6580262 Managerial Economics Chapter 4

8/13/2019 6580262 Managerial Economics Chapter 4

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Outline of the lecture

-Introduction

Statistical estimation of the demand

function Model

OLS estimation techniue

Interpretation of the results !esting


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!he preceding chapter de"eloped the theory ofdemand# including the concepts of priceelasticity# income elasticity# and cross-elasticityof demand$

% manager &ho is contemplating an increase inthe price of one of the firm's products needs to(no& the impact of this increase on:

)*+ uantity demanded ),+ total re"enue

)+ profits


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.hat uestions should themanager ans&er/

- Is the demand elastic# inelastic# or unit elastic&ith respect to price o"er the range ofcontemplated price increase/

-.hat &ill happen to demand if consumerincomes increase or decrease as a result of aneconomic e0pansion or contraction$

Managers face these types of pro1lemse"eryday &hether in a profit-see(ing enterprises#not-for-profit organi2ations or go"ernments$


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E0ample

.hat &ill 1e the impact of cigarette ta0eson my uantity demanded of my product/

.hat effect &ill a tuition increase ha"e onlocal state uni"ersity re"enues/

!hese are the types of uestions theempirical in"estigation attempt to ans&er


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Statistical Estimation of theDemand 3unction

Econometrics is a collection of statisticaltechniues a"aila1le for testing economictheories 1y empirically measuring relationships

among economic "aria1les$ !he measurement of economic relationships is a

necessary step in using economic theories andmodels to o1tain estimates of the numerical

"alues of "aria1les that are of interest to thedecision ma(er$


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!he estimation of a demand function usingeconometric techniues in"ol"es the

follo&ing steps - Identification of the "aria1les

- Collection of the data

- 3ormulation of the demand model

-Estimation of the parameters

- De"elopment of forecasts )estimates+1ased on the model


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Identification of the "aria1les

%s discussed in the pre"ious chapter# thedemand function may 1e "ie&ed as therelationship 1et&een the uantity demanded )thedependent "aria1le+ and se"eral independent

"aria1les$ !he first tas( in de"eloping a statistical demand

model is to identify the independent "aria1lesthat are li(ely to influence uantity demanded$

!hese might include factors such as the price ofthe good in uestion# price of competing orsu1stitute goods# population# per capita income#and ad"ertising and promotional e0penditures$


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Collection of the data

Once the "aria1les ha"e 1een identified#the ne0t step is to collect data on the"aria1les$ Data can 1e o1tained from a

num1er of different sources$


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3ormulation of the model

!he ne0t step is to specify the form of theeuation# that indicates the relationship1et&een the independent "aria1les and

the dependent "aria1le$


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E0ample

!he linear model# &hich is the mostcommon form of estimation euation:

++++=332211

XXXY


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Linear Model

# 5*#5,#5#6 are the parameters of the

model and 6 is the error term$

!he error term is included in the model toreflect the fact that the relationship is notan e0act one# i$e$# the o1ser"ed demand"alue may not al&ays 1e eual to thetheoretical "alue$


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Interpretation of the "alue of 5

!he "alue of each 5 coefficient pro"idesan estimate of the change in uantitydemanded associated &ith a one-unit

change in the gi"en independent "aria1le#holding constant all other independent"aria1les$


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Interpretation of the "alue of 5

Y

X

E

Y

X

X

Y

E

D

D

2

2

2

.

.

=

=


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Simple Linear 7egression Model

!he analysis in this section is limited to thecase of one independent and onedependent "aria1le )t&o-"aria1le case+#

&here the form of the relationship 1et&eenthe t&o "aria1les is linear$


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Estimating the simple linearregression coefficients

=

=

=

=

=

=

n

i

i

n

i

ii

n

i

i

ii

n

i

nXX

YXnYX

b

XX

YYXX

b

1

22

1

1

2

1

)(

))((


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E0ample *

Sher&in-.illiams company is attempting tode"elop a demand model for its line of e0teriorhouse paints$ !he company's chief economist

feels that the most important "aria1les affectingpaint sales )8+ )measured in gallons+ are:

)*+ promotional e0penditures)9*+ )measured indollars+$ !hese include e0penditures on

ad"ertising )radio# !# and ne&spaper+# in-storedisplays and literature# and customer re1ateprograms$


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E0ample*

),+ Selling price )9,+ )measured in dollars pergallon+$

)+ Disposa1le income per household )9+)measured in dollars+

!he chief economist decides to collect data onthe "aria1les in a sample of ten company salesregions that are roughly eual in population$Data on paint sales# promotional e0penditures#

and selling prices &ere o1tained from thecompany's mar(eting department$ Data ondisposa1le income )per capita+ &as o1tainedfrom the ;ureau of La1or Statistics


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%ns&er

8< a = 1$9

8< *,>$?@ = >$449

Interpretation: !he coefficient of 9 )>$44+indicates that for one-unit increase in 9)A*#>>> in additional promotional

e0penditures+# e0pected sales )8+ &illincrease 1y >$44 )9*#>>>+< 44 gallonsin a gi"en sales region$


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Bsing the regression euation toma(e predictions

