 So, welcome to the third session of Managerial Economics, we are on the first model of managerial economics which deals with the introduction and fundamentals to the managerial economics. So, if you remember in the last class, we just started our discussion about the marginal analysis. So, in today's class, we will talk about the marginal and incremental analysis first. Then we talked about a model of any typical economy, how it works, what are the different sectors, how the flows works between two different sectors. And then we will focus on the basic tools of economic analysis and optimization techniques. So, coming to the marginal analysis, as we discussed in the last class, marginal is always a unit change in any of this variable, whether the variable is cost, whether the variable is revenue, whether the variable is utility. Whenever there is a change in the output, what is the corresponding change in the cost, what is the corresponding change in the revenue or what is the corresponding change in the utility? That is the marginal cost, marginal revenue and marginal utility. So, marginal cost is the change in the total cost because there is a change in the output. Marginal revenue is the change in the total revenue because there is a change in the the output, output leads to revenue and marginal utility is the change in the utility because the consumer consumes one more unit of the output or one more unit of the product. So, marginal cost or marginal revenue or marginal utility is the total cost utility revenue of the last unit of output. Whatever the total cost, whatever the total utility or whatever the total revenue of the last unit of output, that is the marginal cost, marginal revenue and marginal utility of the corresponding unit. So, if you consider, if you need to identify what is the marginal cost of any unit, then it is the total cost of any unit minus total cost of n minus 1 minute, where n is the number of unit of output. So, this marginal cost of any unit is nothing but the total cost, whatever comes in the last unit of the last unit of the output. So, we know that profit is the difference between the revenue and the cost. Whenever there is a change in the total revenue due to unit change in the output, that we get as the marginal revenue. So, mathematically we can find this by taking the first order derivative of total revenue function with respect to Q, that is output and geometrically this is the slope of the total revenue curve. Similarly, change in the total cost coming from the unit change in the output that gives us the marginal cost and mathematically we find marginal cost by change in the total cost with respect to change in the Q or that is output. So, similarly the geometrically the slope of the total cost curves gives us the marginal cost curve. We will take an example to understand all this concept particularly with respect to marginal analysis. So, if you look at in the table this is a hypothetical table. So, this is not a information relevant to the real world. So, there are six units of output 1, 2, 3, 4, 5, 6. The second columns gives us the total revenue, the third columns gives us the marginal revenue, the fourth column gives us the total cost, the fifth column gives us the marginal cost and the last column gives us the profit. As we know the total revenue is equal to sorry total profit is equal to total revenue minus total cost. So, if you look at if there is when there is one unit of output the total revenue is 20,000 and total cost is 4000. So, the profit comes as 16,000. Since there is only one unit there is no marginal revenue associated with this unit of output. When there are two units of output the total revenue is 34,000, marginal revenue is 14,000. Now, how this marginal revenue comes? This marginal revenue is the difference between the second unit and the first unit. Similarly, what is the total cost? Total cost is 8,000. Here the cost is fixed because for one unit if it is 4000 and for the second unit it is 2 units it is 8,000 then the cost remain constant. So, if you look at we will get a constant marginal cost because the per unit cost is remain constant. So, in this case total cost is 8,000 for two units and marginal cost is 4000. Now, how we get this marginal cost is 4000? That is the difference between the second unit of output the cost associated with the second unit of output and cost associated with the first unit of output. So, this is 8,000 minus 4000. So, marginal cost comes to 4000. In this case how we will find out the profit? The profit is total revenue is 34,000 and total cost is 8,000. So, 34,000 minus 8,000 that gives us 26,000 as the profit for second unit or the two units of the output. Similarly, for the third unit of output total revenue is 42,000, marginal revenue is 8,000. How we get the marginal revenue here? The difference between the third unit of the total revenue and second unit of the total revenue. So, 42,000 minus 34,000 that gives us 8,000 as the marginal revenue for the third unit of the output or the for three units of the output. Now, what is the cost over here? Considering unit cost remain constant for three units it should be 12,000. So, if it is for one unit it is 4000 for the third unit it is 12,000 where there are three units of output. Marginal cost again it is same because it is the difference between the third unit of the total cost and second unit of the total cost. So, 12,000 minus 8,000 that gives us 4000. Now, what is the profit over here? Total revenue minus total cost so that comes to 30,000 that is 42,000 minus 12,000 that is 30,000 as the desired activity level. I will talk about the desired activity level bit later once we understand the table. Now, for the fourth unit total revenue is 46,000. How we will find out what is the marginal revenue associated with the fourth unit? That is the difference between the total revenue of fourth unit and the third unit. So, total revenue of fourth unit is 46,000, total revenue of third unit is 42,000. So, 46,000 minus 42,000 that gives us 4000 which is the marginal revenue associated with the fourth unit of output. Total cost it is 16,000 for one unit is 4000. Considering this as a fixed cost for the fourth unit this is 16,000. Marginal cost 4000 the difference between the cost associated with the fourth unit and the third unit. So, marginal cost is 4000, marginal revenue is 4000. Coming to