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D e m a n d Elasticities from a Discrete
Choice Model: The Natural
Christmas Tree M a r k e t
George C. Davis and Michael K. Wohigenant
Key .words: Christmas trees, discrete choice, multinomial logit, sample selectivity.
Estimating market demand elasticities from crosssectional or survey data is becoming commonplace in ag¡
economics. If the commodity is divisible, then a demand system may be
estimated (e.g., Heien and Pompelli). If there
are missing observations, as in the National Food
Consumption Survey, then a Tobit model may
be estimated (e.g., Cox and Wohlgenant, Thraen,
Hammond, and Buxton). However, if the commodity is indivisible at the consumer level, the
appropriate model is a discrete choice one. In
this case, estimating market-level demand elasticities is more challenging. We demonstrate here
a method for estimating market-level demand
elasticities from a discrete choice model. The
technique is applied to a cross-sectional survey
of the U.S. natural Ch¡
tree market, for
which there presently exists no known estimated
market demand elasticities. 1
Expected market demand for a commodity
George Davis is an assistant professor in the Department of Agricultural Economics and Rural Sociology, University of Tennessee,
Knoxville, and Michael Wohlgenant is a professor in the Department of Agricultural and Resource Economics at North Carolina
State University.
Appreciation is expressed to Kerry Smith, Ann McDermed, David
Dickey, Dick Per¡
Wally Thurman, David Eastwood, and an
anonymous referee for helpful comments. We are especially gratefui to R. O. Herrmann for providing the cross-sectional data set
used in estimating Ch¡
tree demand parameters.
Review coordinated by Steven Buccola.
I Most demand studies of the natural Christmas tree market have
been consumer surveys and only descriptive summary statistics are
reported (e.g. see April 1990 American Christmas Tree Journal).
The one exception is Hamlett et al., which is referenced and discussed later in the paper. However, Hamlett et al. did not estimate
demand elasticities.
consumed in dichotomous fashion is a simple
function of individual consumers' probabilities
of consuming the commodity. Since these probabilities are functions of prices, elasticities are
easily derived. We use McFadden's nested multinomial logit (NMNL) model to estimate individual consumers' probabilities of consuming the
commodity. Natural Christmas trees are heterogeneous and have varying quality characteristics, so we employ the Theil-Houthakker model
to derive the indirect utility specification for estimating the NMNL model. The NMNL model
is chosen because the consumer's first-level decision is to decide whether or not to consume
the commodity (i.e. display a tree). The secondlevel decision is to decide which type of commodity to consume: a natural or an artificial tree.
The probability of consuming a commodity
can be estimated at each level. In the example
presented here, the first-level estimated is the
conditional probability of displaying a natural
tree. The second-level estimated is the unconditional probability of displaying any tree. From
these two probability estimates, the unconditional probability of displaying a natural Christmas tree can be derived. Once these probabilities are estimated, aggregate expected demand
across all consumers is obtained and market demand elasticities are derived.
Theoretical Framework
In the Theil-Houthakker model, quality characteristics are manifold. The quantity of the
Amer. J. Agr. Econ. 75 (August 1993): 7 3 0 - 7 3 8
Copyright 1993 American Agricultural Economics Association
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A procedure is demonstrated for estimating market-level d e m a n d elasticities from
household data suffering from sample selectivity problems. The procedure uses
M c F a d d e n ' s nested multinomial logit model and is applied to the natural Christmas tree
market. Results indicate that the own-price elasticity of natural Christmas trees is
- 0 . 6 7 4 and the cross price elasticity of natural Christmas trees with respect to artificial
Christmas trees is 0.188. These are the first k n o w n demand elasticity estimates for the
natural Christmas tree market.
Natural Christmas Tree Market
Davis and Wohlgenant
(1)
V,u = XuAu + X,,,13,,,
1 = natural (n), artificial (a);
u = yes (y), no (n).
