V 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
t 2 t-2 t 2 0.25 4. 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...
Exponential Functions
A quantity that increases or decreases by a fixed factor per unit of time is said to increase or decrease exponentially. If this fixed factor is a. this leads to the study of the exponential function defined by where a and A are positive constants. Note that if f t Aa', then f t 1 Aar Aa a1 af t , so the value of at time t 1 is a times the value of at time t. If a gt 1. then is increasing if 0 lt a lt 1, then is decreasing. Because 0 Aa A, ye can write f t f 0 a'. Exponential functions appear...
Linear Models
Linear relations occur frequently in applied models. The relationship between the Celsius and Fahrenheit temperature scales is an example of an exact linear relation between two variables. Most of the linear models in economics are approximations to more complicated models. Two typical relations are those shown in Example 2.12. Statistical methods have been devised to construct linear func tions that approximate the actual data as closely as possible. Let us consider a very naive attempt to...
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...
Example 219
Some other economic examples of linear functions are the following demand and supply schedules Here a and b both positive are parameters of the demand function D, while a and ft both positive are parameters of the supply function. Such functions play an important role in quantitative economics. It is often the case that the market for a particular commodity, such as a specific brand of 3 -inch computer diskettes, can be represented approximately by linear demand and supply functions. The...
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...
Set Theory
If you know set theory up to the hilt, and no other mathematics, you would be of no use to anybody. If you knew a lot of mathematics, but no set theory, you might achieve a great deal. But if you knew just some set theory, you would have afar better understanding of the language of mathematics. I. Stewart 1975 In daily life, we constantly group together objects of the same kind. For instance, we refer to the university faculty to signify all the members of the academic staff at the university....
Example 45
Consider a firm producing some commodity in a given period. Let C x cost of producing x units R x revenue from selling x units tc x R x C x profit from producing and selling x units We call C' x the marginal cost at x , R' x the marginal revenue, and n' x the marginal profit. Economists often use the word marginal in this way in order to signify a derivative. Other examples of the derivative in economics include the following. The marginal propensity to consume is the derivative of the...
Solution 1
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...
Polynomial Division
One can divide polynomials in much the same way as one divides numbers. Consider first a simple numerical example 2735 -r 5 500 40 7 2500 235 200 Hence, 2735 5 547. Note that the horizontal lines instruct you to subtract the numbers above the lines. You might be more accustomed to a different way of arranging the numbers, but the idea is the same. Consider next -x3 4x2 -6 c-2 - c3 2x2 You can omit the boxes, but they should help you to see what is going on. We conclude that -x3 4.x2 x 6 x 2 x2...
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...