 So, in this session, we will continue our discussion on relationship between the total cost, total revenue, profit and loss, the break-even analysis, what we discussed in the last session also and we will see what is the profitable and non-profitable range of output, how the business activity is planned on the basis of profit and loss. So, if you remember in the last class, we talked about the long-run cost and output relationship, generally how the long-run cost is derived from the short-run cost curves and how both of them they are related in which case short-run cost curve is used and in which case the long-run cost curve is used. Then we introduced the break-even analysis specifically in case of linear cost and revenue function, when linear cost and revenue follow a straight line. So, the break-even point is one where the total cost is equal to total revenue, beyond that it is a profitable range of output because total revenue is more than total cost and before this it is a non-profitable range because the total cost is greater than the total revenue and the break-even point is one where the total cost is equal to total revenue. So, we will to start with, we will continue again our discussion on the linear cost function, we will just look at the algebra behind the linear cost and revenue function specifically in case of break-even analysis. Then we will discuss about the non-linear cost and revenue function, then we will do the contribution analysis and then finally, we will discuss about the learning curve which is the again the background is on the shape of the long-run average cost curve. Generally, the practice is that we follow that there is a economies of scale because of which the average cost is decreasing, but learning curve is the alternate method to understand or the expand that why long-run average cost is decreasing. So, to start with we will look at the what is the algebra behind the break-even analysis. So, as we know that the at the break-even point the total revenue is equal to total cost. So, as we know total revenue is P multiplied by Q and total cost has two parts that is total fixed cost and total variable cost. Now, this total variable cost is alternatively we can say this is the average variable cost multiplied by the quantity and total cost is total fixed cost and in spite of DVC if you write average variable cost multiplied by Q, then this is this comes as total fixed cost and in place of total variable cost we use average variable cost multiplied by Q. So, in this case we can write this as the if QB is the break-even level of output if it is a breaking level available output putting this total revenue is equal to total cost total revenue is since it is PQ. So, QB by P and total cost is total fixed cost plus average variable cost and in case of this Q since QB is the break-even level of output we will use QB. So, QB by P multiplied by P is now simplify this again this is AVC QB equal to total fixed cost. Again simplifying this we will get if you take QB is out that is P minus AVC which is equal to total fixed cost or we can say QB is equal to total fixed cost divided by P minus AVC. So, if we know or if the producer know what is the level of TFC what is the level of TVC and what is the level of P then we can find out find out the quantity that is the break-even level of quantity through this that is QB is equal to TFC by P minus AVC. So, the algebra behind this is if you know the fixed cost if you know the average variable cost and if you know the price of it generally you can find out the find out the break-even level of output. But when it come to break-even analysis specifically in case of linear cost and revenue function it is applicable only if the cost and revenue are linear. So, in case of linear cost and revenue function the total cost and total revenue are straight line the intersect only at one point dividing the whole range of output as two parts that is profitable and non profitable. Like in the previous graph we saw that this is our total fixed cost this is total variable cost this is total revenue cost and the total cost starts from here and this is the total revenue cost total cost. So, this is the break-even level of output because total revenue is equal to total cost and this divides the entire range of output into two level that is non profitable and profitable. This is possible only if it is a linear total revenue function or linear total cost function because they intersect each other only at one point and that is the reason clearly we can divide that this is the profitable range of output and this is the non profitable range of output. But if it is not a case of linear then the possibility is that they intersect more than twice or may be more than once and in this case we it is difficult to find out what is the profitable range of output and what is the non profitable range of output. So, what is the implication for this if it is a linear cost cost and revenue function we get two range of output profitable range of output and non profitable range of output. But the implication for this that the whole output beyond the break-even level is profitable right because the point beyond which total revenue is equal to total cost the total revenue is more than total cost beyond all the level which is of which implies from a linear cost and revenue function. But when it comes to the real life this is not the fact as the conditions are different due to changing price and cost. So, it is not possible that the you get a when it is a case of linear cost and revenue function. So, if you look at the graph beyond this point we say any level of output is profitable right. So, implication of linear cost and revenue function is that beyond the break-even level any level of output is profitable. But in the real life the fact is condition are different due to the changing price and