1.1 Time Warner – Memo 13 by Petek Demircioglu
To: Sales Manager, Region 3
From: Regional Manager, Region 3
Re: Recession Question
One of your MBA interns sent the attached spreadsheet in a response to a question I asked
regarding the potential impact of the 2.85 percent decline in income that has been forecasted for
our Region 3 market. This is nuts – do they not teach people how to speak in words anymore?
What do the numbers mean? Can you help me out? All I need to know is this: If income declines
by 2.85 percent as forecasted, how much will we have to cut price in order to maintain our
existing base of customers?
1.2 The Data
SUMMARY
OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R
Square
Standard
Error
Observations
0.978181049
0.956838164
0.951442934
0.185874756
19
ANOVA
df
Regression
Residual
Total
2
16
18
Coefficients
Intercept
Natural
Logarithm of
Price
Natural
Logarithm of
Income
SS
12.25460682
0.552790798
12.80739762
Standard
Error
MS
F
6.127303409
0.034549425
177.3489263
t Stat
P-value
Significance
F
1.20448E-11
Lower 95%
Upper 95%
0.453076785
2.765629855
0.163824087
0.871921192
-5.409796561
6.315950132
-2.136516553
0.113880332
-18.7610671
2.55888E-12
-2.377932071
1.895101035
0.90170075
0.262286975
3.437840363
0.003379215
0.345677205
1.457724296
2. Related Economics Theories &Analysis
2.1.1 Regression analysis theory1
The above seen data is the regression analysis of the relation between the dependent variable
(Quantity demanded) and two independent variables (Price & Income).A regression analysis is a
collective name for techniques for the modeling and analysis of numerical data consisting of
values of a dependent variable (also called response variable or measurement) and of one or
more independent variables (also known as explanatory variables or predictors).1
Let X be the independent variable and Y be the dependent. Also, suppose there is some data on
the relation between them. Suppose when the values of X and Y are plotted they appear as points
A,B,C,D,E as shown in Figure 1.
Series 1
5
D
A
4.5
4
3.5
3
2.5
Series 1
2
B
1.5
1
C
0.5
E
0
0
1
2
3
4
5
6
Figure 1
The points obviously do not lie on a straight line. Our aim here is to find a smooth curve or line
that does a good job of approximating the points. Suppose that there is a linear relation between
Y and X, but there is also some random variation in the relationship. Mathematically this would
imply that the true relationship between X and Y is: Y=a+bX+e where a and b are unknown
parameters and e is a random variable (an error term) that has zero mean. Note that for any line
drawn through the points in the above figure, there will be some discrepancy between the actual
points and the line. The deviations between the actual points and the line are given by the
distance of the dashed lines since the line represents the expected, or average ,relation between Y
and X, these deviations are analogous to the deviations from the mean used to calculate the
variance of a random variable. A regression line is the line that minimizes the squared deviations
between the line (expected relation) and the actual data points. These values of a and b, which
frequently are denoted by â and b̂ are called parameter estimates and the corresponding line is
called the least squares regression. The least squares regression line for the equation:
Y=a+bX+e
is given by
Y=â+b̂X
The parameter estimates â and b̂, represent the values of a and b that result in the smallest sum of
squared errors between a line and actual data2.
2.1.2 Regression analysis for the case:
In Time Warner’s case the parameter estimates that are calculated with excel are as shown
below:
Coefficients
Intercept
Natural Logarithm of Price
Natural Logarithm of
Income
0.453076785
-2.136516553
0.90170075
If we put the outcomes to the least squares regression line we obtain the following demand
function (the numbers are rounded to the 5th decimal place):
ln Qd=0.45308-2.13652 lnP+0.9017 lnM
Where Qd is the quantity demanded, P is the price and M is the income.
2.2 Evaluating the statistical significance of estimated coefficients
Coefficients
Intercept
Natural
Logarithm
of Price
Natural
Logarithm
of Income
Standard
Error
t Stat
P-value
Lower 95%
Upper 95%
0.453076785
2.765629855
0.163824087
0.871921192
-5.409796561
6.315950132
-2.136516553
0.113880332
-18.7610671
2.55888E-12
-2.377932071
1.895101035
0.90170075
0.262286975
3.437840363
0.003379215
0.345677205
1.457724296
The above given coefficients are parameter estimates, estimates of the true, unknown
coefficients. For every given different data set, different estimates of the true coefficients would
be obtained. Below under the ‘’a’’ section an explanation of the terms to evaluate the statistical
significance is given while in the ‘’b’’ sections the applications of them to the case is presented.
