## Tutorial on Optimal Choice Models

#### Introduction

Mainstream economics, or what is called neo-classical economic theory, is built upon an idea called "constrained maximization". Sometimes constrained maximization is also called "optimization". Whatever you call it, you've been doing it for much of your life even though you didn't realize it. Whenever you try to "stretch your budget" or you insist on "getting the most for your money", you're engaging in constrained maximization. In this tutorial we are going to explore some of the basic mathematics and logic involved in a constrained maximization problem.

Economists use the concept of constrained maximization to build models of how people make economic decisions. Some of these models are used to explain how consumers like you and I choose to spend our money - what economists call consumer choice theory. Other constrained maximization models are used to explain the decisions business firms make when making short-run production plans. What is essentially the same logic or model is also used to explain how many workers a firm will hire in a competitive market.  All of these types of decisions are essentially similar.  They involve somebody trying to maximize an objective (some benefits) in a situation where they face some sort of limitation or constraint (costs).  If this sounds kind of like "unlimited wants but having limited resources", it's no accident.  These models do a good job of modeling the choice when faced with an economic problem.   But before we get into an example of how this thinking works, we need to explain a very important concept: marginal vs. total amounts.

#### Understanding Marginal vs Total vs. Average Amounts

All of these models are focused on a simple question:  How much should a consumer/firm choose to consume/buy/sell/use/produce?   In other words, what's the best quantity  of some good?  For example, we could be discussing a consumer's choice of "what's the best quantity for me to purchase?", or  a firm's choice of "what's the most profitable quantity of this good for us to produce?".   We are always interested in finding how much is best, or in other words, finding the optimal Q.  What Q represents will depend upon the specific problem.  It could be quantity of pizzas purchased, or quantity of workers hired, or quantity of books printed, or quantity of hours worked.

The Quantity of a good is measurable and we can put a number to each quantity.  But each quantity of a good (say a doughnuts), also comes with two other concepts that can also be counted: benefits and costs. For the moment, let's not think about how we might measure the benefit of doughnuts. The issue of how we might measure or quantify the benefit from consuming a good is a sticky issue in economics. It involves the issue of utility measurement, which you will read about in your book. For the moment, let's think about the costs of the doughnuts. Thinking about the costs is more intutive at first.

Let's suppose you go to the bakery to purchase some doughnuts.  You can purchase as many as you wish (and can afford).  Let's say the bakery has established the following prices: doughnuts are \$0.80 each for small quantities, but they are \$9.00 per dozen.  We are measuring the costs of the doughnuts using dollars.  So now we have two variables that are associated with each other:  the quantity of doughnuts and the costs of those same doughnuts.  Measuring the quantity of doughnuts is simple enough: count them. But measuring the costs of any particular quantity of doughnuts is a bit trickier. There are three ways could measure the cost of the doughnuts. In other words for some particular quantity of doughnuts, say 12, we could give three different numbers that describe the cost of 12 doughnuts. These three ways of measuring or expressing the cost of 12 doughnuts are called the total, the average, and the marginal amount. In this example, they would be like this for a quantity of 9 doughnuts. The Total cost would be \$7.20.  The total is how much you paid for all units together - the total bill. But the average cost would be \$0.80 because each of those 9 doughnuts cost you \$0.80. The average is what each of the units contributed to total cost. Mathematically, average = total / quantity.   Now you may observe that the "average cost" is the same as the price - this is true only when the price is always fixed and the same for all quantities.  We will see that won't be true for 12 dougnuts in a moment.  Finally, we would have a marginal cost.  The marginal amount is how much the incremental unit (the marginal unit) contributes to the total. You can generally consider the next unit to be the marginal unit. In this case of 9 doughnuts, the incremental increase in cost when we added the 9th doughnut to our order was \$0.80. In other words, the total for 9 doughnuts is \$0.80 greater than the total for 8 doughnuts.

Let's look at these cost measures in a table for different quantities of doughnuts.

