Mathematical Economics Assignment

Problem- The duopoly model describes a market with two firms only. They produce the same product and the supply is Q = q₁ + q₂. The demand for this product is of a linear form

p(Q) = a - bQ

where p is the price of the product. The firms have different production costs given by

C_i(q_i) = c_i + d_iq_i, i = 1, 2.

Because the marginal cost d_i must be lower than maximal market price a, so the following condition should be fulfilled: d_i < a.

The profit of the firm i is

Π_i = q_ip(Q) - C_i = q_i(a - bQ) - (c_i + d_iq_i).

Bischi and Naimzada (1999) proposed the following model of duopoly with the bounded rationality principle. Both the firms knows only their own profit and adjusts the supply to the changes of its profit

q?_i(t) = α_iq_i(t)(∂Π_i(q₁9t), q2(t)/∂qq_i),        i = 1, 2,

where α₁ is a positivepositive speed of adjustment.

Hence, the duopoly model is given by a system of differential equations

q?₁(t) = α₁q₁(t)[a - d₁ - 2bq₁(t) - bq₂(t)]                                                 (1)

q?₂(t) = α₂q₂(t)[a - d₁ - bq₁(t) - 2bq₂(t)].                                                (2)

Exercise-

For the duopoly model (1)-(2)

• find all critical points,
• find the type of these critical points.

Computer exercise

Choose values of model parameters: e.g. α₁ = α₂ = 1, a = 10, b = 1, d₁ = 2, d₂ = 3. You can assume your own parameter values set, remembering the economic meaning of this parameters (all these parameters should be positive).

Draw the phase portrait of the duopoly system with following elements

• the vector field using the function flowField,
• the nullclines using the function nullclines,
• a few trajectories to illustrate the behaviour in neighbourhoods of critical points using the function trajectory.

Literature

Bischi, G.I., Naimzada, A., 1999, Global analysis of a dynamic duopoly game with bounded rationality. Dynamic Games and Applications, vol. 5, Birkhouser (chapter 20).