summarize the key economic principles and concepts in each of the assigned chapters.

Chapter 3

Marginal Analysis for

Optimal Decisions

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1-1

Learning Objectives

â– Define several key concepts and terminology related to

marginal analysis

â– Use marginal analysis to find optimal activity levels in

unconstrained maximization problems and explain why

sunk costs, fixed costs, and average costs are irrelevant

for decision making

â– Employ marginal analysis to find the optimal levels of

two or more activities in constrained maximization and

minimization problems

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3-2

Optimization

â– A persons decision is optimal if it leads to

the best outcome under a given set of

circumstances. This is accomplished by

using Marginal Analysis.

â– A person needs to determine the benefit

of a changing activity and compare it with

the cost associated with the change in

activity.

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3-3

Optimization

â– Tactical Decisions

â– Strategic Decisions

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3-4

Optimization

â– An optimization problem involves the

specification of three things:

~ Objective function – what is to be maximized

or minimized (profit, satisfaction, value).

~ Activities or choice variables that determine

the value of the objective function â€“

production level for profits.

~ Any constraints that may restrict the values of

the choice variables â€“ such as cost.

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3-5

Optimization

â– We have discrete and continuous choice

variables.

â– A Discrete Choice Variable can only take

on specific integer values.

â– A Continuous Choice Variable can take

on any value between two points.

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3-6

Optimization

â– Maximization problem

~ An optimization problem that involves

maximizing the objective function

â– Minimization problem

~ An optimization problem that involves

minimizing the objective function

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3-7

Optimization

â– Unconstrained optimization

~ An optimization problem in which the decision

maker can choose the level of activity from an

unrestricted set of values.

â– Constrained optimization

~ An optimization problem in which the decision

maker chooses values for the choice

variables from a restricted set of values (such

as total costs).

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3-8

Choice Variables

â– Activities or choice variables determine

the value of the objective function

â– Discrete choice variables

~ Can only take specific integer values

â– Continuous choice variables

~ Can take any value between two end points

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3-9

Marginal Analysis

â– Analytical technique for solving

optimization problems that involves

changing values of choice variables by

small amounts to see if the objective

function can be further improved

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3-10

Net Benefit

â– Net Benefit (NB)

~ Difference between total benefit (TB) and total

cost (TC) for the activity

~ NB = TB â€“ TC

â– Optimal level of the activity (A*) is the

level that maximizes net benefit

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3-11

Optimization

â– The optimal level of activity does not

generally result in the maximization of

benefits.

â– The optimal level of activity in an

unconstrained maximization problem

does not result in the minimization of total

costs.

â– Only marginal benefits and marginal costs

are relevant in the decision process.

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3-12

Optimal Level of Activity

(Figure 3.1)

Total benefit and total cost (dollars)

TC

4,000

D

B

â€¢

2,310

F

â€¢

â€¢ Dâ€™

3,000

â€¢

â€¢

G

TB

2,000

NB* = $1,225

C

â€¢

1,085

1,000

â€¢ Bâ€™

â€¢Câ€™

0

200

A

350 = A*

600

700

1,000

Level of activity

Net benefit (dollars)

Panel A â€“ Total benefit and total cost curves

M

1,225

1,000

â€¢câ€™â€™

â€¢

â€¢

600

0

dâ€™â€™

200

350 = A*

â€¢

600

Level of activity

A

fâ€™â€™

1,000

NB

Panel B â€“ Net benefit curve

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3-13

Marginal Benefit & Marginal Cost

â– Marginal benefit (MB)

~ Change in total benefit (TB) caused by an

incremental change in the level of the activity

â– Marginal cost (MC)

