Is the slope of the short-run total cost curve equal to the slope of the short-run variable cost curve at every output?

The slope of the short-run total cost curve is equal to the slope of the short-run variable cost curve at every output.

In the short run, production involves both fixed cost and variable cost. Total cost (TC) is the sum of fixed cost (FC) and variable cost (VC). Mathematically, this could be written as:

• TC=FC+VCTC=FC+VC

Fixed cost does not vary with the quantity of output, but variable cost does. Let QQ denote the quantity of output, then we have:

• dFCdQ=0dFCdQ=0

• dVCdQ>0dVCdQ>0

The slope of a cost curve is the derivative of the cost function with respect to quantity, therefore, we have:

• dTCdQ=dFCdQ+dVCdQdTCdQ=dFCdQ+dVCdQ

Since dFCdQ=0dFCdQ=0, it follows that :

• dTCdQ=dVCdQdTCdQ=dVCdQ

That is, the slope of the total cost curve is the same as the slope of the variable cost curve at every output level.