X,, is a vector of explanatory variables influencing an upper-level decision: to display or not
display a tree. Xh, is a vector of explanatory
variables influencing a lower level decision, i.e.,
tree type. Au and 13~uare conformable parameter
vectors. Using (1) in the random utility model,
McFadden (p.84) shows that for this specific
decision tree structure, the probability of displaying a natural tree can be specified as the following nested multinomial logit model (NMNL):
II(n, y) :
exp[X,,y/9,,y + X y A y -
AIy]
exp X,,A,, + exp[XvAy + (1 - A)/v]
ly is called the inclusive value defined as ly =
ln(exp X,yO,,y + exp XayO,,y), with Oty =/3~y(1 A) -1, where I = n, a. By the laws of conditional
probability, (2) may also be written as II(n, y)
= II(y) II(n y). These probabilities are
(3)
exp[XyAy + (1 - A)ly]
(4)
Sn
Sn
19, = E ¡
y ) = E (Ig(y). (-lg(nly).
g=l
g=l
/9, is the sample expected market demand, Su is
the number of households in the sample, and
IIg(n, y) is the expected probability that the gth
household will display a natural tree. Assume
that the group of Sn households may be treated
as representative of the entire market. Then expected total demand may be obtained by taking
the total number of households in the population
(Pn) and dividing it by the number of households in the sample (SH) and multiplying the resulting number by the expected sample market
demand. Mathematically, expected market demand is D e = Ne 9 D, and Ne = PuSft I. From
this equation it is easy to see that the population
demand elasticity equals the sample market demand elasticity, assuming that Ne is not affected
by the variable in question. The population market demand elasticity with respect to an arbitrary variable x is then
SH
(5)
~Tx = E 0 Ilg(n, y). x__L
g: l
ax~
Ds"
Econometric Approach
(2)
II(y)
timate expected individual demand and then aggregate over consumers to get market-level
demand.
The quantity of trees displayed by the gth
household is a Bernoulli variable; so the sample
expected demand over all Sn households is
=
exp XnA n + exp[XyAy + (1 - A)Iy]
II(n [y) -
exp X,y6)~y
exp ly
Equation (3) is associated with the upper-level
decision (yes or no) and is the probability of displaying a tree. Equation (4) is associated with
the lower-level decision (natural or artificial) and
is the conditional probability of displaying a natural tree. Given these probabilities, we can es-
The estimation procedure is a two-stage sequential one providing consistent parameter estimates (Maddala, p. 70). In the first-stage, the
lower tier, equation (4), is estimated a s a
switching regression (Lee 1978, Lee and Trost).
Structural parameters in this tier are estimated
by the Amemiya principle (Amemiya 1978a),
which Lee (1986, pp. 349-351) showed to be
more efficient than the two-stage probit (logit)
procedure used by Lee (1978). In the secondstage, parameters from the first-stage are used
to construct the inclusive value which becomes
a regressor in the second-stage, equation (3).
Once equations (3) and (4) have been consistently estimated, market demand elasticities are
derived from equation (5).
First-Stage Estimation
The objective of the first-stage estimation is to
obtain consistent estimates of the structural pa-
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commodity consumed can be written as q = f ( p ,
he) and the price can be written as p = g(hc),
where he is a vector of household characteristics
including income. Prices are a function of
household characteristics because the price represents quality differences caused by heterogenous commodity aggregation, and hr acts as a
proxy for household preferences over unobservable quality characte¡
(Cox and Wohlgenant; Hanemann; Houthakker). The implied
indirect utility function for the ith choice is then
Vi = vi(p, h,) = v;(hr If the commodity analyzed were divisible, the analysis would be
straightforward. But because the choice variable
is dichotomous, the random utility model must
be employed.
Let the indirect utility function be approximated by the linear form
731
732
Amer. J. Agr. Econ.