~ Change in total cost (TC) caused by an

incremental change in the level of the activity

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3-14

Marginal Benefit & Marginal Cost

Change in total benefit ï„TB

MB =

=

Change in activity

ï„A

Change in total benefit ï„TC

MC =

=

Change in activity

ï„A

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3-15

Using Marginal Analysis to Find

Optimal Activity Levels

â– If marginal benefit > marginal cost

~ Activity should be increased to reach highest

net benefit

â– If marginal cost > marginal benefit

~ Activity should be decreased to reach highest

net benefit

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3-16

Using Marginal Analysis to Find

Optimal Activity Levels

â– Optimal level of activity

~ When no further increases in net benefit are

possible

~ Occurs when MB = MC

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3-17

Unconstrained Maximization with

Discrete Choice Variables

â– Increase activity if MB > MC

â– Decrease activity if MB < MC
â– Optimal level of activity
~ Last level for which MB exceeds MC
~ To make the optimal decision for a discrete
choice variable, decision makers must increase
activity until the last level of activity is reached
for which marginal benefit exceeds marginal
cost.
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3-18
Irrelevance of Sunk, Fixed, and
Average Costs
â– Sunk costs
~ Previously paid & cannot be recovered
â– Fixed costs
~ Constant & must be paid no matter the level
of activity
â– Average (or unit) costs
~ Computed by dividing total cost by the
number of units of activity
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3-19
Irrelevance of Sunk, Fixed, and
Average Costs
â– Decision makers wishing to maximize the
net benefit of an activity should ignore
these costs, because none of these costs
affect the marginal cost of the activity and
so are irrelevant for optimal decisions
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3-20
Constrained Optimization
â– The ratio MB/P represents the
additional benefit per additional dollar
spent on the activity
â– Ratios of marginal benefits to prices of
various activities are used to allocate a
fixed number of dollars among activities
â– It is marginal benefit per dollar spent
that matters in decision making.
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3-21
Constrained Optimization
â– To maximize or minimize an objective
function subject to a constraint
~ Ratios of the marginal benefit to price
must be equal for all activities
~ Constraint must be met
MBA MBB MBC
MBZ
=
=
... =
PA
PB
PC
PZ
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3-22
Summary
â– Marginal analysis is an analytical technique for solving
optimization problems by changing the value of a choice
variable by a small amount to see if the objective
function can be further improved
â– The optimal level of the activity (A*) is that which
maximizes net benefit, and occurs where marginal
benefit equals marginal cost (MB = MC)
~ Sunk costs have previously been paid and cannot be recovered;
Fixed costs are constant and must be paid no matter the level of
activity; Average (or unit) cost is the cost per unit of activity;
these 3 types of costs are irrelevant for optimal decision making
â– The ratio MB/P denotes the additional benefit of that
activity per additional dollar spent (â€œbang per buckâ€)
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3-23
Chapter 4
Basic Estimation Techniques
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1-1
Learning Objectives
â– Set up and interpret simple linear regression equations
â– Estimate intercept and slope parameters of a regression
line using the method of leastâ€squares
â– Determine statistical significance using either tâ€tests or
pâ€values associated with parameter estimates
â– Evaluate the â€œfitâ€ of a regression equation to the data
using the R2 statistic and test for statistical significance
of the whole regression equation using an Fâ€test
â– Set up and interpret multiple regression models
â– Use linear regression techniques to estimate the
parameters of two common nonlinear models: quadratic
and logâ€linear regression models
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4-2
Basic Estimation
â– Parameters
~ The coefficients in an equation that determine
the exact mathematical relation among the
variables â€“ for the cost function to be useful
for decision making we must know the values
of the parameters.
â– Parameter estimation
~ The process of finding estimates of the
numerical values of the parameters of an
equation
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4-3
Regression Analysis
â– Regression analysis
~ A statistical technique for estimating the
parameters of an equation and testing for
statistical significance.
~ Regression analysis uses data on economic
variables to determine a mathematical
equation that describes the relationships
between the economic variables.
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4-4
Regression Analysis
â– Dependent variable
~ Variable whose variation is to be explained
â– Explanatory variables
~ Variables that are thought to cause the
dependent variable to take on different values
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4-5
Simple Linear Regression
â– True (Actual) regression line relates
dependent variable Y to one explanatory
(or independent) variable X
Y = a + bX
~ Intercept parameter (a) gives value of Y where
regression line crosses Y-axis (value of Y when X
is zero)
~ Slope parameter (b) gives the change in Y
associated with a one-unit change in X:
b = ï„Y ï„X
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4-6
Regression Analysis
â– If Y = 2500 + 10X
â– Then
