Dynamiczny oligopol z lepkimi cenami - mimuw.edu.plmiekisz/wiszniewska.pdf · Dynamiczny oligopol z lepkimi cenami Agnieszka Wiszniewska-Matyszkiel, Marek Bodnar i Fryderyk Mirota

Dynamicznyoligopol z lepkimi

cenami

AgnieszkaWiszniewska-

Matyszkiel,Marek Bodnar iFryderyk Mirota

The model

Literature

Previous research

Infinite horizonMaximum Principle

Open loop NashequilibriumPhase diagram

Results

Feedback NashequilibriumResults

GraphicalillustrationNumber of firms

Speed of adjustment

Asymptotic behaviour

ConclusionsConclusions cont.

Dynamiczny oligopol z lepkimi cenami

Agnieszka Wiszniewska-Matyszkiel,Marek Bodnar i Fryderyk Mirota

Instytut Matematyki Stosowanej i MechanikiUniwersytet Warszawski

10. wrzesnia 20156 Forum Matematykow Polskich PTM,

Warszawa


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Plan of presentationThe model

Literature

Previous research

Infinite horizon Maximum Principle

Open loop Nash equilibriumPhase diagramResults

Feedback Nash equilibriumResults

Graphical illustrationNumber of firmsSpeed of adjustmentAsymptotic behaviour



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The model

I N identical firms with cost ci(q) = q2

2 + cq;I inverse market demand P(q1, . . . , qN) = A −

∑qi (price

resulting from firms’ decisions).

I So we have a game with N players, each of them withset of strategies R+ and payoffsΠi(q1, . . . , qN) = P(q1, . . . , qN)qi − ci(qi).

I The basic concept Nash equilibrium – a profile ofstrategies such that strategy of each player maximizeshis/her payoff given strategies of the remaining players.

Application of this concept to this specific model – is calledby economists Cournot oligopoly (so let’s call itCournot-Nash equilibrium).We obtainI equilibrium production level qCN

i = A−cN+2 ;

I equilibrium price pCN = 2A+NcN+2 .


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∑qi (price

resulting from firms’ decisions).I So we have a game with N players, each of them with

set of strategies R+ and payoffsΠi(q1, . . . , qN) = P(q1, . . . , qN)qi − ci(qi).



i = A−cN+2 ;



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∑qi (price





i = A−cN+2 ;



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∑qi (price




Application of this concept to this specific model – is calledby economists Cournot oligopoly (so let’s call itCournot-Nash equilibrium).

We obtainI equilibrium production level qCN

i = A−cN+2 ;



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∑qi (price





i = A−cN+2 ;



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∑qi (price





i = A−cN+2 ;



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Another way of modelling this situation by economists –perfect competition – each firm maximizes his/her payoff,treating price as independent of his/her choice.If we treat these N firms as competitive firms, we obtainI equilibrium production level qComp

i = A−cN+1 ;

I equilibrium price pComp = A+NcN+1 .

What if prices do not adjust immediately (menu costs etc.)?


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Another way of modelling this situation by economists –perfect competition – each firm maximizes his/her payoff,treating price as independent of his/her choice.If we treat these N firms as competitive firms, we obtainI equilibrium production level qComp

i = A−cN+1 ;

I equilibrium price pComp = A+NcN+1 .

What if prices do not adjust immediately (menu costs etc.)?


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Model cont.

I Sticky price equation p(t) =

s(P(q1(t), . . . , qN(t)) − p(t)) = s(A −∑N

i=1 qi(t) − p(t))for s > 0 – measuring speed of price adjustment;

I Firms consider dynamic optimization problems: firm imaximizes over qi , Ji

0,x0(q1, . . . , qN) =

=∫ ∞

0 e−ρt(p(t)qi(t) − cqi(t) −

qi(t)2

2

)dt ,

where ρ > 0, given strategies of the remaining players.I So we have a differential game.

Two formulations:I open loop strategies: qi are measurable functions of

time;I feedback strategies: qi are functions of price; in all

above definitions qi(t) is replaced by qi(p(t)).


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Model cont.

I Sticky price equation p(t) =s(P(q1(t), . . . , qN(t)) − p(t)) = s(A −

∑Ni=1 qi(t) − p(t))

for s > 0 – measuring speed of price adjustment;

I Firms consider dynamic optimization problems: firm imaximizes over qi , Ji

0,x0(q1, . . . , qN) =

=∫ ∞


qi(t)2

2

)dt ,






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Model cont.


