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**InDyTeRRoD** (**In**dustrial **Dy**namics and **Te**chnological **R**egimes and the **R**ole of **D**emand) is an evolutionary model designed to explore industrial dynamics in alternative technological regimes. The model incorporates entry/exit mechanisms, innovation, imitation, competition, strategic learning and firm growth in an innovative industry.

This section explains the formal model that **InDyTeRRoD** implements. The information provided here should suffice to re-implement the same formal model in any sophisticated enough modelling platform. We use bold red italicised arial font to denote *parameters* and *initial conditions* (i.e. variables that can be set by the user), and we use bold green italicised arial font to denote the name of the corresponding *slider* or *box* in the interface of the model above. All *sliders* can be changed at run-time with immediate effect on the dynamics of the model (except initial conditions).

The figure below summarizes the main interactions among the variables in the model; it is a helpful reference to look at whilst reading the explanation of the model.

**Overview of the interactions among the variables in the model.** The variables that are more closely related to costs and prices are placed on the left, whilst the variables more related to the processes of innovation and performance improvement appear towards the right. A solid black arrow from X to Y denotes that an increment in variable X implies an immediate increment in variable Y. A dashed black arrow implies that the positive influence is not immediate, but it is delayed one time step. A solid red arrow from X to Y denotes an immediate negative influence, i.e. that an increment in variable X implies an immediate decrement in variable Y.

As an example, identify the following two negative feedback loops in the figure. The first one favors the stability of the variable performance *x _{i}*, whilst the latter favors the stability of competitiveness

- Higher R&D productivities
*z*have a positive effect on performance_{it}*x*in the next period, but -ceteris paribus- this increase in performance_{it+1}*x*diminishes the gap that separates it from the maximum performance_{it+1}*x*, and thus lowers R&D productivity^{max}_{t+1}*z*in the next period._{it+1} - Similarly, greater levels of competitiveness
*γ*have a direct positive effect on market shares_{it}*s*in the next period, which lead to higher prices_{it+1}*p*, which in turn imply lower competitiveness_{it+1}*γ*in the next period._{it+1}

In order to simplify the model's presentation, we classify our assumptions into six subsections: "The Firms' Competitiveness", "Demand Transformation", "Production", "R&D Spending", "Product Innovation" and "Exit and Entry of firms".

At any time there are *n _{t}* firms (indexed in

*p _{it}* = (1 +

Thus, firm *i*'s unit profit is

*π _{it}* =

Regarding performance *x _{i}*, we will establish below how firms improve their products through R&D-based technological innovations. For now, given the vector (

*γ _{it}* = (1-

where *x _{t}* = ∑

This formula captures the fact that consumers value both high levels of performance and low prices. The subjective relativity implied by the terms "high" and "low" is quantified using the average across the different products, whilst the trade-off between performance and price is regulated by parameter *α* (the *price/performance-sensitivity* of demand).

Production and growth are demand-driven. Regarding the demand-side of the market, we consider that the global demand (*Q ^{d}*) is constant and equal to 1. As

*Q ^{d}_{it}* =

If we consider that the consumers interact among themselves, observing each other and disseminating information regarding prices and performances of different products, we can assume that there will be a gradual process of demand transformation. That is, consumers will withdraw their demand from certain firms and pass it on to others with a higher level of competitiveness *γ _{it}*. Fatas-Villafranca et al. (2011) propose explicit evolutionary microfoundations to capture this kind of processes, and obtain a typical replicator dynamics expression. Along these lines, and drawing also on Metcalfe (1998), we propose that this process of demand transformation can be represented by:

(*s _{it+1}*-

where *γ _{t}* = ∑

It is then clear that those firms with higher than average levels of competitiveness will tend to capture a greater proportion of the demand.

Let us see how production and growth fit the evolution of demand. Starting out from a supply-demand equilibrium for each firm *Q ^{d}_{i0}* =

*Q ^{s}_{it}* =

where *K _{it}* is firm

*g ^{k}_{it}* ≡ (

That is, the growth of physical capital *K _{it}* -and, therefore, of the output- in each firm fits the growth of its demand in such a way that, at all times,

We assume that firms spend on R&D activities a proportion *r _{i}* ∈ [0,1] of the profits obtained in the previous period:

*R _{it+1}* =

Clearly, *r _{it}* is a firm-specific operating routine. According to Silverberg and Verspagen (2005), deciding the most convenient level of has traditionally been considered as an uncertain strategic choice. Therefore, instead of assuming that

*r _{it+1}* =

where *r _{t}^{*}* denotes the R&D routine of the most profitable firm at time

Thus, firm *i*'s unit cost will be:

*c _{it}* = 1 +

Along the lines in Nelson (1982), we assume the following equation to model the rate of improvement in each firm product performance:

(*x _{it+1}*-

where *z _{it}* can be identified with the productivity of R&D, and is determined as follows:

*z _{it}* =

where *Φ* ∈ [0,1] is the *knowledge-spillover-rate*, *x ^{max}_{t}* is the maximum performance in time

