 module 97 chapter consumer behavior and the topic in the expenditure function we are going to study the various properties and one property that we have studied in our previous module and here the property that is called the Schaeffer lemma. This property is a very unique, unique means because it entails that given by Ronald Schaeffer and on the name of Ronald Schaeffer that whenever there is a consumer he will select a very unique amount of the commodity to have the minimum expenditure. If I say it in simple words then it is that one of our Ronald Schaeffer in the name of which he is present in his book the theory of cost and production functions in which there is a Saref that Saref if it is a production set up because this topic is more production set up related then these details when we do production analysis it will come in very detail and if a Saref whether it is within its consumer analysis which we talk about it is above its indifference curves then in both the forms when it is to be spent on an expenditure then that unique point or unique value it selects its own and that unique value of its commodity which enables it to reach the limit of its expenditure where it cannot go further than Saref then that is a very unique point that is why I said that Schaeffer lemma is a very unique concept and if we say here in our book chapters it is that this Schaeffer lemma is a property of our expenditure function and it is also proved in the expenditure function through the distant formulas and that is a variant of the envelope theorem now if we look at it then we have a difference in the amount of the money and the change in the amount of the expenditure with respect to the prices if we check it then the formula that comes in front of us is that which we have seen many times in our substitution effect and the equations of the sluts here we have a unique form which is that there should be an ecstatic amount and that ecstatic amount is the very unique amount that will enable the consumer to have the minimum expenditure and plus the other equation that we are going to see here and when we solve that equality so we reach those things that until and unless expenditure function will not exhibit the property of Schaeffer lemma it will not be possible to have that minimum level of the expenditure that is why when we have to keep the utility constant and we have to decide if the price of a commodity it varies and when we say very it may either rise or either it may decline so keeping in view whatever the change there is we have to decide with the change in this price what will be the change in the our constraint and here the constraint we have to decide that with respect to the change in the price we come up this that it is equal to the utility multiplied by the change in the commodity x and that x steric it means that x steric is that unique amount of the commodity with the change in that price p steric that provides him that amount that he will be able to attain with that minimum expenditure the given level of the utility so if we look at the partial derivative of the expenditure function if we derive that we have with respect to the price of either price of x and if we want it then we can check with the price of y and means from any other unit so what gives us this aspect so this is basically compensated demand curve with that commodity price we reach that derivation so basically compensated demand function derivation when we reach that we require that the minimum expenditure should be there and if we see the basic requirement of compensated demand function is that utility should be kept constant and under that utility our budget setup we will compensate it then if we use that aspect then the expenditure function will insist on the property of chefford lemma if we look at it in the form of graph so on x axis if we look at it we have shown the prices of the commodity means it can be the initial price and on the right side it is going to increase so when there is increase in the prices and on the y axis this is the amount of m and if you want instead of this amount you can use this as e because the same amount has to be spent for the expenditure to attain the utility and here this utility will be kept constant and this if it is the price of the other goods it is also has to be kept constant so when only the price of this commodity that is moving on the right side this if it is only x with the increase in this x this total expenditure function it is going to rise and rise and rise so when it is going to increase and we are going to differentiate this expenditure function or this with the change in the price we are going to ascertain that there should be at least any very unique particular point that here we are going to say x i prime or we can say that it is x i where this compensated demand function will be equally compensated by the setup of these two price compensation so for this p i prime if we say this will be the unique point if we will consider this p i double prime then this will be the unique point and likewise if there will be any other third price then we can have an other third level of the unique point that if we have to change price from here to here and here so for each respective price whatever will be the required utility level we have to attain we will be requiring a very unique amount of the commodity of either that can be shown by x prime or that can be shown by x tarry that unique point has to be ascertain thanks