 Welcome to the fourth session of managerial economics. Basically, we are on the first model of managerial economics which talks about introduction and fundamental to managerial economics. So, if you remember in the last class, we just discussed about the functional relationship between the economic variables, how they are related, what are the different form to represent them and then we discussed some of the important economic function like demand function, bivariate demand function and multivariate demand function. So, today's session will focus on the different type of function gets used in a typically in a demand function. How to measure a slope and what is its use in the economic analysis? Different method to analyze the slope, find out the slope or measurement of the slope. Then derivative of the various function and in the next session will basically take the optimization technique and constant optimization. So, till now all our discussion, if you look at it just focuses the demand function, but apart from the demand function in there are certain other topics also where we generally use the relationship between two variable in a functional form like production function which represents the relationship between the inputs like labor and capital with the output. We talk about the cost function where it is basically the relationship between the output and the cost of the production associated with that. Then we talk about the total revenue function and it represents the combined function of quantity, produce and price function based on the demand function. And sometime also we talk about a profit function and this is the profit is basically as you know it is a difference between the total revenue and total cost function. So, whenever there is a change in the total revenue and wherever there is a change in the total cost, it generally affects the profit. So, profit function is basically the relationship between the profit, revenue and cost. Then we will discuss what are the general forms of function used in the economic analysis. So, one way we are clear that we use a functional form to understand the relationship between two types of variable and the variable typically in this case it both all the variables they are the economic variables. So, there are three type of function we used in analyzing the relationship between the variable. One is linear function, second one is the non-linear function and third is the polynomial function. Linear function is used when the relationship between the dependent and independent variable remain constant, non-linear function is used where the relationship between the independent variable and dependent variable is not constant rather it changes with the changes with the change in the economic variable and polynomial function is getting used particularly those who have the various terms of measure for the same independent variable. So, we will check all this function in more details by taking individually. So, in a linear function the relationship is linear, the change in the dependent variable remain constant throughout for a one unit change in the independent variable irrespective of the level of the dependent variable. Whatever the change in the independent variable the change in the dependent variable is remain constant in case of a linear function. Suppose, we are taking a demand function which says that q x is equal to 20 minus 2 p x. What does it signify? Each one rupee change in price the demand for commodity changes by 2 units because if you look at look at the second term of this functional form there is minus 2 p x. So, one rupee change in the price the demand for the commodity changes by 2 units. When you represent graphically the linear demand function is always a straight line because the change in the dependent variable remain constant, constant out for one unit change in the independent variable. So, this is just a hypothetical way to understand the linear demand where in the vertical axis we are taking price and horizontal axis we are taking quantity. So, if you look at when the price is changing the quantity demanded is also changing. So, initially when the price is 2 dollar the quantity demanded is 100 units. When the price increase from 2 unit 2 dollar to 3 dollar the quantity decreases by 100 units to 50 unit. So, if you look at the demand curve at each point it gives a price and quantity combination. Here quantity demanded is the dependent variable whenever there is a change in the price that leads to change in the quantity demanded also. And if you look at in the percentage wise also when the price changes from 2 dollar to 3 dollar it is 50 percent change in the price. And when the quantity demanded decreases its decreases from 100 to 50 again this is a 50 percent decrease in the quantity demanded. So, 50 percent increase in the price is leading to 50 percent decrease in the quantity demanded. And at this point the relationship between these two variables linear as it is constant whenever the price point change from one to another. Then we will discuss the nonlinear demand function where the relationship between the dependent and independent variable is not constant it changes with the change in the level of independent variable. So, in the previous case the way we are discussing that 50 percent change in the price will bring 50 percent change in the quantity demanded. However, in case of nonlinear demand function the unit of change it may not constant with each change in the price. When the price change from 1 dollar to 2 dollar or 2 dollar to 3 dollar that may not necessarily the change the same kind of change in the level of the quantity demanded. So, if you are taking a nonlinear demand function that is dx which is a function of price x. So, here x is the product px is the price of x and dx is the quantity demanded of x. Taking the functional form in a nonlinear dx is dx is a px to the power minus p. Here a and b they are the constant minus p is the exponent of variable px constant a is the coefficient of variable px. If you simplify it further maybe you are taking a number term over here suppose dx is 32 px to the power minus 2 or maybe we can take just a reciprocal of this that 32 minus px square. So, in this case the demand function produce a nonlinear or curvilinear demand function demand curve. It means it is not a straight line the change in the independent variable is not constant throughout whenever there is a change in the price. So, this is example of a nonlinear demand schedule that how it changes when there is a change in the price. So, when price is 1, quantity demanded is 32, price is 2, quantity demanded is 8, price is 3, quantity demanded is 3.5. Similarly, for 4, 5 and 6 if you look at the trend the quantity demanded is going on decreasing when the price is increasing. But here the point is not to establish a negative relationship or inverse relationship between the price of x and dx. The point what we are discussing here is that with each change in the price point