UNIVERSITY OF MAIDUGURI CENTRE FOR DISTANCE LEARNING

Transcription

1 UNIVERSITY OF MAIDUGURI CENTRE FOR DISTANCE LEARNING ECO 317: Mathematical Economics ( Units) Course Facilitator: Dr. M. O. Lawan

2 ECON 317: MATHEMATICAL ECONOMICS Unit: STUDY GUIDE Course Code/ Title: ECO 317: Mathematical Economics Credit Units: Timing: 6hrs Total hours of Study per each course material should be twenty Six hours (6hrs) at two hours per week within a given semester. You should plan your time table for study on the basis of two hours per course throughout the week. This will apply to all course materials you have. This implies that each course material will be studied for two hours in a week. Similarly, each study session should be timed at one hour including all the activities under it. Do not rush on your time, utilize them adequately. All activities should be timed from five minutes (5minutes) to ten minutes (10minutes). Observe the time you spent for each activity, whether you may need to add or subtract more minutes for the activity. You should also take note of your speed of completing an activity for the purpose of adjustment. Meanwhile, you should observe the one hour allocated to a study session. Find out whether this time is adequate or not. You may need to add or subtract some minutes depending on your speed. You may also need to allocate separate time for your selfassessment questions out of the remaining minutes from the one hour or the one hour which was not used out of the two hours that CDL, University of Maiduguri, Maiduguri ii

3 ECON 317: MATHEMATICAL ECONOMICS Unit: can be utilized for your SAQ. You must be careful in utilizing your time. Your success depends on good utilization of the time given; because time is money, do not waste it. Reading: When you start reading the study session, you must not read it like a novel. You should start by having a pen and paper for writing the main points in the study session. You must also have dictionary for checking terms and concepts that are not properly explained in the glossary. Before writing the main points you must use pencil to underline those main points in the text. Make the underlining neat and clear so that the book is not spoiled for further usage. Similarly, you should underline any term that you do not understand its meaning and check for their meaning in the glossary. If those meanings in the glossary are not enough for you, you can use your dictionary for further explanations. When you reach the box for activity, read the question(s) twice so that you are sure of what the question ask you to do then you go back to the in-text to locate the answers to the question. You must be brief in answering those activities except when the question requires you to be detailed. In the same way you read the in-text question and in-text answer carefully, making sure you understand them and locate them in the main text. Furthermore before you attempt answering the (SAQ) be iii CDL, University of Maiduguri, Maiduguri

4 ECON 317: MATHEMATICAL ECONOMICS Unit: sure of what the question wants you to do, then locate the answers in your in-text carefully before you provide the answer. Generally, the reading required you to be very careful, paying attention to what you are reading, noting the major points and terms and concepts. But when you are tired, worried and weak do not go into reading, wait until you are relaxed and strong enough before you engage in reading activities. Bold Terms: These are terms that are very important towards comprehending/understanding the in-text read by you. The terms are bolded or made darker in the sentence for you to identify them. When you come across such terms check for the meaning at the back of your book; under the heading glossary. If the meaning is not clear to you, you can use your dictionary to get more clarifications about the term/concept. Do not neglect any of the bold term in your reading because they are essential tools for your understanding of the in-text. Practice Exercises a. Activity: Activity is provided in all the study sessions. Each activity is to remind you of the immediate facts, points and major informations you read in the in-text. In every study session there is one or more activities provided for you to answer them. You must be very careful in answering these activities because they provide you with major facts of the iv CDL, University of Maiduguri, Maiduguri

5 ECON 317: MATHEMATICAL ECONOMICS Unit: text. You can have a separate note book for the activities which can serve as summary of the texts. Do not forget to timed yourself for each activity you answered. b. In-text Questions and Answers: In-text questions and answers are provided for you to remind you of major points or facts. To every question, there is answer. So please note all the questions and their answers, they will help you towards remembering the major points in your reading. c. Self Assessment Question: This part is one of the most essential components of your study. It is meant to test your understanding of what you studied so you must give adequate attention in answering them. The remaining time from the two hours allocated for this study session can be used in answering the self- assessment question. Before you start writing answers to any questions under SAQ, you are expected to write down the major points related to the particular question to be answered. Check those points you have written in the in-text to ascertain that they are correct, after that you can start explaining each point as your answer to the question. When you have completed the explanation of each question, you can now check at the back of your book, compare your answer to the solutions provided by your course writer. Then try to grade your effort sincerely and honestly to see your v CDL, University of Maiduguri, Maiduguri

6 ECON 317: MATHEMATICAL ECONOMICS Unit: level of performance. This procedure should be applied to all SAQ activities. Make sure you are not in a hurry to finish but careful to do the right thing. e-tutors: The etutors are dedicated online teachers that provide services to students in all their programme of studies. They are expected to be twenty- four hours online to receive and attend to students Academic and Administrative questions which are vital to student s processes of their studies. For each programme, there will be two or more e-tutors for effective attention to student s enquiries. Therefore, you are expected as a student to always contact your e- tutors through their addresses or phone numbers which are there in your student hand book. Do not hesitate or waste time in contacting your e-tutors when in doubt about your learning. You must learn how to operate , because ing will give you opportunity for getting better explanation at no cost. In addition to your e-tutors, you can also contact your course facilitators through their phone numbers and s which are also in your handbook for use. Your course facilitators can also resolve your academic problems. Please utilize them effectively for your studies. Continuous assessment The continuous assessment exercise is limited to 30% of the total marks. The medium of conducting continuous assessment may be vi CDL, University of Maiduguri, Maiduguri

