Instructor’s Manual Essential Mathematics for Economic Analysis rd
3
edition
Knut Sydsæter Arne Strøm Peter Hammond
For further instructor material please visit:
www.pearsoned.co.uk/sydsaeter
Preface This instructor’s manual accompanies Essential Mathematics for Economic Analysis, 3rd Edition, FT/Prentice Hall, 2008. Its main purpose is to provide instructors with a collection of problems that might be used for tutorials and exams. Most of the problems are taken from previous exams and problem sets at the Department of Economics, University of Oslo, and at Stanford University. We have endeavoured to select problems of varying difficulty. Many of them are very routine, but we have also included some problems that might challenge even the best students. The number in parentheses after each problem indicates the appropriate section of the text that should be covered before attempting the (whole) problem. The chapters before Chapter 10 in this manual do not follow the chapters in the book, but are rather ordered according to popular areas in which exams are given. For each chapter we offer some comments on the text. Sometimes we explain why certain topics are included and others are excluded. There are also occasional hints based on our experience of teaching the material. In some cases, we also comment on alternative approaches, sometimes with mild criticism of other ways of dealing with the material that we believe to be less suitable. Chapters 1 and 2 in the main text review elementary algebra. This manual includes a Test I (page 100), designed for the students themselves to see if they need to review particular sections of Chapters 1 and 2. Many students using our text will probably have some background in calculus. The accompanying Test II (page 102) is designed to give information to both the students and the instructors about what students actually know about single variable calculus, and about what needs to be studied more closely, perhaps in Chapters 6 to 9 of the text. Oslo and Coventry, July 2008 Knut Sydsæter, Arne Strøm, and Peter Hammond Contact addresses:
[email protected] [email protected] [email protected]
Version 1.0 21072008 1394 © Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
Contents 1 Elementary Algebra. Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Functions of One Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Implicit Differentiation. Linear and Quadratic Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5 Single-Variable Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 7 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 8 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 9 Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 10 Interest Rates and Present Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 11 Functions of Many Variables. Elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 12 Tools for Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 13 Multivariable Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 14 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 15 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 16 Determinants and Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 17 Linear Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Answers to the problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Test I (Elementary Algebra) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Test II (Elementary Mathematics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Answers to Test I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Answers to Test II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
© Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
CHAPTER 1
ELEMENTARY ALGEBRA. EQUATIONS
1
