multi-marketcontact, when demands curves recognize substitution and complementarity of differentproducts.

In terms of the analysis of the dynamics of the multi-market Cournot problem, wefound only a few papers [39–42], focusing on different products and scope.

In this context, this research deals with the analysis of stability and chaos of multi-market competition in the presence of economies and diseconomies of scale, extending thisway the analysis of the dynamics of the oligopolistic competition. Thus, we model themain characteristics of the retail competition today, analysing the dynamics and stabilityof this particular dynamic system, and we compare these results with the stability analysisof traditional small business that cover the same demand, the classic Pu’s formulation.

The main hypothesis of the paper is that non-linear cost structures in multi-marketsetting are important sources of instability in the game outcome. Particularly, we study

2

the stability of a multi-market Cournot-Nash equilibrium with global economies of scale,that is, the scale level that is related to the total production of firms, in all markets,as opposed to local economies of scale presented at each store individually or linearproduction structures. In this setting, the internal organization of a firm may affect itsperformances over the markets and the global equilibrium [43]. For example, multi-marketfirms that buy their products in a centralized manner, storing them in a distributioncenter, to be redistributed afterwards to their retailers store in all markets usually operatethis way to obtain economies of scale in the process of buying and distribution. In thispaper, we assume this type of centralized structure where companies takes advantage oftheir size, under economies of scale, that allow them to decrease the cost structure [44].

This papers is organized as follows: in section 2 classical models of the Cournotproblem are described and extended to the multi-market framework. In the section 3 aMulti-market-Cournot problem is presented, considering interrelated cost structures andeconomies of scale. In section 4, the study of the stability of the system is addressed andgeneralized for the duopoly case. In the fifth section, the complex dynamics of the gamefor different numbers markets and values of the scale parameter are shown by path graph-ics and bifurcation diagrams. Finally, the main conclusions for this work are presented.

2 Baseline: Single market oligopoly modelsThe well-known Cournot-Theocharis model [45–48] proposed a Cournot Oligopoly modelwith inverse lineal demand function and constant marginal cost, that is:

P = a−∑

qi (2.1)

Ci = ciqi (2.2)

where i = 1, . . . , N identify the player, P is the price of the good, C is the total cost andq is the quantity.

The profit is obtained subtracting the revenues by total cost:

πi = qiP(q1, . . . , qN

)− ciqi

The optimization problem (maximization of profit) arrives to:∗qi= a− ci

2 −∑k 6=i

qk

2 (2.3)

With solution (Cournot-Nash equilibrium point):

∗qi=

a− ci +N(c− ci

)N + 1

3

with c = (1/N)∑

i ci

In order to transform the static game 2.3 into a dynamic one, the Cournot or naivestrategy is used (see Cournot [1] and Puu [2])1. Thus, the long run map as an iterativeprocess is given by:

qi(t+ 1) = a− ci

2 −∑k 6=i

qk(t)2 (2.4)

The dynamical system of N reaction functions defined by 2.4 has a stable equilibrium(fix point) for n ≤ 2 . For n = 3 the equilibrium is neutrally stable (stationary oscillations)and for n ≥ 4 the equilibrium becomes unstable.

A generalization of the Cournot-Theocharis problem was developed by Fisher [49].This research allows to work with increasing or decreasing returns to scale (i.e., economiesor diseconomies of scale). In order to get this result, we need to add a non-linear term tothe traditional linear cost function:

P = a−∑

i

qi

Ci = ciqi + d(qi)2

Thus, the profit takes the following form:

πi = qiP(q1, . . . , qN

)− ciqi − d

(qi)2 (2.5)

Some restrictions for avoiding non-negativity of outputs, price, profits and marginalcost are considered: c > 0, a ≥ c and d > −1/2.

Thus, the maximization of the profit (eq. 2.5) leads to the best response of the ith-firm:

∗qi=

a−∑k 6=i

qk − ci

2 (1 + d) (2.6)

Then, the Cournot-Nash equilibrium is given by:1In this strategy, it is assumed that a player at time of the decision making process, it takes into

account the rival’s previous output, e.g, managers use past information and supposes it doesn’t changeat this moment.

