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# The examples below are based upon the work by: # Shone,R (2003) Economic Dynamics- Phase Diagrams and their Economic Application # and was implemented in Maple by Marcus Davidsson (2008) davidsson_marcus@hotmail.com # Nash Equilibrium in a Static Duopoly Cournot Model restart; # Reaction function firm 1 # Total Cost for firm 1 is given by TC_1:=3*q1;

LCQqJiIiJCIiIkkjcTFHNiJGJUYl

# The price is given by P:=9-(q1+q2);

LCgiIioiIiJJI3ExRzYiISIiSSNxMkdGJkYn

# Profit for firm 1 is given by TR-TC=q1*P-TC_1 profit_1=q1*P-TC_1;

L0kpcHJvZml0XzFHNiIsJiomSSNxMUdGJCIiIiwoIiIqRihGJyEiIkkjcTJHRiRGK0YoRigqJiIiJEYoRidGKEYr

# where q2 is the expected production of firm 2 (keept constant) # solving for the optimal value of q1 that maximize profits for firm 1 (assuming firm 2 keeps its output constant) diff(%,q1);

LyIiISwoIiInIiIiKiYiIiNGJkkjcTFHNiJGJiEiIkkjcTJHRipGKw==

A_1:=isolate(%,q1);

L0kjcTFHNiIsJiIiJCIiIiomI0YnIiIjRidJI3EyR0YkRichIiI=

plot(rhs(%), q2=0..100,labels=[q2, q1],color=green,thickness=3,view=[0..8, 0..8] );

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

# Reaction function firm 2 # Total Cost for firm 2 is given by TC_2:=3*q2;

LCQqJiIiJCIiIkkjcTJHNiJGJUYl

# The price is given by P:=9-(q1+q2);

LCgiIioiIiJJI3ExRzYiISIiSSNxMkdGJkYn

# Profit for firm 2 is given by TR-TC=q1*P-TC_2 profit_2=q2*P-TC_2;

L0kpcHJvZml0XzJHNiIsJiomSSNxMkdGJCIiIiwoIiIqRihJI3ExR0YkISIiRidGLEYoRigqJiIiJEYoRidGKEYs

# where q1 is the expected production of firm 1 (keept constant) # solving for the optimal value of q2 that maximize profits for firm 2 (assuming firm 1 keeps its output constant) diff(%,q2);

LyIiISwoIiInIiIiSSNxMUc2IiEiIiomIiIjRiZJI3EyR0YoRiZGKQ==

B_1:=isolate(%,q1);

L0kjcTFHNiIsJiIiJyIiIiomIiIjRidJI3EyR0YkRichIiI=

plot(rhs(%), q2=0..100,labels=[q2, q1],color=red,thickness=3,view=[0..8, 0..8] );

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

# Nash Equilibrium A_1;

L0kjcTFHNiIsJiIiJCIiIiomI0YnIiIjRidJI3EyR0YkRichIiI=

B_1;

L0kjcTFHNiIsJiIiJyIiIiomIiIjRidJI3EyR0YkRichIiI=

rhs(A_1)=rhs(B_1);

LywmIiIkIiIiKiYjRiUiIiNGJUkjcTJHNiJGJSEiIiwmIiInRiUqJkYoRiVGKUYlRis=

isolate(%,q2);

L0kjcTJHNiIiIiM=

subs(q2=rhs(%),A_1);

L0kjcTFHNiIiIiM=

# The output for each firm is 30 with(plots): A_2:=plot(rhs(A_1), q2=0..100,labels=[q2, q1],color=green,view=[0..100, 0..50] ): B_2:=plot(rhs(B_1), q2=0..100,labels=[q2, q1],color=red, view=[0..50, 0..100]): C_2:=textplot([2,2.2.1,"Nash_Equilibrium"],align={above,right}): display({A_2,B_2,C_2},labels=[q2, q1],view=[0..8, 0..8], thickness=3 );

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# IsoProfit Curve Firm 1 restart; TC_1:=3*q1;

LCQqJiIiJCIiIkkjcTFHNiJGJUYl

P:=9-(q1+q2);

LCgiIioiIiJJI3ExRzYiISIiSSNxMkdGJkYn

# The profit for firm 1 is given by profit_1:=q1*P-TC_1;

LCYqJkkjcTFHNiIiIiIsKCIiKkYmRiQhIiJJI3EyR0YlRilGJkYmKiYiIiRGJkYkRiZGKQ==

# Firm 1 isoprofit curve is given by plot3d(profit_1,q1=0..6,q2=0..6,axes = boxed, style=patchcontour, color=grey, contours=40,labels=[q1, q2, Profit_1]);

6)-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%Q#q16"Q#q26"Q)Profit_16"-%+LIGHTMODELG6#%%NONEG-%%GRIDG6';$""!""!$""'""!;$""!""!$""'""!X,I)anythingG%*protectedG6"6"[gl'!%"!!#\bm":":00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003FF70000000000003FF60000000000003FF50000000000003FF40000000000003FF30000000000003FF20000000000003FF10000000000003FF00000000000003FEE0000000000003FEC0000000000003FEA0000000000003FE80000000000003FE60000000000003FE40000000000003FE20000000000003FE00000000000003FDC0000000000003FD80000000000003FD40000000000003FD00000000000003FC80000000000003FC00000000000003FB00000000000000000000000000000BFB000000000000040060000000000004005000000000000400400000000000040030000000000004002000000000000400100000000000040000000000000003FFE0000000000003FFC0000000000003FFA0000000000003FF80000000000003FF60000000000003FF40000000000003FF20000000000003FF00000000000003FEC0000000000003FE80000000000003FE40000000000003FE00000000000003FD80000000000003FD00000000000003FC00000000000000000000000000000BFC0000000000000BFD0000000000000400F800000000000400E000000000000400C800000000000400B00000000000040098000000000004008000000000000400680000000000040050000000000004003800000000000400200000000000040008000000000003FFE0000000000003FFB0000000000003FF80000000000003FF50000000000003FF20000000000003FEE0000000000003FE80000000000003FE20000000000003FD80000000000003FC80000000000000000000000000000BFC8000000000000BFD8000000000000BFE200000000000040140000000000004013000000000000401200000000000040110000000000004010000000000000400E000000000000400C000000000000400A000000000000400800000000000040060000000000004004000000000000400200000000000040000000000000003FFC0000000000003FF80000000000003FF40000000000003FF00000000000003FE80000000000003FE00000000000003FD00000000000000000000000000000BFD0000000000000BFE0000000000000BFE8000000000000BFF00000000000004017C000000000004016800000000000401540000000000040140000000000004012C0000000000040118000000000004010400000000000400E000000000000400B80000000000040090000000000004006800000000000400400000000000040018000000000003FFE0000000000003FF90000000000003FF40000000000003FEE0000000000003FE40000000000003FD40000000000000000000000000000BFD4000000000000BFE4000000000000BFEE000000000000BFF4000000000000BFF9000000000000401B0000000000004019800000000000401800000000000040168000000000004015000000000000401380000000000040120000000000004010800000000000400E000000000000400B0000000000004008000000000000400500000000000040020000000000003FFE0000000000003FF80000000000003FF20000000000003FE80000000000003FD80000000000000000000000000000BFD8000000000000BFE8000000000000BFF2000000000000BFF8000000000000BFFE000000000000C002000000000000401DC00000000000401C000000000000401A40000000000040188000000000004016C00000000000401500000000000040134000000000004011800000000000400F800000000000400C0000000000004008800000000000400500000000000040018000000000003FFC0000000000003FF50000000000003FEC0000000000003FDC0000000000000000000000000000BFDC000000000000BFEC000000000000BFF5000000000000BFFC000000000000C001800000000000C005000000000000C0088000000000004020000000000000401E000000000000401C000000000000401A00000000000040180000000000004016000000000000401400000000000040120000000000004010000000000000400C0000000000004008000000000000400400000000000040000000000000003FF80000000000003FF00000000000003FE00000000000000000000000000000BFE0000000000000BFF0000000000000BFF8000000000000C000000000000000C004000000000000C008000000000000C00C000000000000C0100000000000004020E00000000000401F800000000000401D400000000000401B0000000000004018C00000000000401680000000000040144000000000004012000000000000400F800000000000400B000000000000400680000000000040020000000000003FFB0000000000003FF20000000000003FE20000000000000000000000000000BFE2000000000000BFF2000000000000BFFB000000000000C002000000000000C006800000000000C00B000000000000C00F800000000000C012000000000000C01440000000000040218000000000004020400000000000401E000000000000401B8000000000004019000000000000401680000000000040140000000000004011800000000000400E000000000000400900000000000040040000000000003FFE0000000000003FF40000000000003FE40000000000000000000000000000BFE4000000000000BFF4000000000000BFFE000000000000C004000000000000C009000000000000C00E000000000000C011800000000000C014000000000000C016800000000000C0190000000000004021E000000000004020800000000000401E400000000000401B8000000000004018C00000000000401600000000000040134000000000004010800000000000400B800000000000400600000000000040008000000000003FF60000000000003FE60000000000000000000000000000BFE6000000000000BFF6000000000000C000800000000000C006000000000000C00B800000000000C010800000000000C013400000000000C016000000000000C018C00000000000C01B800000000000C01E4000000000004022000000000000402080000000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# we can see that firm 1 highest profit occures when q1=3. This is the output firm 1 # would produce if firm 1 was a monopoly # If we rotate the above diagram we get plot3d(profit_1,q1=0..6,q2=0..6,axes = boxed, style=patchcontour, color=grey, contours=40,labels=[q1, q2, Profit_1],orientation = [-90,0] );

