Quantitative Demand Analysis B. Other Demand Elasticities

1. Cross Price Elasticity

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1. Cross Price Elasticity (continued) 2. Income Elasticity

3. Advertising Elasticity

C. Point price elasticities and demand functions 1. Linear Demand functions 2. Logarithmic Demand

Lecture ______________________________________________________

B. Other Demand Elasticities. Elasticity is a sensitivity measure that may be of interest with respect to any independent variable. Some other important elasticities can be readily calculated from a demand function.

1 Cross Price Elasticity of Demand (continued) c. Uses:

i. To assess responses of competitor's actions.

ηXY > 0 good are substitutes.

ηXY < 0 goods are compliments.

ηXY = 0 goods are unrelated

ii. Forecasting. Suppose you sell fish dinners. A close rival also sells the same, and lowers his price 20%. If the cross price elasticity of demand is 2, how much will you lose in terms of sales?

2 = (%∆Qx)/-20. thus, - 40% is the answer.

iii. More complicated interactions. Continuing with the above example, Suppose your own price elasticity is -3. How much of a price increase would restore your sales?

-3 = 40/%∆P. Implies 13.33% price decrease.

2. Income elasticity:

a. Definition: The percentage change in demand brought about by a 1% change in income on aggregate demand.

ηI = (∆Q/∆I)(I/Q)

via the arc price estimate, this is calculated as:

= [(Q1-Q0)/(I1-I0)][(I1 +I0)/(Q1+Q0)]

Point and percentage change calculations are also parallel to those for price elasticity.

b. Use: To assess effects of changes in underlying economic conditions - If 0 < ηI < 1, the good is noncyclical. Ex: Foods, shoes, gasoline.

- If ηI > 1 We will say the good is cyclical. Ex: Autos, housing, luxury goods.

- If -1< ηI < 0 the good is inferior

- If ηI < -1 the good is countercyclical (Note: We don’t talk much about countercyclical goods. There are few)

c. Applications:

i) Assessing susceptibility to economic conditions. With normal goods, firms with a big income elasticity of demand will grow quickly. Example, Income elasticity of demand for autos is 3. Such goods are also sensitive to decreases in aggregate income.

ii) Forecasting: Suppose income elasticity for cigarettes is .6. A 5% increase in personal income could be expected to increase demand by :

.6 x .05 = .03 or 3%.

3. Other Demand Elasticities. Elasticity is a sensitivity measure that may be of interest with respect to any independent variable. Some other important elasticities can be readily calculated from a demand function. To illustrate, we introduce one final elasticity measure, Advertising Elasticity

a. Definition. The percentage change in Quantity induced by a one percent change in advertising expenditures.

ηA = (∆Q/∆A)(A/Q) - Using the arc elasticity formula

= [(Q1-Q0)/(A1-A0)][(A1 +A0)/(Q1+Q0)

A point advertising elasticity parallels the statement of the other elasticities.

b. Uses: It is one of the variables the firm can control, and can use to respond to changes in the things it cannot control (such as the price of related goods, income etc.) Notice that advertising elasticity should be positive. If not, it is an advertising campaign that is detracting from sales.

c. Examples.

-Suppose that the income elasticity of demand is .5, and the advertising elasticity of demand is .2. If income falls by 2%, how much would a firm have to increase

advertising to make up for the difference?

(%∆Q)/(%∆I) = .5

If %∆I = -2, then %∆Q = -1. If ηA= .2, then 1/(%∆A) = .2, or

%∆A = 5.

-Suppose a competitor lowers the price of a good by 20% and that ηA = 1, ηXY = 2 and η= -3. How much must you lower price in order to keep sales constant? How much must the firm increase advertising in order to keep sales constant?

C. Point Price Elasticities and Demand Functions. Given the underlying demand function, we are able to make more precise elasticity calculations. We will consider first how to draw inferences from linear demand specifications. This will be followed by inferences from a second popular demand specification, a logarithmic specification.

1. Elasticity and Linear Demand Estimates. Consider the following demand function (say Q = mugs of beer in local restaurants), Pf = the price of meals, A = Advertising expenditures)

Q = -50P + 20I -5Pf + .1A

Where I is measured in 000's of dollars. and where I = $10,000, Pf = $10, A = $500, and, say, P = $2. Then Q = 200 - 50(2) = 100

- Point price elasticity is dQ/dP = -50

η = -50(2)/100 = -1.