7 ECON 317: MATHEMATICAL ECONOMICS Unit: through online testing, Tutor Marked test or assignment. You may be required to submit your test or assignment through your . The continuous assessment may be conducted more than once. You must make sure you participate in all C.A processes for without doing your C.A you may not pass your examination, so take note and be up to date. Examination All examinations shall be conducted at the University of Maiduguri Centre for Distance Learning. Therefore all students must come to the Centre for a period of one week for their examinations. Your preparation for examination may require you to look for course mates so that you form a group studies. The grouping or Networking studies will facilitate your better understanding of what you studied. Group studies can be formed in villages and township as long as you have partners offering the same programme. Grouping and Social Networking are better approaches to effective studies. Please find your group. You must prepare very well before the examination week. You must engage in comprehensive studies. Revising your previous studies, making brief summaries of all materials you read or from your first summary on activities, in-text questions and answers, as well as on self assessment questions that you provided solutions at first stage of studies. When the examination week commences you vii CDL, University of Maiduguri, Maiduguri

8 ECON 317: MATHEMATICAL ECONOMICS Unit: can also go through your brief summarizes each day for various the courses to remind you of main points. When coming to examination hall, there are certain materials that are prohibited for you to carry (i.e Bags, Cell phone, and any paper etc). You will be checked before you are allowed to enter the hall. You must also be well behaved throughout your examination period. CDL, University of Maiduguri, Maiduguri viii

9 ECO 317 : MATHEMATICAL ECONOMICS UNITS: INTRODUCTION TO THE COURSE This lecture is concerned with the definition of the subject matter: Mathematical Economics and the Methods of its analysis. We will discuss various definitions of the subject matter of Mathematical Economics. Every society has a definite system of production and distribution relations or the economic basis, the economic system of the society, which rests on the property relations. Mathematical economics studies the relationship of production with optimization, maxima and minima. The topic essentially deals with optimization, minimization and maximization, Langrange multiplier and linear programming. 9

10 ECO 317 : MATHEMATICAL ECONOMICS UNITS: STUDY SESSION 1: MARGINAL FUNCTIONS 1.1 INTRODUCTION Mathematics plays two basic roles in economics there is the direct and indirect role. An example of the direct role is the comparison of two or more equilibrium situations (Comparative Statics) which invariably involves relations between increments of quantities and directions of change. Comparative Statics analysis in economics are often quite complicated to deal with unless with the aid of mathematical tools. Also, some problems of monopoly, duopoly and oligopoly have proven to be most amenable to mathematical techniques. Now you may be wondering what on earth differentiation has got to do with economics. In fact, we cannot get very far with economic theory without making use of calculus. Most mathematical models used in economics and other disciplines are based on ordinary differential equations, partial differential equations, and integral calculus. Numerical methods for solving these equations are primarily of two types: i) The first type approximates the unknown function in the equation by a simpler function, often a polynomial or piecewise polynomial (Spline) function, chosen to closely follow the original equation. The finite method is the best known approach of this type. ii) The second type of numerical method approximates the equation of interest, usually by approximating the derivatives or integrals in the equation. The approximating equation has a solution at a discrete set of points, and this solution approximates that of the original equation. Such numerical procedures are often called finite difference methods. In this section, we concentrate on three main areas which illustrate its applicability; revenue and cost, production, savings and consumption. 1. LEARNING OUTCOMES At the end of the topic, you should be able to; i) Calculate marginal revenue and marginal cost 10

11 ECO 317 : MATHEMATICAL ECONOMICS UNITS: ii) iii) iv) Derive the relationship between marginal and average revenue for both monopoly and perfect competition, Calculate marginal product of labour, State the law of diminishing marginal productivity using the notation of calculus, v) Calculate Marginal Propensity to Consume and Marginal Propensity to Save. 1.3 IN TEXT REVENUE AND COST Total Revenue is defined by PQ, where P denotes the price of a good and Q denotes the quantity demanded. In practice, we usually know the demand equation which provides a relationship between P and Q. For example, if P = 100 Q Then TR = PQ = (100 Q)Q = 100Q Q The formula can be used to calculate the value of TR corresponding to any value of Q. not content with this; we are also interested in the effect on TR of a change in the value of Q from some existing level. To do this, we introduce the concept of Marginal Revenue (MR). The Marginal Revenue of a good is defined by; MR = ()TR Q Marginal Revenue is the derivative of total revenue with respect to demand 11

12 ECO 317 : MATHEMATICAL ECONOMICS UNITS: For example, the Marginal Revenue function corresponding to is given by TR = 100Q Q ()TR Q = 100 4Q If the current demand is 15, say, then MR = 100 4(15) = 40 Marginal Revenue is sometimes taken to be the change in Total Revenue TR brought about by one unit increase in Q. it is easy to check that this gives an acceptable approximation to Marginal Revenue MR, although it is not quite the same value obtained by differentiation. e.g. Substituting Q = 15 into the total revenue function considered previously gives TR = 100(15) (15) = 1050 An increase of one unit in the value of Q produces a total revenue: TR = 100(16) (16) = 1088 This is an increase of 38, which according to non calculus definition is the value of MR when Q is 15. This compares with the exact value of 40 obtained by differentiation. It is instructive to give a graphical interpretation of these two approaches. Fig. 1.1 TR Tangent B TR curve A ΔTR ΔQ Q0 Q0 + 1 Q 1

13 ECO 317 : MATHEMATICAL ECONOMICS UNITS: In figure 1.1, the point A lies on the TR curve corresponding to a quantity Q 0. the exact value of MR at this point is equal to the derivative; ()TR Q And so is given by the slope of the tangent at A. the point B lies on the curve but corresponds to a one unit increase in Q. The vertical distance from A to B therefore equals the change in TR when Q increases by one unit. The slope of the chord joining A and B is; ()TR Q In order words, the slope of the chord is equal to the value of MR obtained from the non calculus definition. Inspection of the diagram reveals that the slope of the tangent is approximately the same as that of the chord joining A and B. in this case the slope of the tangent is slightly the larger of the two but there is not much in it. We therefore see that the one unit increase approach produces a reasonable approximation to the exact value of MR given by; ()TR Q Example If the total revenue function of a good is given by; 100Q Q Write down the expression for the marginal revenue function. If the current demand is 60, estimate the change in the value of TR due to a two () unit increase in Q Solution If TR = 100Q Q Then MR = When Q = 60 ()TR Q = 100 Q 13