6(-I%GRIDG6$%*protectedGI(_syslibG6"6&;$""!""!$""'""!;$""!""!$""'""!X,I)anythingG%*protectedG6"6"[gl'!%"!!#\bm":":00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003FF70000000000003FF60000000000003FF50000000000003FF40000000000003FF30000000000003FF20000000000003FF10000000000003FF00000000000003FEE0000000000003FEC0000000000003FEA0000000000003FE80000000000003FE60000000000003FE40000000000003FE20000000000003FE00000000000003FDC0000000000003FD80000000000003FD40000000000003FD00000000000003FC80000000000003FC00000000000003FB00000000000000000000000000000BFB000000000000040060000000000004005000000000000400400000000000040030000000000004002000000000000400100000000000040000000000000003FFE0000000000003FFC0000000000003FFA0000000000003FF80000000000003FF60000000000003FF40000000000003FF20000000000003FF00000000000003FEC0000000000003FE80000000000003FE40000000000003FE00000000000003FD80000000000003FD00000000000003FC00000000000000000000000000000BFC0000000000000BFD0000000000000400F800000000000400E000000000000400C800000000000400B00000000000040098000000000004008000000000000400680000000000040050000000000004003800000000000400200000000000040008000000000003FFE0000000000003FFB0000000000003FF80000000000003FF50000000000003FF20000000000003FEE0000000000003FE80000000000003FE20000000000003FD80000000000003FC80000000000000000000000000000BFC8000000000000BFD8000000000000BFE200000000000040140000000000004013000000000000401200000000000040110000000000004010000000000000400E000000000000400C000000000000400A000000000000400800000000000040060000000000004004000000000000400200000000000040000000000000003FFC0000000000003FF80000000000003FF40000000000003FF00000000000003FE80000000000003FE00000000000003FD00000000000000000000000000000BFD0000000000000BFE0000000000000BFE8000000000000BFF00000000000004017C000000000004016800000000000401540000000000040140000000000004012C0000000000040118000000000004010400000000000400E000000000000400B80000000000040090000000000004006800000000000400400000000000040018000000000003FFE0000000000003FF90000000000003FF40000000000003FEE0000000000003FE40000000000003FD40000000000000000000000000000BFD4000000000000BFE4000000000000BFEE000000000000BFF4000000000000BFF9000000000000401B0000000000004019800000000000401800000000000040168000000000004015000000000000401380000000000040120000000000004010800000000000400E000000000000400B0000000000004008000000000000400500000000000040020000000000003FFE0000000000003FF80000000000003FF20000000000003FE80000000000003FD80000000000000000000000000000BFD8000000000000BFE8000000000000BFF2000000000000BFF8000000000000BFFE000000000000C002000000000000401DC00000000000401C000000000000401A40000000000040188000000000004016C00000000000401500000000000040134000000000004011800000000000400F800000000000400C0000000000004008800000000000400500000000000040018000000000003FFC0000000000003FF50000000000003FEC0000000000003FDC0000000000000000000000000000BFDC000000000000BFEC000000000000BFF5000000000000BFFC000000000000C001800000000000C005000000000000C0088000000000004020000000000000401E000000000000401C000000000000401A00000000000040180000000000004016000000000000401400000000000040120000000000004010000000000000400C0000000000004008000000000000400400000000000040000000000000003FF80000000000003FF00000000000003FE00000000000000000000000000000BFE0000000000000BFF0000000000000BFF8000000000000C000000000000000C004000000000000C008000000000000C00C000000000000C0100000000000004020E00000000000401F800000000000401D400000000000401B0000000000004018C00000000000401680000000000040144000000000004012000000000000400F800000000000400B000000000000400680000000000040020000000000003FFB0000000000003FF20000000000003FE20000000000000000000000000000BFE2000000000000BFF2000000000000BFFB000000000000C002000000000000C006800000000000C00B000000000000C00F800000000000C012000000000000C01440000000000040218000000000004020400000000000401E000000000000401B8000000000004019000000000000401680000000000040140000000000004011800000000000400E000000000000400900000000000040040000000000003FFE0000000000003FF40000000000003FE40000000000000000000000000000BFE4000000000000BFF4000000000000BFFE000000000000C004000000000000C009000000000000C00E000000000000C011800000000000C014000000000000C016800000000000C0190000000000004021E000000000004020800000000000401E400000000000401B8000000000004018C00000000000401600000000000040134000000000004010800000000000400B800000000000400600000000000040008000000000003FF60000000000003FE60000000000000000000000000000BFE6000000000000BFF6000000000000C000800000000000C006000000000000C00B800000000000C010800000000000C013400000000000C016000000000000C018C00000000000C01B800000000000C01E40000000000040220000000000004020800000000000401E000000000000401B0000000000004018000000000000401500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# IsoProfit Curve Firm 2 restart; TC_2:=3*q2;

LCQqJiIiJCIiIkkjcTJHNiJGJUYl

P:=9-(q1+q2);

LCgiIioiIiJJI3ExRzYiISIiSSNxMkdGJkYn

# Profit for firm 2 is given by TR-TC=q1*P-TC_2 profit_2:=q2*P-TC_2;

LCYqJkkjcTJHNiIiIiIsKCIiKkYmSSNxMUdGJSEiIkYkRipGJkYmKiYiIiRGJkYkRiZGKg==

# Firm 2 isoprofit curve is given by plot3d(profit_2,q1=0..6,q2=0..6,axes = boxed, style=patchcontour, color=grey, contours=40,labels=[q1, q2, Profit_2]);