â– Y will change on average 10 units for
every 1 change in X
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4-7
Simple Linear Regression
â– Regression line shows the average or
expected value of Y for each level of X
â– True (or actual) underlying relation
between Y and X is unknown to the
researcher but is to be discovered by
analyzing the sample data
â– Random error term
~ Unobservable term added to a regression model to
capture the effects of all the minor, unpredictable
factors that affect Y but cannot reasonably be
included as explanatory variables
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4-8
Fitting a Regression Line
â– The purpose of a regression analysis is:
â– 1. To estimate the parameters (a and b)
of the true (actual) regression line
(population regression line).
â– 2. to test the estimated values of the
parameters to see if they are statistically
significant.
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4-9
Fitting a Regression Line
â– Estimating â€œaâ€ and â€œbâ€ is equivalent to
fitting a straight line through a scatter of
data points on a graph.
â– The data collected is on both the
dependent and explanatory variables.
â– The objective of regression analysis is to
find the straight line that best fits the
scatter of data points (sample regression
line).
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4-10
Fitting a Regression Line
â– Time series
~ A data set in which the data for the
dependent and explanatory variables are
collected over time for a single firm
â– Cross-sectional
~ A data set in which the data for the
dependent and explanatory variables are
collected from many different firms or
industries at a given point in time
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4-11
Fitting a Regression Line
â– The sample regression line is only an
estimate of the true regression line.
Naturally, the larger the size of the
sample, the more accurately the sample
regression line will estimate the true
regression line.
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4-12
Fitting a Regression Line
â– Method of least squares â€“ used for
Regression Analysis
~ A method of estimating the parameters of a
linear regression equation by finding the line
that minimizes the sum of the squared
distances from each sample data point to
the sample regression line (you want as tight
a fit as you can get).
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4-13
Fitting a Regression Line
â– Parameter estimates are obtained by
choosing values of a & b that minimize
the sum of squared residuals
~ The residual is the difference between the
actual and fitted/predicted values of Y: Yi â€“
Å¶i
~ Equivalent to fitting a straight line through a
scatter diagram of the sample data points â€“
in order to minimize the sum of the squared
residuals.
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4-14
Fitting a Regression Line
â– The sample regression line is an
estimate of the true (or population)
regression line
Ë†
YË† = aË† + bX
~Where aÌ‚ and bÌ‚ are least squares estimates
of the true (population) parameters a and b
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4-15
Sample Regression Line
(Figure 4.2)
S
Sales (dollars)
70,000
S
60,000
Sii =
= 60,000
60,000
ei
50,000
20,000
10,000
â€¢
â€¢
40,000
30,000
â€¢
Sample regression line
Åœi = 11,573 + 4.9719A
â€¢
= 46,376
SÌ‚i Åœ
=i 46,376
â€¢
â€¢
â€¢
A
0
2,000
4,000
6,000
8,000
10,000
Advertising expenditures (dollars)
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4-16
Unbiased Estimators
â– The estimates aÌ‚ & bÌ‚ do not generally
equal the true values of a & b
~ aÌ‚ & bÌ‚ are random variables computed using
data from a random sample
â– The distribution of values the estimates
might take is centered around the true
value of the parameter
â– An estimator is unbiased if its average
value (or expected value) is equal to the
true value of the parameter
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4-17
Unbiased Estimators
â– We now turn to the task of testing
hypotheses about true (actual) values of
â€œaâ€ and â€œbâ€, which are unknown to the
researcher, using information contained in
the sample to see if there is a significant
relationship or simply a relationship based
on randomness.
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4-18
Statistical Significance
â– Do the coefficients mean anything? Do
they statistically explain the change in the
dependent variable?
â– The farther away from 0 (+ or -) the
estimated coefficient is, the estimated
coefficient is considered Statistically
Significant.
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4-19
Statistically Significant
â– If Y is indeed related to X, the true value
of the slope of the parameter will be either
a positive or negative number.
â– Remember, if the coefficient is the change
in Y divided by the change in X, and the
coefficient is zero, then no change in Y
will occur when there is a change in X.
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4-20
Statistically Significant
â– We need to test for the statistically
significance of a parameter because we
do not know the true value of the
parameters. i.e. they are a random
sample.
â– Because of randomness in sampling,
different samples will result in different
values.
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4-21
Statistically Significant
â– The estimator of the coefficient is
Unbiased if the average value of the
estimator is equal to the true value of the
parameter.
â– Unbiased does not mean that any one
estimate of the coefficient is close to the
true value. Unbiased means that in
repeated samples, the estimates tend to
be centered around the true value.
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4-22
Statistical Significance