August 1993
rameters associated with (4). If prices of natural
and artificial trees were observed for the entire
sample, the structural parameters of (4) could be
estimated by probit or logit methods; however,
because of the selectivity problem, the econometric approach required to recover the structural parameters is more complicated.
Lee (1978) shows that the lower-tier (firststage) problem may be written as
(6)
P. = X6. + u.
(7)
P . = X 6 a -4- u a
(6.2)
P . = X 6 . - o-~. ¡
+ oo,(~ . - w.) + v.
= 2.r. +,.
Pa ~- X 6 a 71- O'~a l~a -- O'ljn(l/~a -- Wa) 71- Va
(7.2)
=Z, Fa+e,.
where Z = [x, }~l], En = [6Tn, --0"~.] T, F a = [6•
Or,~a]T .
L -- OlXl Al- 02X2 "JI- 0 3 3 X 3 + 04X 4
+ 05X5 + 06(Pn - P a )
+ e.
PI is the nt • 1 vector of prices, X is the n t •
kt mat¡ of exogenous variables (defined later),
and Ic is the criterion function for the conditional tree-type decision (le = 1 if a natural tree
is displayed and L = 0 if an artificial tree is
displayed). In terms of probabilities, equation
(8) is equivalent to equation (4). 8~ are conformable parameter vectors and the error terms are
distributed as uf
n(0, o"2) a n d e ~ n(0, Gre2) ,
l = a, n. We further assume that e is independent of u, and Ua because e n and P~ represent
market-level prices, but u, and u, are assumed
to be mutually dependent. 2 Variables xi through
x5 are household characte¡
variables in this
model. The parameters associated with these
variables are differences in marginal utilities associated with natural and artificial trees. That is,
OŸ = (1 - ,~)-l[flay i -- ~nyi], i = 1, - . . , 5, because xi through xs are constant across alternatives, but 06(1 - A) = i~.y6 = --l~ny6 because tree
price differs across alternatives (Maddala, p. 42
foomote 4).
To estimate the structural parameters of this
lower tier, first substitute (6) and (7) into (8) to
form the following switching regression:
Variable O'~lis the covariance between ~ and
l, where l = n, a. Parameter wt is the appropriately defined inverse Mill's ratio and the hats
represent the substitution of the reduced-form
parameter estimates into these ratios. The parameters of (6.2) and (7.2) can be estimated by
OLS to provide consistent estimates. However,
as Lee, Maddala, and Trost show, inference will
be biased because the errors are heteroskedastic;
therefore, the parameters should be estimated by
GLS. Having obtained estimates of the reducedform choice function parameters/I, and parameter estimates of F, and Fa, one can use the
Amemiya principle to estimate consistently the
structural parameters in (8).
The Amemiya principle exploits the overidentifying rest¡
between the reduced-form
parameters (H), price parameters (6a, 6,), and
structural parameters O. In our model this overidentified parameter system is
(9)
II = J O ~ + 06(6, - 6 J .
l i is the m • 1 true reduced-form parameter
vector, J is an m • 5 matrix of ones and zeros,
Os = (0~, 02, 03, 04, 0s), and 06 ate the parameters from (8). (6, - 6a) is the difference between parameters of the natural tree price equation and the annualized artificial tree price
equation. Note this equation can be rewritten as
(6.1)
P.=X6.+u.,lc=l
iff XHo--~>-
(9.1)
(7.1)
P a = X6,. + ua, l c = O
iff XIIo--I <
where
where ~ = o--l[e + 06(u. - uo)], o- is the standard deviation of ~ so ~ ~ n(0, 1), and X I I represents the reduced form of the right hand side
of (8). II is then estimated by the logit procedure.
Once II is estimated, it is used to form the
appropriate inverse Mill's ratios to correct for
2 This assumption is not necessary and could be relaxed without
changing any of our empirical results. In making this assumption,
however, the variance-covariance formulas are slightly simplified.