cost and that leads to the fact that the cost and revenue function may not be linear they may non-linear the cost function and the revenue function is non-linear. Because of the fact that the in case of real life there is a changing price which leads to change in the cost changing price of inputs and changing price of raw materials which leads to the variation in the cost and which leads to the possibility that the cost and revenue function are non-linear. So, the non-linearity arises due because of average variable cost and the price vary with the variation in the output. Since the average variable cost changes due to change in the price which vary with the variation in the output and as a result total cost may increase at the increasing rate and total revenue may increase at the decreasing rate. Since the there is a non-linearity due because of average variable cost and price change with the variation in the output that leads to the possibility that total cost increase at the increasing rate and total revenue increase at the decreasing rate. So, at some stages of output total cost exceeds the total revenue, but in case of linear cost how it was happening it was like after the break even level the total revenue is always greater than the total cost, but in case of non-linear since total revenue will increase at the decreasing rate and total cost will increase at the increasing rate at least at some stage of output the possibility is that the total cost will increase the total revenue. So, in this case maybe we get two break even point that is the one break even point when total cost is equal to total revenue and the possibility is that another break even point where again the total cost is equal to total revenue which limits the profitable range of output and determine the lower and upper limit of output. So, it is not that the profitable range of output is unlimited rather it is the it is defined the lower and upper limit of the profitable range of output. So, there is a need to pretest there is a need to verify the validity of the linearity of cost and revenue function before assuming that the cost or the revenue is the linear. So, in this case there is a need of pretest there is a need of verification before taking the cost and revenue as the linear function. So, what happens in case of non-linear there is two break even point there is not only one break even point two break even point and that decides the limit upper limit and lower limit of the ranges of the output. So, let us find out the graph for the non-linear cost and revenue function in case of a break even analysis this is total cost and revenue this is output suppose this is total fixed cost this is total revenue. So, this is total cost this is total revenue and this is total fixed cost. So, here the total fixed cost line it shows that the fixed cost at OF this is fixed cost and the vertical distance between total cost and total fixed cost it measures the total variable cost because this is the total fixed cost and total cost is always the summation of the total variable cost and total fixed cost. So, the vertical distance between the total cost and total fixed cost that gives us the total variable cost. The curve total revenue shows that the total sales or total revenue at different output level at the different price and the vertical distance between the total revenue and total cost measure the profit or loss for various level of output. So, the vertical distance between total revenue and total cost that will give you the various level of output. So, corresponding to that we get two break even level of output one is q 1 and second one is q 2. So, we can say this is p 1 and this is p 2. So, total revenue and cost intersect to each other at two different point one at the point p 1 second at the point p 2 where the total revenue is equal to total cost. Before p 1 before the level p 1 the total cost is greater than total revenue. So, this is the non profitable range of output beyond p 1 any level of output up to p 2 this is the profitable range of output. But like in case of linear cost and revenue function it is not unlimited the profitable range of output is non limited because we get another break even point at p 2 which leads to the fact that beyond this point total cost is greater than total revenue and again this range is non profitable range. So, in case of non linear cost function we get two break even level and which also identify the lower limit and upper limit of profitable range of output. So, q 1 is the beginning of the profitable range of output and q 2 is the end of the profitable range of output. This is the lower limit of profitable range of low level of output where the profit can be achieved, this is the higher level of output where the profit can be output. So, this represent lower and upper break even point p 1 is the lower break even point p 2 is the upper break even point. And for the whole range of output between and OQ 1 and corresponding to the this and this Q 1 and Q 2 is the breakeven point corresponding to this output level the total revenue is greater than total cost. So, this is the profit, this is the lower breakeven point, this is the upper breakeven point. So, if the firm is producing more than OQ 1, then or less than OQ 2, they are making the profit. So, if the firm is producing more than OQ 1, it should be more than OQ 1, the Q should be more than OQ 1 and less than OQ 2, then only the firm is making the profit. So, the output level should be more than OQ 1 and less than OQ 2, then only the firm is making the profit and producing less or more than these limits will give rise to the losses. So, basically the essential difference between the linear and non-linear breakeven analysis is in case of linear breakeven analysis the profitable range of output is unlimited, but in case of non-linear analysis there is a limit