Modelling Descentralized Interaction in a Monopolistic Competitive Market Nicol´as Garrido∗ Department of Economics. Universidad Catolica del Norte - Chile March 3, 2006

Abstract This paper explores how the interaction of consumers’ and firms’ optimal choices with imperfect information in a monopolistic competitive market institution, leads them to stay in a subspace of a space of states. The first part of the paper studies the model using Markov mean field. In the second part, the model is investigated using computational specifications with two purposes: first, to explore how the results depend on an exogenously established price rule and second, to study the interpretation of transition rates parameters. Key Words: Local Interaction - Markov Mean Field - Evolutionary Algorithms Random Networks - Institutions. JEL classification: C61, D43, D71, D83

1

Introduction

In this paper, we assume that an economy or a market jumps through a countable infinitum space of states. Every state is identified by a tuple of observable variables. The market cannot move from a given state to any other state. Instead of this, there is a connected graph that links all the states specifying the possible changes in the observable variables that the agents of the market are able to cause within a given institutional framework. Agents have incentives to move the economy from one state to another according to the utility or profit that they obtain in the current state. Indeed, when an agent is not in his most desirable state, he tries to make decisions that improve his payoff. The farthest he is from the desirable state the higher the intensity to move the market toward the desirable one. The interaction of all the incentives, within the institutional framework, makes the economy travel through the network of states, drawing different trajectories on the graph. The aim of this paper is double. On the one hand it describes the states in which an economy spends more time, when it is inhabited by selfish agents, having imperfect information and interacting within a monopolistic competitive market. On the other hand, it explores whether the resulting steady states change as consequence of two variations; first, a change in the firm learning algorithms and the release of the monopolistic competitive price rule, and second a different specification of the agent ”intensities”. The first part of the paper is studied using Markov random fields, whereas the second part, is explored with computational models. ∗

email: [email protected]

1

Following North (1990), individual choice and the institutional constraints are integrated in order to study how the agents are channeled through the space of states. In our model, a monopolistic competitive market is the institutional framework in which agents with imperfect information and bounded procedural capacities attempt to make optimal choices. In recent years there has been an increasing use of Markov random fields to analyze stochastic interaction in the social sciences. Some references are Aoki (1998, 2000), Aoki and Shirai (2000), Blume (1995) , Blume and Durlauf (2000, 2002) and Wolfgang (2000). The main concept is that interaction among agents produces aggregate effects, i.e. mean fields, that affect their individual behavior. Thus, there is a feedback between the individual behavior and the aggregated result obtained as consequence of the individual interaction. The interaction using mean fields has also been combined with network theory. Dalle (1997), assuming that there is a network of heterogeneous agents, shows through the use of Gibbs random fields, how, according to the network externalities among the neighbors, different landscapes of technology are found in an industry. Instead of having a network of states, Kauffman et al. (2000) introduce a technology network (i.e. technological landscape) where each node has an economic value according to the efficiency of the technology that represents. In this landscape it is modeled a technological search, driven by the decisions made for optimizing labor efficiency firms, constrained by the search cost. The framework presented here is similar to Currie and Metcalfe (2001) because firms employ relatively simple behavioral routines, without making conjectures about other firms and using only internally available information. Moreover, firms make decision on more than one variable; price and production capacities. The introduction of a graph to connect the state of the economy is not a necessary formalism to solve the model. However, it is useful to makes clear that, due to the bounded rationality of the agents and physical and institutional constraints in the economy it is not possible to jump from one state to any other state. The work is organized as follows: in the next section, based on Wolfgang(2000) the model is presented explaining how firms and consumers act in the market. Afterward, using the master equation, the mean trajectory is presented as a dynamical system and the steady state solution is found. In the next section, using the same model as in the previous section, a simulated model with agents is constructed to explore the behavior of the market when an institutional constraint is lifted. Finally, the conclusions are presented.

2

Model

There are N consumers and F firms buying and selling respectively an homogeneous perishable good. Every time that a consumer goes shopping he buys one unit of the good from one of the F active brands in the market. Consumers are indistinguishable, therefore the state of the demand is characterized by the number of consumers buying from the different firms. On the other hand the state of the supply is described by two vectors; current prices and produced quantities. Defining the vector n = {n0 , n1 ...nF }, with N =Fi=0 ni , as the distribution of consumers buying in the different firms, the vector p = {p1 , ..., pF } as the current prices in the market and q = {q1 , ..., qF } as the production of the firms, it is possible to describe the state of the market through the following state vector of observable variables, V = {n, p, q}

(1)

The state vector V can take an infinite number of states, however, in reality is observed just 2

a small sample of them. This sample of most visited states can be computed as consequence of the specification of the stochastic transitional dynamics that move the system through the whole space of states. This dynamic in turn, is determined as consequence of the satisfaction that the agents, either firms or consumers, have in a given state. An agent, entrepreneur or consumer, translate his optimal bounded rational decision into an adaptive rule. The rule is adaptive because drives an agent from the current less desirable state toward his most desirable one in a finite number of steps. The rule is applied by the agent with a probability proportional to the distance between his optimal state and the current one. In other words, dissatisfied agents are trying to apply their adaptive rule to improve their state with high frequency. Every agent attempts to reach his most desirable state. However, there is no social agreement on which is the social desirable state. Accordingly, the interaction among all the agents results in a state (or a subset of states) where the intensities of all the agents to leave it are compensated. For instance, assume that the states of a market is described by the values of one price. In this market, the ”equilibrium” price, i.e. the state where the market stay lockedin, is the state where the incentives to reduce the price by the consumers and to increase the price by the firms is compensated. In the following subsections the transitional probabilities and the adaptive rules that determines the dynamics of the demand, the price and the quantity vectors are specified. Afterwards, we identify the most visited states of the market.