(I = H O + r/
O = [05, 06] T, and
= [j, (~. _ 91
,7 = ( ¡
~-
o6
( ~ i - 8,) T 9
With the estimates of II and (6, - 6,), O can
now be estimated by OLS, or more efficiently
by GLS, because ~/is heteroskedastic. In an appendix (available on request from the authors)
we give the formula for var(7/) = X, and if a
consistent well-behaved estimate of X can be
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(8)
the selectivity bias which exist in (6.1) and (7.1).
Lee (1978) and Amemiya (1978a) show the following two equations must be estimated:
Davis
and Wohlgenant
Natural
obtained, GLS estimation of O will be BLUE.
With the parameter estimates from (9.1) and some
algebra, the inclusive value /~ is next constructed and used in the estimation of the second-stage, equation (3).
Second-Stage Estimation
Tree Market
733
Our study contrasts with Hamlett et al. in two
ways. First, Hamlett et al. assumed the yes-no
and tree type decisions were independent and
used a sequential probit model to estimate the
probability of displaying a natural tree. We assume the yes-no and tree type decisions are interrelated and use the nested logit model to capture the interaction. Second, Hamlett et al.
focused on forecasting whereas our interest lies
in estimating price parameters. We amend the
Hamlett et al. variable list to include prices.
The price paid for a natural tree (P,) was taken
directly from the survey. However, the price the
consumer pays for an artificial tree (w) is not
the price that is compared to that of a natural
tree. Artificial trees are durable goods, so the
price of an artificial tree must be converted to a
user cost or service flow price. This price (Pa)
is constructed following Hausman,
Pa = wr(1 + r)-l[1 - (1 + r)-q] -~
where 60 is the price paid for an artificial tree,
r is the individual discount rate (assumed to equal
0.10), and q is the expected life of an artificial
Christmas Tree Survey Data
The data set is a survey of 558 households
in the Washington, D.C., northern Virginia,
southem Maryland, and Philadelphia areas. The
survey was conducted by phone in January 1986
by the Agricultural Economics Department at the
Pennsylvania State University.
Of the 558 households surveyed, 147 (26%)
did not display a tree while 411 (74%) did. There
were 214 (38%) households which displayed
(bought) a natural tree and 197 (35%) households which displayed an artificial tree. 3 Of the
total number of questions asked (105), 39 were
household characteristic questions and 66 were
alternative attribute questions which did not
overlap across alternatives. With 105 questions,
there are a large number of potential model
specifications. For our analysis, we adopt the
particular specification employed by Hamlett et
al. Table 1 shows the variable list and description. In this paper, Xv = X,, = X, = (xi, x6, x 7,
X8, X9, XI0, XI l),
Xny
-~- X a y
~-
Xlv
=
( x i , x2, x 3,
x4, xs, P,, - Pa), and X = (x~ . . . . , x19).
3 It is important to distinguish between the purchase decision and
the display decision. By modeling the display decision, we allow
the household to own both types of trees, but only display one type
of tree. Some households may display both natural and artificial
trees, but these representa small portion of the sample (4%). Those
who did display both tree types were treated as natural tree users,
Because display and purchase are for the natural tree the same decision, we can construct demand elasticities from this model.
Table 1.
Variable Description
Variable
Desc¡
x,yl = xoyl = xyl
Xnl
Xny 2 =
~
Xay 2 = X 2
Xny 3 = Xay 3 = X 3
Xny 4 =
Xay 4 -~- X 4
Xny 5 ~
Xay 5 ~
X5
Xy2 = Xn2 = X6
Xy 3 ~
Xn3 = X 7
Xy 4 ~
Xn4 ~
X8
Xr5 = x,5 = x9
Xy6 = xn6 = Xlo
Xy 7 ~
Xn7 =
Xy 8 ~
Xn8 ~-
Xl l
XI2
xr9 = xn9 = x l s
X14
XI5
Xi6
Xi7
x~8
xi9
p.
