## Relative versus Absolute Extremum

If the objective function is a constant Junction, as in Fig. 9. la, all values of the choice variable * will result in the same value of y, and the height of each point on the graph of the function (such as A or B or C) may be considered a maximum or, for that matter, a minimum or, indeed, neither. In this case, there is in effect no significant choice to be made regarding the value of x for the maximization or minimization ofy. In Fig. 9.1ft, the function is strictly increasing, and there is...

## Comparative Statics and the Concept of Derivative 124

6.1 The Nature of Comparative Statics 124 6.2 Rate of Change and the Derivative 125 The Derivative 126 Exercise. 6.2 127 6.3 The Derivative and the Slope of a Curve 128 Left-Side Limit and Right-Side Limit 29 Graphical Illustrations JO Evaluation of a Limit 131 Formal View of the Limit Concept 133 Exercise 6,4 I35 6.5 Digression on Inequalities and Absolute Values 136 R ules of I nequaiiftes 136 AbsoIure Values andInequalides 137 Solution of an Inequality 138 Exercise 6.5 139 1he.itrems...

## Further Topics in Optimization 402

13.1 Nonlinear Programming and Kuhn-Tucker Conditions 402 Step 1 Effect oj Non negativity Restrictions 403 Step 2 Effect of Inequality Cons mints 404 interpretation oj the Kuhn-Tucker Conditions 408 The n - Va r table, m - Co ns train t Case 409 Exercise 13.1 411 13.2 The Constraint Qualification 412 Irregi t f amies a t Bo uiukt jy Poin ts 412 The Constraint Qualification 415 Linear Constraints 416 Exercise 13,2 41 War-Time Rationing 418 Peak-Load Pricing 420 Exercise 13,3 423 13.4 Sufficiency...

## Economic Models

2.1 ingredients of a Mathematical Model 5 Variables, Cons arils, and Parameters 5 Equations and Identities 6 Set Notation 9 Relationships between Sets 9 Operations on Sets 11 Laws of Set Operations 12 Exercise 2.3 14 Ordered Pairs 15 Relations and Functions 16 Exercise 2 A 19 Constant Functions 20 Polynomial Functions 20 Rational Functions 21 Nonalgebraic Functions 23 A Digression on Exponents 23 Exercise 2,5 24 2.6 Functions of Two or More Independent Variables 25

## Matrices as Arrays

There are essentially three types of ingredients in the equation system (4.1). The first is the set of coefficients ai the second is the set of variables X , , xf) and the last is the set of constant terms du , dm. If we arrange the three sets as three rectangular arrays and label them, respectively, as A, x, andd (without subscripts), then we have As a simple example, given the linear-equation system fu'i + 3X2 + .*3 22 X +4X2-2XI 12 4xi - x2 + 5.v3 10 Each of the three arrays in (4.2) or...

## Exercise

Examine the comparative-static properties of the equilibrium quantity in (7.15), and check your results by graphic analysis. 2. On the basis of (7.18), find the partial derivatives a > 7to*, and ar a J. Interpret their meanings and determine their signs, 3. The numerical input-output model (5.21) was solved in Sec. 5.7, (a) How many comparative-static derivatives can be derived (b) Write out these derivatives tn the form of (7.23') and (7.23).

## Partial Derivatives

Where the variables xt (i 1, 2,are all independent of one another, so that each can vary by itself without affecting the others, if the variable x-. undergoes a change Ax while X2> ,xn all remain fixed, there will be a corresponding change in y, namely, Ay. The difference quotient in this case can be expressed as If we take the limit of Ay Axi as Ax( -> 0. that limit will constitute a derivative. We call it the partial derivative oiy with respect to X , to indicate that all the other...

## Constant Functions

A function whose range consists of only one element is called a constant function. As an example, we cite the function which is alternatively expressible as v 7 or fix) 7, whose value stays the same regardless of the value of x, In the coordinate plane, such a function will appear as a horizontal straight line. In national-incomc models, when investment 1 is exogenously determined, we may have an investment function of the form S100 million, or o, which exemplifies the constant function.

## Rules of Differentials

A straightforward way of finding the total differential dy, given a function is to find the partial derivatives j and and substitute these into the equation But sometimes it may be more convenient to apply certain rules of differentials which, in view of their striking resemblance to the derivative formulas studied before, arc very easy to remember, Let k be a constant and u and v be two functions of rhe variables and .12. Then the following rules are valid ' Rule I dk 0 (cf. constant-function...

## Extremum of a Function of One Variable

The expansion of a function into a Taylor (or Maclaurin) series is useful as an approximation device in the circumstance that 0 as n oc, but our present concern is with its application in the development of a general test for a relative extremum. Taylor Expansion and Relative Extremum As a preparatory step for that task, let us redefine a relative extremum as follows A function fix) attains a relative maximum (minimum) value at v0 jf f(x) - (.r< j) is negative (positive) for values ofx in the...

## Vectors as Special Matrices

The number of rows and the number of columns in a matrix together define the dimension of the matrix. Since matrix A in (4.2) contains m rows and n columns, it is said to be of dimension m x n (read um by ). It is important to remember that the row number always precedes the column number this is in line with the way the two subscripts in aSJ are ordered. In the special case where m h, the matrix is called a square matrix thus the matrix A in (4.4) is a 3 x 3 square matrix. Some matrices may...