2.1

Demand

The demand is composed by N consumers that periodically buy the unique homogeneous good available in the economy. We can identify the consumers in two groups: consumers with and without a good. Having the ownership means that the consumer has a unit of the good produced by one of the firms in his shelf. In the other case, if the shelf is empty the good has to be replaced, so the consumer is ready to buy a unit the next time that he goes shopping. Thus, in any instant of time the vector n = {n0 , n1 , . . . , nF } describes the number of consumers with ownership of a good and the number of consumers that are shopping. The former is represented by nj , i.e. the number of consumers that have in their shelves the good from the brand j, and n0 represents the number of consumers ready to go shopping. The vector n characterize the state of all the consumers in the market; in other words N =Fj=0 nj . There are two different transitional dynamics to be specified on the demand side; first, how a consumer decide from which firm to buy given that they have his shelf empty, and second how a consumer “consume” the good in his shelf. In other words, it has to be explained how consumers move from n0 to nj and from nj to n0 , where j = 1..F . 2.1.1

How consumers buy

Each consumer i has a stochastic utility function Ui defined over the good offered by the firms. Following Anderson, De Palma and Thisse (1992), the utility of the consumer i defined over the good of firm j is defined as, Ui,j = ui,j + εi,j

(2)

where ui,j corresponds to the deterministic utility of the individual and εi,j is the portion of the individual behavior that can not be appreciated by the modeler. It could be interpreted as idiosyncratic differences within the population of consumers N 1 . Without loss of generality 1

Blume and Durlauf (2002) attribuite this randomness to some form of bounded rationality.

3

we assume homogeneity in the deterministic component of the utility, so it is possible to avoid the sub index i in the previous specification. The probability that an individual chosen at random from N buys from the firm j is given by µ ¶ PF (j) = Pr Sj = max Sf f =1..F

where Sj = uj − pj is the surplus that the consumer obtains when buying from j, pj is the price of firm j and Pr (Sj = maxf =1..F Sf ) represents the probability that the surplus of the firm j is the maximum surplus among all the firms in the market. Considering that the good sold by the firms is homogeneous, it is possible to normalize the deterministic utility for all the brands, such that uj = 0 ∀j. Theorem 2.2 in Anderson, De Palma and Thisse (1992), proves that the random utility with appropriate assumptions on εj , generates a multinomial logit model where the probability that a consumers, randomly chosen, buys from firm j at price pj is given by e−pj PF (j) = PF −pf f =1 e

(3)

PF (0, j) defines the transition probability that a consumer buys one unit from the firm j, given the vector of prices p. We do not assume neither, inertial demand nor switching cost. Accordingly, for a given consumer, the probability of buying from a firm j is independent from the brand that the consumer previously had. Moreover, notice that all the firms have a positive demand. Indeed, while firms fixing low prices have high demand, there are firms with high prices having a positive demand. Following Kelly (1979) the probabilistic transition rate is the probability generated per unit of time to the occupation of a new state. The relationship between the transition rate ω ,where ωk,j is the transition rate from j to k and and transition probabilities is P (j, k) = ωk,j j PF ωj = k=1 ωk,j . Using equations (3), the transition rate for a consumer of moving from the state of not having the ownership of a good to buy it from firm j is, ωj,0 = υe−pj . Considering that there are n0 agents without the good the transition rate becomes, c ωj0 = υe−pj no

where υ represents the intensity in an instant of time at which the consumers n0 , decide to go shopping. To make simple the interpretation of υ it can be considered as the inverse of a measure of the period of time that a consumer wait until he goes shopping. Thus, longer the period of time, lower is the intensity of υ, therefore there is a lower probability that c can be the consumers in n0 will go shopping in a given instant of time. The value of ωj0 interpreted as the realized demand for the firm j. 2.1.2

How consumers consume

As time goes by, consumers exhaust the unit that they have in their shelf so they have to go shopping again. In other words they move from nj to n0 with j = 1..F. It is assumed that consumers have the good in their shelves during a period of time ρ0 , so that ρ = 1/ρ0 represents the number of consumers that loose the ownership of the good in a unit of time. According to this the transition rate is given by, 4

c ω0j = ρnj

P c can be interpreted as the number of consumers that, in an instant of Therefore, Fj=1 ω0j time, consume the good from their shelves.

2.2

Firms

Firms operate in the market according to price and production plans that they revise periodically. The revision are due to differences between their plans and the results obtained from the market. When the difference between plans and results is high, firms revise their plans and adopt corrective actions frequently. If the difference is negligible, firms are lazy to correct it. Firms revise their plans asynchronically, in the sense that if firm j decide to revise its production plan, it does not mean neither, it revises its price plan nor that other firms revise their production plans. It is also assumed that firms adjust their plans slowly, in the sense that they can change either price or production decision just through the possible path defined by local network of states; there are not shortcuts between two states that are not linked. Even though this assumption seems to be restrictive, this is not necessarily the case, because the further a firm is from its desirable state the higher the frequency with which it decides to revise its plans. 2.2.1

Price dynamics

The firm’s profit is given by, Πj = pj qj − c(qj ) where c(qj ) and pj qj represent the cost function and the revenue for the firm j respectively. We assume that c (0) = 0 and c0 (·) = c¯ > 0. Even though firms can survive in the market having different prices, changes on the price of a firm affect the demand of other firms. This characterize the dynamic of the monopolistic competitive market interaction. Moreover, as in the long run firms prevent new entries, every firms follows the plan of equalizing price to average cost, pj = c¯(qj )2 . Thus the adaptive rule to update the price is: when the price is below the average cost increases it, in the opposite case reduces it. In any case the price is adjusted in one unit3 . On the other hand, a firm revises its price with a frequency proportional to both: the difference between its price and average cost, and its price. A firm is reactive to change its price if it realizes that it is far from the average cost because if the price is too high, the high profits could produce new entries. If the price is below the average cost, the firm is having negative profits, therefore it has to change its price soon. The additional awareness to revise firms’ plans when prices are high capture the idea that firms having high prices have a low demand, therefore they have to be ready to change when there is a deviation from the plan very soon. Thus, the frequency with which firm j wants to revise its plan is given by, Xj = (¯ cj (qj ) − pj )pj The sign of Xj specifies the direction in which the price has to be adjusted. 2

Notice that the average cost and the marginal cost in this case coincide. This assumption about the size of the price adjustment ∆p = 1 does not affect the general result. The size of the jumps can be reduced below one but greather than zero without modifying the main results. 3

5

Any price revision by firm j can be from pj to pj + 1 in the case that Xj > 0 or to pj − 1 when Xj < 0. Formally the transition rate for the price revision is given by, p ωj+ = βXj ϑ(Xj )

p ωj−

(4)

