Generalized Demand Densities for Retail Price Investigation

The paper introduces generalized demand densities as a new and effective way of conceptualizing and analyzing retail demand. The demand density is demonstrated to contain the same information as the demand curve conventionally used in economic studies of consumer demand, but the fact that it is a probability density sets bounds on its possible behavior, a feature that may be exploited to allow near-exhaustive testing of possible demand scenarios using candidate demand densities. Four such demand densities are examined in detail. The Household Income demand density is based on the assumption that a person’s maximum acceptable price (MAP) for an item is proportional to his household after-tax income. The Double Power demand density allows the mode to be located anywhere in the range between zero and the highest MAP possessed by anyone in the target population. The two-parameter, Rectangular demand density, the simplest model that a retailer may employ, has the useful feature that it may be matched relatively easily to any unimodal demand density and hence may act as its approximate proxy. The Kinked demand density is derived from the kinked demand curve sometimes used as a relatively uncomplicated way of conceptualizing the effects of oligopoly. The central measures of each of these demand densities are derived: mean price, mode, median, optimal and, when appropriate, the mean of the matched Rectangular demand density. In a further result arising from the use of demand densities, it is shown that stable trading at the kink price will not occur if the demand curve is kinked and convex.


Introduction
While a demand curve may be used to investigate retail prices and how they are set [1][2][3], it has been demonstrated [4] that there are advantages in recasting the information into a probability density for maximum acceptable price (MAP) or "demand density". The "demand density" allows investigations of the optimal price to proceed in a natural and convenient way. The fundamental restriction on any probability density, namely that its integral over all values must equal unity, turns out to be a particularly useful feature, making the demand density a feasible tool for exploring situations where data are sparse. Here a finite number of demand densities may be employed to provide a near-exhaustive coverage of possible price preferences. The paper will begin by explaining the equivalence between the demand density curve and the demand curve conventionally shown in economics textbooks. It will show how the price optimization procedure based on demand density produces the same answer as the graphical method based on the demand curve. Having established this equivalence, the paper will go on to consider four demand densities that have been found useful in assessing how retail prices may be set, deriving key properties of  the Household Income demand density, where MAP is proportional to household income after tax.  the Double Power demand density, where an appropriate choice of the four defining coefficients allows the mode to be located anywhere in the range between zero and the highest MAP possessed by anyone in the target population. believes he could charge before sales become negligible.  the Kinked demand density, derived from the kinked demand curve introduced independently by both Hall and Hitch [5] and Sweezy [6] as a relatively simple way of conceptualizing the effects of oligopoly. The clear perspective on the optimization procedure promoted by the use of the demand density rather than the demand curve has allowed the correction of a misapprehension concerning the kinked demand curve. The location of the optimal price when that curve is convex is found not to be located at the kink price.
The usefulness of the demand density as a model for retail demand has been found previously not to be greatly compromised when the underlying probability density for MAP is approximated by a Rectangular demand density [4]. Therefore an analytical procedure will be presented that allows a Rectangular demand density to be matched to a general, continuous and unimodal demand density.
Note: upper case letters will be used in the paper to denote the name of each demand density for clarity and emphasis.

General Equations
A retailer will need to offer a price common to all, but will face a differentiated market, with different people having a different MAP for the same good. As noted in [4], the term, "uniconsumer", might be used to denote a consumer prepared to buy one but only one item if the price is right. Then a person, a "multiconsumer", who will buy more than one item may be represented, as far as his economic behavior is concerned, as multiple, identical uniconsumers. In the rest of the paper we shall use the word, "consumer", in place of the more exact "uniconsumer", simply to make it less cumbersome to read. Let n be the number of consumers in the target population prepared to pay at least p, i.e. having a MAP of p, for the good, so that: Assuming a constant variable cost per item, v , and letting the fixed costs be , the retailer's profit will be: Since n is a function of MAP, the maximizing condition, 0   dp d , may be written formally as: Provided the rate of change, dp dn , in the number, n, of people in the target population prepared to pay at least p for the good is non-zero, the maximizing condition of Thus differentiating Equation (2) gives the profitmaximization condition as . Thus the profit maximizing condition has the form: where the marginal revenue is given by The bracketed term, (n), emphasizes that the price, p, is related to the number, n, of consumers prepared to pay at least that amount.
The fraction of the target population of consumers prepared to pay at least p for the good is     N p n p S  . Differentiating that expression with respect to n gives: Moreover, N dS dp dn dS dS dp dn dp 1   (9) Substituting from Equation (9) into Equation (7) gives the marginal revenue as: The price, p, is related to the number, , willing to pay that price or more for the good by the probability distribution for MAP, p, or demand density, Figure 1. Meanwhile the fraction of consumers prepared to pay price p or more is given by: where m is the highest MAP for anyone in the target population, the maximum price anyone is prepared to pay. Equation (24) matches Equation (67) derived from the direct optimization procedure explained in Section 4.

