REVIEW OF MATHEMATICAL CONCEPTS

Transcription

1 REVIEW OF MATHEMATICAL CONCEPTS 1 Variables, functions and slopes A variable is any entity that can take different values such as: price, output, revenue, cost, etc. In economics we try to 1. Identify the variables. Measure them 3. Study their behavior in relation to other variables to understand how and why these variables change 4. Find the conditions that will lead to the optimal values of the variable. We express the change in the variable in response to a change in another variable in term of a function A function refers to the relationship between one (dependent) variable s value and the value of one or more (independent) d variable(s). Functions are expressed in functional forms The functional relationship can be expressed in schedules (tables), graphs, or equations 3 The general form of an equation looks as the following Y ƒ(x) Where, Y is the dependent variable, and X is the independent variable. Independent variable is the variable whose value is determined independently of any other variable under consideration Dependent variable is the variable whose value depends on some other variable. 4 To explain the components of the linear function equation Y a + bx Where, Y is the dependent variable, X is the independent variable, a is the intercept, and b is the coefficient of X, which measures the slope of the line. 5 The slope of the line measures the rate of change of the dependent variable with respect to a change in some independent variables. Slope Y Y Y1 X X X1 Since much of economic analysis is based on the incremental or marginal analysis of variables, slope of the function is very important in economics. It tells us what would be the change in the dependent variable as a result of change in the independent variable. 6 1

2 Example: TR P x Assume P is known so TR f () If P 5, Tabular form of the function TR Graphical representation of the function TR Equation form of the function Y a + bx TR a + b TR + 5 TR 5 From the above table, graph, and equation, the intercept a;and and the coefficient slope b TR / (15-1) / (3-) 5/1 5 TR is expected to change by $5 for every one unit change in. If the relationship between dependent and independent variables is linear (as in this case), the change is represented by the slope. Most of economic analyses attempt to study how a change in one or several variables affect the change in another variable and in what direction. 9 1 For example, P R and C π. So the change in price ultimately results in change in profit, which may be positive or negative depending on other factors. P (demand function) R (TRf function) C (TC function) π (π function) Marginal and incremental Analysis Marginal analysis refers to 1. The change in dependent variable as a result of a one unit change in an independent variable (discrete change definition). The consideration of a small change around some given point (continuous change definition). 11 1

3 Many economic decisions rely on marginal analysis, such as hiring additional worker, purchasing additional machines, etc. However, many other decisions are also taken in incremental way. Incremental analysis is used when total change is considered if P and as a result by 1 and TR by then Marginal Revenue TR/ /1 Incremental Revenue 13 Functional Forms For the purpose of illustration and simplicity we will often use a linear function, but there are many instances when a linear function is not the proper representation for changes between variables. So we use non-linear functions. For example, most common cost functions are non-linear functions, and non-linear TR functions are also common, 14 Most famous non-linear functions in economics are quadratic functions and cubic functions Linear (straight line) function: Y a + bx uadratic function: Y a + bx + cx Cubic Function: Y a + bx + cx + dx 3 Consider the following demand schedule and demand curve 15 Demand Schedule P d P Both the demand schedule and demand curve show a linear relationship between P and. f (P) a + bp demand function Intercept: when P, a 7 Coefficient: when P by one ( P -1), by 1 ( 1), b ( / P) (1) / (-1) -1 Therefore, this demand function could be written as 7 1P demand function (1) But TR is a function of, not of P TR f () 18 3

4 So we have to express P in terms of in order to eliminate P From equation (1) 1 P 7 P 7.1 inverse demand function () Substituting in TR equation, TR P x (7-.1) TR 7.1 TR equation (3) As we can see, a linear demand function results in a nonlinear TR function (quadratic equation) TR (1) () (3) (4) (5) (6) (7) 19 TR If the independent variable is raised to a third power we will have a cubic function such as TC In both cases (quadratic, cubic) the curves will not be a straight line. TR 7 1 TC There are other forms used in economics such as a. Exponential: Y X a TC Y 3 X 4 4

