## Review

Convergence general solution homogeneous form initial-value problem integrating factor particular solution steady-state value 1. Describe the two-step procedure for obtaining ihe general solution to Ihe complete, autonomous, linear, first-order differential equation. 2. Explain what is meant by the steady state of a linear, first-order differential equation. 3. Under what conditions is the particular solution equal to the steady-state solution 4. State and explain the necessary and sufficient...

## Info

Since is homogeneous of degree k, we have f (,VA1. SX2 SXn ) SkJ (a-1. ,V2 xn) Writing each .va, as z, and differentiating both sides with respect to s gives 3 , ' f dz2 , , 3 for. + 7-- --1----+ 7--7T- 'f(Xf,X7-----a ) iJft + fjx2 H-----1- fnx ksklf x ,x2 x ) Since this condition holds for any .s > 0. it holds for I which implies that f x + f7X2 + + f x kf(xi, a2. -,x ) (11.12) Euler's theorem slates that if is homogeneous of degree k. then multiplying the marginal product of each input ' by...

## Intermediate Value Theorem

In this section we present a straightforward theorem, the intermediate-value theorem and show that it can be very powerful in the study of equilibrium, which is one of the most important concepts in economics. The particular application we make is a very simple one. The rtnge of applications, however, is in fact very broad. Suppose that the function )' -- f(x) is continuous on the interval fa, J. b > a. It follows that the function must take on every value between f(a) and ( > ). which are...

## Introduction to Comparative Statics

The Simple Keynesian Model of Income Determination Let ) denote the value of the aggregate supply of goods and services in the economy. Since this accrues as sales revenue to firms who then pay it out as incomes to suppliers of inputs, including labor, we also refer to Y as national income. The aggregate demand for goods and services has two components consumption demand C and investment demand I. We take as exogenous, but C is determined by Ihe consumption Junction where the constant c is the...

## J

Derivative ', in the neighborhood of a stationary value. The curvature of the function f determines the slope of its derivative '. In figure 6.16(a). the function is strictly concave in the neighborhood of a . This implies that in figure 6.16(d). at x-values below a*, we have . (a) > 0. while at x-values above x*. f x) < 0. The slope of ' is the second derivative (a), and so we have J'(x') < 0. Since, in this case, x* yields a maximum of the function, we have Theorem 6.2 If '(x* i 0, and...

## J 10xeZxdx Vojc I 0dx

Ft is also useful to know how to differentiate with respect to some parameter that affects either the limits of an integral or the integrand. The economic applications of these techniques are primarily in the field of dynamic analysis (see chapter 25), and it is traditional to use the variable t. which represents time, as the variable of integration. We will usex as the parameter which may affect either the integrand or the limits of integration. First, consider the case in which only the upper...

## Oiq

Figure 24.5 Phase diagram tor example 24.14 Figure 24.5 Phase diagram tor example 24.14 for which the solutions are -1 I 2 and r2 1 2. Since the roots are of opposite sign, the steady-state solution is a saddle-point equilibrium. Step I Determine the motion of yBegin by graphing the yi isocline setting y, 0 to find the isocline gives the horizontal line y 2. Next, we note that y i < 0 below this isocline (when v2 < 2) and v, > 0 above the isocline (when > '2 > 2). The appropriate...

## JlU Unlli u

E(u u ) E U U2) ( ) where is the identity matrix of order n. The assumption above about the errors simply states that they are pairwise uncorrelated, since E(u,u ) 0, for all i j- They also have the same variances. If we further assume that the joint distribution of these n errors is normal, then they will be independent, sincc for the case of normality, lack of correlation implies independence, and vice versa. Then we say that the u,s are independently and identically distributed, or i.i.d and...

## JU tj F ritvij a 0 f2x Ai a XtA x2 a

If we assume that the functions and g possess continuous first and second derivatives, and that the determinant then we can apply the implicit function theorem. This amounts to saying we can solve for the endogenous variables as differentiable functions of the exogenous variable in the neighborhood of the optimal point, so the value of the function in the same neighborhood is and V is known as the value function for the maximization problem. The value function expresses directly the idea that...

## K

The reason for calling this function linear is obviously that its graph is a straight line. The steepness of the line is determined by the absolute value of a Taking two -values, we can write where Ay is read ihe change in y, and likewise for Ax. The ratio Ay Ax is called the slope of the line and so a is the slope coefficient. Note that (he line y ax is fully determined once a is chosen. In equation (2.7) y is often referred to as the dependent variable and x as the independent variable. This...

## Lcb

The determinant of the coefficient matrix of the homogeneous system is ( -< 5 a), which is negative. We therefore know immediately that the steady-state equilibrium is a saddle point. Bv theorem 24.2, the solutions to the system of differential equations in (25.7) and (25.8) aie where r and r2 are the eigenvalues or roots of the coefficient matrix in equation (25.9), Ci and C2 are arbitrary constants of integration, and k and K are the steady-state values of the system, and serve as...

## Km

The steady-state solutions are y ( 1 und yi - 2. The complete solutions then are Construct the phase diagram for this system. First, determine the motion for _V . Setting v, 0 gives the y isocline as the horizontal line yj - 2 in figure 24.7. If v > 2, then y, < 0. so above the isocline, yi is decreasing below the isocline, yi is increasing. Second, determine the motion for y2. The y2 isocline occurs along the line v V + I. This is a straight line in figure...

