Frijns, J.M.G.

Hempenius, A.L.

English

Research Papers in Economics

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1978
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subfaculteit der econometrie
RESEARCH MEMORANDUM
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TILBURG UNIVERSITY
DEPARTMENT OF ECONOMICS
Postbus 90135 - 5000 LE Tilburg
NetherlandsK.u.B.
g1~8LIOTHEEK
TcL~uRGRESEARCH MEMORANDUM
THE PRICE-DEPENDENCE OF THE VARIANCE OF
DIIdAND FUNCTIONS
by
A.L. Hempenius and J. Frijns.
~ p~Js.s.~ A-c.t.~
Katholieke Hogeschool Tilburg
Hogeschoollaan 225
Tilburg
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:. L:CONTENTS
1. Introduction
2. Durable goods
3. Non-durable goods
4. Conclusion
REFERENCES.- 1 -
1. INTRODUCTION
In several papers and books ( e.g. Hempenius [ 2] , Iwai [ 3] , Mills
[ SJ , Smith ( 8] , Zabel [ 10) ) the classical monopoly model has been analy-
sed for the case of stochastic demand. For the mathematical expectation
of the random variable demand it is obvious that one requires the usual
properties of a market demand function, in particular that it is a de-
creasing function of price. For the way in which the dispersion of de-
mand ( around expected demand) depends on the price, there is not such an
obvious requirement. However, something must be postulated, e.g, that
there is no dependence on price; see e.g. Mills [5J.
In the present note the properties of the dispersion as a func-
tion of price will be investigated. Two types of goods will be distin-
guished: goods demanded in very small quantities, to be called durable
goods, and other goods, to be called non-durable goods.
'1'ho c1uantiLy demanded will be denoted by q, with q a non-ne~ative
random variable with expectation h(p), the expected demand at price p.
By assumption h(p) is a decreasing function of p. The quantity demanded
ma,y be written as
(1) q - h(P) t v
where v- q-h(p) will be interpreted as disturbance term. In general the
distributions of q and v depend on the price p, implying that the vari-
ances of q and v depend on p, i.e. one has heteroscedastic demand and
disturbance. In the sequel it is analysed what type of heteroscedasticity
may be expected for durable goods (Section 2) and non-durable goods (Sec-
tion 3).- 2 -
2. DURABLE GOODS
Durable goods are bought in very small quantities at a time.
Suppose that, within the time unit used for measuring demand, an indi-
vidual consumer either buys one unit of a certain durable good or he
does not buy the good. Denote the quantity demanded for consumer i(i - 1,
...,N) by Xi and the probability of buying by ni, with ni depending on
the market price p. Then it holds:
(2) yi - E(Xi) - ni (0 ~ ni ~ 1)
(3) ai - var(Xi) - ni(1-ni)
Market demand is defined as q- EXi. The expected value of mar-
ket demand, E(q), is denoted by h(p). As h(p) only depends on p~),ni will
be made dependent on p alone. Expected market demand thus is:
N
(k) h(P) - E ~ri(P)
i-1
As it is reasonable to assume that the probability of buying, ni, is a
decreasing function of p, it holds that expected individual and market
demand, ui and h(p), respectively, are decreasing functions of p.
In order to analyze heteroscedasticity one has to know the vari-
ance of q, to be denoted by a2. Assuming independent buyíng behaviour,
i.e. assumming that buyers do not react on each other's behaviour but
almost exclusively on the price, one has at price p:
(5) a2(P) - var(q)
N 2 N
- E a. - E n.(1-n.)
i-1 1 i-1
1 1
1) Market demand is considered under cet. par. conditions, viz. for given
incomes, etc.- 3 -
Whether a2 decreases or increases if p increases, depends on
da? dn.
( 6) áP -( 1 - 2n i) áP c 0 for tr
i~ 2
~ 0 for
ni ~ 2
for i- 1,...,N. It thus depends on the frequency distribution of the
ni(i - 1,...,N) on the interval [0,1] what the sign of da2~dp is.