% regression euation can 1e used toma(e predictions concerning the "alue of8# gi"en any particular "alue of 9$ !his is

done 1y su1stituting the particular "alue of9# namely 9p# into the sample regression

euation


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E0ample

Suppose one is interested in estimatingSher&in-.illiam's paint sales for ametropolitan area &ith promotional

e0penditures eual to A*@#>>> )i$e$#9p$44)*@+

8*$>4@ gallons or ,>*#>4@ gallons


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E0ample

9p> or A>>#>>>

7emar(: 9p falls outside of the series of

o1ser"ations for &hich the regression line&as calculated$

!hus# 1ecause of the a1o"e remar(# &ecannot 1e certain that the prediction ofpaint sales 1ased on the regression model&ould reasona1le$


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Standard Error of the estimate

!he error term eiis defined as the

difference 1et&een the o1ser"ed andpredicted "alue of the dependent "aria1le$

!he standard de"iation of the error termsis calculated as:

2

)(

2

1

2

1

=

=

==

n

bxay

n

e

S

n

i

ii

n

i

i

e


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!he standard error of the estimate )Se+

can 1e used to construct predictioninter"als for 8$ %n appro0imate @ percentprediction inter"al is eual to

eSY 2


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7eturning to our pre"ious e0ample

)7.22(2045.2012 =e

SY

*@@$44? to ,4$4 )that is# from *@@#44? gallons to ,4#4 gallons+$


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!esting

F>: 5 ) Go relationship 1et&een 9 and8+

Fa: 5H> ) linear relationship 1et&een 9and 8+


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!esting

!here are t&o &ays of doing the testing:

+ Calculate the t statistic and compare it tothe critical "alue

4+ Bse the p-"alue techniue


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Correlation Coefficient

2

1

2

1

)()(

))((

=

=

=

n

i

ii

n

ii

yyxx

yyxxr


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!he correlation coefficient measures the degreeto &hich t&o "aria1les tend to "ary together$

Correlation analysis is useful in e0planatory

studies of the relationship among economic"aria1les$ !he information o1tained in thecorrelation analysis can then 1e used as a guidein 1uilding descripti"e models of economic

phenomena that can ser"e as a 1asis forprediction and decision ma(ing$


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!he "alue of the correlation coefficient ranges from =* for the t&o "aria1les &ithperfect positi"e correlation to -* for t&o

"aria1les &ith perfect negati"e correlation$


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!he Coefficient of Determination

It measures the proportion of the "ariationin the dependent "aria1le that is e0plained1y the regression line )the independent

"aria1le+$ !he coefficient of determination ranges

from > ) &hen none of the "ariation in 8 ise0plained 1y the regression+ to *) &hen allthe "ariation in 8 is e0plained 1y theregression$


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E0ample

If r,$@* ) from the Sher&in-.illiam'scompany e0ample+$

Interpretation: the regression euation# &ith

promotional e0penditures as the independent"aria1le# e0plains a1out @, percent of the"ariation in paint sales in the sample$

7emar(: In the t&o-"aria1le linear regression

model# the coefficient of determination is eualto the suare of the correlation coefficient# i$e$#r,$@*@+,$


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3-ratio

It is used to test &hether the estimatedregression euation e0plains a significantproportion of the "ariation in the dependent

"aria1le$ !he decision is to reJect the null hypothesis of no

relationship 1et&een 9 and 8 ) that is# noe0planatory po&er+ at the ( le"el of significance

if the calculated 3-ratio is greater than the 3(#*#n-,"alue o1tained from the 3-distri1ution$


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E0ample

If 3


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%ssociation and Causation

!he presence of association )correlation+does not necessarily imply causation$


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Multiple Linear 7egression

% functional relationship containing t&o ormore independent "aria1les is (no&n as amultiple linear regression model$

+++++=mm

XXXY ......2211


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7egression !echniues


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Estimating Coefficients

Consider a small restaurant chain speciali2ing infresh lo1ster dinner$ !he 1usiness has collectedinformation on prices and the a"erage num1er of

meals ser"ed per day for a random sample ofeight restaurants in the chain$ !hese data aresho&n 1elo&$ Bse regression analysis toestimate the coefficients of the demand function

Kd< a =1$ ;ased on the estimated euation#calculate the point price elasticity of demand atthe mean "alues of the "aria1les$


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!he Least-Suares regressionestimation

XbY

X

a

X

YYXX

b

i

i

=

=

2(

))((

)


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Estimating the demand for lo1stersdinners using the OLS

*?>a

-4$?@1

-*?@4>

>4-,>*4*>>

>>>@**>@

->-*>*>

->-,>**,>

-,>4-,*>*4**>

-4@-*@*@

-,>4,-*>*>

>*-*>*@*>>

)i-)1ar++)Ki-K)1ar++)i-)1ar++,i-)1ar+Ki-K)1ar+iKi


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Estimating the demand for lo1sterdinners

Bsing the ordinary Least Suaresregression# &e find the estimates of thedemand for lo1sters$

.e can no& use our results to determinethe point elasticity of demand for lo1sters$


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.ritten assignment

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6580262 Managerial Economics Chapter 4

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