Econ 4001 assignment assume that b lt 0 alpha gt 0 gamma gt

Question

ECON 4001 Assignment-

Q1: Consider the following system of differential equations:

x^• = x + y + t                                                                                                        (1)

y^• = - x + 2y                                                                                                         (2)

Find the complete solution to the system of differential equations.

Q2: Consider the following system of differential equations:

x^• = 2x - x² - y + 2                                                                                                (3)

y^• = x - y                                                                                                               (4)

(a) Find all steady-states of this system of equations. For each of them determine if it is locally stable, unstable or a saddle point.

(b) Draw a phase diagram in the y - x space with y on the vertical axis and x on the horizontal axis. Draw the phase diagram for the system in the first quadrant, that is for x > 0 and y > 0. Clearly explain how you obtain the x^• = 0 curve and the y^• = 0 curve. Justify the slope, rate of change of the slope, etc...Separate the phase diagram in (4) different zones and explain using arrows how x and y move in each zone. Justify the direction of the arrows. Identify the paths and explain if they converge to the steady-state or not. Draw possible trajectories to the steady-state if is stable or away from the steady-state if it unstable.

Q3: We have the following model of market equilibrium:

p^• = α(q^D - q^S)                                                                                                       (5)

N^• = γ (p - c^-)                                                                                                         (6)

where N is the number of firms in the industry and where:

q^D = A + Bp                                                                                                            (7)

q^S = (F + Gp)N                                                                                                        (8)

Assume that b < 0; α > 0; γ > 0; F > 0;G > 0; c^- > 0:

(a) Find the steady-state of the economic system.

(b) Examine the stability of the economic system. Hint: proceed as follows. Linearize the nonlinear equations around the steady-state and determine if the steady-state is stable.

(c) Examine how very small or very large values of could affect the stability of market equilibrium. Explain why.

Q4: Consider the following system of differential equations:

y^•₁ = 2y₁ + y₂/2 - 10                                                                                             (9)

y^•₂ = 7y₁/2 - y₂ + 8                                                                                               (10)

(a) Solve the homogeneous system of differential equations.

(b) Find the steady-state.

(c) Find the complete solution.

(d) Is the steady-state stable? Justify your answer. If it is a saddle point, find the equation of the saddle path.

(e) If y₁(0) = 1 for what value of y₂(0) will the economic system converge to the steady-state?

(e) Draw a phase diagram in the y₂ - y₁ space with y₂ on the vertical axis and y₁ on the horizontal axis. Clearly explain how you obtain the y^•₁ = 0 curve and the y^•₂ = 0 curve. Justify the slope, rate of change of the slope, etc...Separate the phase diagram in (4) different zones and explain using arrows how y₁ and y₂ move in each zone. Justify the direction of the arrows. If applicable identify the saddle path.