[Music] welcome back in this video we are going to talk about SK primarily we will talk about return to scale and time scale and we will also talk about return to school hope so let us start with return to scale we have seen that production technology is represented by an isoquant it can also be represented by a production function and this is what we are writing for Simplicity we are taking two inputs and one output let us see what happens if all the inputs are scaled up by a factor of Lambda for example so from let us say we had at X1 X2 not and the not indicates that I'm taking a particular value X1 and X2 they indicate that they are variable so initially this is the amount of input that we are using and we move to Lambda X1 KN and Lambda X2 so all the inputs are scaled up by factor of Lambda what happens so now the total output should be given by Lambda X1 KN comma Lambda X2 and there are four possibilities one that it is equal to Lambda F X1 kn comma X2 that is a possibility number one case number case number two is that this happens to be greater than Lambda FX1 KN comma X2 we can also say that this gives us the output y not which is y we can say this is X1 not X2 so we can say we this is equal to Lambda Y and this is the first case and this is equal to Lambda Y is the second case here the output is more than Lambda * of earlier output and the third possibility is the third possibility is that this is lamb this output is now less than than Lambda X1 KN FX1 KN X2 KN so the production is less than Lambda times of the earlier situation and these are the three basic cases but there is fourth possibility if we bring one more thing that if these three are true for any X1 KN and X2 KN okay if let us say this is for all X1 KN X2 KN it means any input combination that we use and this is the way production function behaves so these are the three cases the fourth case would come that some time the production function exhibit this one some other time it exhibits this one and some other time it exhibits this one or any combination of two so these are four different possibilities so what's happening basically we are scaling up input by a factor of Lambda and now we are interested in seeing that how does it change output whether whether the output gets scaled up by the same factor if output gets scaled up by the same factor the first case it means F Lambda X1 KN comma Lambda X2 KN is equal to Lambda F of X1 X2 not it means that exactly Lambda times we get more output and this is true for all input combination this is called called constant return to scale return to scale in short we can say RS it is constant return to scale if this happens to be greater than when output gets scaled up more than by a factor of Lambda then we call it for we call it increasing return to scale what we have to keep in mind let me write here CRS IRS increasing return to scale and this has to be true for all the input combination and the third case when output gets scaled up by less than the factor of Lambda then it is called decreasing return to scale and this is V RS so these are the three cases have name four cases four fourth case what we can say that the production function exhibit constant return to scale for some range increasing return to scale for some other range or decreasing return to scale for different range so there we will have to bring a word that this kind of return to scale we are seeing for a particular range and this is what return to scale is now let us try to understand what is the logic so let us un start with constant return to SK you have a production process you can replicate what it's doing it's taking X1 KN X2 KN and giving you y KN let us say you replicate the process exactly like this X1 1 not X2 KN and you get because you already have this process which gives you y not output you can replicate and now so what happens 2 X1 KN 2 X2 KN gives you 2 y the replication argument works really well for the constant return to scale exactly the process is getting copied but there are two problems here if we do this go for this replication argument Lambda can take one integer values how about fraction but we can do fraction also we can say that we can say that rather than this producing on daily basis this we will do it for only half a day okay so what what we are basically assuming that here we have the Assumption of additivity we can add and do the production but second assumption is divisibility that infraction things can be produced sometime we can achieve it by changing the time scale but sometime we will run into trouble it doesn't make sense to talk about half a car but it makes sense to talk about a car per day and a car for half a day so again depending on the context we can figure out the logic but the logic is very simple it's replication argument now let us talk about increasing return to scale how can we get increas ining return to scale so let us say that you are using let me give you an example that you are using uh steel seat to make a box and so you can make one by one by one let's say it's in a square fet or it can be square meter you know you can make a box okay and so how much you will get let's say it's in meter 1 M Cub of volume you will get how much of seat you are using because it is steel seat and this box will have six side and each side will have length of 1 M and wi of 1 M so total you are using 6 M square of seat now let us say you cut the seat differently and you make a box of 2 multili by 2 ultied by two how many units of volume you will get you will get 8 m Cub how much of a space you are using how much of the seat you are using again six side and each side has two multiplied by 2 and so the total square area is going to be 24 square m so your input is going up by four time but your output is going up by 8 time so this can be one of the reason that you have increasing return to scale so once you at larger scale better production technology becomes available that you can use it at very small scale you may not able to use it another logic could be that we can say the probabilistic efficiency when you are dealing at larger scale the probability is smaller that