Quantity competition in product-variety networks

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Quantity Competition in Product-Variety Networks



Abstract

competing firm modeling network

The aim of this paper is to show how pricing in Cournot competition affects the market in network models. The model presented in the article Uschev and Zenou (2018) was chosen as the network model. The main innovations of this model are the combination of two main approaches to modeling networks - spatial and monopolistic. It is also worth noting the modification of the Bonahich centrality formula, which allows us to take into account the influence of the number of competing firms on each other. In the process, the Uschev and Zenou (2018) model was solved in a Cournot competition. Optimal prices and output were obtained, and the influence of network parameters such as the spatial discount factor and the rescaled substitutability parameter on the equilibrium price was described. In addition, the option of a model without a network effect, as well as public welfare, was considered. The most important element of the work is the proof that for small values of the degree of the firm's node, the prices for Cournot are higher than for Bertrand.



1. Introduction

Networks or social networks are an effective tool for analyzing social interaction within groups. Social interactions on networks are formalized through graphs, where the vertices of the graph are agents (individuals / organizations), and the edges of the graph reflect the relationship between the agents.

In the context of the presented work, pricing in social networks will be considered. A large role in the formation of product prices is played by the consumption of this product among economic agents. In the case of networks, social interaction between agents is expressed in the exchange of information about the consumed product, which includes the price and quality of the product. The information obtained allows consumers to benefit from network externalities, in turn, the task of the company providing the product is to maximize profits through pricing.

The aim of this paper is to show how pricing in Cournot competition affects the market in network models. The model presented in the article Uschev and Zenou (2018) was chosen as the network model. According to Ushchev and Zenou (2018) approach to modeling pricing on ad networks comes down to two main approaches:

1) Spatial, which was presented in the Hotteling article(1929) and later modified Lancaster (1966)

2) Monopolistic, proposed by Chamberlin (1933) and in subsequently formalized in Spence (1976) and Dixit and Stiglitz (1977)

Despite the highlighted approaches to modeling in the article Ushchev and Zenou (2018) has proposed a model that combines data approaches. The key features of this article are that the authors we moved away from the standard definition of a network and characterized vertices as the ideal position of the agent within the network.

It is known that Bertrand prices competition is more efficient than Cournot quantity competition. However network structure is a flexible tool that could generate various outcomes that are impossible in basic monopolistic competition. Majority of articles on network competition models never consider Cournot competition outcomes. The aim of my work is to show how pricing in Cournot competition affects the market in network models.



2. Related literature

The literature on microeconomic approaches to pricing on social networks can be divided as follows:

1) the approach to modeling is spatial or monopolistic. [Hotteling(1929); Lancaster (1966); Chamberlin (1933); Spence (1976); Dixit and Stiglitz (1977); Ushchev and Zenou (2018)]

2) on the influence of network externalities - positive or negative. [Jullien (2001); Katz and Shapiro (1985); Banerji and Dutta (2005); Bloch and Querou (2008)]

3) the ratio of agents to the consumption of neighbors-positive or negative. [Ghiglino and Goyal (2008); Ballester, Calvу-Armengol and Zenou (2006); Tan (2006), Bloch and Querou (2008); Mookherjee, Napel, and Ray (2008).]

4) by the extent to which agents possess information about networks - complete or incomplete. [Hendricks,Piccone and Tan (1995); Sundararajan (2006); Galeotti et al. (2006); Jackson and Yariv (2005), Galeotti and Vega-Redondo (2006), Weinstein and Yildiz (2007);]

The article Katz and Shapiro (1985) was the basis for a large number of works on positive network externalities. In their work, Katz and Shapiro described the condition for the existence of a network effect in a situation where the utility that an individual receives from consumption depends on the number of other agents consuming a product of the same brand, or a similar (compatible) product.

Pricing on networks can be considered from the point of view of models with positive network externalities. In Jullien (2001), author considers a case in which the reputation of firms is taken into account, which is formed by limiting consumer strategies, as well as the use of price discrimination, which leads to consumers not being able to choose the only optimal firm, thereby increasing competition between firms. Adding such conditions leads to mixed results - on the one hand, the model becomes representative, the average equilibrium price decreases (due to the fact that consumers can not choose the only optimal firm for themselves), on the other hand, the market becomes less stable.

Banerjee and Dutta (2005) model a situation in which interaction between groups of agents is via a graph, where each vertex represents a group of agents, and the edges of the graph characterize two-way interaction between groups. The work of Banerjee and Dutta (2005) challenges the conclusions of Katz and Shapiro (1985), suggesting that the advantages of the network are limited only among groups of related agents. The analysis also shows that some forms of networks do not allow segmentation, in which firms divide the market and make a positive profit.

