 So, in continuation to our session and theory of production and cost, we are going to cover a few more concepts today in today's session. So, if you remember in the last couple of session, we are just discussing about the different type of production analysis. We started with short run production analysis and we discussed through the law of diminishing return and then again we started the return to scale that is the long run analysis of production. And there we check that how the scale differ with respect to change in the input and proportionately the change in the output. Then we discussed the case of producer equilibrium or so called the least cost input combination with the help of the two concepts that is isoquant and isocost and through which how they reach the or how the firms or how the producer they reach the equilibrium. Then we discussed about the expansion path and economic region of production which talks about basically which one is the feasible region where the two inputs can be substituted one to another and that is the efficient region because by in by producing or by using less of input the producer is producing the desirable output. So, in today's session we will see what are the different kind of production function mainly and mainly we talk about the cup Douglas production function which is used more in economic analysis. Then again we will continue our discussion optimal input combination through a graphical representation how the graphical representation in case of a maximization of output and minimization of cost. And again we will see that when there is a change in the input price whether it is the input price of the capital or input price of the labor how it changes. Then we will talk about numerical examples related to the law of diminishing return and return to scale. How generally this the firm uses this law of diminishing returns or return to scale empirically whether it is really works that marginal product gets decreases and then it reaches the negative and whether there is a evidence of increasing decreasing and constant return to scale. To start this today's discussion we will see that all the production function they are based on the assumption it is not that we can just formulate a production function just taking a functional form which talks about the relationship between the input and output rather the production function they in order to formulate a production function we need to assume certain thing. And what are the general assumption over here there is perfect divisibility of both inputs and outputs. So inputs are divisible and output also divisible two factor of production generally we use you do not use more than two factors like if you look at there are number of factor of production like labor, capital, time, raw materials, technology and entrepreneurship. But for all these analysis whether it is short run whether it is long run we generally use only capital and labor as the input not any other inputs in the production process. Then we are assuming that both the factor inputs that is labor and capital they are substitute to each other but they are in a limiting sense there is no unlimited substitution or they are not close they are not perfectly substitute to each other. Like if you remember the if it is perfectly substitute then the output can be produced either with the help of capital or with the help of labor. But in this case we are assuming that certain amount of both the inputs are necessary in the production process the production cannot be run only on the basis of the input or only on the basis of the capital. Then technology is given technology cannot change maybe at least in the short run in the long run it can be changed and also we assume that there is a inelastic supply of fixed factor in the short run and that is the reason short run there are few factors those are considered as fixed. And in specific sense when we are taking the case of two inputs here generally capital is fixed and there is inelastic supply and whenever there is a increase in the output or whenever there is a need to increase the production output generally the labor gets changed in order to increase the output because there is a inelastic supply of fixed factor in the short run. So, in that context when we in the economics literature there are two main type of production function are used one cup dog less production function and second one is the cost and elasticity of substitution production function that is CES production function. We mainly use typically in economics literature either cup dog less production function or cost and elasticity of substitution popularly known as CES production function. Today, we will focus more on the cup dog less production function because this is mostly used in case of the economic analysis. Cup dog less production function takes the form of Q that is output which is a which is a k to the power a and l to the power b where a and b they are the positive fraction and k and l is the k is the capital and l is the level over here. So, Q is the a k to the power a and alternately we can take this as l is to the power 1 minus a because a plus b has to be equal to 1. So, if a plus b is equal to 1 then alternately we can formulate this production function as Q is equal to capital a k to the power a and l to the power 1 minus a. Now, what are the properties of cup dog less production function? Firstly, the multiplicative form of power function can be transformed into a log linear form like log Q is equal to log a plus small a log k and b log b. So, in logarithmic form the function becomes simple to handle and can be empirical