of profitable range of output. Beyond that producing more before that producing less will lead to the loss. So, if you look at this is the loss because total cost is greater than total revenue, this is also loss because the total cost is greater than total revenue. This is the profitable range of output where the total revenue is greater than total cost. Then we will come to the one more type of analysis may be in relation to this breakeven analysis that is called as the contribution analysis. So, till the time here we are considering the total revenue total cost to understand the breakeven analysis, but in case of contribution analysis we are not taking the total cost total revenue, we are taking the incremental revenue and incremental cost of the business activity. So, contribution analysis is the analysis of incremental revenue and incremental cost of business activity and breakeven charts can also be used to measuring the contribution made by business activity towards covering the fixed cost. So, through contribution analysis we will use some breakeven charts and the what is the role of breakeven charts here the role of breakeven charts over here is to measure the contribution made by the business activity towards the covering the fixed cost and in the graph always the variable cost is plotted below the fixed cost. So, fixed cost are constant addition to the variable cost. Total cost line will run parallel to the variable cost line because the change in the total cost is depend on the change in the total variable cost and the contribution is the difference between the total revenue and variable cost arising out of the business decision. So, the difference between the total revenue and total cost gives us the profit and contribution is strictly defined as the difference between the total revenue and the total variable cost. So, this is how the contribution analysis if you look at the graph the total revenue curve which TR starts from the origin. Total cost is the summation of the variable cost and the fixed cost that is starting at a point in the y axis which includes the fixed cost. Variable cost is starting from the origin. The total cost and the difference between the total cost and the variable cost gives us the fixed cost and which is alternately also known as the may be the difference between the total fixed cost and total variable cost and total cost. The difference between the total revenue and total cost is profit and the difference between the total revenue cost and the total cost is profit and the difference between the total revenue and the total variable cost is known as contribution. So, if you look at this contribution is nothing, but also the fixed cost at the breakeven level. So, breakeven level is corresponding to if you look at the graph breakeven level is corresponding to the point Q, where total revenue is equal to the total cost and the variable cost is VC and the difference between the total revenue and total variable and the variable cost gives us the contribution and the difference between the total revenue and total cost gives us the profit. So, if you look at this graph previously, OQ is the breakeven level of output and the contribution equals to the fixed cost. Below the output OQ to the total contribution is less than fixed cost that is amount of loss below this the contribution is less than fixed cost that is the reason this is the amount of the loss and beyond this point the contribution is more than the fixed cost and that is the reason if you look at this the case of the profit that is the contribution exceed fixed cost and the difference is the contribution towards the profit resulting from the business decision. So, beyond the before the breakeven level of output total contribution is less than fixed cost. So, this is the amount to loss and beyond output OQ that is beyond the breakeven level of output contribution exceed the fixed cost and this is the difference in the contribution towards the profit resulting from the business decision. So, one is before breakeven level of output the contribution is less than fixed cost that is the reason this leads to loss and the other point is beyond the breakeven level of output which exceeds the fixed cost the contribution exceed the fixed cost and this is the difference in the contribution towards the profit resulting from the business decision. But if you look at the contribution over time over time period under review is plotted in order to indicate the commitment that the management has made to the fixed expenditure and to find the level of output from which it will be recover and profit will begin to emerge. So, over a time period contribution over a time period under the review is plotted in order to indicate the commitment that the management has made to fixed expenditure because there is a commitment even the output leads to profitable output or not still they have to incur a certain amount of the expenditure and to find the level of output from which it will be recover and profit will begin to emerge that will look from the contribution. So, if you look at the graph here. So, we will just draw a graph that how the contribution emerge when there is a commitment when the management decides to or management has the commitment to made to the fixed expenditure and there we need to find the level of output from which level of output the fixed cost whatever the contributed before that can be recover and the new level of profit can be generated. So, we will take a total revenue curve here as a straight line this is the fixed cost to make it simplify we are not adding the variable cost here. So, this is fixed cost and contribution on the y axis and output and the x axis. So, this is Q and this is the contribution, but beyond the breakeven level since Q is the breakeven level of output breakeven level of output O Q is the breakeven