Relation of MC to AVC and ATC

Total cost is the sum of fixed cost and variable cost. Dividing each term by total output, we derive a similar relationship between averages: Average total cost equals average fixed cost plus average variable cost. In symbols, we could write *ATC*(*Q*) = *TC*(*Q*)/*Q* = *FC*/*Q* + *VC*(*Q*)/*Q*. (Note that fixed cost is independent of output, while total cost and variable cost vary with—are functions of—output.)

To determine the relationship of marginal cost to *AVC *and *ATC*, we begin by finding the slope of the average total cost function. Using the quotient rule, = + . We can simplify this by noting that d*FC*/d*Q* = 0 and dividing both numerator and denominator by *Q* to obtain = . Consider the first term in the numerator, . This is simply marginal cost. The term in parenthesis is recognizable as Average Total Cost. Hence, the slope of the average total cost function can be written as = . Consequently, we see that average total cost is decreasing if *MC* is less than *ATC*, *ATC* is increasing if *MC* exceeds *ATC*, and *ATC* achieves its minimum if *MC* = *ATC*. (The first-order condition for a minimum is that the slope equals zero.)

Likewise, the slope of *AVC* is easily found to be = , leading to similar conclusions regarding the relationship between marginal cost and average variable cost. The latter reaches its minimum at the output at which marginal cost equals average variable cost; average variable cost will be falling (rising) if marginal cost is less than (greater than) average variable cost.