The consumption of agents in the network is not always positively reflected (perceived) by neighboring agents. In Ghiglino and Goyal (2008), the authors analyzed how welfare, consumption, and equilibrium prices change when agents have private funds and trade in the market, resulting in an increase in the consumption of one agent negatively affecting the utility of another. The authors determined that a parameter such as vertex centrality (an indicator that determines the number of edges that the vertex in question connects to) directly affects pricing within the network and the amount of good consumed by the agent. The more central the node (the agent), the higher its consumption and lower the cost of consumption itself, or, the further away from the center of the node, the higher its prices for good and the less its consumption. The background used in Ghiglino and Goyal (2008) contrasts with the work of Kuhn et al. (2008) and Luttmer (2005), which showed that the positive shock received by households (agents) does not have a negative effect on agents who have not experienced a positive shock.

The article Ghiglino and Goal (2008), is related to the works of Ballester, Calvу-Armengol and Zenou (2006), Tan (2006), Bloch and Querou (2008); Mookherjee, Napel, and Ray (2008). Tan (2006) examines the influence of social networks on the General equilibrium model and considers specific forms of networks, such as the star shape, while Ghiglino and Goyal consider arbitrary networks.

Ballester, Calvу-Armengol and Snow (2006) consider a game on a network and deduce the relationship between Nash equilibrium and network centrality. The key point of this article is how the structure of the social network and heterogeneity of funds affect pricing and the utility received by agents. Ballester, Calvу-Armengol and Zenou concluded that the Nash equilibrium in such games is proportional to Bonacich centrality, meaning that the aggregate equilibrium increases with increasing network density and size. At the same time, Ghiglino and Goyal (2008) argue that the usual centrality of the graph also determines the equilibrium in the game. Here it is worth clarifying that bonahich's centrality takes into account the number of connections that not only the c qentral agent has, but also neighboring agents associated with It.

In Bloch and Querou (2008), authors study the optimal discriminatory prices of firms for consumers who face price externalities relative to their neighbors. In their work the authors come to two main conclusions:

-subject to positive consumer externalities, prices are higher at the nodes that have a greater degree. In other words, the agent at the top with the highest degree has a greater demand for the product.

-provided relative to price externalities, prices are higher in the nodes connected with vertices with the smallest degree. In this case, firms use uniform prices for the entire network

Mookherjee, Napel, and Ray (2008) study a model in which families spend a portion of their income on educating children who will earn wages in the future. The incentive to invest in human capital depends on the average salary in the local social network. The main results obtained by the authors are reduced to the existence of an equilibrium in which communities with low and high human capital arise as a result of complementarity - in the acquisition (exchange) of human capital between such communities, by comparing expected wages.

In the works described above, agents had complete information, thus benefiting from the entry of new agents into the network. Hendricks, Piccone and Tan (1995) investigated a situation in which a monopoly firm dictates the price of air tickets between any pair of cities depending on the number of passengers, while passengers do not have information about the price of tickets until a certain date. In the article presented, the authors built a network model in which the graph vertices were connected to each other and were cities, so the network structure and prices were endogenous. The authors concluded that equilibrium prices are formed when the network size is equal to n-1 (where n is the number of cities). This conclusion is true under the condition that the ticket prices are formed depending on the distance between the cities

In Sundararajan (2006), agents interact with each other within a small (local) network, they are interested in joining new agents in the network, but do not directly benefit from this, but the decisions of two unrelated agents A and B affect the entire network as a whole. The author concludes that symmetric Bayes-Nash equilibria can be ranked by Pareto depending on the probability of accepting a new agent into the local network. In other words, the probability of purchasing a product in such a game depends on the number of agents accepted into the local network.

When considering pricing in terms of incomplete information, it is worth paying attention to the article by Galeotti et al. (2006). In the model of Galeotti et al. (2006) agents have partial information about the network structure, that is, the agent only knows about its own number of connections in the network (its own degree). The authors created a unique model of incomplete information, which examines the effects of positive and negative externalities, as well as assesses the impact of strategic additions and strategic replacements. The conclusions of this article are comparable to Sundararajan (2006), namely that the decision to purchase a product is proportional to the number of agents within the network. In addition to the considered works on games on networks with incomplete information, it is worth paying attention to the following articles: Jackson and Yariv (2005), Galeotti and Vega-Redondo (2006), Weinstein and Yildiz (2007).



3. Model setup

In presented model there exists firms and consumers. Firms produces goods within geographic area that is presented via graph. Consumer solve optimization problem by choosing goods that are ideal for them or outside goods. This model is based on Uschev and Zenou (2018) article.

