Monopoly Part III: Multiple Local Equilibria and Adaptive Search

Transcription

1 FH-Kiel University of Applie Sciences Prof Dr Anreas Thiemer, 00 Monopoly Part III: Multiple Local Equilibria an Aaptive Search Summary: The information about market eman neee by a profit maimizing monopolist is rather more comple an costly than that neee by a "price taker" in a perfect competitive market: The monopolist must know at least the entire eman curve instea of just one point on it In tet book cases the eman curve is a simple ownwar sloping straight line But in her pioneering work Joan Robinson 9 has alreay remarke that "cases of multiple equilibrium may arise when the eman curve changes its slope, being highly elastic for a stretch, then perhaps becoming relatively inelastic, then elastic again" This may happen if the eman is a composite of the eman from several subgroups of consumers, with ifferent elasticities an specific reservation prices Puu (997) further assume, that the monopolist oesn't know the eman function Hunting for a profit maimum, the firm follows an aaptive search proceure just using the information about recent reaction of profits to supply changes Such an output rule may cause perioic cycles an chaotic behaviour of the monopolistic supply This worksheet allows you to eplore the comple ynamics of Puu's moel in epth A Thiemer, 00 monopolymc,60006

2 I The case with one prouct Basic assumptions The (inverse) eman function is p( ) α 0 α α α where p enotes commoity price, enotes quantity emane an α i represents (positive) constants: α α 7 0 α 6 00 α 0 This function is own-sloping an thus invertible provie that the following conition hols (=): α < α α = Because the total revenue is R() = p(), the marginal revenue must be: R' ( ) p( ) vereinfachen To provie an opportunity of multiple equilibria, the conition for an upwar slope at the point of inflection of the marginal revenue curve must hol (=): 8 α α < α = Net we efine the total cost as a function of the quantity supplie: Κ ( ) β 0 β β β As positive constants we take: β 0 0 β β 0 β 0 Therefore, the marginal cost K/ must be: Κ' ( ) Κ ( ) 0 The profit function is compute by Ψ ( ) p( ) Κ ( ) vereinfachen A Thiemer, 00 monopolymc,60006

3 The first orer conition (FOC) of a profit maimum is Ψ' ( ) 0 We etermine the solution FOC for this equation: Compute the marginal profit: Ψ' ( ) Ψ ( ) Collect the coefficients of this polynom: coeff Ψ' ( ) koeff, Solve the polynom: FOC nullstellen( coeff) FOC = 47 Ψ FOC = 8 Let's check for the secon orer conition: Compute the secon erivative of the profit function: Ψ'' ( ) 4 Ψ ( ) 8 Search for local maima where Ψ'' FOC < 0 : Ψ'' FOC = 06 A Thiemer, 00 monopolymc,60006

4 Now take a look to the figures below to verify the results from above: ma 6 0, ma ma 000 Profit function Ψ( ) Deman, marginal revenue & cost p ( ) R' ( ) 4 Κ' ( ) A Thiemer, 00 4 monopolymc,60006

5 Introucing aaptive search Assume that the monopolist oesn't know more than a few points on the eman function This information woul be of local character an short lifetimethe monopolist might not even know that globally there are two istinct profit maima Given this, the monopolist follows a simple search algorithm (in the vein of Newton) for the maimum of the unknown profit function: He estimates the ifference of profits from the last two visite points an Then he uses a given step length δ > 0 to move his supply in the irection of increasing profits: step size: δ 6 initial values: time Τ maimum of perios compute: Τ 00 Ψ Ψ δ time Τ 0 Time series of supplie quantity time The search proceure from above escribes a ifference equation of secon orer We can ecompose this in a system of two first orer equations by efining: y y time Τ y time y time Ψ y time Ψ y time δ y time A Thiemer, 00 monopolymc,60006

6 Hence, we raw the trajectory for this iterate map: Trajectory y time Fie points, cycles an chaos Not unepecte, the maima an the minimum of the profit function are fie points of the iterate map given by y time an Ψ' 0 Of course the minimum must be an unstable fie point To fin out the stability of the maima we reformulate the iterate map to use Mathca's symbolic processor: fxy (, ) Y g( X, Y) Y δ Ψ ( Y) Ψ ( X) Y X Now we are able to evaluate the Jacobian: Jacobian( X) X fxy (, ) X g( X, Y) Y fxy (, ) Y g( X, Y) ersetzen, Y X vereinfachen 0 δ X 6X 8 A Thiemer, 00 6 monopolymc,60006