2.2.1.a Standard error: If we took several samples of the same thing we would, of course, be
able to compute several means, one for each sample. If we computed the standard deviation of
these sample means as an estimate of their variation around the true but unknown population
mean that standard deviation of means is called the standard error. Standard error measures the
variability of sample means.2
2.2.1.b In Time Warner’s case the standard error of each estimated coefficient is a measure of
how much each estimated coefficient would vary in regressions based on the same underlying
true demand relation, but with different observations.
2.2.2.a A confidence interval gives an estimated range of values which is likely to include an
unknown population parameter, the estimated range being calculated from a given set of sample
data.3
2.2.2.b In Time Warner’s case we can see from the table the upper and lower bounds of the 95
percent confidence interval for each coefficient. Take the coefficient of income for instances. We
know from the table that the best estimate for it is 0.90170075 and we are 95 percent confident
that the true value of it lies between 0.345677205 and 1.457724296.The same applies to other
two coefficients too.
2.2.3.a.1 T-Statistic: After an estimation of a coefficient, the t-statistic for that coefficient is the
ratio of the coefficient to its standard error. That can be tested against a t distribution to
determine how probable it is that the true value of the coefficient is really zero.4
2.2.3.a.2 Rule of thumb for the t test: At the 95% confidence level, if the absolute value of a tstatistic is greater than or equal to 2 ,we would conclude that the unknown true coefficient is
significantly different from zero and the parameter estimate is a good estimate for it.5
2.2.3.b In Time Warner’s case we can see from the table that while price and income parameter
estimates’ absolute value are significantly greater than 2 (approximately 18.8 and 3.4) intercept
parameter estimate’s absolute value is less than 2.This implies that we can be confident that the
true parameter of income and price are not zero.
2.2.4.a P-Value: The p-value is a probability, computed using the test statistic, that measures the
support(or lack of support)provided by the sample for the null hypothesis.6
2.2.4.b The p-values are a much more precise measure of statistical significance. For instance the
p-value 0.003379215 for the estimated coefficient of income means that there is only a 34 in
10,000 chance that the true coefficient of income is 0. Likewise the chance of the true coefficient
of price being 0 is very, very low while the chance of the true coefficient of intercept is 87%
which is considerably high. This implies that the estimated coefficient of price and income are
highly significant.
2.3. Evaluating the overall fit of the regression line
2.3.1 Terms and theories
2.3.1.1 R= Coefficient of Multiple Correlation is the positive square root of R-squared.7
2.3.1.2 R2 = Coefficient of Multiple Determination is the percent of the variation in the y-variable
that is explained by the x-variables and the model8.
Every sample has some variation in it (unless all the values are identical, and that's unlikely to
happen). The total variation is made up of two parts, the part that can be explained by the
regression equation and the part that can't be explained by the regression equation.
The ratio of the explained variation to the total variation (R2) is a measure of how good the
regression line is.9 It is computed as the ratio of the sum of squared errors from the regression
(SSregression) to the total sum of squared errors (SStotal).
R2=Explained variation
/ Total variation=SSregression / SStotal
The value of an R2 ranges from 0 to 1.The closer the R2 to 1, the ‘’better’’ the overall fit of the
estimated regression equation to the actual data. A problem with it although is that it cannot
decrease when additional explanatory variables are included in the regression. Sometimes it is
very close to 1 merely because the number observations are small relative to the number of
estimated parameters. This can provide a misleading indicator of the goodness of fit of the
regression line.10
2.3.1.3 R-squared adjusted is the version of R-squared that has been adjusted for the number of
predictors in the model. R-squared tends to overestimate the strength of the association,
especially when more than one independent variable is included in the model.11
R̅2=1-(1-R2)*(n-1)/(n-k) where n is the total number of observations and k is the number of
estimated coefficients. The difference n-k represents the residual degrees of freedom after
conducting the regression.12
2.3.1.4 S = Standard Error = Standard Error of the Estimate is the average squared difference of
the error in the actual to the predicted values.13
2.3.1.5 Observations= Number of observations in the sample.14
2.3.1.6 The F-Statistic: Provides a measure of the total variation explained by the regression
relative to the total unexplained variation. The greater the F-statistic the better the overall fit of
the regression line through the actual data.15
2.3.2 Analysis for the case
In Time Warner’s case the analysis of the below shown data is:
ANOVA
df
Regression
Residual
Total
SS
2
16
18
12.25460682
0.552790798
12.80739762
MS
F
6.127303409
0.034549425
177.3489263
Significance
F
1.20448E-11
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.978181049
0.956838164
0.951442934
0.185874756
19
From table above we can see that sum of squared errors is 12.25461 while the total sum of
squared errors is 12.8074.Thus the R2 is 12.25461/12.8074=0.956838164.This means that the
demand function we have established above (ln Qd=0.45308-2.13652 lnP+0.9017 lnM) explains
approximately 95.684% of the total variation in sales across the sample of 19 observations. Here,
since 0.956838164 is a number that is very close to 1 the overall fit of the estimated regression
equation to the actual data is considered good. But is it misleading from a number of
observations vs. number of estimated parameters? To answer this question we have to look at the
adjusted R2 value as defined above. We know that the sample size is 19 and that we have 3
parameters so the residual degree of freedom is 19-3=16.With 16 degrees of freedom the
adjusted R2 =1-(1-0.956838164)*(19-1)/(19-3)= 0.951442934.The difference between the R2 and
the adjusted R2 is 0.956838164-0.951442934=0.00539523 which is significantly little means
that the R2 value is strong based on sample size and not estimated numbers.