∑Ni=1 qi(t) − p(t))

for s > 0 – measuring speed of price adjustment;I Firms consider dynamic optimization problems: firm i

maximizes over qi , Ji0,x0

(q1, . . . , qN) =

=∫ ∞


qi(t)2

2

)dt ,

where ρ > 0, given strategies of the remaining players.

I So we have a differential game.





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Model cont.


∑Ni=1 qi(t) − p(t))



(q1, . . . , qN) =

=∫ ∞


qi(t)2

2

)dt ,






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Model cont.


∑Ni=1 qi(t) − p(t))



(q1, . . . , qN) =

=∫ ∞


qi(t)2

2

)dt ,






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I So two kinds of Nash equilibria:open loop and feedbackwith different tools to obtain them

—Pontriagin maximum principle and Bellman equation,respectively).

I Usually in dynamic games those two equilibria aredifferent,unlike in optimal control problems.

I The reason: in feedback equilibrium a player in hisoptimization problem takes into account the fact that theother players’ strategies take the state variable (price inthis model) into account,unlike in the open loop (decision at every time instantdecided at the beging of the game, whatever thetrajectory of price is).

I We are interested in symmetric Nash equilibria (andprove that in open loop there are no asymmetric ones).

I We calculate the entire trajectories of prices andstrategies not only steady states.


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I So two kinds of Nash equilibria:open loop and feedbackwith different tools to obtain them—Pontriagin maximum principle and Bellman equation,respectively).






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I Usually in dynamic games those two equilibria aredifferent,

unlike in optimal control problems.I The reason: in feedback equilibrium a player in his

optimization problem takes into account the fact that theother players’ strategies take the state variable (price inthis model) into account,unlike in the open loop (decision at every time instantdecided at the beging of the game, whatever thetrajectory of price is).




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I The reason: in feedback equilibrium a player in hisoptimization problem takes into account the fact that theother players’ strategies take the state variable (price inthis model) into account,

unlike in the open loop (decision at every time instantdecided at the beging of the game, whatever thetrajectory of price is).




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I We are interested in symmetric Nash equilibria

(andprove that in open loop there are no asymmetric ones).



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I We calculate the entire trajectories of prices andstrategies

not only steady states.


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Literature

I R. Cellini, L. Lambertini, 2004, Dynamic Oligopoly withSticky Prices: Closed-Loop, Feedback and Open-LoopSolutions, Journal of Dynamical and Control Systems10, 303-314.

I C. Fershtman, M. I. Kamien, 1987, Dynamic DuopolisticCompetition with Sticky Prices, Econometrica 55,1151-1164.

I M. Simaan, T. Takayama, 1978, Game Theory Appliedto Dynamic Duopoly with Production Constraints,Automatica 14, 161-166.

I A. Wiszniewska-Matyszkiel, M. Bodnar, F. Mirota, 2014,Dynamic Oligopoly with Sticky Prices: Off-Steady-StateAnalysis, Dynamic Games and Applications, DOI10.1007/s13235-014-0125-z.


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I No exhausitive analysis of the open loop case.

I Main reason – problems with infinite horizon Pontriaginmaximum priciple.

I ”Everybody knows that Pontriagin maximum generallydoes not hold in infinite time horizon”

I Even the core relations do not have to be fulfilled.I The situation with the terminal condition (discounted

costate variable tends to 0) is even worse.I Even if we assume that both are fulfilled – very

unpleasant calculations.

I Appropriate version of Maximum Principle proven in2012

I S. Aseev, V. Veliov, 2012, Maximum principle forinfinite-horizon optimal control with dominating discount,Dynamics of Continuous, Discrete and ImpulsiveSystems 19, 43-62.


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I No exhausitive analysis of the open loop case.I Main reason – problems with infinite horizon Pontriagin

maximum priciple.

I ”Everybody knows that Pontriagin maximum generallydoes not hold in infinite time horizon”







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maximum priciple.I ”Everybody knows that Pontriagin maximum generally

does not hold in infinite time horizon”







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does not hold in infinite time horizon”I Even the core relations do not have to be fulfilled.

I The situation with the terminal condition (discountedcostate variable tends to 0) is even worse.

I Even if we assume that both are fulfilled – veryunpleasant calculations.




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does not hold in infinite time horizon”I Even the core relations do not have to be fulfilled.I The situation with the terminal condition (discounted

costate variable tends to 0) is even worse.

