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Stackelber Game Model with Sticky Prices

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El Mercado de las Telecomunicaciones en M´xico: e Un modelo din´mico de competencia. a Mar´ Fernanda Porras Jacobo ıa Germ´n Villar Argueta a Centro de Investigaci´n y Docencia Econ´micas o o
22 de noviembre del 2013

Resumen

¿Qu´ precios se fijar´n en la industria de las Telecomunicaciones en M´xico dada e a e la reciente apertura comercial? Esta pregunta se analiza en el contexto de un juego secuencial, en el cual la empresa l´ ıder sol´ ser un monopolio. No obstante ıa con la reforma al mercado sus seguidoras iniciar´n una competencia secuencial a con precios r´ ıgidos. Los resultados muestran que no hay un estado estacionario ya que debido a las rigideces en precios y a rendimientos a escala crecientes.

1

´
Indice
1. Introducci´n o 3

2. Literatura Relacionada: El Modelo de Fujiwara

4

3. El Modelo: Mercado de las Telecomunicaciones

5

3.1. Costos Distintos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

3.2. Con Costos Id´nticos e Iguales a Cero . . . . . . . . . . . . . . . . . . . e 7

3.3. Comparaci´n con Equilibrios Est´ticos . . . . . . . . . . . . . . . . . . o a

8

4. Resultados

8

4.1. Costos Distintos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

4.2. Con Costos Id´nticos e Iguales a Cero . . . . . . . . . . . . . . . . . . . e 9

4.3. Comparaci´n con Equilibrios Est´ticos . . . . . . . . . . . . . . . . . . o a

10

5. Conclusiones

11

6. Ap´ndice A e 14

7. Ap´ndice B e 20

2

1.

Introducci´n o La industr´ de las Telecomunicaciones en M´xico ha sufrido recientes modificaıa e ciones en su regulaci´n permitiendo una apertura comercial mayor. Esto se debe a que, o en 1976, en este mercado se dio un monopolio al darle el T´ ıtulo de Concesi´n a Tel´foo e nos de M´xico (Telmex) para explotar el servicio de conferencia telef´nica. No obstante, e o no son permitidas las pr´cticas monop´licas en M´xico; por lo que recientemente se a o e busc´ regular y abrir el mercado. o Con la reciente apertura y regulaci´n del mercado de las Telecomunicaciones o se esperar´ una entrada de firmas al mercado, lo que a su vez generar´ una mayor ıa ıa competencia en precios y en cantidad para ganar la demanda del mercado. Por lo tanto, resulta pertinente preguntarse, ¿c´mo se fijar´n los nuevos precios en un mercado m´s o a a competitivo en las Telecomunicaciones en M´xico? Se espera que los precios terminen e convergiendo en el estado estacionario y adem´s cuando ambas firmas tengan costos a marginales nulos se dar´ una competencia reduciendo precios hasta llegar al equilibrio a de Bertrand.

Para poder contestar esta pregunta, se utiliz´ una modificaci´n del modelo de o o
Fujiwara (2006) [1] en los costos que tienen las firmas. Los costos de la firma l´ ıder (es decir, el que sol´ ser monopolio) siempre ser´n cero, mientras que para la empresa ıa a seguidora (o entrante) ser´n decrecientes y luego ser´n cero. No obstante, los precios a a no camb´ instant´neamente, este modelo incorpora la r´ ıan a ıgidez de estos al venderse las l´ ıneas telef´nicas v´ contratos con una duraci´n particular, por lo que estos precio o ıa o podr´n cambiar con el tiempo seg´n su grado de rigidez. a u

3

Los resultados obtenidos permiten ver como la apertura comercial al dividirse las conseciones generan una mayor competencia, aunque no un equilibrio din´mico a estacionario. Esto se debe a que al abrir la entrada, si no hay costos para ninguna empresa o son iguales, por la existencia de precios r´ ıgidos podr´n no llegar al equilibrio a de Bertrand y no mantener ning´n equilibrio a lo largo del tiempo, pero s´ habr´ baja u ı a en los precios, sin tener que llegar, tal vez, al equilibrio de Bertrand. Cuando se tienen costos distintos, resulta l´gico pensar que la firma m´s eficiente, en este caso la l´ o a ıder, puede robarse el mercado, no obstante por las rigideces de precios puede que esto no suceda. Aunado a esto, el tipo de costos de la empresa seguidora motivaban a que tuviera rendimientos crecientes, por lo que buscar´ producir infinito hasta conseguir ıa que sus costos fueran iguales a cero.

