Producers and Profit Maximization

In economic theory, people play two main roles in the market. As we have just seen, they are consumers. The other main role is producers. For producers, the economic problem is to maximize profits. The key decisions are which outputs to produce, how much of each output to produce, and which inputs to use to produce the outputs. We will take these decisions one at at time.

How Much Output?

Let us return to Josh's lawn mowing business, and focus on the decision of how many lawns to mow each day. To make things simple, suppose he already has leased a lawnmower and a pickup truck. Also, to avoid confusing the output decision with Josh's labor-leisure choice, let us assume that all of the labor is done by two hired workers.

Now, Josh has to decide how many hours for which to hire the workers, which in turn will determine how many lawns his company mows. For example, if each worker works 5 hours, then the total hours worked will be 10 and the total number of lawns mowed will be, say, 9. The key factors in Josh's decision will be worker productivity, the wage rate, and the price he can charge for mowing lawns.

Suppose that Josh receives $18 for each lawn that his workers mow. Finally, suppose that his workers' lawnmowing productivity is as follows.

Total Hours Worked Per Day | Lawns Mowed Per Day | Cost ($11 per hour) | Revenue ($18 per lawn) |
---|---|---|---|

8 | 7 | $88 | $126 |

10 | 9 | $110 | $162 |

12 | 11 | $132 | $198 |

14 | 13 | $154 | $234 |

16 | 14 | $176 | $252 |

18 | 15 | $198 | $270 |

20 | 16 | $220 | $288 |

22 | 16 | $242 | $288 |

How many hours should Josh have his employees work? We look at profit, which is revenue minus cost, to try to find a maximum. The maximum profit is at 14 hours of work, mowing 13 lawns, with revenue of $234, cost of $154, and profit of $80.

Another way to find the point of maximum profit is for Josh to hire additional hours until the next hour will add less in revenue than in cost. When we start at 12 hours of work, the next two hours produce two more lawns mowed, which gives Josh $36 in additional revenue compared with $22 in marginal cost. However, when the employees work a total of 14 hours (7 hours apiece), the next two-hour increment produces only one more lawn mowed, which means only $18 in revenue. This is less than the cost of two additional hours, which is $22. So, he should stop after 14 hours.

Another way of putting this is that a firm should increase its output as long as the marginal cost of producing additional output is less than the price of output. In short, we stop increasing output when price equals marginal cost.

Which Outputs?

Suppose that two partners are just starting a business in the field of email management solutions for corporations. They are in the process of discussing the strategy for their new venture. One partner is a computer wizard, named Cool. The other partner is a marketing and sales expert, named Slick.

A difficult decision that the partnership faces is whether it should try to make money by developing a product or by selling consulting services. Cool thinks that the company should focus on product development. "We'll get a lot more leverage out of my computer skills if we can sell a product," Cool says.

"I don't agree," Slick replies. "Products are really hard to sell. It means I'll be wasting a lot of time giving presentations to companies that are not ready to make up their minds. I say we should start out doing consulting. It's much easier to get a decision on a consulting contract."

Cool and Slick continue to argue. Cool does not like consulting, because it uses up a lot of his time and has limited revenue potential. Slick likes consulting because the cost of sale is lower. How can this issue be resolved?

The firm's goal is to maximize profits. To do so, Cool and Slick are going to have to make some estimates about the the technology and about prices.

The technology issue concerns what they can produce with their inputs, which consist of technical effort and sales effort. The price issue concerns what the prices are for a product and for consulting services.

Slick and Cool each have 2000 hours a year that they can spend working. Slick says, "For every 400 hours I put into selling consulting services, I can generate 6 consulting contracts. Each contract is worth $8000. So I can generate 30 contracts in a year, for a total of $240,000 a year in revenue."

"That's great," Cool replies. "But each contract takes 10 hours of my time, and I don't have 3000 hours. So the most we can do in consulting is 2000 hours, or $160,000 worth. What if you put all of your time into product sales?"