things would go wrong another logic for probabilistic efficiency could be that you have a larger customer base and on average their behavior is more stable so it can help you in a way to achieve efficiency and you have this increasing return to SK uh SK scale how about decreasing return to scale so their logic is that you may be writing that Y is equal to X1 comma X2 but there is a possibility there is a third factor that you have not accounted for like let us say that I'm teaching in a class and the classroom size is 100 and I saying that the output depends on how much of time I have put in and as long as that room is not filled I can keep on increasing the number of the student in that room without considering the importance of that room but once we reach to number 100 we have to increase that room also and for that in production process we will have to make room for X3 but when we are doubling here X1 and X2 we are not doubling X3 so at one particularly it so happens that you reach to a level that by doubling all the inputs which you consider all the inputs but may there may be some hidden inputs that you are not considering and they start playing a role and rather than getting output scaled up by a factor of Lambda your scaling up is not that high you get less than Lambda like for example a manager is dealing with a factory they reach a level that he cannot cannot manage people any anymore or if you know so there it would become quite difficult to scale up Beyond a particular label and you get decreasing return to scale so this is the logic that you get for constant return to scale that's replication increasing return to scale access to better production technology at larger level and decreasing return to scale missing of some of the inputs on on the description of production function so this these Logics would justify different kind of return to scale in reality what we encounter is the fourth case that any production function we do deal with they exhibit constant over some range increasing over another range and decreasing over another range so that we have to keep in mind in the last last video we were talking about the cost function and we figured out that cost function when the cost of input is W1 W2 and you want to produce y your cost function comes out to be this and so and we saw many of the production function in which this became equal to as multiplication of one so you can very well understand that here CRS technology is working really well that if you increase all the inputs by factor of Lambda your output is going up with a factor of Lambda remember the three cases that we had seen let us look at linear technology that is ax1 bx2 and let us see what happens when you use Lambda type of X1 and Lambda type of x x 2 basically you get Lambda a X1 + B X2 and this is equal to Lambda y so exactly output also goes up by Lambda similarly you have leonti production technology and you have minimum of a X1 bx2 let us say input goes up by Lambda time Lambda X1 Lambda X2 what happens to Output the new output is minimum of a Lambda X1 B Lambda X2 and we can take Lambda common because it's available in both term we get Lambda minimum of a X1 bx2 and again we get Lambda y so here also constant return to scale here also constant return to scale the third case was that X1 to the power Alpha X2 to the power beta here also now input is going up by factor of Lambda and this is Lambda X1 to the power Alpha Beta X2 to the power sorry here we should have Lambda this is what we get so we get Lambda to the power Alpha + beta X1 to the power Alpha X2 to the power beta and this is coming out to be Lambda Alpha + beta X1 to the power Alpha X2 to the power beta that's what we get and this is basically this is basically I can say that this is nothing but y so if Alpha + beta is equal to 1 this is again equal to Lambda Y and we get in case of Lambda + beta equal to 1 constant return to scale if Lambda + beta is greater than 1 then of course this output the new output is going to be more than more than Lambda Y and so in that case we will get increasing return to scale and if Lambda + beta is less than one we will get decreasing return to scale using the same logic so the reason we were getting in this particular form because we were using constant return to scale technology and in this case y can simply come out we can say this is a new function because we have to differentiate here and if we want to obtain the average cost it's going to be independent of label of output so we can say in case of CRS average cost remains constant let us look what happens in case of increasing return to scale and decreasing return to scale as what happens in increasing return to scale let me rewrite it that F Lambda X1 Lambda X2 gives you more than of Lambda FX1 X2 for all values of X1 and X2 but it means that if you are using Y if you want to produce y using X1 KN and X2 KN your cost is going to be W1 X1 KN and W2 X2 but if you multiply X1 KN with a factor of Lambda your cost is not you know your your output is going Beyond Lambda y if you want to produce just Lambda y you have to use less than Lambda proportion of input okay your increase in input is going to be less than of Lambda proportion will give you more than Lambda Y no more than Lambda Y and so therefore what happens in case of average cost when you take C of W1 W2 y ided by y it keeps on decreasing in case of increasing return to that's the logic and if you have decreasing return to scale it means if you have decreasing return to scale what you need to do you need to scale up your input by more than a factor of Lambda to produce more Lambda Factor more of output and therefore in this case this average cost keeps on we can say keeps on increasing increasing in the case of PRS please understand that when you obtain this this is optimally done here I am writing I am just mix using a particular combination but logic Remains the Same it's very much possible when you scale up when you scale up input by factor of Lambda you have access to better technology in case of increasing return to scale so you may you must be using less of all the inputs and that's why