## Quotient Rule

The derivative of the quotient of two functions, f(x) g(x)t is d fix) f(x)g(x)-f(x)g x) dx g(x) g2(x) in the numerator of the right-hand expression, we find two product terms, each involving the derivative of only one of the two original functions. Note that f(x) appears in the positive term, and g'(x) in the negative term. The denominator consists of the square of the function g(x)i that is, g1ix) lg(x) 2. d x x + (x + 1)2 (x + 1)2 d f 5x S(x2 -5x(2x) _50 x2) Tx l TTj (x2 + 1)2 (x2 + l)2 d...

## Exponential and Logarithmic Functions

The Mh-derivative test developed in Chap. 9 equips us for the task of locating the extreme values of any objective function, as long as it involves only one choice variable, possesses derivatives to the desired order, and sooner or later yields a nonzero derivative value at the critical value Jt0. In the examples cited in Chap. 9, however, we made use only of polynomial and rational functions, for which we know how to obtain the necessary derivatives. Suppose that our objective function...

## Chain Rule

If we have a differentiable function r (y), where y is in turn a differentiable function of another variable x, say. y (*), then the derivative of r with respect to x is equal to the derivative of z with respect to y9 times the derivative of y with respect to x. Expressed symbolically, This rule, known as the chain rule, appeals easily to intuition. Given a Ax, there must result a corresponding Ay via the function y g(x), but this A> ' will in turn bring about a Az via the function z ( v)....

## Gradient Vector

All the partial derivatives of a function (xuxj, xn) can be collected under a single mathematical entity called the gradient vector, or simply the gradient, of function grad (*,, x2l , * ) - ( 1, 2 fn) where J dy Ox,. Note that we are using parentheses rather than brackets here in writing the vector. Alternatively, the gradient can be denoted by V f.r ,xj xit), where V (read del) is the inverted version of the Greek letter A. Since the function has n arguments, there arc altogether n partial...

## Equilibrium Analysis in Economics

The analytical procedure outlined in Chap. 2 will first be applied to what is known as static analysis, or equilibrium analysis. For this purpose, it is imperative first to have a clear understanding of what equilibrium means. Like any economic term, equilibrium can be defined in various ways. According to one definition, an equilibrium is a constellation of selected interrelated variables so adjusted to one another that no inherent tendency to change prevails in the model which they...

## Optimization A Special Variety of Equilibrium Analysis

When we lirst introduced the term equilibrium in Chap. 3, we made a broad distinction between goal and nongoal equilibrium. In the iatter type, exemplified by our study of market and national-income models, the interplay of certain opposing forces in the model e.g., the forces of demand and supply in the market models and the forces of leakages and injections in the income models dictates an equilibrium state, if any. in which these opposing forces are just balanced against each other, thus...

## Optimization A Special Variety of Equilibrium Analysis 220

9.1 Optimum Values and Extreme Values 221 9.2 Relative Maximum and Minimum First-Derivative Test 222 Relative versus A bsolute Extremum 222 First-Derivative Test 223 Exercise 9.2 226 93 Second and Higher Derivatives 227 Derivative of a Derivative 227 Interpretation of the Second Derivative 229 A n Application 23 i Attitudes toward Risk 231 Exercise 9.3 233 Necessary versus Sufficient Conditions 234 Conditions for Profit Maximization 235 Coefficients of a Cubic Tohil-Cost Function 238...

## Simultaneous Equation Approach

The analysis of model (8.32) was carried out on the basis of a single equation, namely, (8.35). Since only one endogenous variable can fruitfully be incorporated into one equation, the inclusion of P* means the exclusion of Q*. As a result, we were compelled to find (dP*fdYo) first and then to infer (dQ*fdYo) in a subsequent step. Now we shall show how P* and Of can be studied simultaneously. As there are two endogenous variables, we shall accordingly set up a two-equation system. First,...

## Comparative Static Analysis of General Function Models

The study of partial derivatives has enabled us, in Chap. 7, to handle the simpler type of comparative-static problems, in which the equilibrium solution of the model can be explicitly stated in the reduced form. In that case, partial differentiation of the solution will directly yield the desired comparative-static information. You will recall that the definition of the partial derivative requires the absence of any functional relationship among the independent variables (say so that x, can...

## Economic Meaning of the Hawkins Simon Condition

For the two-industry case, the Leontief matrix is The first part of the Hawkins-Simon condition, B > 0, requires that Economically, this requires the amount of the first commodity used in the production of a dollar's worth of the first commodity to be less than one dollar The other part of the condition. 1 > 0, requires that (1 )(I 22) '12 21 > 0 f A thorough discussion can be found in Akira Takayama, Mathematical Economics, 2d ed., Cambridge University Press, 1985, pp. 380-385. Some...

## Derivation of the Rule

Given an equation system Ax d, where A is n x the solution can be written as provided A is nonsingular. According to (5.15), this means that Equating the corresponding elements on the two sides of the equation, vc obtain the solution values The Y, terms in (5.17) look unfamiliar. What do they mean From (5.8), we see that the Laplace expansion of a determinant ,4 by its first column can be expressed in the form fl i Cn . If we replace the first column of A by the column vector d but keep all the...

## Relationships between Sets

When two sets are compared with each other, several possible kinds of relationship may be observed. If two sets S and happen to contain identical elements, , 2,7, and & (2, tf, 7, then S and S2 are said to be equal (S S2). Note that the order of appearance of the elements in a set is immaterial. Whenever we find even one element to be different in any two sets, however, those two sets are not equal. Another kind of set relationship is that one set may be a subset of another set. If we have...

