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Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay
Lecture – 6

Then we will move to our next topic and that is basic tools of economic analysis and optimization technique.

Now, what is the learning objective or session outlines of this topic? We will first look at what is the functional relationship between the economic variables and then we will discuss some important economic functions. Then we will see slope and its use in the economic analysis and derivatives of various functions, optimization techniques and finally, how we do optimization with a constant.

So, now coming to the relationship between economic variables. Now, what we consider as an economic variable? Any economic quantity, value or rate that varies on its own or due to change in its determinants is an economic variable. Any economic quantity or value or the rate, the variables rate, any variable whether its value or rate that changes due to its own or due to change in the determinants of each is an economic variable.

So, when the variable changes the value due to its own value or due to some other factors, those are considered as economic variables. We can take the example as demand for a product whether it is 10 units or 12 units or 13 units, every time it is changing a value. The demand is not constant. So, this is an economic variable.

Price of the product, wage rate, and advertising expenditure these are few more examples of the economic variable, where the value get changed either due to own factors or due to change in the determinants, that is the factors affecting the demand for the product.

Suppose you take an example like why there is a change in the product price or why the price of goods increases, when the cost of production increases. Suppose you take the case of this marker, the cost of production is 10 rupees. So, price is on the basis of 10 rupees. When you add a normal profit and a tax with this, it becomes the market price forthis marker. Suppose the market price of this marker is 13 rupees and out of this, cost of production is 10.
So, what is the determinant of this price of this marker? The cost of production. Now, on what basis there will be an increase in the market price of this marker? When there will be an increase in the cost of production. Suppose, the increase in the cost of production has become from 10 rupees to 11 rupees. So, the market price given all other factors, the value of all other factors remains constant, and the market price of this marker will go up by 1 rupee. So, if it is 13 rupees, now it is 14 rupees. So, product price in this case, the product price is changing due to change in the value of its determinants. So, this is one example of the economic variable.

Now, all these economic variables are interrelated and interdependent. All economic variables they are not independent but they are interdependent and they are interrelated. This implies that a change in one variable cause a change in the value of other related variables. If they are interrelated or interdependent, when value of one variable changes, generally that leads to change in the other variable. Suppose, we take an example of price and quantity of a product. If you take the same example, that is price of marker, earlier the price of marker was 13 rupees. Due to change in the cost of production, the price of marker is 15 rupees. Now, price and quantity of product, they are interrelated. The price is more. Now, if it is from 13 to 15 rupees, few customers who cannot afford to pay 15 rupees for that, they will not buy this product. So, this increasing price affects the quantity of the product what is getting sold in the market.

So, price increases leads to decrease in some quantity of product that is getting sold in the market. So, if we look at price and quantity of product, they are interrelated. Because of that, when there is a change in the price or when there is a change in the value of one variable, that leads to change in the value of the other variable.

In this case, typically the price of marker gets changed and that leads to change in the quantity of the products getting sold in the market. Similarly, income and consumption expenditure. Suppose, if your income is more, you consume more and you spend more. If income is less, you spend less. So, if you look at income and consumption expenditure, they are interrelated. So, value of one gets changed, due to change in the value of the others.

Similarly, interest rate and demand fund. If the interest rate is less, more people go for loan. If the interest rate is high, there is at least decrease in the demand for loans because the interest rates are on the higher side. So, economic variables are interrelated and they are interdependent. When there is a change in the value of one variable, that leads to change in the value of other variables because both of them are interdependent and interrelated.
Now, what are the different kinds of economic variable? Variables are classified on the basis of economic variables. So, the first category is dependent and independent variables. The value of this variable depends on the value of other variable in case of dependent variable. Independent variables are those where the value of these variables changes on their own or due to some exogenous factors.

So, dependent variable is one where the value of this variable is always dependent on the value of the other variable. Independent value is the value of this variable changes due to their own or may be due to some exogenous factors, but not due to change in some other variable.

So, if you take the example of computer price and demand for computers. Here, demand for computers is dependent, computer price is independent because demand for computers is dependent on the computer price. When there is an increase in the computer price, it leads to decrease in the demand for computers. When there is a decrease in the computer price, that leads to increase in the demand for computers.

So, in this typical case, the computer price is the independent variable and demand for computer is the dependent variable. Similarly, there is an increase in the petrol price. If you look at nowadays, there is an increase in the petrol price. Why there is an increase in the petrol price because there is a hike in the import oil price.