= β(−Xj )ϑ(−Xj )

where the function ϑ(X) is 1 in case that Xj > 0, and 0 otherwise, β measure the intensity of the transition rates, according to the same interpretation of the parameters υ and ρ in the consumer analysis. Thus, if firms wait an infinitum period of time to adjust their plans, this mean β = 0. In this case, prices remain the same. 2.2.2

Production dynamics

The revision in the production plans is done according to differences between the production and the demand; higher the excess or the short demand of the firm, higher will be the probability of making revisions in the production plan. Formally, Zj = (dj − qj )qj c n . where dj is the demand that the firm receives at any time, dj = ωj0 0 As in the case of price revision, it is assumed that when the level of production is higher firms are more sensitive to make an adjustment in their production plan. According to this, the transition rates are given by

q ωj+ = αZj ϑ(Zj )

q ωj− = α(−Zj )ϑ(−Zj )

where α is the intensity parameter that represents how sensitive firms are to a revision of the production plan and the function ϑ(X) is the same as before. Assuming that β is higher than α, means that to an equal difference in the price and the production plan firms are more willing to revise their prices.

2.3

Most visited nodes of the network of states

The transitions rates constitute the basic ingredient for the equation of probabilistic motion of the system through the network of states. The state vector V in (1) characterizes the system in a specific instant of time, whereas the transition rates ω c , ω p and ω q specify the change in probability that the state vector has of moving from a given state V to a neighbor V 0 . A neighborhood of V in the network is defined according to the three dimensions that the state vector describes: demand, price and production. Given an initial state V = {n, p, q} a demand neighborhood V n is the state vector in which n = {n0 ± 1, ..., nj ∓ 1} with j ∈ [1, F ] and the vector describing price and production are exactly the same as in V. In other words a demand neighbor of V is the state in which one consumer either moves from having the ownership to go shopping or vice versa. In the same way a price or production neighborhood of V is the state in which one firm increases or decreases its price or production by one unit. Thus, the dynamics of the model, is specified in terms of the transition probabilities of the outflows and inflows transitional probabilities from the reference state V. Transitions from V 6

toward any of the possible neighbors are outflows and the transition rates coming from the neighbors toward the reference state are inflows. The probabilistic transitions rate between neighbor states, either outflows or inflows, capture the micro dynamic of the model and how the aggregate state affects it, i.e. mean fields. For instance the multinomimal model of demand, is the mean field that aggregate the interaction of the decision of the consumers and firms. Indirectly this field will affect the production plans of the firms and the decisions that the consumers make to buy a good. The main assumption to obtain the solution of the model is that there is a distribution that describes the probability of observing at time t the system in a given state V. Such a probability satisfies,

X

Pr(V, t) > 0 Pr(V, t) = 1

V ∈S

where the summation is over all the possible states in the space of state S. The equation that represents the dynamic evolution of the probability in the system, is the master equation,4 ¡ ¢ X ¡ ¢ d Pr(V, t) X = inf lows V 0 → V − outf lows V → V 0 dt 0 0 V

V

0

where V represents all the neighbors’ state of V . Thus, the system is in the stationary state when the flow of probability of leaving is equal to the flow of probability of reaching the state V . The stationary state solution of this equation gives information about the most probable state or subset of states. If we denote the stationary state probability as Pr∗ , the subset of most visited spaces V ∗ is the solution to arg maxV ∈S Pr∗ (V ) . This subset of states represents the demands, prices and quantities that are found with higher probability in our monopolistic competitive market. The specification of the master equation for our model is given by: 4

For more details on the master equation see Chapter 10, of Wolfgang (2000)

7

∂ Pr(V, t) ∂t

=

F X

c ω0j Pr ({(n0 − 1, n1 , ..., nj + 1, ..., nF ), p, q}, t) −

j=1

+

F X

+

c ωj0 Pr ({(n0 + 1, n1 , ..., nj − 1, ..., nF ), p, q}, t) −

+

p ωj+ Pr({n, (p1 , ..., pj − 1, ..., pF ), q}, t) −

+

F X

p ωj− Pr({n, (p1 , ..., pj + 1, ..., pF ), q}, t) −

+

p ωj− Pr ({n, p, q}, t) +

p ωj+ Pr ({n, p, q}, t) +

j=1 q ωj+ Pr({n, p, {q1 , ..., qj − 1, ..., qF }}, t) −

j=1 F X

c ω0j Pr ({n, p, q}, t) +

j=1

j=1 F X

F X j=1

F X

j=1 F X

c ωj0 Pr ({n, p, q}, t) +(5)

j=1

j=1 F X

F X

F X

q ωj− Pr ({n, p, q}, t) +

j=1 q ωj− Pr({n, p, {q1 , .., qj + 1, ..., qF }}, t) −

j=1

F X

q ωj+ Pr ({n, p, q}, t)

j=1

The first and second rows of the differential equation represent the flow of probability between the reference state V = {n, p, q} and all its possible demand neighborhoods. For instance, the first term on the right side of the first row, is the inflow probability that the system moves from a demand neighbor into the state V. On the other hand, the second term represents the outflow probability that the system moves from V to a neighbor state. The third and fourth lines of the equation represent the flow of probability between a price neighbor state V p and the state of reference V. Finally the fifth and sixth rows represent the flow of probability among quantity neighbor states and the reference state V.

2.4

Master Equation Solution

Suppose that it is possible to produce a bundle of experiments using the model previously defined with different initial conditions. Accordingly, suppose that we collect information about the evolution of V for each experiment. The solution to (5) represents the mean trajectory thorough the graph of states that the market would follow over all the experiments. We use the quasi-mean value equations5 to describe the mean trajectory through the following dynamical system, 5

In the appendix A there is an explanation of the problems related with the solution of the master equation together with the derivation of the quasi-meanvalue equation.

