Dynamic Cournot Oligopoly with Output Adjustment costchrystie/The Book of...

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Dynamic Cournot Oligopoly with Dynamic Cournot Oligopoly with Output Adjustment CostOutput Adjustment Cost

by by Chrystie T. BurrChrystie T. Burr

underunderDr. Dr. FerencFerenc SzidarovszkySzidarovszky

AgendaAgenda

1.1. Intro Intro –– Classical Cournot OligopolyClassical Cournot Oligopoly2.2. Intro Intro –– Variant: Output Adjustment CostVariant: Output Adjustment Cost3.3. Best ResponsesBest Responses

1.1. Continuous Output Adjustment CostContinuous Output Adjustment Cost2.2. Discontinuous Output Adjustment CostDiscontinuous Output Adjustment Cost

4.4. Equilibrium ConditionEquilibrium Condition1.1. Continuous Output Adjustment CostContinuous Output Adjustment Cost2.2. Discontinuous Output Adjustment CostDiscontinuous Output Adjustment Cost

5.5. ConclusionConclusion

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Classical Cournot OligopolyClassical Cournot Oligopoly

Researches into the Mathematical Principles of Wealth (1838)Several firms produce a homogeneous productOutputs are chosen simultaneous by firmsThe goal is to choose the output quantity that will maximize its own profit

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Let:Let:Firm 1 produces output Firm 1 produces output xx1 1 and and CC11((xx11) is the cost ) is the cost Firm 2 produces output Firm 2 produces output xx2 2 and and CC22((xx22) is the cost ) is the cost

Price is Price is PP((xx11 + x+ x22))

Profit of firm 1 is Profit of firm 1 is PP((xx11 + x+ x22))xx11 -- CC11((xx11) ) Profit of firm 2 is Profit of firm 2 is PP((xx11 + x+ x22))xx22 -- CC22((xx22) )

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Classical Cournot DuopolyClassical Cournot Duopoly

Cournot equilibrium = Nash equilibriumCournot equilibrium = Nash equilibrium

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* * *1 1 2 1 1 2 1

* * *2 1 2 1 1 2

( , ) ( , ),

( , ) ( , ),

x x x x x X

x x x x x X

Π ≥ Π ∀ ∈

Π ≥ Π ∀ ∈

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Finding Nash equilibrium: use best response Finding Nash equilibrium: use best response functionsfunctions

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Finding Nash equilibrium: use best response Finding Nash equilibrium: use best response functionsfunctions

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VaraintVaraint: With Output Adjustment Cost: With Output Adjustment Cost

When the optimal output increases from one When the optimal output increases from one time period to the next, additional cost is time period to the next, additional cost is required.required.

This is called output adjustment costThis is called output adjustment cost

Denoted by Denoted by KK

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Output Adjustment CostOutput Adjustment Cost

Two components: marginal cost and fixed costTwo components: marginal cost and fixed cost

K K = = ωω((xxtt -- xxtt--11) + ) + ττ

Case 1: Continuous OAC (Case 1: Continuous OAC (ττ = 0)= 0)Profit = Profit = xx((AA -- BsBs) ) --αα--ββx x -- ωω((xxtt -- xxtt--11))

Case 2: Discontinuous OAC (Case 2: Discontinuous OAC (ττ > 0)> 0)Profit = Profit = xx((AA -- BsBs) ) --αα--ββx x -- ωω((xxtt -- xxtt--11)) -- ττ

MC FC

Best Response Best Response with Continuous Output Adjustment Costwith Continuous Output Adjustment Cost

Πk = -Bxk2 + (A – Bsk – βk)xk - αk Πk = -Bxk

2 + (A – Bsk – βk – ωk)xk – αk – ωkxkt-1

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Best Response Best Response with Continuous Output Adjustment Costwith Continuous Output Adjustment Cost

(i)

(ii)

(iii-a)

(iii-b)

(iii-c)

Best Response Best Response with Discontinuous Output Adjustment Costwith Discontinuous Output Adjustment Cost

Πk = -Bxk2 + (A – Bsk – βk)xk - αk Πk = -Bxk

2 + (A – Bsk – βk – ωk)xk – αk – ωkxkt-1 - τk

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(i)

(ii)

(iii-a-2)

(iii-b-2)

(iii-b-1)

1t kk kL x B

τ−− >Case 1:

(iii-c)

(iii-b-3)


(i)

(ii)

(iii-a-2)

(iii-b-1)

1t kk kL x B

τ−− <Case 2:

(iii-c)(iii-a-1)

(iii-a-3)