4

(a− ci

)(1 + 2d) +N

(c− ci

)4d2 + 2 (N + 2) d+N + 1

Hence, the naive dynamics takes the form:

qi(t+ 1) =

a−∑k 6=i

qk(t)− ci

2 (1 + d) (2.7)

Finally, this dynamical system becomes stable when (N − 3) /2 < d . Thus, if thereare two players the game has a stable equilibrium if d > −1/2.

The approaches revised above (Teocharis and Fisher), were design to model a single-market oligopoly problem (for example the rivalry between “mom-and-pop” stores). Inthis case, both the prices and the costs don’t depends upon the behavior of the players inother markets, because they did not consider the case of large corporations, with multipleoperations in various locations.

In the next section, and taking as baseline the models presented above, we will developa multi-market Cournot model with economies of scale that will allow us to describe themodern retail competition.

3 The ModelLet us consider a multi-market oligopoly where N single-product firms compete in Mmarkets. All markets are very far away, so there not exist arbitrage possibilities. Eachmarket has its own price (from a linear demand function). Then, if qi

j is the quantity ofthe ith company at the market j and Qj the market supply, the selling price at the jth

market is given by,

Pj = aj −∑

i

qij = aj −Qj (3.1)

We assume a centralized managerial structure, where the production costs dependsof the total outputs of each firm. Besides, due both their size and specialization thecompany can have economies of scale. So we have:

Ci = ci∑

j

qij + d

∑j

qij

2

= ciQi + d(Qi)2 (3.2)

with ci greater than zero. Depending of the value of d the companies operates undereconomies of scale (d < 0) or diseconomies of scale (d > 0). The allowable range for theparameter d will be analyzed later.

5

The square form of the production cost was used previously for other scholars in asingle-market context [6, 49–51]. However, for this multi-market scheme we use the non-linearity for to couple the costs and to enable the existence of economies (diseconomies)of scale. This is a realistic approach for the retails firms, because they produce or buy inlarge scale.

Under this cost structure, ci > 0 and dQimax > −ci/2 , where Qi

max is the maximumlevel of production of firm i. The theoretical maximum production (extreme case) isachieved when a company becomes a monopoly in all the markets (i.e. Qj = qi

j ,∀j). AsPj ≥ 0, we have

∑j aj > Qi

max, and then we have 2d > ci/∑

j aj .Thus the profit function of each firm depends upon each market price, the quantity

sold by the firm in that particular market, and the total cost of the firm, that considersthe sum of the global cost and the local cost:

πi =∑

j

qijPj − Ci (3.3)

=∑

j

qij

(aj − bj

∑i

qij

)− ciQi − d

(Qi)2 (3.4)

The managerial decision of each firm is to choose the quantity qij that maximizes its

profits. In other words, the ith company which produces a total output of Qi divided overthe M different markets according to the output vector Qi =

{qi

1, qi2, . . . , q

iM ,}needs to

fix the optimal allocation given by∗Qi=

{ ∗qi

1,∗qi

2, . . . ,∗qi

M

}, where each one of the

∗qi

j is asolution of theM×N simultaneous equations which comprising the first order conditionsof this game given by:

∂πi

∂qij

= aj −Qj − qij − ci − 2dQi = 0 (3.5)

Defining the residual market supply for i as Qij = Qj − qi

j , and the participation ofthe firm i in other markets different to j as Qi

j = Qi − qij , the equilibrium quantity take

the value:

∗qi

j=aj − Qi

j − ci − 2dQij

2 (1 + d) (3.6)

Clearly, the allocation decision depends on the decision of the other players on thismarket, and also depends on the participation of the firm in other markets (or the totalfirm supply). This results is consistent with the Cournot intuition and consistent toprevious results for the single-market problem2 [20, 49, 52].

2When Qi = 0

6

In order to keep the game on rails, we have that∗qi

j is a maximum if and only if d > −1.Also the marginal profit for zero output must be non-negative [49], so aj ≥ ci,∀i. Thisconditions joined with the previous restrictions arrives to:

d > − 12m (3.7)

The profit of each firm at the equilibrium point is given by:

πi =∑

j

( ∗qi

j

)2

+ d

∑j

∗qi

j

2 (3.8)

The Jacobian matrix at the equilibrium is given by:

J =

∂∗Q1

∂Q1∂∗Q1

∂Q2 · · ·∂∗Q1

∂QN

∂∗Q2

∂Q1∂∗Q2

∂Q2 · · ·∂∗Q2

∂QN

... . . . ...