Este trabajo est´ organizado de la siguiente manera: En la segunda secci´n explica a o el modelo de Fujiwara y los resultados obtenidos por esta literatura. Posteriormente se presentar´ el modelo propuestos y en la secci´n cuatro se presentar´n los resultados; a o a de esta forma, poder concluir en la ultima secci´n.
´
o

2.

Literatura Relacionada: El Modelo de Fujiwara
El modelo de Stackelberg permite revisar las acciones dominantes de los jugadores

si no juegan simult´neamente. En el trabajo de Fujiwara (2006) [1] busca caracterizar a el equilibrio de este juego en un duopolio din´mico debido a que si no se considera la a temporalidad puede llegar a existir inconsistencias.

El trabajo busca aplicar un modelo donde la firma lider maximiza sus beneficios para que esta se convierta en la funci´n de reacci´n del seguidor. Posteriormente se o o utilizar´ un factor de descuento para resolver el problema y encontrar un equilibrio a 4

consistente.

Posteriormente analiza cu´les son las diferencias entre un modelo con din´mico y a a est´tico, probando que el din´mico lleva a la obtenci´n de mayores beneficios y c´mo, a a o o a su vez, los precios bajan cuando pueden maximizar a lo largo del tiempo.

Fujiwara (2006) [1] muestra con su trabajo como una optimizaci´n intertemporal o en un juego secuencial permit´ una mejor acci´n por parte de los jugadores, o al menos, e o la empresa l´ ıder se encuentra en una mejor posici´n si cambia a lo largo del tiempo. o Adem´s de esto, el modelo de Fujiwara logr´ encontrar un punto estacionario lo que a a permite hacer la comparaci´n entre equilibrios m´s f´cil. o a a

3.
3.1.

El Modelo: Mercado de las Telecomunicaciones
Costos Distintos
Esta modificaci´n al modelo de Fujiwara permite explicar porque existir´ una o ıa

competencia en los precios con las firmas entrantes y sobre todo como al inicio la nueva firma tiene que hacer inversiones por cada nuevo cliente que con el tiempo se reducir´n a tendiendo a cero. El inter´s de estudiar esta parte recae en saber si los costos que e pueden tener al iniciar podr´ resultar una limitante a la entrada por parte de las ıan nuevas firmas, o simplemente estas no ser´ problema alguno, en la fijaci´n de los ıan o precios. Tambi´n permitir´ explicar si la empresa l´ e a ıder tiene algun beneficio extra por ser la l´ ıder. Los costos de la firma 2 estar´n dados por la funci´n a o

C(x) = ln(x2 + 1) − 1
5

la cual ser´ siempre decreciente, pero tambi´n siempre mayor o igual a cero. De esta a e forma nos permitir´ modelar unos costos marginales que ir´n reduci´ndose seg´n vaya a a e u expandi´ndose la compa˜ia. e n

Se plantea el siguiente problema de maximizaci´n para las firmas, recordando que o se resolver´ primero el problema de la firma seguidora (firma 2) para poder utilizarla a como funci´n de reacci´n para la firma lider. El problema que se debe resolver para la o o firma entrante es:

m´x = px2 (t) − ln(x2 (t) + 1) a x2 (t)

s.a. p = a − x1 (t) − x2 (t)

Una vez resuelto este problema, la firma 1 podr´ maximizar sus beneficios tomana do en cuenta la mejor respuesta del la firma entrante. Al problema al que se enfrenta la firma es:


m´x a x1 (t)

0

e−rt px1 (t)dt

s.a. p = s(a − p − x1 (t) − r(x1 (t))
˙
Donde, la s representa el nivel de flexibilidad que tienen los precios, a es un par´metro a arbitrario estrictamente mayor a cero y r es la tasa de inter´s la cual debe cumplir con e r > 0.