14 ECO 317 : MATHEMATICAL ECONOMICS UNITS: MR = 100 (60) = 0 If Q increases by units ΔQ = and the formula ΔTR MR ΔQ Shows that the change in total revenue is approximately 0 = 40 A two unit increase in Q therefore leads to a decrease in TR of about 40. The simple model of demand, assumed that price P, and quantity Q, are linearly related according to an equation; P = aq + b Where the slope a is negative and the intercept b is positive. A downward sloping demand curve such as this corresponds to the case of monopolist (A single firm or group of firms forming a cartel) is assumed to be the only supplier of a particular product and has control over the market price. As the firm raises the price, demand falls. The associated total revenue function is given by TR = (aq + b)q = aq + bq An expression for marginal revenue is obtained by differentiating TR with respect to Q to get: MR = aq + b 14

15 ECO 317 : MATHEMATICAL ECONOMICS UNITS: Fig 1. b AR -b/a -b/a Q -b MR Fig. 1.3 TR -b/a -b/a Q The average revenue AR is defined by 15

16 ECO 317 : MATHEMATICAL ECONOMICS UNITS: TR AR Q Since TR = PQ AR PQ P Q For perfect competition P = b TR = PQ = bq MR = b Fig 1.4 AR b MR Cost Q So far, we have concentrated on the total revenue function. Exactly the same principle can be applied to other economic functions. For instance, we define the marginal cost, MC by; ()TC MC Q 16

17 ECO 317 : MATHEMATICAL ECONOMICS UNITS: Marginal cost is the derivative of total cost with respect to output Using simple geometric argument, we can see that if Q changes by a small amount ΔQ, then the corresponding change in TC is given by; ΔTC MC ΔQ change in total cost Marginal cost change in output in particular, putting ΔQ = 1 gives ΔTC MC e.g. If the average cost function of a good is AC = Q Q Find an expression for MC. If the current output is 15, estimate the effect on TC of a three unit decrease in Q. Solution We first need to find an expression for TC using the given formula for AC. Now we know that the average cost is just the total cost divided by Q, that is; AC = TC Q Hence TC = (AC)Q 13 = Q 6 Q Q TC = Q + 6Q + 13 In this formula, the last term 13 is independent of Q so it must denote the fixed costs. The remaining part Q + 6Q is the variable cost. Differentiating gives; 17

18 ECO 317 : MATHEMATICAL ECONOMICS UNITS: MC = Q L = 4Q + 6 Note that the fixed costs are constants and have no effect on the marginal cost. When Q = 15 MC = 4(15) + 6 = 66 Also, if Q decreases by 3 units, then ΔQ = 3 Δ(TC) = MC ΔQ = 66 ( 3) = 198 So, TC decreases by 198 units approximately PRODUCTION FUNCTION A firm is a technical unit in which commodities are produced. Its entrepreneur decides how much of and how one or more commodities will be produced, and gains the profit and bears the loss which results from his decision. An entrepreneur transforms inputs into outputs, subject to the technical rules specified by his production function. The difference between his revenue from sale of outputs and the costs of his inputs is his profit, if positive, or his loss; if negative. The entrepreneur s production function gives mathematical expression to the relationship between the quantities of inputs he employs and the quantities of outputs he produces. The concept is perfectly general. A specific production function may be given by a single point, a single continuous or a discontinuous function, or a system of equations. The focal points in the study of production functions include the following; i) The homogeneity of the function, ii) iii) iv) Strict convexity of the related isoquants, Euler s theorem, The elasticity of substitution. 18

19 ECO 317 : MATHEMATICAL ECONOMICS UNITS: Consider a simple production process in which an entrepreneur utilizes two variable inputs (X 1 and X ) and one or more fixed inputs in order to produce a single output (Q). His production function states the quantity of his output (q) as a function of the quantities of his variable inputs (x 1 and x ); Q = f(x 1, x ) This is assumed to be a single valued continuous function with continuous first and second-order partial derivatives. The production function is defined for nonnegative values of the input and output levels. Negative values are meaningless within present context. The domain of the production function may not include the entire nonnegative quadrant, and may differ from case to case. The production function normally is assumed to be increasing, i.e. the f 1 > 0 within its domain. It is assumed to be a regular strictly quasi-concave function when output is maximized or cost minimized and strictly concave function when profit is maximized. The output Q of any production process depends on a variety of inputs known as the factors of production. These include land, capital, labour and enterprise. For simplicity, we restrict our attention to capital and labour. Capital K, denotes all man-made aids to production such as buildings, tools and plant machinery. Labour, L, denotes all paid work in the production process. The dependence of Q on K and L may be written Q = f(l,k), where: Output Q is assumed to be a function of labour L and capital K. in the short run, the inputs K can be assumed to be fixed so Q is then only a function of one input L. The variable L is usually measured in terms of the number of worker hours. Motivated by our previous work, we define the Marginal Product (MP L ) by; MP L = Q L Marginal Product of labour is the derivative of output with respect to labour 19

20 ECO 317 : MATHEMATICAL ECONOMICS UNITS: MP L gives the approximate change in Q that result from using one or more unit of L. Example If the production function is Q = 300 L 4L Where Q denotes output and L denotes the size of workforce, calculate the value of MP L when; a) L = 1 b) L = 9 c) L = 100 d) L = 500 Discuss the implication of the result. Solution If Q = 300 L 4L Then Q MP L L = 300L 1/ 4L 1 1 = L = 150L 1/ L When a) L = 1 MP L = b) When L = 9 0

