This online support page will be updated frequently...since many browsers don't recognize changes, please refresh this page each time you visit or you might not see the updates!

   You will find reviews, assignments, answers, and links to sites of interest, including the course website discussion page. When you first log in you'll be prompted to set up an account. If you don't have a current email address, make up something like "bob@bob.net". Our conference is titled Econ 3600. If you have any problems logging in, make sure to let me know.

   Week One

   It will be a good idea for you to read Chapter 0 ASAP. This chapter reviews algebraic concepts and will help bring you up to speed on math-jargon. I'll assume that you are comfortable with this material unless you let me know.

   The first topic we study involves linear models. It's a good place to begin because these models are simple as well as being very useful in many applications. Now you might be thinking, "why is simple a good thing when the world is so complex?" I'm sure there are many answers to this question, and in our first meeting I'll ask you to help me answer it. But one thing for sure is that it's a good idea to build up to complexity in stages. In addition to pedagogical reasons, simplicity is kind of elegant. Many economists appreciate microeconomic theory because of this reason alone. Linear models are often very good approximations to reality. At the Economic Time Series Page you can view many economic variables plotted against time (thus the name Time Series). As you browse through these, some variables tend to trend upwards. If we stretched a rubber band along the data (do this on your screen!), we would find that some series fit the rubber band well. But other might not (for example, nominal GDP). As we become comfortable with using linear models we will find out ways to "straighten" time series so that even wildly curved data can be modeled using linear methods.

   In class we'll also talk about models graphically. Since you are Econ majors, you are VERY familiar with graphical presentations. I like graphs a lot because pictures convey so much information in a very concise way. If you have a chance, check out The Elements of Graphing Data by William S. Cleveland, AT&T Bell Laboratories, Murray Hill, New Jersey, 1994 (ISBN: 0-9634884-1-4). (William Cleveland is one of the world's top researchers in statistics.)

   Week Two

   As you know from last week, one of the major economic applications of linear models is in the context of systems of equations. In micro theory, market equilibrium occurs when demand equals supply (Qd = Qs). For a single market we have a demand curve and a supply curve. If the law of demand holds (the slope is negative) and if the quantity supplied is either fixed or is directly related to price (a positive slope) we are assured of an equilibrium price. Intro texts don't talk much about general equilibrium which is a theory that prices in all markets move to equilibrate the entire economic system.

   Thinking about big sets of simultaneous systems of equations will motivate our discussion of matrices and linear algebra in Chapter 2. Linear algebra is important in macro, micro, and econometrics. Essentially we'll develop a notation that facilitates the solution to big systems. Here's an example. If you look in almost any intro econometrics text, discussions are pretty scary and difficult to understand. I think this is largely because authors at this level use SIGMA notation, ie, "for i equal 1 to n, sum some function of i, blah, blah, etc." It's easy to get lost in the notation and distracted from the concepts. By spending just a few weeks on linear algebra, we can more easily talk about econometrics in a non-intimidating way.

   This week we will become fluent with basic matrix operations including addition, scalar multiplication, matrix multiplication, and inversion. We won't spend too much time on finding matrix inverses (Sections 2.3 & 2.4) by hand since that's what a computer is for. But we will do a few, just so you convince yourself that it might even be worth waiting for a dial-up connection to the University of Utah system to get access to a computer. BTW, not all good ISP's are listed on the U's link to Utah Internet Service Providers. (If you have some suggestions, please post on our webboard.)

   Week Three

   This week we continue working with matrices. I've prepared a couple of documents that explain how to use Microsoft Excel to solve systems of equations. They are:

   Now we look at one of the most important applications of linear models: linear programming. Linear programming is a subset of mathematical programming where a decision maker chooses values of decision variables in order to maximize or minimize an objective function, subject to constraints. Linear programming assumes that both the objective function and the constraints are linear. One of the nice features about this set-up is that a solution to the problem is guaranteed, if it exists.

    I was pleasantly surprised to find out that the "Solver" add-in in Excel can do both linear and non-linear programming, though the explanation was a bit hard to follow. But once I got the hang of it, setting up a problem was easy. We'll do this in class tonight beginning with a simple production problem. Take a look at the Excel spreadsheet: LinearProgram.xls and explore the values in the highlighted cells. If you cannot download it, you can view LinPro instead. In this example we have a manufacturer who produces two products (Type I and Type II). The manufacturer earns $3.00 profit from each Type I produced and $4.00 from each Type II produced. This translates into the objective function: Profit=$3X + $4Y where X,Y represents each unit of Type I and II respectively. Now let's introduce two constraints that are imposed by available time on two machines (1 and 2). Each unit produced requires two hours on machine 1 and there are a total of 10 hours available. This is the Machine 1 constraint: 2X + 2Y <=10. Additionally each Type I product requires 1.5 hours on Machine 2 and each Type II product requires 2 hours on Machine 2. We assume there are 10 hours available on Machine 2 so this production constraint translates to: 1.5X + 2Y < = 10. Before moving on, here's how to summarize our decision maker's problem:

• Choose values of X & Y to
• Maximize Profit: 3X + 4Y
• Subject to
  • 2X + 2Y < = 10
  • 1.5X + 2Y < =10
  • X,Y >=0

   Here is a link to another linear programming problem for you to try.