the quantity demanded is the change in the quantity demanded is not remain constant. The change in the quantity demanded with respect to each price point is become different and this is a typical feature of a nonlinear demand curve. And when you plot this in a graph we generally get a curvilinear relationship which is in the form of a graph which is in the form of a curve we do not get a line, we do not get a straight line which is generally the representation of a linear demand curve. So, this is the graphical representation of a demand curve and if you look at the different point of the demand curve the change in the quantity demanded is not remain same. So, when the if you are taking here the p is the price which is represented in the vertical axis and q is the quantity which is represented in the horizontal axis. When the price is changing from 100 to 80 the quantity demanded is increasing and again when it is decreasing from 80 to 20 the quantity demanded is again increasing. But if you look at the change in the price point from 100 to 80 and the corresponding change in the quantity demanded from may be 10 to 12 that is not remain constant. And with the next change in the price point from 80 to 20 the there is a significant amount of change in the quantity demanded that is from 12 unit to 50 unit. So, in case of a nonlinear demand curve even if the demand changes along with the change in the price point, but there is always a difference in the amount of change at different kind of price point. The third kind of function generally used in the economic analysis is polynomial function and what is polynomial function? The function contains many terms of the same independent variable are called the polynomial function. So, we consider a short run production function here where output is a function of the labour and output is represented as q and labour is represented as l. So, putting it in a functional form q is the function of l over here. The polynomial function takes a takes different type of functional form sometimes it is take a quadratic function sometimes it takes a q b function sometimes it is take a power function. So, taking the example of the same short run production function where q is the output l is the labour and a b c d are constant associated with the different coefficients. It takes a quadratic function it takes a form by cubic function or it takes a function of the power function. So, when it is become a quadratic function q is equal to a plus b l minus c l square your a b c are constant. When you take a cubic function then it is a plus b l plus c l square minus d l q where again a b c are the constant associated with the coefficient. When it takes a power function here it is a l to the power b here a and b is the constant and b is the coefficient associated with variable l. So, polynomial function may take a quadratic function polynomial function may take a cubic function and polynomial function also takes a power function. When you represent this polynomial function graphically with all this three type of function whether it is quadratic cubic and function. So, graphically if you look at a cubic function how this cubic function when the polynomial function takes a cubic function. Suppose we take l over here l is the labour q is the output. Now, the cubic function takes a this type of shape. Now, what is this curve? This curve is total product curve and total product is dependent on the output and the labour. So, if you are taking q over here and l over here cubic function take a form which is may be not a straight line not exactly a curve it follows a different kind of change at each change in the l. So, how this q and l they are related here l is the independent variable and q is the dependent variable. So, l is the whenever there is a change in the l that will bring change in the q. So, in this case this in a case of a cubic function l changes along with this q changes, but the change in the q is not constant with each change in the labour. Now, if you now if you take a case of a quadratic. So, with the same short run production function we take l over here in the x axis and q over here in the y axis. Now, it is a quadratic. So, just follow a there is no cyclical function over here or there is no much fluctuation here and this total product curve is this and this is a typical example of a quadratic function. Now, how graphically when you represent the power function of the power function of the polynomial function. So, the power can take any value the coefficient associated with b or the coefficient b associated with l it can take any form. So, if you remember the power function is q is equal to this is a to the l to the power b. So, b can take a value which is equal to 1 b can take a value which is less than 1 or b can take a value which is greater than 1. So, in this case if you represent graphically again taking the same formulation that here it is labour here it is output. When we get the value of b equal to 1 it is a straight line the total product curve is a straight line. When b is less than 1 we get a this kind of shape and when b is greater than 1 we get a this type of shape. So, if you take a power function in case of a polynomial function the power associated with the coefficient b can take any value may be it is sometime it is 1 sometime it is less than 1 and sometime it is greater than 1. So, it is whether it is 1 whether it is less than 1 and greater than 1 when you represent that graphically or when you represent that geometrically this is the shape what we get for the different kind of function. So, polynomial function takes the quadratic function polynomial function also can be represented through cubic function and polynomial function also represent through a power function. And each time the value of b changes in time each time the graphical representation changes depend on the value of the coefficient. Now, how do you find what is the degree of a polynomial function? So, degree of a polynomial function if you taking a functional form which is q which is equal to a plus b l minus c l square the highest power is 2. So, polynomial this is a polynomial function of degree 2 and a polynomial function of power 2 is also called a quadratic function. So, in order to identify what is the polynomial function it is always the highest power associated in this functional form. So, in this case the highest power is 2. So, the polynomial function of degree 2 we can say is having this functional form. So, polynomial function of power 2 is also this is also called a quadratic function. Let us take one more example in term of a cubic function. So, here what is the functional form? The functional form is q is equal to a plus b l plus c l square minus d l q. Here the highest power associated with the coefficient is 3 and this is a polynomial function of degree 3. A function of power 3 is also called cubic function. So, in the previous example in the functional form the highest power is 2. So, that is why it was a quadratic