## L

Subject to i g .t(r), y(i), a(0) Vo The Hamiltonian for this problem is the usual We wish to maximize H with respect to the control variable y. but now subject to the inequality constraint on y. To do this, we form the ordinary Lagrangean function which incorporates the inequality constraint, where 0 is the Lagrange multiplier. The necessary conditions now are The important differences that arise are in equations (25.44) and (25.45). In equation (25.44) we have the necessary condition for...

## L5rw

1 ' 2f C2v T72cos(jt 4) - CrN T72sin T 4) - 2C When the solution is required to satisfy given initial conditions, the constants C and Co take on specific values. Example 24.23 Find the values of C and C2 that make the solution to the difference equation system in example 24.20 satisfy v0 2 and xq 1. Setting - 0 in the solutions given in example 24.20 and setting yo 2 and .Vu I gives Solving these two equations now tor Ct and C2 gives, alter some simplification, C, -8 and C2 8. In chapters 18 to...

## Lim V 1 1246858 r ri I fr

Which is 78 of the entire present value, compared to the error of 2.1 if the interest rate were 8 (from the calculation above). Example 3.19 Suppose that a stream of equal payments of amount I 0.000 per yea is to continue in perpetuity. At the interest rate of 6 compute (i) (he present value of this entire stream of benelils (ii) (he present value of the benefits beginning at the end of (he 50th year (iii) the present value of the first 50 years of benefits

## Limit Of A Sequence

A sequence is said lo have the limit L if, for any e > 0. however small, there is some value N such that L < e whenever n > N. Such a sequence is said to be convergent, and we write its limit as lini,,-. a L. In less formal language, the definition above states that a sequence has a limit L provided that all values of the sequence beyond some term can be made as close to L as one wishes (i.e., the condition a - L < t can be met for as small a positive number i as one likes by choosing a...

## Linear First Order Differential Equations

In the next three chapters we explain techniques for solving and analyzing ordinary differential equations. We do not attempt to provide exhaustive coverage of the topic but instead focus on the types of differential equations and techniques of analysis that are most common in economics. We begin in this chapter with linear, first-order differential equations. In the next chapter we turn to an examination of nonlinear, first-order differential equations, and in the chapter after that we examine...

## Linear Second Order Differential Equations

Until now we have confined our analysis of differential equations 10 those of the first order. In this chapter we will examine linear, second-order differential equations with constant coefficients. We focus our attention on the autonomous case in section 23.1 and consider a special nonautonamous case in section 23.2. We begin by explaining how to solve a linear, autonomous, second-order differential equation. The linear, autonomous, second-order differential equation (constant coefficients and...

## Linear Systems in nVariables

Although graphing solutions and ifnding solutions by simple substitution is tine for systems of equations with only two variables, we need to develop other procedures to find solutions for general systems of linear equations. These procedures often involve generalizations of some of the alternative solution methods that we have already referred to subtracting equations and multiplying equations by a constant. As we develop these methods, we will illustrate with 3-variable systems first and...

## M injx m2x2 0 0pM wJ

- r X - r2x > 0, p* > 0. 0 The Lagrange multipliers are ususally referred to as dual variables in linear programming. The key point is their interpretation as the shadow prices of the input constraints. At the optimal solution, the value of A.*, p*, or p' gives the increase in revenue the firm would earn if it acquired a little bit more of the respective inputs and allocated that optimally between the outputs. The last three conditions also tell us that if a shadow price is positive, all of...

## Mathematics for Economics

Michael Hoy John Livernois Chris McKenna Ray Rees Thanasis Stengos The MIT Press Cambridge, Massachusetts London, England 2001 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. This book was set in Helvetica and Times Roman by Interactive Composition Corporation. Printed and...

## Matrix Transposition

A very useful operation in matrix algebra is that of transposition. The transpose of a matrix A, is the matrix in which the rows of the original matrix A become columns and the columns of A become rows. The transpose is denoted by Ar. The transpose matrix, AT, is the original matrix A with its rows and columns interchanged. This implies that A' will have its dimensions reversed when compared with A. Example 8.14 Find the transpose of the 2x3 matrix 4, given by Since A is 2 x 3. its transpose...

## Maxfxix st ati 2

Consists of deriving the following first-order conditions to find the critical point(s) of the Lagrange function C xuXi, X) (A-I, .vi) + .tiUi, A I (A-, A,') + > .> ,( A'f. Ai ) 0 h(xl x ) + X*glOtJ.JT ) 0 OX 2 How can we be sure that for some arbitrarily given problem there is always a unique solution point at a tangency of the Constraint curve and contour of , and that the Lagrange procedure always works in that it delivers this point Exactly what role is played by the assumptions on the...

## Mc Ac

Figure 13.11 Total-, average-, and niarginal-eost curves with constant returns to scale Figure 13.11 Total-, average-, and niarginal-eost curves with constant returns to scale Figure 13.12 Total-, average-, and marginal-cost curves for diseconomies of scale y > > . The cosl function is said to exhibit diseconomies of scale. Figure 13.12 illustrates. (Refer to section 11.5 for a more comprehensive discussion of returns to scale.) Cost-Minimization with the CES Production Function In chapter...

## N a yWi

Combined with y' ' + Y I this gives implicitly die optimal outputs. 7. Hint L'se the first-order conditions to solve for Kir, u y), A'(r, w, v) and .(r. mj. y ) Insert these into die Lagrangean and differentiate with respect to y. Consider that the optimized Lagrangcan has the same value as the cost function for all v.