fifth unit total revenue is 42,000, total cost is 20,000. For fifth unit for one unit is 4000, so 5 units total cost is 20,000. Now, what is the marginal revenue and marginal cost? Marginal revenue is the difference between the fifth unit of total revenue and fourth unit of total revenue. So, this is 48,000 minus 46,000 so that comes to 2000. And what is the marginal cost? It is constant, the difference between the cost associated with the fifth unit of output and fourth unit of output. So, marginal cost comes to 4000 for fifth unit. Then it comes to sixth unit, sixth unit the total revenue is 49,000. How to find out the marginal revenue? Again the difference between the sixth unit and the fifth unit. So, in that case if you look at there is a difference of 1000 over here as the marginal revenue. Cost is 24,000 given 4000 as the per unit cost when it were using 6 units. So, this becomes 24,000. So, the marginal cost is the difference between the cost associated with the sixth unit and fifth unit. So, that comes to 4000. Now, what is the profit in the case of fifth unit and sixth unit? For profit is 28,000 in case of fifth unit that is 48,000 minus 20,000. And for sixth unit this is the difference between the total revenue 49,000 and the total cost 24,000 where the profit is 25,000. So, this is a hypothetical scenario where we are getting whatever the number of unit of output we are getting total revenue, we are getting total cost, we are getting marginal revenue, we are getting marginal cost. From the difference between the total revenue and total cost we are getting the profit. Now, for any producer what should be the desired activity level and what should be the absolute activity level? Now, what is the difference between the desired activity level and absolute activity level that on the basis of the profit and on the basis of the what is the value of marginal revenue and marginal cost. So, if you look at the cost remain constant, marginal cost remain constant, marginal revenue is going on decreasing. It started in started with 14000 and it reached to 1000 and marginal cost remain constant because the per unit total cost is remain constant that is at 4000. So, in the first case the producer is getting profit as 16000, second case it is 26000, third case it is 30000, fourth case it is 30000. Now, between third unit and fourth unit, one is desired activity level and second one is the absolute activity level. Now, why this is a desired activity level? If you look at in the third unit, still the marginal revenue is greater than the marginal cost. It means the per unit revenue by the third unit is more than the per unit cost associated with the third unit. So, still it is profitable for the producer to go further because by producing one more unit, he is getting a profit same level of profit, but the marginal revenue is still greater than the marginal cost. So, when he is operating in the fourth unit, the marginal revenue is 4000 and marginal cost is also 4000. So, this is the point the marginal revenue equal to marginal cost. If the producer is going beyond fourth unit, then the revenue is decreasing cost remain constant. So, the marginal revenue is less than the marginal cost. What does it imply? It implies that whatever the last unit revenue by producing one more unit of output, whatever the revenue generated at the last unit that becomes less as compared to the whatever the cost incurred by the last unit. So, marginal revenue is less than marginal cost. Now, what happens in case of second unit or third unit? In case of second unit and third unit, the marginal revenue is greater than the marginal cost. It means still there is a scope for the producer to produce more because the per unit revenue generated in the last unit is more than the per unit cost associated with the last unit. So, unit 4 is the point where the marginal cost is equal to marginal revenue. Any unit above this, the marginal revenue is greater than marginal cost. Any unit below this, the marginal cost is greater than marginal revenue. So, the choice between the producer is to operate whether in the third unit or whether it in the fourth unit. So, using marginalist principle, we will see which one is the ideal level, whether it is third unit or whether it is the fourth unit. So, to maximize the profit, output should be increased up to the point where marginal revenue is equal to marginal cost. So, if you are going by this policy or this rule, this marginal revenue equal to marginal cost that is at the fourth unit where the marginal revenue is equal to 4000 and the marginal cost is equal to 4000. So, at this point, marginal revenue is equal to marginal cost. So, even if the profit remains same between unit 3 and unit 4, the revenue is more in case of fourth unit and going by the marginalist principle, we have to maximize the profit. The output should be increased up to the point where marginal revenue is equal to marginal cost. So, in this case, fourth unit is that level of output what the producer should produce to maximize the profit. So, one possibility when marginal revenue is equal to marginal cost, there are two other possibilities where we are getting that at some point or at any level of output, either marginal revenue is greater than marginal cost and the marginal or the marginal cost is greater than the marginal revenue. Now, we will check when we are encountering the position or we are encountering the possibility where marginal revenue is greater than m c and when the marginal revenue is less than m c. What does it imply when marginal revenue is greater than m c? It means the last unit of output increased revenue more than the increased cost. So, this is profitable for the producer to produce more because the last unit of output is generating more revenue than the cost. Now, what is the other possibility? Marginal revenue is less than marginal cost. It means the last unit of the output increased cost more than its increased revenue. So, the cost incur in the last unit is more than the revenue generated. So, it is not profit, it may not be the profitable for the producer to go beyond this or produce at this level because they are not generating extra revenue rather they are generating extra cost and whatever the extra revenue they are generating that is less than the extra cost. So, marginalist principle is always marginal revenue is equal to marginal