Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

efined as the transposed

(A.15)

It.

unique inverse, denoted

(A.16)

= I.

non-singular matrix)

pute AA-1 and A -1 A

Mathematical Appendix

Assuming that the indicated inverses exist (and the matrices A and B are thus non-singular), we find the following properties:

A.2.5 Cramer's Rule

Suppose we have a linear system of n equations in n unknowns:

anxi + a12x2 + " • + ainxn =

anxi + a22X2 + • • • + a2nxn = b2

(A.18)

anixi + an2x2 + • • • + annxn =

where au are the coefficients, bi are the exogenous variables, and xi are the endogenous variables. We can write this system in the form of a single matrix equation as:

Provided the coefficient matrix A is non-singular (so that IA l 0 0) the solution of the matrix equation is:

Instead of inverting the entire matrix A we can find the solutions for individual variables by means of Cramer's Rule (which only involves determinants):

Mathematical Appendix

where 1A1 1 is the determinant of the matrix A1 which is obtained by replacing column j of A by the vector of exogenous variables, for example A 1 is:

If the vector b consists entirely of zeros we call the system homogeneous. If IA I 0 0 then the unique solution to the matrix equation is the trivial one: x = A -1 b = 0. The only way to get a non-trivial solution to a homogeneous system is if the coefficient matrix is singular, i.e. if IA I = 0. In that case Cramer's Rule cannot be used. An infinite number of solutions nevertheless exist (including the trivial one) in that case. Take, for example, the following homogeneous system:

Clearly, AlI= 4 — 4 = 0 so the system is singular (row 2 is two times row 1). Nevertheless, both the trivial solution (x 1 = x2 = 0) and an infinite number of nontrivial solutions (any combination for which x 1 + 2x2 = 0) exist. Intuitively, we have infinitely many solutions because we have a single equation but two unknowns.

A.2.6 Characteristic roots and vectors

A characteristic vector of an n by n matrix A is a non-zero vector x which, when premultiplied by A yields a multiple of the same vector:

where A is called the characteristic root (or eigenvalue) of A. By rewriting equation (A.25) we find:

which constitutes a homogeneous system of equations which has non-trivial solutions provided the determinant of its coefficient matrix, AI — A, is zero:

This expression is called the characteristic equation of A. For a 2 by 2 matrix the characteristic equation can be written as:

where tr(A) and IA l are. I Hence, for such a matro possesses two roots:

These roots are distinct real (rather than comply IA < 0). For an n by n with n roots, A1 , A2, characteristic roots are:

Xi = tr(A) 117_1 Ai = IA

Associated with each chan is unique up to a const (A.26). If a matrix has follows:

P-1AP = A <#. A

where P is the matrix \\ diagonal matrix with chui tion is useful in the conte and below.

Intermezzo

Eigenvalues, eigenvt.‘ defined as:

The characteristic eq, characteristic roots ar. with Al is obtained b\

—410 [

"Pd by replacing column

(A.23)

homogeneous. If IA 1 0 0 I one: x = A -l b = 0. The c'em is if the coefficient cannot be used. An the trivial one) in that

1.

(A.24)

2 is two times row 1). finite number of noncist. Intuitively, we have n but two unknowns.

vector x which, when

I

(A.25)

By rewriting equation

(A.26)

-hich has non-trivial c, - A, is zero:

(A.27)

r a 2 by 2 matrix the

• -a21

(A.28)

Mathematical Appendix

where tr(A) and IA 1 are, respectively, the trace and the determinant of matrix A. Hence, for such a matrix the characteristic equation is quadratic in A and thus possesses two roots:

These roots are distinct if the discriminant, [tr(A)] 2 - 41A1, is non-zero. They are real (rather than complex) if the discriminant is positive (this is certainly the case if 1A 1 < 0). For an n by n matrix the characteristic equation is an n-th order polynomial with n roots, —1, —2, • • • , An which may not all be distinct or real. Some properties of characteristic roots are: -

Associated with each characteristic root is a chara eristic ve or (or eigenvector), which is unique up to a constant. The characteristic vector x(i) associated with Xi solves

where P is the matrix with the characteristic vectors, x(i) , as columns and A is the diagonal matrix with characteristic roots, X i , on the principal diagonal. Diagonalization is useful in the context of difference and differential equations—see Chapter 2 and below.