   Transportation problems are non-trivial and generalize into many other production/distribution problems. I've included three links below that discuss ways to formulate a shipping problem. We examine warehouses in Nevada, Illinois, and New Jersey which ship products to retail distribution sites in California, Colorado, Texas, and New York. One link is the Word document (if you have Word), one is the accompanying Excel spreadsheet which "solves" the problem, and the other discusses the problem using HTML(Java). For the mid-term exam you should expect a similar problem and I'll ask you to formulate the problem, but not actually solve it.

   Weeks Four/Five

   Chapters 4 and 5 in the text review non-linear functions that we will frequently use. Instead of spending time on those chapters I want you to use them as a reference. In class we now skip to Chapter 9 on derivatives. Section 9.1 explains how to evaluate the limit of a function (see definition p. 580). In this section you'll find that a limit either exists or not. Fortunately most of the functions we use in economics are "smooth" or "well-behaved" so limits do exist. This is fortunate because we use the concept of a limit to define the derivative which is a function, derived from another function (the parent), that tells us about the rate of change of the parent function (see Section 9.3). Finding derivatives is quite simple and next week we will focus on Sections 9.4 through 9.9. As way of warm-up, you'll find the following assignment useful.

   
You should be comfortable with finding simple derivatives. Good practice problems are on page 631+. You should be able to do exercises 1-30. Note the text distinguishes between dy/dx and f'(x) in interpreting the slope of the line tangent to the curve and the instantaneous rate of change. (Assignment Three.)

   Week Six

    This week we use calculus to solve assorted optimization problems. For example:

1. By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, the cardboard may be turned into an open box. If the cardboard is 16" long and 10" wide, find the dimensions of the box that will yield the maximum volume.
2. A company requires that its hash containers have a capacity of 36 cubic inches, have the form of right-circular cylinders, and be made of tin. Determine the radius and height of the container that requires the least amount of metal.
3. The estimated monthly profit realized by a company for manufacturing and selling X units of its product is:

   P(X) = -.04X^2 + 240X – 10,000

   How many units should be produced in order to maximize profits?

4. Postal regulations specify that a parcel may have a combined length and girth of no more than 100 inches. Find the dimensions of the cylindrical package of greatest volume that may be sent through the mail.
5. You wish to manufacture a closed rectangular box that has a volume of 4 cubic feet. The length of the base will be twice as long as its width. The material for the top and bottom of the box costs 30 cents per square foot. The material for the sides costs 20 cents per square foot. Find the dimensions of the least expensive box that can be built.

   Week Seven

   Although the functions x^2 and 2^x look similar, there's a huge difference in their behaviors. At x=2, both functions evaluate to 16, but by the time we reach x=10, x^2 evaluates at 100 and 2^x reaches 1024. One thing we have learned is that by exploring the derivative we can better understand the basic shape and key features of the function. The derivative of x^2 is 2x, but the derivative of 2^x is (ln(2))2^x! Now...what is ln(2)? Remember that a logarithm is tied to the base of an exponential function. In economics we use "e," defined as the limit as x grows large of (1 + 1/x)^x. The derivative of e^x is e^x. (The derivative of an exponential function is: d(blah^x)/dx = (ln(blah))blah^x; ln(e) = 1; the derivative of a logarithmic function, logbaseblah(x) is (ln(baseblah))(1/x), so d(ln(x)) = 1/x.) Some exercises we will go over are on page 764 & 771, but are a bit obscure in the text -- so don't get frustrated at first reading. Just remember that almost always in economics we use e as the base since e relates to growth and compounding. So here are some simple examples, look over them and you'll get the hang of it:

• d(e^x) = e^x
• d(e^(x+2)) = e^(x+2)
• d(e^some_function_of_x) = e^some_function_of_x * d(some_function_of_x)
• d(ln(x)) = 1/x
• d(ln(some_function_of_x)) = (1/(some_function_of_x)) * d(some_function_of_x)

   Important!The second mid-term is next Wednesday. I've put together a handout on partial derivatives. The HTML is here or you can download it as a Word Document. Here is the link to the Sample Mid-Term.

