 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 sessions, 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 concept 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 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 Cub 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. So 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 this 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 and there is an 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 cop 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 cop dog less production function or cost and elasticity of substitution popularly known as CES production function. Today we will focus more on the cop dog less production function because this is mostly used in case of the economic analysis cop 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 h 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 cop 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 cop dog less production function can be transformed into a log linear form and why it generally what is the benefit if it is getting transformed into a log linear form? It becomes simple to handle and when we are doing a empirical analysis using the cop dog less production function then this is easy to handle and using linear regression technique we can empirically estimate the cop dog less production function. Then secondly, the second property of cop 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 cop dog less function. So, if a plus b is equal to 1 the production function is homogenous 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 cop dog less function. So, if a plus b is equal to 1 then this the production function is homogenous 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 cop dog less 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 labor 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 a 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 inverted 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 labor 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 labor capital as constant and the value of output coefficient with respect to labor is coming out to be b. So, elasticity coefficient of capital elasticity coefficient for capital keeping labor as constant it is a the same procedure can be applied to find out the elasticity coefficient with respect to labor and the elasticity coefficient of output for labor is coming to the l and the value of it will come as the b. So, properties of Cub Devil's 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 distributed 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 Cobb-Douglas 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 Cobb-Douglas 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 MPL is A Q by L and MPK is 1 minus A Q by K. So, considering this as a considering this as a Cobb-Douglas 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 Cobb-Douglas 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 Cobb-Douglas production function then marginal rate of technical substitution of L for K is MPL by MPK 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 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 CS production function or any other form of production function we can in the similar way we can derive the basic concept used in 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 spending 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 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, this is so if this is the production function that is where x is the function of labour 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, whenever we need to maximize something minimize something with a 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 point. 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 if you remember all the first order condition if it is a maximization or may be 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 isocond this represents the slope of isocost. Since the cost and is given in the form of isocost 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 isocost has to be equal to the slope of the isocond. So, the ratio of marginal product of labor and capital gives us the slope of the isocond and the ratio of input prices that is W and R that gives us the slope of the slope of the isocost. 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 isocost has to be equal to the slope of the isocond. 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 take the second order derivative with respect to labor and secondary 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 factors is to negative. We need to take the second order 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. And 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. First 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 we 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 isocort 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 they are 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 and 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. Let us look at the case where if there is a change in the input prices how it affects the least cost input combination. So, change in input price if you look at it affect the optimal combination of inputs at different magnitude depending on the nature of the input price change. So, if all input price change in the same proportion. So, it bound to happen the optimal combination of inputs has to change if there is a change in the input price either in the different magnitude or in the same magnitude depends on the nature of the input price change. So, if all input price changes in the same proportion the relative prices of input that is the slope of the budget constant or they may remain unaffected. So, if all input prices changes in the same proportion the relative prices of inputs also if you look at the moves in the same proportion and those are remain unaffected, but when the input prices changes at a different rate in the same direction or the opposite direction or the price of only one input changes while the price of other input remain constant the relative price of input will change. Input price when it change in the different rate in the same direction different rate in the opposite direction input price of one changes other remaining constant the relative price of input will change. This change in the relative input out price change in both in the input combination and the level of output as a result of substitution effect of change in the relative prices of input. So, this change in the relative input prices changes both the input combination and the level of output. So, whenever there is a change in the input prices it affects the input combination and the level of output as a result of substitution effect of change in the relative price. So, a change in the relative price of inputs either in the same or the opposite would imply that some input have become cheaper in relation to the others. So, cost minimization firm attempts to substitute relatively cheaper inputs for more expenses one refers to the substitution