6(-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%Q#q16"Q#q26"Q)Profit_26"-%%GRIDG6';$""!""!$""'""!;$""!""!$""'""!X,I)anythingG%*protectedG6"6"[gl'!%"!!#\bm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# we can see that firm 2 highest profit occures when q2=3 This is the output firm 2 # would produce if firm 2 was a monopoly # If we rotate the above diagram we get plot3d(profit_2,q1=0..6,q2=0..6,axes = boxed, style=patchcontour, color=grey, contours=40,labels=[q1, q2, Profit_2],orientation = [-90,0]);

6(-I%GRIDG6$%*protectedGI(_syslibG6"6&;$""!""!$""'""!;$""!""!$""'""!X,I)anythingG%*protectedG6"6"[gl'!%"!!#\bm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# Both ContourPlots Combined restart; with(plots): A:=plot3d(q1*(9-q1-q2)-3*q1,q1=0..6,q2=0..6,axes = boxed, style=patchcontour, color=grey, contours=50,labels=[q1, q2, Profit_1],orientation = [-90,0], transparency=1 ): B:=plot3d(q2*(9-q1-q2)-3*q2,q1=0..6,q2=0..6,axes = boxed, style=patchcontour, color=grey, contours=50,labels=[q1, q2, Profit_2],orientation = [-90,0],transparency=1 ): display({A,B});

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restart: with(plots): with(Statistics): A:=contourplot(q1*(9-q1-q2)-3*q1,q1=0..3,q2=0..3,axes = boxed, color=black, contours=[4,5],labels=[q1, q2], transparency=0 ): B:=contourplot(q2*(9-q1-q2)-3*q2,q1=0..3,q2=0..3,axes = boxed, color=black, contours=[4,5],labels=[q1, q2], transparency=0 ): C:=PointPlot([2], xcoords=[2], thickness =20,color=black, symbol=solidbox, symbolsize = 15): unprotect(D): D:=PointPlot([1.36], xcoords=[1.68], thickness =20,color=black, symbol=solidbox, symbolsize = 15): unprotect(D): E:=PointPlot([1.69], xcoords=[1.34], thickness =20,color=black, symbol=solidbox, symbolsize = 15): unprotect(D): F:=textplot([2.1,2,"Nash_Equilibrium"],align={above,right}): G:=textplot([0.7,1.8,"Collusion : Advantage Firm 2"],align={above,right}): H:=textplot([1.7,1,"Collusion : Advantage Firm 1"],align={above,right}): display({A,B,C,D,E,F,G,H});

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# At the Nash equlibrium both firms produce 2 units of output, which means that q1=2 and q2=2 # Firms 1 profit in the Nash equilibrium is given by profit_1= subs({q1=2,q2=2},q1*(9-q1-q2)-3*q1);

L0kpcHJvZml0XzFHNiIiIiU=

# Firms 2 profit in the Nash equlibrium is given by profit_2= subs({q1=2,q2=2},q2*(9-q1-q2)-3*q2);

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# However firm 1 and firm 2 can achive higher profits by starting to cooperate (collusion).

# Collusion: Advantage firm 1 # For example if firm 1 convince firm 2 to reduce its produce to q2=1.4 then # firm 1 can decrease its production to q1=1.7 and still make a higher profit # Profit firm 1 ( q2=1.4 and q1=1.7 ) profit_1= subs({q2=1.4, q1=1.7},q1*(9-q1-q2)-3*q1);

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# Profit firm 2 ( q2=1.4 and q1=1.7 ) profit_2= subs({q2=1.4,q1=1.7},q2*(9-q1-q2)-3*q2);

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# Collusion: Advantage firm 2 # For example if firm 2 convince firm 1 to reduce its produce to q1=1.3 then # firm 2 can decrease its production to q2=1.6 and still make a higher profit # Profit firm 1 ( q1=1.3 and q2=1.6 ) profit_1= subs({q1=1.3,q2=1.6},q1*(9-q1-q2)-3*q1);

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# Profit firm 2 ( q1=1.3 and q2=1.6) profit_2= subs({q1=1.3,q2=1.6},q2*(9-q1-q2)-3*q2);

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# In both cases the profit of firm 1 and 2 is higher with collusion than without. # This means that there exist incentives for collusion # Dynamic Two Firm Model With Constant Marginal Cost (Shone, 2003) restart: # In the static model we assumed that firm 1 maximised its profits under the assumption that firm 2 keept # its output constant. Now we assume that firm 1 maximize its profits under the assumption that firm 2 keeps # output at the same level as output in the previous period t-1 ###################### # Note also that the marginal cost is constant over time TC:=3*q;

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MC:=diff(TC,q);

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# Since MC=3 it means that MC is constant and does not depend on the quantity produced q. ###################### # Profit firm 1 is given by profit1 = TR1 - TC1 profit1(t)= q1(t)*(9-q1(t)-q2(t-1))-3*q1(t);

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# Profit firm 2 is given by profit2 = TR2 - TC2 profit2(t)= q2(t)*(9-q1(t-1)-q2(t))-3*q2(t);

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# Firm 1 maximize its profits by setting the partial derivative with respect to q1(t) equal to zero diff(q1*(9-q1-q2(t-1))-3*q1,q1);

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q1(t)=solve(%,q1);

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# Firm 1 maximize its profits by setting the partial derivative with respect to q2(t) equal to zero diff(q2*(9-q1(t-1)-q2)-3*q2,q2);

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q2(t)=solve(%,q2);

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# Our two equations of motion are therfore given by q1(t) = 3-(1/2)*q2(t-1);

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q2(t) = 3-(1/2)*q1(t-1);

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######################## # We can solve for the generalization in Maple rsolve({q1(t)=3-(1/2)*q2(t-1),q2(t)=3-(1/2)*q1(t-1)},{q1(t),q2(t)});

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# we note that since (-(1/2)^t and (1/2)^t are smaller than 1. They will become zero over time evalf(seq(((-1/2))^t,t=0..10),3);

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# note that we oscillating between - and + but over time there is no such thing as - and + zero evalf(seq(((1/2))^t,t=0..10),3);

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# This means that both equations will converge to q1*=2 and q2*=2 over time q1(t)=2;

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q2(t)=2;

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# This means that we have a stable eqilibrium with fixed point 2 ######################## # We can now solve for the trajectories for the two equations above with brutal force # We assume that firm 1 start from a monopoly situation [q1(0)=3, q2(0)=0] # period 0 q1(0):=3;

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q2(0):=0;

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#period 1 q1(1):=3-0.5*q2(0);

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q2(1):=3-0.5*q1(0);

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#period 2 q1(2):=3-0.5*q2(1);

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q2(2):=3-0.5*q1(1);

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#period 3 q1(3):=3-0.5*q2(2);

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q2(3):=3-0.5*q1(2);

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#period 4 q1(4):=3-0.5*q2(3);

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q2(4):=3-0.5*q1(3);

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#period 5 q1(5):=3-0.5*q2(4);

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q2(5):=3-0.5*q1(4);

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#period 6 q1(6):=3-0.5*q2(5);

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q2(6):=3-0.5*q1(5);

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#period 7 q1(7):=3-0.5*q2(6);

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q2(7):=3-0.5*q1(6);

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#period 8 q1(8):=3-0.5*q2(7);

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q2(8):=3-0.5*q1(7);

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#period 9 q1(9):=3-0.5*q2(8);

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q2(9):=3-0.5*q1(8);

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#period 10 q1(10):=3-0.5*q2(9);

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q2(10):=3-0.5*q1(9);

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# We can now plot the result # Dynamic path for q1 q1_:=[seq([t,q1(t)],t=0..10)];