## Nonlinear First Order Differential Equations

In chapter 21 we saw that we could apply a single solution technique to solve any first-order differential equation that is linear. When the differential equation is nonlinear, however, no single solution technique will work in all cases. In fact only a few special classes of nonlinear, first-order differential equations can be solved at all. We will examine two of the more common classes in section 22.2. Even though solutions are known to exist for any nonlinear differential equation of the...

## Numbers

The most basic and familiar kinds of numbers are natural numbers, the elements of the set They arise naturally in counting objects of all kinds. What does it mean to count a set of objects, say a pile of dollar bills' When we count dollar bills, we lake each element in the set of dollar bills and pair it wilh an element of Z+, starting with 1 and moving successively through the sel. When we have exhausted the elements of the first set (of dollar bills), the element of Z, that we have reached in...

## Obtaining the Determinant and Inverse of a 3 x 3 Matrix

In section 9.1 we obtained the determinant of a 2 x 2 matrix. Below we will derive the determinant of a 3 x 3 matrix and then obtain the determinant of a general square matrix of dimension n x n. The determinant of a 3x3 matrix A will be composed of all the elements of A. However, as we will see, the expression of the determinant of A will be reduced to particular expressions involving the determinants of certain 2x2 submatrices 9.2 OBTAINING THE DETERMINANT AND INVERSE OF A 3 , 3 MATRIX 371 of...

## Obtaining the Inverse of a 3 x 3 Matrix

Having obtained the determinant of a 3 x 3 matrix, it is quite straightforward to obtain its inverse matrix. Below we will present the steps that one follows to obtain the inverse of a matrix of order 3 and illustrate the method bv means of an example. Step 1 Corresponding to the elements ( , of A, we obtain the cofactors C1(, ' 1.2, 3 j 1,2, 3. Then we form a matrix in which each element a,, is replaced by the corresponding cofactors C,,, given below as 9 2 OBTAINING THE DETERMINANT AND...

## Optimization of Functions of nVariables

The idea of optimization is fundamental in economies, and the mathematical methods of optimization underlie most economic models. For example, the theory of demand is based on the model of a consumer who chooses the best ( most preferred) bundle of goods from the set of affordable bundles. The theory of supply is based on the model of a lirm choosing inputs in such a way as to minimize the cost of producing any given level of output, and then choosing output to maximize profit. Rationality and...

## Optimization of Functions of One Variable

Many economic models are based on the idea that an individual decision maker makes an optimal choice from some given set of alternatives. To formalize this idea, we interpret optimal choice as maximizing or minimizing the value of some function. For example, a firm is assumed to minimize costs of producing each level of output and to maximize profit a consumer to maximize utility a policy maker to maximize welfare 01 the value of national output', and so on. it follows that the mathematics of...

## Optimization over an Interval

The discussion of first- and second-order conditions in sections 6 I and 6.2 has dealt exclusively with the unconstrained case, in which a solution to the problem can be anywhere on the real line. Often in economics, however, this is unacceptably general. For example, in problems in which firms choose outputs (as in several examples in the previous section ) or consumers choose goods, we cannot assume that negative quantities are possible. In other problems, it may be reasonable to place an...

## P

Simplifying, using C> An. and solving for C gives Substituting this expression into the solution for fji(t) and then using it in the solution for c(t) gives the optimal path of consumption chosen by the individual Figure 25.5 Optima consumption path. c(t), and bank account path. ,v( ). when x T) v0 and p < r Figure 25.5 Optima consumption path. c(t), and bank account path. ,v( ). when x T) v0 and p < r An interesting case arises when b jr(). which requires the bank account at time T to...

## P Q

2.5 PROOF, NECESSARY AND SUFFICIENT CONDITIONS 61 There are several ways in which these statements can be read, and it is useful to spell these out P is a sufficient condition tor Q Q is a necessary condition for P > is sufficient for means that the truth of P guarantees the truth of Q. Q is always true when P is true. It follows that if Q is not true, then P cannot be true. Thus it is necessary that Q is true for P to be true. In other words, P can be true only if Q is true. We illustrate...

## P [ B paa

Which gives a system of reduced-form equations for the prices. The following 3x3 example illustrates and shows how to solve for a particular price (rather than the entire vector of prices) using Cramer's rule. Example 9.22 Consider the markets for coffee, tea, and sugar. These goods are related in demand. since the first two are often substitutes for each other while the third is often complementary with each of the other two goods. Ignoring any supply side links (which are. in any case,...

## Power Exponential and Logarithmic Functions

When a number a is multiplied by itself n times, we write a. where n is called the exponent. This leads to the rules of exponents Intuitively, we may think of n as an integer, but in fact n could be any real number. The power function takes the form (Note that the rectangular hyperbola is a special form of the power function with b I. The linear function is also a special case with b I. The quadratic may be thought of as the sum of two power functions. ) Figure 2.26 shows two power functions...

## Present Value of a Stream of Payments

Earlier we saw how the sequence P V, V ( 1 +r)' (see equation 3.3) represents the present value of an amount of money V received t periods into the future. In many economic settings we need to compute the equivalent present value of a series (i.e., the sum total) of such amounts. For example, a mortgage or other long-term loan represents a current sum of money loaned to ar, individual or institution in return for a stream of future payments. Thus, if an individual makes annual payments at the...