"Well," Slick says, "that means you put all of your time into product development, so we have a really cool product. In that case, I could sell maybe 1000 licenses for $150 each, or a total of $150,000. We're better off with consulting."

"Not so fast," says Cool. "What if we compromise? I don't develop quite as fancy a product, but I make myself available to spend some time doing consulting."

After a while, they arrive at the following estimates of what would happen under alternative allocations of time.

Cool's 2000 Hours | Slick's 2000 Hours | Product Licenses Sold | Consulting Contracts Sold | Total Revenue | ||
---|---|---|---|---|---|---|

Product Development | Consulting | Product Sales | Consulting Sales | ($150 per license) | ($8000 per consulting contract) | |

2000 | 0 | 2000 | 0 | 1000 | 0 | $150,000 |

1400 | 600 | 1600 | 400 | 770 | 6 | $163,500 |

800 | 1200 | 1200 | 800 | 480 | 12 | $168,000 |

200 | 1800 | 800 | 1200 | 140 | 18 | $165,000 |

0 | 2000 | 0 | 2000 | 0 | 20 | $160,000 |

From the standpoint of maximizing revenue, the best approach is for Cool to spend 800 hours on product development and 1200 hours on consulting. Slick will spend 1200 on product sales, and 800 hours on consulting sales.

What would be the optimal allocation of each consulting contract were worth $8500 and each product license were worth $150? What if a consulting contract is worth $16,000 and a product license is worth $300?

"The profit-maximizing allocation depends on the *relative* price of the two outputs, not on their absolute prices." Comment.

We say that by shifting hours from product development to consulting, Cool and Slick can transform their output from product licenses to consulting contracts. Going from the first row in the table to the second row, product licenses decrease by 230 and consulting contracts increase by 6. The transformation ratio, of 6/230, is greater than the price ratio, of $150/$8000, so on the margin the transformation is profitable. Similarly, from the second row to the third row, the transformation ratio is 6/290, which is still greater than the price ratio. However, moving from the third row to the fourth row, the transformation ratio is 6/340, which is less than the price ratio. On the margin, it does not pay to transform 340 site licenses into 6 consulting contracts.

In general, we say that the producer sets the marginal rate of transformation between two outputs to be equal to the relative price of those two outputs.

Choice of Inputs

We already encountered the issue of the choice of inputs when we discussed substitution. The problem is to choose the optimal levels of input to produce a given level of output.

For example, suppose that Cool and Slick hire employees to do consulting. An expert programmer charges $80 an hour, and a novice programmer charges $25 an hour. The more expert hours we use, the fewer novice hours we need. The problem is to choose the optimal mix to complete two consulting contracts. The combinations that can do this are:

Expert hours ($80) | Novice hours ($25) | Total Cost |
---|---|---|

40 | 600 | $18,200 |

50 | 400 | $14,000 |

80 | 300 | $13,900 |

100 | 240 | $14,000 |

200 | 100 | $18,500 |

The cost-minimizing combination is 80 hours of experts with 300 hours of novices. Once again, we can see find the answer by looking at marginal changes.

When we increase expert hours from 40 to 50, we reduce novice hours from 600 to 400. On average, we save 20 novice hours for each additional expert hour. This is higher than the ratio of the expert wage to the novice wage, so that it pays to add experts.

When we increase expert hours from 50 to 80, we reduce novice hours from 400 to 300. On average, we save 3.3 novice hours per expert hour, and this is slightly higher than the ratio of the expert wage to the novice wage.

When we increase expert hours from 80 to 100, we reduce novice hours from 300 to 240. On average, we save 3.0 novice hours per expert hour, and this is slightly lower than the ratio of the expert wage to the novice wage. Thus, at the margin it does not pay to increase expert hours from 80 to 100.

The way we describe this optimization is to say that the producer sets the marginal rate of substitution between inputs equal to the ratio of the prices of the inputs.