Equivalence between the Two Curves for a Rectangular Demand Density
The demand density, , for a general Rectangular distribution for MAP, p, is given by:  The function may be rearranged to give p explicitly in terms of S: Substituting from Equation (27) into (28) gives the result that is the basis of the straight-line graph often used in economic text books: Thus when a Rectangular distribution is used to represent the MAP, then the demand curve is a downward sloping straight line (Equation (27)), while the marginal revenue curve is also a downward sloping straight line, with twice the gradient (Equation (29)).
Since the optimal price occurs when v r , using Equation (28) and then Equation (26) to eliminate S, we have: So that the optimal price is Assuming that the retailer will expect v and the lower limit of his mental model for the MAP to coincide, so that c a v p c  [4], then the optimal price is simply The same as the mean of the Rectangular demand density for MAP. The result coincides with Equation (10) of [4].

Relating the Demand Density to UK Post-Tax Household Income Percentiles
This Section addresses the problem of relating demand densities to the willingness to pay as measured by UK post-tax household income. The income percentile will be shown to be equivalent to the cumulative probability of a household chosen at random having an income less than the specified amount. The relationship between this cumulative probability and an associated cumulative probability for MAP will be established. Mathematical reasoning then produces the necessary relationship between demand density and probability density for the income of a cohort at a given percentile. Data on income may be available only cumulative form, in which case the demand densities need to be found by numerical differentiation, so that they will emerge as staircase functions. Since the method of matching the Rectangular demand density to the underlying demand density presented in Section 5 relies on the latter being continuous, it is necessary to fit a polynomial to portions of the staircase function, with a quadratic giving adequate accuracy.

Household Post-Tax Income and Income Cohorts
The data on income are often available only in cumulative form. Thus Figure 3 shows the cumulative probability for the post-tax income, x, of a UK couple with no children [5]. The data are presented in the format of the "Modified OECD" equivalence scale, in which an adult couple with no dependent children is taken as the benchmark with an equivalence scale of 1.0. This "equivalised income" is intended to allow comparability between all individuals within the nation.
Let x be a household income level, and let   x F be the cumulative probability of a household, chosen at random, having an income, X, up to x (£/y):  , for the percentage of households having that income, x, or lower: Let that income be called the  -percentile income and let the  -percentile cohort be the collection of people whose household income is less than or equal to this income, x. Now choose an income level, y, less than or equal to x. The conditional probability that a household chosen at random has an income level, X, satisfying given that the household is known to be a member of the y X   -percentile cohort follows from the basic tenets of probability theory: But since x y  , it follows that:   1 Hence, using the notation: Thus, for any two cohorts defined by income levels, and 2 , with associated cumulative percentages,

