## 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

Figure 11.3 Some or the infinite number of palhs to approach a point inK2 Figure 11.3 Some or the infinite number of palhs to approach a point inK2 illustrated in figure 11.3, where the paths marked P . P2 P are clearly just a In section 4.1 we also considered an alternative definition of continuity. In definition 4.4. which states that a function is continuous at a point.v a if, within a small neighborhood of this point (i.e., for points close to x a), the function values J'(x) are close to...

## 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

Figure 11.5 Sections of the function v iy.i ' Note that (d) suppresses Figure 11.6 Effect of r on the derivative function i 5,V '' .> ) Figure 11.6 Effect of r on the derivative function i 5,V '' .> ) of labor falls as more labor is used. Thus this production function conforms to the law of diminishing marginal productivity of an input (see chapter 5, example 5.6). An analogous interpretation follows for the marginal product of input 2. The production function used in example 11.4 is a...

## 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...

## 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...

## 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

## 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...

## 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...

## 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...

## 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 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...

## 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...

## 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 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...

## 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....

## 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 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...

## 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

Where C, e'Assuming that the initial time is t 0 and that the initial values of income and debt are Y ) and A> respectively, we require )'(()) I'd C , Thus the solution to the initial-value problem for equation (21.13) is Substitution of this solution into equation (21.12) gives Although this is actually nonautonomous, it is in a form that can be solved by direct integration. Integrating both sides gives Since 0(0) Do, the value of Ci must be set to (Do - h gYu). Using this value, we have 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...

## 2 I I I oo39

Which is clearly a divergent series, This means that if one is willing to accept the idea that individuals value risky outcomes according to the expected value of the monetary outcome, then this gamble is preferred to a gain of 10 million for certain or any other finite amount, no matter how large. Bernoulli believed this was a ludicrous conclusion and offered a way out of the apparent paradox. An intuitive expUuiation of his argument is as follows. The vulue to an individual of different...

## Solutions by Substitution and Elimination

Perhaps the simplest analytical (rather than diagrammatic) ways ol solving these simple linear systems is by the substitution and elimination methods. Consider again the two equations represented in figure 7.1. These are The analytical solution by substitution is obtained simply by solving one of these equations for either x or v and substituting the result in the other equation. For example, we can solve equation (7.8) for .r to lind a- y + 1 and then substitute this value for x in equation...

## Chapter Review 697

The price of meal is 10 per pound and that of potatoes is I per pound. She has an income of 80. In addition to her budget constraint, she has a subsistence-calorie constraint she must consume at least 1,000 calories. One pound of potatoes yields 20 calories, one pound of meat yields 60 calories. Find her optimal consumption bundle. Now suppose that the pricc of potatoes rises to 1.60 per pound. Find the new optimal bundle. Explain your results, and discuss their...

## C fx i to Ai

Applying the envelope theorem directly gives Thus the Lagrange multiplier measures the rate at which the value function changes when the corresponding constraint is lightened or relaxed slightly. This interpretation of the Lagrange multiplier is of fundamental importance in economic applications of methods of constrained optimization. One implication is immediate if a constraint is nonbinding at the optimum, so that a small lightening or relaxing of it has no effect on the solution, then the...

## Rip x pim p 100 r

Jrip Rip - C p 100, - p - 2.500 - 25, I25p- p2 2.500 Then maximizing with respect to gt gives 7t' p 125 - 2p' 0 giving p 62.50, just as before. A monopolist has inverse demand function p 50 - 2x. The total-cost function is C 20 2.x 0.5x2. What are the profit-maximizing price and output TT .x 50x - 2x - 20 2x 0.5x2 48.r - 2.5.v2 - 20 x' 9.6 p 50 - 2 9.6 30.80 The level of profit at the maximum is then rtix 48 9.6 - 2.5 9.6 f - 20 210.40 See figure 6.10. A monopolist has inverse demand function p...

## Mathematics For Economic Analysis

Part I Introduction and Fundamentais 1.1 What Is an Economic Model 3 2.3 Some Properties of Point Sets in R 33 2.5 Proof, Necessary and Sufficient Conditions 60 Chapter 3 3.1 Definition of a Sequence 67 3.3 Present-Value Calculations 75 Part It Univariate Calculus and Optimization 4.1 Continuity of a Function of One Variable 115 4.2 Economic Applications of Continuous and Discontinuous Functions 125 4.3 Intermediate-Value Theorem 143 The Derivative and Differential for Functions of One Variable...