effect of relative input price change. Because whenever the price of one input changes it becomes cheaper with respect to the other inputs and what the cost minimizing producer or cost minimization firms they do it over here. They generally try to replace the expensive input with respect to the cheaper inputs and this is generally known as the substitution effect of relative input price change. So, this numericals we will just look at later that how the quantity of labor and capital changes before that we will look at the graphical representation that when there is a change in the input prices how it affects the least cost input, how it affects the level of output or how it affects the level of input combination. So, initially this is the ISO cost K L is the ISO cost Q is the output Q 1 is the isoquant and this is the level of output. Now, price of labor input decreases with the help of that now the quantity of the or if you look at the ISO cost will change from K L to K L dash. So, once it changes from K L to K L dash then in this case now what is the new input combination the new input combination is if you look at this becomes L 3 that is L 1 and this is K 2 this is K 1. So, with the change in the input price now the firm will use more of labor and less of capital that is the reason the combination now earlier it is K 1 L 1 now it is K 3, K 2 and L 3. Now, to get to keep it in the same level what the producer will do the may be at least at the same level of output still want to change the input combination now what they will do they will try to draw a parallel line which also tangent at a point which is at this level. So, suppose this is your point E this is your point E 1 this is your point E 2. So, at this case if you look at still at U j say higher level of labor as compared to the previous level, but it is U j also a higher level of capital. So, the movement from this E 2 E 1 is the price effect that is in the form of L 1 to L 3 the movement from may be E 1 to E 2 is budget effect because the producer is trying to keep the income level the real income level of the producer at the same level that is the reason we got a compensated budget line which is may be K dash and L dash which compensate or which may be the reduce the real income of the producer in term of change in the input prices and that leads to a different combination that is E 2. So, E 1 to E 2 E 1 the movement from E 2 E 1 is the price effect which is in term of the labor input because use we can say L 1 and L 3 movement from E 1 to E 2 is that is L 2 to L 3 is because of the income effect because the real income is changing and movement from may be E 2 E 2 is the substitution effect because of the change in the real income. So, if you look at the price effect is the combination of the substitution effect and the income effect. So, if you remember your price effect substitution effect and income effect in case of the consumer theory this is nothing but the counter part of the counter part of the counter part of that in the production theory which talks about the change in the input prices if there is a change in the input prices generally the producer try to substitute that with a cheaper input as compared to expensive input that is the reason they go on using more of that input. So, in this case also the same thing has happened the producer is once the price of labor has gone down the producer has try to optimize it and use more of the labor as compared to the capital and that leads to the combination of the change in the input combination or also the change in the level of output. Now, next we will see the input combination or may be the input combination how it changes or how to find out numerically when the production function is given price of inputs is given like W and R and if the production function is given and to produce a specific level of output how to optimize the cost of production and what should be the minimum cost of production. For that we will just take a example of a we will take a numerical example like Q is equal to 100 K that is to the power 0.5 and L to the power 0.5 this production function is in the form of a cup dog less production function where W is 30 and R is 40. So, W is the price of labor that is 30 rupees R is the price of capital that is 40 rupees and what we need to do here we need to find the quantity of labor and capital that the firm should use in order to minimize the cost of producing of 144 units of output. So, if we look at this is the minimization of case like the second case where the unit of output is given we need to minimize the cost and what is the minimum cost we need to find out that. Then how we will go for this we will use the we will take the help of the Lagrangian multiplier to solve this. So, Q is equal to 100 K 0.5 and L 0.5 and here if you look at then we have W is equal to 30 and R is equal to 40. So, first we will try to find out the composite function. Now, composite function is rupees 30 L plus rupees 40 K plus lambda dash that is Q 0 minus 100 L to the power 0.5 K to the power 0.5. Now, the what we need to do we need to find out the first order condition the first order condition we need to find del z by del L that is 30 minus lambda dash 50 L that is L minus 0.5 K 0.5 has to be equal to 0 or 50 lambda dash L minus 0.5 K 0.5 has to be equal to 30. Suppose this is our equation 1. Now, we need to look at the first order partial derivative or the first order derivative or the partial derivative with respect to the other input. In the first case we have checked it for L. Now, we will check it for K. So, here we need to find out del partial derivative with respect to K that gives us 40 minus lambda dash 50 L 0.5 K minus 0.5 which has to be equal to 0.5. Or 50 lambda dash L 0 to the power 0.5 K to the power minus 0.5 which is equal to 40 let us call it equation 2. Now, to find out the partial derivative with respect to lambda that is Q 0 minus 100 L 0.5 K 0.5 which is equal to 0. So, you can call it 100 L to the power 0.5 K to the power 0.5 which is equal to Q 0 that is equation 3. So, now, if you divide equation 1 by equation 2 suppose equation 1 by equation 2 then this is 30 by 40 equal to lambda dash that is 50 L minus 0.5 K 0.5 K 0.5 which is equal to 0.5 K 0.5 K 0.5 then lambda dash 50 L 0.5 K minus 0.5 simplifying this again. So, if you simplify this then this is 3 by 4 equal to K 0.5 K 0.5 L 0.5 and L 0.5 that comes to 0.5. Comes to 3 by 4 by K by L. So, K is equal to we can say 3 by 4 L or 0.75 L. So, in order to take the first order partial derivative once we get the partial derivative out of the first order derivative with respect to L then the first order derivative with respect to K and then the first order derivative with respect to lambda and then we solve for the value K in term of L. Now, what we have got from all this calculation that is K is equal to 3 by 4 L. Now, we will see how we can substitute the value of K into value of this into the equation and find out the value of K and L because ultimately what we need to find out we need to find out ultimately to produce 144 units of output