Marginal Cost and Marginal Revenue

One of the most direct commerce applications of differentiation is measuring how total cost and total revenue change as the level of output changes.

Marginal Cost (MC)

If $TC(Q)$ is the Total Cost of producing $Q$ units, the Marginal Cost is the rate of change of total cost with respect to output — approximately, the extra cost of producing one more unit:

Example: if $TC = 3Q^2+6Q+30$ (Rs), then $MC = \dfrac{d(TC)}{dQ} = 6Q+6$. At $Q=8$: $MC = 6(8)+6=54$, meaning the 9th unit costs approximately Rs 54 more to produce than the 8th.

Marginal Revenue (MR)

If $TR(Q)$ is the Total Revenue from selling $Q$ units, the Marginal Revenue is the rate of change of total revenue with respect to output:

Total Revenue is generally $TR = P\times Q$, where $P$ is price. If the (linear) demand function is $P = a-bQ$, then:

Notice that $MR$ falls twice as fast as $P$ as $Q$ rises (the coefficient of $Q$ doubles from $-b$ in the demand function to $-2b$ in $MR$) — a standard result worth remembering directly, since it is frequently examined in Business Mathematics and Statistics question papers.

Marginal versus Average Values …