cost and this is the profit maximization principle. And this we are going to follow in case of our managerial economics that in order to maximize the profit there should be equality between the marginal revenue and the marginal cost. So, in the last few minutes we are discussing about the marginal analysis. So, what is the basic understanding about the marginal analysis? What is the change in the revenue? What is the change in the cost or what is the change in the utility? It is total revenue, total cost and total unit when there is a per unit change in the output. So, basically marginal analysis deals with per unit change in the variable. When you take this to a real life example maybe we get some situation we get some examples where per unit change is not possible. The change is not per unit the change is chunk. If it is a per unit change sometimes there is a difficulty in evaluating sometimes difficulty in estimating. So, the particularly in those time period the change is not per unit change is in a chunk. So, in reality variable may not be subject to unit change always and specifically at those situations we need the incremental concept in order to analyze whatever the change and how it affects the other variable due to this change. So, incremental concept is applied when change is not necessarily in term of single unit, but in a bulk unit. So, marginally specifically per unit change and when there is no per unit change in this case we use the term incremental concept for the change in bulk not change in the single unit. Now, it estimate the impact of decision alternatives sometimes some decisions that may not lead to per unit change the change is in term of bulk. So, in that respect incremental analysis estimate the impact of the decision alternatives. Now, we will check what is incremental cost and what is incremental revenue. Incremental cost is the change in the total cost as a result of change in the level of output or investment whatever may be the variable. What is incremental revenue? This is the change in the total revenue resulting from a change in the level of output or the price. So, when there is a change in the level of output when there is a change in the level of price what What is the change in the revenue that is incremental revenue and what is incremental cost when there is a change in the total cost as a result of change in the level of output or the investment. And how the manager decides like marginalist principle, the profit maximizer rule is marginal cost is equal to marginal revenue. Similarly, in case of incremental analysis, how the manager decides whether the decision is profitable or whether the decision is not profitable, managers always decides determines the worth of a decision on the basis of the criteria that is incremental revenue is incremental cost. So, whatever the decision taken the outcome should be that incremental revenue should be greater than the incremental cost because of this typical decision or because of this typical change. So, if you look at marginal also deals with change, incremental also deals with change, marginal analysis deals with change for one unit and incremental analysis deals with change in the in unit change in the bulk not the single unit. So, we will take an example in order to understand the incremental analysis. Suppose, the firms decides that they will go for online selling, they feel that or they adopt a strategy that if they are going for online selling, it will be profitable for their firm. So, this is one kind of decision taken by the firm that they are going for online selling in order to increase the revenue or in order to increase the sales which will increase the revenue. So, after going for online selling, there is an increase in the sales of the firms. Now, what is incremental revenue here?暈 revenue is that there is a increase in the sales of firm due to introduction of online selling. Online selling is the decision due to online selling, there is a increase in the sales of firm that leads to increase in the revenue. So, increase in the total revenue due to increase in the sales of firm that is incremental revenue. Now, what is cost here? Cost of launching the online selling mechanism. And they have taken a decision for introduction of online selling, it involves some amount of cost that increases the total cost of product. So, cost of launching the online selling mechanism is the incremental cost. Increase in the revenue due to sales of firm due to increase in the sales of firm is the incremental revenue. So, if incremental revenue is greater than incremental cost, decision of introducing online mechanism is right. Now, manager on what basis he will take a call that whether he should go with the whether he should continue with the online marketing, online selling or he should stop it. For him the decision criteria is that till the time the incremental revenue is greater than incremental cost, the decision of introducing online mechanism is right and the manager will continue with this decision. If it is not, then if the incremental revenue is not greater than incremental cost, then the decision is not bringing any profit to the firm and the manager will discontinue this online selling. So, incremental analysis the decision rule is incremental revenue should be greater than incremental cost in order to bring the whatever the decision taken by the firm to be profitable. So, as we discussed if you compare between the marginal versus incremental analysis always the marginal analysis is it relates to one unit of output and incremental analysis relates to one managerial decision it may involve multiple unit of output. So, marginally strictly deals with one unit of output and incremental is always deals with the decision which involves more than one unit of output. Then we will come to take a typical model of an economy and we will see what are the different factors. So, what are the different sectors here and how it works or what is the money flow, what is the real flow for each sectors and how generally typical economy works here. So, if you look at in a typical economy there are four sector, one is factor market, second is product market, third is household and fourth one is business firm and fifth one is government sector. So, if you look at if you are not considering market as the sector there are typically three sector one is household, second one