## Continuity

The word continuous is rather common in everyday language. We use it, in particular, to characterize changes that are gradual rather than sudden. This usage is closely related to the idea of a continuous function. Roughly speaking, a function is continuous if small changes in the independent variable produce small changes in the function values. Geometrically, a function is continuous if its graph is connected that is, it has no breaks. An example is indicated in Fig. 6.7. It is often said that...

## Note on Income Distribution and Lorenz Curves

In Section 10.4 it was explained how, if f(r) is the income distribution function for a population of n individuals, then n f fir) dr represents the number of individuals with incomes in the interval a. b see Equation 10.27 , In addition, n J r f(r)dr represents the total income of these individuals see Equation 10.28 . 2This function, its bell-shaped graph, and a portrait of its inventor Carl Friedrich Gauss (1777 1855). appear on the German 10-mark currency note issued in early 1989. A...

## Quadratic Functions

Many economic models involve functions that either decrease down to some minimum value and then increase, or else increase up to some maximum value and then decrease. Simple functions with this property are the general quadratic functions f(x) ax2 + bx +c (a, b, and are c constants, a 0) 3.1 (If a 0, the function is linear, hence, the restriction a 0.) Figure 2.20 of Section 2.4 shows the graph of f(x) x2 3x, which is obtained from 3.1 by choosing a 1, b 3, and c 0. In general, the graph of...

## Limits Continuity and Series

We could, of course, dismiss the rigorous proof as being superfluous if a theorem is geometrically obvious why prove it This was exactly the attitude taken in the eighteenth century. The result, in the nineteenth century, was chaos and confusion for intuition, unsupported by logic, habitually assumes that everything is much nicer behaved than it really is. This chapter is concerned with limits, continuity, and series key ideas in mathematics, and also very important in the application of...

## Example 315 Radioactive Decay

Measurements indicate that radioactive materials decay by a fixed percentage per unit of time. Plutonium 239, which is a waste product of certain nuclear power plants and is used in the production of nuclear weapons, decays by 50 every 24,400 years. We say, therefore, that the half-life of plutonium 239 is 24,400 years. If there are Iq units of plutonium 239 at time t 0, then after t years, there will be I(t) Iq Q)7 24'400 o 0.9999716' units remaining. (Observe that this is consistent with...

## The Natural Logarithmic Function

In Section 3.5, the doubling time of an exponential function f(t) a , with a > 1, was defined as the time it takes for fit) to become twice as large. In order to find the doubling time t*, we must solve the equation ar 2 for t*. In economics, we often need to solve similar problems 1. At the present rate of inflation, how long will it take the price level to triple 2. If the national debt of the U.S. continues to grow at the present proportional rate, how long will it take to reach 10...

## Why Economists Use Mathematics

Economic activity has been part of human life for thousands of years. The word economics itself originates from a classical Greek word meaning household management. Even before the Greeks there were merchants and traders who exhibited an understanding of some economic phenomena they knew, for instance, that a poor harvest would increase the price of com, but that a shortage of gold might result in a decrease in the price of corn. For many centuries, the most basic economic concepts were...

## Ih

Hence, * or * I y f f The two solutions are, therefore, * + and * I - I As demonstrated in Example A.24, we see that 3 can be written as 12(x2-tx- ) 12(x-l)(x + i) 4 Check that this is correct by expanding the right-hand side. We will now apply the method of completing the square to the general quadratic equation A.31 . We begin by taking the nonzero coefficient of x2 outside the parentheses so that A.31 becomes Because a 0 this equation has the same solutions as One-half of the coefficient of...

## Partial Derivatives with Two Variables

When we study a function y fix) of one variable, the derivative fix) measures the function's rate of change as x changes. For functions of two variables, such as z fix,y), we also want to examine how quickly the value of the function changes with respect to changes in the values of the independent variables. For example, if fix, y) is a firm's profit when it uses quantities x and > of two different inputs, we want to know whether and by how much profits increase as x and > are varied....

## The Domain and the Range

The definition of a function is incomplete unless its domain has been specified. The domain of the function defined by f(x) (see Example 2.1) is the set of all real numbers. In Example 2.2, where C(x) QOxJx + 500 denotes the cost of producing x units of a product, the domain was not specified, but the natural domain is the set of numbers 0, 1, 2, x0, where xo is the maximum number of items the firm can produce. If output x is a continuous variable, the natural domain is the closed interval 0....

## First Derivative Test for Extreme Points

In many important cases, we can find maximum or minimum values for a function just by studying the sign of its first derivative. Suppose f(x) is differentiate on an interval I and suppose f(x) has only one stationary point, x c. If FIGURE 9.5 Point * c is a maximum point. FIGURE 9.5 Point * c is a maximum point. FIGURE 9.6 Point x d is a minimum point f'(x) > 0 for all x e.I such that x < c, whereas f'(x) < 0 for all x e I such that x > c, then fix) is increasing to the left of c and...