Assuming that the number of buyers N is large enough to approxi-
mate the frequency distribution of ni at price p by a continuous frequen-
cy distribution over the intervall [0,1J, having distribution function
F(~r), it is possible to write formula's (4) and (5) as
(7) h(P) - N I~ n dF(~r)
(8) a2(P) - N I~ n(1-,r) dF(~r)
A distribution which is much used for the interval [0,1J because
of its great flexibility, is the standard beta distribution ( see e.g.
Johnson and Kotz [4] and Raiffa and Schlaifer [7]) with parameters r and
s, having density function:
(9) f(~) - B(r,s) nr-1(1-n)s-1 (S `- n`- 1)
where r and s are positive functions of p and where B(r,s) is the beta-
function.2) As it is reasonable to assume that the cases n- 0 and n- 1
do not occur, one has to choose r and s in such a way that r~ 1 and s~ 1,
as then f(~r) vanishes for n- 0 and n- 1 and the curve of f(~r) then
decreases monotonously to both sides from the mode (r-1)~(rts-2).
If r- s, then the distribution is symmetric around n- 1~2. The traditio-
nal measure of skewness a3 is c(s-r), with c a positive function of r
and s. If r~ s or r ~ s, then the distribution has negative skewness or
positive skewness, respectively, and the mode is larger or smaller, res-
2) See e.g. Johnson and Kotz [4] for the now following properties.pectively, than n - 1~2.
Introduction of the price is best done by fixing one of the
parameters r or s and by making the other one dependent on p: assume
r a constant and s a function of p. Assuming that the mode decreases
with increasing price, it is required that ds~dp ~ 0 for all p~ 0.
Depending on the value of p, one has values for s smaller or greater
than r and the distribution of ~r is negatively or positively skewed, res-
pectively.
For the beta-distribution of (9) one has for h(p) in (7) and
a2(p) in (8):
(10) h(p) -
Nr
rts
o2(p) - rts)(rfstl)
For dh~dp s negative sign is found:3)
(12)
dh - -Nr ds ~ 0
dP (r~s)2 ~ dP
For da2~dp, one has
(13)
da2 Nr{r(rt1)-s2} ds 2
~ 0 iff s ~ r(rt1)
dp - {r(rt1)t(2rt1)sts2}2 . dP
~ 0 iff s2 ~ r(rt1)
From (13) follows that, for prices implying negative skewness of the
n-distribution (s ~ r) one has da2~dp ~ 0. For prices for which the
distribution has more or less a tail to the right, i.e. has positive
skewness (s ~ r), one has in general da2~dp ~ 0, except for those prices
for which it holds that r ~ s(p) ~ r(rt1 .
3) Alternatively, the assumption that dh~dp ~ 0, implies ds~dp ~ 0 and
thus a decreasing mode with increasing price.- 5 -
So assuming that the individual probabilities of buying follow
a standard beta-distribution, which is hardly restrictive at all, Figure
1 shows the way in which the dispersion of demand depends on the price.
For "low" prices, i.e. prices for which s(p) ~ dr rt1), one has da2~dp ~ 0
and otherwise one has da2~dp ~ 0.
FIGURE 1
3. NON-DURABLE GOODS
P
For non-durable goods one must use the more realistic assumption
that the number of quantity units bought during a time unit, again denoted
by Xi for consumer i, is a random variable that assumes the values 0,1,
2,... which has expectation ui and variance ai. Further the assumption
that dui~dp ~ 0, leads directly to dhjdp ~ 0, as h(p) - E(q), where again
q - EXi.
In order to analyse the dependence of a2 - var(q) on p, one has
to be more specific about the ai - var Xi, as one has a2 - Eai. In
management science a much used distribution for a total demand q isthe Poisson distribution; see e.g. Hadley and Whitin [1]. It seems rea-
sonable to use the Poisson distribution also for the Xi, as the assumedly
independent buying behaviour leads to Poisson distributed total demand.
Denoting the Poisson parameters by ai, it follows from E(Xi) - var (Xi) -
- ai and dui~dp ~ 0 that da2~dp ~ 0. It is not necessary to assume that