estimated using linear regression technique. So, the first property is cup dog less production function can be transformed into a log linear form and why it is generally what is the benefit if it is getting transferred into a log linear form. It becomes simple to handle and when we are doing a empirical analysis using the cup dog less production function then this is easy to handle and using linear regression technique we can empirically estimate the cup dog less production function. Then secondly the second property of cup dog less production function is that power function are homogeneous and the degree of homogeneity is given by the sum of exponent a plus b as in the cup dog less function. So, if a plus b is equal to 1 the production function is homogeneous degree 1 and implies a constant return to scale. So, the power function are homogeneous and the degree of homogeneity is given by the sum of exponents of a and b as in the cup dog less function. So, if a plus b is equal to 1 then this the production function is homogeneous of degree 1 and implies constant return to scale. If a plus b is greater than 1, then it implies a increasing return to scale and if a plus b is less than 1, again it is implies a decreasing return to scale. So, depends upon the value of the exponent in the Cub Douglas production function that is for a and b, that determines that what kind of production function it is and what kind of scale it is bearing on. Thirdly, the a and b represent the elasticity coefficient of output for input k and l respectively. So, the output elasticity coefficient e in respect to capital can be defined as the proportional change in the output as a result of given change in k keeping l constant. So, if you are keeping l constant and if you are trying to find out what is the elasticity coefficient of output for input with respect to capital only, then this is del Q by Q that is the change in the output with respect to change in the capital. So, del k by k and if you simplify this, then this is del Q by del k multiplied by k by Q. So, this is nothing but the elasticity coefficient with respect to input k keeping l as the constant. So, partial elasticity of this production function which is a dependent on capital and labour keeping l as the fixed, the elasticity coefficient with respect to capital is del Q by del k multiplied by k by Q. So, taking the specific production function that is Q is equal to a k to the power a l to the power b with respect to k and substituting the result into the equation, the elasticity of coefficient e k can be derived as del Q by del k that is a then capital a k to the power a minus 1 and l b. So, substituting the value of Q and del Q by del k in the equation elasticity of coefficient with respect to capital says that a a k by a minus 1 l b and in a imported bracket we have k divided by a k l by b and when we simplify this we get it equal to a. So, elasticity coefficient with respect to capital keeping labour as constant the value is equal to a. Similarly, when we find out for the b the same procedure we can follow and we can find out the output coefficient with respect to labour capital as constant and the value of output coefficient with respect to labour is coming out to be b. So, elasticity coefficient of capital elasticity coefficient for capital keeping labour as constant it is a the same procedure can be applied to find out the elasticity coefficient with respect to labour and the elasticity coefficient of output for labour is coming to the l and the value of it will come as the b. So, properties of cup double less production function in continuation of this we have the fourth property and here the constant a and b represent the relative distributive share of input k and l in the total output q. So, fourth property talk about the constant a and b and what it represent. So, basically constant a and b associated with input capital and labour represent the relative distributive share of input k and l in the total output q. So, the share of k in q is given by del q by del k multiplied by k and similarly the share of l in q is del q by del l multiplied by l. So, del q by del k multiplied by k is the share of k in q and share of l in q is the del q by del l multiplied by l. So, the if you look at this del q by del k multiplied by k the first part is talks about the change in the q with respect to change in the k multiplied by the the actual amount of k and the share of l in q is the del q by del l that is the that is the change in the output with respect to labour and multiplied by l. So, fourth property is talks about the constant of a b associated with labour and capital and they generally represent the relative distributive share of input k and l in the total output. So, in continuation with the fourth property the relative share of k in q can be obtained as del q by del k multiplied by k multiplied by 1 by q which comes to a and the relative share of l in q can be obtained as del q by del l multiplied by l 1 by q that comes to b. Finally, the cop dog less production function its general form that is q is equal to k to the power a l to the power 1 minus a implies that at 0 cost there will be 0 production because the value of intercept is or the value of constant is missing here. So, if in the general form if it is q is equal to k to the power a and l to the power 1 minus a it implies that at 0 cost there will be 0 production because the capital a value is missing over here. So, given this cop dog less production function if the production function is a k to the power a l to the power b the average product of l is a p l and k is a p k. So, a p l is