 Quantity of Doughnuts Purchased Total Cost (\$) Average Cost (\$ each) Marginal Cost (\$) 0 0 1 0.80 0.80 0.80 2 1.60 0.80 0.80 3 2.40 0.80 0.80 4 3.20 0.80 0.80 5 4.00 0.80 0.80 6 4.80 0.80 0.80 7 5.60 0.80 0.80 8 6.40 0.80 0.80 9 7.20 0.80 0.80 10 8.00 0.80 0.80 11 8.80 0.80 0.80 12 9.00 0.75 0.20

For quantities between 0 and 11, there are no surprises here.  The total cost = Q times Price.  The average for the first 11 is equal to the price and the marginal for the first 11 is also equal to the price.  But look carefully at the line for 12 doughnuts. The total costs is \$9.00 - what the baker will charge us for a dozen doughnuts (remember the price break for buying a full dozen?).  The average now is \$0.75 because that is the total divided by quantity: \$9.00 / 12 = \$0.75.  Now look at the marginal amount: \$0.20.  What does that mean?  A marginal cost of \$0.20 when the quantity is 12 means that the 12th doughnut only added \$0.20 to our bill.  Once we had already decided to get 11 doughnuts, we could get one more for only another \$0.20.

Both costs and benefits can be measured using any of these three ways: total, average, or marginal.  Which way is most useful depends on what you're trying to calculate or predict.  If you want to know

#### Calculating Marginal vs Total vs. Average Amounts

All three ways of calculating the associated costs or benefits of a quantity of a good, total, average, and marginal, are related to each mathematically.  The simple (non-calculus) ways to figure them are:

 To calculate this: And you have this data for some particular quantity Q: total average marginal Total done Avg * Q = total total = sum of all marginal amounts you need the marginal amounts for all units from 0 to Q., or  if you have the total for Q-1 units, then  total for Q = (total for Q-1) + marginal for Q Average average = total / Q done calculate the total for Q first, then divide that by Q Marginal Two ways: 1. most accurate but depends on having total data for both Q and Q-1 units: marginal for Qth unit = Total-for-Q minus Total-for-Q-1 unit (in other words, how much total increased from Q-1 to the Q unit). 2. less accurate, but if you only have data on say Q units and Q-x units (such as the total for 12 and the total for 10, but you don't know the total for the 11th unit): marginal for Qth unit = [(Total for Q units) - ( Total for Q-x units)] / x calculate the total for Q first, then divide that by Q done

NOTE for Students who have had/are taking calculus:  If Total is expressed as function of Q, as in:  Total = f(Q), then the Marginal amount is the first derivative.  Similarly, if you know the marginal amounts from 0 through Q, you can integrate from 0 to Q to get the total amount. We don't use calculus in the principles course, but I thought you might be interested to see that all that calculus stuff has a lot of practical application in economics.

#### Patterns of Marginal and Total

In our earlier example with the doughnuts, for quantities between 0 and 11, the results weren't very interesting.  The Total kept increasing at a nice, steady rate: \$0.80 for each additional doughnut (up to the 12th). The average was constant: always \$.80. And the marginal was also constant at\$0.80.  Of course once we ordered the 12th doughnut, things changed - the total went up but at a slower rate, the average declined a little, and the marginal declined a lot.  If all situations of costs or benefits were like the first 11 doughnuts - a constant rate of increase in total - these choices would be rather simple and boring.  But most situations are not like this.

In general there are three types of situations we encounter:  constant  marginal amounts, declining marginals, and increasing marginal amounts.  The following table illustrates each of these different situations with some hypothetical numbers. I've left some of the cells empty.  You should calculate and fill-in the blank cells - it's good practic for calculating marginal from total and vice-versa.  You will be doing this in worksheets and on quizzes.

 Quantity A constant marginal A declining marginal An increasing marginal Total Marginal Total Marginal Total Marginal 0 0 - 0 - 0 - 1 10 10 20 20 1 1 2 20 10 36 16 4 3 3 30 10 49 13 9 5 4 40 10 59 10 16 7 5 50 10 65 6 26 10 6 60 10 68 2 39 13 7 70 70 0 17 8 80 68 -2 21

Often times what makes decisions interesing is that the decision-maker faces benefits that have one pattern and costs that have another pattern.  For example, typically consumers face a declining marginal benefit of consuming more and more of the good.  Just think the first doughnut tasted really good and gave you a lot of satisfaction, but the 12th doughnut that ate (in one sitting) probably didn't make you feel better - it may even have made you feel worse (a negative marginal benefit).  Yet consumers typically face constant or increasing marginal costs. In contrast, a business firm looking to produce some good typically experiences increasing marginal costs but decreasing marginal benefits.  Explaining why behind these patterns for consumers and firms is the topic of this and the next few units in the course.