â– Statistical significance
~ There is sufficient evidence from the
sample to indicate that the true value of the
coefficient is not zero
â– Hypothesis testing
~ A statistical technique for making a
probabilistic statement about the true value
of a parameter
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4-23
Statistical Significance
â– Must determine if there is sufficient
statistical evidence to indicate that Y is
truly related to X (i.e., b ï‚¹ 0)
â– Even if b = 0, it is possible that the
sample will produce an estimate bÌ‚ that
is different from zero (Type I error)
â– Test for statistical significance using
t-tests or p-values
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4-24
Statistical Significance
â– A t-test is used to make a probabilistic
statement about the likelihood that a true
parameter value of a coefficient is not
equal to zero.
â– The t-ratio indicates how much
confidence one can have that the true
value of a coefficient is actually larger
than zero. The larger the value the better.
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4-25
Statistical Significance
â– First determine the level of significance
~ Probability of finding a parameter estimate to
be statistically different from zero when, in
fact, it is zero
~ Probability of a Type I Error
â– 1 â€“ level of significance = level of
confidence
~ Level of confidence is the probability of
correctly failing to reject the true hypothesis
that b = 0
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4-26
Statistical Significance
â– The level of significance of a t-test is the
probability that the test will indicate the
coefficient is not equal to zero when in
fact it is equal to zero.
â– For example a level of significance of .01,
.02, .05 assumes a willingness to accept
a 1, 2, or 5 percent probability of finding a
parameter to be significant when it is not
(99%, 98% or 95% confidence level).
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4-27
Performing a t-Test
bÌ‚
â– t-ratio is computed as t =
SbÌ‚
where SbÌ‚ is the standard error of the estimate bË†
â– Use t-table to choose critical t-value with
n â€“ k degrees of freedom for the chosen
level of significance
~ n = number of observations
~ k = number of parameters estimated
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4-28
Performing a t-Test
â– t-statistic
~ Numerical value of the t-ratio
â– If the absolute value of t-statistic is
greater than the critical t, the parameter
estimate is statistically significant at the
given level of significance
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4-29
Using p-Values
â– Treat as statistically significant only those
parameter estimates with p-values
smaller than the maximum acceptable
significance level
â– p-value gives exact level of significance
~ Also the probability of finding significance
when none exists
~ One minus the p-value is the exact degree of
confidence that can be assigned to a
particular parameter estimate.
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4-30
Coefficient of Determination
â– R2 measures the fraction of total variation
in the dependent variable (Y) that is
explained by the variation in X
~ Ranges from 0 (Explains none) to 1 (explains
all)
~ High R2 indicates Y and X are highly
correlated, and does not prove that Y and X
are causally related
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
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F-Test
â– Used to test for significance of overall
regression equation (explained and
unexplained variations)
â– Compare F-statistic to critical F-value
from F-table
~ Two degrees of freedom, n â€“ k & k â€“ 1 (# of
independent variables)
â– If F-statistic exceeds the critical F, the
regression equation overall is statistically
significant at the specified level of
significance
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
4-32
Multiple Regression
â– Uses more than one explanatory variable
â– Coefficient for each explanatory variable
measures the change in the dependent
variable associated with a one-unit
change in that explanatory variable, all
else constant
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manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
4-33
Summary
â– A simple linear regression model relates a dependent
variable Y to a single explanatory variable X
~ The regression equation is correctly interpreted as providing the
average value (expected value) of Y for a given value of X
â– Parameter estimates are obtained by choosing values of
a and b that create the best-fitting line that passes
through the scatter diagram of the sample data points
â– If the absolute value of the t-ratio is greater (less) than the
critical t-value, then is (is not) statistically significant
~ Exact level of significance associated with a t-statistic is its p-value
â– A high R2 indicates Y and X are highly correlated and the
data tightly fit the sample regression line
Â© 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
4-34
Summary
â– If the F-statistic exceeds the critical F-value, the
regression equation is statistically significant
â– In multiple regression, the coefficients measure the
change in Y associated with a one-unit change in that
explanatory variable
â– Quadratic regression models are appropriate when the
curve fitting the scatter plot is U-shaped or âˆ©-shaped
(Y = a + bX + cX2)
â– Log-linear regression models are appropriate when the
relation is in multiplicative exponential form (Y = aXbZc)
~ The equation is transformed by taking natural logarithms
Â© 2016 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or distribution in any
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.
4-35
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