- Point income elasticity is dQ/dI = 20

ηI = 20(10)/100 = 2 - Cross price elasticity is dQ/dPf = -5

ηbf = -5(10)/100 = -.5 - Advertising elasticity is dQ/dA = 0.1

ηA = 0.1(500)/100 = .5

And you can do exactly the same exercises as before.

i) Suppose that GNP increases by 5%. How much could restaurants raise price and keep sales constant?

2 = x/5 implies a 10% increase in sales.

To offset the sales increase, a 10% increase in price would suffice.

ii) Suppose that food prices increase by 10%. What change in advertising expenditures would keep sales constant?

Answer: Food price increases by 10% imply that beer sales decrease by 5% (due to the cross price elasticity of -.5) A 10% increase in advertising expenditures would offset the change.

2. Logarithmic demand. A problem with trying to do exercises with interrelated

elasticities in this context is that the elasticity coefficients change with alterations in the value of the independent variables. It is for this reason that an alternative specification is used frequently. This alternative specification, called a constant elasticity demand curve, has the property that elasticities remain constant.

Q = aP^b1I^b2

If a = 200, b1 = -.3 and b2 =2, then

Q = 200P^-.3I²

Although this looks obtuse, it is in fact pretty useful. Most importantly, it is readily estimated, by taking logarithms:

log Q = log 200 + -.3 log P + 2 log I.

More importantly, the parameters in this case are directly the elasticities. It is for this reason that this is called a constant elasticity demand function. It is a useful

approximation when you wish to assume that elasticity of demand is constant.

(∂Q/∂P)(P/Q) = ab1P^b1-1 I^b2(P/Q)

= b1

With this information we can examine firm responses to own and other effects.

3.. Some Practice Exercises with Demand Functions.

i. 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?

ii. 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?

iii. 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 η = -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

η = 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.

iv. Example 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?

Answer η = -1.7

Lecture 17

REVIEW___________________________________________________:

III. Quantitative Demand Analysis

C. Point price elasticities and demand functions 1. Linear Demand functions 2. Logarithmic Demand

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D. Estimating Demand: Regression Analysis.

1. The Bivariate Case 2. The Multivariate Case

LECTURE_________________________________________________________

D. Estimating Demand: Regression Analysis. Our intention in this section is to learn from where the parameters of a demand function might come. That is, given the relationship

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.

We would like to know how to get values for the parameters A, B1, B2, B3, and B4. Regression analysis is simply a statistical tool that allows us to estimate the magnitude of these parameters. Estimates may be made 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)

1. The Bivariate Case. 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

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

The method of least squares is simply a way to pick the intercept and slope of a line that gives a “best” fit of the points. The idea is simply to 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!)

2. 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

Via such an analysis you would generate a demand function estimate of the type

Yi = 100 + 4X - 2.5Pi

We could, with this linear relationship, do precisely the sorts of elasticity calculations and estimates that we discussed just prior to the first exam.

Lecture 18

REVIEW___________________________________________________:

III. Quantitative Demand Analysis

. Estimating Demand: Regression Analysis.

1. The Bivariate Case 2. The Multivariate Case

(Recall, regression analysis is just an analytical way of picking the intercept and the slope of a line to ‘best fit’ the data, where by the term ‘best fit’ we mean a line that minimizes the sum of squared deviations between observations and the estimated line.

Squaring deviations has the advantages of (a) keeping positive and negative deviations from canceling each other out, and (b) paying particular attention to outliers.

Preview__________________________________________________________

3. Doing regressions. Examples and an analysis of regression output.

4. Interpreting the Significance of Individual Parameter Estimates.

5. Forecasting.

LECTURE_________________________________________________________

3. Doing regressions. Examples and an analysis of regression output.

Regressions are quite easy to do with a spreadsheet.

a. Let’s start with a bivariate case. Suppose we are interested in estimating the relationship between sales of a good (Yi) and advertising expenditures (Xi). We have the following data.

