Demand Estimation
Regression concepts:
1. Regression as fitting a straight line
through a scattergram.
2. Why the need for a formal method
3. Why the minimize least squares rule
instead of something else
4. The infinite number of monkies regression
5. How it is actually done--calculus
Regression tools: t values
1. Recall how to use the normal distribution
to test hypotheses.
2. Choosing the 5% tail, why.
3. Recall the Z value for the 5% tail.
4. For large numbers of observations, the t
distribution is approximately normal.
5. The rule of t > 2; and when it is that
t>1.65 is an acceptable rule.
Regression tools: F distribution and Rsquared
1. Compare: t tests whether a single coefficient
is significant, while F test whether the
equation as a whole is significant.
2. For both t and F most programs now print
out the probability value, too.
3. Rsquared is similar to F in that it measures
the goodness of fit of the entire regression.
4. Rsquared measured the percent of the
variation in the dependent variable that is
explained by the regression.
The regression equation; this is created in
general terms by you from your theory,
then you test it and acquire estimates of
the parameters.
For example:
Y= a + bP + cX + dM + eL + u
Y= a + bP + cX + dM + eL + u
Reading regression output.
The "parameters" are the small lettered terms
Also called the constant and the coefficients.
For example, the term cX means that if the
independent variable X were to increase by
a unit, then the dependent variable would
increase by c units.
Regression concept:
Your hypotheses (whether they are just
hunches, guesses, or implications
carefully derived from theory) are
tested by the estimates of the parameters
and the Rsquared.
For example: Hypothesis 1--Demand
slopes downward. This theory implies that
the parameter b is negative and significant.
Your Demand Estimation Project
"1. Apply Excel's Regression routine to
estimate the demand fundtion for your produce,"
"where Observed demand = Constant + a*OwnPrice
+ b*Income,Percap + c*PriceofZ + d*Quality" ofProduct
+ e*Advertising + u (a random error term)
Then apply the LN function to recalculate all
your 6 variables into logs and redo.
Which version had the better fit?
"Of the two versions of regression you have
done, use the nonlogged to answer these questions."
2. Is the equation significant overall? Which
variables contribute significantly? Are each of
these of the theoretically correct sign?
3. Is the Product Z a substitute product or a
complement for your product?
4. Is your product an economically inferior
product in this market or normal?
"5. If Ownprice=24, Income=45, PriceofZ=12,
Quality=50, and Advertising=37; what is then"
the price elasticity of your product? Would
you be wise to raise your price? Is your
product considered a luxury good by this
market?
Regression Anova Regular Data
Multiple R
R Squared
Adj R Sqd
Std Error
Observations
F Value
Significance
0.9656
0.9320
0.9211
626.30
36
82.8
0.000
Regression Output, Regular Data
Variable
Intercept
Own Price
Income
PriceZ
Quality
Advertsng
Coeff.
305.8
-31.8
27.31
6.11
39.12
9.72
StdError
531.6
3.26
4.54
7.94
2.46
1.61
tValue
0.57
-9.74
6.00
0.76
15.8
6.03
PValue
0.560
0.000
0.000
0.440
0.000
0.000
3. The product Z is a ________because
the coefficient for PriceZ is positive.
4. Your product is a(an)___________, because?
5. First calculate the quantity, Q:
Then find dQ/dP =
It will follow the the price elasticity = ?
Luxury good? Discuss.
You can check part of your work against this.
Regression output, Log/Log Version
Statistic
Multiple R
R Square
Adj RSqr
F value
Value
0.817
0.668
0.612
12.083
pValue
Variable
Intercept
OwnPrice
Income/cap
Priceof Z
Quality
Advertisng
Coefficient
5.08
-0.30
0.16
0.02
0.55
0.27
Std Error
0.92
0.07
0.13
0.066
0.09
0.09
0.000
t Value
5.48
-4.02
1.23
0.44
6.14
2.77
P Value
0.000
0.000
0.228
0.660
0.000
0.009
1. If the cost function for quality is C(Qu) = 11 +10.0*Qu
and if the cost function for advertising is
C(Adv) = 30 + 5.0*Adv; then which would make a
better investment of an extra $100 by the company
2. Would it ever pay to substitute quality with advertising?
Part 2: Some examples of applications of
demand estimates.