is government and third one is business firm. So, household sector is basically deals with providing manpower to the business firm and in return of that they get a goods and services produced by the firm what they purchase. Government sector does a transfer payment to both the household sector and the business firm and in return of that they get a tax and fee from the both the household firm and the business firm. So, let us first analyze the flows between two sector household and the government sector. Now, household provide manpower to the government sector and in return of it they gets the wages and the salaries. Now, what the government sector gets out of it? The government sector gets taxes and fees what is the revenue of the government sector what they gets from the household sector. Now, the third sector is business firm. Now, what is the relationship between the government sector and the business firm? They get a business firm gives a transfer payment to the government and governments pay for their purchases of their goods and services produced by the business firm. And also the business firm gives tax and fees to the government sector which is their revenue. Now, what is the relationship between the households, household sector and the business firm? Households provide the input in term of manpower to the business firm to produce the goods and services. In return of that they get the wages and salary and what is the outcome of the business firm or what is the output of the business firm? They produce goods and services they sell it in the open market and they get a payments for their purchases from the household sector and from the government sector. So, if you simplify it household sector and business sector how they are related? If you are considering it is a two economy there is only household sector and the business sector in case of household sector they provide the manpower to produce goods and services by the business sector. In return of that they get the wages and salaries from them. Business sector use the manpower from the household sector. They utilize that for the production of goods and services what they sell it to the household sector and in return of that they get a payment for their purchases of the goods and services. How household sector and government sector they are dependent? Household sector again provide manpower to the government sector in return of that they get the wages and the salaries. They pay tax and fees to the government which is revenue of the government and government sector provides transfer payment to the household in term of the pension and different type of payments benefit. Now, in order to facilitate this payments of the in order to facilitate the sales of goods and services of business firm to the household and the government sector there is product market. So, business firm after producing the product they send it to the product market in order to sell those goods and services. So, there is a product flow from the business firm to the product market. Product market determines the product price which is consumed by the household. So, household buys the product from the product market there is a product flow of goods and services from the product market to the household. In term of that household gives the payment to the product market which goes finally to the business firm who are the producer of the goods and services. Now, in order to facilitate the input market like in order to produce the product the business firm needs certain factor of production or certain inputs that he gets through the factor market rather than getting directly from the household. So, household provides labour and capital to the factor market which gets used by the business firm in order to produce the product. Now, the factor market determines the factor price since household is providing a factors to the business firm they are getting a factor income which is in term of wage and interest. So, wage is the payment for labour and interest is the payment for the capital. Now, after using the factors labour and capital the business firm gives back the wage and interest to the factor market which finally goes to the household as the factor income. So, how factor market and business firm they are related factor market is facilitating the factor requirement for the business firm getting it from the household and providing it to the business firm. The business firm using the factor provided by the household produce the product give it to the product market and product market is sending this to household what the household buys from the product market and get a payment for it what is finally goes to the business firm. So, if you look at apart from the government sector there are two major factors one is household and second one is business firm. Household provides the factor input to the business firm business firm produce the product and the household again buys it from the product market and gives back the price of the product as the income of the business firm. Similarly, what is the income of the household? The income of the household is that whatever the factor they are providing to the business firm the payment made for that if they providing labour it is wage and if it is providing capital then this is interest. So, if you look at the income of the household become expenses of the business firm and the income of the business firm becomes the expenses of the household. So, all the sector they are interrelated with each other when it comes to the economic activity of the economy. Government sector is there and it is interlinked with both the household sector and the business firm. They provide transfer payment to household sector get the transfer payment from the business firm. They do the purchases from the business firm and make a payment for it. They take the help of the manpower to operate the government sector and in term of that they pay the wages and salaries to the household. And what is the revenue of the government sector? Whatever the tax and fees they get from the household sector and the business firm that becomes the income of the government sector. So, basically there are three sector one is household sector second one is the business firm sector and third one is the government sector and all the three firms are related with each other. There are two market one is factor market it is deals