level of output this is the loss because the contribution is less than the fixed cost and at the point O Q the fixed cost is equal to the contribution and beyond this the fixed cost is less than the less than the contribution and that is why this is the net profit added to the firm beyond the breakeven level of output. So, at the output of level of O Q contribution equal to fixed cost before this the contribution is less than the fixed cost that is why the firm is incurring loss and beyond this point the contribution is more than the fixed cost and that is the reason the firm is getting the profit. Next, we will look at the profit volume ratio. So, profit volume ratio is another useful tool for finding the breakeven point for sales especially for the multipurpose firm. So, this is the breakeven point in the short form is known as BEP that is of sales especially for the multipurpose firm and what is the P V ratio P V ratio is S minus V upon S multiplied by 100 where S is the selling price V is the average variable cost. So, the profit volume ratio is the difference between the selling price and the average variable cost and this is P V ratio is generally used as the breakeven point for the sales particularly in the multipurpose firm. So, if you the selling price is S is equal to 5 unit the average variable cost B is equal to rupees 4 unit then the P V ratio is selling price that is 5 minus variable cost 4 upon 5 multiplied by 100 that is 20 percent. So, we can say that 20 percent is the P V volume ratio then we will take another example to understand this P V ratio. Suppose, we have product then two products since we are saying multipurpose firm. So, A and B will take selling price here and will take variable cost per unit and will take machine hour per unit. So, in case of selling price in case of A this is 2 in case of B this is 2.5 variable cost per unit is 1 in case of A 1.5 in case of B. So, machine hour per unit is 2 in case of A and 1.0 in case of B. Now, to find out the P V ratio for A will take selling price minus variable cost upon selling price multiplied by 100. So, this will 2 is the selling price minus 1 is the variable cost per unit divided by 2 multiplied by 100 that gives to 50 percent. So, 50 percent is the break even level price that is through the P V ratio. So, now to find out the P V ratio for machine hour because if you look at machine hour require is 2 unit. So, in this case to find out the P V ratio for each machine hour P V ratio for A for each machine hour this 50 percent will be divided by 2 that is 25 percent. So, 25 percent is the P V ratio and that leads to the break even sales price of the break even price for the sales. Similarly, how to find out the P V ratio for B? P V ratio for B is again S minus B by S. So, there it comes to 2.5 minus 1.5 by 2.5 multiplied by 100 that comes to 40 percent. So, P V ratio for B will be 40 percent and since this is the machine hour is only 1 this is the total P V ratio in term of the machine hour also. So, in case of A the P V ratio is 25 percent in case of B the P V ratio is 40 percent. So, in this case B will be preferable to A or the in the other word we can say the firm will get more break even or get may be more profit from B as compared to the A because the P V ratio is more in case of B as compared to A and P V ratio is nothing but the difference between the selling price and the selling price and the variable cost of production. So, the larger is the variation more profit to the firm because that finally, lead to the break even price for the sales for the typical product. Then we will discuss about the. So, break even point in the sales value is calculated by dividing the fixed expenses F by the P V ratio. So, B E P sales value is the fixed expenses by P V ratio which is equal to F by S minus B divided up by S. Then we will talk about the margin of safety and margin of safety generally represents the difference between sales at the break even point and the total actual cost. So, the difference between the sales at the break even point and the total actual cost leads to the margin of safety. There are three measures to this margin of safety. One that is profit multiplied by sales by P V ratio, second profit by P V ratio and third margin of safety is S A minus S B divided by S A multiplied by 100 where S A is the actual sales, S B is the sales at B E P. So, we will just take an example to understand this margin of safety. So, here total revenue is 10 Q, total cost is 50 plus 5 Q and S A is 20. So, given the total revenue and total cost function sale at B E P S B can be if you look at. So, total revenue is 10 S B and total cost is 50 plus 5 S B because these are sales at the break even level of output which is equal to Q. Then at the break even point total revenue is equal to total cost. So, substituting the value of total revenue and total cost that is 10 S B is equal to 50 plus 5 S B. So, 10 S B minus 5 S B which is equal to 50. So, 5 S B is equal to 50 and S B is equal to 10. Now, in order to find the margin of safety we know that S B is equal to 10 and S A is equal to 20. So, following the third measure that is S A minus S B by S A multiplied by 100 will give us the margin of safety. Here S A is the actual sales this is the break even sales. So, corresponding to this we will get marginal load of safety is equal to 20 minus 10 by 20 multiplied by 100 which is equal to 50 percent. So, this margin of safety can be increased by increasing the selling price provided that demand of four product is in elastic because if it is elastic then small change in the price is going to get influenced by the consumer and they will change the demand pattern. So, the margin of safety is the difference between the actual sales and the break even sales and break even sales can be find out from total cost and total revenue. So, margin of safety in the last