introduction to Calculus theoretical vs mathematical PDF

Preview

Summary

This file gives you a brief description about introduction to calculus for freshman students joining economics department. it helps to show some overview about the course...

Description

Calculus for Economists

Compiled by: Habtamu Adane

AAU: Department of Economics

CHAPTER ONE 1. INTRODUCTION 1.1.

Theoretical Vs Mathematical Economics

Since mathematical economics is merely an approach to economic analysis, it should not and does not differ from the nonmathematical approach to analysis in any fundamental way. The purpose of any theoretical analysis, regardless of the approach, is always to drive a set of conclusions or theorems from a given set of assumptions or postulates via a process of reasoning. The major difference between “mathematical economics” and “literary economics” lies principally in the fact that, in the former, the assumptions and conclusions are stated in mathematical symbols rather than words and in equations rather than sentences; moreover, in place of literary logic, use is made of mathematical theorems-of which there exists an abundance to draw upon-in the reasoning process. Moreover, mathematical economics draws upon mathematical logics unlike the literary logic, which is drawn by literary economics. At this juncture, you may ask yourself “If the two approaches’ ends are the same-establish valid hypotheses, then why do economists choose mathematical approaches with which many literary people are in phobia?” In fact, it is amazing to see people in horror with mathematics for mathematics is making things simpler and our life less costly. The choice between literary approach and mathematics approach to economic reasoning is like the choice between horse cart and plane for travel from Hawassa to Addis. The time and cost savings associated with plane or cars has driven out the horse cart from the market of travel from Hawassa to Addis. An economist without the tools of mathematics is like a blind person swimming in the middle of an ocean. Till this person gives up, he will continue to struggle till he reaches an island; but his blindness has left him as incapable to identify whether swimming to the north, south, east or west is the shortest distance to an island. But an economist equipped with the tools of mathematics is like a normal person with motorboat or ship depending upon his personal inclination to each. As a result, most economic researchers are extensively using the tools of mathematics to economic reasoning. Specifically mathematical approach has the following advantages: ➢ The language used is more precise and concise; ➢ There are wealth of mathematical assumptions that make things simple and lifeless costly Page | 1

Calculus for Economists

Compiled by: Habtamu Adane

AAU: Department of Economics

➢ By forcing us to state explicitly all our assumptions as a prerequisite to the use of mathematical theorems, it keeps us from the pitfall of an unintentional adoption of unwanted implicit assumptions ➢ It helps us to understand relationships among more than two economic variables simply and neatly with which the geometric and literary approaches are at high probability of risk of committing mistakes.

1.2.

Functions

For calculus is the study of functions, it is necessary to see what a function is. Function, as you might remember it, is a unique mathematical rule that relates one or more variables to determine another variable. It is a special type of relation in which an independent variable (domain) can never be tied with more than one dependent variable (range). It is a relationship between numbers in which to each element in the input (domain), there corresponds exactly one element in the output (range).

For most functions in this course, the domain and range will be collections of real numbers and the function itself will be denoted by a letter such as f . The value that the function f assigns to the number x in the domain is then denoted by f (x) (read as“ f of x ”), which is often given by a formula, such as f (x) = x 2 + 4 .

Figure 1.1 Interpretations of function f (x)

Page | 2

Calculus for Economists

Compiled by: Habtamu Adane

AAU: Department of Economics

Functions are often defined using more than one formula, where each individual formula describes the function on a subset of the domain. A function defined in this way is sometimes called a piecewise-defined function. Here is an example of such a function.

Determining the natural domain of a function often amounts to excluding all numbers x that result in dividing by 0 or in taking the square root of a negative number. This procedure is illustrated in the following example.

Page | 3

Calculus for Economists

Compiled by: Habtamu Adane

AAU: Department of Economics

There are four methods of representing functions. These are: i.

Vein diagram A 1 2 3 4

B 1

1

2

2

3

3

C 1

1

1

2

2

2

3

3

3

4

4

4

4

5

The column which represents initial points of the rays is the column of domain and the column to which the rays are directed is the column of range. In this example, Vein diagram A and B are functions since we don’t have any two range values mapped from a single domain. But, C is not a function since the independent variable [domain] value 1 is tied to output [range] values of 2 and 5. ii.