8

dnj (t) dt

c c = ωj0 − ω0j = ve−pj n0 − ρnj

dn0 (t) dt

=

dpj (t) dt dqj (t) dt

F X

c ω0j −

j=1

F X

c ωj0 =−

j=1

(6)

F X dnj (t) j=1

dt

p p = ωj+ − ωj− = βXj ϑ(Xj ) − β(−Xj )ϑ(−Xj ) q q = ωj+ − ωj− = αZj ϑ(Zj ) − α(−Zj )ϑ(−Zj )

for j = 1..F. Replacing the transition rate with the formula derived previously and given the complementary between the two demand equations, the system which determines the solution is given by, dnj (t) dt dpj (t) dt dqj (t) dt dn0 (t) dt

= ve−pj n0 − ρnj

(7)

= β(¯ cj (qj ) − pj )pj = α(dj − qj )qj = −

F X dnj (t) j=1

dt

which is a system with 3F + 1 equations, that for a relatively big number of firms it becomes hard to deal with.

2.5

Homogeneous Oligopoly Case

In order to simplify the solution of the dynamical system (7) we use the slaving principle 6 to express the system as a function of order parameters. We assume that the total number of consumers N does not growth, the incumbents have production capacities to satisfy all consumers and that firms are capable of adjusting their prices without cost. If firms can adjust prices and productions to the demand, the order parameter of our model is the demand. In other words, prices and productions are fast variables that firms adjust, following the slow changes in the demand. Thus, 0 = β(¯ cj (qj ) − pj )pj 0 = α(dj − qj )qj 6 The slaving principle comes from the synergetic. The idea is that in a dynamic system there are fast and slow variables. The dynamics of the fast variables is driven by the slow variables. Thus, it is possible to analyze the system assuming that the fast variables are in their steady state, and that the stable state is driven by the order parameters of the slow variables.

9

which means that firms set price equal to the average cost and that firms produce exactly what is demanded in the market at t. In other words, pj

= c¯j (qj )

qj

= υe−pj n0

for j = 1..F. Given the fixed number of consumers, the dynamic of the demand equation can be expressed as, F

X dnj (t) = υe−pj (N − ni ) − ρnj dt i=1

which after the normalization of n into x, with

F x i=0 i

= 1, we obtain,

F

X dxj (t) = υe−pj (1 − xi ) − ρxj f or j = 1..F dt

(8)

i=1

Thus, the system of F equations (8) represents the dynamic of the quasi-mean demand for every firm of the monopolistic competitive market of the model under the assumption that price and production decisions are variables that follow the demand. i (t) In steady state (i.e. dxdt = x˙ i = 0 ) the solution to the system is given by, xj =

ρ υ

+

e−pj PF

i=1 e

−pi

f or j = 1..F

(9)

where price is equal to average cost and ρ and υ represent the reciprocal of the periods that consumers have the good in their shelves and the periods that they wait before of going shopping once they exhausted the good in their shelves7 . From equation (9) it is possible to compute the proportion of consumers that are shopping in any period of time, x0 =

ρ/v +

ρ/v PF

i=1 e

(10)

−pi

7

Stability Analysis: We assume that the average cost of every firm is constant. Thus, the Jacobian of the system (8) is given by 2

ψ1 − ρ 6 J(x) ψ1 =6 4 ... ∂(x) ψ1 ·

ψ2 ψ2 − ρ ... ψ2

... ... ... ...

3

ψF 7 ψF 7 5 ... ψF − ρ

where ψi = −ve−pi . Notice that when firms have the same average cost all the ψi are equal. The generic solution of the characteristic polynomial of this matrix is given by λi = −ρ i = 1..F − 1 λF = −

F X

ψi − ρ

i=1

Notice that in any case the first F − 1 roots are real and negatives, therefore the system is asymptotically stable.

10

The proportion of consumers shopping increases with any of these three factors: (i) the time that the consumers have the good in their stack is short, so the parameter ρ is high, (ii) the time that the consumers spend shopping is long, so that v is low, and (iii) when the prices are high. If there is no market, i.e. F = 0, no consumer has a good x0 = 1. In the competitive market, where the number of firms setting the market price is infinitum all the consumers has a good in their stack, x0 = 0. Equation (10) can be used to explain why there is a big number of retailers and few shops selling durable goods. Food is consumed relatively fast, so ρ is high, therefore the population of consumers shopping is high. On the other hand durable goods are consumed during a long period of time, therefore all else equal, there are few consumers buying durable goods. However, the durable case has a caveat: when the cost of producing a particular durable good is reduced, its price fall, therefore the proportion of consumers shopping increases. Thus, the number of shops selling the durable good with low price increase as well. The solution obtained in (9) differs from the standard oligopoly solutions ´a la Bertrand and Cournot, because in steady state prices and production can be different. The asymmetry comes from an unobserved consumer component in his utility function. This stochastic component, specified in (2), represents an idiosyncratic bias in the consumer utility function. Our model differ from spatial models of product differentiation as Salop (1979) because the interaction between firms is not local. In our model, the interaction is among all the firms in the market. Our model is close to the monopolistic competitive literature as in Hart (1985). The interaction is global and there is a low level of strategic interaction, ”in that the strategies of any particular firm do not affect the payoff of any other firm”. [Mas Colell et al. (1995) pp. 400] The model of this paper is close to models of bounded rationality in Sargent (1983), because all the agents use private information. Moreover, the agents do not use any ”proxy” of the market behavior, as the mean demand or the mean price. In all the cases, agents use adaptive rules, and the asymmetric equilibrium is explained as consequence of the interaction among all the agents.

2.6

Heterogeneous duopoly

In this section ρ is assumed to be a proxy of the quality of the goods produced by the firms. We assume that high quality goods stay longer in consumer’s shelves than low quality goods. Assume that in the economy there are two firms producing a good with different quality. The good of firm 1 has higher quality than the good of firm 2, i.e. ρ1 < ρ2 . With these assumptions and using the slave principle, the solution to (7) is given by.

x1 = x2 = x0 =

ρ1 ρ2 v

ρ2 e−p1 + ρ1 e−p2 + ρ2 e−p1

ρ1 ρ2 v

ρ1 e−p2 + ρ1 e−p2 + ρ2 e−p1

ρ1 ρ2 v

+ ρ1

ρ1 ρ2 v e−p2

+ ρ2 e−p1

Assuming that firms do not modify their prices when the quality changes, and setting v = 1, we can compute how the shares changes as the quality of the good of firm 1 improve, i.e. ρ1 falls. 11

dx1 dρ1

= −

dx2 dρ1

=

dx0 dρ1

=

ρ2 e−p1 (e−p2 + ρ2 ) <0 (ρ1 ρ2 + ρ2 e−p1 + ρ1 e−p2 )2

ρ2 e−(p1 +p2 ) >0 (ρ1 ρ2 + ρ2 e−p1 + ρ1 e−p2 )2 ρ22 e−p2 >0 (ρ1 ρ2 + ρ2 e−p1 + ρ1 e−p2 )2

An improve on the quality of the good produced by firm 1 increases the market share of firm 1, reduces the share of firm 2 and the proportion of consumers shopping x0 . Moreover, dx1 dx2 0 notice that dx dρ1 + dρ1 + dρ1 = 0.