In Time Warner’s case the F-statistic is 177.3489263, which is a high value, thus the overall fit
of the regression line through the actual data is considerably good.
The low significance value of F means that there is a very, very low (the probability is
1.20448E-11) chance that the estimated regression model will fit the data purely by accident.
2.4 Calculating the Elasticity of demand:
Now that we have our demand function, we can obtain the income and price elasticity out of it.
ln Qd=0.45308-2.13652 lnP+0.9017 lnM
2.4.1. Terms and theories
2.4.1.1 Elasticity of demand
The degree to which a demand curve reacts to a change in price is the curve's elasticity. To
determine the elasticity of the demand curves, we can use this simple equation16:
Elasticity = (% change in quantity / % change in price)
2.4.1.2 Income Elasticity of Demand
The income elasticity of demand measures the responsiveness of the quantity demanded of a
good to the change in the income of the people demanding the good. It is calculated as the ratio
of the percent change in quantity demanded to the percent change in income. For example, if, in
response to a 10% increase in income, the quantity of a good demanded increased by 20%, the
income elasticity of demand would be 20%/10% = 2.17
2.4.1.2 Formula: Elasticities for log linear demand: When the demand function for good X is
log-linear and given by
ln Qd x=ß0+ßxlnPx+ßylnPy+ßMlnM+ßHlnH
Where ß0 and ßi’s are arbitrary real numbers, Qd x is the quantity demanded of good x, Px is the
price of good x, Py is the price of good y(the sign of the coefficient of lnPy determines whether
goods x and y are complements ore substitutes),M is the income(the sign of the coefficient of
lnM determines whether x is a normal or an inferior good) . 18
2.4.2 Analysis of the case
In Time Warner’s case since the demand function is:
ln Qd=0.45308-2.13652 lnP+0.9017 lnM
According to above given Elasticities formula the Elasticities are as follows:
Own price elasticity: -2.13652
Income elasticity: 0.9017
Which means if we increase the price of good x by 1% ,quantity demanded of good x will
decrease by 2.13652%.Also since the absolute value of own price elasticity is greater than
1,demand for good x is elastic.
Also, the income elasticity value tells us that for every 1% increase in consumer income there
will be a 0.9017% increase in quantity demanded of good x. Also, since the value is positive we
can conclude that the good x is a normal good.
Now back to our question ‘’ If income declines by 2.85 percent as forecasted, how much will we
have to cut price in order to maintain our existing base of customers?’’
To calculate that, first we have to calculate the effect of 2.85% income decline on quantity
demanded.
0.9017=% change in Qdx / -2.85 from there
% change in Qdx=-2.5698
This means that if the income declines by 2.85% quantity demanded of x will decrease by
2.5698%.
To prevent this we have to decrease the price. By how much percent shall we decrease the price
to increase the quantity demanded by 2.5698%? The answer to this question lies below:
-2.13652=2.5698/% change in price from there
% change in price=-1.2028
In order to increase the Qdx by 2.5698% we have to decrease the price by 1.2028%.
3.Conclusion
In order to keep their existing base of customer Time Warner has to cut price by 1.2028% if the
income declines by 2.85% as forecasted.
We can also conclude from the income elasticity value 0.9017 that this good is a normal good
because the income elasticity is greater than 0.Also, since the income elasticity is less than 1, we
know that an increase in income will increase the expenditure on this good by a lower percentage
than the percentage increase in income. When income declines as it is in our case, expenditures
on this good will decrease less rapidly than income.
In addition to that, since the own price elasticity value is -2.13652, we may say that the demand
for this good is elastic, which means in order to increase the total revenue we have to decrease
the price as we do in our case by 1.2028%.