El problema suguiere que la firma entrante va a maximizar dada la demanda y la funci´n de costos, mientras que la firma 1 ya va a incorporar c´mo maximizar´ en o o a este y los siguientes periodos, para as´ maximizar sus beneficios. ı 6

3.2.

Con Costos Id´nticos e Iguales a Cero e Esta modificaci´n al modelo original da una mejor explicaci´n al momento en el o o cual ambas firmas ya no tienen costos; es decir,

C(xi ) = 0

Y se encuentran en la misma situaci´n, en la cual ambas resultan ser eficientes y la o competencia recae unicamente en qu´ precio pueden poner para llevarse una mayor
´
e parte del mercado. El inter´s recae en saber qu´ pasa si el costo de las l´ e e ıneas ya est´ cubierto o alguien m´s incurre en ese gasto (que no sean las firmas) cu´l ser´ el a a a a comportamiento ´ptimo de ambas. o El problema que maximiza los beneficios de la firma seguidora es el siguiente:

m´x = px2 (t) a x2 (t)

s.a. p = a − x2 (t) − x2 (t)
1
Este problema se simplifica con la eliminaci´n de los costos y permite explicar cuando o las dos compa˜ias son igual de eficientes qu´ sucede a lo largo del tiempo y si se n e encuentra un estado estacionario en el equilibrio din´mico. a Por su parte, la firma 1 resuelve el mismo problema que en la secci´n anterior, o lo que cambia es la funci´n de reacci´n dado por el problema de maximizaci´n de la o o o segunda firma1 .


m´x a x1 (t)

1

0

e−rt px1 (t)dt

Los par´metros siguen representando lo mismo que en la secci´n 3.1 a o

7

s.a. p = s(a − p − x2 (t) − r(x1 (t))
˙
1

3.3.

Comparaci´n con Equilibrios Est´ticos o a
El autor propone la comparaci´n entre los equilibrios est´ticos y din´micos para o a a ver cu´l resultaba m´s ´ptimo, por lo que tambi´n aqu´ se buscar´ hacer una compaa a o e ı a raci´n entre ambos equilibrios para corroborar si en el caso de las telecomunicaciones o tambi´n se cumple lo que dice Fujiwara (2006) [1]. Para esto, el autor recomienda suse tituir r = 0 y s = 1 en los optimos; de esta forma, no tendr´s la cuesti´n de tasa de
´
a o inter´s ni precios que cambien parcialmente a lo largo del tiempo. e No obstante, se puede resolver el problema omitiendo la dependencia de t de las variables y volviendo a resolver el problema de competencia a la Stackelberg y de esta forma encontrar los optimos.
´

4.
4.1.

Resultados
Costos Distintos
Al inicio de este trabajo se busca explicar c´mo ser´ los equilibrios y estados o ıan

estacionarios cuando dos empresas compet´ y una ten´ costos decrecientes. No obsıan ıa tante, la realizaci´n del problema mostr´ que si hab´ costos distintos y decrecientes, o o ıan no se encontrar´ un estado estacionario. ıa Resultar´ pertinente preguntarse, por qu´ con costos decrecientes no se enıa e contr´ un equilibrio, esto se debe a que si tienes costos decrecientes tu rendimientos a o escalas son siempre crecientes, por lo tanto lo que buscas es producir infinito, no obstante la condici´n de producir infinito es imposible, ya que no hay suficiente ”dotaci´n.o o o
8

existe una escacez de este. Aun cuando estos resultados soprenden, s´ se puede decir ı que en un contexto de optimizar en el tiempo si tienes costos marginales decrecientes siempre buscar´s producir infinito. Lo que se podr´ estudiar es el caso de costos fijos a ıa decrecientes en el tiempo que no tengan que ver con la cantidad producida.

El resultado de la presente investigaci´n cuando los costos son distintos permite o darse cuenta que a´n cuando se desee tener rendimientos de este tipo, no son los m´s u a optimos a veces porque no existe el estado estacionario; es decir, no hay un equili´ brio sostenible lo que llevar´ a desviaciones. Adem´s puede generar una competencia a a agresiva de precios predatorios para que saquen al d´bil del mercado; no obstante, si la e competencia se hace m´s grande, es decir entran m´s firmas este mecanismo podr´ a´n a a a u albergar competencia entre firmas.

