February 13, February 18, 2002



Monday February 24,2003

Lecture #5

Reading: This evening I will cover the material in Ch. 3, pp. 92-102. Given sufficient time I will also provide some supplementary regression material.

TEST: The class can decide. We can defer the test a week if you like. However, that means that test #1 will not be until after spring break.

Homework - Today, I asked you to do handout #1; 1, 2, 3, 4, 5, 6, 7. I will collect these problems.

Keys: Ch. 3: 3, 4, 5, 6, 12, 13, 14 Handout #1, 1, 2, 3, 4, 5, 6, 7,

Next class: Handout #2, 1, 2, 3, 4

Ch. 3, 7, 8, 10, 15, 16

Review

B. Other Demand Elasticities. We demonstrated how one could calculate a variety of elasticity concepts in a way that paralleled our development of own price elasticity. Together with price elasticity, these allowed examination of yet other problems

Example: Joe Doe, CEO of Doppler Inc. observes the sales of his weather radar printers fall 10% in response to a 5% increase in the price of weather tracking software.

a. What is the implied cross price elasticity of demand? How are radar printers and weather tracking software related? What would be another example of such goods?

b. Suppose that own price elasticity is -.5. Approximately, how much, and in what direction could John adjust the price in order to restore sales quantity to its original level? Would such a response be a profitable?

Example Suppose that the income elasticity for Calaphon aluminum cookware is 1.5, and that the advertising elasticity is 2. Approximately how much, and in what direction could Calaphon adjust its advertising revenues to counteract the effects of a projected 5% decrease in GNP in the coming year?

c. We also learned how to anticipate the effects on revenue of a price change. For example, suppose that you sell coffee and donuts. Daily TR from coffee is $4000 and daily TR from donuts is $3000. Suppose that (= -1.2 for coffee and (donuts, coffee =-1. What would be the change in TR if you cut the price of coffee by 10%?

(TR = [4000(1-1.2) + 3000(-1)}(-.1)

= [-800 -3000](-.1)

= 380

C. Point Price Elasticities and Demand Functions. Then we demonstrated how point-elasticities could be calculated from both linear demand functions, and from logrithmic demand functions.

With this information, you could work problems like the following:

Example Suppose that the demand function for Sorby Floppy disks is of the form

Qx = 600 - 40Px + .2Py + 2I

Where Qx = Hundreds of packages of Sorby Floppy disks sold in the U.S. per week,

Px = the price of a package of the disks.

Py = the price of upgrade Hard Disk Drives for Personal Computers.

I = per capita income (in thousands of dollars).

Suppose that at present

Px = 10

Py = 300

I = 10,000.

a. Calculate the point price elasticity of demand, the point cross price elasticity of demand for computer disks with respect to hard disk drives, and the point income elasticity of demand.

Qx = 600 - 40(10) + .2(300) + 2(10)

= 600 - 400 + 60 + 20

= 280

Thus ( = -40(10)/280 = -1.43

b. Given the above demand relationship, what can you say about the relationship between hard-disk drives and floppy disks? Why?

Answer: The sign on the intercept for hard disk and floppy disks is positive, indicating that an increase in hard-disks will increase floppy disk sales. The products are substitutes.

c. Given the above demand relationships, are computer disks a normal or inferior good? Why?

Answer: The positive sign on the income coefficient indicates that the floppy disks are normal goods. Incidentally, the income elasticity is

( = 2(10)/280 = -0.07.

Thus, the goods are noncyclical normal goods.

d. Could the makers of computer disks increase profits by raising prices? Why or why not?

Answer: You can’t tell. The firm is on the elastic portion of the demand curve. An increase in prices will decrease both revenues and costs. A definitive answer could be given in this case only with information about the cost function.

Also, you can do problems with non-linear equations. Suppose, for example, that the demand function is given by

logQx = 5 - 1.7log Px + .3log S - 3 log Ay

Then what is price elasticity of demand?

Finally, we made some observations regarding the price elasticity of demand and the underlying industry structure.