Chapter 1 Elementary Algebra. Equations The main purpose of Chapters 1 and 2 in the text is to help those students who need to review elementary algebra. (Those who never learned it will need more intensive help than a text of this kind can provide.) We highly recommend instructors to test the elementary algebra level of the students at the outset of the course, using Test 1 on page 100, or something simpler. Reports we have received suggest that instructors who are not used to giving such tests, sometimes have been shocked by the results when they do, and have had to adjust the start of their course accordingly. (In a short test in a basic course in mathematics for economists in Oslo, among 180 students there were 30 different answers to (a + 2b)2 = ?.) But we do feel that reviewing elementary algebra should primarily be left to the individual students. That’s why the text supplies a rather extensive review with many problems. We recommend illustrating power rules (also with negative exponents) with compound interest calculations (as in Sections 1.2 in the text), which are needed by economics students anyway. We often encounter students who have a purely memory based, mechanistic approach to the algebraic rules reviewed in Section 1.3. A surprisingly large number of students seem unaware of how algebraic rules can be illustrated in the way we do in Fig. 1.3.1. We find the sign diagrams introduced in Section 1.6 to be useful devices for seeing when certain products or quotients are positive, and when they are negative. Alternative ways of solving such problems can be used, of course. Mathematics for economic analysis often involves solving equations, which is the topic of Chapter 2. In particular, it is important to train the students in handling equations with parameters, which arise in so many economic applications. (We often see that students being used to equations involving x and y have problems when the variables are Y , C etc.) The examples and problems in Section 2.5 dealing with some types of nonlinear equation which frequently occur in optimization problems, can be postponed until the techniques for solving them are needed. In Chapter 3 we discuss some miscellaneous topics that are not so often featured in exam papers. However, in Chapter 9 of this Manual we give some problems involving sums. Set theory, in our opinion, is not crucial for economics students, except when the need for it arises in their statistics courses. Problem 1-1 (1.2) Find the solution x of the following equations: (a) 52 · 5x = 57 (d) (25)2 = 5x (e) 210 − 22 · 2x = 0 (f) (x + 3)2 = x 2 + 32
(b) 10x = 1
(c) 10x ÷ 105 = 10−2
Problem 1-2 (1.2) (a) A person buys x1 , x2 , and x3 units of three goods whose prices per unit are p1 , p2 , and p3 respectively. What is the total expenditure? (b) A rental car costs F dollars per day in fixed charges and b dollars per mile. How much must a customer pay to drive x miles in 1 day? (c) A company has fixed costs of F dollars per year and variable costs of c dollars per unit produced. Find an expression for the total cost per unit (total average cost) incurred by the company if it produces x units in one year. (d) A person has an annual salary of $L and then receives a raise of p% followed by a further increase of q%. What is the person’s new yearly salary? © Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
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ELEMENTARY ALGEBRA. EQUATIONS
Problem 1-3 (1.3) Which of the following equalities are correct? (a) 35 = 53 (b) (52 )3 = 52 2x + 4 (d) (5 + 7)2 = 52 + 72 (e) = x + 4 (f) 2(x − y) = x · 2 − y · 2 2
3
(c) (33 )4 = (34 )3
Problem 1-4 (1.5)
125−2/3 (b) 5−3
5562 − 5552 Simplify: (a) 1111
(c)
2 1 − 2 3 6
−1 (d)
x α/2 y −β/3 zγ (x −α y 8β/3 z2γ )−1/2
(c)
(p α q −β/2 )2 (p−2α/3 q 4β/3 )−3/2
Problem 1-5 (1.5) Simplify: (a)
896 · 897 − 897 895
(b)
1 1− 1 2
+
1 1− 1 4
+
1 1+ 1 2
Problem 1-6 (1.5) Simplify: (a) 210 (32)−9/5
(b)
√ 52 + 122 − 10
(c)
(a 3c )−1 a 3c a −5c (a 2c )2
(d)
1 1 1 + + 2 1+x 1−x 1−x
Problem 1-7 (1.5) Simplify: (a) 40 − (0.4)−1 +
3 2
+ 4 · 4−1
(b) 64 · 32−3/5
(c)
2 2 8 + − x 2 − 4x x x−4
Problem 1-8 (1.5) The surface area S and the volume V of a sphere of radius r are respectively, S = 4πr 2 and V = 43 πr 3 . (a) Express S in terms of V by eliminating r. (b) A sphere of capacity 100 m3 is to have its outside surface painted. One liter of paint covers 5 m2 . How many liters of paint are needed?
Problem 1-9 (2.2) Solve the following equations: a −2 − b−2 x x (a) −1 for = + −1 a b a −b
x
(b) Y = I + a(Y − (c + dY ))
Problem 1-10 (2.4) If 23x = 16y+1 and 2x = 5y − 2, what is x + y? © Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
for
Y
CHAPTER 2
FUNCTIONS OF ONE VARIABLE
3
Problem 1-11 (2.2) Consider the system of equations (i) Y = C + A∗ − M
(ii) C = aY + b
(iii) M = mY + M ∗
Express Y in terms of A∗ , M ∗ and the constants a, b, and m.