21 ECO 317 : MATHEMATICAL ECONOMICS UNITS: MP L = c) When L = 100 MP L = d) When L = 500 MP L = Notice that the value of MP L declines with increasing L. part (a) shows that if the workfo rce consists of only one person, then to employ people would increase output by 146 approximately. In part (b), we see that to increase the number of workers from 9 10 would result in about 46 additional units of output. This example illustrates the Law of Diminishing Marginal Productivity (sometimes called Law of Diminishing Returns). It states that the increase in output due to a one unit increase in labour will eventually decline. In other words, once the size of the workforce has reached a certain threshold level, the marginal product of labour will get smaller. This is not always so. It is possible for the Marginal Product of Labour to remain constant or to go up to begin with for small values of L. However, if it is to satisfy the Law of Diminishing Marginal Productivity, then there must be some value of L above which MP L decreases. A typical product curve is sketched below which has slope Q MPL L 1

22 ECO 317 : MATHEMATICAL ECONOMICS UNITS: Fig. 1.5 Q bends downwards Bends upward s 0 L 0 L Between 0 and L 0 the curve bends upwards becoming progressively steeper and so the slope function, MP L increases. Mathematically, this means that the slope of MP L is positive, that is; MP L 0 Q Now MP L is itself the derivative of Q with respect to L, so we can use the notation for the second derivative and write this as Q L 0 Similarly, if L exceeds the threshold value of L 0, then the figure 1.5 shows that the product curve bends downwards and the slope decreases. In this region, the slope of the slope function is negative so that;

23 ECO 317 : MATHEMATICAL ECONOMICS UNITS: Q L 0 The Law of diminishing returns states that this happen eventually, that is Q L CONSUMPTION AND SAVING Assuming that national income is only used up in consumption and savings, then; Y = C + S Of particular interest is the effect on C and S due to variations in Y. expressed simply, if national income rises by a certain amount, are people more likely to go out and spend their extra income on consumer goods or will they save it? To analyze this behaviour, we use the concepts of Marginal Propensity to Consume, MPC, and Marginal Propensity to Save, MPS, which are defined by MPC = C Y and S Y Marginal Propensity to Consume is the derivative of consumption with respect to income Marginal Propensity to Save is the derivative of savings with respect to income MPC and MPS are slopes of the linear consumption and savings curves, respectively. At first sight it appears that, in general, we need to work out two derivatives in order to evaluate MPC and MPS. However, this is not strictly necessary. Recall that we can do whatever we like to an equation provided we do the same thing to both sides. Consequently, we can differentiate both sides of the equation Y = C + S With respect to Y to deduce that; 3

24 ECO 317 : MATHEMATICAL ECONOMICS UNITS: Y C S Y Y Y = MPC + MPS Now we are already familiar with the result that when we differentiate x with respect to x, the answer is 1. in this case Y plays the role of x, so Y 1 Y Hence 1 = MPC + MPS This shows that in practice, we need only work on one of the derivatives. The remaining derivative can then be calculated directly from this equation. Example If the consumption function is C = 0.01Y + 0.Y + 50 Calculate MPC and MPS when Y = 30 Solution Here, the consumption function is given so we begin by finding MPC. To do this, we differentiate C with respect to Y. if C = 0.01Y + 0.Y + 50 Then C 0.0Y 0. Y So when Y = 30, MPC = 0.0(30) + 0. = 0.8 To find the corresponding value of MPS, we use the formula MPC + MPS = 1 4

25 ECO 317 : MATHEMATICAL ECONOMICS UNITS: Which gives; MPS = 1 MPC = = 0. This indicates that when national income increases by 1 unit (from its current level of 30) consumption rises by approximately 0.8 units, whereas saving rise by only 0. units. At this level of income, the nation has a greater propensity to consume than it has to save. 1.4 SUMMARY Calculus is used to obtain marginalities in economics. It gives the exact values rather than approximations. Marginal Revenue is the derivative of Total Revenue with respect to demand, ()TR MC Q Marginal Product of Labour is the derivative of output with respect to labour Q MPL L Marginal Propensity to Consume is the derivative of consumption with respect to income C MPC Y Marginal Propensity to Save is the derivative of savings with respect to income S MPS Y MPS + MPC = 1 5

26 ECO 317 : MATHEMATICAL ECONOMICS UNITS: 1.5 SELF ASSESSMENT QUESTIONS (SAQ) FOR STUDY SESSION 1 SAQ 1 SAQ if the demand function is P = 60 Q Find an expression for TR in terms of Q (a) Differentiate TR with respect to Q to find a general expression for MR in terms of Q, Hence, write down the exact value of MR at Q = 50 (b) Calculate the value of TR when; (i) Q = 50 (ii) Q = 5 Find the Marginal Cost given Average Cost Function AC = 100 Q Deduce that one unit increase in Q will always result in a two unit increase in TC irrespective of the current level of output. SAQ 3 If the saving function is given by; S = 0.0Y Y Calculate the value of MPS and MPC when Y = 40 Give the brief interpretation of these results. 6

27 ECO 317 : MATHEMATICAL ECONOMICS UNITS: 1.6 REFERENCE Jacques Ian (003), Mathematics for Economics and Business, Prentice hall, London. Dowlings Edward (000), Introduction to Mathematics for Ecnomics; Shaum s Outline Series, McGraw Hill, International Edition, New York. Allen R.G. (001), Mathematical Economics, Oxford University Press, London 1.7 FURTHER READING Chiang A.C. (001), Fundamental Methods of Mathematical Economics, McGraw Hill book company, New York. Jacques Ian (003), Mathematics for Economics and Business, Prentice hall, London. 7