I Even if we assume that both are fulfilled – veryunpleasant calculations.




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I Before: only the approach ”write the core relations ofthe Pontriagin maximum principle and find the steadystate of the state-costate equation – it may be asolution.

I however, the same authors noticed that it is notLiapunov stable – it was proven to be a saddle point,

I thus, there is no reason to assume that a solution fromany other initial state will converge to it.

I Incomplete analysis in N-players feedback case.


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Assev-Veliov infinite horizon Maximum Principle

I Consider a dynamic optimization problemI Maximize

J0,x0 (u) =

∫ ∞

t=0e−ρtg(t , x(t), u(t))dt ,

where the trajectory x is the trajectory corresponding tocontrol u and it is defined byx(t) = f(t , x(t), u(t)) for t > 0,

x(0) = x0,

TheoremUnder some unpleasant technical assumptions A1–A4core relations (CR) of the Pontriagin maximum principle holdtogether with terminal condition (TC).


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Assev-Veliov infinite horizon Maximum Principle

I Consider a dynamic optimization problemI Maximize

J0,x0 (u) =

∫ ∞

t=0e−ρtg(t , x(t), u(t))dt ,

where the trajectory x is the trajectory corresponding tocontrol u and it is defined byx(t) = f(t , x(t), u(t)) for t > 0,

x(0) = x0,

TheoremUnder some unpleasant technical assumptions A1–A4core relations (CR) of the Pontriagin maximum principle holdtogether with terminal condition (TC).


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(A1) The functions f and g and their partial derivatives withrespect to x are continuous in (x, u) for every fixed tand uniformly bounded as functions of t over everybounded set of (x, u).

(A2) There exist numbers µ, r , κ, c1 ≥ 0 and β > 0 such thatfor every t ≥ 0

(i) ‖x∗(t)‖ ≤ c1eµt and(ii) for every control u for which the Lebesgue measure oft : u(t) , u∗(t) ≤ β, the corresponding trajectoryexists on R+ and ‖ ∂g(t ,y,u∗(y))

∂x ‖ ≤ κ(1 + ‖y‖r ) for everyy ∈ convx(t), x∗(t), where conv denotes the convexhull.

(A3) There are numbers η ∈ R, γ > 0 and c2 ≥ 0 such thatfor every ζ ∈ X with ‖ζ − x0‖ < γ state equation withinitial condition replaced by x(0) = ζ has a solution xζ

defined on R+, such that xζ(t) ∈ X, for all t ≥ 0, and

‖xζ(t) − x∗(t)‖ ≤ c2‖ζ − x0‖eηt .

(A4) ρ > η + r maxη, µ for r , η, µ from (A2) and (A3).


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Assev-Veliov Maximum Principle — 2I Hamiltonian: H(x, t , u, ψ) = g(t , x, u) + 〈ψ, f(t , x, u)〉

I Let (x∗, u∗) be the optimal pair and A1–A4 hold,I then there exists an absolutely continuous costate

variable ψ∗ such that

(i) (CR)I for a.e. t , u∗(t) maximizes the hamiltonian

H(x∗(t), t , u, ψ∗(t)),I ψ∗(t) = −

∂H(x∗(t),t ,u∗(t),ψ∗(t))∂x ,

(ii) (TC)I for every t ≥ 0 the integral

I∗(t) =∞∫t

e−ρw[Z(x∗,u∗)(w)

]−1 ∂g(w,x∗(w),u∗(w))∂x dw,

I where Z(x∗ ,u∗)(t) is the normalised fundamental matrix ofthe following linear systemz(t) = − ∂f(x∗(t),t ,u∗(t))

∂x z(t),

converges absolutely, and

(iii) I∗(t) = [Z(x∗,u∗)(t)]−1ψ∗(t).


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Open loop

Application of the Pontriagin maximum principle tooptimization of player i, given strategies of the remainingplayers qj .I Present value hamiltonian

HPVi (p, t , qi , λi) = pqi − cqi −

q2i

2 + λis(A −∑

j,i qj(t)− qi)

I for redefined costate variable λi(t) := ψ(t)eρt .

I Costate variable – shadow price λi fulfilsI λi(t) = λiρ −

∂HPVi (p(t),t ,qi(t),λi(t))

∂pI with transversality condition λi(t)e−ρt → 0,I and λi(t) > 0 for every t .