4.2.

Con Costos Id´nticos e Iguales a Cero e El caso de los costos id´nticos e iguales a cero, es a´n m´s particular. Como e u a son constantes esperas rendimientos, a su vez, constantes; no obstante, el resultado del problema nos arroj´ que no se encontr´ un equilibrio estacionario. Esto se debe a o o que al existir una competencia en precios y tener costos marginales iguales a cero, las empresas buscar´n una competencia mucho m´s fuerte y podr´ llegar al equilibrio a a ıan Bertrand.

Sin embargo, este equilibrio de Bertrand no resulta sostenible, ya que la funci´n o de precios est´ dada por la cantidad producida del otro; es decir, si uno produce mucho a los precios del contrincante van a bajar, entonces ese empezar´ a bajar m´s los precios a a hasta que lleguen a cero, que ser´ el equilibrio de un juego de Bertrand, no obstante, se ıa trata de un juego secuencial. Si la firma lider primero fija un precio la firma seguidora

9

lo podr´ bajar y en el siguiente periodo la firma lider har´ lo mismo. El problema a a recae en que los precios que se manejan no son flexibles y pueden cambiar en cada periodo, son r´ ıgidos por un contrato que as´ lo establece. De esta manera, el equilibrio ı de Bertrand no resulta sostenible, porque si no entran nuevos compradores al mercado y tienen un contrato que seguir entonces a´n cuando la contrincante baje sus precios u puede que, en caso de existir una penalizaci´n no la cubra, o no genere los suficientes o incentivos para cambiar de proveedor.

La existencia de precios pegajosos en el mercado produce una inconsistencia temporal ya que no permite ajustar a estos cambios en el mercado. Por lo que el equilibrio no ser´ est´tico porque dependiendo en que punto en el tiempo de los contratos se a a encuentren podr´n existir incentivos a desviarse. Adem´s los contratos no tienen la a a misma temporalidad, por lo que no siempre genera la competencia en precios el resultado deseado por las firmas.

4.3.

Comparaci´n con Equilibrios Est´ticos o a
Respecto a los equilibrios est´ticos, se espera que se llegue en el caso de costos a iguales al resultado de Bertrand, esto se debe a que est´s quitando el efecto de los a precios r´ ıgidos por lo que podr´n iniciar poniendo un precio ambas firmas e irlo bajando a conforme vean la respuesta del oponente para ganar toda la demanda. No obstante, esto lleva a que se reduzca cada vez m´s los precios hasta llegar a P = CM g. a En el caso de costos distintos, se sabe que en este caso la empresa l´ ıder resulta la eficiente, entonces al ser esta la eficiente puede poner el precio por debajo del costo marginal del oponente aunque sea en una cantidad m´ ınima ganando as´ todo el mercaı do e incentivando la salida del otro a menos que se vuelva m´s competitivo. Para que el a 10

seguidor se vuelva m´s competitivo tiene que aumentar sus unidades producidas para a que su costo marginal tienda a cero. Pero si no tiene demanda tampoco tendr´ incena tivos a hacerlo generando de esta forma la posibilidad que la empresa l´ ıder se quede como monopolista, sacando a la otra.

Lo anterior permite estudiar como la regulaci´n en materia de Telecomunicacioo nes, a´n cuando no tiene un equilibrio, s´ consigue el objetivo de mayor competencia, u ı aunque esta se prolongue y la desaparici´n del monopolio, ya que permite que en el o tiempo las r´ ıgideces en precios puedan repartir el mercado, consiguiendo mayor poder de mercado en ciertos momentos y reducirlo en otro. De esta forma pueden generar mejores servicios de Telecomunicaciones.

5.

