Student Solutions Manual to Accompany Economic Dynamics in Discrete Time, secondedition

Chapter 1

Deterministic Linear Systems

Exercise 1

(a) Under rational expectations, art . We can express pt as

Solving this equation forward gives

By the transversality condition

we obtain

Substituting mt+1 = ρmt + μ into the preceding equation yields

When ρ ∈ [0, 1), this is the unique stable solution.

(b) From part (a) we can derive that pT = m′. The price level at time t < T satisfies

We can derive

Exercise 3

Let xt = [kt, yt]′.

(a) The steady state is (0, 0). Solving Δxt+1 = (A − I)xt = 0, we get two nullclines

The phase diagram is shown in Figure 1.1. We can find that the two eigenvalues of A are art ; so one is outside the unit circle and the other is inside. Hence the steady state is a saddle point.

(b) Any point on the k-axis is a steady state, and is unstable. The phase diagram is shown in Figure 1.2.

(c) The steady state is (0, 0). Solving Δxt+1 = (A − I)xt = 0, we get two nullclines

Figure 1.1: Phase diagram 1

The phase diagram is shown in Figure 1.3. We find that A has two eigenvalues of +i and −i, which are on the unit circle. Hence the steady state is stable but not asymptotically stable.

Exercise 5

Consider a version of the Dornbusch (1976) model:

Figure 1.2: Phase diagram 2

(a) To derive the steady state of the model, we remove all the time subscripts and derive

(b) We can solve for rt from equation (1.3) and get the following two equations:

Figure 1.3: Phase diagram 3

and

Let us now write the system of equations as the following matrix form:

To obtain a unique stable solution, we need one eigenvalue to be in the unit circle and the other to be outside the unit circle. The characteristic polynomial is

We can find

since all parameters are positive. So the characteristic polynomial has one root outside the unit circle. In order to have the other root within the unit circle, we require

(c) We can calculate the two nullclines:

The phase diagram is shown in Figure 1.4, where the dotted line is the saddle path.

(d) Assume that the economy is initially at the steady state. There is an unexpected permanent increase in m from m to m′ at t = 0. The steady state satisfies

Since art , at the new steady state, p′ > p and e′ > e. Figure 1.5 shows the new steady state and the corresponding new saddle path in red. When the unexpected

Figure 1.4: Phase diagram

increase in money supply arrives, the exchange rate e first jumps up to the new saddle path while the price does not move. Then et and pt move along the new saddle path towards the new steady state. The dynamics are shown in Figure 1.5.

(e) After the money supply increase is announced at time t, the exchange rate et first jumps up while the price does not move. From time t to T, both et and pt increase according to the dynamics of the old system. At time T, et and pt will exactly reach the new saddle path. Then they move along the new saddle path and converge to the new steady state. Figure 1.6 shows the dynamics.

Figure 1.5: Dynamics after a permanent money supply shock

Figure 1.6: Dynamics after a permanent money supply announcement

Chapter 2

Deterministic Nonlinear Systems

Exercise 1

In all three cases the steady state is zero and is non-hyperbolic. All steady states are asymptotically stable, since the derivative of each map in absolute value is less than one in the neighborhood of the