A few additional observations can be made about these three patterns of marginal amounts.  (it's a useful exercise to plot the above numbers on graph paper. Plot Q on the x-axis and each of the other variables on the y-axis, one curve for each of them.  It will help you intuitively understand the next few observations).

• a constant marginal results in a total amount that is always increasing, but increases at a constant rate.  Graphically the marginal will be horizontal (when plotted with Q on the x axis) and the total will look like a linear curve rising upward as Q increases.
• an increasing marginal results in a total amount that starts to really take off - it increases at an increasing rate.  On a graph, the marginal keeps rising, but the total not only rises, but appears to take-off and eventually looks almost vertical.
• a decreasing marginal will, of course, slope downward on a graph. Eventually, a declining marginal amount may turn negative.
• If the marginal amount is positive (marginal > 0), then the total curve will be rising and increasing but at a slower rate - it will appear to be "flattening out and reaching a peak"
• If the marginal amount is zero (0), then the total is neither increasing or decreasing.  The total has reached a peak.
• If the marginal amount is negative, then the total is decreasing at that point.

#### Finding the "Best" Q from Totals and Marginals

If we didn't face some limitations like scarce resources, then the choice would be simple. We want more. We would always answer the best Q is the largest possible Q.  After all, if the item in question is indeed a "good", then we want more.  We want as much as we can get. Why? Because if it's a "good", then it confers benefits upon us.  The good gives us satisfaction or profit or whatever it is we want.  These are the Benefits. We want to maximize our benefits - that is get as much benefit as possible.

But of course, we can't get as much as we want.  We face costs in choosing any particular Q. It's not that we want to truly to minimize costs, since that would mean we wouldn't choose anything at all.  What we want to do is maximize our benefits relative to the costs. In everyday language this is called "getting the most for our money" or "getting the biggest bang for our buck".  In more formal terms, we want to maximize the net benefit.  In other words what we really want is to find the quantity where the Total Benefits - Total Costs is greatest. This is where the constrained maximization model is handy. It provides an easy-to-use way to figure that out.

In the earlier example with doughnuts, I used costs as the variable that was associated with different quantities of the good.  Costs are measured in dollars and they're intuitive to understand.  If you are a consumer buying goods, then costs are measured in dollars, but your benefits aren't necessarily measured in dollars.  As consumers, benefits are hard to objectively measure and quantify in the real world - how do you put a number to the satisfaction that eating last night's dinner gave you?  Conceptually, we can do it by simply assuming there is a  way to measure it.  Economists call your satisfaction "Utility" and it's measured in some hypothetical units call "utils".  You can read more about this problem in your book.  For now, I want to focus on the general nature of any constrained maximization model, so I'll use an example where both costs and benefits can be measured in dollars.

A constrained maximization model, also called an optimization model, is simply a method for figuring out what Q will result the largest net benefit  - the difference between total benefits and total costs.

#### The Key:  Make Marginal Benefit vs Cost Decisions.

We could, of course, just calculate the total benefit for all quantities, calculate the total costs for all quantities, then calculate the net benefit (total benefits - total costs) for all quantities, and then review all our data to find out which Q has the highest net benefit.  This is the hard way.  There is a much easier, simpler, more practical method to find out which Q will yield the largest net benefit.

The simple way is to not try to make a single decison about what is the best quantity.  Instead, make a lot of little, incremental decisions.  In other words, make marginal decisions - decisions about whether 1 more unit of the good would makes things better or worse.  When making marginal decisions it's not necessary to know exact amounts.  What matters is very simple:  is the marginal benefit greater than the marginal cost?  It's not necessary to know the by how much the marginal benefit is greater than the marginal cost - only that marginal benefit is greater than, less than, or equal to the marginal cost.

