Transcript Document

THE MECHANICS OF DEMAND
AND PRODUCTION
FIFTH LECTURE
February 21, 2012
William R. Eadington, Ph.D.
Professor of Economics, College of Business
Director, Institute for the Study of Gambling and Commercial Gaming
University of Nevada, Reno
www.unr.edu/gaming
POSSIBLE SESSIONS WITH THE
GRADER
• Times to consider:
– Tuesday 4:15 to 5:15
– Thursday 4:15 to 5:15
– Thursday 7:00 to 8:00
SIMPLE CALCULUS CONCEPTS
• Consider Y = f(X) = a + b*X + c*X2
In
general, Y = sum of terms of form ai*Xn
• A derivative is the Rate of Change of Y as X
changes, written dY/dX = n*ai*Xn-1 summed over
all terms
• dY/dX = coef*exp.*X(n-1) summed over all terms
• Example: When driving a car, let D = distance
traveled, and t = time from start
• Suppose D = a + b*t + c*t2
• Then dD/dt = b + 2*c*t = velocity
• Then d2D/dt2 = 2*c = acceleration
APPLICATIONS TO CURRENT
WORK
•
•
•
•
•
•
Consider Q = a + b*P as a demand curve
TR = P*Q = a*P + b*P2
MR = dTR/dP = a + 2*b*P
Optimization occurs when dY/dX = 0
Inverse Demand curve: P = c + d*Q
Consider elasticity: Ƞ = percent change in
Q/percent change in P = (dQ/Q)/(dP/P) =
(dQ/dP)*P/Q
PROBLEM OF THE WEEK: 4-30
Japan has 4,350 miles of expressway – all toll roads. In fact, the tolls are so high
that many drivers avoid using expressways. A typical 3 hour expressway trip
can cost $47. A new $12 billion bridge over Tokyo Bay that takes 10 minutes
and costs $25 rarely is busy. One driver prefers snaking along Tokyo’s city
streets for
hours to save $32 in tolls. Assume that the daily demand curve
for a particular stretch of expressway is:
P = 800 yen - .16 Q.
a. At what price-quantity point does this demand curve have a price elasticity
of one?
b. Assume the government wishes to maximize its revenues from the
expressway, what price should it set? And how much revenue does it generate
at this price?
c. Suppose that traffic engineers have determined that the efficient utilization
of this particular toll road is 4,000 cars per day. This traffic level represents an
optimum tradeoff between congestion (with its associated reduction in speeds
and increase in accidents) between expressways and surface roads. If 4,000
cars per day is the socially efficient utilization of the toll road, what price
should be set on the toll road? And how much revenue is collected by the
government?
DEMAND, TOTAL REVENUE, & MARGINAL
REVENUE
Ticket price (in dollars)
$
60
Elastic demand (n > 1)
n=1
30
Inelastic demand (n < 1)
Q
Total revenue (in dollars)
$
6,000
200
Quantity of Theater tickets
Q
OTHER DEMAND INFLUENCES
• Complements versus substitutes
– Cross price elasticity of demand
Q x
Qx1  Qx 2
 xy 
Py
Py1  Py 2
or Ƞxy = dQx/dPy*Px/Qy
CROSS PRICE ELASTICITY
• For substitutes, ηXY > 0
• If the price of Pepsi rises, the
demand for Coke rises
• For complements, ηXY < 0
• If the price of peanut butter rises,
the demand for jelly falls
INCOME ELASTICITY
• Income elasticity of demand
Q x
Qx1  Qx 2
I 
I
I1  I 2
or ȠI = dQx/dI*I/Qx
• Normal goods – demand rises as income
increases (>0)
• Inferior goods – demand falls as income
increases (<0)
• Luxury goods – demand rises more than
proportionately as income increases (>1)
NETWORK EFFECTS
• Demand for a good increases as the number of
users of the good increases => “Winner Take All”
technology?
–
–
–
–
–
Telephone networks v. Voice-Over Internet
Mail: The internet versus the postal system
Microsoft Outlook versus Netscape
Beta-Max versus VHS
High Definition DVD versus Blu-Ray
• Without critical mass, a product will die
– FAX machines
– 3-D TV
PRODUCT LIFE CYCLE:
Consider gaming in Reno
Industry quantity of output
Q
Introduction
Growth
Maturity
Decline
Product life cycle
T
Time
DEMAND ESTIMATION
• Three general techniques
– interviews
• surveys, focus groups, questionnaires
– price experimentation
• track changes in sales when prices change
– statistical analysis
• Cross-section versus time series
• must account for omitted variables and other
issues
INTERPRETING DEMAND
FUNCTIONS
• Mathematical representations of demand
curves.