## Evaluation of a Limit

Let us now illustrate the algebraic evaluation of a limit of a given function q Given q - 2 + v2, find lim q. To take the left-side limit, we substitute the series of negative values -1, - (in that order) for vand find that (2 -f v2) will decrease steadily and approach 2 (because v2 will gradually approach 0). Next, for the right-side limit, we substitute the series of positive values 1, , (in that order) for vand find the same limit as before. Inasmuch as the two limits are identical, we...

## Second Derivative Test

Returning to the pair of extreme points B and E in Fig. 9.5 and remembering the newly established relationship between the sccond derivative and the curvaturc of a curve, we should be able to see the validity of the following criterion for a relative cxtremum Second-derivative test for relative extremum If the value of the first derivative of a function at AJ .yo is ' Uo) 0- then the value of the function at.vo, fix o), will be a. A relative maximum if the second-derivativc value at x is (xo)...

## Higher Degree Polynomial Equations

If a system of simultaneous equations reduccs not to a linear equation such as (3.3 )r or to a quadratic equation such as (3.7) but to a cubic (third-degree polynomial) equation or quartic (fourth-degree polynomial) equation, the roots will be more difficult to find. One useful method which may work is that offactoring the function. The expression x3 - x2 - 4x + 4 can be written as the product of three factors (x - 1), (x + 2), and (x - 2). Thus the cubic equation In order for the left-hand...

## Market Model

First let us consider again flie simple one-commodity market model of (3.1). That model can be written in the form of two equations Q a bP (a, h > 0) demand Q c + dP (c,d> 0) supply These solutions will be referred to as being m the reduced form The two endogenous variables have been reduced to explicit expressions of the four mutually independent parameters a, b, c, and d. To find how an infinitesimal change in one of the parameters will affect the value of P*, one has only to...

## Rules of Differentiation and Their Use in Comparative Statics

The central problem of comparative-static analysis, that of finding a rate of change, can be identified with the problem of finding the derivative of some function y provided only an infinitesimal change in is being considered. Even though the derivative dv dx is defined as the limit of the difference quotient q g( v) as i 0, it is by no means necessary to undertake the process of limit-taking each time the derivative of a function is sought, for there exist various rules of differentiation...

## National Income Model

In piacc of the simple national-incomc model discussed in Chap. 3, let us now work with a slightly enlarged model with three endogenous variables, Y (national income), C (consumption), and T (taxes) C a + fi(Y - T) (a > 0 0 < ff < 1) (7.17) T y+SY (y > 0 Q< < ) The first equation in this system gives the equilibrium condition for national income, while the second and third equations show, respectively, how Cand fare determined in the model. The restrictions on the values of the...

## Implicit Functions

A function given in the form of v (x), say, is called an explicit junction, because the variable v is explicitly expressed as a function of x. If this function is written alternatively in the equivalent form however, wre no longer have an explicit function. Rather, the function (8.17) is then only implicitly defined by the equation (8.17')- When we are (only) given an equation in the form of (8.17'). therefore, the function y f x) which it implies, and whose specific form may not even be known...

## SLM Model Closed Economy

As another linear model of the economy, we can think of the economy as being made up of two sectors the real goods sector and the monetary sector. The goods market involves the following equations C a - -b(l t)Y I d-ei G C0 The endogenous variables are Y, C, T and i (where is the rate of interest). The exogenous variable is Go, while a, d, e, h, and t are structural parameters. In the newly introduced money market, we have where is the exogenous stock of money and k and are parameters. These...

## Finite Markov Chains

A common application of matrix algebra is found in what is known as Markov processes or Markov chains. Markov processes arc used to measure or estimate movements over time. This involves the use of a Markov transition matrix, where each value in the transition matrix is a probability of moving from one state (location, job, etc.) to another state. There is also a vector containing the initial distribution across the various states. By repeatedly multiplying such a vector by the transition...

## Properties of Transposes

The following properties characterize transposes The first says that the transpose of the transpose is the original matrix a rather self-evident conclusion. The second property may be verbally stated thus The transpose of a sum is the sum of the transposes. The third property is that the transpose of a product is the product of the transposes in reverse order To appreciate the necessity for the reversed order, let us examine the dimension conformability of the two products on the two sides of...

## Left Side Limit and Right Side Limit

The concept of limit is concerned with the question What value does one variable (say, q) approach as another variable (say, v) approaches a specific value (say, zero) '1 In order for this question to make Sense, q must, of course, be a function of v say, q ( . )> Our immediate interest is in finding the limit of q as v 0, but we may just as easily explore the more general case of v Nf where N is any finite real number. Then, lim q will be merely a special case of lim q where N 0. In the...

## Leontief Input Output Models

In its static version, the input-output analysis of Professor Wassily Leontief. a Nobel Prize winner,1 deals with this particular question What level of output should each of the n industries in an economy produce, ill order that it will just be sufficient to satisfy the tolal demand for that product The rationale for the term input-output analysis is quite plain to see. The output of any industry (say the steel industry) is needed as an input in many other industries, or even for that industry...

## Determinants and Nonsingularity

The determinant of a square matrix A, denoted by A 9 is a uniquely defined scalar (number) associated with that matrix. Determinants arc defined only for square matrices. Tlic smallest possible matrix is, of course, the I x 1 matrix A a . By definition, its determinant is equal to the single elementan itself (A a a . The symbol an here must not be confused with the look-alike symbol for the absolute value of a number. In the absolute-value context, we have, for instance, not only 5 5,but also -...