## Present Value Calculations

An important economic application of sequences is the determination of the present value of a sum of money to be received at some poini in the future. This computation is the inverse of determining how much money one would have in the future upon investing a certain amount now. Suppose, for example, that one had S90.91 to invest currently at an annual interest rate of 10 . Then the amount of money received at the end of one year would be 90.91 (I 4-0.1) 100. In general. investing X today at an...

## Proof Necessary and Sufficient Conditions

Why should one person ever accept as true a statement made by someone else The usual response would be, Prove it If the statement is a purely factual one, for example, prices have risen. then proof would lake die form of some factual evidence that substantiates the statement. Economics is more often concerned, however, with deductive statements such as the money supply increases, then the price level will rise. which is to say. increases in the money supply lead to inflation. A stronger...

## Properties Of Sequences

Example 3.13 Use the results that and result (iv) of theorem 3.2 to find ( + 3)( r-l) -f 3 n2 - 1 lira ---- --- hm - Iim - - -few n cc n ii oo tl n oc j Another useful application of this theorem concerns the present-value formula developed in section 3.3 Theorem 3.2 provides a proof of the claim that, if r > 0. then PV, - 0 as I -v oo, Since the denominator is a delinitely divergent sequence (if r > 0) and the numerator is a constant, then part (v) of theorem 3.2...

## Q 20 p 05p7 q2 100 2 p2

(a) Comment on the relationship between the three goods on the demand side. fb) What is the nature of any production externality on the supply side (c) Solve for the equilibrium prices and quantities of the three goods. 5. An economy has an IS curve given by r 210-2)' and an LM curve given by r m + y 4. The long-run equilibrium level of output must equal 100. What value of M makes the IS and LM curves intersect at Y - 100 What is the economic interpretation of a situation in which M exceeds...

## QrQ QQr

An orthogonal matrix is a matrix for which its inverse equals its transpose. Theorem 10.3 For the problem in equation (10.8), where A is a symmetric matrix, the eigenvectors that correspond to distinct eigenvalues are pairwise orthogonal and if put together into a matrix, they form an orthogonal matrix. Let qi and q denote the eigenvectors corresponding to A and Then Aqi Aiqi qi Aq, A.,q qi A< 2 X2q2 > q 4q2 X2q q Since A is symmetric we have that f Ai Aj....

## Quadratic Functions

We can write a quadratic function in explicit form as As figure 2.24 shows, this is a useful function in economics because in its convex form, with a > 0. it could be used to depict a typical U-shaped average or marginal-cost curve, while in its concave form, with a < 0, it could depict a typical total-revenue or total-profit curve. (Note that in these examples the domain of the function must be restricted to R+, since negative outputs arc not allowed.) The unique minimum (< n the convex...

## Quantities Magnitudes and Relationships

We can slart by thinking of how we measure things in economics. Numbers represent quantities and ultimately i( is this circumstance that makes it possible to use mathematics as an instrument for economic modeling. When we discuss market activity, for example, we are concerned with the quantity traded and the price at which (he trade occurs. This is so whether the quantity is automobiles, bread, haircuts, shares, or treasury hills. These items possess cardinality, which means lhat we can place a...

## R 106006

(in as of the beginning of the 50th year the present value of 10,000 per year in perpetuity is 166,666.67, as computed in part (i). Since this is in effect received at the end of the 50th year, its current ( i.e as of now) present value must be discounted so that it becomes (iii) The present value of the first 50 years' worth of payments is simply the answer in (i) less that in (ii) 166,666.67 - 9,048.06 157,618.61 The examples above all deal with the problem of determining the present value of...

## R E V I E W

Characteristic equation complete solutions direct method eigenvalues eigenvectors general form global stability homogeneous solutions improper stable node improper unstable node isocline isoscctors local stability particular solutions phase diagram phase plane saddle path saddle point saddle-point equilibrium simultaneous system stable focus stable node steady state substitution method trajectory unstable focus unstable node 1. Explain how the characteristic equation can be derived and how it...

## R i rTi tfluv r i

The rule for multiplying matrices is essentially a generalization of the above reasoning. To multiply matrices, it is not necessary that they be of the same order. The requirement is that the number of columns of the first matrix be the same as the number of rows of the second matrix. Matrices that satisfy this requirement are said ro be conformable for matrix multiplication. Before we present the formal definition of matrix multiplication we will illustrate the idea by means of some examples....

## R1 11 v

If this were not true, then y,+ l would not depend on .v,7 and we could then solve it directly as a single, lirst-order difference equation. Using this to substitute for.v, gives +1 11 y, Vi+2 11 V H-t 4-0 2 2l> 'i -FapiJ22( - Alter simplifying and rearranging this becomes yl+2 - (fl -F 22)Jin-1 + (fl ifl22 - Oi2 21 ).V, 0 which is a homogeneous, linear, second-order difference equation with constant coefficients. Equations like this were solved in chapter 20 theorem...

## Rank of a Matrix

The concept of linear independence or dependence of vectors is closely linked to the concept of nonsingularity or singularity of matrices discussed in chapter y. We now investigate further the nonsingularity (singularity) properties of matrices by introducing the concept of the rank of a matrix. Consider any arbitrary matrix of order m x n. U consists of it column vectors with m elements each, and w row vectors with n elements each. Therefore the n column. m-element vectors belong to R',...