• Example:
QX  10  2 PX  3PY  2 M
d
– Law of demand holds (coefficient of PX is
negative).
– X and Y are substitutes (coefficient of PY is
positive).
– X is an inferior good (coefficient of M is negative).
LINEAR DEMAND FUNCTIONS
AND ELASTICITIES
• General Linear Demand Function and
Elasticities:
QX  0   X PX  Y PY   M M   H H
d
PX
EQX , PX   X
QX
Own Price
Elasticity
EQ X , PY
PY
 Y
QX
Cross Price
Elasticity
M
EQX , M   M
QX
Income
Elasticity
EXAMPLE OF LINEAR DEMAND
•
•
•
•
Qd = 10 - 2P.
Own-Price Elasticity: (-2)P/Q.
If P=1, Q=8 (since 10 - 2 = 8).
Own price elasticity at P=1, Q=8:
(-2)(1)/8= - 0.25
LOG-LINEAR DEMAND
• General Log-Linear Demand Function:
ln QX d   0   X ln PX  Y ln PY   M ln M   H ln H
Own Price Elasticity :  X
Cross Price Elasticity :  Y
Income Elasticity :
M
EXAMPLE OF LOG-LINEAR
DEMAND
• Qd = A*P-2 = A/P2
• ln(Qd) = ln(A) - 2 ln(P)
• Own Price Elasticity: -2
GRAPHICAL REPRESENTATIONS OF
LINEAR AND LOG-LINEAR DEMAND
P
P
D
Linear
D
Q
Log Linear
Q
REGRESSION ANALYSIS
• One use is for estimating demand
functions
• Important terminology and concepts:
– Least Squares Regression model:
Y = a + bX + e.
– Least Squares Regression line: Yˆ  aˆ  bˆX
– Confidence Intervals
– t-statistic
– R-square or Coefficient of Determination
– F-statistic
EXAMPLE
• Use a spreadsheet to estimate the
following log-linear demand function:
ln Qx  0   x ln Px  e
Summary Output
Regression Statistics
Multiple R
0.41
R Square
0.17
Adjusted R Square
0.15
Standard Error
0.68
Observations
41.00
ANOVA
df
Regression
Residual
Total
Intercept
ln(P)
SS
1.00
39.00
40.00
MS
F
3.65
18.13
21.78
Coefficients Standard Error
7.58
1.43
-0.84
0.30
3.65
0.46
t Stat
5.29
-2.80
Significance F
7.85
0.01
P-value
0.000005
0.007868
Lower 95%
Upper 95%
4.68
10.48
-1.44
-0.23
INTERPRETING THE REGRESSION
OUTPUT
• The estimated log-linear demand function
is:
– ln(Qx) = 7.58 - 0.84 ln(Px)
– Own price elasticity: -0.84 or 0.84 (inelastic)
• How good is the estimate?
– F-statistic significant at the 1 percent level
– t-statistics of 5.29 and -2.80 indicate that the
estimated coefficients are statistically different
from zero
– R-square of 0.17 indicates the ln(PX) variable
explains only 17 percent of the variation in ln(Qx)
ESTIMATING DEMAND
THE IDENTIFICATION PROBLEM
Price (in dollars)
$
Three different
equilibriums do not
map out the demand
curve; what do they map out?
S1
P1
S
D1
2
P2
D2
P3
S3
D3
Estimated demand
Q1 Q2
Q3
Quantity
Q
PRODUCTION AND COST
PRODUCTION FUNCTIONS
A production function specifies
maximum output from given inputs:
Q  f ( x1, x2 ,...xn )
Technological efficiency: Maximum Q
for given levels of x1,x2, …, xn; challenge
for the production engineers
DEFINITION OF EFFICIENCY
• Technological efficiency: Extracting the
maximum amount of output from any given
mix of inputs
• Cost efficiency: Among all the
technologically efficient combinations of
inputs to achieve a given level of output,
which can be acquired at least cost
• Implications of efficiency concepts applied to
agriculture: See the documentary FOOD,
INC. (available as streaming video on
NETFLIX)
RETURNS TO SCALE
• The relation between output and a
proportional variation of all inputs
together
• Increasing returns to scale - if inputs
double, output more than doubles
• Decreasing returns to scale - if inputs
double, output less than doubles
• Constant returns to scale- if inputs
double, output doubles
RETURNS TO A FACTOR
• The relation between output and the
variation in only one input, holding all
other inputs constant
• Total product - amount of output, Q,
obtained when an input, L, increases
• Average product Q/L
• Marginal product Q/L