place if you look at the more is the actual sale that is S A that more is the margin of safety. So, it can be increased by increasing the selling price the sales can be increased by increasing the selling price provide the sales are not seriously affected and it can happen only when the demand for product is in elastic because if it is elastic whenever there is change in the change in the price selling price that leads to affect the demand and in one way that will reduce the sales the quantity of the sales. Also this margin of safety can be increased by increasing the production and sales up to the capacity of the plant even by reducing the selling price provided the demand is elastic. So, it can be also increased by increasing the production and sales up to the capacity of the plant even by reducing the selling price and here again the precondition is the demand has to be elastic. So, this for a margin of safety the other method includes reduction in the fixed absence in order to increase the margin of safety. The other methods to increase the margin of safety is to reduction in the fixed expenses, reduction in the variable expenses or having a product mix with a greater share of one that assure a greater contribution per unit which has a higher PV ratio. So, either the margin of safety can be increased by reducing the fixed expenses reducing the variable expenses or using a product mix which gives more contribution that is more than the fixed cost and also a higher PV ratio the profit volume ratio. So, margin of safety can be increased by reducing the expenses both fixed expenses and the variable expenses or greater share of contribution or the greatest value of the PV ratio. So, when it comes to this break even analysis whether it is contribution analysis through the PV ratio or through any other method. If you look at it can be only applied to the single product system or cannot be applied when the cost and price data cannot be determined beforehand. So, there is a limitation to the break even analysis if it is applicable only to the single product system and it can be applied only if the cost and price data is known beforehand. Then we will talk about the concept of a learning curve and if you look at we are going on discussing one fact across this session that this long run average cost curve is u shape and this long this u shape the decreasing part is because of economies of scale. The reduced cost is because of economies of scale and the increasing part is because of diseconomies of scale. But if you look at there may be other reason also to understand that this reduced cost it not only because of economies of scale or the increased cost is not because of diseconomies of scale. There may be few other factors which influence the increasing and decreasing average cost of production when there is the scale of output increases. So, one of the fact here is that the acquire knowledge and experience health firms in reducing the long run average cost curve most continuously. The learn about time to get to work done in a shorter period of time reduce cost of production and increase factor productivity. Like if you take the case of labor if you take the case of capital I think when you the opinion of the firm or the opinion of the learning proponent of learning curve is that when the firm does the same type of production over a period of time they get experience in doing that and that leads to less time that leads to less cost of production and increase the factor of productivity. So, the efficiency of the inputs increases efficiency of the firm increases and that is the reason the cost is decreasing. So, the learning cost the theory behind learning cost is that it is not because of economies of scale or the diseconomies of scale. There is one more fact that the knowledge and experience in doing the same kind of activity they learn through it and that leads to the average cost decreases. Suppose someone knew to operate that machine and if the laborer is operating that machine over a period of time he gain the skill he gain the exposure to operate this operate the machine and he is doing the whatever the time he is requiring time request to operate the machine that has come down and may be also the the productivity of the labor has increased because in the same time he can do something else now. Similarly, if you look at the process itself initially there is a time to fix up or set up the process, but in the long run when you do it over a period of time the process is set up the system is in place and that leads to less time and the inputs become more productive. So, in case of learning curve the opinion is that the acquire knowledge and experience help the firms to reduce the average cost and that leads to the shorter period of time reduce cost of production and increase the factor productivity. And this learning curve is one that is applicable to the cumulative output and it is not the output may be average output the major difference between the long run average cost curve and the learning curve is in case of long run average cost curve we take the average cost of production with the increase in the scale of output, but in case of learning curve we take the cumulative output that is the total product from the beginning of the scale. So, if you look at this curve is showing a decline trend in the. So, if this is the learning curve it is decreasing, but in case of long run average cost curve it is always increases after a point, but in case of learning curve in case of learning curve it the cumulative cost cumulative output in case of cumulative the average cost goes on decreasing there is no shine to increase the output increase the cost of production when the output in the scale of output increases. So, this curve shows a declining trend in long term average cost of production and learning curve is different from