∂∗QN

∂Q1∂∗QN

∂Q2 · · ·∂∗QN

∂QN

(3.9)

where each one of the N ×N entries of J is a M ×M block matrix, that represents thechange of the i-firm’s reaction functions with respect to the outputs of j, with:

∂∗Qi

∂Qk=

∂∗q

i

1∂qk

1

∂∗q

i

1∂qk

2· · · ∂

∗q

i

1∂qk

M

∂∗q

i

2∂qk

1

∂∗q

i

2∂qk

2· · · ∂

∗q

i

2∂qk

M... . . . ...∂∗q

i

M

∂qk1

∂∗q

i

M

∂qk2· · · ∂

∗q

i

M

∂qkM

(3.10)

According to our optimal outputs (Eq. 3.6), we have:

∂∗Qi

∂Qk={− 1

2(1+d)IM×M , for i 6= j

H , for i = j(3.11)

where I is the identity matrix and H is a zero-diagonal matrix with all the off-diagonalentries equal to −d/(1 + d).

7

4 Dynamic Analysis of the EquilibriumFor the dynamic modeling, we use naive expectations. Thus, the game is developed ondiscrete time as follow3:

qij(t+ 1) =

aj − Qij(t)− ci − 2dQi(t)

2 (1 + d) (4.1)

When there are two players competing over M different markets, the long-run mapproposed in 4.1 is defining for the following 2m equations:

q11(t+ 1) =

a1 − q21(t)− c1 − 2d

(q1

2(t) + q13(t) + . . .+ q1

M (t))

2 (1 + d) (4.2)

......

q1j (t+ 1) =

aj − q2j (t)− c1 − 2d

(q1

1(t) + . . .+ q1j−1(t) + q1

j+1(t) . . .+ q1M (t)

)2 (1 + d)

......

q1M (t+ 1) =

aM − q2M (t)− c1 − 2d

(q1

1(t) + q13(t) + . . .+ q1

M−1(t))

2 (1 + d)

q21(t+ 1) =

a1 − q11(t)− c2 − 2d

(q2

2(t) + q23(t) + . . .+ q2

M (t))

2 (1 + d)...

...

q2j (t+ 1) =

aj − q1j (t)− c2 − 2d

(q2

1(t) + . . .+ q2j−1(t) + q2

j+1(t) . . .+ q2M (t)

)2 (1 + d)

......

q2M (t+ 1) =

aM − q1M (t)− c2 − 2d

(q2

1(t) + q23(t) + . . .+ q2

M−1(t))

2 (1 + d)

The nontrivial Cournot-Nash equilibrium point for the previous set of equations (staticsolution) is given by:

3The dynamic proposed in 4.1 can lead to negative outputs without economic sense. Following [53],we will differentiate between two types of trajectories: admissible and feasible. Calling T the map definedby the 2m equations of the game, the set of admissible (S) and feasible points (F ) is defined respectivelyby:

S ={(

q11 , q1

2 , . . . , q1M , q2

1 , q22 , . . . , qM

2)∈ R2M

+ : T n(

q11 , q1

2 , . . . , q1M , q2

1 , q22 , . . . , qM

2)∈ R2M ∀n > 0

}F =

{(q1

1 , q12 , . . . , q1

M , q21 , q2

2 , . . . , qM2)∈ R2M

+ : T n(

q11 , q1

2 , . . . , q1M , q2

1 , q22 , . . . , qM

2)∈ R2M

+ ∀n > 0}

Then, the mathematical results are based on the admissible trajectories. However, in the economiccontext, only the feasible points will be considered.

8

∗qi

j=(aj − ci

)(1 + 2Md) + ck − ci

3 + (2M)2d2 + 8Md

+ 23M (aj − a)

(2Md2 + d

)3 + (2M)2

d2 + 8Md, d 6= − 3

2M ,− 12M(4.3)

Using the equations3.9, 3.10 and 3.11 the Jacobian matrix for the system is definedby:

J2xM =

Mtimes

0 − d1+d · · · − d

1+d

− d1+d 0 · · · − d

1+d... . . . ...