## The Existence of Nonnegative Solutions

In the previous numerical example, the Leontief matrix I - A happens to he nonsingular, so solution values of output variables xj do exist. Moreover, the solution values a* all turn out to be nonnegative, as economic sense would dictate. Such desired results, however, cannot be expected to emerge automatically they come about only when the Leontief matrix possesses certain properties. These properties are described in the so-called Hawkins-Simon condition. To explain this condition, we need to...

## Second and Higher Derivatives

Hitherto we have considered only the first derivative f(x) of a function y f x) now let us introduce the concept of second derivative (short for second-order derivative),, and derivatives of even higher orders. These will enable us to develop alternative criteria for locating the relative extrema of a function. Since the first derivative f' x) is itself a function ofx, it, too, should be differentiate with respect to x, provided that it is continuous and smooth. The result of this...

## The Derivative

Frequently, we are interested in the rate of change of> ' when Ax is very small. In such a case, it is possible to obtain an approximation of Ay Ax by dropping all the terms in the difference quotient involving the expression Ax. In (6.2), for instance, if Ax is very small, we may simply take the term 6*0 on the right as an approximation of Ay Ax. The smaller the value of Ax, of course, the closer is the approximation to the true value of A> r Ax. As Ax approaches zero (meaning that it gets...

## Basic Properties of Determinants

We can now discuss some properties of determinants which will enable us to discover the connection between linear dependence among the rows of a square matrix and the vanishing of the determinant of that matrix. Five basic properties will be discussed here. These arc properties common to determinants of all orders, although we shall illustrate mostly w ith second-order determinants Property I The interchange of rows and columns does not affect the value of a determinant. In other words, the...

## Multiplication of Matrices

Whereas a scalar can be used to multiply a matrix of any dimension, the multiplication of two matrices is contingent upon the satisfaction of a different dimensional requirement. Suppose that, given two matrices A and we want to find the product AB. The eonformability condition for multiplication is that the column dimension of A (the lead matrix in the expression AH) must be equal to the row dimension of 8 the lag matrix). For instance, if the product A B then is defined, since A has ivvo...

## Continuity of a Function

When a function q g(v) possesses a limit as v tends to the point N in the domain, and when this limit is also equal to g(N) that is, equal to the value of the function at v V the function is said to be continuous at A7. As defined here, the term continuity involves no less than three requirements (1) the point A7 must be in the domain of the function i.e., g N) is defined (2) the function must have a limit as v A' i,e., lim g(v) exists and (3) (hat limit must be equal in value to g(N) i.e. lim...

## Solution Outcomes for a Linear Equation System

Our discussion of the several variants of the linear-equation system Ax d reveals that as many as four different types of solution outcome are possible. For a belter overall view of these variants, we list them in tabular form in Table 5.1. As a first possibility, the system may yield a unique, nontrivial solution. This type of outcome can arise only when we have a nonhomogeneous system with a nonsingular coefficient matrix A. The second possible outcome is a unique, trivial solution, and this...

## Inverse Function Rule

If the function y fix) represents a one-to-one mapping, i.e., if the function is such that each value of y is associated with a unique value of x9 the function y will have an inverse function x l( v) (read x is an inverse function of 1). Here, the symbol ' is a function symbol which, like the derivative-function symbol f, signifies a function related lo the function f it does not mean the reciprocal of the function fix). What the existence of an inverse function essentially means is that, in...

## Constant Function Rule

The derivative of a constant function v i, or j x) k, is identically zero, ix., is zero for all values of a . Symbolically, this rule may be stated as Given y f(x) k, the derivative is Alternatively, we may state the rule as Given y f x) k, the derivative is where the derivative symbol has been separated into two parts, d dx on the one hand, and y or (jr) or A on the other. The first part, d dx, is an operator symbol, which instructs us to perform a particular mathematical operation. Just as...

## The Derivative and the Slope of a Curve

Elementary economics tells us that, given a total-cost function C ( ), where C denotes total cost and Q the output, the marginal cost (MC) is defined as the change in total cost resulting from a unit increase in output that is. MC AC AQ. It is understood that A Q is an extremely small change. For the case of a product that has discrete units (integers only), a change of one unit is the smallest change possible but for the case of a product whose quantity is a continuous variable, AQ can refer...

## Evaluating a Third Order Determinant

A determinant of order 3 is associated with a 3 x 3 matrix. Given I1 2233 1l 23 32 + 12 23 M - 12 21 33 4- 1 21 32 - tfi3 22 3i a scalar (5.6) Looking first at the lower line of (5.6). wc sec the value of A expressed as a sum of six product terms, three of which are prefixed by minus signs and three by plus signs. Complicated as this sum may appear, there is nonetheless a very easy way of catching all these six terms from a given third-order determinant. This is best explained diagram-matically...

## The Nature of Mathematical Economics

Mathematical economics is not a distinct branch of economics in the sense that public finance or international trade is. Rather, it is an approach to economic analysis, in which the economist makes use of mathematical symbols in the statement of the problem and also draws upon known mathematical theorems to aid in reasoning. As far as the specific subject matter of analysis goes, it can be micro- or macroeconomic theory, public finance, urban economics, or what not Using the term mathematical...