3

Computational Agent Model

The model of the previous sections, shows the aggregated macro behavior that results as consequence of the probabilistic micro behavior of the agents within a given institutional settings. Two characteristics of our institutional setting identify it with a monopolistic competitive market: first, the interaction is global with a low level of strategic interaction and second, in order to prevent new entries firms set price equal to average cost. In this section we construct a model with computational agents of our previous specification. As in the analytical model, each of the N customers buys and consumes a homogeneous good sold by one of the F firms. Every firm i chooses price and quantity to maximize its profit, πie = pi qie − C(qie ), where qie is the expected demand for the firms and the cost function is C(q) = cq. The dynamic of the system is represented by a set of events that modify the state of the system. The system change when either of the following four types of events happen; (i) a consumer goes shopping, (ii) a firm revises price, (iii) a firm revises the produced quantity, or (iv) a customer consumes the good that he bought previously. In every period t, a number Et of events may occur, with 0 ≤ Et ≤ N + 2F. If the system does not change in a period t, it means that Et = 0. On the other extreme, the maximum number of events occurs in the unlikely case where all these events happen in a period t: each of the n0 (t) consumers buys a good from one of the firms, each of the N − n0 (t) consumers consumes the good that they bought in previous periods, the F firms revise their prices, and the F firms revise their production plans. Notice that the sum of all these events is Et = n0 (t) + (N − n0 (t)) + F + F = N + 2F. In every event an agent, consumer of firm, executes an action or change his state. Events are asynchronics because the agents are not coordinated to make decisions. For instance, when a firm revises its price it does not mean that any other firm has to revise its price. Thus, in the specification of the model to be simulated the following two characteristics have to be described; first, when the events of making an action is triggered and second, how the agents execute the specific action.

3.1

Consumers

Consumers may be in two different states during a simulation; consuming the good that they have in their shelves or shopping a unit of good because they do not have the good anymore. 12

3.1.1

Consuming

The good sold by the firms is perishable. In fact, once a costumer i acquire a good he spends a period of time τic consuming it. Every consumer takes a different period of time to consume its good. This depends on a set of unobservable, for the modeler, characteristics. We assume that the periods to consume a good is independently among the consumers and that in average a consumer spend 1/ˆ ρ periods of time with a good. In order to simulate this behavior we use two variations: first, every time that a consumer i buys a good we draw the value of τic from an exponential distribution with mean 1/ˆ ρ. Thus, the consumer i moves to the group of consumers n0 after τic periods. The second strategy is that in every period, the consumers in nj move to n0 with probability ρˆ. When a consumer having a good from firm j consumes his good, he becomes a consumer ready to go shopping again. This means, that the number n0 of customers ready to go shopping is increased by one unit and the number of customers from a firm j is reduced by one unit. 3.1.2

Shopping

The consumer search in the market before deciding from which firm to buy. Indeed, we assume that the agent i spends τis periods of time searching before of buying the good. We assume that the agents spend in average 1/ˆ ν periods of time searching. We use two strategies to modelling this, according to the strategy follows to model the consuming time. With the first strategy, the amount of periods τis for the agent i is drawn from an exponential distribution with mean 1/ˆ ν . With the second strategy the value of vˆ is used to compute the probability that every agent from n0 stop searching and decide to buy. After the searching process elapses, a consumer buys from firm j with probability Pr(j) = −pj e PF −pi , where pj is the price of firm j. We rationalize this decision as in Section 2.1.1. i=1

e

The parameters ρ,v, ρˆ and vˆ capture the flow intensity of the transitions from consuming to shopping and viceversa. In the analytical model the influence is through the intensity of the transition rates. In the computational model we explore two different alternatives to model the intensity. In the first one, we use the parameters with the same effect of the analytical model: intensity of the transition rate. However, transition rates is an infinitesimal concept hard to be measured in reality. Thus, we use the computational model to explore whether the predicted results of the analytical model change with a different specification of the flow intensity of the transitions from consuming to shopping and viceversa. In the second strategy we specifically model the time. We assume that the parameters are mean times of a distribution. In particular we use a negative exponential distribution8 to model the time that every consumer spend consuming and shopping.

3.2

Firms

We assume that every firm follows a price and produces a quantity that is fixed in its plan. A firm revise its plan according to its performance. Indeed, plans with low performance are revised frequently while good plans are revised with less frequency. We explore two specifications about how firms revise their plans; in the benchmark case, every firm has a plan for the price and a plan for the quantity whereas in the evolutionary model, firms have a single plan for both, price and quantity. 8 The negative exponential distribution applies frequently in the simulation of the interarrival and interdeparture times of service facilities. For more details see Law and Kelton (1982).