28 ECO 317 : MATHEMATICAL ECONOMICS UNITS: STUDY SESSION : FURTHER RULES OF DIFFERENTIATION.1 INTRODUCTION We were introduced to some basic rules of differentiation last year. unfortunately, not all functions can be differentiated using these rules alone. For example, we are unable to differentiate the functions x x (x 3 and x 1 using just the constant, sum or difference rules. The aim of the present section is to describe three further rules which allow you to find the derivative of more complicated expressions. Indeed, the totality of all six rules will enable you to differentiate any mathematical function. Although you may find that the rules described here take you slightly longer to grasp than before, they are vital to any understanding of economic theory.. LEARNING OUTCOMES At the end of this session, you should be able to; i) Use the chain rule to differentiate a function of a function, ii) iii) iv) Use Product Rule to differentiate the product of two functions, Use Quotient Rule to differentiate the quotient of functions, Differentiate complicated function using combination rule..3 IN-TEXT.3.1 THE CHAIN RULE Given a composite function also called a function of a function, in which y is a function of u and u in turn is a function of x, that is y = f(u) and u = g(x), then y = f(g(x)) and the derivative of y with respect to x is equal to the derivative of the first function with respect to u times the derivative of the second function with respect to x. 8

29 ECO 317 : MATHEMATICAL ECONOMICS UNITS: y y y x u x Differentiate the outer function and multiply by the derivative of the inner function Example Consider the function y = (5x + 3) 4 To use the chain rule, let y = u 4 and u = 5x + 3, then y = 4u 3 u and = 10x u x Substituting these values in the chain rule formula y x = 4u3. 10x = 40xu 3 Then express the derivative in terms of a single variable, substitute 5x + 3 for u y x = 40xu3 (5x + 3) 3 With practice, it is possible to perform the differentiation without explicitly introducing the variable u. to differentiate. y = (x + 3) 10 we first differentiate the outer power function to get 10(x + 3) 9 and then multiply by the derivative of the inner function x + 3 which is, so y = 0(x + 3)9 x Example Differentiate 9

30 ECO 317 : MATHEMATICAL ECONOMICS UNITS: (a) y = (3x 5x + ) 4 (b) y = 1 3x 7 (c) y = (1) x Solution The chain rule shows that to differentiate (3x 5x + ) 4, we first differentiate the outer power function to get 4(3x 5x + ) 3 and then multiply by the derivative of the inner function 3x 5x +, which is 6x 5. hence if (b) y = (3x 5x + ) 4 then y x = 4(3x 5x + ) 3 (6x 5) to use the chain rule to differentiate y = 1 3x 7 recall that reciprocal are denoted by negative powers, so that y = (3x + 7) -1 the outer power function differentiates to get (3x + 7) - And the inner function differentiates, 3x + 7, differentiates to get 3. by the chain rule we just multiply these together to deduce that if y = 1 3x 7 then y x = - (3x + 3 7)- (3) = (3x 7) c) to use the chain rule to differentiate y = (1) x 30

31 ECO 317 : MATHEMATICAL ECONOMICS UNITS: recall that roots are denoted by fractional powers, so that y = 1 (1) x the outer power function differentiates to get 1 (1) 1 x And the inner function, 1 + x, differentiates to get x. by the chain rule, we just multiply these together to deduce that if y = = (1) x x (1) x then y 1 (1)() x x 1 x.3. THE PRODUCT RULE If y v u y = uv then u v x x x this rule tells you how to differentiate the product of two functions: multiply each function by the derivative of the other and add Example Differentiate i) y x (x 1) 3 ii) x (6x 1) iii) x y 1 x 31

32 ECO 317 : MATHEMATICAL ECONOMICS UNITS: Solution i) The function 3 x (x 1) involves the product of two simpler functions, namely x and (x+1) 3, which we denote by u and v respectively. (it does notmatter which function we label u and which we label v. the same answer is obtained if u is (x+1) 3 and v is x. you might like to check this for yourself later). Now if then u = x and v = (x+1) 3 u x x and v 6(x 1) x Where we have used the chain rule to find y v u u v x x x v x. By the product rule, x 6(x 1)( 1)() x x = 3 The first term is obtained by leaving u alone and multiplying it by the derivative of v. similarly, the second term is obtained by leaving v alone and multiplying it by the derivative of u. If desired, the final answer may be simplified by taking out a common factor of x(x+1). This factor goes into the first term 3x times and into the second x+1 times. Hence y ( x x 1) [3(x 1)] x x = ( x x 1)(5 x1) ii) The function x (6x 1) involves the product of the simpler functions u = x and v = (6x 1)(6 x1) 1 3

33 ECO 317 : MATHEMATICAL ECONOMICS UNITS: for which u 1 x and v 1 (6 x 1) 6 3(6 x 1) x Where we have used the chain rule to find y v u u v x x x 1 1 v x. By the product rule, = 1 1 x[3(6x 1) ](6 x1)(1) = 3x (6x 1) (6x 1) If desired, this can be simplified by putting the second term over a common denominator (6x 1) To do this, we multiply the top and bottom of the second term by (6x 1) to get 6x 1 (6x 1 ( 6x+1 6x+1 = 6x+1) Hence, y 3(6 x 1) x x (6x 1) 9x 1 (6x 1 ii) at first sight it is hard to see how we can use the product rule to differentiate 33

34 ECO 317 : MATHEMATICAL ECONOMICS UNITS: 1 x x, since it appears to be quotient and not the product of two functions. However, if we recall that reciprocal are equivalent to negative powers, we may rewrite it as x(1 + x) -1 it follows that we can put u = x and v = (1 + x) - Where we have used the chain rule to find y v u u v x x x y x[(1) ](1)(1) x x x x (1) x 1 = 1 1 x v x. By the product rule, If desired, this can be simplified by putting the second term over a common denominator (1 + x) To do this we multiply the top and bottom of the second term by 1 + x to get Hence, 1 x (1) x y x 1 x (1)(1) x x x (1) x (1) x = 1 = (1) x 34