Pa
1
X1
1 if single family dwelling; 0 otherwise
Income category:
1 if under $10,000
2 if between $10,001 and $15,000
3 if between $15,001 and $20,000
4 if between $20,001 and $25,000
5 if between $25,001 and $35,000
6 if between $35,001 and $45,000
7 if between $45,001 and $60,000
8 if over $60,000
Age of household head
1 if white; 0 otherwise
1 if majority of holiday spent away;
0 otherwise
1 if married; 0 otherwise
1 if Christian; 0 otherwise
1 if wreath was hung; 0 otherwise
1 if stockings were hung; 0 otherwise
1 if special holiday meal; 0 otherwise
1 if gifts were exchanged; 0 otherwise
Number of people in household
1 if live in open country; 0 otherwise
1 if live in small town; 0 otherwise
1 if live in suburb; 0 otherwise
1 if live in Philadelphia area;
0 otherwise
1 if live in Maryland; 0 otherwise
1 if live in Washington; 0 otherwise
Price paid for natural tree
Service flow price of artificial t r e e
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In the second-stage, the probability of displaying a tree is estimated. Parameter estimates are
obtained by applying a probit or logit procedure
to (3). Amemiya (1978b) shows that the standard variance of this estimator is heteroskedastic
because it does not take into account that ly
has been estimated. For unbiased inference,
heteroskedasticity should be corrected for by
Amemiya's formula. If it is not corrected, the
t-statistics are biased upward.
We now implement the econometric approach
outlined above to estimate market demand elasticities for natural Christmas trees.
Christmas
734
August 1993
Table 2. Estimates of Reduced Form Parameters of Equation (8)*
Dependent variable
Constant (xt)
Table 2 presents estimates of reduced-form parameters II associated with the reduced-form
c¡
function. Va¡
most influential in
the reduced-form decision are dwelling, income, age, and race. However, little emphasis
should be placed on these reduced-form parameters or their significance, because insignificance of the reduced-form parameters does not
imply insignificance of the structural parameters.
From the parameter estimates in table 2, we
construct the appropriate inverse Mill's ratios and
then estimate (6.2) and (7.2) by OLS. Results
are shown in table 3. As stated earlier, more
efficient estimates of Fn and F a could be obtained by GLS but estimates of the complicated
variances were not well behaved (because the
inequality o-Ÿ_> tr~t, l = a, n was violated and
cannot be imposed). The ill conditioning of these
variance estimators is not uncommon (Lee 1978,
Lee and Trost; Domencich and McFadden). Because we know theoretically that the OLS va¡
ance-cova¡
structure underestimates the true
variance-covariance structure (Lee, Maddala, and
Trost), the t-statistics from OLS will be inflated
asymptotically; hence, some adjustment must be
made. Following Amemiya's suggestion (1985
- 1.075
(1.056)
Dwelling (x2)
0.607
(1.805)
Income (x3)
Household age (x4)
Race (xs)
Spent away (x6)
0.154
(2.413)
-0.023
(2.689)
1.023
(3.012)
0.008
(0.030)
Marriage (x7)
Religion (xs)
-0.411
(1.319)
0.573
(1.531)
Hung wreath (Xg)
Hung stockings (Xlo)
Special meal (x.)
Exchange gifts (xi2)
-0.434
(1.435)
0.242
(0.874)
-0.184
(0.473)
0.050
(0.065)
Number in household (x~3)
Live in country (x14)
Econometric Results
I
Live in small town (x~s)
Live in suburb (x16)
Philadelphia area (x,~)
0.044
(0.482)
0.505
(0.660)
-0.600
(1.202)
0.002
(0.009)
- 0.218
(0.625)
Maryland area (x~s)
Washington area (x~9)
Likelihood function
Sample size
-0.118
(0.314)
-0.157
(0.277)
-212.669
338.00
* Absolute asymptotic t-statistic in parentheses.
p. 370), we use the MacKinnon and White HC2
approximation to the true variance-covariance
matrix to obtain unbiased asymptotic inference.