I Optimal strategyqi(t) ∈ Argmaxqi∈R+

HPVi (p(t), t , qi , λi(t)).


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Open loop



q2i

2 + λis(A −∑

j,i qj(t)− qi)

I for redefined costate variable λi(t) := ψ(t)eρt .I Costate variable – shadow price λi fulfils

I λi(t) = λiρ −∂HPV

i (p(t),t ,qi(t),λi(t))∂p

I with transversality condition λi(t)e−ρt → 0,I and λi(t) > 0 for every t .




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Open loop



q2i

2 + λis(A −∑

j,i qj(t)− qi)

I for redefined costate variable λi(t) := ψ(t)eρt .I Costate variable – shadow price λi fulfils

I λi(t) = λiρ −∂HPV

i (p(t),t ,qi(t),λi(t))∂p

I with transversality condition λi(t)e−ρt → 0,I and λi(t) > 0 for every t .




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Open loop – dynamics of state and costatevariables

I The results of optimization imply that the linep = sλ+ c splits the nonnegative quadrant of (λ, p) intoΩ1 (below) on which qi = 0

and Ω2 (above), on which qi = p − c − λs.I The resulting costate and state equations are:

I λ =

(ρ + 2s)λ − p + c, (λ, p) ∈ Ω2,

(ρ + s)λ, (λ, p) ∈ Ω1,and

I p =

Ns2λ − (N + 1)sp + As + Ncs, (λ, p) ∈ Ω2,

−sp + As, (λ, p) ∈ Ω1.


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and Ω2 (above), on which qi = p − c − λs.

I The resulting costate and state equations are:

I λ =

(ρ + 2s)λ − p + c, (λ, p) ∈ Ω2,

(ρ + s)λ, (λ, p) ∈ Ω1,and

I p =

Ns2λ − (N + 1)sp + As + Ncs, (λ, p) ∈ Ω2,

−sp + As, (λ, p) ∈ Ω1.


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I λ =

(ρ + 2s)λ − p + c, (λ, p) ∈ Ω2,

(ρ + s)λ, (λ, p) ∈ Ω1,and

I p =

Ns2λ − (N + 1)sp + As + Ncs, (λ, p) ∈ Ω2,

−sp + As, (λ, p) ∈ Ω1.


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I λ =

(ρ + 2s)λ − p + c, (λ, p) ∈ Ω2,

(ρ + s)λ, (λ, p) ∈ Ω1,

and

I p =

Ns2λ − (N + 1)sp + As + Ncs, (λ, p) ∈ Ω2,

−sp + As, (λ, p) ∈ Ω1.


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I λ =

(ρ + 2s)λ − p + c, (λ, p) ∈ Ω2,

(ρ + s)λ, (λ, p) ∈ Ω1,and

I p =

Ns2λ − (N + 1)sp + As + Ncs, (λ, p) ∈ Ω2,

−sp + As, (λ, p) ∈ Ω1.


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Open loop – phase diagram of (λ, p)

λ

p

c

sλ+ c(ρ+ 2s)λ+ c

A+NcN+1

A

A−cs

Γ1

Γ2

Ω1

Ω2

Figure: Solid red line with vertical bars – λ-null-cline. Solid greenline with horizontal bars – p-null-cline. Dark brown thick line witharrows denotes the stable saddle path. Dashed blue line isp = sλ + c that divides the first quarter into region Ω1 (below thisline) and Ω2 (above it).


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Open loop – results of Pontriagin maximumprinciple

I Given initial condition p0, there exists unique λ0, suchthat the necessary conditions are fulfilled.

I (λ, p) is always at the stable saddle path.I We have global asymptotic stability!I Apparent instability in previous research caused by

either misunderstanding of the concept of costatevariable or omitting the terminal condition – which is apart of necessary condition.

I There exists a unique open loop Nash equilibrium and itis symmetric.

I Let us denote the intersection of the stable saddle pathwith line p = sλ + c by (λ, p).

If p(t) < p then qi(t) = 0, otherwiseqi(t) = p(t) − c − λ(t)s.


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I (λ, p) is always at the stable saddle path.

I We have global asymptotic stability!I Apparent instability in previous research caused by






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I (λ, p) is always at the stable saddle path.I We have global asymptotic stability!

I Apparent instability in previous research caused byeither misunderstanding of the concept of costatevariable or omitting the terminal condition – which is apart of necessary condition.