Problem 1-12 (2.4) Solve the following equations: (a) 10r 2 − 3r + 0.2 = 0
(b)
5P + 6Q = 28 8P − 3Q = 7
(c) (x 2 − 4)(x 2 − 25) = 0
Problem 1-13 (2.5) Solve the following equations (systems): (a) x(16 − x 2 ) = 0
(b) 52p−1 = 125−p
(c)
P + 2Q = 1 P 2 + Q2 = 2
(d)
9x − xy 2 = 0 3x 2 − 2y = 0
Problem 1-14 (4.10) 2
(i) ln(9 · 4) − 2 ln(3 + 3) (ii) ln 32 − 4 ln 2 − eln(ln 2) (iii) eln(e ) − ln 2 (iv) e5 ln 3−ln 2 √ 2 2 (b) Solve for x: (i) ex −3x+1/2 = e (ii) (ex − 1)(ex + 1) = 3 (iii) 8ln x = 64 (iv) x −γ = 2 (v) ln(5 − x 2 ) = ln(x 2 + 4x − 11) (vi) e2t + 3et = 10 (a) Compute
Chapter 2 Functions of One Variable In this chapter of the manual, we collect problems on functions of one variable, referring to chapters 4–6 in the text. In our experience, training students to master functional notation is very important, yet sometimes neglected in introductory texts. In particular, understanding the definition of the derivative requires such an understanding, and in some sense it is more important for economists than the ability to differentiate complicated functions. Economic interpretations of the derivative like in Examples 6.4.3, Problems 6.4.5 and 6.R.4 should be stressed. In order to have students see mathematics applied as soon as possible to economic optimization problems, we consider in Section 4.6 a monopolist with a quadratic cost function who faces a linear (inverse) demand function. The perfectly competitive firm is treated as a special case. It is shown that, in order that the profit maximum for a monopolist should coincide with that for a competitive firm, its output should be subsidized (rather than taxed). (Alternatively, it should have its price regulated.) General polynomials and polynomial division might perhaps be regarded as optional, but power functions are definitely crucial So are exponential and logarithmic functions, of course. © Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
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CHAPTER 2
FUNCTIONS OF ONE VARIABLE
Section 5.1 which examines how changes in a function relate to shifts in its graph, with application to shifting demand and supply curves is useful for economic students. Section 5.3 on inverse functions might be postponed till Section 7.3 on differentiating the inverse. Our treatment of the single variable calculus begins in Chapter 6 and is rather standard. As already indicated, one should emphasis economic interpretations of the derivative. Limits were used to define derivatives informally in Section 6.2. The more careful discussion that is really needed is the topic of Section 6.5, though even this remains rather informal. In fact, Chapter 6 almost completes the inventory of functions of a single variable used in this book, and in most mathematical work in undergraduate economics. The major omission is the family of trigonometric functions discussed in FMEA. In economics, they are used almost exclusively to solve second- and higherorder difference and differential equations, which are topics in FMEA. Problem 2-1 (4.3) Figure 2-1 below shows the graphs of a quadratic function f and a linear function g. Use the graphs to find for which x the following hold: (i) f (x) ≤ g(x) (ii) f (x) ≤ 0 (iii) g(x) ≥ 0. y 6 5 4 3 2 1 −3
−2
−1
−1 −2 −3 −4
y = g(x)
1
2
3
4
x
y = f (x)
Figure 2-1
Problem 2-2 (4.2, 6.4) The cost of producing x units of a commodity is given by C(x) = x 2 + x + 100. (a) Compute C(0), C(100), and C(101) − C(100). (b) Compute the incremental cost C(x + 1) − C(x), and explain in words its meaning. (c) Compute the marginal cost C (x) and C (100). What is the difference between C (x) and C(x+1)−C(x)?
Problem 2-3 (4.6) The price P per unit obtained by a firm in producing and selling Q units of a commodity is P = 52 − 4Q, and the cost of producing and selling Q units is C = Q + 41 Q2 . (a) What is the profit function π(Q)? (b) Find the value of Q that maximizes profits. © Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
CHAPTER 2
FUNCTIONS OF ONE VARIABLE
5
Problem 2-4 (4.5) (a) Consider the demand and supply curves D = 200 − 41 P , S = 20 + 2P . Find the equilibrium price P ∗ , and the corresponding quantity Q∗ . (b) Suppose a tax of $0.25 per unit is imposed on the producer. How will this influence the equilibrium price? ˆ Compute the total revenue obtained by the producer before the tax is imposed (R ∗ ) and after (R).