The fu'st numbers in parentheses give the OLS
t-statistic and the second numbers in parentheses
give the HC2 adjusted t-statistics. Some adjusted t-statistics are lower and some are higher
than the unadjusted t-statistics. The fact that all
are not lower may be due to poor small-sample
properties of the approximation or to the fact
that the HC2 approximation corrects for an unknown form of heteroskedasticity. Although
many of the t-statistics and R2's are low, three
important points should be made. First, like all
economic models, the Theil-Houthakker model,
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tree. The value q was specified by each consumer who bought an artificial tree. As the expected life (q) of an artificial tree decreases, the
service flow price (Pa) of an artificial tree increases. Thus, ceteris paribus, the relative pfice
of a natural tree decreases and the probability of
displaying (buying) a natural tree increases. This
may explain, in part, why an individual may own
an artificial tree and a natural tree, but only display a natural tree.
Missing household characte¡
variables in
the survey are assumed to be generated by a random process (Heckman 1976, 1979; Griliches,
Hall, and Hausman 1978). The 101 missing
household characte¡
observations were deleted, leaving a sample size of 457. On the other
hand, the survey design suggests the missing price
observations are generated by a nonrandom process. This process is the standard selectivity
problem whereby if the individual displayed a
natural tree he only reported the price of a natural tree, and if he displayed an artificial tree
he only reported the price of the artificial tree.
Selectivity bias is corrected for in the estimation.
Amer. J. Agr. Econ.
Davis and Wohlgenant
Table 3.
Natural Christmas Tree Market
Selectivity Bias Carrected OLS Estimation of Tree Prices (Etluatian 6.2 and 7.2)*
Dependent variable
Constant (x,)
P.
R2
1~2
F statistic
Sample size
0.17
0.06
1.58
163
Dwelling (x0
Income (x3)
Household age (x4)
Race (xs)
Spent away (x6)
Marriage (XT)
Religion (xs)
Hung wreath (xg)
Hung stockings (x~0)
Special meal (x~0
Exchange gifts (x~2)
Number in household (x,3)
Live in country (x~4)
Live in small town (x~5)
Live in suburb (x~6)
Philadelphia area (x~7)
Maryland area (x~8)
Washington area (x~9)
Pa
(1.103)
(0.765)
(1.145)
(1. 145)
(0.697)
(1.732)
(1. 118)
(0.515)
(0.977)
(1.030)
(1. 307)
(0.713)
(0.611)
(1.574)
(1.773)
(1.253)
(1.747)
(1.058)
(1.734)
21.313
(0.878)
- 0.383
(0.086)
-0.693
(0.623)
- 1.796
(0.368)
3.426
(0.733)
- 1.784
(1.393)
-0.770
(0.329)
- 1.164
(0.574)
-2.772
(1.084)
0.127
(0.107)
4.615
(1.445)
0.077
(0.456)
-4.893
(0.650)
-4.954
(1.123)
3.870
(1.168)
1.123
(0.555)
0.227
(0.11 l)
-1.346
(0.441)
0.195
(0.354)
--
(0.814)
(0.093)
(0.649)
(0.251)
(0.732)
(1.416)
(0.309)
(0.582)
(1.264)
(0.118)
(1. 557)
(0.479)
(0.630)
(1. 105)
(1. 149)
(0.585)
(0. 129)
(0. 344)
(0. 349)
(0.931)
-21.782
(0.666) (0. 667)
0.26
0.12
1.85
120
* First number in parentheses is absolute OLS t-statistic. Second number in parentheses is absolute asymptotic t-statistic corrected for
heteroskedasticity by the MacKinnon and White HC2 approximation. The sample size of reduced forro parameter estimates, table 2, does
not equal 163 + 120 because some people who actually displayed a certain tree did not report its p¡
does not provide a guide as to which household
characteristics should be included in the price
equations to proxy for unobservable quality
characteristics, so there exists no theoretical reason to remove variables. Second, the low R2's
imply that heterogeneous quality effects are rather
small (Cox and Wohlgenant, p. 914). Finally,
by (9) or (9.1), the parameters of interest are
the differences in parameters of the natural
Christmas tree and the artificial Christmas tree
price equations. The insignificance of the individual parameters is irrelevant, because their
differences can be significant. 4
Table 4 shows the structural parameter estimates of the lower tier obtained from estimating
4 The appropriate t-statistic for the difference between two parameters is t = (8, - 8a)[var(6,) + var(Sa) - 2 cov(6,, 6~)]-I/2.