1. Filling in the unknown market area
Drawing: pizza shops around the city
but several areas uncovered.
2. Advising pricing policy.
Contrast Millie's dress shop
with Acme Cement, Inc.
3. Investigate sensitivities:
a. J.D.Power example
b. Is your product "upscale"?
c. Ethnic tastes
4. Provide guidance to advertising.
5. Court cases and cross-elasticity.
6. "Bads" and tax policy.
a. cigarette and alcohol studies
b. illicit drugs and price elasticity
7. Projections of firm demand.
8. Estimating demand for "free goods".
9. Test one's product's relation to the
business cycle.
How things can go wrong with regression
analysis:
1. Multicolinearity (highly correlated indepents).
2. Serial correlation (affects time series).
3. Heteroscedasticity ("Christmas tree residuals")
4. Omitted variables (sometimes a problem).
How to estimate a curvilinear curve:
A popular method is to start with a CobbDouglas demand function:
Q = APbYc
as an example.
Then convert this to logs
lnQ = lnA + blnP + clnY
The b an c are the price and income elasticities.
Review questions for the midterm:
1. Find the first derivative of
Y = 120 - 2.3X + 33X2 - 21.4X3 + 3.1X5
2. Find the second derivative of the above
function of X.
3. Find the first derivatives of each of the
following:
a. X(3X -1/X)
b. b. 34X/(X2 - 110)
c. G(H) where G=g(H3 - 2H) and H =h(X-2X2)
4. Find the values of X and Y the form an
optimum of the function
Z = 120 + 4.5X -3.4Y - 5.1X2 + 3Y2 - 3XY
6. Set up the LaGrangean function for each
of the following:
a. ACME Inc. wishes to squeeze more
production efficiency into its plant, but the
board of directors insists that expenses not
exceed 1200, when w=15 and r=20.
b. NASA wants to minimize wing stress but
maintain lift of at least 2000 pounds.
7. Find the point elasticity of the following
demand functions:
a. Q = 2000 - 45.5P when the P=33.
b. Q = 1000 - 45.5P + 30Y - 23Pz
when the P= 33, Y=100; and Pz = 14
c. Q = 1000P-1.2Y3.0PZ1.2
8. In which of the above cases would the firm
be advised to raise its price?
9. If price elasticity is greater than one in
absolute value, and then if you lower your
price--do you therefore increase your profits?
10. Define R Squared in your own words.
11. Which axiom is broken when indifference
curves slope upwards?
12. Which axiom is broken when indifference
curves cross?
12. If a regression coefficient for income, Y,
has an insignificant t value but is nevertheless
positive, is the product in question a normal
good?
13. If the following variables are all entered as
independent variables in a regression, which
are likely to be highly colinear?
education, health status, income, smoking
behavior, drinking behavior, air pollution.
14. Define serial autocorrelation in your own
words.
15. Can there be such a thing as spatial
autocorrelation?
16. Under indifference curve analysis, with wellbehaved indifference curves, is it possible for
a consumer to double his income but
nevertheless consumer no more than before
of all the goods?
17. What are the slopes (the expressions for
them) of each of the following curves?:
a. The budget line:
b. b. An indifference curve:
18. Define a cross-price elasticity of demand and
explain how its relation to the quantity demanded
depends on its sign.
19. Does your hypothesis count as valid if its
t value is significant but the F value is not?
20. Suppose you have estimated the following
demand function for your company and it
is easily and highly significant.
Q = 1000 - 42P + 33Y + 10Z
Then suppose that Y =7; and Z=12.
Find the revenue maximizing price and quantity.