Problem 2-5 (4.7) Figure 2-5 displays the graphs of two functions f and g. y
y
x
y = f (x) =
x
ax + b x+c
y = g(x) = px 2 + qx + r Figure 2-5
Specify (if possible) which of the constants a, b, c, p, q, and r are > 0, = 0 or < 0. Problem 2-6 (6.6) Differentiate: (a) y = 2x − 5 (f) y = x 5 − x −5
(b) y = 31 x 9
1 10 (c) y = 1 − 10 x 1 1 (g) y = 41 x 4 + 13 x 3 + 21 52 (h) y = + 3 x x
(d) y = 3x 7 + 8
(e) y =
x−5 10
Problem 2-7 (6.8) Differentiate: (a) y = x 12 (b) y = x − x 2 − x 3 √ (e) y = 2x 2 + 10 (f) h(L) = (La + b)d
(c) f (x) = 21 x 6 − 4x 2 + 3
(d) G(t) =
2t + 1 t2 + 3
Problem 2-8 (6.4) If x(t) denotes the number of litres of petrol left in a car at time t, with time measured in hours, what is the interpretation of x(0) ˙ = −0.75? Problem 2-9 (5.2) ax + b Let f (x) = , where a, b, and c are constants, and c = 0. Assuming that x = a/c, show that cx − a f (f (x)) = x. © Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
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CHAPTER 2
FUNCTIONS OF ONE VARIABLE
Problem 2-10 (6.2) Suppose f (0) = 2.5 and that the derivative of f has the linear graph shown in Fig. 2-10. Find a formula for f (x) and sketch its graph. (First find a formula for f (x).) y 3 2
f (x)
1 −1
1
2
3
x
Figure 2-10
Problem 2-11 (6.9) Mediana is a small nation where the number f (x) of digital TV sets (in millions) in year x is expected to grow according to the formula 1.5 , x≥0 f (x) = 2 − x+1 (a) Find f (x) and f (x). (b) Sketch the graph of f . What is the limit of f (x) as x → ∞? Problem 2-12 (6.9) (a) Let g(x) = 3x 3 − 15 x 5 . Find g (x) and g (x). (b) Check where g is increasing and where it is concave. (c) Prove that g(−x) = −g(x). What does this mean geometrically? (d) Sketch the graph of g. Problem 2-13 (6.7) (a) The function f is defined by f (x) = (b) Show that x 2 − 2x + 4 is never 0.
x2
x for all x. Compute f (−2), f (1), and f (5). − 2x + 4
(c) Compute f (x) and find where f (x) is 0. (d) Let P be the point on the graph corresponding to x = 1. Find the equation for the tangent to the graph at this point. (e) Sketch the graph of f . Problem 2-14 (6.7) Let a, b, m, and n be fixed numbers, where a < b, and m and n are positive. Define the function f for all x by f (x) = (x − a)m · (x − b)n . For the equation f (x) = 0, find a solution x0 that lies between a and b. © Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
CHAPTER 2
7
FUNCTIONS OF ONE VARIABLE
Problem 2-15 (6-11) Consider the production function which, for fixed inputs N and K, depends on the parameter α as follows: F (α) = a
NvKv (N α + bK α )v/α
(a, b, v, N , and K are positive constants)
Use logarithmic differentiation to find an expression for F (α).
Problem 2-16 (6.11) Consider the function f defined by f (x) = x(ln x)2 , x > 0. (a) Calculate f (x) and f (x). (b) Determine where f is increasing and where f is decreasing.