Obviously it is possible for the individual parameters to be insignificant but their difference be significant.
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~o
58.918
(0.902)
- 8.236
(0.662)
-0.322
(0.104)
- 12.486
(1.007)
7.483
(0.569)
-4.875
(1.633)
4.792
(0.861)
1.998
(0.444)
5.428
(0.773)
-2.988
(I .057)
7.673
(0.898)
0.245
(0.522)
-.500
(0.475)
- 11.071
(0.995)
10.598
(1.245)
-6.441
(1.107)
8.475
(1.584)
4.909
(0.624)
- 1.875
(1.432)
-65.537
(0.748)
--
ff~~
735
736
Amer. J. Agr. Econ.
August 1993
Table 4. Structural Parameters of Conditional Probability of Displaying a Natural Tree
(Equation 9.1)*
O~ (Constant)
02 (Dwelling)
03 (Income)
04 (Age)
05 (Race)
F statistic
Sample size
Constant (x0
(1.539)
Spent away (x6)
(4. 256)
Marriage (xT)
(54.486)
Religion (xs)
(11.625)
Hung wreath (x9)
(21.460)
Hung stockings (xt0)
(4.937)
Special meal (xn)
* Because of the way Os and J are defined in (14.1), if a standard
-4.072
(7.453)
-0.989
(5.986)
0.461
(2.361)
0.794
(3.265)
1.035
(5.725)
1.775
(9.1), which is based on the Amemiya principie. The form of the variance of 77 (,~) depends
on the ill conditioned reduced-form variance-covariance matrices, which are forbiddingly complex, so we do not estimate O by GLS. However, we estimate O by OLS and again use the
MacKinnon-White HC2 heteroskedasticity variance-covariance approximation to calculate adjusted t-statistics. All parameters have the expected sign, with dwelling, income, and race
having positive signs and, age and the price difference having negative signs. The first number
in parenthesis is the OLS t-statistic. According
to these biased statistics, the only two significant parameters, at the 5% level are race and
the price difference. Alternatively, the HC2 correction for heteroskedasticity shows that all parameters except 01 are highly significant. These
results highlight the need to correct for heteroskedasticity when using the Amemiya principie.
Table 5 shows the results of the second-stage
estimation. All parameters have the expected sign
with the exception of the inclusive value. As before, the unadjusted t-statistics ate given first,
followed by the Amemiya (1978b) adjusted tstatistics. All parameters are significant when
considering the adjusted t-statistics except for
the inclusive value. The insignificance of the inclusive value suggests that the decision process
may be sequential as assumed by Hamlett et al.
(5 817)
(2 287)
(3 257)
(5 351)
(8.751) (7 615)
0.745
(3.028) (2 931)
Exchange gifts (xi2)
1.476
(6.539)
0.568
(10.114)
-0.040
(0.099)
- 158.186
Household number (xi3)
software package such as SAS is used, one should specify no in-
tercept. The first number in parentheses is the absolute OLS t-statistic. The second number in parentheses is the absolute HC2 corrected t-statistic. Because of the ones and zero elements in J, it was
-2 (1.00(0)O1 - ksi~), to obtain
necessary to make the divisor for trsii,
a nonzero denominator. We also estimated the HC1 MacKinnonWhite va¡
approximation, which uses the degrees
of freedom correction n(n - k) -~. The absolute t-statistics from this
estimator were 1.277, 3.532, 45.221, 9.648, 17.810, 4.097.