## Exercises

For the function u x . a2 5.V 3.t2 a Find the total differential. b Draw the level curve for u 120. c Use the pair of points 12. 20 and 18, 10. lo illustrate thai the MRS 5 3 and derive this result from the total differential in part a . 2. For the. function u xi. .r2 ax gt x2. a Find the total differential. b Draw a representative level curve for u. c Use the expression for the total differential to illustrate that the MRS a b. 3. Use the total differential lo hnd the MRTS for the...

## Ifffl 0 ihI 0Ii ll j o

In this case dry lt 0 and so . ' is concave. Moreover, if is concave this set of conditions must hold. The following examples illustrate how to use the results in theorem 11.9 to determine the concavity convexity properties of a function. Example 11.28 Use theorem 11.9 to determine the convexity concavity property of the function y jci , x2 x x2 l2 defined on x 6 R . The second-order partial derivatives are Since all oi these are negative, we check first for strict concavity of Note...

## Marginal Rate Of Technical Substitution And Monotonic Transformation

Addilively separable function bordered Hessian cross-partial derivatives elasticity of substitution Euler's theorem first-order total differential gradient vector Hessian matrix homogeneous function homothetic function implicit differentiation implicit function theorem indifference curves marginal rate of substitution MRS marginal rate of technical substitution positive monotonic transformation remainder formula second-order total differential Taylor series Young's theorem 2. Why is...

## Bakery Advertises Its Bagels By Noting The Price Per Dozen

Find the slope of each of the following production functions, y f L . Graph he functions and their derivative functions. Give the economic significance of the sign of the slope of the derivative functions i.e whether the derivative is increasing or decreasing in L . a y 10L lt b v 8JLl 3 c y 3LA 2. Find the slope of each of the following production functions, y ' L . Graph the functions and their derivative functions. Give the economic significance of the sign of the slope of the derivative...

## Concavity Convexity Quasiconcavity Quasiconvexity

In our description of some specific functions we used the terms convexity and concavity. Visually the meaning should be clear, but we now present a formal definition. Figure 2.29 shows how we proceed in the case of a concave function. First we must assume that the domain of the function is a convex set. because we want convex combinations of points in the domain to be in the domain. Take any two points x' and x i n the domain of the function and the corresponding function values f x' and f x ....

## Z crfp p

Where p is the input price, and a, b. u. p gt 0. Find the profit-maximizing price and quantity of the input the monopsonist will choose, and compare the analysis to that of the profil-maximizing monopoly. 5. A firm has the production function x f L , where x is output and L is labor input. The linn buys the input in a competitive market. a Assuming the linn sells its output in a competitive market, show that setting output where price equals marginal cost is equivalent to setting labor input...

## By 05x70fx5V2

X. a Given the strictly quasiconcave function y f xt.x2 . sketch a typical level set in each of the following cases i The function is increasing in x and decreasing in xj. ii The function is decreasing in X and increasing in x . iii The function is decreasing in both variables Him First determine which way the curve of the level set must slope, then identify the area that gives the better set, and then find how the curvature must look to make the better set convex. fb Repeat part a , assuming...

## Rule 2 Derivative of a Linear Function fx mx b

If fix nix b, with m and b constants, then fix in. Figure 5.20 Linear demand has a constant slope example 5.5 Figure 5.20 Linear demand has a constant slope example 5.5 This result follows because Ay fix Ax - J' x - mix A.v b- m x b m A.v. Then Ay Ax m. and so lima, o Ay Ax in. For example, the derivative of the function v 3.v 5 is fix 3, The important implication of this result is that for a linear function the rate at which the variahle y changes with respect to a change in x is the same at...

## Unconstrained Maxima and Minima

Given some function i.e., y . , we optimize it by finding a value of x at which it takes on a maximum or minimum value. Such values are called extreme values of the function. If the set of v-values from which we can choose is the entire real line, the problem of finding an extreme value is unconstrained, while if the set of.r-values is restricted to be a proper subset of the real line, the problem is constrained. To begin with we consider only unconstrained problems. We also assume that the...

## Rectangular Hyperbola

A rectangular hyperbola may be written for some positive constant a. The name stems from the fact that every rectangle drawn to the curve has the same area a. Note that the graph of the function in figure 2.25 has two parts, one entirely in the positive quadrant and the other entirely in the negative quadrant. In economics we often restrict x to so only Figure 2.25 Rectangular the upper curve is relevant. As x tends to zero, the curve approaches the y-axis hyperbola asymptotically, and as x...