Problem 2-17 (4.2) Let f (x) = |x|. Which of the following statements are true for all possible pairs of numbers x and y? (a) f (x + y) = f (x) + f (y) (b) f (x + y) ≤ f (x) + f (y) (c) f (xy) = f (x) · f (y) (d) f (2x) = 2f (x) √ (e) f (−2x) = −2f (x) (f) f (x) = x 2 (g) f (−2x) = 2f (x) (h) |f (x) − f (y)| ≤ |x − y|
Problem 2-18 (4.10) All organic material contains the stable isotope carbon 12 and a little of the radioactive isotope carbon 14. The ratio between the quantities of carbon 14 and carbon 12 is constant in living organisms, and seems to have been constant for thousands of years. When an organism dies, its carbon 14 decays according to the law f (t) = f (t0 )e−1.25·10
−4 (t−t ) 0
(∗)
where f (t0 ) is the quantity of carbon at the moment of death t0 , and f (t) is the quantity that is left at time t. f (t) . (This formula is the basis for “radiocarbon dating”, for (a) Show that t0 is given by t0 = t + 8000 ln f (t0 ) which the American W. F. Libby received the Nobel prize in chemistry in 1960.) (b) Helge and Anne Stine Ingstad found several Viking tools on old settlements in Newfoundland. The charcoal from the fireplaces was analysed in 1972, and the percentage of carbon 14 in the charcoal (compared with the content of carbon 14 in fresh wood) was 88.6%. Use the result from (a) to determine when the Viking settlers lived in Newfoundland.
Problem 2-19 (5.3)
x+1 Let the function ϕ be defined by ϕ(x) = ln(x + 1) − ln(x + 2) = ln x+2 (a) Find the range of ϕ. (b) Find the inverse of ϕ. Where is it defined? © Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
for all x ≥ 0.
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CHAPTER 2
FUNCTIONS OF ONE VARIABLE
Problem 2-20 (5.4) The Norwegian income tax schedule for a single person in 2004 was as follows. Income below 23 000 kroner was exempt from tax. Above this lower limit, all income was subject to a 7.8% social security tax. Above 32 900 kroner, the tax rate was 28% (in addition to social security). Above 354 300, a surtax of 13.5% was imposed, and above 906 900, the surtax increased from 13.5% to 19.5%. Define a mathematical function that describes this tax schedule, and sketch its graph.
Problem 2-21 (5.5) A firm has two plants A and B located 60 kilometers apart at the two points (0, 0) and (60, 0). See Fig. 2-21. The two plants supply one identical product priced at $p per unit. Shipping costs per kilometer per unit are $10 from A and $5 from B. An arbitrary purchaser is located at point (x, y). (a) Give economic interpretations of the expressions p + 10 x 2 + y 2 and p + 5 (x − 60)2 + y 2 . (b) Find an equation for the curve that separates the markets of the two firms, assuming that customers buy from the firm for which total costs are lower. (c) Generalize the problem to the case where A = (0, 0) and B = (a, 0), and assume that shipping costs per kilometer are r and s dollars, respectively. Show that the curve separating the markets is a circle, and find its centre and radius. y (x, y)
A (0, 0)
B x (60, 0)
Figure 2-21
Problem 2-22 (6.10)
The extreme-value distribution in statistics is given by F (x) = exp − exp(−x) . (a) Write F (x) in standard form. (b) Compute f (x) = F (x), and write the result in two ways. The function f is called the density function associated with F . Compute f (x).
Problem 2-23 (7.5) Let f (x) = e−1/x (x = 0) with f (0) = 0. Verify by induction that f (k) (x) = x −3k pk (x)e−1/x for x = 0, where pk (x) denotes some polynomial whose degree is 2k − 2. Hence, show that f (k) (0) = 0 for all positive integers k. (For this function, all Taylor polynomials at the origin are identically equal to 0, but the function itself is 0 only at the origin. This example shows that the polynomial approximation to a function may be very inaccurate. For this reason it is very important to estimate the size of the remainder.) 2
© Knut Sydsæter, Arne Strøm, and Peter Hammond 2008
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