(7 453)
Inclusive value (fy)
Likelihood function
Sample size
(3.040)
(9.776)
(0.045)
457
* First number in parentheses is absolute unadjusted t-statistic. Second number is absolute adjusted t-statistic using Amemiya's (1978b)
formula.
Overall, table 5 results indicate that the natural
tree buyer consumes a bundle of complementary
goods important in the Ch¡
celebration.
Market Demand Elasticities
With the results of tables 4 and 5, we can now
use elasticity formula (5) to obtain the following
own-p¡
and cross-price market demand elasticities for natural Ch¡
trees.
Own Price with respect to P,g07,,):
su
"Onn = E
g=l
[~ny6¡ (n' y ) ' ( 1
¡241
-
y)
']- 06 " (-lg(n
y) " (1 - (Ig(n ly)
9(Ig(y)]P.g]" 1~[1
= -0.674
Cross Price with respect
t o Pag
(1"]na):
"O,,a= [~= (Ig(n,Y)'(1-(lg(n]Y))
9 [--~ny6" (1 = 0.188
(-lg(y)
-
06) ] 9 Pag]"
Os l
Downloaded from http://ajae.oxfordjournals.org/ at Penn State University (Paterno Lib) on May 17, 2016
06 (P, - ea)
0.487
(1.185)
0.281
(1.249)
0.170
(0.798)
-0.016
(0.077)
0.832
(3.830)
-0.041
(4.445)
12.84
19
Table 5. Probability of Displaying a Tree
(Equation 3)*
Davis and Wohlgenant
As expected, the demand for natural trees is inelastic and artificial trees are a substitute for natural trees.
Summary and Conclusions
737
supports this concern and indicates that continued decreases in the real price of artificial trees
could cause the share of artificial trees in the
total Christmas tree market to continue to rise.
To conclude, this paper demonstrates the feasibility of using cross-sectional survey data to
estimate market-level p¡
elasticities of commodities consumed in discrete units for which
time series data are lacking. By aggregating appropriately over individual demand estimates,
market demand elasticities can be obtained for
other commodities which exhibit characteristics
similar to Christmas trees.
[Received July 1991; final revision received
October 1992.]
References
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--.
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Our purpose has been to demonstrate a method
for estimating market demand elasticities from
a discrete choice model; the method has been
applied to the natural Christmas tree market. The
discrete choice model used is McFadden's nested
multinomial logit model, in which the household is assumed to decide first whether to disp l a y a tree or not, and second whether to display
a natural or artificial tree. The advantage of this
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The nested multinomial logit model was applied to a cross-sectional data set for selected
locations in the northeastern United States. The
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probability of displaying a natural tree were estimated using Amemiya's principle (1978a). In
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Empirical results indicate that the price difference between natural and (annualized) artificial trees is very important to the decision about
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such as hanging a wreath, hanging stockings,
and exchanging gifts were important in the decision about whether to display a tree.
The empirical estimates from the cross-sectional data were used to derive market-level demand elasticities. Price elasticities of - 0 . 6 7 4
and 0.188 for natural trees with respect to natural tree prices and annualized artificial tree
prices, respectively, are indicated. These are the
only known estimated demand elasticities for
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own-price elasticity indicates that increased harvesting of Christmas trees could have significant
effects on prices and returns to Christmas tree
producers; an industry concern because of increased plantings in recent years. The industry
has also been concerned about the increasing artificial tree share of the total Christmas tree market. The magnitude of the cross-price elasticity
Natural Christmas Tree Market
738
August 1993
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