 Hello and welcome to the session. In this session we are going to discuss average and marginal costs. Let us first discuss average cost. Cost of producing each unit of a commodity is called average cost. The average cost which is denoted by A C now if C is equal to C of X is the total cost of producing and marketing each unit of a commodity, then the average cost is given by A C is equal to total cost upon total quantity of goods produced which is equal to C by X. The average cost represents the cost per unit. Let us take an example. Let the total cost function that is C of X is equal to 2 X cubed plus 4 X squared plus 8 X plus 10 for producing X units of a commodity, then the average cost is We know that the average cost A C is equal to C upon X that is equal to 1 by X into C of X that is 2 X cubed plus 4 X squared plus 8 X plus 10. Therefore, we get 1 by X into 2 X cubed that is 2 X squared plus 1 by X into 4 X squared that is 4 X plus 1 by X into 8 X that is equal to 8 plus 1 by X into 10 that is equal to 10 upon X. Therefore, average cost A C is equal to 2 X squared plus 4 X plus 8 plus 10 upon X. Let us now discuss marginal cost. It is defined as the weight of change of the total cost then X units are produced which is denoted by MC is given by MC is equal to D by DX of C that is equal to DC by DX that is marginal cost is the first derivative of the total cost C with respect to the level of output. It is also defined as instantaneous weight of change in total cost at any level of output the value of DC by DX is equal to N at X is equal to A then for an increase in production by 1 unit from A to A plus 1 the cost of additional unit is N. Consider an example find the marginal cost if the total cost function for producing a unit for commodity is C of X is equal to 2 X cubed plus 4 X squared plus 8 X plus 10. Here the cost function is given as C of X is equal to 2 X cubed plus 4 X squared plus 8 X plus 10 and we know that marginal cost MC is equal to DC by DX which is equal to D by DX of C that is D by DX of 2 X cubed plus 4 X squared plus 8 X plus 10. Which is equal to D by DX of 2 X cubed plus D by DX of 4 X squared plus D by DX of 8 X plus D by DX of 10 which is equal to D by DX of 2 X cubed that is differentiating 2 X cubed with respect to X. 2 X we get 6 X squared plus differentiating 4 X squared with respect to X we get 8 X plus differentiating 8 X with respect to X we get 8 plus differentiating 10 with respect to X we get 0 since derivative of a constant is 0. Therefore the value of marginal cost MC is equal to 6 X squared plus 8 X plus 8. Now we are going to discuss relation between average cost and marginal cost. Let C be the total cost of producing and marketing X units of a commodity then marginal cost MC is equal to DC by DX average cost AC is equal to C by X. Now we have D by DX of average cost that is AC is equal to D by DX of C upon X which is equal to now using the question rule we get X into DC by DX minus of C into DX by DX that is 1 whole upon X squared. Therefore D by DX of AC that is the average cost is equal to 1 upon X into DC by DX minus of C by X. Which can be written as D by DX of AC is equal to 1 by X into DC by DX that is the marginal cost MC minus of C upon X that is the average cost AC. Now 3 cases arise as follows. Case 1 is when the value of marginal cost is greater than average cost then we have marginal cost minus average cost is greater than 0 and D by DX of average cost AC is also greater than 0. And we know that D by DX of average cost is equal to 1 by X into marginal cost minus average cost and if the value of marginal cost minus average cost is greater than 0 this implies that D by DX of average cost is greater than 0. That is AC increases with X AC curve is rising case 2 is when marginal cost is equal to average cost which implies that marginal cost minus average cost is equal to 0 which also implies that D by DX of AC that is the average cost will be equal to 0. Which implies that average cost is constant that is average cost remains constant at all levels of output and the third case is when marginal cost is less than average cost. It implies that marginal cost minus average cost is less than 0 that is the value of D by DX of AC is less than 0. Which implies that average cost decreases with X and average cost curve is falling we should note that D by DX of AC is called marginal average cost and is denoted by N AC. Let us take an example find the marginal cost with the total cost function for producing X unit of commodity is COX is equal to 2X cube plus 4X square plus 8X plus 10. The cost function is given as COX is equal to 2X cube plus 4X square plus 8X plus 10. Now average cost AC is equal to COX is given by 2X square plus 4X plus 8 plus 10. The cost function is given as D by DX of average cost that is AC is equal to D by DX of 2X square plus 4X plus 8 plus 10. That is D by DX of 2X square plus D by DX of 4X plus D by DX of 8 plus D by DX of 10 upon X that is equal to D by DX of 2X square that is differentiating 2X square with respect to X. We get 4X plus differentiating 4X with respect to X we get 4 plus differentiating 8 with respect to X we get 0 plus differentiating 10 upon X with respect to X we get minus of 10 upon X square which is equal to 4X plus 4 minus of 10 upon X square. My marginal average cost is given by 4X plus 4 minus of 10 by X square. This completes our session for the enjoyable session.