a k by l 1 minus a and a p k is a l by k by 1 and similarly we can find out the marginal product for capital and marginal product for labour m p l is a q by l and m p k is 1 minus a q by k. So, considering this as a considering this as a cop dog less production function accordingly the value of the average product for labour average product for capital marginal product for labour and marginal product of capital will change. Similarly, when we are finding out the marginal rate of technical substitution of l for k then taking that specifically the cop dog less production function. So, as we know this marginal rate of technical substitution is the slope of the isoquant and how the slope of the isoquant can be represent this is the ratio of the marginal product of both the inputs. So, in this case the ratio of the marginal product of capital and labour. So, this is when we are finding out marginal rate of technical substitution specifically for the cop dog less production function then marginal rate of technical substitution of l for k is m p l by m p k that is a by 1 minus a k by l. Here we have to note that the marginal rate of technical substitution l by k is the rate at which marginal unit of labour l can be substituted for marginal unit of capital K along a given isoquant without affecting the total output. So, it is like rate of substitution between two inputs without and the even if the input level is changing or the amount of input getting used as from labour and capital is changing still it has to be in the same isoquant. So, the level of output is not changing. Similarly, if you take a C S production function or any other form of production function we can in the similar way we can derive the basic concept using the production analysis like average product marginal product and the marginal rate of technical substitution both for l for k and k for l. Now, if you remember in the last class we talk about the least cost input combination and least cost input combination is one where the slope of the isocost is equal to the slope of the isoquant and this is the point at which the producer or the firm maximizes the output looking at the given constant. So, today we are going to spend may be another spend in detail that how the equilibrium conditions are derived, how we can say that the slope of isocost has to be equal to the slope of isoquant. We will see then we look at the graphical representation and then we will come to the point where the input prices changes and how it generally affects the least cost input combination and how the effects are being captured. So, we will first see that how the equilibrium conditions are derived or may be how the precondition for the least cost inputs are derived and then we will look at into the graphical representation both for the maximization case and for the minimization case. So, let us look at into the equilibrium condition, how the equilibrium conditions are derived. So, if this is the production function that is where x is the function of labor and capital, in this case how we can find out what is the equilibrium condition. Now, here there is a constant that is subject to C bar that is W L plus R K. So, if you remember this is your isocost. Now, this if you can change this constant into this then this is C bar minus W L minus R K which has to be equal to 0. Now, we will whenever we need to maximize something minimize something with respect to a constant in this case we need to use a Lagrangian multiplier. Generally known this as a Lagrangian multiplier method and here what is the Lagrangian multiplier? The Lagrangian multiplier here is x C bar minus W L R K which is equal to 0. Now, what is this Lagrangian multiplier? Generally this is the undefined constant or undefined constant which generally used to maximize or minimize a function because if there is a constant associated with this if there is a constant associated with this we cannot directly maximize the production function and that is the reason we need to take the help of the Lagrangian multiplier method and these are the and the Lagrangian multiplier is the undefined constant which generally used to maximize or minimize a function. So, once we get the Lagrangian multiplier method then we will get the composite function. Composite function is x plus lambda C bar minus W L minus R K which has to be equal to 0. So, this is the composite function using the Lagrangian multiplier method. Now, what is the next job? Next job we need to maximize it and we will see what should be the first order condition and what should be the second order condition in order to maximize or in order to minimize. So, given this as the composite function what should be the first order condition? If you remember all the first order condition if it is maximization or maybe it is a minimization the partial derivative has to be equal to 0. So, here we will take the partial derivative with respect to the undefined constant and we will set them equal to 0 in order to find out the first order condition. So, del Q by del L which is equal to has to be equal to 0 del Q by del K which has to be equal to 0 and del Q by del lambda which has equal to be 0. So, this implies del x by del L plus lambda minus W equal to 0 then this implies del x by del K plus lambda minus R is equal to 0 and this implies C minus W L minus R K which has to be equal to 0. So, from equation first to if you solve for lambda then this comes to del x by del L is equal to del W or then this equal to del x by del L by W and this leads to lambda is equal to this is our marginal product for labor by W. Similarly, from equation two if you find solve