Why marginal revenue must equal marginal cost?

Question

Price  Quantity   TR      TC   Profit   MR    MC
6         0       0       3     -3     -     -
6         1       6       5      1     6     2
6         2       12      8      4     6     3
6         3       18      12     6     6     4
6         4       24      17     7     6     5
6         5       30      23     7     6     6

Consider table above. If my understanding is correct, optimal quantity equals 5, because at this point MR = MC. But after producing 5th widget, profit does not increase. Then why is producing 5 widgets considered to be profit maximizing quantity, but not 4?

Answer 1

The condition $MC=MR$ comes from studying the case where you can produce any quantity, for example, $4.0, 4.1, 4.41, 4.987$ or any fraction of production. In that case (under some general conditions), $MC=MR$ will give you the unique quantity that maximizes profits.

In contrast, in your table, you can only produce whole units, for example, $3, 4, 5$, etc. In that case, $MC=MR$ is an approximate condition to find the optimal profits and usually, it gives you the unique solution. Your example is special in the sense that there are 2 optimal quantities and the $MC=MR$ condition only gives you one such solution, but you can actually choose either.

Answer 2

Literally from the definition of Marginal Cost; it is the cost of producing one additional unit.

As your marginal revenue remains constant, pretty clear that you are going to make the maximum profits as the curves intersect at MC = MR. In the specific case that you consider, YES, you can choose either of 4 units or 5 units to be produced but, as Maarten said, your profits will definitely rise if you consider Optimal Production -2 units VS -1 unit in a general scenario.

This is the simple the reason why we say that MC = MR equilibrium is the optimal point to maximize profits. Doesn't mean you can't reach a similar state of profits before it can happen, just that MC = MR will for sure give you a maximal profit.

Also, emphasis on the point Discrete Quantities. Under a continuous distribution, your profits will keep increasing as long as the next extra unit produced will sell for more than it's production cost; i.e. till MC tends to the value of MR.