13

In the benchmark case firms set a plan and they adapt their current price or quantity to their plan using adaptive rules. In the evolutionary case firms use an evolutionary algorithm to select the best plan. In both cases firms attempt to maximize their profit. 3.2.1

Production Decision in the benchmark case

In this case, a firm has more incentive to adjust its production plan if it is far from the demand that it received during the current period. Thus, the probability that a firm i revises its production plan at the end of the period t increases according to |β(di,t −qi,t )qi,t |, where di,t represents the current demand, qi,t is the quantity produced according to the last production plan, β is a normalization parameter and |·| represent the absolute value of ·. Firms cannot adjust their production instantaneously. When a firm revises his production plan, it adjust the production on one unit according to whether it receives an excess or not of demand. In other words, firms will adjust the production according to the following adaptive rule, qi,t+1 = qi,t + sign(di,t − qt,i ) where sign(.) is the sign function9 . Notice that if the current production is far from the current demand the adjustment on qi are frequent. The distance is measured by the normalization parameter β. If |β(di,t −qi,t )qi,t | ≥ 1 the production is adjusted in every period. 3.2.2

Price Adjustment in the benchmark case

Following the setting of the analytical model, in the benchmark case firms set price equal to average cost. The further the current price is from the current average cost, the higher will be the incentives to adjust the price. Thus a firm j revises its price, at the end of period t, with a probability given by |α(¯ c(dj,t ) − pj,t )pj,t | where dj,t is the demand in the current period, pj,t is the price of the current period, c¯(dj ) is the average cost and α is a normalization parameter. Firms adjust prices according to the following adaptive rule, pj,t+1 = pj,t + sign(¯ c(dj ) − pj ) 3.2.3

Evolutionary Firms: Production and Price Decision made simultaneously

In this case, we assume that firms revise their prices and productions plans every Λ periods of time in order to obtain the maximum profit, πj = min(dj , qj )pj − C(qj ) where the min(dj , qj ) means the minimum between the demand and the production. Every firm has a set of plans L. Every plan has a price and a production decision. A firm explore different plans until it manages to obtain the most profitable one. Every plan produces a profit, as consequence of interacting with the plans of all the other firms. 9

The sign function is defined as, sign(x) =

8 <

1 0 : −1

14

x>0 x=0 x<1

To obtain the best plan that gives the maximum profit every firm evolve its set of plans, discarding low performance plans and using good performance plans to generate new ones. The performance of a plan is measured by its profit. Every firm apply every Λ periods, three genetic operators over its set of plans; selection, crossover and mutation10 . Selection selects the best plans, and crossover and mutation generate a set of new one. Using the evolutionary algorithm to simulate how firms make their price and quantity decisions, introduces two modifications in the original assumptions of the analytical model: on the one hand it changes the structure of neighborhood of one states; a firm can change its price or its production in more than one unit. The neighborhood of one state is now is defined by the mixing matrices in Vose’s model of simple genetic algorithms (see Vose 1999)11 . The second deviation from the original assumption is that firms do not follow the rule of price equal to average cost. With the goal of observing the consequences of this modification of assumptions in the next section we simulate our computational specifications.

3.3

Experiments

We simulate the computational model using the following base line parameters: there are N = 1000 consumers and F = 10 firms interacting during T = 5000 periods. The 5000 periods are in all the cases enough to reach a stable state. The normalization parameters for the price and production revisions are α = β = 0.2. This means that if the difference between the price and the average cost, or between demand and production for a firm is equal or greater than 5, the firm revises his price or production plan with probability 1. For the evolutionary firms we assume that every firm has a set of L = 30 plans. Every Λ = 30 periods each firm evolve its set of plans. 12 In order to explore the effect of heterogeneity in the cost function, two cases are studied: the homogeneous case, where all the firms have the same cost function and the heterogeneous case where there are two groups of firms differentiated by their cost function. In the homogeneous case all the firms have the same cost function C(q) = 2.5q. In the heterogeneous case half the firms, type 1, have the cost function C(q) = 2.5q and half, type 2, C(q) = 3q. The two different cost structures together with the two different model of firms, produce four simulations to be tested, 1. Benchmark firms with homogeneous in their cost function. 2. Benchmark firms with heterogeneous cost function. 3. Evolutionary firms with homogeneous cost function. 4. Evolutionary firms with heterogeneous cost function. According to the analytical results obtained in section 2.3, and using the parameters described in this section, the steady state of the system (i.e. the most probable result observed) for each case is given in table 1. These analytical results are useful as benchmarks to compare the simulated experiments. 10

See Vose (1999) for more details on Genetic Algorithms. The mixing matrices define all the possible states that can be visited as consequence of applying the mutation and crossover operator to the active population of plans that the firms have. 12 We select 25% of the plans using roulette. Moreover mutation is applied with 0.01 probability. 11

15

Homogeneous Quantity Price

7.6% 2.5

Heterogeneous Type 1 Type 2 9.02% 5.4% 2.5 3

Table 1: Proportion of agents consuming a given brand and prices according to the anlytical model Benchmark Evolutionary

Homogeneous itbpF1.6224in1.081in0.3649inFigure itbpF1.6224in1.081in0inFigure

Heterogeneous itbpF1.6224in1.081in0.3649inFigure itbpF1.6224in1.081in0inFigure

Table 2: Proportion of agents shopping and consuming For each of the four experiments we run 250 simulations. Each of the 1000 simulations is started with a different initial conditions. Thus, at time 0 consumers are distributed among firms uniformly and initial prices are set equal to ci (1 + ei ) with ci the average cost of firm i and e an independent draw from a normal distribution N (0, 1). Consumers spend 1/ˆ ρ = 40 periods consuming a unit of the good and they spend searching for a new unit 1/ˆ ν = 10 periods. In section 3.1 we specify two strategies to model the transitions from shopping to consuming and viceversa: the transition rate probability and the modelling of time strategy. Using both strategies we do not distinguish differences between the outcomes. Therefore, in order to reduce the size of the paper, we present only results based on the modelling of time strategy. In other words, ρˆ and νˆ are parameters of a negative exponential distribution. We simulate the periods of time that a consumer spend shopping or consuming through a random draw from a negative exponential distribution with parameters νˆ and ρˆ respectively. In the first row of table 2 we plot the mean quantity in the period 5000 for the benchmark13 . The first bar is the proportion of consumers shopping x¯0 . The next ten bars represent the proportion of consumers that have in their shelves a good of one of the firms. Notice that for the benchmark with homogeneous and heterogeneous costs the results are close to the analytical one. The small fluctuations are consequence of the probabilistic demand. In Table 3 we plot the average price for the benchmark case in the first row again. In both cases, homogeneous and heterogeneous cost structure, the average cost is reached with high precision. The results for the the evolutionary case are reported in the second row of tables 2 and 3. In this case the outcomes require an additional analysis. Although firms use a completely diverse learning mechanism and set simultaneously price and quantity, the quantities of steady state are close to the benchmark, but prices are different. There are two differences between the prices in the benchmark and the evolutionary case: first the prices are higher than the average costs and second, the prices have more variability. 13

Each bar in the plot representsP the mean of 250 simulations. For instance, in the benchmark with homoge250 1 nous cost the first bar is x ¯0 = 250 j=1 x0,5000,j . Where the subindex j counts the simulations.