## Idiosyncrasies of Matrix Algebra

Despite the apparent similarities between matrix algebra and scalar algebra, the ease of matrices does display certain idiosyncrasies that serve to warn us not to borrow'1 from scalar algebra too unqueslioningly We have already seen that, in general, AB BA in matrix algebra. Let us look at two more such idiosyncrasies of matrix algebra. For one thing, in the case of scalars, the equation ah 0 always implies that either a or h is zero, but this is not so in matrix multiplication. Thus, we have...

## Summary of the Procedure

In the analysis of the general-function market model and national-income model, it is not possible to obtain explicit solution values of the endogenous variables. Instead, we rely on the implicit-function theorem to enable us to write the implicit solutions such as Our subsequent search for the comparative-static derivatives such as (dP*fdYo) and then rests for its meaningfulness upon the known fact thanks again to the implicit-function theorem- that the P* andr* functions do possess continuous...

## Different ability of a Function

The previous discussion has provided us with the tools for ascertaining whether any function has a limit as its independent variable approaches some specific value. Thus we can try to take the limit of any function y f(x) as x approaches some chosen value, say, x0. However, we can also apply the limit concept at a different level and take the limit of the difference quotient of that function, Ay Ax, as Ax approaches zero. The outcomes of limit-taking at these two different levels relate to two...

## Vector Space

The totality of the 2-vectors generated by the various linear combinations of two independent vectors u and v constitutes the two-dimensional vector space. Sincc wc arc dealing only with vectors with real-valued elements, this vcctor spacc is none other than R2, the 2-space we have been referring to all along. The 2-space cannot be generated by a single 2-vector, because linear combinations of the latter can only give rise to the set of vectors lying on a single straight line. Nor does the...

## Power Function Rule

The derivative of a power function v f(x) xr' isnx , Symbolically, this is expressed as xn nxn- or f(x) nxn (7.1) Example 1 The derivative of y xMs 3*2. Example 2 The derivative of y x9 is x9 - 9x This rule is valid for any real-valued power of x that is, the exponent can be any real number. But we shall prove it only for ihe ease where n is some positive integer. In the simplest ease, that of n 1, the function is fix) x, and according to the rule, the derivative is The proof of this result...

## Levels of Generality

In discussing the various types of function, wc have without explicit notice introduced examples of functions that pertain to varying levels of generality In certain instances, we have written functions in the form y - 7 y 6x + 4 y x2 - 3x + 1 (etc.) Not only are these expressed in terms of numerical coefficients, but they also indicate specifically whether each function is constant, linear, or quadratic. In terms of graphs, each such function will give rise to a well-defined unique curve. In...

## Inverses and Their Properties

For a given matrix A, the transpose A' is always derivable. On the other hand, its inverse matrix another type of derived matrix may or may not exist. The inverse of matrix A, denoted by A is defined only if A is a square matrix, in which case the inverse is the matrix that satisfies the condition That is, whether is pre- or postmultiplied by A , the product will be the same identity matrix. This is another exception to the rule that matrix multiplication is not commutative. The following...

## The Nature of Comparative Statics

Comparative statics, as the name suggests, is concerned with the comparison of different equilibrium states that are associated with different sets of values of parameters and exogenous variables. For purposes of such a comparison, we always start by assuming a given initial equilibrium state. In the isolated-market model, for example, such an initial equilibrium will be represented by a determinate price P* and a corresponding quantity Q*. Similarly, in the simple national-income model of...

## Sum Difference Rule

The derivative of a sum (difference) of two functions is the sum (difference) of the derivatives of the two functions f(x) g(x) -f(x) j-x(x) f x) g (x) dx dx dx The proof of this again involves the application of the definition of a derivative and of the various limit theorems. We shall omit the proof and. instead, merely verify its validity and illustrate its application. From the function y 14x we can obtain the derivative dyjdx Alx1. But 14x3 Sx* -9x so that y may be regarded as the sum of...

## Variables Constants and Parameters

A variable is something whose magnitude can changc, i.e., something that can take on different values. Variables frequently used in economics includc pricc, profit, revenue, cost, national income, consumption, investment, imports, and exports. Since each variable can assume various values, it must be represented by a symbol instead of a specific number. For example, wc may represent price by P, profit by tt, revenue by R, cost by C national income by F, and so forth. When we write P 3 or C 18,...

## Graphical Illustrations

Let us illustrate, in Fig. 6.2, several possible situations regarding the limit of a function Figure 6.2a shows a smooth curve. As the variable v tends to the value V from either side on the horizontal axis, the variable q tends to the value L. In this case, the left-side limit is identical with the right-side limit therefore we can write lim q L, Figure 6.2a shows a smooth curve. As the variable v tends to the value V from either side on the horizontal axis, the variable q tends to the value...

## Formal View of the Limit Concept

The previous discussion should have conveyed some general ideas about the limit concept. Let us now give it a more precise definition. Since such a definition will make use of the concept of neighborhood of a point on a line (in particular, a specific number as a point on the line of real numbers), we shall first explain the latter term. For a given number , there can always be found a number (L a ) < L and another number (L+ ai) > where u and a2 are some arbitrary positive numbers. The set...

## Differentials and Point Elasticity

To illustrate the economic application of differentials, let us consider the notion of the elasticity of a function. Given a demand function Q f P), for instance, its elasticity is defined as (AQfQ)f(APfP). Using the idea of approximation explained in Fig. 8.1, we can replace the independent change A P and the dependent change A Q with the differentials dP and dQ, respectively, to get an approximation elasticity measure known as the point elasticity of demand and denoted by (i (the Greek letter...