So, in this case, which one is dependent and which one is independent? Petrol price is a dependent variable, because petrol price is related with the value of the import oil price. Whenever there is a change in the import oil price, either increase or decrease in the import oil price, and that leads to change in the value of petrol price. So, if there is an increase in the import oil price, that leads to increase in the petrol price. If there is a decrease in the import oil price, that leads to decrease in the petrol price. So, in this case, petrol price is dependent and input oil price is the independent variable. So, dependent variable is one where the value of that variable is dependent on the other variables. Independent variable is one where it is not dependent on any other variable for its value, rather the value changes due to own or due to the exogenous factor.

The second kind of economic variable is endogenous and exogenous variable. Now, what is endogenous variable? Endogenous variables are those where the value of these variables is determined within the framework of the analysis model. So, if there is a model between price and quantity, the endogenous variable is one where the value of price or value of quantity has to be determined within this specific framework or specific model. Exogenous variables are what the value of these variables are determined outside the framework of the analysis model. So, any exogenous factor or any external factor will decide what is the value of this exogenous variable.

Now, we will take the example of the endogenous and exogenous variable. If you are going to the same petrol price example, domestic oil price is endogenous and international oil price is exogenous variable. So, domestic oil price is dependent on the import oil price. So, in this case, the value of the domestic oil price is decided within the framework from the import oil price. However, exogenous variable is international oil price. International oil price is not strictly on the basis of the import oil price. It has some other factors and the value, those other factors also decide whatever is the international oil price. So, in this case, domestic oil price is the endogenous variable, whose value is determined within the framework and international oil price is the exogenous variable, whose value is decided on the basis of the external factors.

Now, when we analyze the relationship between the variables, we can analyze this or we can present the relationship between these variables through three methods. One is tabular method, second one is functional method and third one is the graphical method.

So, if we are taking the example of price, demand and supply. Suppose, there are three variables. This relationship between price, demand and supply can be presented through a graphical analysis. That is, through a supply curve, and through a demand curve taking quantity in the right axis and price in the left axis. We can do a tabular, where we can find out what is the demand and supply when the price is 1 rupee, 2 rupees, 3 rupees and 4 rupees. So, this is the tabular representation of the relationship between the variables and this is the graphical relationship between these variables and third one is functional, which deals with the cause and effect relationship, which we analyze or which we present through a functional form.

So, in this typical example, when we are deciding the relationship between demand and price, it will take a functional form, which is equal to Q d which is equals to a minus b P, where a and b are constant and P is the price of the product and Q is the quantity demanded for this product. So, relationship between these three variables can be presented through graphical method, through tabular method or through the functional method.

So, tabular and graphical form is useful when number of variables and observations are small. If it is two or three variables, then tabular and graphical form can be used. But if the number of variables is more, specifically in case of economic analysis, all the economic variables are interrelated and interdependent.

So, the number of variables and the number of observations are more. So, in this case, it is always good to use the functional form in order to represent the relationship between these variables. So, most economic problems are complex. It involves large number of variables because they are interrelated and interdependent. In such cases, the economist uses a mathematical tool known as function to express the relationship between the economic variables. So, the tool is functional and we generally call it as a functional representation of relationship between the economic variables. Economic analysis is more useful because there are large numbers of variables. Next, we will see what is a function because function is used to represent the relationship between different economic variables.

So, it is a mathematical tool used for expressing the relationship between economic variable that have a cause and effect relationship. When they are interrelated, if one is cause and other is effect and it is the relationship between different economic variables. It is a mathematical tool. Function is a mathematical tool used for expressing the relationship between the economic variables.

There are two types of functions. One is bi-variable function and second one is the multivariable function. Bi-variable function involves only two variables and multi-variable function has one dependent and more than one independent variable. In case of bivariable function, it has only two variables. One is dependent and another is independent. In case of multi-variable function, there is only one dependent and more than one independent variable.

Now, we will take an example to understand this bi-variate function and multivariate function. If the value of variable X depends on value of variable Y, then the relationship between the two is, Y is a function of X, where Y is the dependent variable and X is the independent variable. So, this is a typical function, which expresses the relationship between Y and X, where Y is the dependent variable and X is the independent variable and Y is the function of X.
Now, taking the example of a demand function. If you consider P is the price of the product and d P as the demand for the product, the demand for the product is always dependent on the price for the product. So, in case of your bi-variate demand function, we are taking that there is only one dependent variable and one independent variable. In this case, we use this function d P, f is a function of P and this is a bi-variate demand function, where the demand for the product is dependent only on price.
Now, suppose we assume that demand for the product is not only dependent on the price. It is also dependent on the income, which is represented through Y, dependent on A, that is advertising expenditure and also depending on the taste and preference of the consumer.

So, in this case, how we represent the relationship between the variable price, demand for the product, income, advertising expenditure, and taste and preference of the consumer through a function. We know that demand for a product is dependent on price for the product, income for the product, advertising for the product and taste and preference for the product. So, demand for the product is a function of price, income, advertising expenditure and taste and preference.