Intermezzo

Eigenvalues, eigenvectors, and matrix diagonalization. Suppose that A is defined as:

A = 6 10 -2 -3

The characteristic equation is 2, 2 — 3A + 2 = - 1)(A -- 2) = 0 so that the characteristic roots are A-= 1 and 2,2 = 2. The characteristic vector associated with Al is obtained by noting from (A.26) that:

(A 11-A)x = 0

([ 01 01 [ 26103 1) [

665

Mathematical Appendix

Any solution for which 2x1 + 4x2 0 will do. Hence, by setting x1 = c (a non-zero constant) we find that x2 = -c/2 so that the characteristic vector associated with Xi is:

x( )[ C -c/2 •

Similarly, for A2 = 2 we find:

Any combination for which 2x1 + 5x2 = 0 will do. Hence, the characteristic vector associated with

X(2)

-2c/5

In the example matrix we have:

from which we verify the result:

It works!

A.2.7 Literature

Basic: Klein (1998, chs. 4-5), Chiang (1984, chs. 4-5), Sydsxter and Hammond (1995, chs. 12-14). Intermediate: Intriligator (1971, appendix B), Kreyszig (1999,

chs. 6-7), and Strang (19 (1985), and Ortega (19b

I

A.3 Implicit Funct

A.3.1 Single equation

Suppose we have the fi interest, y, to one or mor

F(y, xi, .. xm) =

Assume that (a) F has co aF/axi for j = 1,2, ... ,n (A.32). Then according - neighbourhood of [x ) ] variables:

Y = f (Xl, X2, • • • Xrr:'

I

The implicit function is c by fj af /axe, which c.

ay = f = - L,t . ax, F

As an example, consider that ay/ax = -Fx /Fy = -

A.3.2 System of equa

Next we consider the sysi

I

Fl (yi, y2, yn; xi,.

F2 (yi, y2, • • • , yn;

Fn (yi, y2, • • Yn; xl

I

We assume that (a) th. respect to all yi and xi ai

v setting xi c (a characteristic vector

Mathematical Appendix

chs. 6-7), and Strang (1988). Advanced: Ayres (1974), Lancaster and Tismenetsky (1985), and Ortega (1987).

Mathematical Appendix

Then, according to the generalized implicit function theorem there exists an m- dimensional neighbourhood of [4] in which the variables yi are implicitly defined functions of the exogenous variables:

These implicit functions are continuous and have continuous partial derivatives, denoted by afiiaxi, which can be computed as follows:

where J! is the matrix obtained by replacing column i of matrix J by the following vector of partial derivatives:

--8F1 /axi -

-0F2 /ax;

(A.39)

-81'n/ax1 -

Intermezzo

Generalized implicit function theorem. As a example, consider the IS-LM model:

Y = C(Y - T(Y)) + .1(r) + Go

Mo L(r, Y),

where Y is output, C is consumption, I is investment, r is the interest rate, T is taxes. The endogenous variables are r and Y and the exogenous variables are government consumption Go and the money supply Mo. By differentiating

with respect to Go v L

1 - Cy_T(1 -

Ly

The Jacobian determin

Ill Lr [1 - CY-7

where the sign folios% depend negatively on ti sity to consume and th

0 < Cy_T(1 -- Ty) <

(Ly > 0). By Cramer's F

These expressions, o: _

A.3.3 Literature

Basic: Klein (1998, pp. 2 a (1995, pp. 591-593). Adv

A.4 Static Optimiz

A.4.1 Unconstrainedi

Suppose we wish to fn. function:

Y = f (X),

where we assume that th tives. The necessary con

(A.36)

qz there exists an m- v are implicitly defined

(A.37)

ous partial derivatives,

(A.38)

• i x I by the following

(A.39)

sider the IS-LM

the interest rate, xogenous variables differentiating

Mathematical Appendix

with respect to Go we get:

Cy-T(1 - Ty)

Ly Lr

The Jacobian determinant is:

Lr [1 - CY-T(1 - TY)} + IrL < 0,

where the sign follows from the fact that both money demand and investment depend negatively on the interest rate 0 and Ir < 0), the marginal propensity to consume and the marginal tax rate are between zero and unity (so that 0 < Cy_T(1 -- Ty) < 1), and money demand depends positively on output

0). By Cramer's Rule we get the partial derivatives:

These expressions, of course, accord with intuition (see Chapter 1).

A.3.3 Literature

Basic: Klein (1998, pp. 239-245), Chiang (1984, ch. 8), and Sydsxter and Hammond (1995, pp. 591-593). Advanced: De la Fuente (2000, ch. 5).