with sales and purchase of the input and the second market is product market it deals with sales and purchase of the goods and services. So, business firm sell their product through the product market and get their factors through the factor market. How soon sell their factors through the factor market and buy their product from the product market. Now, there are two kind of flows here one is real flows that is the real transfer of goods and services from product market to household and the real transfer of labour and capital from as a factor flow from the factor market to business firm. And second kind of flows is money flows it is the income the real transfer of income from factor market to household in term of factor income and real transfer of income from product market to business firm that is for the from the payment of the product that is made by the household sector. So, there is inter linkages between all these three sectors and there are two markets which facilitates the transaction one is factor market and second one is product market. So, whatever we discussed in the last class and this marginal analysis few opportunity cost few other concept and this marginal and incremental analysis these are the session references. And then we will move to our next topic that is basic tools of economic analysis and optimization technique. Now, what is the learning objectives or session outlines of this topic? We will first look at what are the functional relationship between the economic variables. Then we will discuss some important economic function then we will see slope and it is used in the economic analysis derivative of various function optimization technique and finally, how we do a optimization with a constraint ok. So, coming to the relationship between the economic variable 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 determinant is an economic variable. Any economic quantity or value or the rate the variables rate any variable when its value whether its rate that changes due to its own or due to change in the determinants of each and economic variable. So, when the variable changes the value due to its own value or due to some other factors those are considered as the economic variable. We can take the example that demand for a product whether it is 10 units whether 12 units and 13 units every time it is changing a value the demand is not constant. So, this is a economic variable price of the product wage rate advertising expenditure these are few examples what we consider as the economic variable where the value get changes either due to own factor or due to change in the determinants the factor separating the demand for the product. Suppose, you take the example 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 attacks with this that becomes the market price for this marker and suppose the market price of this marker is 13 rupees 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 increase in the market price of this marker when there will be 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 remain constant the market price of this marker will go up by 1 rupees. So, it will go up to it will go till now if it is 13 rupees to 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 determinant. So, this is one example of the economic variable. Now, all these economic variables are interrelated and interdependent all the economic variables they are not independent they are interdependent and they are interrelated. This implies that a change in one variable causes change in the value of other related variables if they are interrelated and interdependent when value of one variable changes generally that leads to change in the other variable. Suppose, we take the example of price and quantity of a product. If you take the same example 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 the product they are interrelated. If price is more now if it is from 13 to 15 rupees few customer who cannot effort to pay 15 rupees for that they will not buy this product. So, this increase in price affecting also the quantity of the product what is getting sold in the market. So, price increases that leads to decrease in the some quantity of product that getting sold in the market. So, if you 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 and price of marker gets change that leads to change in the quantity of the products getting sold in the market. Similarly, income and consumption expenditure. If your income is more you consume more 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 get changes due to change in the value of the others. Similarly, interest rate and demand for fund if the interest rate is less more people they go for loan. If the interest rate is high there is at least decrease in the demand for a loans because the interest rates are on a higher side. So, economic variables are interrelated they are interdependent. When there is a change in the value of one variable that leads to change in the value of the other variables because both of them they are interdependent and interrelated. Now, what are the kinds of economic variable? Variables are classified on the basis of economic variable. So, the first category is dependent and independent variable. The value of this variable depends on the value of other variable in case of dependent variable and independent variable the value of this variable changes on their own or due to some exogenous factor. So, dependent variable is one where the value of this variable is always dependent on the value of the other variable and independent value is the value of this variable changes due to their own or may be some exogenous factor, but not due to change in some other variable. So, if you take the example of computer price and demand for computer. Here demand for computer 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 that 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 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 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, import oil price is independent variable. So, dependent variable is one where the value of that variable is dependent on the other variables and import oil price also and sorry the independent variable is one where it is not dependent on any other variable for its value, rather the value changes due to own or the exogenous factor. The second kind of economy variable