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plot( q1_, color=red,labels=[t,q1], view=[0..10,1.8..3]);

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# The output of firm 1 q1 will go from q1=3 to approaching the Nash Equilibrium=2 over time # Dynamic path for q2 q2_:=[seq([t,q2(t)],t=0..10)];

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plot( q2_, color=red,labels=[t,q2],view=[0..10,0..2.2] );

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# The output of firm 2 q2 will go from q2=0 to approaching the Nash equilibrium=2 over time # Dynamic path in a 2-d plane with(plots): with(Statistics): points:=[seq([q1(t),q2(t)],t=0..10)];

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A:=plot(points, color=red , view=[0..4,0..2.5 ]): B:=plot(2,y=0..4,color=black): C:=plot([2,y,y=0..4],color=black): unprotect(D): D:=PointPlot([2], xcoords=[2], thickness =20,color=black,symbol=solidbox, symbolsize = 15): E:=textplot([2,2,"Nash Equilibrium"],align={above,right}): display({A,B,C,D,E},labels=[q1,q2]);

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NiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWM2Ii0lJUZPTlRHNiUlKERFRkFVTFRHJShERUZBVUxURyIjNSUrSE9SSVpPTlRBTEclK0hPUklaT05UQUxHLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIkKychIiItJSlCT1VORFNfWUc2IyQiJD8iISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlP2MhIiItJS5CT1VORFNfSEVJR0hURzYjJCIlU00hIiItJSlDSElMRFJFTkc2Ig==

# The output of firm 1 and 2 (q1,q2) will approach the Nash equilibrium=2 over time

# Dynamic Two Firm Model With Non-Constant Marginal Cost (Shone, 2003) restart: # In the static model we assumed that firm 1 maximised its profits under the assumption that firm 2 keept # its output constant. Now we assume that firm 1 maximize its profits under the assumption that firm 2 keeps # output at the same level as output in the previous period t-1 ###################### # Note also that the marginal cost is not constant anymore TC:=3*q^2;

LCQqJiIiJCIiIilJInFHNiIiIiNGJUYl

MC:=diff(TC,q);

LCQqJiIiJyIiIkkicUc2IkYlRiU=

# Since MC=6*q it means MC is not constant but rather depend on the quantity produced q. # The precise nature of the MC in the form of increasing or decreasing MC # depend on the trajectory of q over time ###################### # Profit firm 1 is given by profit1 = TR1 - TC1 profit1(t)= q1(t)*(9-q1(t)-q2(t-1))-3*q1(t)^2;

Ly1JKHByb2ZpdDFHNiI2I0kidEdGJSwmKiYtSSNxMUdGJUYmIiIiLCgiIipGLEYqISIiLUkjcTJHRiU2IywmRidGLEYsRi9GL0YsRiwqJiIiJEYsKUYqIiIjRixGLw==

# Profit firm 1 is given by profit2 = TR2 - TC2 profit2(t)= q2(t)*(9-q1(t-1)-q2(t))-3*q2(t)^2;

Ly1JKHByb2ZpdDJHNiI2I0kidEdGJSwmKiYtSSNxMkdGJUYmIiIiLCgiIipGLC1JI3ExR0YlNiMsJkYnRixGLCEiIkYzRipGM0YsRiwqJiIiJEYsKUYqIiIjRixGMw==

# Firm 1 maximize its profits by setting the partial derivative with respect to q1(t) equal to zero diff(q1*(9-q1-q2(t-1))-3*q1^2,q1);

LCgiIioiIiIqJiIiKUYkSSNxMUc2IkYkISIiLUkjcTJHRig2IywmSSJ0R0YoRiRGJEYpRik=

q1(t)=solve(%,q1);

Ly1JI3ExRzYiNiNJInRHRiUsJiMiIioiIikiIiIqJiNGLEYrRiwtSSNxMkdGJTYjLCZGJ0YsRiwhIiJGLEYz

# Firm 1 maximize its profits by setting the partial derivative with respect to q2(t) equal to zero diff(q2*(9-q1(t-1)-q2)-3*q2^2,q2);

LCgiIioiIiItSSNxMUc2IjYjLCZJInRHRidGJEYkISIiRisqJiIiKUYkSSNxMkdGJ0YkRis=

q2(t)=solve(%,q2);

Ly1JI3EyRzYiNiNJInRHRiUsJiMiIioiIikiIiIqJiNGLEYrRiwtSSNxMUdGJTYjLCZGJ0YsRiwhIiJGLEYz

# Our two equations of motion are therfore given by q1(t) = 9/8-(1/8)*q2(t-1);

Ly1JI3ExRzYiNiNJInRHRiUsJiMiIioiIikiIiIqJiNGLEYrRiwtSSNxMkdGJTYjLCZGJ0YsRiwhIiJGLEYz

q2(t) = 9/8-(1/8)*q1(t-1);

Ly1JI3EyRzYiNiNJInRHRiUsJiMiIioiIikiIiIqJiNGLEYrRiwtSSNxMUdGJTYjLCZGJ0YsRiwhIiJGLEYz

######################## # We can solve for the generalization in Maple rsolve({q1(t)=(9/8)-(1/8)*q2(t-1),q2(t)=(9/8)-(1/8)*q1(t-1)},{q1(t),q2(t)});

PCQvLUkjcTFHNiI2I0kidEdGJiwoKiYjIiIiIiIjRiwpIyEiIiIiKUYoRixGLEYsRiwqJiMiIiRGLUYsKSNGLEYxRihGLEYsLy1JI3EyR0YmRicsKEYqRixGLEYsRjJGMA==

# we note that since (-(1/8)^t and (1/8)^t are smaller than 1. They will become zero over time evalf(seq(((-1/8))^t,t=0..10),3);

Ni0kIiIiIiIhJCEkRCIhIiQkIiRjIiEiJSQhJCY+ISImJCIkVyMhIickISQwJCEiKCQiJCJRISIpJCEkeCUhIiokIiQnZiEjNSQhJFgoISM2JCIkSiohIzc=

# note that we oscillating between - and + but over time there is no such thing as - and + zero evalf(seq(((1/8))^t,t=0..10),3);

Ni0kIiIiIiIhJCIkRCIhIiQkIiRjIiEiJSQiJCY+ISImJCIkVyMhIickIiQwJCEiKCQiJCJRISIpJCIkeCUhIiokIiQnZiEjNSQiJFgoISM2JCIkSiohIzc=

# This means that both equations will converge to q1*=1 and q2*=1 over time q1(t)=1;

Ly1JI3ExRzYiNiNJInRHRiUiIiI=

q2(t)=1;

Ly1JI3EyRzYiNiNJInRHRiUiIiI=

# This means that we have a stable eqilibrium with fixed point 1 ######################## # We can now solve for the trajectories for the two equations above with brutal force # We assume that firm 1 start from a monopoly situation [q1(0)=3, q2(0)=0] # period 0 q1(0):=3;

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q2(0):=0;

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#period 1 q1(1):=evalf(9/8-(1/8)*q2(0),4);

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q2(1):=evalf(9/8-(1/8)*q1(0),4);

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#period 2 q1(2):=evalf(9/8-(1/8)*q2(1),4);

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q2(2):=evalf(9/8-(1/8)*q1(1),4);

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#period 3 q1(3):=evalf(9/8-(1/8)*q2(2),4);

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q2(3):=evalf(9/8-(1/8)*q1(2),4);

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#period 4 q1(4):=evalf(9/8-(1/8)*q2(3),4);

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q2(4):=evalf(9/8-(1/8)*q1(3),4);