Set of order pairs A = (1, 2), (2,3), (2,4), (3,6), (4,0) B = (0,1), (1,2), (4,8), (9,10), (3,0)

iii.

Equations (commonly used in economic researches)

In this example, A is not a function since domain ‘2’ is mapped to more than one element in the range but B is a function. A . Y= x + 4 is a function Domain: x+4  0; i.e.= x : x  −4 and Range : Y  0

Page | 4

Calculus for Economists

Compiled by: Habtamu Adane

AAU: Department of Economics

B . Y2 = x + 1 is not a function since Y can be either Domain : x  -1; Range: Y  iv.

x +1 or - x +1

Graphs(commonly used in economic education)

1.2.1. Types of Functions Generally there are two major types of functions; i.e. i.

Algebraic functions

ii.

Non algebraic function

⇒Algebraic functions

Page | 5

Calculus for Economists

Compiled by: Habtamu Adane

AAU: Department of Economics

Before generalizing what an algebraic function is (are), let’s see each type of algebraic functions. a. Polynomial functions: polynomial function of a single variable is given by the general form: p(x) = a 0 x0 + a1 x1 + a2 x2 + a3 x3 + ... + an xn ai  where  n  whole number

If n = 0, Y will be a constant function like Y = 3 If n = 1, Y will be a linear function like Y = 2x+3 If n = 2, Y will be a quadratic function like Y = 2 + 4 x + x 2 b. Rational functions: A rational function is a function which is the ratio of two Polynomial functions. That is,

R (x) =

p1(x) p2 (x)

c. The third type of algebraic function is a function which is the square root of Polynomial function. NB. An algebraic function is a function which is either polynomial, rational, or the third type of algebraic function. They can be with one variable or n-independent variables. ⇒Non algebraic functions/Transcendental functions There are four types of non-algebraic functions that are commonly used in economics; i.e. i.

Exponential function

ii.

Logarithmic function

iii.

Trigonometric function and

iv.

Incommensurable power functions

In this course you will be introduced with the first two non-algebraic functions and the rest are left for further studies. i.

Exponential Functions

A function of the general form f (x) = bx , where b is a positive number, is called an exponential function. Such functions can be used to describe exponential and logistic growth and a variety of Page | 6

Calculus for Economists

Compiled by: Habtamu Adane

AAU: Department of Economics

other important quantities. For instance, exponential functions are used in demography to forecast population size, in finance to calculate the value of investments, in archaeology to date ancient artifacts, in psychology to study learning patterns, and in industry to estimate the reliability of products.

Note that the domain of exponential function is the set of real numbers and the range of the function is that set of positive real numbers. Graphically, exponential functions look like the following.

Note: Every exponential function passes through (0, 1). Exponential functions obey the same algebraic rules as the rules for exponential numbers. These rules are summarized in the following box.

Page | 7

Calculus for Economists

ii.

Compiled by: Habtamu Adane

AAU: Department of Economics

Logarithmic Functions

Page | 8

Calculus for Economists

Compiled by: Habtamu Adane

AAU: Department of Economics

In calculus, the most frequently used logarithmic base is e. In this case, the logarithm loge x is called the natural logarithm of x and is denoted by ln x (read as “ el en x ”); that is, for x  0 . 1.2.2 Multivariate Functions Multivariate function is a function in which the dependent variable, y, is a function of more than one independent variable. For the case where the dependent variable, which we shall continue to call y, depends on two variables x and z we express the function y = f (x, y) which we read as y is a function of x and z . In this function there are two independent variables x and z . If we have possible values for x

and z we may substitute them to obtain the corresponding value of y. Since there are two independent variables, we may fix one of them, say x , at a particular value and change the other variable, z. This lets us investigate how y changes as z changes. The approach corresponds to comparative statics analysis in economics where economists investigate the effect of changing one

Page | 9

Calculus for Economists

Compiled by: Habtamu Adane

AAU: Department of Economics

variable while other things remain unchanged. We can, of course, also investigate the effects on y of changing x while z is held constant. 1.2.4. Inverse and Implicit Functions Inverse Functions Many important mathematical relationships can be expressed in terms of functions. For example, the demand for a product is a function of the price p. Mathematically, Qd ( p) = 50 − 2 p . In many cases, we are interested in reversing the correspondence determined by a function. Thus, we can express price of a product as a function quantity demanded, Q as below: 1 p( Qd ) = 25 − Qd 2

As this examples illustrates, reversing the relationship between two quantities often produces a new function. This new function is called the inverse of the original function and is usually denoted by f − 1. Note: It turns out that one-to-one functions are the only functions that have inverse functions. A function is one-to-one if no two ordered pairs in the function have the same second component and different first co𝑚𝑝𝑜𝑛𝑒𝑛𝑡𝑠.