dui~dp ~ 0: as long as dh~dp ~ 0, one also has da2~dp ~ 0 for the case
of individual Poisson distributions.
Another interesting case is the one of individual distributions
with constant variation coefficients, say ai~ui - c~ 0. As then ai -
c2 ui, one has from dui~dp ~ 0 that do2~dp ~ 0. The case of constant
variation coefficients and also the case of individusl Poisson distri-
butions are special cases of distributions for which a? - g.(u.) and i i i
d gi~dui ~ 0, as one then has:
2 da? du.
(~~) aP - E d~l . áP ~ 0
i
This result is restrictive in the sense that all individual variances are
assumed to increase with individual mean demand, although the assumption
seems a realistic one for the great majority of consumers; see also Prais
and Houthakker [ 6] and Theil [ 9] .
From this section one may conclude that for non-durable goods
it seems reasonable to expect a dispersion as in Figure 2.
FIGURE 2.- 7
4. CONCLUSION
The above conclusions, conveniently summarized in Figure 1 for durable
goods and in Figure 2 for non-durable goods, hold under ceteris paribus
conditions for other variables influencing demand. F~apirical testing for
a particular kind of heteroscedasticity with respect to the price will
thus, for example, have to take into account shifts in demand because of
changes in incomes.
A more general conclusion from the above might run as follows.
Suppose one wishes to get an insight into the way in which the variance
of s certain economic variable, say Y, depends on variables, say X~,...,Xk,
that influence Y. Then one should start with a, possibly very simple,
stochastic micro-theory for Y, allowing to analyse the dependence of
the variance of Y on one or more of the Xi on the micro level. By aggre-
gation one then tries to obtain conclusions at the macrolevel.
The result of this approach will always be that E(Y~X~,...,Xk) and var
(Y~X~,...,Xk) depend on X~,...,Xk. (As an for estimation purposes con-
venient approximation one often assumes that E(Y~X~,...,Xk) depends
linearly on the Xi.) In the above it has been investigated, by means of
this approach, in what way var(qlp) depends on p in monopoly situations.-8
REFERENCES
[ 1] Hadley G., and T.M. Whitin, Analysis of Znventory Systems, Prentice
Hall, Englewood Cliffs, 1963.
[ 2] Hempenius, A.L., Monopoly with Random Demand, Rotterdam University
Press, 1970
[ 3] Iwai, K., "The Firm in Uncertain Markets and Its Price, Wage and
Employment Adjustments", Review of Economic Studies,
april 1974.
[ 4] Johnson, N.L., and S. Kotz, Distributions Statistics, Continuous
Univariate Distributions 1 en 2, Houghton Mifflin Comp.,
Boston, 1970.
[ 5] Mills, E.S., Price, Output and Inventory, John Wiley and Sons,
New York, 1962.
[ 6] Prais, S.J., and H.S. Houthakker, The Analysis of Family Budgets,
Cambridge University Press, 1971.
[ 7] Raiffa, H., and R. Schlaifer, Applied Statistical Decision Theory,
Harvard University, Boston 1961.
l b] Smith, K.R., "The Effects of Uncertainty on Monopoly Price, Capital
Stock and Utilization of Capital", Journal of Economic
Theory, vol. 1 (t969).
[ 9] Theil, H., Principles of Econometrics, John Wiley 8~ Sons, New York,
1971.
[10 ] Zabel, E., "Monopoly and Uncertainty", Review of Economic Studies,
april 1970.IIIVIIMNIINIIII~MIIIIhINllllllhllll I

Analysis of Znventory Systems,

Applied Statistical Decision Theory,

Monopoly and Uncertainty&quot;, Review of Economic Studies,

Monopoly with Random Demand,

Output and Inventory,

Principles of Econometrics,

The Analysis of Family Budgets,

The Firm in Uncertain Markets and Its Price, Wage and Employment Adjustments&quot;, Review of Economic Studies,

The price-dependence of the variance of demand functions

The price-dependence of the variance of demand functions

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Research Papers in Economics