Abstract
This paper models the interaction of consumers’ and firms’ optimal choices with imperfect information in a monopolistic competitive market institution. The decisions of the agents are modeled with stochastic utility and profit functions. We show that using Markov mean field and an agent based model specification the agents stays most of the time within a subset of the space of states. We explore how our results depends on the exogenously established price rule and we make specific interpretation of the transition rate parameters in the context of the economic problem.
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Notes
- 1.
Blume and Durlauf [6] attribute this randomness to some form of bounded rationality.
- 2.
Notice that the average cost and the marginal cost in this case coincide.
- 3.
This assumption about the size of the price adjustment Δ p = 1 does not affect the general result. The size of the jumps can be reduced below one but greater than zero without modifying the main results.
- 4.
- 5.
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.
- 6.
Stability Analysis: We assume that the average cost of every firm is constant. Thus, the Jacobian of the system (9.22) is given by
$$\displaystyle{ \left [\begin{array}{cccc} \psi _{1}-\rho & \psi _{2} & \ldots & \psi _{F} \\ \psi _{1} & \psi _{2}-\rho &\ldots & \psi _{F}\\ \ldots & \ldots &\ldots & \ldots \\ \psi _{1} & \psi _{2} & \ldots & \psi _{F}-\rho \end{array} \right ] }$$(9.24)where \(\psi _{i} = -ve^{-p_{i}}\). Notice that when firms have the same average cost, this means that all the ψ i are equal. The generic solution of the characteristic polynomial of this matrix is given by
$$\displaystyle{ \lambda _{i} = -\rho \ \ i = 1..F - 1 }$$(9.25)$$\displaystyle{ \lambda _{F} = -\sum \limits _{i=1}^{F}\psi _{ i}-\rho }$$(9.26)Notice that in any case the first F − 1 roots are real and negatives, therefore the system is asymptotically stable.
- 7.
The negative exponential distribution applies frequently in the simulation of the interarrival and interdeparture times of service facilities. For more details see [13].
- 8.
The sign function is defined as,
$$\displaystyle{ \mathit{sign}(x) = \left \{\begin{array}{cc} 1 & x > 0\\ 0 & x = 0 \\ - 1 & x < 1 \end{array} \right. }$$(9.29) - 9.
See [18] for more details on Genetic Algorithms.
- 10.
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.
- 11.
We select 25 % of the plans using roulette. Moreover mutation is applied with 0.01 probability.
- 12.
Each bar in the plot represents the mean of 250 simulations. For instance, in the benchmark with homogenous cost the first bar is \(\bar{x}_{0} = \frac{1} {250}\sum _{j=1}^{250}x_{0,5000,j}.\) Where the subindex j counts the simulations.
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Brida, J.G., Garrido, N. (2014). Modelling Decentralized Interaction in a Monopolistic Competitive Market. In: Pinto, A., Zilberman, D. (eds) Modeling, Dynamics, Optimization and Bioeconomics I. Springer Proceedings in Mathematics & Statistics, vol 73. Springer, Cham. https://doi.org/10.1007/978-3-319-04849-9_9
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