Conclusiones
El presente trabajo analiz´ las repercusiones de la apertura del mercado de Teo

lecomunicaciones en M´xico mediante modificaciones al modelo original de Fujiwara e (2006) [1], cambiando las funciones de costos e incluso anularlas. Esto con el objetivo de representar mejor a esta industria y como se piensa que se comportar´ con la apertura. a En el trabajo de Fujiwara (2006) [1], presenta un modelo de Stackelber din´mia co con precios r´ ıgidos para observar la existencia de un estado estacionario y de esta forma poder arreglar las incosistencias que se presentan si no se toma en cuenta la temporalidad. Esto se debe a que hay diferencias en los equilibrios, los agentes maximizan distinto e incluso tomando en consideraci´n la temporalidad, los agentes lo o hacen mejor.

11

El modelo que se presenta en este trabajo resulta una modificaci´n en los costos. o Esto se debe a que en la industria de las Telecomunicaciones los costos no est´n o se a eliminan en alg´n punto. Para explicar ambos escenarios se propuso un modelo con u costos marginales decrecientes para que estos pudieran explicar que suceder´ si una ıa empresa entra, dado esta apertura, y tiene estos peque˜os costos por l´ n ıneas, equipo, etc... que entre m´s vendan menores ser´n; mientras que la empresa lider ya no tiene a a estos costos porque lleva m´s en el mercado. No obstante, se considera que el mercado a llevar´ a la firma seguidora a tener los mismos costos, de esta forma se propone un sea gundo modelo sin costos marginales, para ver si esto llevar´ a un equilibrio estacionario ıa similar al de Bertrand.

Los resultados obtenidos no fueron lo que se pens´, ya que un resultado que o se pod´ predecir era que iban a llegar a una convergencia en un estado estacionario ıa similar al de Bertrand; no obstante, eso no pas´. Lo que se mostr´ en el trabajo fue que, o o al contrario, con los modelos propuestos no se puede llegar a un equilibrio estacionario, debido al comportamiento funcional. Esto se debe b´sicamente a que en el caso de a costos marginales decrecientes se provocan rendimientos a escala crecientes, por lo que la firma querr´ producir cada vez m´s entonces su stock ser´ enorme y sus costos a a a tender´n a cero para poder ser m´s competitiva y eficientes respecto al contricante. a a
En el caso de los costos nulos e id´nticos, se puede notar que por la rigidez en estos, e los contratos impiden que la competencia en precios de el resultado deseado, ya que al ser manejado por contratos con distintos vencimientos, en ocasiones bajar tu precio no incrementar´ tu demanda y en otra puede incrementarla mucho, depende de cu´ntos a a contratos est´n por vencerse. e Este trabajo concluye que con este tipo de costos el equilibrio no es sostenible en el tiempo, genera incosistencias pero s´ permite la competencia. Se concluye que optimizar ı 12

din´micamente, puede llevar a desviaciones a lo largo del tiempo pero s´ mejora los a ı beneficios en el caso que tal vez nunca lleguen a precio igual a costo marginal, lo que s´ pasar´ en un modelo est´tico. ı ıa a Referencias
[1] Fujiwara, K. (2006) A Stackelberg Model of Dynamic Duopolistic Competition with
Sticky Prices, ”Economics Bulletin”, 12, 12, pp. 1-9.
[2] Mancero, X. Saavedra, E. (2006) Un Modelo de Entrada y Competencia en Telecomunicaciones, Revista de an´lisis econ´mico”, 21, 1, pp. 29 - 57. a o
[3] Dockner, E., S. Jorgensen, N. V. Long and G. Sorger (2000), Differential games in economics and management science, New York: Cambridge University Press.

13

6.

Ap´ndice A e Con Costos Distintos

Primero se resolver´ el problema de la firma 2, para que la firma 1 optimice dadas a las preferencias de este. El problema de la firma dos es:

m´x = px2 − ln(x2 + 1) a x2

s.a. p = a − x1 − x2

Se sustituye la restricci´n en la funci´n que se maximizar´: o o a m´x(a − x1 − x2 )x2 − ln(x2 + 1) a x
2

Encontramos las condiciones de primer orden:

a − x1 − 2x2 −

1
=0
x2 + 1

Entonces encontramos el valor ´ptimo de x2 (1): o x2 =

−(x1 + 2 − a) ± (a − x1 + 2)2 − 8
4

(1)