Example. If firm price elasticity of demand for Axel Hammers is -.75, what can you say about the type of industry in which it competes? What about

-3?

Preview

Chapter 3: Quantitative Demand Analysis: Part 3, Demand Estimation

E. Ways to Collect Data about Demand Parameters.

F. Regression Analysis: The method of least squares.

G. Interpreting Regression Results.

Lecture

E. Ways to estimate demand. Our ultimate intention in discussing demand functions and demand curves is to develop a means of estimating demand. That is, given a particular specification of a demand curve, how can one come up with numbers for the parameters? Three methods are used most typically.

1. Consumer interviews: You could go to a shopping mall and ask people what their response might be to a 5% change in the price of hamburger sandwiches.

a. Advantages: Inexpensive

b. Disadvantages:

Sample Bias: You must be careful to collect evidence from a representative portion of the population.

Response bias: People do not respond very well to hypothetical situations. They make a lot of implied threats (e.g., if the price increased by 5% , I would never purchase a sandwich again!)

2. Market Experiments: Several methods.

a. Direct market experimentation: Find a series of demographically similar, but geographically distinct locations. Then vary the price of the product among the locations.

i: Advantage: Great information

ii. Disadvantages: Can be exceedingly expensive. Due to non-cost related price changes, and to reputational effects.

b. Laboratory experiments: Give consumers money, and have them shop in a simulated store.

i. Advantages:

Again, direct information

Relatively low cost.

ii. Disadvantages:

Response Bias may again be a problem. Consumers know that their behavior is monitored, and may alter purchasing patterns. Consider, for example, the problem of packaging fresh beef (the paper by Hoffman et al.). There is a clear response bias toward trying out the new product.

3. Regression Analysis: Use statistical techniques to estimate Demand from the price and quantity data generated in the sales process. .

a. Advantages

i. Inexpensive

ii. Non-invasive

c. Disadvantages

Estimates may be unreliable or imprecise. (But we can learn to qualify appropriately results)

It is this latter method that will be the focus of our attention.

F. Regression Analysis: An Overview

1. Consider a typical demand function

Y = A + B1X + B2P + B3I + B4Pr

Where

Y = qty. demanded

X = advertising and promotional expenses

P = price of a good

Pr = price of a related (competing) good.

As we know, A, B1, B2, B3, and B4 are all parameters. Our goal is to estimate them. Regression analysis is simply a statistical tool that allows us to estimate the magnitude of the parameters.

2. The bivariate case.

a. Suppose in some simple world, sales are only affected by advertising expenditures. Assume also that the factors are linearly related. Then we have

Y = A + B1X.

Suppose further, however, that this specification is a model - by assumption a simplification from the natural world. Suppose that there is some random error e in our estimate. That is, for each observation i,

Yi = A + B1Xi + ei

ei has a mean of 0.

Graphically

Q

| |

| |

|B |

| |

| |

| |

| |

|}A |

| |

A

This is called a population regression line. (Or the true underlying relationship.)

b. Sample Regression line. Of course, we don't see the underlying population regression line. Rather, we must try to estimate it from available data. The general expression for this sample estimate is

^

i = a + bXi.

That is, given Xi, we pick a and b to estimate i . For example, an estimate might be:

^

i = 2.533 + 1.504Xi

Sales

| ^ |

|* Y |

|* |

|* * |

|1.504 |

|* |

|* * |

|* |

|* |

|} 2.533 |

| |

X = Advertising

4. Some Insight into the method of least squares.

a. The idea: Simply minimize the sum of squared differences

^

between Yi and i. We will call this sum S, or

^

S = Σ (Yi - i )2

S = Σ (Yi -a - bXi )2

To optimize this equation, simply take the derivatives and solve: (but we will skip this derivation!) take FONC, w.r.t. to the intercept and slope parameters:

[pic]

[pic]

_ _

b. Example: Given X and Y, you can easily compute optimal a and b values. Consider the following problem:

