# A First Step

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Economists see the world through a special pair of glasses. It takes practice and concentration to learn how to see things like an economist. The interpretation of reality that is the hallmark of modern economics has been called the economic way of thinking, the economic approach, and the method of economics. Thinking and seeing the world like an economist is the ultimate goal of this book.

You will learn the economic way of thinking by working through many examples. Here is the first one.

## Optimal Allocation of Worker Hours

Suppose that you manage a tech support service for a major software company. You have two types of callers: *Regular* and *Preferred*. Your preferred customers have paid extra money for faster access, which means they expect to spend less time waiting on hold. There are equal numbers of the two types of customers and they call with equal frequency.

Management has given you a fixed number of worker hours per day to answer calls from users needing help. Daily, you have 10 workers, each working 8-hour shifts, and 5 part-time workers (4-hour shifts each); or 100 hours per day in total to support customers calling for help. These 100 hours comprise your *Total Resources*.

When customers call, an automatic message is played asking the caller to input an ID number and the caller is put on hold. The ID number is used to identify the caller as a regular or preferred customer.

Keeping callers on hold creates frustrated, unhappy customers. The callers are already angry since something has gone wrong with the software and they need help. The faster you get support to the caller the better. You keep track of *time waiting* (the amount of time, in seconds, that the typical caller is on hold) and you know that it depends on the number of worker hours available to answer the calls.

To keep things simple, assume typical time waiting = 6000/worker hours allocated. So, say there are 80 worker hours available to answer preferred callers. Dividing 6000 by 80 yields 75, which means the typical hold time is 75 seconds. This leaves 20 worker hours for regular callers, so their hold time is 300 seconds (since 6000/20 = 300). Five minutes is a long time to wait on the phone!

The problem becomes an *economic problem* because you have two types of callers, so you must decide how to allocate your worker hours. When you have to make a decision where you trade-off one thing for another you are doing economics. In this case, the more hours you allocate to one type of caller, the lower that caller’s wait time. That’s the good news.

The bad news is that the fixed amount of caller-support hours means that more time devoted to one type of caller results, by definition, in fewer hours to the other type and, therefore, higher waiting times for the other type.

So the general structure of the problem is clear: You must decide how to allocate scarce support resources (worker hours) to two competing ends. Figure 3 shows a simplified picture of the problem.

## A Complication

It is unclear exactly what preferred customers expect. Do they expect to get help twice as fast or 10 times as fast as regular customers?

To incorporate the fact that the preferred customer merits greater attention, management gives you a *value weight* parameter. The value weight tells you how much more valuable the preferred caller is compared to the regular caller.

We can write the *objective function* as \[TotalTimeWaiting=\frac{6000}{RegHours}+ValueWeight\frac{6000}{PrefHours} \nonumber \]

The objective function says that time spent waiting by a preferred caller is multiplied by a factor that reflects how much more we value the preferred customer’s time. If *ValueWeight* = 1, then preferred and regular callers are equally valuable. Management has decreed that preferred customers’ time is worth twice that of regular customers so *ValueWeight* = 2; you (the call center manager) cannot change this parameter.

So, if you decide to allocate 50 hours each to the regular and preferred customers, then both types of customers will wait 6000/50 = 120 seconds and our objective function will be 120 + 2 x 120 = 360 seconds.

Is there a better allocation, one that yields a smaller total time waiting (adjusted with the value weight), than 50/50? This question, how to allocate 100 worker hours to answering calls from regular and preferred customers in order to minimize value weighted total time waiting, has an answer, called the optimal solution. We have to find it.

## Setting Up the Problem

We will solve this problem by first organizing the information into three separate parts. All optimization problems can be set up the same way, with three components: *goal*, *endogenous variables*, and *exogenous variables*.

The goal is synonymous with the objective function. Endogenous, or choice, variables can be controlled by the decision maker. Exogenous variables are given, fixed constants that cannot be changed by the decision maker. The exogenous variables (sometimes called parameters or independent variables) form the environment under which the decision maker acts.

In the tech support time minimization problem, we can organize the information like this:

- Goal: minimize total time waiting (value weighted)
- Endogenous variables: worker hours allocated to preferred and regular customers
- Exogenous variables: total worker hours and value weight

*STEP* Open the Excel workbook *Introduction.xls*, read the *Intro* sheet, and then go to the *SetUp* sheet to implement the problem in Excel.

This workbook (along with all of the files that accompany this book) is available for download at www.depauw.edu/learn/microexcel. The User Guide has detailed instructions on how to properly configure Excel before downloading and opening these files.

Make sure that you enable macros when you open the file. If the buttons do not work, the most likely suspect is in the security settings.

*STEP* Answer the three questions in column A (below the exogenous variables). Check yourself by clicking the buttons.

## Finding the Initial Solution

Now that we have set up the problem, we can turn our attention to finding the answer, the optimal solution. There are two ways to solve optimization problems:

- Analytical (algebra and calculus) methods
- Numerical (computer) methods

The analytical method uses pencil and paper to write down equations and manipulate them to find the answer. It was the only way available until computers came along and gave us algorithms for finding solutions. Numerical methods rely on testing many trial solutions very quickly and repetitively, converging to the answer. We will ignore the analytical approach in this example and concentrate on showing how Excel’s Solver works.

