 Under the production analysis, we are going to study the concepts describing the technology and here that is the one that is called the technical progress. What is technical progress? It is a very simple thing in the sense that if we compare any activity from its previous activity and over this time frame we see that the increase in the output is more than that has been utilized in the previous then we will say that there is a progress or in other words we say that when the rate of growth or rate of increase in the output due to utilization of certain inputs in this period as compared to the previous period is more than there is a technical progress. So conventionally we will say that the quantity that has been produced through the capital and the labor with the utilization of certain technology now that is the total level of the output in one period and likewise one also equation that was utilized in previous and what will be the change in the level of the output over the time. So when the time is included here we can say that in the equation when we put the notation of the technology through A that A is basically the function of time. And when we explain this in the form of a graph we see that on the x axis we have taken the labor and on y axis we have taken the capital and through the combination of this capital and labor utilization of the technology producer is producing this level of the output. But at the same time we have shown to other isoquants and here these isoquants they are not showing the higher level of the output means in generally the isoquant that is on the right side it shows the higher level of the output but here we have to assume that these three levels of the isoquants they express the same level of the output. So these three graphs they under our assumption show the same level of the output but we see that this arrow shows that from here the producer has shifted inside. So this inside shifting shows that the same level of the output is being produced but with the utilization of less amount of labor mean L1 that is less than L2 and capital K1 that is less than K2. Before the same level of output was possible at this level mean K2 and L1 and at the same time at this combination that was possible through K1 of amount of capital and labor amount L2. But the same level of output is now possible at the point A. So this increase in not only in the output we can say sometime through the saving of the inputs required for the same level of the output it is also related to the technical progress. So over the time we can see that there are the various process in our daily life. If I give you a different example in front of you then if you are using a small series USB you can use it. So if you look at the same USB from 20 years ago then the storage capacity was probably in a few megabytes. But now there are available certain of which you have 30 GB, 64 GB. So when you have available gadgets through technology like this now you can have more level of output from the same level of the input or otherwise we can say we can have same level of output with less number of the input. Similarly, if we get some efficient fuel then if we look at that then our production setup can run very well. So whenever the production efficiency of our inputs increases or the technical efficiency of our inputs increases then we can say that there is a technical progress and if we explain this further then as we have explained the production function earlier then in the same production function if we look at that over the time the changes that come we can take those changes in various components that how many changes are made through technology and how many changes are made through capital and how many changes are made because of labour. So we can further subdivide the total growth or the technical progress in various components which mathematically can help us to solve a lot of problems further. So here we are going to assume one production function that we have taken before that Q is the function of A, the function of T and capital and the labour. Now we are seeing that A is only the function of time or that it represents all the influences that are being used in our output except for capital and labour. And all the changes that are being made over the time in this technology we are expressing them as function of time. And change in technology over the time it is assumed that it will always be possible in the form of positive integer means it will always be in a better form like if we say that an invention of a wheel will come forward as the betterment comes forward. So the particular level of the inputs of either labour or the capital to produce the same level of output will be now less and less and this technological progress will make it possible. So when we take this equation Q that is equal to A function of T and capital and labour and we differentiate this with respect to time then we by taking the first derivative we first differentiate change in technology or the technical part due to change in time keeping other part constant and then keeping A constant and change in the other part of the production function with respect to time. And when we take this this Q actually this shows that if it is that part of function of capital and labour that is shown here we have just substituted we can see that it is Q divided by A so we have substituted this value and likewise A here we have substituted by Q divided by function of K and L. So by substituting these values we have further substituted and coming to this now this equation is further divided by Q. So over the time change divided by the actual amount will give us the growth or over the time we can say it will give us the technical progress. So this aspect that is given by a change in technology due to time divided by actual technology and likewise these aspects that we have taken and when we further take this so we can see that this equation first part is related to technology the second part here we have introduced this K divided by K so when the same figure is added in the numerator and as well as in denominator this will have no effect on the equation but it will provide us the opportunity to solve the equation in a better way. So likewise we added in the part of the labour this labour divided by labour and when we have further summarized this we can see that this part of the technology that we have taken here that we conclude that it is called the growth part due to technology and this is the part that is the total change in output so that is has the notation of G mean growth in the total output and when we're coming to this part of the labour we have shown that this is basically the change in this capital over time and divided by total so this part is only the growth due to change in the capital and multiplied by this aspect and likewise this change in labour over the time divided by actual level of the labour it gives the growth in the labour situation and this is the other aspect mean change in the function or the change in the output due to labour and this labour divided by the function into K and L. So when we take this aspect basically it is the elasticity of output mean change in the total output with respect to labour and this is the change in the output due to the change in the capital. So we put the notation change in the capital due to K and change in the total output due to labour and then the equation is given here in this form. So here this total output change or the growth change is basically that some of the three types of the change is one the change due to technology plus the change due to capital but multiplied by the elasticity of production of output with respect to capital plus change or the growth related to the labour multiplied by the elasticity of output or the elasticity of production with respect to labour. So this summing up of all these three aspects will provide us the total change in the output and this equation it provides us the opportunity that for any change over the time that we are seeing mean if we see the growth in the output. So with the help of this equation we can estimate from its components that this change has come from the increase in labour productivity, the increase in capital productivity and the increase in the share which is only because of technology. So because we can give them the due share only or we can allocate the resource allocation only when we properly estimate that what is the actual share of our total growth. So the growth in output we can break down these aspects and estimate it in its various components. So the changes that the total output due to change in capital and labour will be on one side and in addition to these two, in addition to capital and labour, the rest of the changes are called residuals or we say that that is the change only and only due to the technology. If we look at residuals, basically it is given as a notation in macroeconomics in the name of residuals, it is given as a notation.