Sales Yi Adv Xi Xi2 XiYi

4 1 1 4

6 2 4 12

8 4 16 32

14 8 64 112

12 6 36 72

10 5 25 50

16 8 64 128

16 9 81 144

12 7 49 84

98 50 340 638

_

Y = 98/9 = 10.889

_

X = 50/9 = 5.556

Inserting these values into the above equations generates:

b = 638 - 9(5.556)(10.889)= 1.503571

340 - 9(5.556)2

a = 10.889 - 1.503571(5.556)= 2.535714

Thus, the regression equation is

^

i = 2.533 + 1.504 Xi

c. Regression and EXCEL. Fortunately, we can use a spreadsheet to do many of these calculations for us. In class, we will replicate these columns on EXCEL, and show that the regression package yields exactly the same result. (NOTE: You should also plot these results and illustrate the residuals.

5. Multivariate Regression. The intuition that we have motivated with a single variable readily extends to multiple variables. Although the calculations quickly become very messy, they are easy to do with a computer.

a. The Problem. Consider our original problem, but now include P;

Yi = A + B1Xi + B2Pi + ei

(assume that X and P are independent, and that the average ei is 0) via a regression, we can compute and estimate:

^

i = a + b1Xi + b2Pi

Note, the process is one of minimizing a, b1 and b2

^

Σ( Yi - i )2 as before. In this case

S = Σ( Yi - a - b1Xi - b2Pi)2.

Just as before, we get three equations in three unknowns via this exercise, and we can solve just as before.

b. Example. Using a spreadsheet, suppose we extend our original as follows:

Sales Adv Price

(mill U) Exp

4 1 1

6 2 0.5

8 4 5

14 8 8

12 6 4

10 5 3

16 8 2

16 9 7

12 7 6

Regressing the first column on the second and third generates

a = 2.61, b1=1.75 and b2=-0.36

(Observe that the b1 coefficient changes from before (now it is 1.76 vs 1.5 before. The reason is that the new equation holds constant the effect of price changes.)

Notice that we can do the same exercises that we did with other elasticity estimates. Suppose, for example, that p = 20 and Sales Exp. = 6. Then what is the price elasticity of demand?

Q = 2.61+ 1.765(6) - 0.36(20) = 6

Thus, η = -0.36(20)/ 6 = -1.2.

c. Comments on Multivariate Regression

i. The effects of multiple variables. Increasing the number of independent variables always improves your estimate in the sense that you get a better "fit." (Intuitively, by adding terms you gain extra latitude in trying to minimize the squared differences between observed and predicted data. At the worst, the new estimate will equal the old one. Improvements are possible if tilting the "plane" (relative to the line estimated in 2 dimensions) allows for a reduction in errors. Mathematically, each term is added linearly, and thus is the variable's partial derivative). Nevertheless, it is generally not a good idea to add variables to "maximize the fit," for you can easily add in too many things, disguising possible significant relationships.

ii. Reasons for adding multiple variables

- To evaluate the effect of a predicted independent variable on the dependent variable. To see if this effect is independent of the other independent variable(s).

-. Leaving out variables may bias results (in a statistical sense that will be made clear presently).

Clearly there is a balancing of sorts here. Too many variables detract from the explanatory power determinants that are in fact important. Too few will lead to biased results. I suggest the following

A decision rule: Include all variables for which you have a straightforward reason for including.

iii. Adjusting the Regression Line. In some instances a poor regression fit may arise because the data is non linear. This problem can be addressed in a number of ways. I mention two.

- Additive power terms. If we expect the relationship between Advertising and Sales to increase at an increasing rate, we may wish to add in a squared term, e.g.,

Yi = a + b1Xi + b2Xi2

Here, b2 > 0 implies a convex relationship, while b2 < 0 implies a concave one.

- Log-linear Relationships. As mentioned in the last chapter, logrithmic expressions may also be used to reflect nonlinear relationships. Consider the following relationship

Yi = aXib

In this case b>1 implies that this increases at an increasing rate, and b ................
................

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