for the value of lambda then this is del x by del K is lambda R or x is equal to del x by del K by R and this is since this is m p K by R. After solving for both this lambda from then this comes to m p K by R which is equal to m p L by W leads to m p L by m p K equal to W by R. This is the first order condition for the least input combination because this represents the slope of isochron this represents the slope of isochrist. Since the cost and is given in the form of isochrist and the output is given in this case we can say that the first order condition has to be the point at which the slope of the isochrist has to be equal to the slope of the isochron. So, the ratio of marginal product of labor and capital gives us the slope of the isochron and the ratio of input prices that is W and R that gives us the slope of the slope of the isochrist. So, first order condition for least cost input combination says that at the point of equilibrium or at the point of least cost input combination the slope of the isochrist has to be equal to the slope of the isochron. Next we will see what should be the secondary condition for this maxima or minima with respect to the least input combination. This requires the marginal product curve for both the factors has to be negative. So, del square x del L square and del square x del K square. So, this is what this in order to find out the slope we need to the take the second order derivative with respect to labor and second order derivative with respect to capital. So, this has to be 0 that is del square x by del L square has to be less than 0 and del square x and del K square has to be less than 0. So, second order condition for the least input combination requires the marginal product for both the factors that is capital and labor the marginal product curve for both the factor has to be negative and how we will find out the marginal product curve for both the factor has to be negative. We need to take the second derivative of the with respect to capital and with respect to labor of the composite function and that gives us del square x by del L square for the second derivative for the labor del square x by del K square the second order derivative for the capital. Second order condition says that it has to be negative and that is the reason the second order derivative that del square x del L square has to be less than 0 del square x del K square has to be less than 0. Next we will see how graphically we look at both the input and the maximization case and the minimization case in case of the least cost input. So, what is the essential difference between the maximization and minimization case? In case of maximization case the cost is given and if taking cost is the constant the isocost line is the constant the producer has to maximize the output whereas, in case of minimization case the output is fixed and looking at the fixed output what is the challenge for the producer or the challenge for the firm is to minimize the cost. So, let us look at the maximization case first. So, maximize with maximization of x which is a function of labor and capital with respect to the input prices that is W and R. So, if you look at there are two graphs graph one is where there are three isoquant and the isocost is given and graph two is where the isoquant is there in a different shape and there are two isocost line. So, in case of maximization case what happens this isocost is given and with this isocost the challenge of the producer is to get the maximum level of output and looking at this the consumer will always pick up a combination in Q 2 level of output because the Q 2 level of output can be achieved with the isocost K and L which is given. But in case of second case if you look at the isoquant is taking a shape of concave and which is not possible because in case of concave it is not following the basic rule of the production analysis like if you look at basic rule of a isoquant because even if with the same isoquant they are able to achieve the combination it is not giving the same level of same level of output across all these stages or may be the input combination are different because there when they are moving from one point to another point they are using more of the inputs of both, but they are producing the same level of output which is not the cost efficient or which is not the input efficient. That is the reason in case of maximization case the output level is can be achieved with the the maximum output level can be achieved with the isocost line given or in term of the cost is given. Next we will see the minimization case where the output is given the challenge for the producer is to minimize the cost of production or minimize the input prices with respect to the given level of output. So, these are all isocost these are the point which talk about the cost of production if you are taking any of this combination of input prices and looking at this if you look at if the x bar is the output level that is given then in this case the producer will always look for this that to produce this level of output which one and can be the minimum cost. So, in this case to achieve this level of output k 3 l 3 is the minimum isocost or the minimum cost of production that is the reason they will choose this point as the least cost input combination because to produce the given level of output this is the minimum possible cost. So, in case of minimization case the challenge of the producer the challenge of the firm is to minimize or they will always look for the combination which gives us the least cost to the producer least cost to the firm for a given level of output.