Relating MAP to Income Cohorts
Assume that the maximum amount that people will be prepared to pay for each good is proportional to their income. Thus the maximum any person is prepared to pay, his MAP, p, measured in £, will be proportional to his ability to pay, y , as measured by his post-tax house- where  is a constant of proportionality. The highest MAP, the maximum that anyone in the  -percentile cohort will prepared to pay,    m p , will be dependent on the highest income in that cohort, viz.
is the maximum income earned by anyone in the cohort. Meanwhile, cohort members in any income bracket will have MAPs in the range will be the same as the number with incomes between 1  n and n . Therefore the following relation will hold, between the cumulative probability density for MAP, , and the cumulative probability of income, for the  -percentile cohort: Because both incomes and MAP may both fall to zero but not go below this value, i.e.
, then: 0 It follows that Equation (37) Equating integrands shows that the probability density for MAP for people in the θ-percentile for income is related linearly to the probability density for income in that percentile: emerge as a staircase function: of commodities that are needed and obtained by all will be determined by the attitudes and decisions of those who have household incomes up to a certain percentile, the  th percentile. Those with incomes above the  th percentile will then be price-takers for these goods. Clearly the valuation of some scarcer, desirable goods will require  to be set high, very high for luxury goods such as high-performance sports cars and large residences in central London; the latter, particularly, are generally accepted as being the preserve of the super-rich. The percentage,  , of people determining the price of each commodity may vary according to commodity, and moreover, that percentage may not be known with any precision. To cope with this situation, results may be derived for a range of possible percentages,  , from 51% to 99%, for example. See Table 1.
So that combining Equation (43)  , needs to be found by numerical differentiation, and hence will Applying the procedure to the data points marked in  , resulting from this procedure is strictly unimodal.
The correctness of the procedure may be checked by from an initial condition of 0 0  p , utilizing the coefficients, n , that have been found from Equation (46) and then employing Equation (44):   Figure 4 shows two piecewise continuous probability densities that have been matched over the central section of the distribution. They have been chosen to be quadratics to allow ease of inversion, a convenient property used in the least-squares fitting of a Rectangular distribution. The process of fitting the quadratics will be discussed in the next section.

Smoothing Sections of the Staircase Probability Density Using 2 nd Order Polynomials
The method of matching the Rectangular demand density to the underlying demand density (explained in Section 5 to follow) relies on the latter being continuous. Hence it  is necessary to fit a polynomial to portions of the staircase function, with a quadratic giving adequate accuracy. Let the 2 nd order polynomial approximation to the staircase probability density over the interval mode p p p j   take the form:  The quadratics are, of course, convenient to invert. Rearranging Equation (51) gives: For the data shown in Figure 4, the positive root of the discriminant is needed, so that the general solution for the MAP, p, between and is:

The Properties of the Double Power Demand Density
This Section will derive the properties of the Double Power demand density for the three exhaustive and exclusive cases, namely 1) when the mode is strictly interior to the interval between zero, 2) when the mode is located on the lower boundary of the interval, viz. 0, and 3) when the mode is located on the upper boundary, viz.
m . The properties sought are the central measures characterizing any probability distribution, namely the mode, the median and the mean, and then the optimal price, which becomes a property of a demand density. Furthermore, the parameters, a and b , that define a Rectangular demand density matched to the underlying demand density become additional characteristics of that underlying probability density. These lead to a Rectangular optimal price that is simply the arithmetic average,

When the Mode Is Strictly Interior
For the general Double Power demand density defined by Equation (17), a strictly interior mode will occur when b > 0 and c > 0. Moreover, continuity implies that . Meanwhile it is a property of any probability distribution that its integral over all possible values will be unity. Hence: The mode, mode , occurs at the maximum value of , which will occur when p   in which Equations (56) and (57) have been used to eliminate a and b. Thus the normalised mean, , is given by: which Equation will normally require an iterative, numerical solution.
The optimal price, , will be the solution to Equation (6) of [4]:   The condition for the least-squares fitting of a Rectangular demand density with base coordinates, ( ) and ( ), is derived in Section 5 as Equation (100). This leads, in the case when the Double Power is the underlying demand density, to: 0 ,

When the Mode Occurs at p = 0
When c = 0, Equation (17) The condition for the least-squares fitting of a Rectangular probability density is given by Equation (102) from Section 5, which yields, in the case where 0  a p , Combining Equations (72)  (77) , which will result in the mean value of the Rectangular distribution becoming 0.25 and 0.5 respectively.
The optimal value resulting from the Double Power probability distribution with c = 0 may be found from substituting c = 0, and also into Equation (69): So that a is defined as soon as c and are defined: The mode for this distribution will be . The mean value will be: , then only Equation (102) must be solved for .

Geometrical Considerations
As will be seen, geometrical considerations allow a more robust numerical algorithm to be developed. Referring to Figure 5, since the area under the probability distribution,   p h , defined on ( ), must equal unity, it follows that m p , 0 The area under the Rectangular probability distribution,   p g , defined on ( ), must equal unity also. Hence: b a p p , Eliminating the area, F C  , gives: which means that the integrated error will be zero. Equation (102) implies that Thus combining Equation (108)