## Extension to the Simultaneous Equation Case

The implicit-function theorem also comes in a more general and powerful version that deals with the conditions under which a set of simultaneous equations The generalized version of the theorem states that Given the equation system (8.24), if (a) the functions Fl, Fn ail have continuous partial derivatives with respect to all they and* variables, and if (b) at a point v q> m v o xuu____satisfying (8.24). the following Jacobian determinant is nonzero will assuredly define a set of implicit...

## Solution of a General Equation System

Ifa model comes equipped with numerical coefficients, as in (3.16), the equilibrium values of the variables will be in numerical terms, too. On a more general level, if a model is expressed in terms of parametric constants, as in (3.12), the equilibrium values will also involve parameters and will hence appear as formulas ys exemplified by (3.14) and (3.15). If, for greater generality, even the function forms are left unspecified in a model, however, as in (3.17), the manner of expressing the...

## Note on Jacobian Determinants

Our study of partial derivatives was motivated solely by comparative-static considerations. But partial derivatives also provide a means of testing whether there exists functional (linear or nonlinear) dependence among a set of n functions in n variables. This is related to the notion of Jacobian determinants (named after Jacobi). Consider the two functions V 2x + 3a'2 v2 - 4_Y + 12.Yi.V2 + If wc get all the four partial derivatives and arrange them into a square matrix in a prescribed order,...

## Limitations of Static Analysis

In the discussion of static equilibrium in the market or in the national income, our primary-concern has been to find the equilibrium values of the endogenous variables in the model. A fundamental point that was ignored in such an analysis is the actual process of adjustments and readjustments of the variables ultimately leading to the equilibrium state (if it is at all attainable). We asked only about where we shall arrive but did not question when or what may happen along the way. The static...

## Relationship Between Marginal Cost and Average Cost Functions

As an economic application of the quotient rule, let us consider the rate of change of average cost when output varies. Given a total-cost function C C(Q), the average-cost (AC) function is a quotient of two functions of Q> since AC C(Q) Q, defined as long as Q > 0. Therefore, the rate of change of AC with respect to Q can be found by differentiating AC d C(Q) C'(Q).Q-C(Q).l dQ Q Qi From this it follows that, for Q > 0, Since the derivative C( Q) represents the marginal-cost (MC)...

## The Closed Model

If the exogenous sector of the open input-output model is absorbed into the system as just another industry, the model will become a closed model. In such a model, final demand and primary input do not appear in their place will be the input requirements and the output of the newly conceived industry All goods will now be intermediate in nature, because everything that is produced is produced only for the sake of satisfying the input requirements of the (w + 1) industries in the model. At first...

## Conditions for Nonsingularity

After the squareness condition a necessary condition is already met, a sufficient condition for the nonsingularity of a matrix is that its rows be linearly independent or, what amounts to the same thing, that its columns be linearly independent . When the dual conditions of squareness and linear independence are taken together, they constitute the necessury-and-sufficient condition for nonsingularity nonsingularity o squareness and linear independence . An n x n coefficient matrix A can be...

## Rank of a Matrix

Even though the concept of row independence has been discussed only with regard to square matrices, it is equally applicable to any m x n rectangular matrix. If the maximum number of linearly independent rows that can be found in such a matrix is r, the matrix is said to be of rankr. The rank also tells us the maximum number of linearly independent columns in the said matrix. The rank of an m x n matrix can be at most m or , whichever is smaller. Given a matrix with only two rows or two columns...

## Conditions for Profit Maximization

We shall now present an economic example of extreme-value problems, i.e., problems of optimization. One of the first things that a student of economics icarns is that, in order to maximize profit, a firm must equate marginal cost and marginal revenue, Let us show the mathematical derivation of this condition. To keep the analysis on a general level, wc shall work with the total-revenue function R R Q and total-cost function C C 0, both of which arc functions of a single variable Q. From these...

## First Derivative Test

As a matter of terminology, from now on we shall refer to the derivative of a function alternatively as its first derivative short fox first-order derivative . The reason for this will become apparent shortly. Given a function y f x gt the first derivative ' v plays a major role in our search for its extreme values. This is due to the fact that, if a relative extremum of the function occurs at x Xi t then either 1 ' .to does not exist, or 2 ' -To 0. The first eventuality is illustrated in Fig....

## The Notation

The use of subscripted symbols not only helps in designating the locations of parameters and variables but also lends itself to a flexible shorthand for denoting sums of terms, such as those which arose during the process of matrix multiplication. The summation shorthand makes use of the Greek letter T sigma, for sum11 . To express the sum oLvi, Xj, and x 9 for instance, we may write Chapter 4 Linear Models ami Matrix Algebra 57 which is read as the sum of xj as j ranges from 1 to 3. The symbol...

## Equations and identities

Variables may exist independently, but they do not really become interesting until they are related to one another by equations or by inequalities. At this moment we shall discuss equations only. In economic applications we may distinguish between three types of equation definitional equations, behavioral equations, and conditional equations. A definitional equation sets up an identity between two alternate expressions that have exactly the same meaning. For such an equation, the...