what is the minimum cost because the minimum cost the firm has to incur or what should be the minimum cost on which ISO cost they have to learn. So, if you substitute the value of K into the given production function for 144 units of output that is 144 that is 100 L 0.5 and 0.75 L because K is equal to 0.75 L. So, that comes to 100 L 0.75 which comes to 144 100 0.75 then it comes to 14486.6 that comes to 16.6767. So, 16.67 is the value of L. Now, we to find out K is equal to 3 by 4 L. So, that comes to 12.5. So, 12.5 is the capital and capital is 16.67 is the labour. So, capital and labour we need we got the value of capital and labour. Next we need to find out what is the cost when the capital is labour is 16.67 and capital is 12.5 because ultimately again let me remind it ultimately we need to find out what is the minimum cost of producing this given level of output. So, C is equal to W L plus R K. So, W is 30 L is 16.67 plus R is 40 and K is 12.5. So, that comes to rupees 100 1000.50. So, in order to produce 144 output this is the minimum cost. So, this is the given level of output and this is the minimum cost. So, whether it is a minimization case or maximization case how generally we solve it numerically, we solve it numerically with the help of this last case. So, it is a minimum cost. So, this is the minimum cost. So, it is a minimum cost. So, that is the minimum cost. So, whether it is a minimization case help of this Lagrangian multiplier method, where we take the constant in the form of a Lagrangian multiplier. We formulate a composite function, then we take the first order partial derivative with respect to 0. Simplifying this, that gives us the value of capital and labor. We put the value of capital and labor in the production function equation. We get the exact value of capital and labor. Use this in the cost function and that gives us the minimum cost of producing the given level of output. Now, let us see if you remember in the last class, we talk about the short run production function that is in term of the law of diminishing return. Next, we will see that numerically how we get the value of the three stages, the different stages of production and how we find out the value of average product, marginal product and at what level generally the diminishing marginal return set in and what level labor is the average product of labor is the highest. So, the form produces the output according to the production function that is Q is equal to 100 K L minus L Q. So, the production function is 100 K L minus L to the power Q. Capital is fixed at 10 because this is a short run production function. Now, what we need to do? We need to find out the value of average product. We need to find out the value of marginal product and then maybe we can find out what is the different stages or what is the different level of output or level of labor where the where the producer is achieving the different level of output. So, first we will find out the marginal product of labor. Now, marginal product of labor is del Q by del L. So, that comes to 20, that comes to the case of 20 K L minus 3 L square. So, that comes to 200 L minus 3 L square. Now, to so this is the marginal product of labor. Now, we will find out the average product of labor. Average product of labor is Q by L. So, that comes to 10 K L minus L square which is equal to since K is equal to 10. So, this is 100 L minus L square. So, average product for average product is 100 L minus L square. Marginal product for labor is 200 L minus 3 L square. Now, second one is we need to find out when MPL is maximum. Where the first order partial derivative with respect to L is equal to 0. So, that comes to 200 minus 6 L which is equal to 0. L is equal to 33.33. Now, what is the significance of this level of labor? The significance of this level of labor is that at this level since marginal product of labor is maximum beyond this generally the law of diminishing returns at in if you remember your three stages of production function like how the total product curve. So, initially it is convex, then it is convex and then it is decreasing. So, this corresponding to this our marginal product of labor is maximum because after this the total product of labor is increasing at a decreasing rate and beyond this MPL is maximum and beyond this if you look at then the total product is increasing at the decreasing rate and marginal product is decreasing. So, we can say corresponding to this point MPL is maximum and this is the point where the law of diminishing return set in. And the second point we will look at is when MPL is equal to 0, MPL is equal to 0 when this 200 L minus 3 L square is equal to 0. So, in this case maybe L is equal to if you find out this comes to 66.66 and where this value of L comes the value of L comes at this point because corresponding to this MPL is 0 and TPL is maximum. So, if you remember from till this point this is your point beyond which there is the law of diminishing return set in. Then when a average product of labor is highest average productive labor is highest when DAPL with respect to DL is 0 that comes to L is equal to 50. So, this comes somehow here the average product of labor is maximum when it intersecting the marginal product of labor. So, corresponding to this L is equal to 50. So now, we can say that from 0 to 50 the first stage of production from 50 to 66 unit of labor, second stage of production and beyond 66 unit of labor we have third stage of production. And how we have identified this first stage, second stage and third stage? First stage ends till the point the marginal product is equal to the average product and average product is maximum at that point where marginal product is equal to the average product. Where the second stage ends, second stage ends when the total product level is maximum or the marginal product is 0. So, that achieve at the unit of labor unit at 66 beyond 66 we have the third stages of production. So, depends upon the labor unit we can find out at which level generally the stages of production are decided whether at the first stage, whether at the second stage and third stage. So, first stage is up to a point where the marginal product of a labor is equal to the average product of labor. Second stage is the point where total product of labor is maximum marginal product of labor is 0 and third point is beyond this. So, once we find out the value of labor where marginal product of labor is equal to 0 that is the definition of the second stage. Once we find out the maximum value of APL that gives us the may be the beginning of the second stage and when we find the marginal product of labor is maximum that gives us the point beyond which the law of diminishing return setting. So, in the next session we will talk about the cost of production different types of cost, how the cost function is formulated and what is the logic of different shape of the cost function in the short term and in the long run. So, these are the session references generally these are the references that is being used for the preparation of this particular session.