average cost is falling so we can say that if we want to draw average cost on y AIS and level of output on x axis in case of CRS it looks like this it remain fixed in case of DRS it goes up like this and in case of IRS it goes like and that's what you will have for the average cost the similar kind of graph can be made for total cost so as we have CRS we have y here and here we have total cost and average cost remains same so let us understand how do we obtain the average cost in any particular case average cost is basically at this point we say this particular this is the total cost this is the output so the ratio gives us and this ratio is never changing what it means the total cost is linearly increasing in case of let us say PRS what happens in case of DRS this is CRS in case of DRS as average cost is always increasing and therefore total cost should be increasing at increasing rate it should look like this that is the DRS so let us say at each point you know when you come to this point when you come go to this point your average because you know angle is always increasing because angle is giving you the average cost and so that is the case and in case of IRS it's simple average cost is falling and therefore total cost is increasing at decreasing rate so that should be clear to you in all three cases let us talk about time scale we can talk about short run and long run going back to our original production process that we are using and deliberately we will use labor in place of X1 and capital in place of X2 so we have this production process it helps us in telling the story so if you want to change the label of production can you change labor or Capital immediately in Western World hiring and firing is lot easier than our country they can decide their labor on day-to-day basis so labor can be changed very easily for them while Capital requires investment you have to go to the market buy it there is a lenier process so it takes time to change the capital so therefore we talk about two time scale short run scale is short run scale is when you cannot change at least one factor of production and long run is all factors can be varied can be changed and that's how short run and long run are different so when you are talking about cost remember the cost will be WL plus RK where W is wage paid to the labor R is the rental paid to Capital so when you cannot and let us continue with the argument from Western world that Capital cannot be changed so let us say you have K not label of capital in your firm so you have to use that Capital so your cost is going to be wlrk KN this becomes a fixed cost this is the source of variable cost okay while in the long run you can change both capital and labor so both will contribute towards variable that should be very very clear to you let us look at it graphically you have here you are doing the cost minimization what we will have here is isoc cost lines as well as isop points these are isocost line let us say here you have Capital here you have Labor and you want to produce let us say we have then we have then we have so what we have we can say there are different labels let us say this is y1 this is Y2 this is Y3 for example Y3 is greater than Y2 it's greater than y1 to produce ideally if you can vary both capital and labor you will use L1 amount of Labor K1 amount of capital here you will use L2 amount of Labor and K2 amount of capital and here you will use L3 amount of Labor and K3 amount of capital but let let us say it so happens that your Capital your firm already has K1 level of capital this you cannot vary this is fixed you are not able to change Capital so to produce y1 level of output of course you will produce here to produce Y2 Level of output rather than operating here you will operate here and you can see the cost is higher because at this point the iso cost line that will pass through this point is going to be on in this particular direction and that's therefore the cost is going to be higher and here also the cost this is the iso cost line that will pass through this point so the cost is going to be higher so when you cannot vary one of the factor of production your cost would be higher at best would be equal to when you are allowed to vary all factor of production and you can think about it slightly different ways that when you are allowed to vary all factor of production you are going to choose a combination that produces the output at minimum possible cost so of course you have the freedom when you don't have the freedom you have to pick something else and that increases the cost okay and that's what we are seeing here so in this in the short run rk1 will become rk1 will become the fixed C so that's what you should understand let us quickly talk about return to scope before we close this video that here we are talking about the scale but in the same form you can produce many things like boing not only can supply civil airplanes but it can also Supply defense airplane so when you have more than one production processes and let us say you can produce S1 amount of in output one and S2 amount of output two when you have the case S1 plus S2 the cost happens to be less than S1 and to produce these two things independently we say we have return to scope like a mean as a supply network it can it can add more items much easily at much cheaper cost to supply than most other vendors in the market so return to scope is similar to return to scale but here we are talking about more items in scale we are talking about more output of the same item in return to scope we are talking about different kind of items and when we have cost efficiency when we produce more items we get return to Hope and it may happen it may not happen let us say let me give you an example where it may not happen let us say there is a pet food supplier and they are different mixing uh available and you want to supply more the different kind of pet food and for each time you have to change the machine setting your cost may increase perhaps rather than decreasing if you produce more varieties there so return to scope is not guaranteed it may happen it may not happen thank you