## Review of Fundamentals

In this chapter we give a concise overview of some fundamental concepts that underlie everything we do in the rest of the book. In section 2.1 we present the basic elements of set theory. We then gv> on to discuss the various kinds of numbers, ending with a concise treatment of the properties of real numbers. End the dimensions of economic variables. We then introduce the idea of point sets, beginning with the simplest case of intervals of the real line, and define* their most important...

## Rule 1 Derivative of a Constant Function fx c

The reason that fix) 0 when fix) c is easy to see intuitively by looking at the graph of the function fix) c (see figure 5.18). Regardless of which poini A' is chosen. Ay 0 for any value of Aa. Here Ay - 0 for any size of A.v. Example 5.4 Marginal Revenue Function for a Competitive Firm A competitive firm believes that if it sells more output there will not be a reduction in the market price. The extra revenue generated by producing and selling one more unit...

## Rule 3 Derivative of a Power Function fx xn

For example, the derivative of the function f(x) ,r is fix) 2x, as we denved1 from the definition of the derivalive in section 5.2. The following list of examples illustrates various types of results using this rule. For fix) x3 2,x > 0. we get f'(x) 0 2)x 2' Q 2)xt 2. The function fix) x 2 has derivative fix) 2x 3. The derivative of fix) .v. -v 0. is fix) Ox'1 0 x 0 for .v 0. Since for x 0, x 1. this is clearly the correct result. (Recall that 0 is not...

## Rule 6 Derivative of the Sum of an Arbitrary but Finite Number of Functions hx gx

Ifh(x) v ' l g,tx), then h'(x) S'M) This result also applies to the case where some or all of the operations involve subtraction rather than addition. Rule 6 is a straightforward generalization of rule 5. That is. since he derivative of the sum of two functions is simply the sum of the derivatives of the functions taken separately, then doing this iteratively allows one to establish rule 6. Thus, for example, if h(x) v4 + 8.v2 + 2a. we can treat h(x) as the sum of two functions, fix) (a-4 +...

## Rules of Differentiation

In section 5.2 several examples were presented showing how to derive the derivatives of some simple functions by using the definition of the derivative. It would be tedious, however, to have to do this every time we wanted to find the derivative of a function. Since such derivations have been done for various general classes of functions, we can use these results and so avoid repeating the exercise each time. These rules or methods of finding derivatives are collected below in table format and...

## Rules Of Differentiation 185

Figure 5.25 Marginal revenue foi u monopolist facing the inverse demand function p 40 - 2q Figure 5.25 Marginal revenue foi u monopolist facing the inverse demand function p 40 - 2q Notice the change in sign to positive for the second term. This simply recognizes that in the expression for AR Aq. we treat the reduction in price as a positive value, while in the expression lim. - o AR Aq. die term dp dq is the slope of the inverse demand function which is itself negative (i.e Ap Aq is the slope...

## S

Give the amounts of labor and capital required to produce, respectively, one unit of agricultural, mining, and manufacturing output. (i) Find the primary input requirements ofthe economy when it wishes to produce the final demand vector (ii) Is this final demand vector feasible for this economy if it has available a maximum of 1,200,000 units of labor and 1,700,000 units of capital (hi) Find the set of final demand vectors that are feasible for the economy given the primary input availability...

## Second Order Conditions

We saw in section 6. that the condition f'(x) 0 does not in itself tell us whether x* yields a maximum, a minimum, or a point of inflection of the function . Since this condition is staled in terms of the first derivative of the function it is usually referred to as the first-order condition. We now go on to examine how conditions on the second derivative of a function, namely second-order conditions, can be developed to help us distinguish among the three kinds of stationary value. In figure...

## Second Order Conditions for Constrained Optimization

We saw in the previous section that the first-order conditions for a maximum and a minimum of a constrained problem are identical, as in the unconstrained case, and so it again becomes necessary to look at second-order conditions. One approach to these is global assumptions are built into the economic model to ensure that the objective function and the constraint function(s) have the right general shape. As we will see in section 13.3, it is sufficient for a maximum (minimum) that the objective...

## Secondorder Partial Derivatives 475

The exercise of determining the economic interpretation of first- and seeond-ordcr partial derivatives is illustrated in the following example Example 11.12 Find and interpret the second-order partial derivatives of the Cobb-Douglas production function with two inputs. The general form of the Cobb-Douglas production function with two inputs is > (je,, *-> ) Ax' 4, vi. *2 > o wliere .rj and x2 are input levels, y is the output level, and a, P, A > 0, are technological parameters. We...

## Sequences Series and Limits

Studying sequences and series is the best way to gain intuition about the rather perplexing notions of arbitrarily large numbers (infinity) and inrtnitesimally small (but nonzero) numbers. We gain such understanding by using the idea of the limit of a sequence of numbers. Thus, from a mathematical perspective, this chapter provides very useful background to the important property of continuity of a function, which we will explore fully in chapter4. There are also some interesting economic...

## Sets and Subsets

5 (x x is an even number between I -nd 11 The first way of writing 5 corresponds to definition by property (the should be read as given that) the second to definition by enumeration. The key aspects of this notation are as follows A capital letter denoting the set, here S. A lowercase letter denoting a typical element of the set, here x. Braces, ( ), that enclose the elements of the set and emphasize that we treat them as a single entity. In general, a lowercase letter such as x denotes items...