## Partial Market EquilibriumA Nonlinear Model

Let the linear demand in the isolated market model be replaced by a quadratic demand function, while the supply function remains linear. Also, let us use numerical coefficients rather than parameters. Then a model such as the following may emerge As previously, this system of three equations can be reduced to a single equation by elimination of variables by substitution This is a quadratic equation because the left-hand expression is a quadratic function of variable P gt A major difference...

## Expansion of a Determinant by Alien Cofactors

Before answering this query, let us discuss another important property of determinants. Property VI The expansion of a determinant by alien cofactors the cofactors of a wrong row or column always yields a value of zero. by using its first-row elements but the cofactors we get on C211 ai2 C22 ai3 C23 4 3 i-1 10 2 1 0. More generally, applying the same type of expansion by alien cofactors as described in Example 1 to the determinant A Q11 will yield a zero sum of products as The reason for this...

## Relations and Functions

Our discussion of sets was prompted by the usage of that term in connection with the various kinds of numbers in our number system. However, sets can refer as well to objects other than numbers. In particular, we can speak of sets of ordered pairs to be defined presently which will lead us to the important concepts of relations and functions. In writing a set a, h j, we do not care about the order in which the elements a and b appear, because by definition a, h b, a . The pair of elements a and...

## Rank of a Matrix Redefined

The rank of a matrix A was earlier defined to be the maximum number of linearly independent rows in A. In view of the link between row independence and the nonvanishing of the determinant, we can redefine the rank of an m v n matrix as the maximum order of a non-vanishing determinant that can be constructed from the rows and columns of that matrix. The rank of any matrix is a unique number. Obviously, the rank can at most be m or n, whichever is smaller, bccausc a determinant is defined only...

## Matrix Multiplication

Matrix multiplication is not commutative, that is, As explained previously, even when AB is defined, BA may not be but even if both products are defined, the general rule is still AB BA. 1 0 2 6 1 -1 2 7 3 0 44 6 3 lt -l 4 7 J Let if be 1 x 3 a row vector then the corresponding column vector u must be 3 x 1. The product u'u will be 1 x 1, but the product uuf will be 3 x 3. Thus, obviously, uu uu In view of the general rule AB BA, the terms premidfipjy and postmuitiply are often used to specify...

## Solution of an Inequality

Like an equation, an inequality containing a variable say, x may have a solution the solution, if it exists, is a set of values of which make the inequality a true statement. Such a solution will itself usually be in the form of an inequality. As in solving an equation, the variable terms should first be collected on one side of the inequality. By adding 3 - x to both sides, we obtain 3x-3 3-x gt x 1 3 - x or 2x gt 4 Multiplying both sides by which does not reverse the sense of the inequality,...

## Evaluating an nthOrder Determinant by Laplace Expansion

Let us first explain the Laplace-expansion process for a third-order determinant. Returning to the first line of 5.6 . we see that the value of A can also be regarded as a sum ai three terms, each of which is a product of a first-row element and a particular secomf-ordur determinant. This latter process of evaluating A by means of certain lower-order determinants illustrates the Laplace expansion of the determinant, The three second-order determinants in 5.6 are not arbitrarily determined, but...

## The Quadratic Formula

Equation 3.7 has been solved graphically, but an algebraic method is also available. In general, given a quadratic equation in the form there are two roots, which can be obtained from the quadratic formula where the part of the sign yields x and the - part yields Also note that as long as b2 - Aac gt 0 the values of x and xi would differ, giving us two distinct real numbers as the roots. But in the special case where b2 Aac 0, we would find that jcjf -b 2a. In this case, the two roots share the...

## The Difference Quotient

Sincc the notion of change figures prominently in the present context, a special symbol is needed to represent it. When the variable x changes from the value x0 to a new value X , the change is measured by the difference xi - .to. Hence, using the symbol A the Greek capital delta, for difference to denote the change, we write A x - Also needed is a way of denoting the value of the function f x at various values of x. The standard practice is to use the notation f xt to represent the value of f...

## Equilibrium in National Income Analysis

Even though the discussion of static analysis has hitherto been restricted to market models in various guises linear and nonlinear, one-commodity and multicommodity, specific and general it, of course, has applications in other areas of economics also. As an example, we may cite the simplest Keyncsian national-income model, where Y and C stand for the endogenous variables national income and planned consumption expenditure, respectively, and lo and Co represent the exogenously determined...

## Inverse Matrix and Solution of Linear Equation System

The application of the concept of inverse matrix to the solution of a simultaneous-equation system is immediate and direct. Referring to the equation system in 4.3 , we pointed out earlier that it can be written in matrix notation as where A,x, and d arc as defined in 4,4 Now if the inverse matrix A exists, the premul-tiplicaiion of both sides of the equation 4t17 by A will yield The left side of 4.18 is a column vector of variables, whereas the right-hand product is a column vector of certain...

## Operations on Sets

When we add, subtract, multiply, divide, or take the square root of some numbers, we are performing mathematical operations. Although sets are different from numbers, one can similarly perform certain mathematical operations on them. Three principal operations to be discussed here involve the union, intersection, and complement of sets. To take the union of two sets A and B means to form a new set containing those elements and only those elements belonging to A, or to B, or to both A and B, The...

## Quadratic Equation versus Quadratic Function

Before discussing the method of solution, a clear distinction should be made between the two terms quadratic equation and quadratic function. According to the earlier discussion, the expression P2 4P - 5 constitutes a quadratic function, say, f P . Hence we may write What 3.8 does is to specify a rule of mapping from P to F , such as Although we have listed only nine P values in this table, actually ait the P values in the domain of the function are eligible for listing. It is perhaps for this...