Here's how it works. Imagine your are starting with a quantity of 0.  The marginal decision is: Should you have another unit? In more precise terms: you want to know if you should increase Q from 0 to 1 - whether you should have the first unit. The answer is: if that next unit will add more benefit than it will add to your costs, then the net benefit of the next unit is positive - it will make you better off.  So you say yes, I'll have another.  Then you ask the marginal decision again: Should you have another unit (this time the 2nd unit)? If the net marginal benefit is positive then the 2nd unit will make you better off. You should say yes.  Keep going. Keep asking whether the next unit will make you better off. Once you get to the quantity where the next unit will make you worse off, you say "no!". You stop increasing your consumption.  The point where the marginal cost of the next unit exacty equals the marginal benefit of that unit is the quantity where your total net benefit is maximized.

Let's look at an example in the next table.  In this case I'm going to measure both costs and benefits using dollars. Suppose we have a situation where you have access to buy some particular good at a fixed cost per unit to you: \$5 per unit. You can buy as much as you want and sell it to your customers.  Now let's suppose you know what your customers are willing to pay for this goods (of course, this is the hard part in real life for business people, but let's suppose we know).  Your customers are willing to purchase only one unit each from you.  You have eight customers to whom you could sell this good.  Each of the customers has a different willingness to pay, ranging from one person who will pay \$12 for it down to one cheapskate that will only pay \$1 for it.  Obviously you want to sell to those who will pay the most for it first.  So you develop the following table.  In this table we list the highest paying customers first. We also calculate profit, the net benefit, as \$ collected from selling minus cost 4 paid.

 Quantity you buy and re-sell Benefit Costs Net Benefit (Benefit - Cost) Total \$ Collected Marginal \$ Collected Total Cost Paid Marginal Cost Total Profit Marginal Profit. 0 0 - 0 - 0 - 1 12 12 5 5 7 7 2 23 11 10 5 13 6 3 33 10 15 5 18 5 4 41 8 20 5 21 3 5 47 6 25 5 22 1 6 51 4 30 5 21 -1 7 53 2 35 5 18 -3 8 54 1 40 5 14 -4

Of course, as we mentioned earlier, you could calculate all possible amounts for all possible quantities and then inspect to find which quantity provides the greatest profit.  But there's an easier way:  make a series of marginal decisions.  Start by asking should you sell the first unit? Well that 1st unit will provide an additional \$12 income but it will increase your costs by \$5, so it will add \$7 to your profit. That's \$7 more profit than you have now, so you decide to say yes to the 1st unit.

Now repeat your marginal decision-making for the second unit.  The second unit will add an incremental (marginal) \$11 to income against and additional \$5 of cost, so it will yield an additional \$6 of profit. Sure \$6 of profit is less than you earned on the first unit, but it's an additional \$6.  So you say yes again. Then repeat for the 3rd unit. And then the 4th unit. And the 5th unit. Notice the 5th unit will only yield an additional \$1 in profit, but that's an addition - it's more than you have with 4 units. Keep going. Now the 6th unit is different. The marginal benefit is less than the marginal cost for the 6th unit. You'll lose \$1 on the 6th unit. So you say no to the sixth unit. Staying at a quantity of 5 units will maximize your total profits.  Sure enough, looking at the Total Profit column we find that Total Profit is greatest at 5 units.

#### The model and logic is the same, only the units and names change.

In the next few units we will be studying how consumers make decisions to allocate their budgets and buy goods. We will study how firms make decisions on how much to produce so they can maximize profits. And we will study how firms make decisions on how firms decide how many workers to hire.  All of these situations are essentially this same marginal benefits vs marginal costs model, what we earlier called constrained maximization.  In each case the name for amount being measured will change.  Consumers will consult marginal utility. Firms look at marginal costs and marginal revenue.  Employers consider marginal revenue of product and marginal resource cost.  The names and object are slightly different. But the logic of how to maximize is the same.  This type of model is very widely used in economics. It is used in situations that we won't get to in this course, such as how government's decide which programs are most cost-effective, how politicians decide which legislation to support, and more.

Indeed, this "constrained maximization" model works so well in so many situations, that economists began to research whether other animals' choices can be predicted using the same model.  The verdict: yes.  Economists have run lab experiments with birds, rats, and other animals that are hardly mathematical geniuses - experiments that resemble more what you'd expect from behavioural psychologists. Consisently economists found that even these animals made optimizing choices as if they were doing these calculations.  Obviously the animals aren't actually doing the measuring and calculating, but they make choices as if they were.  The model works to predict behavior.

In the ne