Page | 10

Calculus for Economists

Compiled by: Habtamu Adane

AAU: Department of Economics

Example: Find f − 1 for f ( x) = x − 1 Solution Step 1: Find the domain of f and verify that f is one-to-one. The domain of f is [1,  ). f is oneto-one, hence f − 1 exists. Step 2: solve the equation y = f ( x ) for x y = x −1 y 2 = x −1 x = y2 + 1 Thus, x = f −1( y) = y 2 +1

Step 3: Interchange x and y y = f − 1 ( x) = x2 + 1

Step 4: Find the domain of f − 1. The equation f − 1( x) = x 2 + 1 is defined for all values of x, but this does not tell us what the domain of f − 1 is. Remember, the domain of f − 1must equal the range of f. From the graph of f, we see that the range of f is [0,  ). Thus, the domain of f − 1 is also [0,  ). That is, f − 1( x) = x 2 + 1

x0

Check: For x in [0,  ), the domain of f, we have

f − 1[ f ( x )] = f − 1( x − 1) =( x −1) 2 −1 = x −1 + 1 =x For x in [0,  ), the domain of f, we have

Page | 11

Calculus for Economists

Compiled by: Habtamu Adane

AAU: Department of Economics

f [ f −1 ( x)] = f ( x2 + 1) = ( x2 +1) −1 = x2 = x

x2 = x for any real number x

=x

x = x for x 0

Question: Find f − 1 for each of the following functions:

a.

f ( x) =

x , x− 1

b.

f ( x) =

3 , 4x − 4

x0

x1

c. f ( x) =

x +1 , x+ 2

d. f ( x) =

4 + 3 4x 2

x −2

Implicit Functions So far we have been working with functions in which dependent variable is an explicit function of the independent variables. In other words, all the functions we have studied have the independent variables on the right side and dependent variable y on the left side. For instance, in univariate 2x 2 functions, we have been considered functions like y = 2 x + 1, y = , y = ln( x + 1) and so on x −1 2

y = f (x ) . In multivariate functions, we have seen functions like

and each denoted as 2

y = 2 x21 + 5 x2 , y =

x1 , y = x1e 2x 2 and so on and each denoted as y = f ( x1 , x2 ). x1 + 2 x2

This ideal situation of an explicit function does not always occur in economic models. Frequently, the equations which arise naturally have the independent variables mixed with the dependent variable. In other words, the equations which appear in economic models have both the independent and dependent variables on the left side and a constant on the right side. Such type of functions are said to be implicit functions. The followings are some examples of implicit functions: 4 x + 2 y − 5 = 0, and y 2 − xy + x 2 = 0 are examples of univariate implicit functions (or implicit

function with one independent variable x) and denoted as F ( x , y ) = 0 .

Page | 12

Calculus for Economists

Compiled by: Habtamu Adane

AAU: Department of Economics

2 x12 − 3 x 22 + 4 y − 3 = 0 and 3 x1 x2 + x2 y 2 = 0 are examples of implicit functions with two

independent variables x1 and x2 . Such type of functions are generally denoted as F ( x1 , x2 , y ) = 0 Generally an implicit function with n independent variables will be denoted as F ( x1 , x2 , . . . xn , y) = 0