We can use a spreadsheet to do many the appropriate calculations. In class, we will replicate these columns on EXCEL, and use the regression package to yield the following result (I interpret regression output more fully below, but here are your parameters)

^

i = 2.533 + 1.504 Xi

b. A Multi case. Adding independent variables is straightforward.

Consider our original problem, but now include price. P;

That is, we estimate

Yi = A + B1Xi + B2Pi + ei

as

^

i = a + b1Xi + b2Pi

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

Sales Adv Price

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. Regression Output. A more detailed overview.. In this section, we review regression results. In this course, we will focus on just three aspects of regression results. (a) The “goodness of fit” or R², (b) the standard error of the regression estimate, and (c) the standard errors of the coefficient estimates.

Suppose that you conduct a linear regression of the equation:

Q = βo + β1P + β2 Py + β3A

where P is the price of the good, Py is the price of a substitute good, and X are advertising expenditures. Your data look like the following

Q P Py A

47 4 11 12

51 4.5 15 15

29 7 8 12

42 4.8 12 22

19 12 6 10

72 5 18 18

30 6.8 6 10

41 5.2 9 20

32 6 12 11

47 5 10 21

65 3.7 17 10

Notice that you have a total of 22 observations. Now, if you input this data into EXCEL and run a regression, you will get output in three blocks of rows. At the top, you get

Regression Statistics descriptive measure indicating how well the estimate fits the

regression line. A number closer to 1 indicates a better fit.

Second look at the Standard Error (here 6.15) we will use this later when we talk about forecasting

A second block of information is the following:

Coefficient

s Standard

Error t Stat P-value Lower 95% Upper 95% Lower

95.0% Upper 95.0%

Intercept 24.2067 10.08245 2.400875 0.027378 3.024246 45.38916 3.024246 45.38916

X Variable 2 -2.06908 0.971137 -2.13058 0.04717 -4.10937 -0.0288 -4.10937 -0.0288

X Variable 3 2.620155 0.488371 5.365095 4.24E-05 1.594126 3.646185 1.594126 3.646185

X Variable 4 0.196674 0.291057 0.675724 0.507805 -0.41481 0.808163 -0.41481 0.808163

This information summarizes the descriptive power of individual

coefficients. Observe first that the variables are simply summarized as

“X Variable 1”, “X Variable 2” and etc. You should replace these with your variable titles, for example:

Intercept 24.2067 10.08245 2.400875 0.027378 3.024246 45.38916 3.024246 45.38916

Price -2.06908 0.971137 -2.13058 0.04717 -4.10937 -0.0288 -4.10937 -0.0288

Price of X 2.620155 0.488371 5.365095 4.24E-05 1.594126 3.646185 1.594126 3.646185

Advertising 0.196674 0.291057 0.675724 0.507805 -0.41481 0.808163 -0.41481 0.808163

Now, the second column provides an estimate of the parameter values, for example, our estimated equation is

Q = 24.207 – 2.069 P + 2.62 Px + 0.197 Adv.

Column 3 provides some particularly interesting information, in that this is a measure of the precision of the estimate. It is called the standard error of the parameter estimate βi , or σβ. For those of you who remember some statistics, you may recall from the central limit

theorem, that about 95% of standard errors fall within the range βi

±2σβ

Columns (6) and (7) provide a more precise statement of the bounds of 95% confidence bands about particular parameter

estimates. If these bounds exclude 0, then you may conclude at a 95%

level of confidence that the parameter explains some of the movement in the dependent variable.

4. Interpreting Significance of Parameter Estimates

Column (4) lists the t test statistic for each parameter estimate.

This test statistic is simply the parameter estimate divided by its standard error. The test statistic is just an alternative way to assess whether or not estimates fall within the βi ±2σβbound. Suppose, for example, that the lower bound of the 95% confidence range for an estimate is βi -2σβ =0 implies that

(βi)/ σβ = 2

or that the bounds of the confidence band just equals zero. If the lower bound was greater than zero, for example,

βi -2σβ =1, then

βi /σβ = 1/σβ+2, a number larger than 2.

Oftentimes, you will see the information from your regression output summarized in a write up as follows

∧Q = 24.207 – 2.069 P + 2.62 Px + 0.197 Adv. R² = .82 , n=22 (10.1) (0.97) (0.49) (0.29)

Where the standard errors are written in parentheses below the coefficient estimates.