function of degree 2 and in this typical functional form the highest power is 3 and that is the reason it is called as a cubic function because the function of power is 3. Then we will say we will see what is the degree of polynomial function when there is the when the polynomial function is in term of a power function. So, here in this case the functional form q is equal to a l to the power b. The range of power is between b greater than 1 b is equal to 1 and b less than 1. So, in this case except 0 it can take any power. So, it may be less than 1 in the negative form equal to 1 or it may be greater than 1. So, in this case b taking value of 0 is not possible it takes any other value and this is the example of a power function under the polynomial functions. So, there are three type of function one is linear, second one is non-linear and third one is polynomial. In case of polynomial again polynomial function again we represent in term of a quadratic function in term of a cubic function or in term of a power function. And every time the degree of it changes on the basis of the power associated with the functional form. The highest degree in case of a quadratic function is 2, the highest degree in term of a cubic function is 3 and the highest degree in term of a power function is greater than 0. It may take a negative value or it may take a positive value. Now, how to solve a polynomial function either through the factoring methods or through the quadratic formula and what is the property of a quadratic or a cubic equation when there is more than one solution. So, polynomial function can be solved by factoring methods and quadratic formula and by both these methods it can be solved and property of quadratic and cubic equation that it has more than one solution. Now, solving this quadratic and cubic equation we have two methods one is factoring method and second one is the quadratic formula. Now, we will check one by one what is factoring method and what is cubic equation or what is the quadratic formula to solve this cubic equation. So, we will take a function that is y is equal to x square plus x minus 12. So, in the factoring method what is the first step? The first step is we have to set y is equal to 0. So, taking this x square plus x minus 12 is equal to 0. Now, what is the second step? We have to factor the equation. So, this x square plus x minus 12 can be also represented with x square plus 4x minus 3x minus 12 which is equal to 0. Now, simplifying again this takes x plus 4 and x minus 3 which is equal to 0. We get two value of x over here if you simplify one is x is equal to 4 the second is x is equal to sorry minus 4 and x equal to 3. If you look at minus 4 has no meaning in economic analysis. So, basically you will go with the value of positive value that is x is equal to 3 and we solve this functional form with the value of x which is equal to 3. So, even if we are getting two value one is minus and second one is plus and typically since we are applying this in an economic analysis there is no maybe significance when we get a negative value of any variable. So, that is the reason we are ignoring the first value of x which is minus 4 and we are going with the second value of x which is equal to 3. So, this is so if you look at this solution is through factoring method. So, this is the solution with the solution of a polynomial function by using the factoring method. Now, we will check the second one through the quadratic formula. Now, what happens in case of a quadratic formula? The quadratic equation is set equal to 0 that is the first step and the equation is again the equation is factor for obtaining the two value of the variables x and y following the formula that is minus b plus minus b square minus 4 a c divided by 2 a. So, let us see how we can solve a polynomial function through the quadratic formula. Now, what is the functional form over here? The functional form is y is equal to x square plus x minus 2 l which is equal to 0 because what is our first step? First step is to set the quadratic equation or whatever the functional form we have to set that equal to 0. Now, what is the implication of this equation? Now, x is equal to how to factor it again to get the value of x because the first step is always to set it is equal to 0 and second we factor out this equation in order to get the two values of the variable x and y or if there is only one value the value of the x. So, if you are following then this is this minus b plus minus b square minus 4 a c by 2 a. Now, what is the implication of this equation? a is equal to 1 this formula taking this equation a is equal to 1 b is equal to 1 and c is equal to minus 12. So, this is the formula to factor out the equation this is the second step. The first step is to set this equal to 0 that is x square plus x minus 12 is equal to 0 and from this equation we get the value that is a is equal to 1 b is equal to 1 and c is equal to minus 1. Now, we will substitute this value of a b and c in case of in the quadratic formula. So, this is x is equal to minus 1 plus minus 1 square 4 a c and 2 a. So, this is 2 and 1. So, x is equal to minus 1 plus minus 49 root divided by 2 which you simplify again this minus 1 plus 7 by 2. So, in the previous case once we have identified the value of a is equal to 1 b is equal to 1 and c is equal to minus 12 we will put this value into the quadratic formula and we are getting this is the value of x. Now, this is 1 plus minus it means x is having 2 values here which satisfy the quadratic equation because this is minus was plus minus 7 which is divided by 2. So, we will get 2 value of x and it satisfy the quadratic equation. Now, so if we take the first value that is x is equal to minus 1 plus 7 by 2. So, we get x is equal to 3 and if we take second value that is minus 1 minus 7 divided by 2 then we get x is equal to minus 4. So, we are getting 1 negative value we are getting 1 positive value even if this is negative still it is satisfying the quadratic equation. So, we have 2 value 1 is positive 1 is negative. So, basically we ignore the value which has some negative which is with negative sign and we always go for the positive sign value because it makes some sense in the economic analysis when we go for the positive value. So, if you remember the previous solution what we did through the factoring method also we got 2 value of x 1 is 3 and second one is minus 4. So, whether you solve the quadratic or cubic equation or whether you solve the polynomial function either by taking factoring method or by taking the quadratic formula you always get 2 values of x and on the basis of the value whatever we get for x on that basis we decide which value to take for the further analysis and which value to ignore over here. So, polynomial function can be solved by using by 2 methods one is the factoring methods and second one is the quadratic formula and we get same value for the x taking any specific formula either through the factoring method or through the quadratic formula.