A.4 Static Optimization

A.4.1 Unconstrained optimization

Suppose we wish to find an optimum (minimum or maximum) of the following function:

where we assume that this function is continuous and possesses continuous derivatives. The necessary condition for a (relative) extremum of the function at point

669

Mathematical Appendix

X = X0 is

To test whether f(x) attains a relative maximum or a relative minimum at x = xo we compute the second derivative. The second-order sufficient condition is:

where f(.) is continuous and possesses continuous derivatives. The first-order necessary conditions for a relative extremum are:

the second-order sufficient conditions we define the Hessian matrix of second-order derivatives, H:

Young's theorem we know that fii = fii so the Hessian matrix is symmetric. We define the following set of principal minors of H:

Then, provided the first-order conditions hold at a point [x7, x°, .., x°], the secondorder sufficient condition for f (4) to be a relative maximum is:

lib I < 0, I H2 I > 0, . (— IHnl > 0, (A.47)

whilst for a relative minimum the condition is:

See Chiang (1984, pp. 337-353) for the relation between concavity—convexity of f 0 and the second-order conditions.

A.4.2 Equality constra

We focus on the case ‘.. straint. As in the unconst constraint is given by:

where c is a constant. \\ L derivatives. The Lagran,

where A is the Lagran0 _ extremum are:

Li = 0, i = 1,

LA = 0,

(A.41)

minimum at x xo ,-gt condition is:

(A.42)

(choice variables):

tives. The first-order

(A.44)

respect to xi. To study t matrix of second-order

(A.45)

partial derivatives. By -frix is symmetric. We

' • fin

" f2n

is:

(A.47)

(A.48)

7nncavity—convexity of

Mathematical Appendix

A.4.2 Equality constraints

We focus on the case with multiple choice variables and a single equality constraint. As in the unconstrained case, the objective function is given by (A.43). The constraint is given by:

where c is a constant. We assume that g(.) is continuous and possesses continuous derivatives. The Lagrangian is defined as follows:

where A is the Lagrange multiplier. The first-order necessary conditions for an extremum are:

(A.51)

LA = 0,

where A aL/axi and LA aL/ax are the partial derivatives of the Lagrangean with respect to xi and A, respectively. To study the second-order conditions we formulate a so-called bordered Hessian matrix, denoted by H:

_ gn fnl fn2 " • • • • fnn _

The bordered Hessian consists of the ordinary Hessian but with the borders made up of the derivatives of the constraint function (gi). We define the following set of principal minors of H:

whilst for the conditions for a relative constrained minimum are:

If there are multiple constraints then additional Lagrange multipliers are added to the Lagrangian (one per constraint) and the first-order condition for each Lagrange

671

Mathematical Appendix

multiplier, Al , takes the form LAi aLiaa.; = 0. See Chiang (1984, pp. 385-386) for the appropriately defined bordered Hessian for the multi-constraint case.

Interpretation of the Lagrange multiplier

We now return to the single constraint case in order to demonstrate the interpretation of the Lagrange multiplier in the optimum. Using the superscript "0" to denote optimized values, we can write the optimized value of the Lagrangian as:

Next, we ask the question what happens if the constraint is changed marginally. Obviously, both A,13 and x? are expected to change if c does. Differentiating (A.56) we get:

where we have used the necessary conditions for an optimum (CA = Li = 0 for i = 1, 2, , n) to get from the first to the second equality. Recall that the constraint holds with equality (c = g(.)) so that A° measures the effect of a small change in c on the optimized value of the objective function f (.). For example, if the objective function is utility and c is income, then A° is the marginal utility of income.

A.4.3 Inequality constraints

We now briefly study some key results from non-linear programming. We first look at the simplest case with non-negativity constraints on the choice variables. Then we take up the more challenging case of general inequalities. We focus on first-order conditions and ignore some of the subtleties involved (like constraint qualifications and second-order conditions).

Non- negativity constraints

Suppose that the issue is to maximize a function y = f (x) subject only to the nonnegativity constraint x > 0. There are three situations which can arise. These have been illustrated in Figure A.1 which is taken from Chiang (1984, p. 723).

Panel (a) shows the case we have studied in detail above. The function attains a maximum for a strictly positive value of x. We call this an interior solution because the solution lies entirely within the feasible region (and not on a boundary). The constraint x > 0 is non-binding and the first-order condition is as before:

f'(xo) = 0. (interior solution)

Panels (b) and (c) deal with two types of boundary solutions. In panel (b) the function happens to attain a maximum for x = xo = 0, i.e. exactly on the boundary of the

feasible region. In panel

f'(xo) =-- 0 and I

Finally, in panel (c) we al f (x) continues to rise have:

f' (xo) 0 and

I These three conditions,

solutions, can be combi I

f' (x0) 0, xo

There are two key things A.1, we can safely ex, _