Regression analysis can be applied to any industry for creating a demand function. Once you
create your demand function you can calculate any quantity demanded according to the increase
or decrease in the various variables like price, income etc. Estimating the quantity demanded
would be very useful for sales estimates and therefore production planning and optimal use of
resources.
There is not much to say about the application of this method to various industries, because this
is the core thing for every business. Without estimating the demand by using former data how
would a business owner decide on what quantity to produce and what quantity of capital and
labor to use.
4. Questions:
1) By how much percent should the consumer income increase or decrease in order to the firm to
keep its customer base if they want to increase price by 1.5%?
a) increase 3.5542 b) decrease 3.5542 c)increase 1.7654 d)decrease 2.7645
Solution:
Own price elasticity: -2.13652
-2.13652=percentage change Qdx/1.5
Percentage change Qdx=-3.2048
Income elasticity: 0.9017
0.9017=3.2048/ percentage change M
Percentage change M=3.5542
2) Write the regression line equation (for the demand function) for the following values:
Coefficients
Intercept
Natural
Logarithm
of Price
Natural
Logarithm
of Income
Standard
Error
t Stat
P-value
Lower 95%
Upper 95%
0.5
2.765629855
0.163824087
0.871921192
-5.409796561
6.315950132
-2.67
0.113880332
-18.7610671
2.55888E-12
-2.377932071
1.895101035
0.86
0.262286975
3.437840363
0.003379215
0.345677205
1.457724296
a) Qd=0.5 -2.67 P+0.86 M b) ln Qd=0.5 -2.67 lnP+0.86 lnM c) ln Qd=-2.67 +0.5 lnP+0.86 lnM
d) The above given information is not enough to solve this problem
Solution:
ln Qd=0.5 -2.67 lnP+0.86 lnM
3) Find the own price and income elasticity for the demand function in question 2.
a) Own price elasticity:-2.67, income elasticity: 0.86
b) Own price elasticity:-0.86, income elasticity: 2.67
c) Own price elasticity:-2, income elasticity: 0.88
d) Own price elasticity:-3.5, income elasticity:-2.3
Solution: a
Own price elasticity: -2.67
Income elasticity: 0.86
4) Find the degree of freedom of the regression for the below given data if there are 3 parameters
estimated.
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.787818
0.857838
0.761442
0.185874
21
a) 5
b) 10
c) 13
d) 18
Solution: d
Degree of freedom=observations-number of estimated coefficients=21-3=18
5) If demand is elastic to increase the total revenue we have to:
a) Decrease the price
b) Increase the price
c) Increase income
d) Pay fewer taxes
Solution: a
References
1
Michael R.Baye, Managerial Economics and Business Strategy, p: 96-97
2
North Carolina State University Statistics Department
http://faculty.chass.ncsu.edu/garson/PA765/normal.htm
3
4
5
Valerie J. Easton and John H. McColl's Statistics Glossary v1.1
http://economics.about.com/od/economicsglossary/g/tstat.htm
Chaman L. Jain, Regression Analysis: Modeling & Forecasting, p: 22
6
David Ray Anderson, Dennis J. Sweeney, Jim Freeman, David Ray Anderson, Thomas A.
Williams, Eddie Shoesmith Statistics for business and economics, p: 298
7
Middle Tennessee University Statistics Department
http://mtsu32.mtsu.edu:11308/regression/level3/indicator/useexcelinterp.htm
8
Middle Tennessee University Statistics Department
http://mtsu32.mtsu.edu:11308/regression/level3/indicator/useexcelinterp.htm
9
http://people.richland.edu/james/lecture/m170/ch11-rsq.html
10
Michael R.Baye, Managerial Economics and Business Strategy, p: 100-101
11
Middle Tennessee University Statistics Department
http://mtsu32.mtsu.edu:11308/regression/level3/indicator/useexcelinterp.htm
12
Michael R.Baye, Managerial Economics and Business Strategy, p: 101
13
Middle Tennessee University Statistics Department
http://mtsu32.mtsu.edu:11308/regression/level3/indicator/useexcelinterp.htm
14
Middle Tennessee University Statistics Department
http://mtsu32.mtsu.edu:11308/regression/level3/indicator/useexcelinterp.htm
15
Michael R.Baye, Managerial Economics and Business Strategy, p: 102
16
Investopedia, http://www.investopedia.com/university/economics/economics4.asp
17
Wikipedia, http://en.wikipedia.org/wiki/Income_elasticity_of_demand
18
Michael R.Baye, Managerial Economics and Business Strategy, p: 93