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#period 5 q1(5):=evalf(9/8-(1/8)*q2(4),4);

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q2(5):=evalf(9/8-(1/8)*q1(4),4);

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# We can now plot the result # Dynamic path for q1 q1_:=[seq([t,q1(t)],t=0..5)];

Nyg3JCIiISIiJDckIiIiJCIlRDYhIiQ3JCIiIyQiJUo1Rio3JEYlJCIlLTVGKjckIiIlJCIlKzVGKjckIiImRjQ=

plot( q1_, color=red,labels=[t,q1], view=[0..5,0.9..3]);

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

# The output of firm 1 q1 will go from q1=3 to converging to the equilibrium point q1=1 over time # Dynamic path for q2 q2_:=[seq([t,q2(t)],t=0..5)];

Nyg3JCIiIUYkNyQiIiIkIiUrdiEiJTckIiIjJCIlVykqRik3JCIiJCQiJWgqKkYpNyQiIiUkIiUpKioqRik3JCIiJiQiJSs1ISIk

plot( q2_, color=red,labels=[t,q2],view=[0..5,0..1.1] );

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# The output of firm 2 q2 will go from q2=0 to converging to the equilibrium point q2=1 over time # Dynamic path in a 2-d plane with(plots): with(Statistics): points:=[seq([q1(t),q2(t)],t=0..5)];

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A:=plot(points, color=red , view=[0..4,0..2.5 ]): B:=plot(1,y=0..4,color=black): C:=plot([1,y,y=0..4],color=black): unprotect(D): D:=PointPlot([1], xcoords=[1], thickness =20,color=black,symbol=solidbox, symbolsize = 15): E:=textplot([1,1,"Equilibrium"],align={above,right}): display({A,B,C,D,E},labels=[q1,q2]);

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# The output of firm 1 and 2 (q1,q2) will therefore approach the fixed point 1 over time # Dynamic Three Firm Model With Constant Marginal Cost (Shone, 2003) restart; # In the static model we assumed that firm 1 maximised its profits under the assumption that firm 2 keept # its output constant. Now we assume that firm 1 maximize its profits under the assumption that firm 2 and firm 3 keep their output at the same level as output in the previous period t-1 ###################### # Note also that the marginal cost is constant over time TC:=3*q;

LCQqJiIiJCIiIkkicUc2IkYlRiU=

MC:=diff(TC,q);

IiIk

# Since MC=3 it means that MC is constant and does not depend on the quantity produced q. ###################### #The profit functions are given by profit = TR - TC profit1(t)= q1(t)*(9-q1(t)-q2(t-1)-q3(t-1))-3*q1(t);

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profit2(t)= q2(t)*(9-q2(t)-q1(t-1)-q3(t-1))-3*q2(t);

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profit3(t)= q3(t)*(9-q3(t)-q1(t-1)-q2(t-1))-3*q3(t);

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# Profit Maximization diff(q1*(9-q1-q2(t-1)-q3(t-1))-3*q1,q1);

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q1(t)=solve(%,q1);

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diff(q2*(9-q2-q1(t-1)-q3(t-1))-3*q2,q2);

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q2(t)=solve(%,q2);

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diff(q3*(9-q3-q1(t-1)-q2(t-1))-3*q3,q3);

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q3(t)=solve(%,q3);

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# Our three equations of motion are therfore given by q1(t) = 3-(1/2)*q2(t-1)-(1/2)*q3(t-1);

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q2(t) = 3-(1/2)*q1(t-1)-(1/2)*q3(t-1);

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q3(t) = 3-(1/2)*q1(t-1)-(1/2)*q2(t-1);

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######################## # We can solve for the generalization in Maple rsolve({q1(t)=3-(1/2)*q2(t-1)-(1/2)*q3(t-1),q2(t)=3-(1/2)*q1(t-1)-(1/2)*q3(t-1),q3(t) = 3-(1/2)*q1(t-1)-(1/2)*q2(t-1) },{q1(t),q2(t),q3(t)});

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# we note that since ((1/2)^t is smaller than 1. They will become zero over time evalf(seq(((-1/2))^t,t=0..10),3);

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# This means that the above equations are reduced to {q1(t) = -(1/2)*(-1)^t+3/2+(1/3)*(-1)^t*q3(0), q2(t) = -(1/2)*(-1)^t+3/2+(1/3)*(-1)^t*q3(0), q3(t) = -(1/2)*(-1)^t+3/2+(1/3)*(-1)^t*q3(0)};

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# we note that (-1)^t is equal to -1 irregardless of the value of t which means that q1(t) = 2-(1/3)*q3(0), q2(t) = 2-(1/3)*q3(0), q3(t) = 2-(1/3)*q3(0);

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# Because we have (-1)^t we will have a oscillation between two values # This means that we have an unstable eqilibrium ######################## # We can now solve for the trajectories for the three equations above with brutal force # We assume that firm 1 start from a monopoly situation [q1(0)=3, q2(0)=0, q3(0)=0] # period 0 q1(0):=3;

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q2(0):=0;

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q3(0):=0;

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#period 1 q1(1):=3-0.5*q2(0)-0.5*q3(0);

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q2(1):=3-0.5*q1(0)-0.5*q3(0);

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q3(1):=3-0.5*q1(0)-0.5*q2(0);

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#period 2 q1(2):=3-0.5*q2(1)-0.5*q3(1);

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q2(2):=3-0.5*q1(1)-0.5*q3(1);

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q3(2):=3-0.5*q1(1)-0.5*q2(1);

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#period 3 q1(3):=3-0.5*q2(2)-0.5*q3(2);

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q2(3):=3-0.5*q1(2)-0.5*q3(2);

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q3(3):=3-0.5*q1(2)-0.5*q2(2);

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#period 4 q1(4):=3-0.5*q2(3)-0.5*q3(3);

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q2(4):=3-0.5*q1(3)-0.5*q3(3);

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q3(4):=3-0.5*q1(3)-0.5*q2(3);

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#period 5 q1(5):=3-0.5*q2(4)-0.5*q3(4);

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q2(5):=3-0.5*q1(4)-0.5*q3(4);

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q3(5):=3-0.5*q1(4)-0.5*q2(4);

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#period 6 q1(6):=3-0.5*q2(5)-0.5*q3(5);

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q2(6):=3-0.5*q1(5)-0.5*q3(5);

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q3(6):=3-0.5*q1(5)-0.5*q2(5);

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#period 7 q1(7):=3-0.5*q2(6)-0.5*q3(6);

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q2(7):=3-0.5*q1(6)-0.5*q3(6);

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q3(7):=3-0.5*q1(6)-0.5*q2(6);

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#period 8 q1(8):=3-0.5*q2(7)-0.5*q3(7);

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q2(8):=3-0.5*q1(7)-0.5*q3(7);

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q3(8):=3-0.5*q1(7)-0.5*q2(7);

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#period 9 q1(9):=3-0.5*q2(8)-0.5*q3(8);

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q2(9):=3-0.5*q1(8)-0.5*q3(8);

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q3(9):=3-0.5*q1(8)-0.5*q2(8);

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#period 10 q1(10):=3-0.5*q2(9)-0.5*q3(9);

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q2(10):=3-0.5*q1(9)-0.5*q3(9);

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q3(10):=3-0.5*q1(9)-0.5*q2(9);

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# We can now plot the result # Dynamic path for q1 q1_:=[seq([t,q1(t)],t=0..10)];

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plot( q1_, color=red,labels=[t,q1] );

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

# The output of firm 1 q1 will go from q1=3 to oscillating between 2 and 1 over time # Dynamic path for q2 q2_:=[seq([t,q2(t)],t=0..10)];