There are N firms that produce N different varieties, each firm i=1,…,N producing one variety . Geographic area in this model is presented via matrix . G is the adjacency matrix of network (N,G), G keeps track of degree of substitutability between neighboring varieties in the network. Links between varieties is kept in matrix elements i.e in . Link between i and j varieties if it is exists is equal to 1 (varieties are direct substitutes), otherwise 0. G is defined as symmetric, undirected matrix with zeroes on the main diagonale.

I use standard network definitions in this paper. A walk in network (N,G) refers to a sequence of with step equal to at the same time length of walks refers to number of such sequence. The geodesic distance is the shortest length between i and j. Another way to find the number of walks between i and j varieties with length equal to s is to raise G in sth power. shows all walks in network (N,G) with length s. More precisely, cell of shows the number of walks between i and j with length s. The set of neighbors (here direct substitutes) of node (here variety) in network (N, G) are denoted by

3.1 Consumers

In this model nodes represents the position firm in the network or the location of good that is produced by firm. Nodes are connected with edges that shows the degree of substitutability of each variety that has link. The key feature of Uschev and Zenou (2018) is that nodes are also represents each consumer “ideal variety”. Therefore, the number of consumers directly connected with the number of firms (nodes).

3.2 .1 Preference

As it was mentioned before, each consumer “ideal variety” is located in . Utility function of consumer is represented in the following way:

Where is the outside good, is the amount of good i that consumer will buy knowing that k is preferred good. The term represents the quantity of that will be chosen to be bought be consumer via consumer-specific willingness to pay parameter i.e proximity of other varieties to the ideal variety of consumer k. The term captures the love for variety via parameter . is substitutability effect.

Analyzing the utility function independent parameters should be considered in more details. Consumers willingness to pay function is defined in the following way:

Where is spatial discount (counterpart of the transportation costs) factor . Parameter exponentially decays with the distance between variety i and k.

Cross-term substitutability effect term is explained via parameter that represents connection between market segmentation parameter and inverse measure of product differentiation parameter , i.e if , then

3.2.2 Individual demand

The next formula will express linear equation system of utility function for every consumer in vector-matrix form:

Where is scalable substitution parameter. is Nx1 vector, is Nx1 vector, is a transpose vector of and I is identity NxN matrix.

Condition for (3) being strictly concave in is expressed in the following way:

Where is the lowest eigenvalue of G and is the biggest respectively.

The equation (4) is result of Lemma 1 derived by Uschev and Zenou(2018), see p. 244, Proof of Lemma 1.

Consumer solve optimization problem by the value of from budget constraint (5) in (3).

Where is consumers k income and is a price vector.

The inverse demand of consumer k for variety :

(5)

Or in vector-matrix form:

(6)

Computing individual demand from (6):

(7)

From (7), we can obtain , matrix B is well-defined see Uschev and Zenou (2018), p 232, eq 8.

B can be expanded in power series if :

Power series (9) could be rewritten in terms of Bonachich sign-alternating variety measure:

Where u is ant Nx1 vector. can be rewritten in the following way:

As it shows in (11) number of walks s exponentially decrease the weight of higher order cycles. Following Uschev and Zenou (2018) - is just the number of neighbors (i.e. direct substitutes) of variety I while is the number of triangles involving i, i.e. the number of couples of direct substitutes, which are also substitutes for each other. In other words, represents the firm-specific measure for toughness of competition for firm i. When firms prime (length) is not odd, the model suggests that, two close firms are direct substitutes of each other good, however, when 3^rd firm enter the market, competition burden relaxes, i.e firm i's competitors also compete with each other.

Equation (11) has a clear interpretation: it is the Slutsky matrix for individual demand. Changes in demand will look like:

Formula (12) shows that demand for varieties is more (less) sensitive to changes of price in higher (lower) centrality in the product-variety network.

From Bonachich (1987) centrality measure of the network looks differently from (9):

Uschev and Zenou modified bonachich centrality, so that as a result, having many direct links (degrees) reduces centrality, but if the links themselves have many links, then centrality is augmented. In other words, “my enemy's enemy is my friend”

3.2.3 Aggregate demand

Calculating vector of aggregate demand from individual demand (7):

(13)

Plug (13) in (7):

Defining the average willingness to pay parameter

Knowing that , we get:

Where N is NxN node degree matrix. Combine (16) and (14) we get aggregate demand

From (17) we obtain inverse aggregate demand:

In vector-matrix (18):

3.3 Firms

Each firm maximizing their profit via solving the optimization problem. Firms profit in Cournot competition:

Firms profit is defined by:

(22)

First order condition for firm optimization problem:



3.4 Equilibrium

First of all, to calculate equilibrium price, the dependence of p from x should be found:

(26)

(26) shows how equilibrium price relates with equilibrium output. Next step is to insert equation (24) in (26) to get the equilibrium price.

(p*,X*) is the unique interir Nash-equilibrium in cournot price competition when see Uschev and Zenou (2018) p. 244.