So, this is the example of a multivariate demand function, where there are four independent variables and one dependent variable. Here, the dependent variable is d P and it is dependent on four independent variables. That is, P ,Y ,where P is the price of the product, Y is the income of the product, A is the advertising expenditure associated with the product and T is the taste and preference of the consumer for the product.

So, there are two types of functions. One is bi-variate and the other is multivariate. Bivariate essentially deals with two variables and multivariate deals with one dependent variable and number of independent variables.

Now, how do we specify a function? on the basis of the nature of the relationship. How both of them are related? Whether they are positively related or whether they are negatively related? Second is on the basis of quantitative measure of the relationship or the degree of relationship, if they are positive. If they are negative, up to what extent. How we can quantify the degree of relationship? That is, on that basis we can specify a function.

Generally, we use a regression technique for specification and quantification. Now, look at this example. Suppose, we take a demand function, which is 500 – 5P. What are the different implications of this demand function or how we can analyze this demand function? When the price is 0 and demand is equal to 500 units because the intercept value is 500. So, the first implication is at 0 price and demand is equal to 500 units. There is a negative 5P. So, negative source. There is an inverse relationship between price and demand.

This nature of relationship between price and demand is inverse. The value 5 implies that, for each 1 rupee change in the price, demand changes by 5 units. So, 1 rupee change in the price leads to 5 units change in the demand. So, this is the degree of relationship between the price and quantity demanded.
So, at 0 price, demand is equal to 500 units. So, when you get the product for free, the demand is 500 units. What is the significance of this minus? This shows the nature of the relationship between two variables. Nature of relationship is inverse. There is an inverse relationship between the price and the demand and 5 implies that, for each 1 rupee change in the price, the demand change is by 5 units. So, if you look at it, there is 5 times change in quantity demanded, when there is a onetime change in the price. This is the quantification of the relationship or the degree of the relationship.

Now, what is the general form of a demand function? The general form of a demand function is Q x is equal to a minus b P x where Q x is the quantity of X, P is the price of X and a and b are the constant. So, constants in a function are called the parameters of the function. What is the role of these parameters? The parameters of the function specify the extent of relationship between the dependent and independent variable.

So, this a and b, they will specify what is the extent of relationship between the dependent and independent variable. They will talk about the nature of the relationship and the degree of relationship between dependent and independent variable.

So, taking this demand function, Q X is equal to a minus b P X, here constant a gives the limit of Q X, when P X is equal to 0 and b is the coefficient of variable P X, which measures the change in the Q X as a result of change in the P x. So, this is basically the change in the Q X, which is equal to minus b and the change in the P x.

So, in the previous example, if you remember, d was equal to 500 minus 5 P. So, 500 was the value of a, which gives the limit of Q X, when P X is equal to 0. So, when price was equal to 0, 500 was the quantity demanded and b is the coefficient of the variable P x. So, if you look at it, in the previous example 5 P. So, 5 P is the value of b, which is the coefficient of variable P X, which measures the change in the Q X as a result of change in the P X. which was 5 times because the change in the Q X was 5, which we can get through the value of b and change in the P X is 1. So, in the previous

Managerial Economics Prof. Trupti Mishra S. J. M. School of Management Indian Institute of Technology, Bombay
Lecture - 7

Welcome to the fourth session of managerial economics. Basically, we are on the first module of managerial economics, which talks about introduction and fundamentals 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, and what are the different forms to represent them. Then we discussed some of the important economic functions like demand function, bi-variate demand function and multivariate demand function.

So, today’s session we will focus on the different types of function that gets used typically in a demand function, how to measure a slope and what is its use in the economic analysis, different methods to analyze the slope and find out the slope or measurement of the slope. Then derivative of various functions and in the next session, we will basically take the optimization technique and constant optimization.

So, till now, all our discussions, if you look at it just focuses on the demand function. But apart from the demand function, there are certain other topics also where we generally use the relationship between two variables in a functional form, like production function which represents the relationship between the inputs like labour and capital with the output.

We talked about the cost function, where it is basically the relationship between the output and the cost of the production associated with that. When you talk about the total revenue function, it represents the combined function of quantity produced and price function is based on the demand function. Sometimes also we talk about a profit function. This is the profit basically, as you know it is the 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. The variable, typically in this case, all the variables are economic variables. So, there are three types of function we use in analyzing the relationship between the variables. 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 dependent and independent variable remains constant. Non-linear function is used where the relationship between the independent variable and dependent variable is not constant, but changes with the changes in the economic variable. Polynomial function represents those functions that have various terms of measure for the same independent variable.