35 ECO 317 : MATHEMATICAL ECONOMICS UNITS:.3.3 THE QUOTIENT RULE If y = u v u u v y then x x v x v This rule tells you how to differentiate the quotient of two functions: bottom times derivative of top, minus top time derivative of bottom, all over bottom squared Example Differentiate x (a) y 1 x (b) 1 x y x 3 Solution (a) In the quotient rule, u is used as the label for the numerator and v is used for the denominator, so to differentiate x 1 x We must take u= x and v = 1 + x for which u v 1 and 1 x x 35

36 ECO 317 : MATHEMATICAL ECONOMICS UNITS: by the quotient rule v u u v y x x x v (1)(1)(1) x x (1) x = 1 x x (1) x = 1 = (1) x Notice how the quotient rule automatically puts the final expression over a common denominator. Compare this with the algebra required to obtain the same answer using the product rule in part (iii) of the previous example. (b) The numerator of the algebraic fraction 1 x x 3 is 1 + x and the denominator is x 3, so we take u = 1 + x and v = x 3 for which u x and x By the quotient rule v u u v y x x x v v 3x x 36

37 ECO 317 : MATHEMATICAL ECONOMICS UNITS: ()()(1)( x x3) x x 3 () x 3 4 x x 3x 3x 3 () x x 3x 4 x () x 3.4 SUMMARY Chain rule states that if y is a function of u which itself is a function of x, then y y u x u x In Product rule, if y = uv, then; y v u u v x x x Quotient rule states that if y = u v then v u u v y x x x v 37

38 ECO 317 : MATHEMATICAL ECONOMICS UNITS:.5 SELF ASSESSMENT QUESTIONS (SAQ) FOR STUDY SESSION SAQ 1: Use the Chain Rule to differentiate (a) y = (x + 3) 10 (b) y = (x + 3x + 5) 3 (c) y = 1 7x 3 (d) y = 1 x 1 (e) y = (8x 1) SAQ : Use the Product Rule to differentiate (a) y = x (x + 5) 3 (b) y = x 5 (4x 5) (c) y = x 4 ( x 1) SAQ 3: Use the quotient rule to differentiate x (a) y = x 4 (b) y = x 1 x (c) y = x 3 ( x 1) 38

39 ECO 317 : MATHEMATICAL ECONOMICS UNITS:.6 REFERENCE Backhouse J.K. & Houldsworth (1997), Pure Mathematics, A first Course, Longman Hong Kong. Allen R.G. (001), Mathematical Economics, Oxford University Press, London Dowlings Edward (000), Introduction to Mathematics for Ecnomics; Shaum s Outline Series, McGraw Hill, International Edition, New York..7 FURTHER READING Jacques Ian (003), Mathematics for Economics and Business, Prentice hall, London. Chiang A.C. (001), Fundamental Methods of Mathematical Economics, McGraw Hill book company, New York. 39

40 ECO 317 : MATHEMATICAL ECONOMICS UNITS: STUDY SESSION 3: OPTIMIZATION OF ECONOMIC FUNCTIONS 3.1 INTRODUCTION The economist is frequently called upon to help a firm maximize profits and levels of physical output and productivity, as well as to minimize cost, levels of pollution, and the use of scarce natural resources. In this topic, we will be shown how the techniques of calculus can be used to find coordinates of the turning point of a parabola. The principal objective of optimization theory is to develop systemic and efficient techniques for the determination of the locations and values of all relative extrema and subsequently to determine the absolute extrema. The beauty of this approach is that it can be used to locate the maximum and minimum points of any economic function. At a stationary point, the tangent to the graph is horizontal and so has zero slope. Consequently, at a stationary point of a function; f () x f '' () x = 0 (See figure 3.1 below) 3. LEARNING OUTCOMES At the end of this topic, you should be able to i) Use the first order derivatives to find the stationary points of a function, ii) Use the second order derivatives to classify the stationary point of a function, iii) Find the maximum and minimum points of an economic function. 3.3 IN TEXT STATIONARY POINTS Calculus was originally used by astronomers to predict planetary motion. If a graph of the distance traveled by an object is sketched against time then the speed of the object is given by the slope since this represents the rate of change of distance with respect to time. It follows that if 40

41 ECO 317 : MATHEMATICAL ECONOMICS UNITS: the graph is horizontal at some point, then the speed is zero and the object is instantaneously at rest, which is stationary. Stationary points are classified into three types a) Local Maxima b) Local Minima, and c) Stationary points if reflection Fig. 3.1 y B H D E F C G A 0 x At a local maximum (sometimes called a relative maximum) the graph falls away on both sides. Points B and E are the local maxima for the function sketched in figure 3.1. the word local is used to highlight the fact that although these are the maximum points relative to their locality or neighbourhood they may not be the overall or global maximum. In figure 3.1 the highest point on the graph actually occurs at the right hand end, H, this is not a stationary point since the slope is not zero at H. At a local minimum (sometimes called a relative minimum) the graph rises on both sides. Points C and G are the local minima in figure 3.1, again it is not necessary for the global minimum to be 41

42 ECO 317 : MATHEMATICAL ECONOMICS UNITS: one of the local minima. In figure 3.1, the lowest point on the graph occurs at the left hand end A which is not a stationary point. At a stationary point of inflection the graph rises on one side and falls on the other. The stationary points of inflection in figure 3.1 are labeled D and F. these points are of little value in economics, although they do sometimes assist in sketching graphs of functions. Maxima and minima on the other hand are important. The calculation of maximum points of revenue and profit function is clearly worthwhile. Likewise, it is useful to be able to find minimum point of average cost functions. Two obvious questions remain, how do we find stationary points of any given function and how do we classify them? The first question is easily answered. As mentioned earlier, stationary points satisfy the equation. f ' () x = 0 It can be shown that if a function has a stationary point at x = a, then if f f '' () '' () x > 0 then f(x) has a minimum at x = a x < 0 then f(x) has a maximum at x = a Therefore, all we need to do is to differentiate the function on a second time and to evaluate this second order derivative at each point. A point is a minimum if this value is positive and a maximum if this value is negative. For = a means no information about the stationary point. To summarize the method for finding and classifying stationary points of a function f(x), is as follows; Step 1: solve the equation f ' () x 0 To find the stationary points, x = 0 f x '' () Step : if f '' () x > 0 then the function has a minimum at x = a f '' () x < 0 then the function has a maximum at x = a 4