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Feedback Nash equilibria – The Bellmanequation

If a C1 function Vi fulfilsI the Bellman equation

ρVi(p) = supqi≥0 pqi−cqi−q2

i2 +V ′i (p)s(A−

∑j,i qj(p)−qi)

I with the terminal condition Vi(p(t))e−ρt → 0 for everyadmissible trajectory of prices

then

I Vi is the value function of player i given strategies of theremaining players;

I qi(p) ∈

Argmaxqi≥0pqi − cqi −q2

i2 + V ′i (p)s(A −

∑j,i qj(p) − qi)

defines optimal strategy of player i given strategies ofthe remaining players.


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i2 +V ′i (p)s(A−

∑j,i qj(p)−qi)


then


I qi(p) ∈


i2 + V ′i (p)s(A −




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i2 +V ′i (p)s(A−

∑j,i qj(p)−qi)


then


I qi(p) ∈


i2 + V ′i (p)s(A −




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i2 +V ′i (p)s(A−

∑j,i qj(p)−qi)


then


I qi(p) ∈


i2 + V ′i (p)s(A −




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Symmetric feedback Nash equilibrium – resultsI The game is linear-quadratic, so assume quadratic

value function, identical for all players, and calculate thecoefficients.

I It does not work – for p below some p equilibriumproduction turns out to be negative.

I So change the candidate for equilibrium strategiesbelow p to 0 and the value function correspondigly.

I Check the terminal condition to exclude one ofsolutions.

I The value function isVi(p) =

kp2

2 + hp + g for p ≥ p = c+sh1−sk ,

(A − p)−ρs (A − p)

ρs

(k p2

2 + hp + g)

otherwise.

for unique k , h and g, with k > 0.I Production at Nash equilibrium is

qi(p) =

p − c − s(kp + h) if p ≥ p,

0 otherwise,


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kp2

2 + hp + g for p ≥ p = c+sh1−sk ,

(A − p)−ρs (A − p)

ρs

(k p2

2 + hp + g)

otherwise.


qi(p) =

p − c − s(kp + h) if p ≥ p,

0 otherwise,


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kp2

2 + hp + g for p ≥ p = c+sh1−sk ,

(A − p)−ρs (A − p)

ρs

(k p2

2 + hp + g)

otherwise.


qi(p) =

p − c − s(kp + h) if p ≥ p,

0 otherwise,


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kp2

2 + hp + g for p ≥ p = c+sh1−sk ,

(A − p)−ρs (A − p)

ρs

(k p2

2 + hp + g)

otherwise.


qi(p) =

p − c − s(kp + h) if p ≥ p,

0 otherwise,


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kp2

2 + hp + g for p ≥ p = c+sh1−sk ,

(A − p)−ρs (A − p)

ρs

(k p2

2 + hp + g)

otherwise.

for unique k , h and g, with k > 0.

I Production at Nash equilibrium is

qi(p) =

p − c − s(kp + h) if p ≥ p,

0 otherwise,


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kp2

2 + hp + g for p ≥ p = c+sh1−sk ,

(A − p)−ρs (A − p)

ρs

(k p2

2 + hp + g)

otherwise.


qi(p) =

p − c − s(kp + h) if p ≥ p,

0 otherwise,


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Open and closed loop production

t

q

open loopfeedback

qComp

qCN

t t 1 2 3 4 5 6

0.5

1

1.5

Figure: Open loop and feedback equilibria for the same initialprice, for A = 10, c = 1, ρ = 0.15, s = 0.2, N = 10; staticCournot-Nash and competitive production levels for comparison.

Correction by effect caused by dependence of other players’strategies on price in the feedback case!


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Open and closed loop production

t

q

open loopfeedback

qComp

qCN

t t 1 2 3 4 5 6

0.5

1

1.5

Figure: Open loop and feedback equilibria for the same initialprice, for A = 10, c = 1, ρ = 0.15, s = 0.2, N = 10; staticCournot-Nash and competitive production levels for comparison.

Correction by effect caused by dependence of other players’strategies on price in the feedback case!