is endogenous and exogenous variable. Now, what is endogenous variable? Endogenous variable is the value of this, the value of this variable 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. And exogenous variable is what? The value of this variable 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 on the import oil price. It has some other factor and the value of the other factor also decide whatever 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 which 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 this variable through three methods. One is tabular method, second one is functional method and third one is graphical method. So, if you are taking the example of price, demand and supply, suppose there are three variables. This relationship between this price, demand and supply, we can present through a graphical analysis, through a supply curve, 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 rupees, when the price is 2 rupees, when the price is 3 rupees and when the price is 4 rupees. So, this is the tabular representation of the relationship between the variable. This is the graphical relationship between the relationship between this variable 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 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 three variable 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 a two or three variables, then tabular and graphical form can be used, but if it is the number of variables are more specifically in case of economic analysis, if you look at 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 always good to use the functional form in order to represent the relationship between this variable. So, most economic problems are complex. It involves a large number of variables because they are interrelated and interdependent. And in such cases, economists use a mathematical tool known as function to express the relationship between the economic variable. So, the tool is functional and we generally called it a functional representation of relationship between this relationship between the economic variable. For economic analysis, it is more useful because there are large number 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 a relationship between different economic variable, 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 bivariable function and second one is the multivariable function. Bivariable function involves only two variables and multivariable function one dependent and more than one independent variable. In case of bivariable function it has only two variables one is dependent another is independent. In case of multivariable function there is only one dependent and more than one independent variables. Now, we will take an example to understand this bivariate function and multivariate function. If the value of variable x depends on value of variable y then the relationship between two each 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 express the relationship between y and x where y is the dependent variable and x is the independent variable and y is a function of x. Now, taking the example of a demand function if you consider p is the price of the product and dp is the demand for the product the demand for the product is always dependent on the price for the product. So, in case of a bivariate demand function when we are assume when we are taking that there is only one dependent variable and one independent variable. In this case we use this function dp is a function of p and this is a bivariate 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 depend 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, 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 variable and one dependent variable. Here, the dependent variable is D P and it is dependent on four independent variable that is P, Y that is P is price of the product, Y is the income of the product, A is the advertising expenditure associated to the product and T is the taste and preference of the consumer for the product. So, there are two types of functions, one is bivariate, another 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, whether they are negatively related and second is on the basis of the quantitative measure of the relationship or the degree of relationship. If they are positive, they are negative up to what extent, what is the, 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 minus 5 P. What are the different implication of this demand function or how we can analyze this demand function? When the price is 0, demand is equal to 500 units because the intercept value is 500. So, the first implication is at 0 price, demand is equal to 500 units. There is a negative 5 P. So, negative source, there is a inverse relationship between the price and demand. This is the nature of relationship between price and demand is inverse and the value 5 implies that for each 1 rupees change in the price demand change by 5 units. So, 1 rupees change in the price leads to 5 unit 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 at free, the price the demand is 500 units. What is the significance of this minus that shows the nature of relationship between two variables and the nature of relationship is inverse. There is a inverse relationship between the price and the demand and 5 implies that for each 1 rupees change in the price demand change by 5 units. So, if you look at there is a 5 time change in the, if you have quantity demanded, when there is a 1 time 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 p p x, where q x is the quantity of x, p is the price of x and a and b are the constant. So, constant in a function are called the parameters of the function and what is the role of this parameters? The parameters of the function specify the extent of relationship between the demand and 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, in the if you look at 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, when the in the previous example when there is a 1 time change in the price that leads to 5 times change in the quantity demanded. So, there are few other function like demand function and other thing like production function cost function that we will discuss in the next session. And for this specific part like basic optimization technique and basic economic analysis followed this managerial economics d and d by d of 7th addition. Thank you.