5 b. Logarithm: Y log X Y c. Reciprocal: Y 1/X Y X 5 X 6 Continuous vs. Discreet We will assume for the purpose of the analysis that all economic variables are related to each other in a continuous fashion although many of these variables are discrete such as people, output, and machines. 7 USING CALCULUS Calculus: is a mathematical technique that enables one to find instantaneous rate of change of a continuous function. That is, instead of finding the rate of change between two points ( Y/ X), we can find it at any given point on the function. It can be applied only if the function is continuous Calculus is actually a slope-finding technique. Calculus helps obtaining optimum values of function (maximum or minimum). 8 Finding the slope for linear function is easy because the slope is the same at any point and between any two points. It is equal to the coefficient b. So, no need for calculus. Calculus (slope-finding) is important if the function is nonlinear because slope is different at every point. Using calculus, it is possible to measure the slope exactly at one point. The slope here is a measure of the change in Y relative to a very small change in X. To know how very small, we need to use the 9 concept of derivative Thus, the derivative (the slope) is a measure of Y relative to a very small X. It is written as: dy/dx lim Y/ X x Derivative turns out to be the slope of a line that t is tangent t to some given point on a curve. 3 5

6 Rules of differentiation Differentiation is the process of determining the derivative of a function, i.e. finding dy/dx when Y ƒ(x). There are 6 main rules to find a derivative: 1. Derivative of Constant The derivative of a constant is zero. Derivatives measure the rates of changes. A constant by definition never change in value. Thus, the change of a constant is zero. dy/dx (or y') of a constant is zero. if Y a, "a" is constant, then dy/dx If Y 1, then dy/dx Derivative of Power function If Y ax n where a is constant and n is the exponent (power), then dy/dx (n)(a) X (n-1) Y 1X 3, dy/dx 3(1)X (3-1) 3X If X 5, dy/dx 3 (5) 75 A very small change in X results in a rate of change of Y equal to Derivative of Sum and Difference Sum: U g(x) and V h(x). U and V are functions of X, then: If Y U + V, dy/dx du/dx + dv/dx The derivative of the sum is the sum of the derivatives. U 3X,V 4X 3, then Y U + V 3X + 4X 3, and dy/dx 6X + 1X 34 3.Derivative of Sum and Difference Difference: If Y U V, where U g(x) and V h(x), then dy/dx du/dx - dv/dx The derivatives of the difference is the difference of the derivatives U X, V 4 X, then Y U V X (4 -X) X 4 + X, and dy/dx X Derivative of Product Y UV where U and V are functions of X, then, dy/dx U (dv/dx) + V (du/dx) The first times the derivative of the second plus the second times the derivative of the first. U 5X, V 7 X, then Y UV 5X (7 X), and dy/dx 5X (-1) + (7 X) (1X) - 5X + 7X -1 X 7X - 15 X 36 6

7 5. uotient Rule: If Y U / V, then dy dx V(dU / dx) U(dV / dx) V Y (5X( -9)) / 1X, then dy (1X )(5) (5X 9)(X) 5X 1X + 18X 4 dx (1X ) 1X 18X 5X 4 1X 18 5X 3 1X 6. Chain Function Rule If Y ƒ(u) and U ƒ(x), then dy/dx dy/du * du/dx Y U 3 + 1, U X ; so, Y (X ) 3 + 1, and then dy/du 3 (X ), and du/dx 4X dy/dx 3 (X ) * 4X 3(4X 4 ) * 4X 48X Economic Application Slope If demand function is 7 1P find the slope Slope b d/dp -1 So, no matter what is the value of P, with respect to P is always the slope b -1 That is why there is no need for calculus in linear equations. 39 Marginal Revenue: If demand function is 1.4P, find MR function. We have to find TR function but since TR is a function of we should first find the inverse demand function P f ().4P 1 P (1/.4) (1/.4) 5 5 TR P x (5 5 ) 5 5 MR dtr/d Marginal Cost: If TC , find the MC function. MC dtc/d Partial Derivatives and Multivariate Functions Multivariate functions have more than one unknown variable. So the maximization procedure is different than for single variable equations Most economic variables depend on more than one economic variable. Y f (X, Y, W, Z,.) For example, demand equation d f (P, I, P s, N,..)