Se debe verificar para qu´ casos x2 > 0, de lo contrario significar´ que la firma no e ıa producir´ ıa:
± (a − x1 + 2)2 − 8 > (x1 + 2 − a)

14

Caso 1 x1 + 2 − a > 0
+ (a − x1 + 2)2 − 8
En este caso, lo que es necesario que se cumpla es:

(a − x1 + 2)2 − 8 > (x1 + 2 − a)2

Al desarrollar la ecuaci´n se obtiene que: o x1 < a − 1

No obstante, la condici´n inicial dice que x1 > a − 2, por lo tanto, dada esta o condici´n s´lo se cumplir´ en el intervalo (a − 2, a − 1). Como no aplica para todos los o o a valores de x1 no se considera este caso.
Caso 2 x1 + 2 − a < 0
− (a − x1 + 2)2 − 8
En este caso, lo que es necesario que se cumpla es:

(a − x1 + 2)2 − 8 > (x1 + 2 − a)2

Al desarrollar la ecuaci´n se obtiene que: o x1 < a − 1

15

No obstante, la condici´n inicial dice que x1 > a − 2, por lo que la condici´n inicial o o ya est´ estableciendo que no ser´ negativo por lo que es un caso posible. a a

Caso 3 x1 + 2 − a > 0
− (a − x1 + 2)2 − 8
En este caso, lo que es necesario que se cumpla es:

(a − x1 + 2)2 − 8 < (x1 + 2 − a)2

Al desarrollar la ecuaci´n se obtiene que: o x1 > a − 1

No obstante, la condici´n inicial dice que x1 > a − 2, por lo que la condici´n inicial o o ya est´ estableciendo que no ser´ negativo por lo que es un caso factible. a a

Caso 4 x1 + 2 − a < 0
− (a − x1 + 2)2 − 8
En este caso, lo que es necesario que se cumpla es:

(a − x1 + 2)2 − 8 < (x1 + 2 − a)2

16

Al desarrollar la ecuaci´n se obtiene que: o x1 > a − 1

No obstante, la condici´n inicial dice que x1 < a − 2, lo que muestra que este caso o no ser´ factible, ya que dada la condici´n inicial no puede darse la condici´n de no a o o negatividad.
Ahora se resuelver´ el problema de la firma 1, s´lo recordando que los dos casos a o factibles son 2 y 3 que es:


m´x a x1 (t)

0

e−rt px1 (t)dt

s.a. p = s(a − p − x1 (t) − r(x1 (t)))
˙
Sustituimos la funci´n de reacci´n, para obtener la restricci´n de la siguiente mao o o nera: p = s(a − p − x1 (t) −
˙

−(x1 (t) + 2 − a) ± (a − x1 (t) + 2)2 − 8
4

s p = (3(a − x1 (t)) − 2(2p − 1)
˙
4

)

((a − x1 (t) + 2)2 − 8))

Para esto se calcula el Hamiltoniano: s H = px1 + λ (3(a − x1 (t)) − 2(2p − 1)
4

(a − x1 (t) + 2)2 − 8))

Encontramos las condiciones de optimalidad:
−(a − x1 (t) + 2)

δH s = 0 = p + λ (−3 δx1 4

((a − x1 (t) + 2)2 − 8)

δH
˙
˙
= −λ + λr → λ = λ(r + s) − x1 δp 17

)

(2)

(3)

s p = (3(a − x1 (t)) − 2(2p − 1)
˙
4

((a − x1 (t) + 2)2 − 8))

l´ λ(t) = 0 ım (4)

(5)

t→∞

Despejo x1 de la ecuaci´n (2): o 4p (a − x1 + 2)2 − 8 + sλ(−3 (a − x1 + 2)2 − 8

−(a − x1 + 2) = 0

4p (a − x1 + 2)2 − 8 + sλ(−3 (a − x1 + 2)2 − 8 ± (a − x1 + 2) = 0
4p (a − x1 + 2)2 − 8 − 3λs (a − x1 + 2)2 − 8 ± sλ(a − x1 + 2) = 0
4p (a − x1 + 2)2 − 8 − 3λs (a − x1 + 2)2 − 8 =

sλ(a − x1 + 2)