## Functions of Two or More Variables

We begin with the following definition A function of two variables x and y with domain D is a rule that assigns a specified number fix, y) to each point (x, y) in D. Consider the function that, to every pair of numbers (x, y), assigns the number 2x + x2y3. The function is thus defined by What are (1,0), (0, 1), (-2, 3), and (a + 1, b) Solution (l, 0) 2 - 1 + l2 03 2, (0,1) 2 - 0 + 02 l3 0, and ( 2,3) 2(-2) + (-2)2 33 -4+ 4 27 104. Finally, we find f(a +1, b) by replacing x with a +1 and y with...

## Integrals of Unbounded Functions

We turn next to improper integrals where the integrand is not bounded. Consider first the function x I -fx, with x e 0, 2 . See Fig. 11.5. Note that x oo as x 0 . The function is continuous in the interval h. 2 FIGURE 11.5 The height of the domain is unbounded, but y 1 Vx approaches the y-axis so quickly that the total area is finite. for an arbitrary fixed number h in 0, 2 . Therefore, the definite integral of over the interval h. 2 exists, and 2y x 2-J2-2 Jh The limit of this expression as h...

## Applications of Exponentials and Logarithms

Suppose that f t denotes the stock of some quantity at time t. The ratio f' t f t is the relative or proportional rate of increase of the stock at time t. In many applications, the relative rate of increase is a constant r. Then f' t rf t for alii 8.21 Which functions have a constant relative rate of increase Functions of the type f t Aert certainly have, because fit Aren rf t . We claim that there are no other functions having this property. Suppose that g is any function satisfying g' t rg t...

## Y jc2 lx 15 b yx c xpx

If Y is a function of K, and AT is a function of r, find the formula for the derivative of Y with respect to t at t to. 5. If y F K and K h t , find the formula for dY dt. 6. Compute dx dp for the demand function x b -Jap c a, b, and c are positive constants where x is the number of units demanded, and p is the price per unit, with P gt c a. 7. If h x f x2 , find a formula for h'ix . 8. Let sit be the distance in kilometers traveled by a car in t hours. Let B s be the number of liters of...

## Partial Derivatives in Economics

This section considers a number of economic examples of partial derivatives. Example 15.20 Consider an agricultural production function Y F K, L, 7 , where Y is the number of units produced, K capital invested, L labor input, and T the area of agricultural land that is used. Then dY 3 K F'K is called the marginal product of capital. It is the rate of change of output Y with respect to K when L and T are held fixed. Similarly, dY dL F and dY dT F'T are the marginal products of labor and of land,...

## Geometric Representations of Functions of Several Variables

This section considers how to visualize functions of several variables, in particular functions of two variables. An equation such as f x,y c in two Variables x and y can be represented by a point set in the plane, called the graph of the equation. In a similar way, an equation g x, y, z c in three variables x, y, and z can be represented by a point set in 3-space, also called the graph of the equation. For a discussion of 3-space, see Section 12.3. This graph consists of all triples x, y. z...

## Example 222

A person has m to spend on the purchase of two commodities. The prices of the two commodities are and lt per unit. Suppose x units of the first FIGURE 2.40 , y 2x y. lt 4 . commodity and y units of the second commodity are bought. Assuming one cannot purchase negative units of x and gt the budget set is as in 1.7 in Section 1.7. Sketch the budget set B in the -plane. Find the slope of the budget line px qy m, and its points of intersection with the two coordinate axes. Solution The set of...

## 100 J1 11100 111 y

So the total present value of the three payments, which is the total amount A that must be deposited today in order to cover all three payments, is given by 1000 1500 2000 A 77 7 The total is approximately 900.90 1217.43 1462.38 3580.71. Suppose now that h successive payments a , , an are to be made, with a being paid after 1 year, lt 22 after 2 years, and so on. How much must be deposited into an account today in order to have enough savings to cover all these future payments, given that the...

## W jtj f iA

If x 3 7 and y 1 14, find the simplest forms of these fractions x x-y 13 2 -3y a. x y b. c. --d. - 5. Reduce the following expressions by making the denominators rational xVy gt v Vx h - y x I -jx 1 a. I---3 u. --- -- c. -------- x x-I 2t 1 2t - 1 x 2 2-x x2-A 7. Prove that x2 2xy - 3y2 x 3y t y , and then simplify x2 Ixy 3 y- x y x 3y 8. Simplify the following expressions 9. Simplify the following expressions 1 xP-v 1 x -P a b 10. Reduce the following fractions 25a2b2 x2 y2...

## Compound Interest and Present Discounted Values

Equation 8.21 , f' t rf t for all t, has a particularly important application to economics. After t years, a deposit of K earning interest at the rate p per year will increase to see Section A.l, Appendix A . Each year the principal increases by the factor 1 r. Formula 1 assumes that the interest is added to the principal at the end of each year. Suppose instead that payment of interest is offered each half year, but at an interest rate p 2. Then the principal after 1 2 year will have increased...

## Present Discounted Value of a Continuous Future Income Stream

Section 6.6 discussed the present value of a series of future payments made at specific discrete moments in time. It is often more natural to consider revenue as accruing continuously, such as the proceeds from a large growing forest. Suppose that income is to be received continuously from time t 0 to time t T at the rate of t dollars per year at time t. We assume that interest is compounded continuously at rate r. Let P t denote the present discounted value of all payments made over the time...

