FH-Kiel Universitf Alie Sciences Prof. Dr. Anreas Thiemer, 00 e-mail: anreas.thiemer@fh-kiel.e A Simle Exchange Economy with Comlex Dynamics (Revision: Aril 00) Summary: Mukherji (999) shows that a stanar iscrete tatonnement rocess within the context of a very simle exchange economy (two goos, two ersons with Cobb-Douglas utility functions) exhibits comlex ynamics of the rice ajustment. This worksheet gives you the numerical tools to exlore the henomenon of erio oubling bifurcation an chaos in this moel. A.Thiemer, 00 mukherji.mc, 6.0.006
. Imortant efinitions A, B: iniviuals : rice of goo x relative to goo y E : equilibrium rice x, y : absolute rices of x an y u A, u B : utilitf iniviual A an B x, y: quantities of goos x o, : enowments of goos Z: excess eman of goo x. Basic assumtions The references of iniviual A are given by: u A ( x, y, α) x α. y α with 0< α < The references of iniviual B are given by: u B ( x, y, β) x β. y β with 0< β< We efine the rice of goo y as a numeraire: y Thus the relative rice is written as: x y x Iniviual A ossesses the enowment (x o, 0) an B has the enowment (0, ). Therefore, the buget constraints are x. o x. y for iniviual A x. y for iniviual B A.Thiemer, 00 mukherji.mc, 6.0.006
3. The exchange equilibrium With these buget constraints the first orer conitions of utility maximization yiel the eman functions for goo x of iniviual A an B: x A, α, x o x u A x, x. o x., α auflösen, x α. x o x B, β, x u B x, x., β auflösen, x β. Now we summarize the eman behaviour by the excess eman function Z ( ) for goo x: Z, β, α, x o, x B, β, x A, α, x o x o β. α. x o x o The market is in equilibrium if Z( ) = 0. Hence the unique equilibrium rice is etermine by: E β, α, x o, Z, β, α, x o, auflösen, β. x. o ( α ) 4. Introucing ajustment ynamics Consier the stanar ajustment on rices in isequilibrium (the "tatonnement") i i γ. Z i, β, α, x o, where γ> 0 is some constant see of ajustment.we can rewrite this equation as an iterate ma: y f ( ) γ. Z, β, α, x o, γ. o β. α. x o x o First orer coniton gives: f ( ) auflösen, γβ.. γβ.. A.Thiemer, 00 3 mukherji.mc, 6.0.006
Insert the ositive solution into the secon orer conition: f ( ) ersetzen, γβ... γ. β. γβ.. 3 Because the secon erivate becomes ositive we know that f() attains a minimum value at ' γβ.. given by f γβ.. vereinfachen γ.. β. x. o γβ.. α. x. o γβ.. γβ..... or more simlifie (by han an not by Mathca): f' ( ). γβ.. γ. ( α ). x 0 In orer to guarantee ositive rices f( ' ) > 0 must hol. Defining Κ γ. ( α ) x o β.. this is ensure if Κ < 4. A.Thiemer, 00 4 mukherji.mc, 6.0.006
5. Some roerties of the ajustment ynamics For the roofs of the following cite claims, see Mukherij (999). Claim : Κ < E is locally stable for the rocess f(). Claim 3: For < Κ <.5 there exists a stable -cycle. Let Κ n enote the critical value of Κ where a n cycle is born; then Κ an Κ.5. Using the Feigenbaum constant F const can be aroximate by: 4.669060909 the value of κ lim n Κ n κ F. const Κ Κ F const κ =.636 Claim 4: For Κ ( 3.0, 3.6) the ma f() exhibits toological chaos. Claim 5: For Κ = 5/9, the ma f() exhibits ergoic chaos; in aition there exists Κ such that f() exhibits ergoic chaos. 6. Numerical Exlorations To exlore the behaviour of the attractors for ifferent values of K, Mukherji (999,.745) fixes the values of all arameters excet the ajustment coefficient γ with β. an ( α ). x o 6 so that Κ 36. γ. Then the iterate ma takes the articular form: f (, Κ ) 6 Κ 36 Notice, that uner these arameter restrictions the value of the equilibrium rice is ineenent from Κ: E f (, Κ ) auflösen, 6 A.Thiemer, 00 5 mukherji.mc, 6.0.006
Choosing a numerical value for Κ, the iterate ma is rawn in the following figure: Κ 5 9 max 0,.0.. max Iterate Ma an Equilibrium E 0.8 fκ (, ) 0.6 0.4 0. 0 0 0. 0.4 0.6 0.8 Given an initial value 0 of the first rice offer an using the Κ-value from above, the ajustment rocess is comute for T max erios. 0.5 T max 50 i 0.. T max i f i, Κ 0.6 Price Ajustment rice 0.4 0. E 0 0 0 40 60 80 00 0 40 time A.Thiemer, 00 6 mukherji.mc, 6.0.006
This ajustment rocess may be also resente by a cobweb lot. Cobweb Plot of the Price Ajustment 0.5 E 0.4 (time+) 0.3 0. 0. 0 0 0. 0. 0.3 0.4 0.5 iterate ma f(,k) (time) = (time+) trajectory (time) A.Thiemer, 00 7 mukherji.mc, 6.0.006
Observe the bifurcations by lotting the Feigenbaum iagram an the Lyaunov exonent. Use these figures to choose K such that you obtain stable cycles of ifferent erioicitr irregular cycles of. Resolution of grah: RES 3 (,,.., 0) Range of lotte values: Κ bottom.9 Κ to 3.6 bottom 0 to.5.5 Feigenbaum iagram.5 rice 0.5..4.6.8 3 3. 3.4 3.6 bifurcation arameter Κ A.Thiemer, 00 8 mukherji.mc, 6.0.006
Lyaunov exonent..4.6.8 3 3. 3.4 3.6 L 3 bifurcation arameter Κ Note: Positive values of the Lyaunov exonent inicate chaotic behaviour of! Literature: Anjan Mukherji: A Simle Examle of Comlex Dynamics. In: Economic Theory, vol.4 (999), 74-749. A.Thiemer, 00 9 mukherji.mc, 6.0.006