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Open and closed loop price

t

p

open loop

feedback

pComp

pCN

t t 1 2 3 4 5 6

1

2

3

4


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Open and closed loop Nash equilibria as thenumber of firms increases

t

q

open loop, N = 2

open loop, N = 10

feedback, N = 2

feedback, N = 10

1 2 3 4 5 6

0.5

1

1.5

2

2.5


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Aggregate production as the number of firmsincreases

t

Nq

open loop, N = 2

open loop, N = 10

feedback, N = 2

feedback, N = 10

1 2 3 4 5 6

1

2

3

4

5

6

7

8


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Price as the number of firms increases

t

popen loop, N = 2

open loop, N = 10

feedback, N = 2

feedback, N = 10

1 2 3 4 5 6

1

2

3

4

5


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Open and closed loop Nash equilibria as thespeed of adjustment increases

t

qopen

loop, s

=0.2

5

open

loop

,s=

0.9

feedback, s

=0.2

5

feedback,s=

0.9

qComp

qCN

1 2 3 4 5 6

0.5

1

1.5


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Price as the speed of adjustment increases

t

p

open loop, s = 0.25open

loop, s = 0.9

feedback

, s =0.25

feedback, s = 0.9

pComp

pCN

1 2 3 4 5 6

1

2

3

4


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Steady state at Nash equilibria as a function ofnumber of firms

N

q

2 5 10 15 20

0.5

1.0

1.5

2.0

2.5

N

N

NNNNNNNNNNNNNNNNN

– open loop, qOL,∗N – feedback, qF,∗

N

p

2 5 10 15 20

2

3

4

5

1.3

N

N

NN

NN N N N N N N N N N N N N N

– open loop, pOL,∗

N – feedback, pF,∗

Figure: Dependence of the asymptotic (as t → +∞) of theproduction level (in the left-hand side panel) and the price level (inthe right-hand side panel) in the Nash equilibrium on the numberof firms N.


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Difference between feedback and open loopNash equilibrium steady state production

N

qF,∗ − qOL,∗

2 5 10 15 20 25 30

0.05

0.10

0.15

0.20

0.25

0.30

H HH

H H H H H H H H H H H H H H H H H H H H H H H H H H

NN

N

NN

NN

N N N N N N N N N N N N N N N N N N N N N N

••

•

•••• • • • • • • • • • • • • • • • • • • • • • •

H – s = 0.1 – s = 0.25N – s = 1• – s = 5

– s = ∞

Figure: Dependence of the difference qfeed,∗ − qol,∗ on the numberof firms N for various values of the price stickiness.


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Steady state at Nash equilibria as a function ofspeed of adjustment

s

q

open loop

feedback

qComp

qCN

1 2 3 4 5

1.5

1.6

1.7

1.8

s

p

1 2 3 4 5

2.8

3.0

3.2

3.4

3.6

3.8

4.0open loop

feedback

pComp

pCN

Figure: Dependence of the asymptotic (as t → +∞) of theproduction level (in the left-hand side panel) and the price level (inthe right-hand side panel) in the Nash equilibrium on the pricestickiness s.


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Conclusions

I Introduction of price stickiness and consideringoligopoly model as a dynamic model, allows prices toremain below their static Cournot oligopoly level (evenat the steady state).

I Production is zero for prices below some level, then it isa strictly increasing function of price and it is strictlyincreasing in time.

I Feedback production at each time instant is greaterthan analogous open loop production, unless they areboth 0 (only at the beginning of the game).

I Reason – a ”corrective” effect in the feedback case –since strategies of the others are increasing functions ofprice.

I Feedback price is less than open loop price from thefirst moment at which feedback production is positive.


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Conclusions







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Conclusions







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Conclusions







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Conclusions







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Conclusions continued

I Both open loop and feedback solutions are stable.

I As speed of price adjustment s → 0, both feedback andopen loop equilibrium production and price tend to theirstatic competitive equilibrium levels;

I while as s → ∞, then open loop equilibrium productionand price tend to their static Cournot oligopoly levels,while feedback – are between Cournot and competivelevels.


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I Both open loop and feedback solutions are stable.I As speed of price adjustment s → 0, both feedback and

open loop equilibrium production and price tend to theirstatic competitive equilibrium levels;



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I Both open loop and feedback solutions are stable.I As speed of price adjustment s → 0, both feedback and

open loop equilibrium production and price tend to theirstatic competitive equilibrium levels;



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Thank you for your attention!

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Dynamiczny oligopol z lepkimi cenami - mimuw.edu.plmiekisz/wiszniewska.pdf · Dynamiczny oligopol z lepkimi cenami Agnieszka Wiszniewska-Matyszkiel, Marek Bodnar i Fryderyk Mirota

Text of Dynamiczny oligopol z lepkimi cenami - mimuw.edu.plmiekisz/wiszniewska.pdf · Dynamiczny oligopol z...