## Simultaneous Systems of Differential and Difference Equations

It is common in economic models for two or more variables to be determined simultaneously. When the model is dynamic and involves two or more variables, a system of differential or difference equations arises. The purpose of this chapter is to extend our single equation techniques to solve systems of autonomous differential and difference equations. 24.1 Linear Differential Equation Systems We begin with the simplest case a system of two linear differential equations and solve it using the...

## Single Equation Models and Multiple Equation Models

Although sometimes the problem we are trying to analyze may be captured in a single-equation model, there are many instances where two or more equations are necessary. Interactions among a number of economic agents or among different sectors of the economy typically cannot be captured in a single equation, and a system of equations must be specified and solved simultaneously. We can extend our earlier example to illustrate this. Consider first the demand and supply of two goods. We denote the...

## Sma iUj A it

Next, choose a set of w, e .v, _ i, .v, values to generate the largest possible value for the Riemann Sum that is. choose ft), such that (< u,) > (x) for all x e xi-1, a, . Let us refer to these values as & > , and the sum they generate as the upper For our example, it is easy to see that Lire values ( are found by choosing the leftmost point of each subinterval, while the values < > , are found by choosing the rightmost point of each subinterval (see figure 16.4). The result is...

## Solution by Row Operations

The idea behind finding solutions by row operations is to transform a given system of equations into another with the same mathematical properties and hence the same solution. The aim is to transform a system in such a way as to produce a simpler system which is easier to solve. Three types of operations are permitted to transform a system 1. Multiply an equation by a nonzero constant. 2. Add a multiple of one equation to another. In practice, not all of these operations may be required to...

## Some Advanced Topics in Linear Algebra

In this chapter we consider diree important advanced topics in matrix algebra vector spaces, eigenvalues, and quadratic forms. All play important roles in a variety of contexts in economic theory and in econometrics. Vector spaces enable us to talk about distance between points, and linear dependence between vectors. They are therefore closely linked to die study of systems of linear equations of chapter 7. Eigenvalues play an important role in determining the stability properties of dynamic....

## Some Properties of Point Sets in R

In section 2.2 we saw diat a point on the real line always corresponds to a real number. We now place two real lines at right angles so that they intersect at the number 0, as in figure 2.9. Any point in the coordinate system formed by these two lines can be defined as an ordered pair of numbers (.r, y) by assigning to x the real number vertically below it on the horizontal axis, and assigning to y the real number horizontally across from it on the vertical axis. The figure shows some examples....

## Some Special Matrices

A number of matrices have particular properties that are often found to be useful in studying systems of equations. A square matrix A of any order is idempotent if Idempotent matrices play a very important role in statistical distribution theory. We will see examples of them in later chapters when we discuss quadratic torms. Example 8.18 Verify that A below is idempotent It is only necessary to verify that A A A. (Why ) 1 6 -1 3 1 6 -1 3 2 3 -1 3 1 6 -1 3 1 6 1 6 -1 3 1 6 -1 3 2 3 -1 3 1 6 -1 3...

## Statics and Dynamics

In introductory treatments of economics the time dimension c> t the problem is often ignored, or supressed for simplicity. In reality, the time dimension is always present. In the examples of market equilibrium already discussed, we should think of the quantities as flows per period of time, so that qD is the quantity demanded of a good per period, however short or long that period may be in terms of calendar time. Models where we explicitly or implicitly consider a situation within a single...

## Systems of Linear Equations

In chapter 2 we defined a linear function as one lhat takes the form for known constants a and b. and where x is the independent variable which takes on values over some specified domain, and y is the resulting value of the function at each .v-value. We also know that by taking specific values of .v, we can draw the graph of .v and y in a two-dimensional picture. The graph is a straight line- hence the phrase linear function. There are many examples of functions in economics which can be...

## Taylor Series Formula and the Mean Value Theorem

Recall (hat the differential, dy f' x)dx. can be used to provide an approximation to the change in the y variable, dy Ay, for a given change in the x variable, dx Ax (see figure 5.45). As we saw in section 5.2, the percentage error from using dy as an approximation to the actual change in y. Ay, can be made arbitrarily small if we are willing to consider changes in a that are made arbitrarily small. However, we are not always satisfied with ihe restriction that A.v be small and for nouitv...

## The Completeness Property of K

We have already indicated that the set of real numbers R has the property that there are no gaps. That is, between any two points (numbers) on the real line, every point is occupied by a number that is either a rational number or an irrational number. In other words, corresponding to each point on the real line there is a real number, and vice versa. This is known as the completeness property, and we can express this property formally by considering the concepts of the greatest lowest bound and...

## The Concave Programming Problem

We can write the concave-programming problem in a simple form as max f x ,x-> ) s.t. gU,.x2) > 0, .t ,x-> > 0 (15.1) The name concave programming was used to distinguish this type of problem from that of linear programming, which, as we will see in section 15.2. is a special case of concave programming. The word concave appears because the functions ' and are assumed to be concave. We discuss the reasons for this below. We also assume the functions to be differentiable. Note also the...

## The Derivative and Differential for Functions of One Variable

The purpose of the derivative is to express in a convenient way how a change in the level of one variable (e.g x) determines a change in the level of another variable (e.g y). Much of economics is in fact concerned with just this sort of analysis. For example, we study how a change in a firm's output level affects its costs and how a change in a country's money supply affects the rate of inflation. Although expressing the relationship between * and y as a function y f(x) does capture this idea...