Notice that we use the capital letter F to denote implicit function and a small letter f to denote explicit function. For instance, we use y = f ( x1 , x2 ) and F ( x1 , x2 , y ) = 0 to de note an explicit and implicit function with two independent variables x1 and x2respectively. 1.2.5. Monotonic and Homogeneous Functions A monotonic function is a function which is either strictly increasing or strictly decreasing in its domain. 1. A function y = f (x ) is strictly (or monotonic) increasing on the interval (a, b) if its graph moves upward from left to right on that interval. More precisely, a function f is strictly increasing on (a, b) if f ( x1 )  f ( x2 ) whenever x1  x2 on (a b) . For instance, the function in figure (a) is strictly increasing. 2. A function y = f (x ) is strictly (or monotonic) decreasing on the interval (a, b) if its graph moves downward from left to right on that interval. More precisely, a function f is strictly decreasing on (a b ) if f ( x1 )  f ( x2 ) wheneverx1  x2 on (a b) .For instance, the function in figure (b) is strictly decreasing because the graph moves down ward from left to right. 3. A function y = f (x ) is neither monotonic increasing nor monotonic decreasing if it is strictly increasing on one interval and strictly decreasing on other interval in its domain (see Figure c). The function in Figure c is neither monotonic increasing nor monotonic decreasing since it is strictly increasing on the interval (0 b ) and strictly decreasing on the interval (b ) Y

Y

Y

X

Figure a

X

Figure b

X

Figure c

| 13

Calculus for Economists

Compiled by: Habtamu Adane

AAU: Department of Economics

Homogeneous Functions A function y = f (x 1 ,x 2 ,...x n ) is homogeneous with degree k if

f (tx1, tx2 ,...tx n ) = t k f (x1, x 2 ,...x n) = t k y In other words, a function f is homogeneous with degree k if it satisfies the property that when all the independent variables are changed by same proportion or amount t, the dependent variable y changes by tK. Example: Show that the following are homogeneous or not. If so, find the degree homogeneity.

a ) y = f (x1 ,x2 ) = x13 + 3x12x 2 + 3x1x 22 + x 23 Solution f (tx1 , tx 2 , ) = (tx1) 3 + 3(tx1) 2 (tx 1) + 3(tx1)(tx 2 ) 2 + (tx 2 ) 3 = t 3 x13 + t133 x12 x1 + t 33 x1 x22 + t 3x23 = t 3 ( x13 + 3 x12 x2 + 3 x1x 22 + x 23) = t3 f ( x1 , x2 ) Thus, from the above definition f is homogeneous with degree 3

b) y = f ( x1, x 2 ) = 2x 2 + 7 x1x 2 + 6x 22 solution f (tx 1 , tx 2 ) = 2(tx1) 2 + 7(tx1)(tx 2 ) + 6(tx 3 ) 2 = t 2 (2x12 + 7x1x2 + 6x 22 ) = t 2 2x12 + 7x 1x 2 + 6x 22 ) = t f (x1 , x2 ) Thus, f is homogeneous with degree 1

c ) y = f (x1 , x2 ) = 3x12 + 6x 2 solution f (tx1, tx2 ) = 3(tx1) 2 + 6(tx 2 ) = 3t 2 x12 + 6tx2 = t (3tx12 + 6x2 ) Thus, since t cannot completely factor out, f is not homogeneous.

Page | 14

Calculus for Economists

Compiled by: Habtamu Adane

AAU: Department of Economics

1.2.5. Application of Functions in Economics Functions, in economics, have diverse and versatile importance. To come up with neater theoretical results, it is good to model relationships in terms of mathematical functions. Though functions are used almost in all specializations of economics extensively, in this course you will be introduced with the most common microeconomic and macroeconomic functions. Detailed acquaintance with these functions is left for your “life in economics”. The most common microeconomic functions are demand functions, supply functions, production function, cost functions, revenue functions, profit functions, pollution functions, and other natural resource functions. The most common macroeconomic functions are consumption, saving, investment, and aggregate production functions. There are also, a lot of other economic functions, but we don’t discuss them here due to time limit we have. For the sake of introduction, let’s take some examples of economic functions. Definitions

Demand Function: Q d = f ( p ) = a − bp Supply function :Q s = f (p ) = a + bp Martket equilibrium :Q s = Q d Revenue function : R =( number of sold) ( unit price) = pQ Cos t function : C = f (Q ) =VC + FC = mQ + b , where b is constant Profit function :  = R (Q ) -C (Q ) = Revenue - cost Break − even point:  = R (Q ) - C (Q ) = 0 Marginal Analysis: In economics and business the term marginal stands for a rate of change. Marginal analysis is an area of economics concerned with estimating the effect on quantities such as cost, revenue, and profit when the level of production is changed by a unit amount. For example, marginal revenue, marginal profit, marginal cost, etc. Examples 1. Find the equilibrium point for the supply function S(p) = 3p + 50 and the demand function D(p) = 100 -2p.

Page | 15

Calculus...