Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

stable, however, because ) = Q(oo) = 0) but the h of the shock terms, i.e. thus "path dependent"

ter ' y.

,-nlosz (1985, p. 355) r the following simple

(LM)

(Fisher)

(IS)

Mathematical Appendix

Non-negative roots (A1 = 0 < A.2)

We now assume that A in (A.96) satisfies I A I = A l A2 = 0 and tr(A) = A l + A2 > 0 so that Al = 0 and A2 = tr(A) > 0. For this hysteretic case the analysis in subsection A.6.3 is relevant. The general solution in Laplace transforms is obtained by setting Al = 0 in (A.115):

(A.121)

Let us once again assume that the shock is temporary and has a Laplace transform for i = K, Q so that:

Equation (A.121) can then be rewritten as:

= -0 and A2 = 0 -A variables (so that ry boost in aggregate he methods developed

permanent effect on

As in the outright stable case (see (A.120)) parameters of the shock path determine the ultimate long-run result.

Intermezzo

Current account dynamics. Consider the simple representative-agent model of a small open economy suggested by Blanchard (1985, p. 230). There is no

1° In going from the first to the second line we use (A.112), note that (A.105) implies A.21 = adj A (A2) — adj A(0), and recall that adj A(0) = —adj A.

Mathematical Appendix

capital and labour supply is exogenously fixed (at unity) so that output, Y, and the wage rate, W = Y, are exogenous. The model is:

C(t) [r(t) - a} C(t)

F(t) r(t)F(t) + TAT (t) C(t),

where F is net foreign assets, and C and r are, respectively, consumption and the exogenous interest rate. As is well known, a steady state only exists in this model if the steady-state interest rate equals the rate of time preference,

After loglinearizing the model around an initial steady state we obtain:

where wF aF/Y = C/Y - 1 is the initial share of foreign asset income in national output, F(t) adFIY, and F(t) a dF/Y. The Jacobian matrix on the right-hand side has characteristic roots Al = 0 and X2 = a and it is assumed that F is the predetermined variable and C is the jumping variable. Now consider a temporary change in the world interest rate, 1. (t) = e- Rt for R > 0 and t > 0. By using (A.113) and making the obvious substitutions we obtain the jump in consumption:

a <0.

(ot + R)(1- + (0F)

In a similar fashion, the long-run results can be obtained by using (A.124):

In the impact period the household cuts back consumption to boost its savings. In the long run both consumption and net foreign assets are higher than in the initial steady state (provided wF --a I (a + R) in the initial steady state).

A.6.5 Literature

The most accessible intermediate sources to the Laplace transform method are to be found in the engineering literature. Kreyszig (1999, ch. 5) and Boyce and DiPrima (1992, ch. 6) are particularly illuminating. An advanced and encyclopedic source on Laplace transforms is Spiegel (1965). Judd (1982, 1985, 1987a, 1987b) was the

first to apply the methc note the close link

I

A.7 Difference Eq

Although continuous- . often work with mock, this category as does the (1958)-Diamond (19 transform method. Thu the Laplace transform avoid unnecessary dupduced. The student shot discrete-time setting a.. z-transform method are (1998) apply the tech:

A.7.1 Basic methods

The basic first-order

+ ayt = b,

where a and b are c. (non-homogeneous). L to (A.64). Just as for two steps. In step 1 %, homogeneous part of 0 The general solution is To solve the homoge

function for which y, Substituting this trial

Hence, the con

yt = A( - a)t .

To find the particu' constant). Substitutii _ can be solved for difference equation is t

b

yr = A( - a)t +

This expression is the C

so that output, Y, and

- consumption and ate only exists in this f time preference, i.e. if 1s4. 3dy state we obtain:

, reign asset income in Jacobian matrix on the

.4 and it is assumed that able. Now consider a for R > 0 and t > 0. we obtain the jump in

KI by using (A.124):

on to boost its savings. s are higher than in nitial steady state).

form method are to be and Boyce and DiPrima

-encyclopedic source 1987a, 1987b) was the

Mathematical Appendix

first to apply the method to saddle-point stable perfect foresight models, and to note the close link with welfare evaluations along the transition path.

