Original The Price Dispersion of Consumer Products

Presented is an analytic dynamic model of the price dispersion of consumer products. The theory is based on the idea that sellers offer product units for a profit maximizing price, denoted p m . Product units not sold at p m are called excess units. Based on the conservation equation of offered units, it can be shown that the stationary price distribution of consumer products consists of a Dirac-delta peak at p m surrounded by a fat-tailed Laplace distribution from the excess units. A good quantitative agreement with empirical data can be obtained with a fit of the two free parameters of the theory.

Independent of the microeconomic origin of the price dispersion, an analytic model is presented that yields a price distribution equivalent to distributions found in empirical studies (Kaplan & Menzio, 2014). The key idea of the model is that for a time interval ∆t, a consumer product has a unique profit maximizing price, denoted p m . The "law of one price" applies if all supplied product units can be sold at this price. However, there are units supplied, but not sold at p m , called excess units. They generated a current of available units on the price scale. This flow is the origin of the price dispersion in this model.
Based on the conservation equation of available units it is shown analytically that, under market equilibrium conditions and a random drift of the excess units, the price dispersion has the form of a Laplace distribution with a central peak at p m .
The remainder of the paper is organized as follows. The first chapter presents the general framework for the description of the price distribution. Based on the conservation equation of supplied units and three simplifying assumptions, the stationary price distribution is established in the subsequent chapter.
After a comparison with empirical data the paper completes with a conclusion.

General Framework
The scope of this paper is to establish the basic equations for the price distribution of a consumer product. The probability density function (pdf) of sold units is defined as: where y(t,p) is the number of sold units per unit time at time step t in a price interval p and p+dp . The total unit sales reads (Note 1): Further introduced is the number of available units z(t,p) at time step t in the price interval p and p+dp.
It is governed by the following conservation equation: The first term on the right-hand side of the equation determines the supply rate s(t,p) of units at time step t in the price interval p and p+dp. It quantifies the number of units entering the market per unit time at price p. The next term considers the sales process. It decreases the number of available units at p by the unit sales rate y(t,p). Units can also withdraw from the market. This happens for example for non-durable consumer products if units increase the expiry date. This contribution is proportional to the current number of offered units z(t,p) and a rate χ. However, since unsold units increase the costs, the rate χ must be very small. It is considered to be sufficiently small to be neglected in further considerations. The last term in this relation takes the flow of units on the price scale into account. It is governed by the current j(t,p) (Note 2).

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The total number of offered units is defined by: and the total supply rate is: A time interval ∆t is chosen such that the profit r(t,p) of the consumer product has a maximum at a price p m with: The derivation of the price distribution P y (p) is based on four assumptions. The first two assumptions specify the price dependent supply rate s(t,p): Sellers try to maximize their profit by supplying all units at p m . The supply rate s(t,p) of the product is therefore located at profit maximum.
This assumption implies that: The distribution of supplied units can be approximated by a Dirac -function (Note 3): If all units are sold during ∆t at p m , the price distribution reduces to (Note 4): However, if this is not possible, sellers offer units for a price pp m . The number of these excess units is denoted z ex (t,p). Therefore, the total number of offered units consists in general of two contributions: Equivalently to eq.(10) the total unit sales can be written as: is the total number of sold units per unit time at p m and ) ( t y ex are the unit sales at pp m .
The stable price distribution from the excess sold units is denoted ) ( p P ex y . The stationary price distribution of a product consists in this model of two components: where 0a1 is a free parameter that quantifies the magnitude of excess units. The first term of eq.(13) stems from units sold at p m and the second term is due to the excess units. The remainder of the paper aimed at deriving ) ( p P ex y for a0.