## The Diagonalization of a Square Matrix

Once we obtain the eigenvalues of a matrix A as solutions to the characteristic equation, we proceed to obtain the corresponding eigenvectors. This leads us to a very important result whereby matrix A is transformed to a diagonal matrix. This result is known as the spectral decomposition of a square matrix. Example 10.11 For the malrix A of example 10.10. tind the eigenvectors corresponding to the characteristic roots A i 5 and 0. For A 5, substituting into equation (10.8) yields

## The Eigenvalue Problem

In previous chapters we examined the solution of a system of n linear equations formulated as where A is an n x n matrix of coefficients, x is an n x I vector of unknowns and b is an n x I vector of constants. In this section we investigate the solution to an alternative problem formulated as where A is a known square matrix of order n x n, q is an unknown -element column vector, and is an unknown scalar. This problem, known as the eigenvalue problem, arises in many situations in economics and...

## The Envelope Theorem

In the comparative-statics analysis of constrained maximization and minimization problems, it is often helpful to use an approach based on the envelope theorem, instead of, or a.s well as that based on the implicit function theorem. Thus suppose that we have the problem max (.vi, a) s.t. (Jt ,x2 a) < 0 where a is an exogenous variable. The Lagrange function for this problem is (xi, jtj, a ix) U , a) + Xg(x .x a)

## The Existence of Equilibrium

The simple result of the intermediate-value theorem is often very useful when trying to prove that a special value of an economic variable exists. Consider the simple partial equilibrium model of demand and supply with p representing price, y representing quantity, y Dip) representing the demand function, and y Sip) representing the supply function. An equilibrium price for this model is defined as a price that clears the market. That is, p p > 0 is an equilibrium price if D(p') Sip' ). We...

## The Gauss Jordan Elimination Method of Computing the Inverse Matrix

Rbe method for computing Ihe inverse matrix that we will describe in ihis section Is based on the application of the so-called elementary row operations that we saw in chapter 7 as a way of obtaining the solution of a system of simultaneous equations. A nonsingular matrix A of order n can be reduced to by a series ol elementary row or column operations defined below. An elementary row operation involves any of the following three cases 2. Adding a multiple A of one row to another 3. Multiplying...

## The Inverse of an n x n Matrix and Its Properties

To obtain the inverse of a matrix of order n. we follow the same steps outlined in the previous section for the case of a 3 x 3 matrix. From a computational point of view, obtaining the cofactors becomes a fairly complicated task. For example, in the case of a matrix of order 4 given below, _ ill) il l a > < 324 obtaining any minor, say M , would involve solving for the determinant of a 3 x 3 matrix, since

## The Keynesiari Multiplier

An important component of the traditional Keynesian macroecononnc model used to explain the importance of government fiscal policy is the multiplier. The basic idea is that if there is an increase in exogenous expenditure in the economy, say government spending, then a multiplier effect ensues so that the ultimate impact on economic activity (GNPi is greater than the initial expenditure. For example, suppose that the government initiates additional expenditure of Sl(X) million for increased...

## The Linear Second Order Difference Equation with a Variable Term

When the term b, is not constant, then the linear, second-order difference equation is nonautonomous. The method of solving a nonautonomous difference equation still involves adding together the solution to the homogeneous form and a particular solution to the complete equation. However, wc can no longer rely on using the steady-state solution as a particular solution since it no longer exists necessarily. Even when b is constant, a steady-state solution does not exist when 1 + 1 +tfj 0. In...

## The Order Properties of M

First we define two important subsets of R. The set R++ c R consists of the strictly positive real numbers with the characteristics that (i) TRL f. is closed under addition and multiplication. (ii) For any a e R. exactly one of the following is true The set R+ R++ U 0 is the set of nonnegative real numbers. Diagrammatically, the set R++ is the right half of the real line in figure 2.7, excluding zero, while R_ is that half including zero. We may similarly identify the left half of the real...

## The Real Numbers and Their Properties

The union of the sets of rational and irrational numbers is the set of real numbers. We think of the set of real numbers. R, as extending along a line to infinity in each direction having no breaks or gaps, as in figure 2.7. We refer to this as the real line. The properties of R define the basic operations that we can carry out on the elements of R. Consider three (not necessarily distinct) elements of R a. b. and c. We can postulate the following properties 1. Closure If a, h e E, then a + b e...

## To the Student

In any course, you will need to attempt exercises, and other material, that have not been covered in class. The only way to learn mathematics and economics is to do mathematics and economics. We have provided many examples and exercises in order to encourage independent study. The aim of this book is to present the key mathematical concepts that most frequently prove helpful in analyzing economic problems. However, the approach we take is not simply to provide a recipe book of results and...

## UL pp UWK2 Jw r D

Clearly, in the ease of a Cobb-Douglas production function, an increase in the wage unambiguously reduces the demand for capital. Comparative Statics for Constrained Optimization Problems The comparative-statics methods we have developed and illustrated so far do not allow us to handle the comparative-statics analysis of constrained optimization problems. This is because the smallest such problem would involve two choice variables and one constraint, giving a system of three tirst-order...

## V

Figure 5.34 Elasiiciiy changes along a linear demand curve Another functional form which is frequently used to express the relationship between price and quantity demanded is the so-called constant elasticity demand function Consider first the particular case with ft I (i.e., v a p), constant unitary elasticity of demand. For this demand function i( is easy to see that total sales revenue is the same at any price (pv a). This is illustrated in figure 5.35. Since total sales revenue is constant...