Ny03JCIiIUYkNyQiIiIkIiM6ISIiNyQiIiMkIiN2ISIjNyQiIiQkIiV2PSEiJDckIiIlJCIldiQqISIlNyQiIiYkIid2bz4hIiY3JCIiJyQiJ3ZWKSohIic3JCIiKCQiKXY9Iyo+ISIoNyQiIikkIil2JDQnKiohIik3JCIiKiQiK3ZvLykqPiEiKjckIiM1JCIrdlZCISoqKiEjNQ==

plot( q2_, color=red,labels=[t,q2],view=[0..10,0..2] );

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

# The output of firm 2 q2 will go from q2=0 to oscillating between 2 and 1 over time # Dynamic path for q3 q3_:=[seq([t,q3(t)],t=0..10)];

Ny03JCIiIUYkNyQiIiIkIiM6ISIiNyQiIiMkIiN2ISIjNyQiIiQkIiV2PSEiJDckIiIlJCIldiQqISIlNyQiIiYkIid2bz4hIiY3JCIiJyQiJ3ZWKSohIic3JCIiKCQiKXY9Iyo+ISIoNyQiIikkIil2JDQnKiohIik3JCIiKiQiK3ZvLykqPiEiKjckIiM1JCIrdlZCISoqKiEjNQ==

plot( q3_, color=red,labels=[t,q3],view=[0..10,0..2] );

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# The output of firm 3 q3 will go from q3=0 to oscillating between 2 and 1 over time # Dynamic path in a 3-d space with(plots): points:=[seq([q1(t),q2(t),q3(t)],t=0..10)];

Ny03JSIiJCIiIUYlNyUkRiRGJSQiIzohIiJGKDclJCIkXSIhIiMkIiN2Ri5GLzclJCIlXUEhIiQkIiV2PUY0RjU3JSQiJl03IiEiJSQiJXYkKkY6Rjs3JSQiJ11pPyEiJiQiJ3ZvPkZARkE3JSQiKF03LiIhIickIid2VikqRkZGRzclJCIpXWk6PyEiKCQiKXY9Iyo+RkxGTTclJCIqXTd5KyIhIikkIil2JDQnKipGUkZTNyUkIitdaSFSKyMhIiokIit2by8pKj5GWEZZNyUkIitDSiY+KyJGWCQiK3ZWQiEqKiohIzVGaG4=

pointplot3d(points,axes=BOXED, color=black,thickness=2,connect=true,labels=[q1,q2,q3]);

NigtJStBWEVTTEFCRUxTRzYlUSNxMTYiUSNxMjYiUSNxMzYiLSUqQVhFU1NUWUxFRzYjJSRCT1hHLSUnQ1VSVkVTRzYjNy03JSQiIiQiIiEkIiIhIiIhJCIiISIiITclJCIiJCIiISQiIzohIiIkIiM6ISIiNyUkIiRdIiEiIyQiI3YhIiMkIiN2ISIjNyUkIiVdQSEiJCQiJXY9ISIkJCIldj0hIiQ3JSQiJl03IiEiJSQiJXYkKiEiJSQiJXYkKiEiJTclJCInXWk/ISImJCIndm8+ISImJCIndm8+ISImNyUkIihdNy4iISInJCIndlYpKiEiJyQiJ3ZWKSohIic3JSQiKV1pOj8hIigkIil2PSMqPiEiKCQiKXY9Iyo+ISIoNyUkIipdN3krIiEiKSQiKXYkNCcqKiEiKSQiKXYkNCcqKiEiKTclJCIrXWkhUisjISIqJCIrdm8vKSo+ISIqJCIrdm8vKSo+ISIqNyUkIitDSiY+KyIhIiokIit2VkIhKioqISM1JCIrdlZCISoqKiEjNS0lLE9SSUVOVEFUSU9ORzYkJCEkSSchIiIkIiRdJyEiIi0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiIhISIiLSUqVEhJQ0tORVNTRzYjIiIj

# Which is the situation with oscillation # Dynamic Three Firm Model With Non-Constant Marginal Cost (Shone, 2003) restart; # In the static model we assumed that firm 1 maximised its profits under the assumption that firm 2 keept # its output constant. Now we assume that firm 1 maximize its profits under the assumption that firm 2 and firm 3 keep their output at the same level as output in the previous period t-1 ###################### # Note also that the marginal cost is not constant anymore TC:=3*q^2;

LCQqJiIiJCIiIilJInFHNiIiIiNGJUYl

MC:=diff(TC,q);

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# Since MC=6*q it means MC is not constant but rather depend on the quantity produced q. # The precise nature of the MC in the form of increasing or decreasing MC # depend on the trajectory of q over time ###################### #The profit functions are given by profit = TR - TC profit1(t)= q1(t)*(9-q1(t)-q2(t-1)-q3(t-1))-3*q1(t)^2;

Ly1JKHByb2ZpdDFHNiI2I0kidEdGJSwmKiYtSSNxMUdGJUYmIiIiLCoiIipGLEYqISIiLUkjcTJHRiU2IywmRidGLEYsRi9GLy1JI3EzR0YlRjJGL0YsRiwqJiIiJEYsKUYqIiIjRixGLw==

profit2(t)= q2(t)*(9-q2(t)-q1(t-1)-q3(t-1))-3*q2(t)^2;

Ly1JKHByb2ZpdDJHNiI2I0kidEdGJSwmKiYtSSNxMkdGJUYmIiIiLCoiIipGLEYqISIiLUkjcTFHRiU2IywmRidGLEYsRi9GLy1JI3EzR0YlRjJGL0YsRiwqJiIiJEYsKUYqIiIjRixGLw==

profit3(t)= q3(t)*(9-q3(t)-q1(t-1)-q2(t-1))-3*q3(t)^2;

Ly1JKHByb2ZpdDNHNiI2I0kidEdGJSwmKiYtSSNxM0dGJUYmIiIiLCoiIipGLEYqISIiLUkjcTFHRiU2IywmRidGLEYsRi9GLy1JI3EyR0YlRjJGL0YsRiwqJiIiJEYsKUYqIiIjRixGLw==

# Profit Maximization diff(q1*(9-q1-q2(t-1)-q3(t-1))-3*q1^2,q1);

LCoiIioiIiIqJiIiKUYkSSNxMUc2IkYkISIiLUkjcTJHRig2IywmSSJ0R0YoRiRGJEYpRiktSSNxM0dGKEYsRik=

q1(t)=solve(%,q1);

Ly1JI3ExRzYiNiNJInRHRiUsKCMiIioiIikiIiIqJiNGLEYrRiwtSSNxMkdGJTYjLCZGJ0YsRiwhIiJGLEYzKiZGLkYsLUkjcTNHRiVGMUYsRjM=

diff(q2*(9-q2-q1(t-1)-q3(t-1))-3*q2^2,q2);

LCoiIioiIiIqJiIiKUYkSSNxMkc2IkYkISIiLUkjcTFHRig2IywmSSJ0R0YoRiRGJEYpRiktSSNxM0dGKEYsRik=

q2(t)=solve(%,q2);

Ly1JI3EyRzYiNiNJInRHRiUsKCMiIioiIikiIiIqJiNGLEYrRiwtSSNxMUdGJTYjLCZGJ0YsRiwhIiJGLEYzKiZGLkYsLUkjcTNHRiVGMUYsRjM=

diff(q3*(9-q3-q1(t-1)-q2(t-1))-3*q3^2,q3);

LCoiIioiIiIqJiIiKUYkSSNxM0c2IkYkISIiLUkjcTFHRig2IywmSSJ0R0YoRiRGJEYpRiktSSNxMkdGKEYsRik=

q3(t)=solve(%,q3);