the conventional long run average cost curve as long run average cost curve gives the cost of plant wise production learning curve gives the average cost of cumulative output the total output right from the beginning of a production of the commodity. Now, we will say we will take a small numerical to understand this learning curve or we will do the we will estimate the unit of the learning. So, if you take a cost function where c is equal to a q to the power b where c is the average cost and q is the average cost for unit of output q, a is the cost of first unit of output and b is the rate of decrease in rate of decrease in average cost with cumulative increase in output. The value of b is negative because decrease in q will always increase the or may be decrease in the cost with cumulative increase in the output the greater the value of b the faster is the decrease in the average cost. So, this learning curve this c a q b can be converted into a logarithmic form that is c a q to the power b. So, this can be log c is equal to log a plus b log q. So, b gives the slope of the learning curve, b gives the slope of the learning curve, a is the intercept. So, if you look at this if you take this that log c is equal to 5 minus 0.4 log 100 suppose if assign the value of q is equal to 100, b is equal to 0.4 and a is equal to 5 then this equation will be log c is equal to 5 minus 0.4 log 100 and since log of 100 is then we can again reframe this equation as log c is equal to 5.0.4 multiplied by 2 and which is equal to 5 minus 0.8 which is equal to 4.2. So, if you take a anti log of 4.2 that comes to 15849 and if you summarize this then unit cost of 100 unit of output is equal to 15849 is the cost of production. So, similarly you can find out for any level of output that is you can find out the cost for any level of output simply changing the value of q and getting the value of log a and log b. So, if you summarize whatever we have discussed today taking this break even analysis learning curve and p v ratio the business manager must plan for the long run administration of the cost revenue and profit and because in the long run firm will be in a position to expand the scale of production by increasing all inputs. So, the it is a kind of scenario analysis it is a kind of long term horizon how the firm because the firm's decision firm's business decision depends upon the cost and revenue because that gives us the profitable and non profitable range of output. So, in the long run with the increase in output the total cost of production first increases at the decreasing rate then at an increasing rate and as a result the long run average cost initially decreases until the optimum utilization of new plant capacity and then it began to increase and the cost and output relation always follow a laws of return to scale and that is the reason the long run average cost curve is show a u shape. So, the decreasing part of the long run average cost curve is because of economies of scale and long run increasing part is because of this economies of scale firms are assumed to plan their production activity much better if the level of production for which total cost and total revenue break even is known and this implies the profitable and non profitable range of output. So, we analyze the break even analysis for both linear and the non-linear total cost and revenue analysis. So, the break even analysis or the profit contusion analysis is the analytical technique used in studying the relationship between the total cost, total revenue, total profit losses over a range of stipulated output and basically this is a technique of previewing the profit prospect and tool of profit planning. So, if you remember in case of linear cost and revenue we get one break even level of output where the profitable range of output is unlimited, but in the real life this is not possible to get a unlimited profitable range of output that brings the non-linearity in the total revenue and total cost function and in case of non-linear total revenue and total cost function we get two break even level of output. So, rather than getting a profitable range of output we get a upper limit and lower limit for the profitable range of output. Then we discuss about the PV ratio and the PV ratio which is specifically deals and the contribution analysis which is specifically deals with the incremental revenue and the incremental cost. Incremental revenue and incremental cost comes to the fact where we cannot do the analysis with the marginal cost and marginal revenue where the per unit change is not possible. In this case the contribution is the guiding factor for the contribution is the guiding factor for the business manager to decide the range of output. So, in case of contribution analysis beyond the before the break even analysis before the break even level of level of output the contribution is less than fixed cost that is that the firm incur loss, but in case of beyond the break even level of output the contribution is more than fixed cost and that leads to the profit. So, here in case of incremental analysis the guiding factor is the contribution and that helps the manager to decide whether to go for that range of output or not. And finally, we discuss about the learning curve which is alternate to the economies of scale as the reason behind the decreasing cost of production. And here it is different from the long run average cost curve because the learning curve which one where the average cost goes on decreasing for the cumulative output and that achieved through the experience that achieved through the productivity of the factor input. So, we will continue our discussion on this cost again typically on economies and disadvantages of scale in the next session. And for preparing this session these are the session references that is being exclusively followed for this.