  Week Eight

   Probability & Statistics. If you ask most students who have taken a class in stat their likely reaction will be something very very unpleasant. So why did Penthouse (The Madonna Issue) rank Prob&Stat one of the most important classes for college students to take? That's what we'll find out in the next few weeks.  (Read Chapters 7 and 8.)

   We begin by looking at data and trying to figure out ways to discover what, if anything, data can tell us. Sometimes just categorizing data in convenient ways is important...that's data summary. A simple example uses the data set: {1,4,5,21,3,1}. So...tell me about this set? First there are six observations. The data values range from 1 to 21. Most of the numbers range between 1 and 5. One of the numbers is much larger than the rest. If we sort the numbers from low (min) to high (max) we have the set: {1,1,3,4,5,21}. The two "most" middle values are 3 & 4. The "middle" of these two numbers we might call the median which we could say is the average of 3 & 4 or 3.5. What does the median tell us? Well, if we wanted just one number to tell us what are data are "like" we could use the median. So are six numbers could be represented by just one: 3.5. What about the average (or mean) of our six? It is 1+1+3+4+5+21 divided by six or 5.83. For this data set what is a better number to represent what are data are like: 3.5 or 5.83? I'd go with 3.5. Why? And why do you think the mean is larger than the mean? What number (if any) occurs more often than any other? Answer: 1. That's called the mode of our data set. So we have five statistics that we can use to describe a list of numbers. What is a statistic? It's a function of the data.

   Now it's your turn: Due Wednesday.

• Find the min, max, mean, median, and mode of the ages of each member of your family
• Find the min, max, mean, median, and mode of the prices of what you most recently purchased at the grocery store
• Take your pulse each hour from eight AM to eight PM. Find the M,M,M,M,&M.
• What might the set of six numbers (above) represent in the real world?
• If you found out that the data set we were using was supposed to represent the ages of seven patients seen by a pediatrician between the hours of 08:30 and 09:00 on February 12, what might you think happened?
• What is the median value for this set of numbers: {3,4,1,3,2,5,1}? Don't forget to sort the numbers first!
• If you change the 5 to a 50 in the set above, does the median change? Does the mean change?
• Robust statistics are somewhat insensitive to outliers. How would you describe what an outlier is? Do you think the mean or the median is more robust? Why?

  Let's continue with our example from above using the modified data set {1,4,5,2,1,3,1} with seven observations of children's ages.  Realistic data are typically more complicated.  For example, these data can be matched with the child's name, sex, weight, and so on.  Multivariate data are best arranged in a table, or matrix.   Here's our new data set:

Name	Age	Weight	Sex
Chris	3	20	M
Ann	4	12	F
Tony	1	8	M
Alice	3	18	F
Robert	2	12	M
Liz	5	18	F
Paul	1	8	M

  Notice that we now have four variables and seven observations.  Two of the four variables are quantitative and two are qualitative.  We can also categorize variables into two types: nominal and interval.  Interval variables "vary" along a sensible scale.  So age and weight are interval variables.  Nominal variables are used to classify groups and order doesn't matter.  Here is a way to present the same data using the nominal variable MALE.  The variable takes on the value 1 when the observed sex is male and 0 otherwise.   Here's something to notice:  our sex variable takes on two possible values: Male or Female, but we only need one variable to indicate sex.   This example might make it clearer. 

Name	Age	Weight	Male
Chris	3	20	1
Ann	4	12	0
Tony	1	8	1
Alice	3	18	0
Robert	2	12	1
Liz	5	18	0
Paul	1	8	1

 

   Let's say that with our data we also track the marital status of the child's parents.  For convenience, let's say the parents are either married, divorced, or other -- three "states."  I know when I first started doing statistics I would code these as something like 1, 2, or 3 for 1 being married, 2 divorced, and 3 other.  That's fine, but there's a better way, like this:

Name	Age	Weight	Divorced	Married	Sex
Chris	3	20	0	1	M
Ann	4	12	0	1	F
Tony	1	8	0	0	M
Alice	3	18	1	0	F
Robert	2	12	1	0	M
Liz	5	18	0	0	F
Paul	1	8	0	1	M

   Can you tell from this data what the marital status of Liz and Tony's parents are?  Yes, it's "Other" since they are neither divorced nor married.   Note that the marriage state nominal variable takes on three values yet we only need two variables to completely describe it.  For "readability" it is sometimes nice to add another column: OTHER. 

Here are some questions: