NationalIncome Model ISLM
A typical application of the implicit-function theorem is a gcneral-functional form of the IS-LM model.1 Equilibrium in this macroeconomic model is characteri7ed by an income level and interest rates that simultaneously produce equilibrium in both the goods market and the money market. A goods market is described by the following set of equations Y is the level of gross domestic product GDP , or national income. In this form of the model, Y can also be thought of as aggregate supply. C, , G,...
Preface
This book is written for those students of economics intent on learning the basic mathematical methods that have become indispensable for a proper understanding of the current economic literature. Unfortunately, studying mathematics is, for many, something akin to taking bitter-tasting medicine absolutely necessary, but extremely unpleasant. Such an attitude, referred to as math anxietyhas its roots we believe largely in the inauspicious manner in which mathematics is often presented to...
Chapter 20
20.1 The Nature of Optimal Control 631 Illustration A Simple Macroeconomic Model 632 20.2 Alternative Terminal Conditions 639 Fixed Terminal Point 639 Horizontal Terminal Line 639 Truncated Vertical Terminal Line 639 Truncated Horizontal Terminal due 640 Exercise 202 643 203 Autonomous Problems 644 dfetime Utility Maximization 645 Exhaustible Resource 647 Exercise 2Q,4 649 Neoclassical Optimal Growth Model 649 The Current- Value llamiltonian 65 Cons one ting a Phase Diagram 652 Analyzing the...
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...
FirstDerivative 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....
Exercise 86
1T let the equilibrium condition for national income be S Y T V - Co St T V gt 0 V V gt V where S, Yt Ttlt and C stand for saving, national income, taxes, investment, and government expenditure, respectively. Al derivatives are continuous. a interpret the economic meanings of the derivatives 5', 7', and '. b Check whether the conditions of the implicit-function theorem are satisfied. If so, write the equilibrium identity. c Find rf dCo and discuss its economic implications, 2. Let the demand...
Exercise 92
1. Find the stationary values of the following check whether they are relative maxima or minima or inflection points , assuming the domain to be the set of all real numbers 0 y -2x3 8x 7 b y c y 3x2 3 d y 3 2-6x 2 2. Find the stationary values of the following check whether they are relative maxima or minima or inflection points , assuming the domain to be the interval 0, oo a y x3 - 3x 5 b c y- 4.5x2 -6a 6 3. Show that the function y - x 1 jx with x 0 has two relative extrema, one a maximum...
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 NationalIncome 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 LinearEquation 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 FirstOrder 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,...
McGrawHill 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....
Relations and Functions 1
Since any ordered pair associates ay value with an x value, any collection of ordered pairs any subset of the Cartesian product 2.3 will constitute a relation between y and .v. Given an x value, one or more y values will be specified by that relation. For convenience, wc shall now write the elements of x y generally as x, y rather than as a, b , as was done in 2.3 where both x andy are variables. The set , y y 2a is a set of ordered pairs including, for example, 1, 2 , 0, 0 , and -1, -2 . It...
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...
TwoCommodity Market Model
To illustrate the problem, let us discuss a simple model in which only two commodities are related to each other For simplicity, the demand and supply functions of both commodities arc assumed to be linear In parametric terms, such a model can be written as Qj Go ai P - -a2P2 g h0 b P b2P2 where the a and b coefficients pertain to the demand and supply functions of the first commodity, and the a and ft coefficients are assigned to those of the second. We have nol bothered to specify the signs...
Solution by Elimination of Variables
One way of finding a solution to an equation system is by Successive elimination of variables and equations through substitution. In 3.1 , the model contains three equations in three variables. However, in view of the equating of and Qs by the equilibrium condition, we can let Q Qd Qs and rewrite the model equivalcntly as follows thereby reducing the model to two equations in two variables. Moreover, by substituting the first equation into the second in 3.2 , the model can be further reduced to...
Laws of Set Operations
Prom Pig. 2.2, it may be noted that the shaded area in diagram a represents not only A U B but also B U A. Analogously, in diagram b the small shaded area is the visual representation not only of A fi B but also of BPjA When formalized, this result is known as the commutative law of unions and intersections These relations are very similar to the algebraic laws a -f h h -H a and a x h h x a. To take the union of three sets A, B, and C, we first take the union of any two sets and then union the...
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...
A 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....
Nonalgebraic Functions
Any function expressed in terms of polynomials and or roots such as square root of polynomials is an algebraic function. Accordingly, the functions discussed thus far are all algebraic. However, exponential functions such as y bx, in which the independent variable appears in the exponent, are nonalgebraic. The closcly related logarithmic functions, such as y log , x, are also nonalgebraic. These two types of function have a special role to play in certain types of economic applications, and it...
Set Notation
A set is simply a collection of distinct objects. These objects may be a group of distinct numbers, persons, food items, or something else. Thus, all the students enrolled in a particular economics course can be considered a set, just as the three integers 2, 3, and 4 can form a set. The objects in a set are called the elements of the set. There are two alternative ways of writing a set by enumeration and by description. If wc let S represent the set of three numbers 2, 3, and 4, we can write,...