− d1+d − d

1+d . . . 0︸ ︷︷ ︸

− 12(1+d) 0 · · · 0

0 − 12(1+d) · · · 0

... . . . ...0 0 · · · − 1

2(1+d)

Mtimes− 1

2(1+d) 0 · · · 00 − 1

2(1+d) · · · 0... . . . ...0 0 · · · − 1

2(1+d)

0 − d1+d · · · − d

1+d

− d1+d 0 · · · − d

1+d... . . . ...

− d1+d − d

1+d 0

(4.4)

The characteristic equation of 4.4 takes the form:

(λ− 1

22d+1(1+d)

)m−1 (λ− 1

22d−1(1+d)

)m−1 (λ+ 1

22(m−1)d+1

(1+d)

)(λ+ 1

22(m−1)d−1

(1+d)

)= 0 (4.5)

Thus, we have four4 different eigenvalues:

λ1 = −2 (m− 1) d+ 12 (1 + d) (4.6)

λ2 = −2 (m− 1) d− 12 (1 + d)

λ3,...,m+1 = 2d+ 12 (1 + d)

λm+2,...,2m = 2d− 12 (1 + d)

The local stability of equilibrium is achieved only if each eigenvalue is within the unitcircle. According to 4.6 this is fulfilled when:

4If we fix M = 1, we have only the two first eigenvalues.

9

−0.4 −0.2 0 0.2 0.4 0.6123456789

1011121314151617181920

d

m

Figure 4.1: Stabililty zone (blue lines) for a duopoly depending of the number of marketsand the values for d.

− 1

2m < d <1

2 (m− 2) , m 6= 2

d > −14 , m = 2

(4.7)

By 4.7 is clear that stability of the duopoly game depends on both the scale parameter (d)and the number of markets (m). In the fig. 4.1 the relationship betweenm and d, in orderof to arrive to the stability of the game is shown. When m=1,2 the equilibrium becomesunstable only if d>-1/(2m). However, if m>3, we need to put an upper bound for thestability condition. We see that when the number of markets increases, the stability isachieved only if d tends to zero.

5 Numerical simulationsUsing numerical simulations we can see how the complex dynamics of the equilibriumdepends on the scale parameter (d), using a scheme of duopolists competing on threemarkets (2 × 3 game). The results of our model (blue) will be compared with the basemodel (red) proposed by Fisher (see section 2) under identical parameter values.

In the figs. 5.1 and 5.2 we show the dynamic of the quantity allocated by player 1 inmarket 1 according to several values of d.

For d = −0.1 and d = 0.2 (figs. 5.1b and 5.2a) we have that the equilibrium undernaive expectations reaches the equilibrium in both models.

When d = 0 (fig. 5.1c), the models no longer have advantages/disadvantages ofscale production, being both the same schemes that the Cournot-Teocharis approach (seesection 2).

10

As seen in the fig. 5.2b, for the critical point d = 1/2, the dynamic goes to station-ary oscillations (neutrally stable) for the retail competition while for the single-marketapproach arrives quickly to the equilibrium .

For the case of figs. 5.1a and 5.2c the dynamic of the Fisher model remains stable butin our model lead to a chaotic behavior.

The figure 5.3 show the complex dynamic of the the quantity q11 by means of bifur-

cation diagrams, using values of d from the interval [−0.17; 0.52] . The decentralizedmodel shows stability of the equilibrium for all the values examined. By contrast, in ourmodel the chaos is achieved outside the critical points (d < −1/6 and d > 1/2, as it waspredicted by the analytical results of the duopoly case, see eq. 4.7). In the stability zone(fig. 5.3b), we can observe how the centralized managerial decision takes advantage interms of production when the company operates under economies of scale in relation tothe disaggregated model; while at diseconomies of scale, the production performance ofthe local model are less affected than the multi-market scheme.

6 ConclusionsIn this work, we analyze the stability of a multi-store retail competition model. Specifi-cally, we model an oligopoly system with multi-market competition. The model considersthe impact of the firm’s economies or diseconomies of scale. In one extreme, the multi-store retail maintain a centralized decision making process, optimizing global economiesof scale due to its global size. On the other hand, we have local oligopolistic competition,where the same demand is served by different firms that compete in only one market.Our model confirms the fact that economies and diseconomies of scale make the Cournotequilibrium very unstable for certain values of the scale parameter of the producers. Ad-ditionally, the number of markets, as expected, tends to contribute to this instability.One very interesting further research is to expand the multi-market oligopolistic modelto a multi-product setting.