4. Change in

4.1.1 Analytical results

Proposition 1. There exists a positive limiting value , when , .

In the other words Cournot prices reacts in the same way as Bertrand prices when increases. Increase in means that competition becomes tougher and this simultaneously reduce the prices.



5. Change in

5.1.1 Analytical results

Proposition 1. There exists a positive limiting value , when , .

The meaning of this increasing of spatial discount factor leads to tougher competition but lower prices.



6. Network vs non-network effect

Network approach in any network-based model is based on spatial discount factor. When spatial discount factor is equal to 0, then model will be pure monopolistic competition. In monopolistic competition Bertrand prices become monopolistic prices. Knowing that Cournot price are always lower than monopolistic prices in the monopolistic competition models. We can conclude that in the non-network models Cournot prices are lower than Bertrand prices,



7. Welfare analysis

This chapter is devoted to the welfare analysis. First of all we need to calculate consumer surplus. To calculate it we need utility function of representative consumer and inverse market demand. Inverse market demand:

Representative consumers preferences:

Aggregate consumer surplus:

The next step is to calculate aggregate profit:

Finally social welfare is defined as sum of aggregate profit and aggregate consumer surplus:



Knowing that B is positive-define, social welfare will be maximized in p=0. Let us calculate equilibrium social welfare using the Cournot prices:

Since i.e Cournot price decays with the length of walk between varieties, we should use i=2, because other primes will slightly affect equilibrium price.

Plugging equilibrium price in W, it makes possible to calculate social welfare numerically using this formula:

Chamberline and Hotteling networks with N=5 and .



8. Bertrand prices lower than Cournot

Proposition 3. There exists positive threshold value , such that Bertrand prices are lower than Cournot one.

Proof of this proposition could be find in Appendix A proof 3.

Let's generate random matrix N=6 and compare Bertrand and Cournot prices:

As it was shown in proposition 3, there exists such cases when Cournot price if bigger than Bertrand.



9. Conclusion

In this paper, we have solved the network model with Cournot competition. In the course of the work, optimal equilibrium values of prices and goods were found. It was also shown that the price change from specific network parameters in the Cournot model is similar to the Bertrand model. It is important to point out that it has been proved that at low values of the , the Bertrand prices are lower than the Cournot prices. It is worth mentioning that the reverse case is possible already at the boundary values of , but this case has not been analyzed in the course of this work.



Appendix A

Proof 1. Change in

G is positively definite, that means that diff X will be negative. B is positively definite only if ,

only satisfied, when and is the positive minimum solution of the following equation:

is minimum of such solution otherwise

Proof 2



Where

Since both sides of that equation are differentiated in , it follows the implicit function theorem that if and only if and equals the minimum of the following equation:

is minimum of such solution otherwise

Proof 3

In Uschev and Zenou (2018) Bertrand prices are equal to:

, where D is diagonal matrix with on diagonals.

Bertrand prices could be rewritten using equation (11):

, where represents i's firm degree i.e the number of nearest firms with length of walk equal to 2. That means that matrix D may be presented like: , where N is the degree matrix of network G.

At this point Bertrand prices can be rewritten in the following way:

In this paper Cournot prices are defined in the following way:

Now it is possible to calculate difference of

Such result could be obtainable only if is small and products are differentiated.



Appendix B

import numpy as np

from numpy.linalg import eig

def MaxEigenvalue(B):

y = eig(B)

return max(y[0])

N=6

G = np.random.randint(0,1+1,size=(N,N))

G_symm = (G + G.T)/2

I=np.identity(N)

delta=MaxEigenvalue(G)

delta2=(delta*delta)/4

B=np.linalg.inv(I+delta*G)

degree = np.diag(np.sum(G, axis=1))

print(degree*delta2)

This code generates a random symmetric undirected graph G of a given size N. On the next step code calculates largest eigenvalue of G matrix, which is a key element of the overall code, as is the basis matrix N, which, in turn, is a matrix of degrees of a node or displays the number of firms with which a particular firm competes with a number which corresponds to its position on the main diagonal. The code can be modified by adding (2) with the specified values б,ц, but in the context of determining the price difference, this can be omitted since the parameter of the willingness to pay is always positive



Related literature

1. Ballester, Corallio, Antoni Calvу-Armengol, and Yves Zenou (2006). “Who's Who in Networks. Wanted: The Key Player.” Econometrica, 74, 1403-1417.

2. Bloch, Francis, and Nicolas Querou (2008). “Pricing in Networks.” Working paper, Brown University and Queens University, Belfast

3. Banerji, A. and B. Dutta (2005), ”Local Network Externalities and Market Segmentation,” mimeo., Delhi School of Economics and University of Warwick.

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