Any firm *i* whose capital *K _{it}* falls below a minimum quantity

P(*E _{t}*=1) = (1-

where *E _{t}* is a random variable that represents the number of new entrants (either 0 or 1) at time

Additionally, we suppose that the new entrant will have an initial capital given by *K _{n+1}* ∼ U(0,

*initial-n-of-firms*(*n*): Number of firms in_{0}*t*=0. Initially, the market is shared equally among the existing firms, i.e. the market share for each firm at time*t*=0 is*s*= 1/_{i0}*initial-n-of-firms*.*initial-performance*(*x*): Initial product performance of all firms in_{i0}*t*=0.*initial-r&d-over-profit*(*r*): Initial proportions of the profits that each firm devotes to R&D spending in_{i0}*t*=0.

*price/performance-sensitivity*(*α*): Sensitivy of demand to prices (in the trade-off between product performance and price).

*learning-rate*(*β*): Rate at which firms approach the R&D routine of the most profitable firm.*sigma*(*σ*): Standard deviation of the random variable*ε*∼ N(0,_{it}*σ*), which models possible imprecisions in the observation of the target competitor's routine.

*knowledge-spillover-rate*(*Φ*): Knowledge spillover rate.*u-max*(*u*): Maximum value of the random variable_{Max}*u*∼ Λ(_{it}*u*=0,_{min}*u*=0,_{mode}*u*), which introduces stochasticities in the R&D productivity_{Max}*z*._{it}

*min-capital*: Minimum capital required to stay in the market.*entry-barriers*(*λ*): Parameter that controls the difficulty to enter the market. If*λ*=1, no firm can enter the market.*max-entry-capital*: Parameter used to control the maximum capital with which new firms enter the market. New entrants will have an initial capital given by*K*∼ U(0,_{n+1}*max-entry-capital*) where U denotes the continuous Uniform probability distribution. Note that capitals are normalized t make sure that they all add up to one.

*Setup*: Creates the initial number of firms with the parameters and initial conditions set in the*sliders*and*boxes*.*Go once*: Pressing this button will run the model one time step only.*Go*: Pressing this button will run the model until this same button is pressed again.

*plot-period*: Plots are updated every*plot-period*time steps.*report-from-time*: Monitors (except for time) are updated from time step*report-from-time*onwards.

*time*: Time in the model.*leadership shifts*: Number of leadership shifts since time step*report-from-time*.*entrants*: Number of entrants since time step*report-from-time*.*rel. perf. of leader*: Relative performance of the leader, i.e. (*x*/^{max}_{t}*x*). The leader is the firm with the highest market share. Reported only from time step_{leader t}*report-from-time*onwards.*market share of leader*: Market share of the leader. Reported only from time step*report-from-time*onwards.*mk. sh. edge of leader*: Market share edge of the leader, i.e. difference between the market share of the leader and the next firm with the greatest market share.Reported only from time step*report-from-time*onwards.

*Herfindahl index*: Time series of the Herfindahl index.*Number of firms*: Time series of the number of firms.*Market share*: Histogram of the market shares*s*._{it}*Competitiveness*: Histogram of firms' competitiveness*γ*._{it}*Price*: Histogram of firms' prices*p*._{it}*Relative performance*: Histogram of firms' performances divided by the highest performance at the time, i.e.*x*/_{it}*x*.^{max}_{t}*Firms' life*: Time series of firms' lifespan. Each row represents a firm. Each new entrant is added as a row placed just above the previous rows. A line is drawn in each row as long as the corresponding firm is alive. When the firm exits the market, its corresponding line stops.*Age distribution*: Histogram of firms' ages.*R&D over profits*: Histogram of firms' proportion of profits that devote to R&D spending:*r*._{it}

InDyTeRRoD (Industrial Dynamics, Technological Regimes and the Role of Demand) is an evolutionary model designed to explore industrial dynamics in alternative technological regimes.

Copyright (C) 2011 Isabel Almudí, Francisco Fatás & Luis R. Izquierdo

This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You can download a copy of the GNU General Public License by clicking here; you can also get a printed copy writing to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.

Contact information:

Luis R. Izquierdo

University of Burgos, Spain.

e-mail: lrizquierdo@ubu.es

This program has been designed by Isabel Almudí, Francisco Fatás & Luis R. Izquierdo, and implemented by Luis R. Izquierdo.

**Fatas-Villafranca, F., Saura, D. and Vazquez, F.J. (2011)**. A Dynamic Model of Public Opinion Formation. Journal of Public Economic Theory, 13 (3), 417-441.**Metcalfe, J.S. (1998)**. Evolutionary Economics and Creative Destruction. Routledge. London.**Nelson, R.R. (1982)**. The Role of Knowledge in R&D Efficiency. Quarterly Journal of Economics, 97 (3), 453-470.**Silverberg, G. and Verspagen, B. (2005)**. Evolutionary Theorizing on Economic Growth. In Dopfer K. (ed) The Evolutionary Foundations of Economics. Cambridge University Press. Cambridge, UK.