The Rules

It is now time to bring our cost and revenue information together to establish guidelines for a firm attempting to maximize profit.   What you will see here is there are a few ways to get to the answer, and you should be able to get to the answer using each of these methods.  It's here where you will discover the logic behind the MR = MC rule.

To demonstrate the logic of profit maximization and the 'rules,' we will use our example of Chris and his tutoring.  Below is the table containing Cost and Revenue data for Chris, data that you have already seen.  What is new are the last two columns.  Total Profit is simply Total Revenue - Total Cost.  The profit of a grade of 36 is $270 [$288 - $18].  Marginal Profit (MP), using the now familiar concept of marginal, is defined as the change in profit associated with increased output.  The formula would be the change in profit divided by the change in output (MP = DP/DQ).  At an output of 46 points, marginal profit is 2.9.  This means that for every point improvement in the grade from 36 to 46 profit increases by $2.9 [ (299-270)/(46-36) = $29/10 = $2.9]. If you look at marginal revenue and marginal cost @ 46 points you find that MR = $3.40 and MC = $.50.  If you subtract MR - MC you have $3.40 - $.50 = $2.90.  What you have 'proved' is that Marginal Profit = Marginal Revenue - Marginal Cost. 

The same information is also presented graphically.  

Now, what about the profit maximizing level of output?   If you look at the Total Profit column, it is easy to pick off the maximum profit grade.  Total Profit is highest at an output level of 4 - a result that you will see as the peak on the Total profit graph.  

There is, however, another way to get to this conclusion. We can look at Marginal Cost and Marginal Revenue columns. If we decide to increase grades from 10 to 22, for each one point increase in the grade, revenue will increase by $8.17 and costs will increase by $.42.  The decision to use the extra hour will increase profit by $7.75 [ $8.17 - $.42 = $7.75].  It is clearly in the interest of Chris to improve his grade from 10 to 22. In fact Chris will increase profit by increasing grades as long as marginal revenue is greater than marginal cost. Similarly, if marginal revenue is less than marginal cost, the Chris should reduce grades to expand profit.  

The same result is obtained by looking at the marginal profit column.  If marginal profit is positive, this simply says that by using this hour of tutoring time, the profit derived from the studying increases.  Chris should continue to expand the use of the tutor as long as marginal profit is positive.

The result is the same using all approaches.  The optimal number of points would be 46.  After that the additional costs are greater than the additional revenues.

The question once again is: are there any generalizations that we could make concerning the output decisions of firms based on our analysis of Chris?  The answer is yes.  Based on what we observed in this example, a firm should continue to alter output until Marginal Revenue = Marginal Cost (Marginal Profit = 0). As long as MR > MC, output should be expanded and when MR < MC, output should be reduced.

Revenue, Cost, and Profit

The other important result here is that the level of fixed costs does not influence the optimal output choice.  This is easy to see since any changes in the level of fixed cost does not influence marginal cost and since the decision rule depends upon marginal cost, then fixed cost must be unimportant.  This is a result that you will see abused many times. Just think about how often you will hear something like: "I have already spent $xx on the project so we must complete it."  Be wary of these type of statements. What you care about is how much additional revenue and cost will be associated with the decision to continue, not how much has already been spent.  What has already been spent can be considered to be sunk costs and they should not influence your decision. 

The logic of profit maximization is specified below. 

Logic of Profit Maximization: implications of decision to raise output

1. Profit = Revenue - Cost
2. Revenue: depends on demand
3. Cost: depends on input use and costs
4. Marginal revenue (MR) is the additional revenue of the decision
5. Marginal Cost (MC) is the additional cost of the decision
6. Marginal Profit (MP) = MR - MC is the additional profit of the decision
7. MP > 0 means that the decision will increase profit
8. MP < 0 means that the decision will decrease profit
9. MR = MC  (MP = 0) means the maximum profit choice
10. Since FC does not affect MC, then FC does not affect optimal choice

Bottom Line: Any firm that wants to maximize profit should choose an output level - or a price for their product - in such a way that the follow the Optimal Choice Rule

Marginal Revenue (MR) = Marginal Cost (MC)

A variation on the rule is:

MR > MC - expand output since the expansion will increase revenue more than cost

MR < MC - reduce output since the reduction will decrease costs more than revenue