## 2dfe

We now have the question Is ' a true (local) maximum We answer this by taking and so t is a maximum. Noie that since T(i) is always negative, T(t) must be a strictly concave function (see ligure 6.19). This concave shape has interesting economic implications. As the tax rate goes up. the equilibrium quantity bought and sold goes down. Initially the increase in tax rate outweighs the effect of reduced quantity and generates increased tax revenue, but after a point this ceases to be true reduced...

## Ayy yn e lim

There are three things to note about this formula As Ap > 0. the two points in figure 5.33 (.4 and B) converge to a single point ( I), hence the term poin' elasticity. If we are referring to the impact of the change in the (own) product's price on demand we generally refer simply to elasticity of demand. We could, of course, measure other elasticities such as the > mpact of a change in income on quantity demanded or the impact of a change in the price of some other good on quantity...

## V2 2y i 2v

The homogeneous form of this difference equation is identical to the one solved in example 20.4. Using that solution plus the steady-state solution gives the general solution to the complete equation. The steady-state solution is obtained as the solution to y - 2y + 2y 10 which gives y 10. The general solution to the complete equation then is II the solution to a second-order difference equation is required to satisfy two specified initial conditions, the values of the constants must be set...

## VU e3J13cos j 11 sin j

Although the substitution method works well for systems of two differential equations. it can become cumbersome for larger systems. The following direct approach to solving a system of linear differential equations circumvents this limitation. A linear system of n autonomous differential equations is expressed in matrix form as where A is an n x i matrix of constant coefficients, b is a vector of n constant terms, y is a vector of n variables, and y is a vector of n derivatives. Write the 2x2...

## VW C C2i2

This solution and its derivative in the expression for y lead to the following solution for y We found fhe solution to the homogeneous form of the system of two linear, first-order differential equations in definition 24.1. We turn our attention now lo the task Of finding particular solutions to that system so that we can add together the homogeneous and particular solutions to obtain the complete solutions. The particular solution we always look for in the case of autonomous differential...

## W Ii

We first show that if < 0 for all i, then d2y < 0. Since dx2 > 0 for any dx, and fa < 0 for all i, then every term in d2y is negative or zero, which proves the sufficiency part of the theorem (i.e., if all f < 0 then d2y < 0). We now need to show that the claim that d2y < 0 for any vector dx requires that J , < 0 for all i. First, suppose the contrary that is, suppose that one of the fa > 0. Without loss of generality, suppose that f > 0. Then choose a vector dx with dxi 0...

## What Is an Economic Model

At its most general, a model of anything is a representation. As such, a model differs from the original in some way such as scale, amount of detail, or degree of complexity, while at the same time preserving what is importani in the original in its broader or most salient aspects. The same is true of an economic model, though unlike model airplanes, our models do not take a physical form. Instead, we think of an economic model as a set of mathematical relationships between economic magnitudes....

## Y

To verify that this is a solution, check that it makes the difference equation true. To do this, first note that the solution implies that y,+ i 2'+l. But 2'+l is equal to 2'(2) but since y, 2', our solution says that y,+i 2y,. which is the same as the difference equation. where C is any arbitrary constant. To check that this. loo. is a solution, note that it implies that y,+i C2'+l. Bui writing 2,+l as 2'(2) makes this y(+, C2'(2). Using y, 2' makes this y,+( - 2.v,, which again is the...

## Y c

Write out the difference equation for aggregate income and solve it. What restriction must be placed on B to ensure that income converges monotonically to the steady-state equilibrium What is the short-run (one period) and the long-run (steady state) impact of an increase in on aggregate income S. Solve the difference equation for the modified cobweb model for the parameter values given in exercise 6, assuming that 0 0.6. 18.2 The General, Linear, First-Order Difference Equation If and h in the...

## Y yi yv

In theorem 23.4 we use this result to pull together all we have derived so far in this chapter. Theorem 23.4 The complete solution to the linear, autonomous, second-order linear differential equation (23.1) (constant coefficients and a constant term) is v(f) C,i'r ' + C-> enJ 4- if r, > 2 y(l) Qtf' + iC-> en + ifr, r- r a2 hi ifr y(l) c (A i cos VI + Ao sin irti - it roots are complex numbers where r . f2 i 2 J a - 4a2 2, h - a 2. and y J 4a2 - a 2. A Price-Adjustment Model with...

## Y2ye2

Linear, First-Order Difference Equations In the nexl three chapters we introduce some elementar)' techniques lor solving and analyzing the kinds of difference equations that are common in economics. We begin in this chapter with linear, first-order difference equations. In the next chapter we introduce nonlinear, first-order difference equations, including the famous logistic equation used extensively in the study of rlwos. In chapter 20 we examine linear, second-order difference equations....

## Behavior and Equilibrium

As we have just seen, a further step in building an economic model is to identify the behavioral equations, or the equations that describe the economic environment, and to identify the equilibrium conditions. In the simple supply-and-demand example above, the behavioral equations are the demand and supply functions describing the relationships between the endogenous variables and exogenous variables. The equilibrium condition determines what the values of the endogenous variables will be. In...

## Economic Applications of Continuous and Discontinuous Functions

There arc many natural examples of discontinuities from economics, In fact economists often adopt continuous functions to represent economic relationships when the use of discontinuous functions would be a more literal interpretation of reality. It is important to know when the simplifying assumption of continuity can be safely made for the sake of convenience and when it is likely to distort the true relationship between economic variables too much. Our first example illustrates a class of...