## More on Concave and Convex Functions

So far convexity and concavity have been defined only for functions that are twice differentiate. An alternative geometric characterization of convexity and concavity suggests a more general definition that is valid even for functions that are not differentiate. It is also easier to extend this new generalized definition to functions of several variables. Function is called concave convex if the line segment joining any two points on the graph is never above below the graph. These definitions...

## Examples of Quadratic Optimization Problems

Much of mathematical economics is concerned with optimization problems. Economics, after all. is the science of choice, and optimization problems are the form in which choice is usually expressed mathematically. A general discussion of such problems must be postponed until we have developed the necessary tools from calculus. Here we show how the simple results from the previous section on maximizing quadratic functions can be used to illustrate some basic economic ideas. Consider a firm that is...

## Convex and Concave Functions and Inflection Points

What can be learnt from the sign of the second derivative Recall how the sign of the first derivative determines whether a function is increasing or decreasing fix gt 0 on a, b lt gt fix is increasing on ia, b 1 fix lt 0 on a, b lt gt f x is decreasing on a. b 2 The second derivative fix is the derivative of fix . Hence fix gt 0 on ia, b fix is increasing on ia, b 3 fix lt 0 on a, b fix is decreasing on ia, b 4 The equivalence in 3 is illustrated in Fig. 9.14. The slope of the tangent, fix , is...

## Solution

a x2 -4x 3 ix2 -4x 3 ix2 - 4x 4 -4 3 ix - 2 2 - The expression x 2 2 1 attains its smallest value, which is 1, at x 2. b 2x2 40x - 600 -2ix2 - 20x - 600 -2ix2 - 20x 100 200 - 600 -2ix - 10 2 - 400 The expression 2ix 10 2 400 attains its largest value, which is -400, at x 10. x2 2 C 1 - - i x l 2-3 The expression j x l 2 3 attains its smallest value, which is 3, at x -1. A useful exercise is to solve the three cases in Example 3.1 by using the expressions set out in 3.4 directly, substituting...

## Example 314 Compound Interest

A savings account of SK that increases by p interest each year will have increased after t years to see Section A.1 of Appendix A . According to this formula. 1 K 1 earning interest at 8 per annum p 8 will have increased after t years to 1 8 100 ' 1.08' 2 Table 3.1 indicates how this dollar grows over time TABLE 3.1 How 1 of savings increases with time TABLE 3.1 How 1 of savings increases with time After 30 years, 1 of savings has increased to more than 10, and after 200 years, it has grown to...

## Set Operations

Sets can be combined in many different ways. Especially important are three operations union, intersection, and the difference of sets, as shown in Table 1.2. Notation Name The set consists of AuB A union B The elements that belong to at least An S A intersection B The elements that belong to both A B A minus B The elements that belong to A, AU B x x e A or x B AH B x x A andx B A B x x A and x B Let A 1,2,3,4,5 and B 3,6 . Find A US, A n A B, and B A. Solution AUB 1.2,3,4,5,6 , A n 3 , A B...

## Peter J Hammond

PRENTICE HALL, Upper Saddle River, New Jersey 07458 Library of Congress Cataloging-in-Publication Data Mathematics for economic analysis Knut Sydsaster, Peter J. Hammond, p. cm. Includes bibliographical references and index. ISBN 0-13-583600-X 1. Economics, Mathematical. 2. Economics-Mathematical models. I. Hammond, Peter J. H Title. HB135.S888 1995 Production Editor Lisa Kinne Acquisitions Editor J. Stephen Dietrich Copy Editor Peter Zurita Cover Designer Maureen Eide Manufacturing Buyers...

## Deductive vs Inductive Reasoning

The three methods of proof just outlined are all examples of deductive reasoning, that is, reasoning based on consistent rules of logic. In contrast, many branches of science use inductive reasoning. This process draws general conclusions based only on a few or even many observations. For example, the statement that the price level has increased every year for the last n years therefore, it will surely increase next year too, demonstrates inductive reasoning. Owners of houses in California know...

## Power Functions

Consider the power function defined by the formula We know the meaning of xr if r is any integer that is. r 0, 1 2,____In fact if r is a natural number, xr is the product of r x's. Also if r 0, then xr 1 for all x 0, and if r n, then xr 1 x for x 0. In addition, for r 1 2, xr x 2 y x, defined for all x gt 0. See Section A.2 of Appendix A. This section extends the definition of xr so that it has meaning for any rational number r. Here are some examples of why powers with rational exponents are...

## Solving Equations

We shall now give examples showing how using implication and equivalence arrows can help avoid mistakes in solving equations like that in Example 1.6. Find all x such that 2x - l 2 - 3x2 2 - Ax . Solution By expanding 2x l 2 and also multiplying out the right-hand side, we obtain a new equation that obviously has the same solutions as the original one 2x - l 2 - 3x2 2 i - 4 4x2 - 4x 1 - 3x2 1 - 8 Adding 8 t 1 to each side of the second equality and then gathering terms gives the equivalent...