7 AcknowledgmentsAid from the Fondecyt Program, project Nº 1131096, is grateful acknowledged by MarceloVillena. Axel Araneda thanks as well Faculty of Engineering & Science of the UniversidadAdolfo Ibáñez for financial support.

References[1] Antoine-Augustin Cournot. Recherches sur les principes mathématiques de la théorie

des richesses. chez L. Hachette, Paris, 1838.

[2] Tönu Puu. Chaos in duopoly pricing. Chaos, Solitons & Fractals, 1(6):573–581,1991.

11

2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

600d=−0.20

t

q1 1(t

)

s2 4 6 8 10 12 14 16 18 20

0

100

200

300

400

500

600d=−0.20

t

q1 1(t

)

(a) d = −0.2

2 4 6 8 10 12 14 16 18 2030

40

50

60

70

80

90

100d=−0.10

t

q1 1(t

)

2 4 6 8 10 12 14 16 18 2030

40

50

60

70

80

90

100d=−0.10

t

q1 1(t

)

(b) d = −0.1

2 4 6 8 10 12 14 16 18 2030

35

40

45

50

55

60

65

70

75d=0.00

t

q1 1(t

)

2 4 6 8 10 12 14 16 18 2030

35

40

45

50

55

60

65

70

75d=0.00

t

q1 1(t

)

(c) d = 0

Figure 5.1: Dynamic of q11 (solid line) and Cuornot equilibrium (dots), for different values

of d, under the proposed retail competition (blue) and the Fisher approach (independentsellers) (red); with a1 = 200, a2 = 150, a3 = 100, c1 = 20, c2 = 40.

12

2 4 6 8 10 12 14 16 18 2030

35

40

45

50

55

60

65d=0.20

t

q1 1(t

)

2 4 6 8 10 12 14 16 18 2030

35

40

45

50

55

60

65d=0.20

t

q1 1(t

)

(a) d = 0.2

2 4 6 8 10 12 14 16 18 2025

30

35

40

45

50d=1/2

t

q1 1(t

)

2 4 6 8 10 12 14 16 18 2025

30

35

40

45

50d=1/2

t

q1 1(t

)

(b) d = 0.5

2 4 6 8 10 12 14 16 18 2015

20

25

30

35

40

45

50

55

60d=0.55

t

q1 1(t

)

2 4 6 8 10 12 14 16 18 2015

20

25

30

35

40

45

50

55

60d=0.55

t

q1 1(t

)

(c) d = 0.55

Figure 5.2: Dynamic of q11 (solid line) and Cuornot equilibrium (dots), for different values

of d, under the proposed retail competition (blue) and the Fisher approach (red); witha1 = 200, a2 = 150, a3 = 100, c1 = 20, c2 = 40.

13

−0.17 −0.168 −0.166 −0.164 −0.162 −0.16

0

2

4

6

8

10

12

14

16

18

20x 10

4

d−0.17 −0.168 −0.166 −0.164 −0.162 −0.16

78.1

78.2

78.3

78.4

78.5

78.6

78.7

78.8

78.9

79

79.1

d

q1 1

(a) d ∈ [−0.17;−0.16]

−0.1 0 0.1 0.2 0.3 0.40

50

100

150

200

250

300

350

d

q1 1

(b) d ∈ [−0.16; 0.45]

0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.5246.5

47

47.5

48

48.5

49

d

q1 1

(c) d ∈ [−0.45; 0.52]

Figure 5.3: Bifurcation diagrams for the quantity of the player 1 over market 1 using theproposed retail competition model (blue) and the Fisher approach (red) with a1 = 200,a2 = 150, a3 = 100, c1 = 20 and c2 = 40.

14

[3] Tönu Puu. Complex dynamics with three oligopolists. Chaos, Solitons & Fractals,7(2):2075–2081, 1996.

[4] Tönu Puu. The chaotic duopolists revisited. Journal of Economic Behavior & Or-ganization, 33(3):385–394, 1998.

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