Ly1JI3EzRzYiNiNJInRHRiUsKCMiIioiIikiIiIqJiNGLEYrRiwtSSNxMUdGJTYjLCZGJ0YsRiwhIiJGLEYzKiZGLkYsLUkjcTJHRiVGMUYsRjM=

# Our three equations of motion are therfore given by q1(t) = 9/8-(1/8)*q2(t-1)-(1/8)*q3(t-1);

Ly1JI3ExRzYiNiNJInRHRiUsKCMiIioiIikiIiIqJiNGLEYrRiwtSSNxMkdGJTYjLCZGJ0YsRiwhIiJGLEYzKiZGLkYsLUkjcTNHRiVGMUYsRjM=

q2(t) = 9/8-(1/8)*q1(t-1)-(1/8)*q3(t-1);

Ly1JI3EyRzYiNiNJInRHRiUsKCMiIioiIikiIiIqJiNGLEYrRiwtSSNxMUdGJTYjLCZGJ0YsRiwhIiJGLEYzKiZGLkYsLUkjcTNHRiVGMUYsRjM=

q3(t) = 9/8-(1/8)*q1(t-1)-(1/8)*q2(t-1);

Ly1JI3EzRzYiNiNJInRHRiUsKCMiIioiIikiIiIqJiNGLEYrRiwtSSNxMUdGJTYjLCZGJ0YsRiwhIiJGLEYzKiZGLkYsLUkjcTJHRiVGMUYsRjM=

######################## # We can solve for the generalization in Maple rsolve({q1(t)=(9/8)-(1/8)*q2(t-1)-(1/8)*q3(t-1),q2(t)=(9/8)-(1/8)*q1(t-1)-(1/8)*q3(t-1),q3(t) = (9/8)-(1/8)*q1(t-1)-(1/8)*q2(t-1) },{q1(t),q2(t),q3(t)});

PCUvLUkjcTFHNiI2I0kidEdGJiwyKiYjIiIqIiM1IiIiKSMhIiIiIiVGKEYuRjFGK0YuKigjRi4iIiRGLkYvRi4tSSNxM0dGJjYjIiIhRi5GLiooRjRGLkYvRi4tRiVGOEYuRi4qKEY0Ri5GL0YuLUkjcTJHRiZGOEYuRi4qKEY0Ri4pI0YuIiIpRihGLkY2Ri5GMSooIyIiI0Y1Ri5GQEYuRjtGLkYuKihGNEYuRkBGLkY9Ri5GMS8tRj5GJywyRipGMUYrRi5GM0YuRjpGLkY8Ri4qKEY0Ri5GQEYuRjtGLkYxKihGREYuRkBGLkY9Ri5GLkY/RjEvLUY3RicsMkYqRjFGK0YuRjNGLkY6Ri5GPEYuRkpGMUZGRjEqKEZERi5GQEYuRjZGLkYu

# we note that since (-(1/4)^t and (1/8)^t are smaller than 1. They will become zero over time evalf(seq(((-1/4))^t,t=0..10),3);

Ni0kIiIiIiIhJCEkXSMhIiQkIiREJyEiJSQhJGMiRiskIiQiUiEiJiQhJHgqISInJCIkVyNGMyQhJDUnISIoJCIkYCJGOCQhJCJRISIpJCIkYSohIio=

# note that we oscillating between - and + but over time there is no such thing as - and + zero evalf(seq(((1/8))^t,t=0..10),3);

Ni0kIiIiIiIhJCIkRCIhIiQkIiRjIiEiJSQiJCY+ISImJCIkVyMhIickIiQwJCEiKCQiJCJRISIpJCIkeCUhIiokIiQnZiEjNSQiJFgoISM2JCIkSiohIzc=

# The equations will therefore converge to q1*=(9/10)=0.9, q2*=(9/10)=0.9 and q3*=(9/10)=0.9 and over time q1(t)=0.9;

Ly1JI3ExRzYiNiNJInRHRiUkIiIqISIi

q2(t)=0.9;

Ly1JI3EyRzYiNiNJInRHRiUkIiIqISIi

q3(t)=0.9;

Ly1JI3EzRzYiNiNJInRHRiUkIiIqISIi

# This means that we have a stable eqilibrium with fixed point 0.9 ######################## # We can now solve for the trajectories for the three equations above with brutal force # We assume that firm 1 start from a monopoly situation [q1(0)=3, q2(0)=0, q3(0)=0] # period 0 q1(0):=3;

IiIk

q2(0):=0;

IiIh

q3(0):=0;

IiIh

#period 1 q1(1):=evalf(9/8-(1/8)*q2(0)-(1/8)*q3(0),4);

JCIlRDYhIiQ=

q2(1):=evalf(9/8-(1/8)*q1(0)-(1/8)*q3(0),4);

JCIlK3YhIiU=

q3(1):=evalf(9/8-(1/8)*q1(0)-(1/8)*q2(0),4);

JCIlK3YhIiU=

#period 2 q1(2):=evalf(9/8-(1/8)*q2(1)-(1/8)*q3(1),4);

JCIlcyQqISIl

q2(2):=evalf(9/8-(1/8)*q1(1)-(1/8)*q3(1),4);

JCIlMSopISIl

q3(2):=evalf(9/8-(1/8)*q1(1)-(1/8)*q2(1),4);

JCIlMSopISIl

#period 3 q1(3):=evalf(9/8-(1/8)*q2(2)-(1/8)*q3(2),4);

JCIlRiEqISIl

q2(3):=evalf(9/8-(1/8)*q1(2)-(1/8)*q3(2),4);

JCIlbiopISIl

q3(3):=evalf(9/8-(1/8)*q1(2)-(1/8)*q2(2),4);

JCIlbiopISIl

#period 4 q1(4):=evalf(9/8-(1/8)*q2(3)-(1/8)*q3(3),4);

JCIlNCEqISIl

q2(4):=evalf(9/8-(1/8)*q1(3)-(1/8)*q3(3),4);

JCIlKioqKSEiJQ==

q3(4):=evalf(9/8-(1/8)*q1(3)-(1/8)*q2(3),4);

JCIlKioqKSEiJQ==

#period 5 q1(5):=evalf(9/8-(1/8)*q2(4)-(1/8)*q3(4),4);

JCIlJioqKSEiJQ==

q2(5):=evalf(9/8-(1/8)*q1(4)-(1/8)*q3(4),4);

JCIlJioqKSEiJQ==

q3(5):=evalf(9/8-(1/8)*q1(4)-(1/8)*q2(4),4);

JCIlJioqKSEiJQ==

# We can now plot the result # Dynamic path for q1 q1_:=[seq([t,q1(t)],t=0..5)];

Nyg3JCIiISIiJDckIiIiJCIlRDYhIiQ3JCIiIyQiJXMkKiEiJTckRiUkIiVGISpGLzckIiIlJCIlNCEqRi83JCIiJiQiJSYqKilGLw==

plot( q1_, color=red,labels=[t,q1],view=[0..5,0.8..3] );