7 Loss of stability occurs when the Jacobian is unitary Thus, we get as the critical value δ c of the step with: δ c ( X) Jacobian( X) auflösen, δ 0 X 6X δ c FOC = Plotting the phase variable against the parameter δ in a Feigenbaum iagram, we see the alternative fie points coeist at δ< δ c After that they are replace by cycles (Note: This cycles are of perio 4 with one value taken on twice as often as are the other two!) Chaos takes over once this cycles lose stability Puu (997, p -6) proves that this happens if the value of the step parameter ecees 488 At first there appears chaotic bans, with each its own basin of attraction After a certain point the attractors merge in a single one To run the bifurcation plot you must enter a resolution number RES (Note: High resolution is very time consuming) Resolution (RES=,,,0): RES 0 range of δ: δ bot δ top range of : bot 0 top 6 6 Feigenbaum iagram 4 0 parameter of step with A Thiemer, 00 7 monopolymc,60006

8 4 Attractors The net figure plots attractors of this system into the phase plane Change δ using the ifferent values from above to see fie points, peroic cycles an two coeistent chaotic attractors merging to a single one Note the symmetry of the shapes of the attractors For symmetric systems any attractors that are not symmetric in themselves come pairwise, forming together a symmetric picture δ 8 6 Attractors 4 y A Thiemer, 00 8 monopolymc,60006

9 II The case with two complementary proucts Suppose the monopolist supplies two goos an y that are complementary in consumers perferences Both goos have eman functions similar to the one in case I But now the proucts of sales volumes y respectively y enter as a positive term in both eman functions This moels the "coupling effect" ue to complementary Puu uses the same cost function K(+y) from above, stating the moifie profit function Π(,y) with the "coupling" parameter κ as: Π (, y, κ) Ψ ( ) Ψ ( y) κ y y Again it will be assume, that the firm hunts with an aaptive search proceure for the profit maimum To simulate this behaviour we nee the graient of profits: Π (, y, κ) Π (, y, κ) κ y y Π y (, y, κ) y Π (, y, κ) 8 4 y 9 y y κ y Given a small value of κ (for eample 000) we obtain a profit function with four ifferent local maima: κ 00 Profit function Iso profit lines profit profit A Thiemer, 00 9 monopolymc,60006

10 To start the aaptive search proceure we set step size: δ 7 initial values: 9 y time Τ maimum of perios compute: Τ 0000 y time δπ, y time, κ y time δπ y, y time, κ time 00 8 Supply of an y 6 y time t start 000 time t start Τ time Attractor 6 y time 4 y , A Thiemer, 00 0 monopolymc,60006

11 Now run this system with increasing step length δ First you will fin coeistent stable fie points After their loss of stability these fie points are replace by perioic cycles an then chaotic attractors You will also fin coeistent cycles an chaotic attractors at once It epens on the initial conitions embee in ifferent basins of attraction, to which one the process settles Here are some eamples of parameter combinations you shoul try (initial conitions are always 9 an y ): Case I: Lower coupling of the proucts with κ 000 δ fie point in NW δ cycle with perio in NW δ perio oubling δ chaotic attractor ("two stripes" in NW) δ chaotic attractor ("prouct set" in NW) δ 6 chaotic attractor ("prouct set" in SW) δ 7 chaotic attractor (epansion of the "prouct set" with sparse points) δ 8 chaotic attractor (epane "prouct set" with sparse points only in NW) Case II: Meium coupling of the proucts with κ 0006 δ chaotic "Henon"-like attractor in NW δ 4 perioic cycles in SW δ chaotic attractor in SW δ 7 chaotic attractor from SW epaning to SE an NW Case III: Strong coupling of the proucts with κ 00 δ 7 strange attractor in NE δ 4 fie point in SW Literature: Puu, T: Nonlinear Economic Dynamics Berlin/Heielberg 997, - Robinson, J: The Economics of Imperfect Competition n e, Lonon 969 A Thiemer, 00 monopolymc,60006

12 ORIGIN A Thiemer, 00 monopolymc,60006