## Continuous Time First Order Differential Equations 475

15 1 First-Order Linear Differential Equations with Constant Coefficient and Constant Term 475 The Homogeneous Case 476 The Nonhomogeneous Case 476 Verification of the Solution 478 Exercise 5.1 479 Th e Dyn amie Stability of Equ ilihnum 481 An Alternative Use of the Model 482 Exercise 15.2 483 153 Variable Coefficient and Variable Term 483 The Homogeneous Case 484 The Nonhomogeneous Case 485 Exercise 15.3 486 15.4 Exact Differential Equations 486 Exact Differential Equations 486 Method of...

## Matrix Algebra versus Elimination of Variables

The economic models used for illustration above involve two or four equations only and thus only fourth or lower-order determinants need to be evaluated. For large equation systems, higher-order determinants will appear, and their evaluation will be more complicated. And so will be the inversion of large matrices. From the computational point of view, in fact, matrix inversion and Cramer's rule are not necessarily more efficient than the method of successive eliminations of variables. However,...

## McGraw Hill Irwin

MJNDAMF.NTAL MLT110DS OF MATHEMATICAL ECONOMICS Published by MeGraw-HilJ Irwin, a business unit ofThc McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY, 10020. Copyright 2005, 1984. 1974, 1967 by The McGraw-Hill Companies, Inc. All rights reserved. No part of this publication may be rcproduccd or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent ofThe McGraw-Hill Companies, Inc., including, but not...

## Necessary versus Sufficient Conditions

The concepts of''necessary condition and sufficient condition are used frequently in economics, it is important that we understand their precise meanings before proceeding further. A necessary condition is in the nature of a prerequisite Suppose that a statement is true only if another statement q is true then q constitutes a necessary condition of p. Symbolically, we express this as follows which is read as p only ifqor alternatively, ii p, then qJt is also logically correct to interpret 5.1...

## Constructing the Model

Since only one commodity is being considered, it is necessary to include only three variables in the model the quantity demanded of the commodity Qd the quantity supplied of the commodity gT , and its price P . The quantity is measured, say, in pounds per week, and the price in dollars. Having chosen the variables, our next order of business is to make certain assumptions regarding the working of the market. First, we must specify an equilibrium condition- something indispensable in an...

## Functions of Two or More Independent Variables

Thus far, we have considered only functions of a single independent variable, i fix . But the concept of a function can be readily extended to the case of two or more independent variables. Given a function a given pair of a and v values will uniquely determine a value of the dependent variable z. Such a function is exemplified by z ax r by or z a a x 4- ciix1 h y bny2 Just as the function y f x maps a point in the domain into a point in the range, the function g- will do precisely the same....

## Polynomial Functions

The constant function is actually a degenerate case of what are known as polynomial functions. The word polynomial means muUilerm and a polynomial function of a single variable x has the general form V 0 rtj.v azx2 H-----H aflx' 2,4 in which each term contains a coefficient as well as a nonnegative-integer power of the variable a. As will be explained later in this section, we can write x and 1 in general thus the first two terms may be taken to be a0x and a x , respectively. Note that, instead...

## Special Case Absorbing Markov Chains

Now, let us extend the model by adding a third option Employees can exit he company, with PAz probability that a currentchooses to exit ' Puf probability that a current 5 chooses to exit E At this point, we will add the following assumptions where P , and P v. are the probabilities that an employee who is currently m E will go to A y B, or , respectively In other words, nobody who leaves the company ever returns. It is also implied by these restrictions that our company never replaces employees...

## Rational Functions

In which y is expressed as a ratio of two polynomials in the variable x, is known as a rational function. According to this definition, any polynomial function moist itself be a rational function, because it can always be expressed as a ratio to 1, and 1 is a constant function. A special rational function that has interesting applications in economics is the function which plots as a rectangular hyperbola, as in Fig. 2.8J. Since the product of the two variables is always a fixed constant in...

## Digression on Exponents

In discussing polynomial functions, we introduced the term exponents as indicators of the power to which a variable or number is to be raised. The expression b2 means that 6 is to be raised to the second power that is, 6 is to be multiplied by itself, or 62 6 x 6 36. In general, we define, for a positive integer nr and as a special case, we note that x x. From the general definition, it follows that for positive integers m and n, exponents obey the following rules Rule I xm x x for example, jc3...

## Ordered Pairs

2. 4 4.4 4-- With this visual understanding, we are ready to consider the process of generation of ordered pairs. Suppose, from two given sets, x 11,2 and v 3, 4 . we wish to form all the possible ordered pairs with the first element taken from set and the second element taken from set 1. The result will, of course, be the set of four ordered pairs 1 3 7 1,4 2, 3 . and 2, 4 . This set is called the Cartesian product named after Descartes , or direct product, of the sets x and y and is denoted...

## The Meaning of Equilibrium

T Fritz Machlup, Equilibrium and Disequilibrium Misplaced Concreteness and Disguised Politics ' Economic journal, March 1958, p. 9. Reprinted in F. Machlup, frsays on Economic Semantics, Prentice Hall Inc., Englewood Cliffs, N. ., 1963. in essence, an equilibrium for a specified model is a situation characterized by a lack of tendency to change. It is for this reason that the analysis of equilibrium more specifically, the study of what the equilibrium state is like is referred to as statics....