[Music] welcome back in this video we are going to talk about cost function and scheme we just finished doing cost minum and to talk about cost function we will consider the cost minimization that we have learned let's do a quick recap of what we have learned the cost minimization problem is minimize W1 X1 where X1 is amount of input one W1 is cost of one unit of input one plus W2 X2 and W2 again is the price of one unit of input 2 X2 is amount of input and we have to vary X1 and X2 such that we are able to produce at least Y amount of of course I'm talking about all these issues in two Dimension because two Dimension is easier to visualize one can also bring n Dimension to it of course there your problem is going to be it's going to be replaced by summation of wxi such that y has to be less than or equal to X1 XX the problem remains so when you solve this particular problem and we have solved for at least three cases we will keep on considering two dimensional problem three cases that we have solved is the linear technology and there our isoquant or the production function is given by a X1 + B X2 and we figured out out that the cost that minimizes the production of Y unit of output can be written as the minimized cost is equal to W1 by a comma W2 by B and the minimum can see minimum and this multiplied by y but there was one more step involved we figured out that optimal amount of X1 star was in this particular case came out to be W1 / a multiplied by y if W1 by a happens to be less than W2 by B I might have used A1 and A2 but you should be used to this that these are par parameters like A1 and a and this is equal to zero if W1 a happens to be greater than W2 by and it can be any you know it can be any if Yus B X2 star if W1 by a is equal to W2 and this is what we have figured out so I'm not going to do it for all technology you can go back to previous videos and see what we have done in each of the cases what we need to realize that when we solve this problem we get X1 star that is the optimal amount of input one we get it as a function of W1 W2 and Y and similarly we get x2 as X2 star r as a function of W1 W2 y all the time remember what we have we have a model and this is not same as the firm we have a model which is given certain inputs and here input is price of input one price of input two and the amount of output that has to be produced and it gives us the minimized cost so these three are the only external Factor how much X1 is going to be used how much X2 is going to be used is determined within the model and therefore X1 star has to be a function of these exogenous factor or parameter here this is W1 W2 and Y and so we can say look at here X1 star it is function of W1 and Y so we can always say that multiplied by W2 to the^ Z this is a general statement that X1 star and X2 star they have to be function of W1 W2 and this is called conditional conditional input demand function let us understand all four word of course it's functions because we are talking about two different cases of so this is a function because X1 is is expressed in terms of W1 W2 and Y X2 is again expressed in terms of W1 W2 and Y therefore it is a function why it is a demand remember last time we encountered demand there was a rent seller he had the owner of apartment complex and he wanted to rent his apartment and we saw there were are different buyers and we figured out their demand function like we have talked about seller in so much of detail we are going to talk about buyers also in so much of details pretty soon so it's like demand of these input who is demanding not a consumer but producer who buys these X1 star and X2 star from Market to produce the output so it's a demand that is is made by seller these are input that's why it's input and why do we say it's conditional because we say that at least Y unit of output has to be produced that is the condition and therefore the whole thing is called conditional input demand function that should be clear to you now let us understand one more thing before we move further let us understand you can use in any amount of X1 and any amount of X2 to produce y the only condition that is required is that y that you can produce using X1 and X2 amount of output has to be less than but it's very much possible that this is not the minimized cost the way it is written you may not have technical efficiency and you may you will definitely not have econom IC so this we say it's a cost and that depends on the prices of inputs in the market and the amount that you use in the market so we can say cost is a function of X1 X2 W1 and W2 of course we keep on assuming that we have only two inputs but we are going to go beyond cost we can talk about minimized cost minimized cost and this is W1 star X1 sorry not W1 star just W X1 star and W2 X2 star W1 and W2 are coming from Market but this is not any amount of input this is the optimal input that you are using that helps you minimize the cost so X1 star and X2 star thems are function of W1 W2 and Y W1 W2 and Y so this minimized cost we can use another color this minimized cost we can write minimized cost is a function of only W1 W2 and Y how come because W1 appears here W2 appears here and X1 star is nothing but aun function of W1 W2 and Y and so is X2 and therefore minimized cost can be written as a function of W1 W2 and Y and this minimized cost is called cost function it depends on input prices and it also depends on the amount of output that you want to produce the minimum amount of output this is what cost function is so I urge you to understand the difference between any cost and cost function cost function in itself includes the idea of optimization the cost has already been minimized that you have to okay that should be very very clear to you let us look at the cost function that we got in case of leonti function as well as in case of C Douglas function so if leonti function happens to be Y is equal to minimum of a X1 comma B X2 your cost function let us say C star which is a function of W1 W2 and Y happens to be W1 by a + W2 by B multiplied by Y and similarly if your cost function happens to be X1 to the^ Alpha and X2 to the power beta we had obtained the cost function that looks like Alpha by Beta to the power multiplied by W1 this [Music] W2 so if you pay attention see there is a unique thing that is happening that in linear technology in minimized cost Y is coming out in leonti technology Y is coming out as if if you want to see the average cost how will you obtain the average cost you will divide the total cost by Y and all three cases we will see that in the in the case of linear technology Y and Y will get cancelled let me write here linear technology logy average cost of at Y is going to be minimum of W1 / a [Music] W2 here minimum cost is again going to be minimum average cost is again going to be W1 / a W2 divided by and let us consider a special case here where Alpha + beta is equal to 1 this can be replaced by just Y and the average cost again is going to be C star W1 W2 yed by Y and it is going to be a function of just W1 and W2 it not depend I will so can we say average cost always is independent of Y that would be a big mistake so let us try to understand that why we are getting total cost as a linear function of Y and our standing of understanding of scale will help us understand this particular phomen me