I would like you to be able to interpret the following from this information:

a) The variability of individual parameter estimates. There is (roughly) a 95% chance that the true value of any coefficient estimate is within ± 2σβ of the estimate.

For example, the price coefficient is in the interval –2.069 – 2(0.97) and –2.069 +2(0.97), or -4.09 to -.129. Thus the price coefficient is, significantly less than 1 with a 95% probability (alternatively, you might observe that -2.069/.97 = -2.13

Lecture 19

REVIEW___________________________________________________:

III. Quantitative Demand Analysis

. Estimating Demand: Regression Analysis.

3. Doing regressions. Examples and an analysis of regression output.

Preview__________________________________________________________

4. Interpreting the Significance of Individual Parameter Estimates.

5. Forecasting.

IV. Chapter 5. The Production Process and Costs A. Introduction:

B. The Production Function.

LECTURE_________________________________________________________

4. Interpreting the significance of individual parameter estimates.

Example #1. If I give you the estimated equation

∧Q = 24.207 – 2.069 P + 2.62 Px + 0.197 Adv. R² = .82 , n=22 (10.1) (0.97) (0.49) (0.29) SER = 2.2

I might ask whether or not price was a significant explainer of sales. You could evaluate this by adding and subtracting 2 times the std. error of the regression to the parameter estimate. If the interval doesn’t include zero then at a 95% level of confidence, an inverse relationship exists between price and quantity.

-2.069 + 2(.97) = -.129 - 2.069 – 2(.97) = -4.009

Result: Yes, price is inversely related to quantity. On the other hand, consider advertising

.197+ 2(.29) = .1125 .197 – 2(.29) =-.383

This interval includes zero, so we cannot conclude that there is a direct relationship between advertising and quantity at a 95% level of confidence.

Comments on Multivariate Regression. Obviously, when constructing a demand relationship, you have some choice as to which variables to include. 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.) 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.

Question: In the above, if Advertising does not significantly affect sales, should we delete it from the analysis? No, because advertising may still be important, and deleting imwe might introduce bias.

In general, the appropriate approach is to include all variables for which you have a straightforward reason for including.

5. Forecasting. One can also get a feel for the precision of a forecast by using the SER Given independent variable values one can construct an approximate 95% forecast interval by adding and subtracting 2* the MSE of the regression to the point estimate.

For example, example 1 above, suppose P = 2, Px = 1 and Adv =10.

Then

Q = 24.207 – 2.069(2) + 2.62 (1) + 0.197 (10) = 24.659

An approximate 95% confidence interval about the estimate would be 24.659 + 2(2.2) = 29.059

24.659 - 2(2.2) = 20.259

Your estimates get worse the further away you get from the mean of the sample. Thus, observations out of the range of the sample are very speculative. (Example: Can you forecast future sales, given price and advertising expenses? It depends on the relationship between your proposed price and advertising expenditures, and those you've observed in the past.)

Also, your forecasts will be worse, to the extent that there is any expectation that the future may be different from the past (For example, a forecast of cigarette sales would be very considerably more variable than a 95% confidence interval would suggest were there some substantial possibility that cigarette sales would be outlawed next year.

Some additional examples:

Example #2: Suppose you conduct a regression with n=9 data points, and generate the following results:

Qi= 10 - .5Pi + .1 Ai

(5) (.03) (.2) n = 9, R² = .83 SER = .5 - Interpret R²,

- Does price explain movement in sales?

- IF p=4 and A=10, make a 95% confidence interval for sales Example #2. Consider a regression with the log of data.

lnQi= 50 - .2 ln Pi + .12ln Ai (20) (.3) (.08)

n = 12, R² = .68, - Interpret R²

-. Does price alone explain movement in sales, Qi?

Notice finally, that we can make one further interesting insight with a log linear regression. Observe that η=-.2 Notice that two standard deviations about -.2 are not necessarily greater than zero. However, this interval does not include -1. Thus, we can conclude that we are on the inelastic portion of the demand curve.

Lecture 20

REVIEW___________________________________________________:

III. Quantitative Demand Analysis

. Estimating Demand: Regression Analysis.

4. Interpreting the Significance of Individual Parameter Estimates.

5. Forecasting.

Preview__________________________________________________________

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