Benchmark Evolutionary

Homogeneous itbpF1.6224in1.081in0inFigure itbpF1.6336in1.139in0inFigure

Heterogeneous itbpF1.6224in1.081in0inFigure itbpF1.6423in1.1268in0ptFigure

Table 3: Firm’s Prices 16

The additional variations is explained because in the optimization algorithm there is a population of firms exploring different plans. This process of searching converge to select the best plans. Although after 5000 periods there are few plans left to be explored and the best plans have been selected, the mutations from the genetic algorithm keeps introducing new combinations of price and quantity. Thus, this is translated translated into the observed variation. The intuition is that even when the market is in equilibrium, there are entrepreneurs searching for new plans. This exploration introduces the additional variation. The increase in the prices is explained by the (low) strategic interaction between the firms. In the evolutionary case, firms do not follow the rule of price equal to average cost. In this case there is a fix number of firms maximizing their profits. The demand is not infinitely inelastic. Instead, every firm has a ”local” elasticity consequence of the global interaction and of the assumption that consumers do not switch to the firm offering the lowest price. Indeed, in the evolutionary case the maximization of profit for every firm is when they set their marginal revenue equal to their marginal cost. Using our cost function we have, µ ¶ 1 pi 1 + = ci f or i = 1..F (11) εi where pi is the price, εi is the price elasticity and ci is the marginal cost for firm i. µ ¶ νe−pi P We use the analytical model to compute the elasticity εi = −pi 1 − . −pj F ρ+ν

j=1

e

We are not able to provide a close form for the F prices that solve (11). However, using the same parameters of the computational model we obtain numerical solutions for the prices. When the cost structure is the same for all the firms, the homogeneous case, firms set price pi = 3.51. The mean of prices obtained from the simulations is p¯ = 3.39. When the cost functions are heterogeneous, the price for the firms with low cost is pi = 3.51, whereas the price for the firms of high cost is pi = 4.01. The mean prices that we obtain for every group with the evolutionary firms, are p= 3.24 and p¯ = 3.93 for the low and high ¯ cost respectively. Notice that the price values are close in every case. The small difference between the theoretical values and the simulated ones are due to the variation introduced by the optimization algorithm.

4

Conclusion

The aim of this paper is double: first, we present a model where consumers and firms interact according to their bounded rational adaptive rules and to the constrain established by an institution. We aim to capture the massive interaction that is present in the market through an stochastic model. The second goal of the paper is to construct a computational model with the same assumptions of the analytical one. This comparison of methods to modelling the same problem allows us to understand better some features of the analytical model. We learnt from the analytical model that the time that consumers spend shopping and consuming is helpful to approximate the population of consumers shopping. When the goods are homogeneous, either one of this two effects: short period of time that consumers spend consuming a good or the long periods that they spend searching (shopping) for a good produce the same result: the proportion of consumers shopping increases. Moreover, if the prices of the firms are raised, all else equal, there will be a higher proportion of consumer shopping. 17

The price effect is explained by the consumers gathering more information about the goods when prices are high. Using the analytical model we also describe the effect of increasing the quality of the good of a firm. Thus, in a duopoly, if a firm increases the quality of its good there is an increase in the proportion of consumers buying it. The increase is explained with both, a reduction in the proportion of consumers buying the other good, and a reduction in the proportion of consumers doing shopping. From the computational models we learnt two things. First, the result of the analytical model is conditioned by the price equal to average cost rule. This rule is exogenously introduced in the analytical model with the assumption that firms do it to prevents the entry of new firms. In the simulated model, when we released that institutional constraint, the results change. This time, as the number of firms is fixed and each firm optimizes they set prices higher than the average cost. We shown that as consequence of introducing evolutionary firms the outcomes of the simulation are close to an equilibrium where all the firms set their marginal revenue equal to marginal cost. The second thing that we learnt from the implementation of the computational model is about the interpretations of the parameters ρ and ν. Using the computational models, we validate the interpretation of those parameters as periods of time that consumers spend either consuming or shopping. Moreover, we model the time evolution as random draws from a negative exponential distribution, using ρ and ν as parameters. This last result, within the assumptions of our analytical model, allows us to estimate the proportion of customers doing shopping, consuming and buying using market prices and the mean period of time that a consumer spend consuming and shopping.

18

References [1] Anderson Simon, Andre De Palma and Jacques-Francois Thisse (1992). Discrete Choice Theory of Product Differentiation. Cambridge (MA) MIT Press. [2] Aoki, Masanao. (1998). Simple Model of Asymmetrical Business Cycles: Interactive dynamics of a large number of agents with discrete choice. Macroeconomic Dynamics, 2, 1998, 427-442. [3] Aoki, Masanao and Y. Shirai. (2000). A new look at the Diamond search model: Stochastic cycles and equilibrium selection in search equilibrium. Macroeconomic Dynamic, 4, 2000, 487-505. [4] Aoki, Masanao (2000). Cluster Size Distributions of Economic Agents of Many Types in a Market. Journal of Mathematical Analysis and Applications, 249, 2000, 35-52. [5] Blume, Lawrence (1995). Population Games in in The Economy as a Complex Evolving System II edited by Arthur, W. B., Durlauf, S. and Lane, D. Addison-Wesley, Redding, MA. December. [6] Blume, Lawrence and Steven Durlauf. (2002). Equilibrium Concepts for Social Interaction Models. International Game Theory Review 5(3) 2003 193-209. [7] Blume, Lawrence and Steven Durlauf. (2000). The Interactions-Based Approach to Socioeconomic Behavior, in Social Dynamics, edited by S. Durlauf and H. P. Young. MIT and Brooking Institution Press. [8] Currie, Martin and Stan Metcalfe (2001). Firms routines, customer switching and market selection under duopoly. Journal of Evolutionary Economics, 11: 433-456, 2001. [9] Dalle, Jean-Michel (1997). Heterogeneity vs. externalities in thechnological competition: A tale of possible technological landscape. Journal of Evolutionary Economics, 7:395-413, 1997. [10] Hart, O. D. (1985). Monopolistic competition in the spirit of Chamberlin: A general model. Review of Economic Studies. 52: 529-46. [11] Kauffman Stwart, Jos´e Lobo and William Macready. (2000). Optimal search on a technology landscape. Jorunal of Economic Behavior & Organization, Vol. 43, 141-166. [12] Kelly F.P (1979). Reversibility and Stochastic Networks. John Wiley & Sons. Press. [13] Law A. and W. Kelton. (1982). Simulation Modeling and Analysis. New York: McGrawHill Book Company. [14] Mas Colell Andreu, Michael Whinston and Jerry Green. (1995). Microeconomic Theory. Oxford University Press. [15] North, Douglass (1990). Institutions, Institutional Change and Economic Performance. Cambridge University Press. [16] Salop, S. (1979). Monopolistic Competition with Outside Goods. Bell Journal of Economics 10: 141-56. [17] Sargent, Thomas (1993). Bounded Rationality in Macroeconomics. Oxford University Press. 19