16p2 ((a−x1 +2)2 −8)−9s2 λ2 ((a−x1 +2)2 −8)−12psλ((a−x1 +2)2 −8) = s2 λ2 (a−x1 +2)2
(a − x1 + 2)2 (16p2 − 9s2 λ2 − 12psλ − s2 λ2 ) − 8(16p2 − 9s2 λ2 − 12psλ) = 0
(a − x1 + 2)2 4(4p2 − 2s2 λ2 − 3psλ) = 8(16p2 − 9s2 λ2 − 12psλ)
(a − x1 + 2)2 =

2(16p2 − 9s2 λ2 − 12psλ)
(4p2 − 2s2 λ2 − 3psλ)

x1 = a + 2

2(16p2 − 9s2 λ2 − 12psλ)
(4p2 − 2s2 λ2 − 3psλ)

(6)

Sustituyo (6) en (3) y (4)

˙ λ = λ(r + s) − (a + 2

2(16p2 − 9s2 λ2 − 12psλ)
)
(4p2 − 2s2 λ2 − 3psλ)

18

(7)

2(16p2 − 9s2 λ2 − 12psλ) s )−2(2p−1) p = (3(−2±
˙
4
(4p2 − 2s2 λ2 − 3psλ)

2(16p2 − 9s2 λ2 − 12psλ) 2
)) − 8))
(4p2 − 2s2 λ2 − 3psλ)
(8)

((a − (a + 4

Se resuelve la matriz Jacobiana para revisar si existe una estabilidad de punto:2
(−18s2 λ−12ps)(4p2 −2s2 λ2 −3psλ)+(4s2 λ+3ps)(16p2 −9s2 λ2 −12psλ)

˙ δλ δλ

= (r + s) ±

˙ δλ δp



δp
˙
δλ

s
= ( 4 )(±3( (−18s

(a−(a+4

2(16p2 −9s2 λ2 −12psλ)
(4p2 −2s2 λ2 −3psλ)

(8p−3sλ)(s2 λ2 )
2(16p2 −9s2 λ2 −12psλ)
(4p2 −2s2 λ2 −3psλ)

2 λ−12ps)(4p2 −2s2 λ2 +3psλ)+(4s2 λ+3ps)(16p2 −9s2 λ2 −12psλ)

2(16p2 −9s2 λ2 −12psλ)
))(
(4p2 −2s2 λ2 −3psλ)

((a−(a+4

2(16p2 −9s2 λ2 −12psλ)
(4p2 −2s2 λ2 −3psλ)

s
= ( 4 )(±3(

(8p−3sλ)(s2 λ2 )
2(16p2 −9s2 λ2 −12psλ)
(4p2 −2s2 λ2 −3psλ)

)

)

2(16p2 −9s2 λ2 −12psλ) 2
)) −8)
(4p2 −2s2 λ2 +3psλ)

2(16p2 −9s2 λ2 −12psλ)
))(
(4p2 −2s2 λ2 −3psλ)

(a−(a+4 δp ˙ δp )

2(16p2 −9s2 λ2 −12psλ)
(4p2 −2s2 λ2 −3psλ)
2 λ−12ps)(4p2 −2s2 λ2 −3psλ)+(4s2 λ−3ps)(16p2 −9s2 λ2 −12psλ)
(−18s

−4p
((a−(a+4

(8p−3sλ)(s2 λ2 )
2(16p2 −9s2 λ2 −12psλ)
(4p2 −2s2 λ2 −3psλ)

)

)

2(16p2 −9s2 λ2 −12psλ) 2
)) −8)
(4p2 −2s2 λ2 −3psλ)

Cuando se escoge el signo de arriba:


˙ δλ  δλ




δp
˙
δλ

>0

˙ δλ δp

>0

δp
˙
δp



> 0
>0




La matriz es positiva, por lo tanto el equilibrio encontrado no es estacionario por lo que habr´ divergencias a lo largo del tiempo. a 2

Debido a lo extensas de las derivadas no se pondr´n en forma matricial. a 19

Cuando se escoge el signo de abajo:


˙ δλ  δλ




δp
˙
δλ

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