43 ECO 317 : MATHEMATICAL ECONOMICS UNITS: f '' () x = 0 then the point cannot be classified using the available information. Example Find and classify the stationary a) f(x) = x 4x + 5 b) f(x) = x 3 + 3x 1x + 4 Solution a) f(x) = x 4x + 5 Differentiate at once f ' () x = x 4 and then differentiate the second time f '' () x = Step 1; the stationary points are the solutions of the equation f ' () x = 0 So we need to solve x 4 = 0 x = 4 x = 4/ = step : to classify this point, we need to evaluate f '' () In this case, f '' () x = For all values of x so in particular f '' () =. this number is positive. 43

44 ECO 317 : MATHEMATICAL ECONOMICS UNITS: Fig 3. y f(x)=x 4x (,1) 0 x So the function has a minimum at x = y = () () + 5 = 1. so the minimum point has coordinates (,1). A graph of f(x) is shown in figure 3.. b) in order to use steps 1 &, we need to find the first and second order derivatives of the functions. f(x) = x 3 + 3x 1x + 4 differentiating once gives f ' () x = 6x + 6x 1 And differentiating a second time gives f '' () x = 1x + 6 Step 1: the stationary points are the solution of the equation f ' () x = 0 So we need to solve 6x + 6x 1 = 0 44

45 ECO 317 : MATHEMATICAL ECONOMICS UNITS: Divide all through by 6 Factorizing, x - 1 x x + x x = x(x + ) -1(x + ) x 1) = 0 or (x + ) = 0 x = 1 or x = - this function has shown that we have stationary pints at x = - and x = 1 step II: to classify these points we need to evaluate f '' ( ) = 1(-) + 6 = -18 f '' ( ) and f(1). Now we have Now, This is negative, so there is a minimum at x = - when x = - y = (-) 3 + 3(-) 1(-) + 4 = 4 So the maximum point has coordinates (-, 4) f '' (1) = 1(-1) + 6 = 18 This is a positive, so there is a minimum at x = 1 when x = 1 y = (-1) 3 + 3(-1) 1(-1) + 4 = -3 so the minimum point has coordinates (1,-3) 45

46 ECO 317 : MATHEMATICAL ECONOMICS UNITS: Fig. 3.3 (-, 4) y f(x)=x 3 +3x +1x+4 4 x (1, -3) 3.3. SHORT-RUN PRODUCTION FUNCTION The task of finding the maximum or minimum values of a function is referred to as OPTIMIZATION. This is an important topic in mathematical economics. Example A firm s short run production function is given by: Q = 6L 0.L 3 where L denotes the number of workers. a) find the size of the workforce which maximizes output and hence sketch a graph of this production function, b) Find the size of the workforce which maximizes the average product of labour. Calculate MP L and AP L at this value of L. what do you observe? Solution a) in the first part of this example, we find the value of L which maximizes Q = 6L 0.L 3 46

47 ECO 317 : MATHEMATICAL ECONOMICS UNITS: Step 1: at a stationary point Q = 1L 0.6L = 0 L This is a quadratic equation and so we could use the formula to find L. however, this is not really necessary in this case because both terms have common factor of L and the equation may be written as L(1 0.6L) = 0 It follows that either L = 0 or 1 0.6L = 0 That is, the equation has solutions L = 0 or 1/0.6L = 0 Step : it is obvious on economic ground that L = 0 is a minimum and presumably L = 0 is the maximum. We can of course check this by differentiating a second time to get Q L When L = 0 = 1L 1.L Q = 0 > 0 L Which confirms that L = 0 is a minimum. The corresponding output is given by Q = 6(0) 0.(0) = 0 As expected, when L = 0 Q = -1 < 0 L Which confirms that L = 0 is a maximum. The firm should therefore employ 0 workers to achieve a maximum output Q = 6(0) 0.(0) 3 =

48 ECO 317 : MATHEMATICAL ECONOMICS UNITS: We have shown that the minimum points on the graph has coordinates (0,0) and the maximum has coordinates (0, 800). There is no further turning point so the graph of the production function has the slope in figure 3.4 Fig. 3.4 Q (0, 800) Q=6L -0L 3 (0, 0) 30 L b) in the second part of this example, we want to find value of L which maximizes the average product of labour. The average product of labour AP L = Q/L this sometimes is called labour productivity. Since it measures the average output per worker AP L = 6L 0.L L 3 = 6L 0.L Step 1: At stationary point () AP L 0 L 48

49 ECO 317 : MATHEMATICAL ECONOMICS UNITS: So 6 0.4L = 0 Which has solution L = 6/0.4 = 15 Step : to classify this stationary point we differentiate a second time to get () AP L L which shows that it is a maximum The labour productivity is therefore greatest when the firm employs 15 workers. The corresponding labour productivity AP L is 6(15) 0.(15) = 45 In other words, the largest number of goods produced per worker is 45. Finally, we are invited to calculate the value of MP L at this point. To find an expression for MP L we need to differentiate Q with respect to L to get MP L = 1L 0.6L When L = 15 MP L = 1(15) 0.6(15) = 45 We observe that at L = 15, the value of MP L and AP L are equal. Marginal product of Labour = Average Product of Labour COST FUNCTIONS Example The demand function of a good is P + Q = 30 and the total cost function is TC = 1 / Q + 6Q + 7 a) Find the level of output which maximizes total revenue b) Find the level of output which maximizes profit. Calculate MR and MC at this value of Q, what do you observe? 49

