PROFIT MAXIMIZATION

Q P TR

0 100 0 

1 90 90 

2 80 160 

3 70 210 

4 60 240 

5 50 250 

6 40 240 

I will continue using the demand example, which was --

We will use a new cost example based on two numbers:

Fixed Cost = $80Variable Cost per Unit = $40

Those numbers imply --

Q AFC vcpu ATC

0 #N/A 40 #N/A

1 80 40 120

2 40 40 80

3 26.67 40 66.67

4 20 40 60

5 16 40 56

6 13.33 40 53.33

Let’s combine the two tables

Q P TR AFC vcpu ATC

0 100 0  ### 40 ###

1 90 90  80 40 120

2 80 160  40 40 80

3 70 210  26.67 40 66.67

4 60 240  20 40 60

5 50 250  16 40 56

6 40 240  13.33 40 53.33

Question: Is it possible for this firm to make a positive profit?

Put another way, the question is, “Should this firm be in business?”

Q P TR AFC vcpu ATC

0 100 0  ### 40 ###

1 90 90  80 40 120

2 80 160  40 40 80

3 70 210  26.67 40 66.67

4 60 240  20 40 60

5 50 250  16 40 56

6 40 240  13.33 40 53.33

Answer: Yes, the firm can make a positive profit. The easiest way to see it from these numbers is by noting that there exists some output for which Price is greater than Average Total Cost. If price exceeds ATC, there is a positive profit: Each unit is sold for more than it cost to make.

So, yes, the firm should be in business.Next question, what price/quantity combination maximizes profit?

Can this firm make a positive profit?

Q P TR AFC vcpu ATC TC Profit

0 100 0  ### 40 ### 80 -80

1 90 90  80 40 120 120 -30

2 80 160  40 40 80 160 0

3 70 210  26.67 40 66.67 200 10

4 60 240  20 40 60 240 0

5 50 250  16 40 56 280 -30

6 40 240  13.33 40 53.33 320 -80

What price/quantity combination maximizes profit?Let’s add two more columns on the end.

From this it appears that output of 3, at a price of $70 is the best choice and indeed it is.

The question is, “What makes that price/quantity combination the best.”

Q P TR TC Profit

0 100 0  80 -80

1 90 90  120 -30

2 80 160  160 0

3 70 210  200 10

4 60 240  240 0

5 50 250  280 -30

6 40 240  320 -80

OK, let’s get rid of some columns for a minute.

We want to find a rule for maximizing profit.Consider the choice to produce 3 units instead of 2. That decision caused revenue to rise, cost to rise and profit to rise. Since we want profit to rise this was a good choice.

Now consider the choice to produce 4 units instead of 3. That decision caused revenue to rise, cost to rise and profit to FALL! Since we want profit to rise this was a bad choice.

In what way was the decision to go from Q = 2 to Q = 3 different from the decision to go from Q = 3 to Q = 4? In both cases revenue and cost both rose, but profit rose in one case and fell in the other case.

Q TRChange

of TR TCChange

of TC Profit

0 0  80 -80

1 90  90 120 40 -30

2 160  70 160 40 0

3 210  50 200 40 10

4 240  30 240 40 0

5 250  10 280 40 -30

6 240  -10 320 40 -80

OK, let’s try looking at the changes of Revenue and Cost.

We can now get the rule for profit maximization:

You should increase output if __________________________________Revenue will rise more than cost will rise

You should reduce output if __________________________________Revenue will fall less than cost will fall

We arrive at a conclusion

You should operate at the point at which the change of revenue from producing one more unit is as close as possible to the change of cost from producing one more unit.

This is such an important conclusion that we will introduce one of economics’ favorite words. The word is marginal and it used to describe the amount by which something changes as you do more or less of an activity.

The term marginal cost (MC) describes the additional cost to produce one more unit.

The term marginal revenue (MR) describes the additional revenue received from selling one more unit.

Let’s look at the terms MC and MR.

So what we’ve called “vcpu” will now be renamed MC for “marginal cost.”

In the simple cost example we’ve been using, marginal cost is what has been called “variable cost per unit.”

This is because vcpu is the amount needed to produce an extra unit. Notice you can ignore fixed costs – they do not change when you produce more and we are now only concerned about the changes of cost. In order to produce an extra unit, you only need the cost of wages and materials to make that unit. Land and capital are already paid for.

Marginal revenue is a bit trickier. Let’s look at our demand example again.

Q P TR MR

0 100 0 

1 90 90 

2 80 160 

3 70 210 

4 60 240 

5 50 250 

6 40 240 

We will introduce an odd trick. When we write marginals, we will write them between the lines. Think of marginal revenue as the change of revenue as you go from one quantity to another.

90

70

50

30

As we go from Q = 0 to Q = 1, Total Revenue changes by $90 (from 0 to 90). Marginal Revenue is $90.

As we go from Q = 1 to Q = 2, Total Revenue changes by _______160 – 90 = $70

As we go from Q = 2 to Q = 3, Total Revenue changes by _______210 – 160 = $50

As we go from Q = 3 to Q = 4, Total Revenue changes by _______240 – 210 = $30

10 As we go from Q = 4 to Q = 5, Total Revenue changes by _______250 – 240 = $10

-10 As we go from Q = 5 to Q = 6, Total Revenue changes by _______-$10 (falls by $10)

Q P TR MR MC

0 100 0 

1 90 90 

2 80 160 

3 70 210 

4 60 240 

5 50 250 

6 40 240 

90

70

50

30

10

-10

Recall that the marginal cost of producing each item is $40.

40

40

40

40

40

40

Start at Q = 0 and ask, “Should I produce the next unit?”

It will raise my revenue by $90 and raise my costs by $40. No matter how well I was doing I’ll be doing $50 better if I produce the next one.

So, Yes, I should produce Q = 1.

Now ask, “Should I produce the next one? Go to Q = 2?”

Again since MR > MC, the answer is, “Yes.”

When does the answer become, “No?”

When you consider producing the 4th unit. At this point the next unit will add more to your cost than it will add to your revenue. Don’t do it!

So the profit maximizing quantity is ______.

The profit maximizing price is _______.

Q = 3

P = $70

GRAPHING

D = PMR

P*

Q*

$/Q

Q

MC

Making it easier:Get rid of MR

P* is half-way between MC and the intercept of the Demand curve

D = P

P*

Q*

$/Q

Q

MC

Demand interceptDemand intercept

Q* occurs where MC = MR