Exercises

1. For each of the following differential equation systems, determine whether the system is a stable or unstable node, saddle point, stable or unstable focus, or center:

2. Solve the following linear differential equation system, draw the phase diagram, and find the equation for the saddle path.

3. Solve the following linear differential equation system, draw the phase diagram. and find the equation for the saddle path. If ,V| (0) = 8. what value must be chosen fory2<0) to ensure that the system converges to the steady state?

4. Consider the following nonlinear market equilibrium model. Price adjusts to excess demand as given by p = - qs)

and the number of firms in the industry adjusts to excess profits according to

where Qb = u + bp is the demand function and the supply function is

Assume that b < 0 and all other parameters (a. y. a, F, G. c) are greater than zero.

Show that if y is not too large, the equilibrium is a stable node, but that if y is large enough, the steady state could be a stable focus.

5. In the Dornbusch model, use the following parameter values: a = 1, v = I, ii = 4/3, b = 1/3, y = 3.4, m = 3. r* = 0.1, and u = 4. Solve for the steady-state price and exchange rate, p and c>: solve the differential equation system. It p(0) - 4, to what value must <?(0) "jump" to put the economy on the saddle path?

24.3 Systems of Linear Difference Equations

The techniques for solving systems of linear difference equations arc similar to those developed in section 24.1 for solving systems of linear differential equations. Wc therefore provide a briefer coverage of this topic.

Definition 24.6

The general form for a system of two I coefficients and terms is

inear difference equations with constant

As usual with linear difference or differential equations, the solutions to llie complete equations are equal to the sum of the homogeneous solutions and particular solutions to the complete equations. We begin by solving the homogeneous form.

The Homogeneous Solutions

Begin by putting the difference equation system into its homogeneous form. This gives

We now show two approaches to solving this homogeneous system of difference equations. The first is the substitution method; the second is the direct method.

The Substitution Method

In this approach we use substitution to reduce the system of two tirst-order difference equations to a single second-order difference equation. From equation (24.32) wc know that y,+2 = + l + «12*,+ ) Using equation (24.33) to substitute for 1. we write

which is a second-order difference equation. Since it still depends on .v(, we use equation (24.32) again but this time to get the following expression:

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