8 Maximization Procedure 1.Take partial derivative of each unknown variable..set each partial derivative equal to zero. 3.Solve the resulting system of simultaneous equations for all unknown variables. The partial derivative, Y/ X, measures the marginal change in Y associated with a very small change in X, holding constant all other factors. 43 if -1P + 5I +P s + 4N Where, uantity demanded; P Price of the good; I Customers income; P s Price of substitute goods; N Number of Customers To know the change in as a result of a change in one of the independent variables, take the partial derivative of with the independent variable and held all other variables constant (the derivative of a constant is zero) / P -1P ; / N 4 Suppose TR f(x,y) where X sales of X and Y sales of Y, and given: TR 8X X - XY 3Y + 1Y, Find X * and Y * that maximize TR, and find the maximum TR First, find partial derivatives: TR/ X 8 4X Y (treat Y as a constant). TR/ Y -X 6Y + 1 (treat X as constant). Second, set partial derivatives equal to zero and solve for X * and Y * : X Y Y 8 4X, and 1 X 6Y X 1 6Y Thus, Y 8 4 (1 6Y) Y Y -3Y -3 Y* 3 / Substituting into X equation, X* 1 6(13.913) 16.5 Therefore, the maximum TR (TR * ) is: TR * 8(16.5) (16.5) 16.5(13.913) 3(13.913) + 1(13.913) Second Derivative So far we have discussed only the first derivative (first- order condition) (dy/dx or y ), But it is important in many economic applications to find the second derivative (second-order order condition) (dy /d X, or y ) Y X 4 dy/dx 8X 3 First-order Condition dy /d X 4 x Second-order Condition 47 Maximum and Minimum Values of a Function The main objective of managerial economics is to find the optimal values of key variables. This means finding the best value under certain conditions. In economic analysis, finding the optimum means finding either the maximum or minimum value of a variable. 48 8

9 Y Y A X B X 49 5 At both pints of A and B, dy/dx To determine whether the optimal value is at maximum or minimum: 1.Take the first derivative of the equation;.to determine the extreme of the function, set dy/dx equal to zero and solve for the optimal value of the independent d variable (*) that maximizes or minimizes the objective function 51 To distinguishing maximum from minimum, find the second derivative of the equation (d Y/dX ). 1.If d Y/dX < (if slope changes from positive to negative), the value is a maximum..if d Y/dX > (if slope changes from negative to positive), the value is a minimum. Taking the second derivative tells you whether the extreme is a maximum or a minimum. 5 More Examples 1. TR Maximization If 45 16P, what is the * and price P* that maximize TR? First find the inverse demand function: P (45/16) (1/16) So, TR equation is: TR P (45/16) (1/16) dtr/d MR (45/16) 1-1 (1/16) -1 (45/16) (/16) (45/16) (1/8) 53 Set MR equation (dtr/d) equal to zero, which means the last unit produced will not add any additional revenue dtr/d (45/16) (1/8) * (45/16) x (8/1) 5 P* (45/16) (1/16)(5) To check whether * and P* are in max or min conduct the second derivative d Y/dX - 1/8 < maximum Maximum TR (45/16)(5) (1/16)(5) MR (45/16) (1/8)(5) 54 9

10 8 P TR 3164 MR D TR. Profit Maximization Suppose P, and TC , find * and P* that will maximize profit. Since π TR TC, first find TR equation. So, Inverse demand function: P 17 1 TR P (17 1) 17 1 π TR TC (17 1 ) ( ) dπ/d 17 * 17/ 4.86 P* 17-1 (4.86) 13.4 d π/d - < maximum profit OR 57 π TR TC dπ/d (dtr/d) (dtc/d) dtr/d dtc/d MR MC * 17/ 4.86 To check second-order condition For TR: d TR/d - < maximum TR For TC: d TC/d + > minimum TC uadratic Equation if you can t find the roots, you can use the equation below to solve for the roots: If Y ax + bx + c, then b ± b 4 ac X a Suppose TR 5; and TC What is the level of output that will maximize profit? 59 π TR TC dπ/d Re-arrange to fit the general formula of a quadratic function Y ax + bx + c So, dπ/d The values of that set this quadratic function equal to zero can be found using the formula for the roots of a quadratic function 6 1

11 X b ± b a 4 ac In our example: a -.3; b +6; c -1 6 ± 6 ± 4 6 ± (.3)( 1) 6 ± 36 1 (.3).6 1 * ( ) / * ( ) / Both quantities satisfy the first order conditions but only one satisfies the second order condition d π/d substitute 1 * and * For 1 *: d π/d -. 6(1.84) > For *: d π/d -. 6(18.16) < Thus, only * satisfies maximization conditions 6 Classical production function (Cobb-Douglus) AL α K β Transfer into log log log A + α log L + β log K