A.7 Difference Equations

Although continuous-time models are quite convenient to work with, economists often work with models formulated in discrete time. Most RBC models fall under this category as does the class of overlapping-generations models in the Samuelson. (1958)-Diamond (1965) tradition. In this section we briefly introduce the z- transform method. This method plays the same role in discrete-time models that the Laplace transform method performs in continuous-time models. In order to avoid unnecessary duplication, only the basic elements of the z-transform are introduced. The student should be able to "translate" the insights obtained above to the discrete-time setting after reading this section. Extremely lucid expositions of the z-transform method are Ogata (1995) and Elaydi (1996). Meijdam and Verhoeven (1998) apply the techniques in an economic setting.

where a and b are constant parameters. If b = 0 (0 0) the equation is homogeneous (non-homogeneous). Equation (A.125) can be seen as the discrete-time counterpart to (A.64). Just as for the continuous case we can solve the difference equation in two steps. In step 1 we solve the complementary function, yF, which solves the homogeneous part of (A.125). In step 2 we then look for the particular solution, yr. The general solution is then given by Yr = yt +

To solve the homogeneous part of the difference equation we are looking for a function for which yr± i lyt = -a which suggests that a good trial solution is yt = Aat. Substituting this trial in (A.125) and setting b = 0 we obtain Aat [a + a] = 0 or a = - a . Hence, the complementary function is:

yF = A( - a)t (A.126)

To find the particular solution we first try the simplest possible guess, yr = k (a constant). Substituting this trial solution into (A.125) we find (1 + a)k = b which can be solved for k provided a 0 -1: k = b/(1 + a) . The general solution of the difference equation is thus:

This expression is the discrete time counterpart to (A.70).

695

Mathematical Appendix

Whereas a zero coefficient necessitates a different trial for the particular solution in the continuous time case, the same holds for the discrete-time case when the coefficient is minus unity. If a = -1 we use the trial solution yr = kt, after which we find that k = b so that the general solution is:

Initial conditions can be imposed just as for the continuous-time case. Suppose that yo is some given constant. Then we obtain from (A.127) that A = yo - b/(1 + a) and from (A.128) that A = yo.

Just as in the continuous-time case, there exists a very convenient transformation method for solving difference equations. We now briefly explain how this z-transform method works.

A.7.2 The z-transform

Suppose we have a discrete-time function, ft , which satisfies ft = 0 for t = -1, -2, . The (one-sided) z-transform of the function is then defined as follows: 11

Provided the sum on the right-hand side converges, Z{ ft , z} exists and can be seen as a function of z. The region of convergence is determined as follows. Suppose that ft satisfies:

Then the infinite sum in (A.129) converges provided:

and diverges if the inequality is reversed. Together, (A.130) and (A.131) imply that (A.129) converges—and Z{ ft , z} exists—in the region VI > R ("heavy discounting"). In the region Izi < R , on the other hand, discounting is "light" and Z{ft , z} does not exist. R is referred to as the radius of convergence of Z{ ft , z}.

By comparing (A.84) and (A.129) we cannot help but notice the close relation that exists between the Laplace transform and the z-transform. Indeed, assuming that f (t) in (A.84) is continuous we obtain by discretizing Gff , sl = Ecttc, e - st ft . By setting z = es we obtain (A.129). See also Elaydi (1996, p. 254).

Table A.2. G

ft

I fort = 0 0 for t = 1,2

1

at

at-, tat-, at - a - b

Here are some exai

otherwise). Then Zt:

I

1

1 - 1 _ " a

provided Izi > 1. Nu., (and ft = 0 otherwise)

1

Z{ft , z} Z(ar ,

1

1 - a/ z

I

provided VI > lat. In Table A.2 we II,

verify that both the fc are correct.

The z-transform ha calculations with the that in each case

t = -1, -2, . . .