The Derivation of the Price Distribution of Excess Units
The derivation starts by rewriting eq.(3) in the form (Note 5): where the growth rate (t,p) of available units is introduced by (Note 6): The evolution of the total number of offered units becomes: with the mean growth rate of supplied units: The mean growth rate can be obtained from: For further use the price scale is separated into three price regions: p<p m , p=p m and p>p m . Based on eq.(18), mean growth rates can be established for the corresponding price regions as follows.
For p=p m : For p<p m : The second statement of the model assumes that the market is in a stationary state (Note 7): For the considered time interval t total unit sales equal total supply rate, With eq.(17) this assumption is equivalent to: From the definitions eq.(19)-eq.(21) follows: This assumption has two consequences: A) It implies that the drift of excess units takes place in both directions of the price scale away from p m with equal chance. The effective current of excess units can be written as: where The stable distribution of excess units (z ex (t,p)/t=0) is governed with eq. (27)  and an integration constant C. From the symmetry of the distribution (assumption iii)) follows C=C.
Taking advantage from the fact that the total number of excess units is ex z , we obtain for the stationary distribution of excess units: The stationary probability density function (pdf) of offered excess units has the form: and becomes with eq.(35) (Note 11): with the standard deviation: This result suggests that the price distribution of excess units is governed by a Laplace distribution www.scholink.org/ojs/index.php/jepf Journal of Economics and Public Finance Vol. 7, No. 4, 2021 7 Published by SCHOLINK INC.

The Price Distribution of Sold Units
Based on assumption i) all units are supplied at p m and with eq.(8) is s(p)=0 for p≠p m . Therefore, eq. (15) suggests that the unit sales rate of excess units y ex (p) can be written in the stationary state as: where we used eq.(11) and eq.(41).

Comparison with Empirical Data
The model is compared with empirical data of a comprehensive investigation of prices of consumer products performed by Kaplan and Menzio (2014). They studied data from the Kilts-Nielsen Consumer transactions by 50,000 households. The panel covers over 1.4 million goods in 54 geographical markets for the time period [2004][2005][2006][2007][2008][2009]. The investigators aggregated the data of products into four different categories of a good: 1.) UPC: A good is the set of products with the same barcode (Universal Product Code: UPC).
2.) Generic Brand Aggregation: According to this definition, a good is the set of products that share the same features, the same size and the same brand, but may have different UPC's. Since the KNCP assigns the same brand code to all generic brands (regardless of the retailer), this definition collects all generic brand products that are otherwise identical.
3.) Brand Aggregation: According to this definition, a good is the set of products that share the same features and the same size but may have different brands and different UPCs.
4.) Brand and Size Aggregation: In this case a good is the set of products that share the same features but may have different sizes, different brands and different UPCs.
After scaling the data by the mean price of a product they aggregated the data to price distributions of the corresponding definitions of a good. Since the first three definitions are closest to that of a consumer product used in this model, we confine here to a comparison with empirical data of the first three definitions.
For this purpose, the empirical data reproduced from Kaplan and Menzio in terms of a pdf are transformed into the corresponding cumulative distribution function (cdf). It is defined by: The cdf of eq.(42) reads: back into the pdf and plotted in Figures 1-3 as fat lines together with the empirical price data (squares).
A good quantitative coincidence of the empirical data can be obtained with this two-parameter fit (Note 13). In view of the presented model, the first definition of a good (UPC) is closest to the model assumptions. It has a pronounced central peak surrounded by a Laplace distribution as suggested by the model. The central peak disappears (decreasing factor a) and the standard deviation slightly increases with a broader definition of a good.

Conclusion
The paper establishes a model for the price dispersion of consumer products. The presented model suggests that the price dynamics of a product is governed by a profit maximizing p m . Excess units, a broader definition of a good, the profit maximizing price p m can no longer be viewed as unique, but has to be described by a distribution. A broader definition of a good broadens therefore the price www.scholink.org/ojs/index.php/jepf Journal of Economics and Public Finance Vol. 7, No. 4, 2021 distribution and the central peak disappears, accompanied with an increase of the standard deviation.
Note that the model is applicable only, if time-dependent variations of the profit maximizing price p m (t) are small during the time of investigation ∆t, compared to the mean drift velocity v of excess units, ∆p m (t)/∆t<<|v|. It can be expected that the pressure to sell product units increases with a shorter mean offering time τ, hence |v|~1/τ. Therefore, the presented model rather applies to non-durables than to durables (Note 14).