50 ECO 317 : MATHEMATICAL ECONOMICS UNITS: Solution We are given demand equation as P + Q = 30, we have to find an expression for TR and then apply the theory of stationary points in our usual way. TR = PQ The demand equation is P + Q = 30 can be rearranged to get P = 30 Q Hence TR = (30 Q)Q = 30Q Q Step 1: At stationary point () TR Q () TR Q 0 30 Q So, 30Q Q = 0 Which has solution Q = 30 = 15 Step : to classify this point, we differentiate a second time to get () TR Q This is negative, so TR has a maximum at Q = 15 b) in the second part of this example we want to find the value of Q which maximizes profit. To do this we begin by determining an expression for profit in terms of Q. once this has been done it is then a simple matter to work out the first and second order derivatives and so to find and classify the stationary points of the profit function. The profit function is defined by 50

51 ECO 317 : MATHEMATICAL ECONOMICS UNITS: π = TR TC from part (a) TR = 30Q Q We are given the total cost function 1 TC = Hence, Q Q = 30Q Q Q 6Q 7 (30)( Q Q 6 Q7) Q = Q Q Step 1: At stationary point 0 Q So, -3Q + 4 = 0 Which has solution Q = 4/3 = 8 Step : to classify this point we differentiate a second time to get Q 3 This is negative, so π has a maximum at Q = 8. infact, the corresponding maximum profit is; 3 π = (8) 4(8)

52 ECO 317 : MATHEMATICAL ECONOMICS UNITS: Finally, we are invited to calculate the marginal revenue and marginal cost at this particular value of Q. to find expression for MR and MC, we need only differentiate TR and TC respectively. If Then, TR = 30Q Q MR = So when Q = 8 If ()TR Q = 30 Q MR = 30 (8) = 14 TC = MC = 1 So when Q = 8 Q Q ()TC Q = Q MC = = 14 We observe that at Q = 8, the values of MR and MC are equal. In this particular example, we discovered that at a point of maximum profit; Marginal Revenue = Marginal Cost 5

53 ECO 317 : MATHEMATICAL ECONOMICS UNITS: 3.4 SUMMARY At stationary point the tangent to a graph is horizontal and has zero slope. f ' () x 0 f f '' () '' () x > 0 then f(x) has a minimum at x = a x < 0 then f(x) has a maximum at x = a This method for finding and classifying stationary points of a function f(x) is; i) Solve the equation ii) Differentiate the function [ f ' () x 0 ] to find the stationary points x = 0 iii) Find the second derivative to determine whether minimum or maximum. 53

54 ECO 317 : MATHEMATICAL ECONOMICS UNITS: 3.5 SELF ASSESSMENT QUESTIONS (SAQ) FOR STUDY SESSION 3 SAQ 1: find and classify the stationary points of the following functions. Hence, sketch their graphs (a) y = 3x + 1x 35 (b) y = -x x 36x + 7 SAQ : A firm s short run production function is given by Q = 300L L 4 Where L denotes the number of workforce; Find the size of the workforce which maximizes the average product of labour and verify that at its value of L MP L = AP L SAQ 3: (a) (b) The demand equation of a good is given by P + Q = 0 and the total cost function is Q 3 8Q + 0Q + find the level of output which maximizes total revenue, Find the maximum profit and the value of Q at which it is achieved. Verify that at this value of Q, MR = MC 3.6 REFERENCES Allen R.G. (001), Mathematical Economics, Oxford University Press, London Chiang A.C. (001), Fundamental Methods of Mathematical Economics, McGraw Hill book company, New York. 3.7 FURTHER READING Lewis J.P. (1985), An Introduction to Mathematics for Students of Economics, Macmillan & Co. Ltd, London 54

55 ECO 317 : MATHEMATICAL ECONOMICS UNITS: STUDY SESSION 4: LANGRANGE MULTPLIERS The method of finding the relative or constrained maxima and minima (or relative extrema) of a function of two or more variables subject to side relations or constraints can be generalized. This general method is called Langrange s method of solution by undetermined multipliers or the method of Langrangian multipliers. This method relies on using the side relations (or constraints) to express one of the variables in the original function (sometimes called the objective function) in terms of the other variables. 4.1 LEARNING OUTCOMES At the end of this session, you should be able to i) Use the method of Langrange multipliers to solve constrained optimization problems ii) Give an economic interpretation of Langrange multipliers, iii) Use Langrange multipliers to maximize a Cobb-Douglas production function subject to a cost constraint iv) Use Langrange multipliers to show that when a firm maximizes output subject to a cost constraint, the ratio of marginal product to price is the same for all inputs. 4. IN TEXT 4..1 CONSTRAINED OPTIMIZATION We now describe the method of Langrange multipliers for solving constrained optimization problems. This is the preferred method since it handles non linear constraints and problems involving more than two variables with ease. It also provides some additional information which is useful when solving economic problems. To optimize an objective function f(x,y) subject to a constraint 55

56 ECO 317 : MATHEMATICAL ECONOMICS UNITS: φ(x,y) = M we work as follows Step 1: Define a new function g( x, y,)(,) f x[( y,)] M x y Step : Solve the simultaneous equation g 0 x g 0 y g 0 For the three unknowns, x, y, and λ. The basic steps of the method are straightforward. In step 1, we combine the objective function and constraint into a single function. To do this, we first arrange the constraint as M φ(x, y) And multiply by the scalar (i.e. number), λ (the greek letter for lamda ). This scalar is called the Langrange multiplier. Finally we add on the objective function to produce the new function g( x, y,)(,) f x[( y,)] M x y This is called the Langrange function. The right hand side involves the three letters x, y, and λ, so g is a function of three variables. In step we work out the three first order partial derivatives g g g,, x y 56