Property 7 Multipli. aZ{f , z}.

for the particular solution - to-time case when the ion yr kt, after which

(A.128)

^uous-time case. Suppose ) that A = yo — b/(1 + a)

y convenient transforma- P briefly explain how this

ft = 0 for t = —1, — ed as follows: 11

(A.129)

z) exists and can be seen d as follows. Suppose that

(A.130)

(A.131)

and (A.131) imply that R ("heavy discounting"). "I'crht" and Z{ft ,z} does z ,z).

I

se relation that exists between %..84) is continuous we obtain ce also Elaydi (1996, p. 254).

Mathematical Appendix

Property 8 If ft and gt both have a z-trans form then we have for any constants a and b that:

By using properties (P10) and (P11) we obtain the subsidiary equation in a few steps:

0 = [z2 Z{xt , z} – z2x0 – zxi] + 3 [zZ{xt, z} – zxo] + 2Z{xt , z} (z2 + 3z + 2) Z{xt , z} = z2x0 + zx1 + 3zxo = z

Inverting (A.133) yields the solution in the time domain:

for t = 0, 1, 2, ...

This example is—of course—rather unexciting apart from the fact that it gives us a hint as to the stability properties of difference equations. Asymptotic stability of (a system of) difference equations is obtained if the roots lie inside the unit circle, i.e. terms like --Ea are (un) stable if la < 1 (la' > 1).

A.7.4 The saddle-patt

We now consider the fol (A.96)):

where gK,t and gQ,t are Si element Sii . Taking the z-

A (z – 1

Z{Kt , z}

ZtQl , zi

[

where A(z –1) (z – 1)1 real and that –1 < A1 < (so that Ko is given) whim Since A(z –1)-1 = adj .•

[z – (1 – AO] [ ZiKi

Zio%

To ensure saddle-point si side of (A.137) must bcil.

for Qo:

[(1 + Az : adj A(A2) (1 +

By rewriting (A.138) we

12 We write the system in a also re-express (A.135) as: g

Kt+1 A.

[ Qt+1

where A* / + A. The ch:.- (1993, p. 65) gives the condi

br any constants a and b

(P9)

(P10)

(P11)

(P12)

(P13)

(P14)

(A.132)

nuation in a few steps:

, z}

(A.133)

A.134)

c fact that it gives us bymptotic stability of inside the unit circle,

Mathematical Appendix

A.7.4 The saddle-path model

We now consider the following system of difference equations (by analogy with (A.96)):

[

where gK,t and gQ,t are shock terms (possessing a z-transform) and A has typical element 84. Taking the z-transform of (A.135) yields:

[

where A(z — 1) (z — 1)/ — A. We assume that the characteristic roots of A are both real and that —1 < Al < 0 and A2 > 0. 12 As before, Kt is deemed to be predetermined (so that K0 is given) whilst Qt is a non-predetermined variable (so that Qo can jump). Since A (z —1)-1 = adj A(z —1)/[(z — (1— A 1 ))(z — (1 + A2 ))1 we can rewrite (A.136) as:

To ensure saddle-point stability the denominator and numerator on the right-hand side of (A.137) must both go to zero as z goes to 1+ A.2. This furnishes the expression for Qo:

By rewriting (A.138) we finally obtain:

12 We write the system in a form which emphasizes the close analogy with (A.96). Of course, we can also re-express (A.135) as:

(1993, p. 65) gives the conditions for saddle-point stability.

699

Mathematical Appendix

Similarly, the general expression for the solution can be written as:

z - (1 + Az)

where the analogy with (A.115) should be obvious. In the appendix to Chapter 15 equations (A.139)-(A.141) are used to solve the impulse-response functions for the unit-elastic RBC model with technology shocks.

A.7.5 Literature

Basic: Klein (1998, ch. 13), Chiang (1984, chs. 16-17), Sydsxter and Hammond (1995, ch. 20). Intermediate: de la Fuente (2000, chs. 9-11). Advanced: Azariadis (1993), Elaydi (1996), and Ogata (1995).