NictJSdDVVJWRVNHNiM3KDckJCIiISEiIiQiI0khIiI3JCQiIzUhIiIkIiVENiEiJDckJCIjPyEiIiQiJXMkKiEiJTckJCIjSSEiIiQiJUYhKiEiJTckJCIjUyEiIiQiJTQhKiEiJTckJCIjXSEiIiQiJSYqKikhIiUtJSVWSUVXRzYkOyQiIiEhIiIkIiNdISIiOyQiIikhIiIkIiNJISIiLSUrQVhFU0xBQkVMU0c2Jy1JI21pRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNjVRInQ2Ii8lJ2ZhbWlseUdRITYiLyUlc2l6ZUdRIzEwNiIvJSVib2xkR1EmZmFsc2U2Ii8lJ2l0YWxpY0dRJXRydWU2Ii8lKnVuZGVybGluZUdRJmZhbHNlNiIvJSpzdWJzY3JpcHRHUSZmYWxzZTYiLyUsc3VwZXJzY3JpcHRHUSZmYWxzZTYiLyUrZm9yZWdyb3VuZEdRKFswLDAsMF02Ii8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdNiIvJSdvcGFxdWVHUSZmYWxzZTYiLyUrZXhlY3V0YWJsZUdRJmZhbHNlNiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWM2Ii1JI21pRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNjVRI3ExNiIvJSdmYW1pbHlHUSE2Ii8lJXNpemVHUSMxMDYiLyUlYm9sZEdRJmZhbHNlNiIvJSdpdGFsaWNHUSV0cnVlNiIvJSp1bmRlcmxpbmVHUSZmYWxzZTYiLyUqc3Vic2NyaXB0R1EmZmFsc2U2Ii8lLHN1cGVyc2NyaXB0R1EmZmFsc2U2Ii8lK2ZvcmVncm91bmRHUShbMCwwLDBdNiIvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XTYiLyUnb3BhcXVlR1EmZmFsc2U2Ii8lK2V4ZWN1dGFibGVHUSZmYWxzZTYiLyUpcmVhZG9ubHlHUSZmYWxzZTYiLyUpY29tcG9zZWRHUSZmYWxzZTYiLyUqY29udmVydGVkR1EmZmFsc2U2Ii8lK2ltc2VsZWN0ZWRHUSZmYWxzZTYiLyUscGxhY2Vob2xkZXJHUSZmYWxzZTYiLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lLG1hdGh2YXJpYW50R1EnaXRhbGljNiItJSVGT05URzYlJShERUZBVUxURyUoREVGQVVMVEciIzUlK0hPUklaT05UQUxHJStIT1JJWk9OVEFMRy0lJkNPTE9SRzYmJSRSR0JHJCIjNSEiIiQiIiEhIiIkIiIhISIiLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIkKyQhIiItJSlCT1VORFNfWUc2IyQiJD8iISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlP08hIiItJS5CT1VORFNfSEVJR0hURzYjJCIlZ00hIiItJSlDSElMRFJFTkc2Ig==

# The output of firm 1 q1 will go from q1=3 to converging to the equilibrium point q1=0.9 # Dynamic path for q2 q2_:=[seq([t,q2(t)],t=0..5)];

Nyg3JCIiIUYkNyQiIiIkIiUrdiEiJTckIiIjJCIlMSopRik3JCIiJCQiJW4qKUYpNyQiIiUkIiUqKiopRik3JCIiJiQiJSYqKilGKQ==

plot( q2_, color=red,labels=[t,q2],view=[0..5,0..1] );

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

# The output of firm 2 q2 will go from q2=0 to converging to the equilibrium point q2=0.9 # Dynamic path for q3 q3_:=[seq([t,q3(t)],t=0..5)];

Nyg3JCIiIUYkNyQiIiIkIiUrdiEiJTckIiIjJCIlMSopRik3JCIiJCQiJW4qKUYpNyQiIiUkIiUqKiopRik3JCIiJiQiJSYqKilGKQ==

plot( q3_, color=red,labels=[t,q3],view=[0..5,0..1] );

NictJSdDVVJWRVNHNiM3KDckJCIiISEiIiQiIiEhIiI3JCQiIzUhIiIkIiN2ISIjNyQkIiM/ISIiJCIlMSopISIlNyQkIiNJISIiJCIlbiopISIlNyQkIiNTISIiJCIlKioqKSEiJTckJCIjXSEiIiQiJSYqKikhIiUtJSVWSUVXRzYkOyQiIiEhIiIkIiNdISIiOyQiIiEhIiIkIiM1ISIiLSUrQVhFU0xBQkVMU0c2Jy1JI21pRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNjVRInQ2Ii8lJ2ZhbWlseUdRITYiLyUlc2l6ZUdRIzEwNiIvJSVib2xkR1EmZmFsc2U2Ii8lJ2l0YWxpY0dRJXRydWU2Ii8lKnVuZGVybGluZUdRJmZhbHNlNiIvJSpzdWJzY3JpcHRHUSZmYWxzZTYiLyUsc3VwZXJzY3JpcHRHUSZmYWxzZTYiLyUrZm9yZWdyb3VuZEdRKFswLDAsMF02Ii8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdNiIvJSdvcGFxdWVHUSZmYWxzZTYiLyUrZXhlY3V0YWJsZUdRJmZhbHNlNiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWM2Ii1JI21pRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNjVRI3EzNiIvJSdmYW1pbHlHUSE2Ii8lJXNpemVHUSMxMDYiLyUlYm9sZEdRJmZhbHNlNiIvJSdpdGFsaWNHUSV0cnVlNiIvJSp1bmRlcmxpbmVHUSZmYWxzZTYiLyUqc3Vic2NyaXB0R1EmZmFsc2U2Ii8lLHN1cGVyc2NyaXB0R1EmZmFsc2U2Ii8lK2ZvcmVncm91bmRHUShbMCwwLDBdNiIvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XTYiLyUnb3BhcXVlR1EmZmFsc2U2Ii8lK2V4ZWN1dGFibGVHUSZmYWxzZTYiLyUpcmVhZG9ubHlHUSZmYWxzZTYiLyUpY29tcG9zZWRHUSZmYWxzZTYiLyUqY29udmVydGVkR1EmZmFsc2U2Ii8lK2ltc2VsZWN0ZWRHUSZmYWxzZTYiLyUscGxhY2Vob2xkZXJHUSZmYWxzZTYiLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lLG1hdGh2YXJpYW50R1EnaXRhbGljNiItJSVGT05URzYlJShERUZBVUxURyUoREVGQVVMVEciIzUlK0hPUklaT05UQUxHJStIT1JJWk9OVEFMRy0lJkNPTE9SRzYmJSRSR0JHJCIjNSEiIiQiIiEhIiIkIiIhISIiLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIkcSYhIiItJSlCT1VORFNfWUc2IyQiJD8iISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlXUwhIiItJS5CT1VORFNfSEVJR0hURzYjJCIlZ00hIiItJSlDSElMRFJFTkc2Ig==

# The output of firm 3 q3 will go from q3=0 to converging to the equilibrium point q3=0.9 # Dynamic path in a 3-d space with(plots): points:=[seq([q1(t),q2(t),q3(t)],t=0..5)];

Nyg3JSIiJCIiIUYlNyUkIiVENiEiJCQiJSt2ISIlRio3JSQiJXMkKkYsJCIlMSopRixGMDclJCIlRiEqRiwkIiVuKilGLEY1NyUkIiU0ISpGLCQiJSoqKilGLEY6NyUkIiUmKiopRixGPUY9

pointplot3d(points,axes=BOXED, color=black,thickness=2,connect=true,labels=[q1,q2,q3],view=[0..3,0..1,0..1]);

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

# The output of firm 1,2,3 (q1,q2,q3) will therefor approach the fixed point 0.9 over time JSFH Legal Notice: The copyright for this application is owned by the authors. Neither Maplesoft nor the authors are responsible for any errors contained within and are not liable for any damages resulting from the use of this material.. This application is intended for non-commercial, non-profit use only. Contact the authors for permission if you wish to use this application in for-profit activities.