So, we will check all this functions in more detail by taking them individually. So, in a linear function, the relationship is linear. The change in the dependent variable remains constant throughout for 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 remains constant in case of a linear function.

Suppose you are taking a demand function, which says that Q x is equal to 20 minus 2 P x. What does it signify? For each 1 rupee change in price, the demand for commodity changes by 2 units. Because, if you look at the second term of this functional form, there it is minus 2 P x. So, for 1 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 remains constant for one unit change in the independent variable.

So, this is just a hypothetical way to understand the linear demand. In the vertical axis, we are taking the price and in the horizontal axis, we are taking quantity. So, if you look at it, 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 increases from 2 dollar to 3 dollar, the quantity decreases by 100 units to 50 units.

So, if you look at the demand curve, at each point it gives a price and quantity combination. Here, the quantity demanded is the dependent variable. Whenever there is a change in the price, that leads to change in the quantity demanded also. If you look in the percentage wise also, when the price changes from 2 dollar to 3 dollar, there is 50 percent change in the price. When the quantity demanded decreases, it 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. At this point, the relationship between these two variables is linear. As it is constant, the price point changes from one to another.

Then, we will discuss the non-linear 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, 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 may not constant with each change in price. When the price changes from 1 dollar to 2 dollar or 2 dollar to 3 dollar, that may not necessarily be the same kind of change in the level of the quantity demanded.

So, if you are taking a non-linear demand function, that is d x which is a function of price x. So, here x is the product, P x is the price of x and d x is the quantity demanded of x. Taking the functional form in a non-linear, d x is a P x to the power minus P. Here, a and b they are the constants. Minus P is the exponent of variable P x and constant a is the coefficient of variable P x.

If you simplify it further, may be you are taking a number term over here. Suppose, d x is 32 P x to the power minus 2. Maybe, we can take just a reciprocal of this 32 minus P x square. So, in this case, the demand function produces a non-linear or curvilinear demand curve. It means it is not a straight line. The change in the independent variable is not constant throughout whenever there is change in the price.

So, this is an example of a non-linear demand schedule that shows how it changes when there is a change in the price. So, when price is 1, quantity demanded is 32. When price is 2, quantity demanded is 8 and when it 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 d x. The point what we are discussing here is that, with each change in the price point, the change in the quantity demanded does not remain constant. The change in the quantity demanded with respect to each price point becomes different.

This is a typical feature of a non-linear demand curve. When you plot this in a graph, we generally get a curvilinear relationship, 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. If you look at the different points in the demand curve, the change in the quantity demanded does not remain the same. So, if you are taking the p here which is the price, it 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. 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 does not remain constant. With the next change in the price point from 80 to 20, there is a significant amount of change in the quantity demanded, that is from 12 units to 50 units.

So, in case of a non-linear demand curve, even if the demand changes along with the change in the price point, there is always a difference in the amount of change at different price point.

The third kind of function generally used in the economic analysis is polynomial function. What is polynomial function? The function that contains many terms of the same independent variable are called polynomial function. So, we consider a short term 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 different types of functional forms such as quadratic functions, cubic functions, and power functions.

So, taking the example of the same short run production function, where Q is the output, L is the labour and a, b, c and d are constants associated with the different coefficients. It takes a quadratic function or it takes the form of a cubic function or it takes the function of the power function. So, when it is becomes a quadratic function, Q is equal to a plus b L minus c L square, where a, b, and c are constants. 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, and c are the constants associated with the coefficient. When it takes as power function here, it is a L to the power b, where a and b are the constants and b is the coefficient associated with variable L.

So, polynomial function may take a quadratic function or a cubic function or a power function. You can represent this polynomial function graphically, with all these three types of function, whether its quadratic cubic and function. So graphically, if you look at a cubic function, when the polynomial function takes a cubic function, suppose we take L over here, L is the labour and Q is the output. Now, the cubic function takes this type of shape. Now, what is this curve? This curve is the 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 takes a form which may be not a straight line and not exactly a curve. It follows a different kind of change at each change in the L. So, how this Q and L are related here? L is the independent variable and Q is the dependent variable. So, whenever there is a change in L, that will bring in change in Q. So, in this case of a cubic function, L changes when Q changes. But the change in the Q is not constant with each change in the labour.

Now, take a case of a quadratic. So, with the same short run production function, we take L in the x axis and Q in the y axis. Now, it is a quadratic. So, you just follow it. There is no cyclical function over here and there is no much fluctuation here. Total product curve is this and this is a typical example of a quadratic function.

Now, graphically we represent 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 can take any form. So, if you remember the power function is Q, it is equal to a L to the power b. So, b can take a value which is equal to 1, less than 1 or greater than 1. So, in this case, if you represent graphically again by taking the same formulation, here it is labour and here it is output. When we get the value