A.8 Dynamic Optimization

In this section we present the key results from optimal control theory as they are used in this book. We focus on infinite-horizon maximization problems in continuous time and gloss over second-order conditions. Discrete-time problems are solved in the text by making use of the Lagrangian methods discussed above in this appendix. Intriligator (1971, pp. 346-348) shows the link between the method of Lagrange multipliers and optimal control theory.

A.8.1 Unconstrained

The proto-typical optimal control problem encountered in economics takes the following form. The objective function is defined as:

where x(t) is the state variable, u(t) is the control variable, e- Pt is the discount factor, and t is time. The state and control variable are related according to the following

state equation:

x(t) = f [x(t), u(t), t] .

The state equation thus d initial condition for the 5

x(0) = xo,

where xo is a given co:. objective is to find a time the objective function the initial condition (A.. To solve this probler

following form:

F [x(t), u(t), t]

where A(t) is the co-st.. multiplier encountered i furnishes the follow 1.. 0

07-I = 0, au(t)

=ago'

=ax(t)aH .

The first condition sa- . Hamiltonian is max...

the state variable, whai co-state variable.

An equivalent way ui

Hamiltonian, which is ch

Rea] = F

where ,u(t) X(t)ea expressed in terms of

a7-cc = 0, au(t)

arc x(t) = a 14.(0'

ii(t) - Mt) = o.

If there are 11 state . except, of course, that

_ tten as:

(A.141)

{SIK,t, 1 + A2}

ZIggt, + A21

%.2)

-,endix to Chapter 15

.sponse functions for the

;vdsxter and Hammond

► . Advanced: Azariadis

Introl theory as they are fion problems in con- - ete-time problems are s discussed above in this between the method of

in economics takes the

(A.142)

is the discount factor, ording to the following

The state equation thus describes the equation of motion for the state variable. The initial condition for the state variable is given by:

where xo is a given constant (e.g. the accumulated stock of some resource). The objective is to find a time path for the control variable, u(t) for t E [0, 00], such that the objective function (A.142) is maximized given the state equation (A.143) and the initial condition (A.144).

To solve this problem one formulates a so-called Hamiltonian which takes the following form:

where )1/4.(t) is the co-state variable which plays the role similar to the Lagrange multiplier encountered in static optimization problems. The Maximum Principle furnishes the following conditions (for t

The first condition says that the control variable should be chosen such that the Hamiltonian is maximized, the second condition gives the equation of motion for the state variable, whilst the third equation gives the equation of motion for the co-state variable.

An equivalent way of solving the same problem is to work with the current-value Hamiltonian, which is defined as follows:

If there are n state variables and m controls then the same methods carry over except, of course, that x(t) [xi (t), , xn (t)] and u(t) [ui (t), , um (t)] must be

701

Mathematical Appendix

where ) 1 (t) is the co-state variable corresponding to the state variable xi (t), j = 1, m, and i = 1, , n.

A.8.2 (In)equality constraints

Suppose the problem is as in (A.142)-(A.144) but that there is an additional constraint in the form of:

(A.156)

where c is some constant. Suppose furthermore that there is a non-negativity constraint on the control variable, i.e. u(t) > 0 is required. The way to deal with these inequalities is to form the following current-value Lagrangian:

(A.157)

where 9 (t) is the Lagrange multiplier associated with the inequality constraint (A.156). The first-order conditions are now:

Equation (A.158) gives the Kuhn-Tucker conditions taking care of the nonnegativity constraint on the control variable. The second equation gives the Kuhn-Tucker conditions for the inequality constraint (A.156). Finally, (A.160) and (A.161) give the laws of motion of, respectively, the state variable and the co-state variable.

A.8.3 Second-order conditions

The second-order sufficient conditions are given by Chiang (1992, p. 290).

A.8.4 Literature

Basic: Klein (1998, ch. 15 Intriligator (1971, chs 11 chs 12-13). Advanced: t, and Chow (1997).