*STEP* Click the Data tab (in the Ribbon across the top of the screen), then Solver (in the Analysis group) to bring up the Solver dialog box (as in Figure 4). If Solver is not available, then use the Add-in Manager to install it. Use Excel’s Help if you are having trouble or visit support.office.com.

Note that necessary information is already entered. The objective cell is the (value weighted) total time waiting, the changing variable cells (the endogenous variables) are the worker hours devoted to the regular and preferred customers, and the constraint is that the sum of the worker hours not exceed the 100 hours you have been given.

*STEP* Click the button to find the solution to the problem. Click the button in the Solver Results dialog box to accept Solver’s solution and put the optimal solution in the *SetUp* sheet.

Congratulations! You, the call center manager, have just used Solver (a numerical methods approach to optimization) to optimally allocate your scarce resources. We can check Solver’s answer for plausibility, noting that it makes sense that preferred callers have more hours allocated to them because they are more valuable. Later, we will see that we can solve this problem using analytical methods and if the two approaches give the same answer, we can be confident that we do indeed have the best solution.

## Comparative Statics

We have found the initial solution, but we are usually much more interested in a follow up question: How will the optimal solution change if the environment changes?

*Comparative statics* is a shorthand way of describing the following procedure: Change an exogenous variable, holding the other parameters constant, and track how the optimal solution changes in response to the shock.

Like finding the initial solution, comparative statics can be done via analytical (algebra and calculus) and numerical (computer) methods. The Comparative Statics Wizard (CSWiz) add-in was used to explore how the optimal allocation of total worker hours would change if worker hours were increased by 10 hours. The CSWiz add-in will be introduced later and you will learn how to do your own comparative statics analyses. For now, we will focus on what it produces.

*STEP* See the results of the comparative statics analysis by going to the *CS1* sheet.

Cells A1:D15 in the *CS1* sheet were produced by the CSWiz add-in. It is easy to see that increased total worker hours are allocated to regular and preferred customers in a stable pattern. Every additional hour of total resources, holding value weight (the only other exogenous variable in this simple problem) constant, produces an increase of 0.586 hours allocated to preferred customers. The chart below the data (row 16) shows the linear relationship. Usually, economists want to determine the relationship between optimal endogenous and exogenous variables.

## Summary: Introducing Optimization

This chapter used an example to show how Excel’s Solver can find the optimal solution. It introduced the basics of optimization, including the three parts of every optimization problem:

- Goal (or objective function),
- Endogenous variables,
- Exogenous variables.

As you work with this book, you will learn how to use analytical methods to solve optimization problems. You will also learn how to do comparative statics analysis via analytical and numerical methods.

This introductory example was completely prepared for you. All you had to do was click a few buttons. Future problems will gradually relax the Excel environment, giving you ever more freedom to make decisions and thereby learn what to do. The ultimate goal is for you to be able to set up and solve problems yourself.

## Exercises

- Suppose Management decides that preferred customers are three times as important as regular customers, so that the
*ValueWeight*= 3. With 100 workers hours, what is the optimal solution? Describe your procedure and report the optimal values of*PrefHours*and*RegHours*. - Compared to the initial solution, when
*ValueWeight*= 2, what is the change in the number of hours allocated to preferred customers? - The percentage change in
*ValueWeight*is 50% (from 2 to 3). What is the percentage change in the number of hours allocated to preferred customers?

## References

Each section ends with references and resources for further study. A citation for the epigraph (lead quotation) of the chapter is provided. References may also contain citations documenting sources used, additional information on the history of a concept or person, and suggestions for further reading.

The epigraph to this chapter is found on page 16 of the second edition of *An Essay on the Nature and Significance of Economic Science* by Lionel Robbins. This book was originally published in 1932 and the second edition is available online at www.mises.org/books/robbinsessay2.pdf. Robbins rejects old definitions of economics based on content (the study of business and work) and argues for a definition of economics based on methods used: optimization and comparative statics. Robbins made the definition of economics (in the epigraph to this chapter) famous, but he includes a footnote that cites various precursors who used a similar description of economics.

For more on Robbins, visit www.econlib.org/library/Enc/bios/Robbins.html. Econlib says that Robbins’ Essay is “one of the best-written prose pieces in economics.”

Nobel laureate Gary Becker’s *The Economic Approach to Human Behavior* (first published in 1976) has a classic introductory chapter on the meaning of the economic approach and applies economic analysis to such non-standard topics as discrimination, crime, and marriage. Becker’s statement, “what most distinguishes economics as a discipline from other disciplines in the social sciences is not its subject matter but its approach” (p. 5), greatly extends the scope of economics.

Modern economics pays little attention to its own history and how we got to be where we are today. The epigraphs in this book highlight important contributions and individuals (like Robbins and Becker) in the development of modern economic theory. Remember to experiment by clicking and searching items that catch your eye.

In Spring 2012, I videotaped my Intermediate Microeconomics classes at DePauw University. They are about an hour long and are freely available at www.depauw.edu/learn/microexcel/videos.htm. The introduction lecture covers material from this chapter.