[18] Vose, Michael (1999). The Simple Genetic Algorithm: Foundation and Theory. MIT Press. [19] Wolfgang, Weidlich (2000). Sociodynamics: a systematic approach to mathematical modeling in the social sciences. Harwood Academic Publishers.

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A

Appendix: Quasi-Mean Value Equation as Solution to the Master Equation

Whereas the dynamic solution obtained to deterministic dynamics problems using Hamiltonians is a well determined path of evolution of the variable, in the case of stochastic dynamical systems the solution behaves in a completely different manner. In a stochastic system the future evolution of the system given that it is in the state i at time t is not unambiguously defined. In this appendix, based on Wolfrang (2000), (see also Kelly (1979) and Aoki (2000)) we describe a single trajectory of a system and we investigate the properties of its stochastic evolution in which it traverses probabilistically a path through nodes (i.e. states) in the stochastic network. The set of probabilistically visited states is denoted as stochastic trajectory. This approach is better suited for social systems where there is no chance of having a big number of trajectories to describe the complete behavior of the momentum of its underlying distribution. Instead of this, the researcher only dispose of one or at best of a few comparable systems to describe. It is assumed that between two any states there is a path with positive probability, i.e. a non zero joint probability, which means that the system is ergodic or that the graph is connected (not necessarily full connected). In other words, starting from any arbitrary node i the system can reach after a period of time the node j and at the same time with positive probability the system will return from j to i, sooner or later. Now suppose that there is a large ensemble of systems, B ≫ 1. Each of them is moving probabilistically along a stochastic trajectory. Notice that the assumption does not mean that the large ensemble is really available. In order to compute the number of systems found in the node i denoted by B(i), at any time, it has to be described the number of system hopping into i from other nodes j and the mean period of time that a system stay in i. PLThe systems in state i hopping per unit of time into the state i from neighboring j is j=1 ω(i|j)B(j), where L represents all the possible nodes, other than i, in the system and ω(i|j) represent the transition rate of going from j to i. Notice that in a local network, where the network is not full connected, all the nodes j that are not directly connected to i have ω(i|j) = 0. To derive the mean staying time in i, τi let’s introduce the probability p(τ ) that is the probability of a system arriving at i in t = 0, and stay there until time τ. Note that p(0) = 1. It is stated that p(τ + δτ ) = p(τ ) − δp(τ ) where δp(τ ) is the probability to have been in cell i until time τ times the probability to make a transition into any other cell in the time interval (τ, τ + δτ ).i.e. δp(τ ) = p (τ ) δτ ω(i), with ω(i) =

L X j=1

using both equation we obtain that14 Replacing in the previous equation p(τ + δτ ) − p(τ ) = −p(τ )ω(i) δτ 14

Notice that dτ ' δτ

21

ω(j|i)

therefore, dp(τ ) = −ω(i)p(τ ) dτ as we can observe greater τ lower the probability at staying in the state i. The solution to the differential equation with the initial condition p(0) = 1 is, pi (τ ) = e−w(i)τ Now it is possible to calculate the mean time, which is equal to15 τi = So, the number of system in state i, B(i) can now be expressed as

1 w(i)

L

B(i) =

1 X w(i|j)B(j) w(i) j=1

or what is the same L X

ω(j|i)B(i) =

j=1

L X

ω(i|j)B(j)

j=1t

which is the master equation in the case that system is stationary. Indeed the time dependent equation is given by, L

L

j=1

j=1

X dB (i; t) X = ω(j|i)B(i; t) − ω(i|j)B(j; t) dt Thus, notice that starting from the description of the hopping process and the mean time that the system waits in one state it is obtained the master equation in both versions; the stationary and the path dependent. Moreover B can be normalized to be interpreted as the frequency with which the system is observed in a given state. Now the system is analyzed with more detail, indeed a given state i in the previous notation can be interpreted as the configuration n = {n0 , n1 ...nF }, thus B (i) = B (n) which is considered as the frequency can be interpreted as the probability16 P (n) . In order to derive the quasi-mean value equations that gives information about the quasi mean value of n, the state of a particular variable after a period of time δt is studied. Thus, note that the value of the variable nj after a period of time δt can be expressed as, (nj + dnj ) = (nj + 1) δti ωji (n) + (nj − 1) δti ωij (n) +nj [1 − δt (i ωji (n) +i ωij (n))] 15

(12) (13)

By the mean time definition

τ¯i

= −∞ 0 τ

dp (i) dτ dτ

=

∞ 0 τ δp (i)

=

∞ − [τ pi (τ )]∞ 0 +0 p (i) dτ =

16

1 ω (i)

Notice that from now on, the values of i and j do not express state of the system but indexes of the variables within the vector of state n.

22

The first term in the right side represent the fraction that has changed from nj to (nj + 1) , the second term represents the fraction that has moved from nj to (nj − 1) and the last term is the fraction that stays in the value of nj . Doing algebra over (12), and having δt ' dt it is obtained dnj =i ωji (n) −i ωij (n) (14) dt This represent the exact equation of motion of the quasi-meanvalues n. These values agree with the approximate mean values in the case of a unimodal probability distribution. In the case of multinomial probability distribution, equation (14) characterizes the mean evolution of a localized cluster of stochastic systems.

23