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Equilibrium Condition Equilibrium Condition with Continuous Output Adjustment Costwith Continuous Output Adjustment Cost

Equilibrium is reached Equilibrium is reached iffiff for all for all kk, , xxkk** is the is the

best response of firm k, that isbest response of firm k, that is

* *0 00

k k k kk kx x x xk kx x= + = −

∂Π ∂Π≤ ≤∂ ∂

Xk*

Equilibrium Condition Equilibrium Condition with Continuous Output Adjustment Costwith Continuous Output Adjustment Cost

Ex. Ex. GGiven iven A A = 20= 20, B , B = 1= 1, L, L11 = L= L22 = 10, = 10, ββ11 = β= β22 = = ωω11 = ω= ω22 = 5, the equilibrium condition becomes= 5, the equilibrium condition becomes

1 2 1,2 1 210 ( ) 15 ( )x x x x x− + ≤ ≤ − +

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Equilibrium Condition Equilibrium Condition with Discontinuous Output Adjustment Costwith Discontinuous Output Adjustment Cost

Very difficult to study the equilibrium Very difficult to study the equilibrium condcond..--

1.1. IInfinitely many nfinitely many equilirbiaequilirbia

2.2. The discontinuity exemplified by the jump in The discontinuity exemplified by the jump in the production level the production level –– profit function.profit function.

Therefore the use of computer simulation is Therefore the use of computer simulation is needed.needed.

SimulationSimulation

Windows deployable program Windows deployable program

Based on Matt Based on Matt Dabkowski’sDabkowski’s Duopoly Basin Duopoly Basin CalculatorCalculator

Iterate all possible initial production levelsIterate all possible initial production levels to to find the corresponding equilibria find the corresponding equilibria

The number of steps taken to reach The number of steps taken to reach equilirbiumequilirbium is then categorized by colors.is then categorized by colors.

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Algorithm Algorithm IInitializenitialize

for (yfor (y00 = = y_miny_min; y; y00 <=<=y_maxy_max; y; y00 = = yy00 + + stepsizestepsize){){for (xfor (x00 = = x_minx_min; x; x00 <=<=x_maxx_max; x; x00 = = xx00 + + stepsizestepsize){){

Then Determine whether firm x changes its production level Then Determine whether firm x changes its production level in the next time periodin the next time period by comparing the profits generated by by comparing the profits generated by the vertices of the two parabolas.the vertices of the two parabolas.

If converged, stop. If converged, stop. If not, update. If not, update. CCount ++ount ++

}}}}

Simulation ResultsSimulation Results

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Simulation ResultsSimulation ResultsA = 20, B = 5, L1 = L2 = 10, β1 = β2 = ω1 = ω2 = 1, τ1 = τ2 = 1

Sensitivity AnalysisSensitivity Analysis

Stage 1: No jump is assumed while the Stage 1: No jump is assumed while the parameters , parameters , AA, , BB, , ωωxx = = ωωyy and and ββxx = = ββyy are are systematically varied.systematically varied.

Stage 2: Discontinuity in the additional Stage 2: Discontinuity in the additional cost function is applied. Only cost function is applied. Only ττ varies.varies.

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Sensitivity Analysis (Stage 1)Sensitivity Analysis (Stage 1)A = 20


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Sensitivity Analysis (Stage 1)Sensitivity Analysis (Stage 1)B = 0.3


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Sensitivity Analysis (Stage 1)Sensitivity Analysis (Stage 1)β = 3


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Sensitivity Analysis (Stage 1)Sensitivity Analysis (Stage 1)L = 4


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Sensitivity Analysis (Stage 1)Sensitivity Analysis (Stage 1)ω = 0.00001

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Sensitivity Analysis (Stage 1)Sensitivity Analysis (Stage 1)ω = 0.001

Sensitivity Analysis (Stage 1)Sensitivity Analysis (Stage 1)ω = 1

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Sensitivity Analysis (Stage 2)Sensitivity Analysis (Stage 2)τ = 0.1


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SummarySummary

NN--firm single product oligopolies with firm single product oligopolies with production adjustment cost were examined.production adjustment cost were examined.

The best response function is always The best response function is always decreasing, not necessarily continuous, decreasing, not necessarily continuous, and might have two different values. and might have two different values.

This leads to infinitely many equilibriaThis leads to infinitely many equilibria

Simulation study shows the sensitivity and Simulation study shows the sensitivity and convergence rate of the equilibrium set.convergence rate of the equilibrium set.

Q & AQ & A

Dynamic Cournot Oligopoly with Output Adjustment costchrystie/The Book of...

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