Master student
Izvekov N.Y.
National Research Tomsk Polytechnic University, Russia
Analysis of
properties of optimization pricing models
The aim of this work is the research of properties of
the original models of determination of optimal consumer prices using the price
elasticity of goods.
N.Y. Izvekov
and T.V. Kalashnikova derived several models of optimal pricing in their
article «Using the price elasticity in the procedures of optimization of
financial indicators for a retail chain» [1]. Now it is proposed to develop that
approach and research properties of those models.
Legend:
– optimal changes of the
consumer price, 
– price elasticity coefficient, 
– last consumer price, $; 
– quantity of sales at the last consumer price, 
– optimal consumer price, $; 
– trade margin at the last consumer price.
The
maximum turnover is achieved at the point 
[1].
Thus,
the turnover at the optimal point is defined as:
According
to the accepted limits the price elasticity coefficient 
takes the following values: 
.
The
limit of the function 
is calculated:
.
So, the
horizontal asymptote passes through the point 
. It follows when the value of the price elasticity coefficient 
decreases, the graph of this
function converges to the horizontal asymptote at the point 
. This points to the fact that when the models of turnover maximization is
used, the optimal consumer price can
not be less than the half of the last «closed» consumer price (price preceded
the current one).
The
maximum gross margin is achieved at the point 
[1].
The gross
margin at the optimal point is defined as:
Further
transformations can be made:
The
optimal price in both models is defined as follows:
Thus, the
determinants of optimal prices are found:
· from the
point of view of maximizing turnover:
1) the last consumer
price;
2) price elasticity
coefficient.
· From the point of
view of maximizing gross margin:
1) the last consumer
price;
2) price elasticity
coefficient;
3) current purchasing
price (or trading margin).
The expression
for the optimal changes in a consumer prices is converted for further
investigations of properties of the obtained gross margin function:
.
In such
a way, optimal changes in a consumer price from the point of view of maximizing
the gross margin is defined as follows:
where 
– optimal change in a consumer
price in relation of maximizing the gross margin; 
– optimal change in a consumer
price from the point of view of maximizing the turnover; 
– coefficient of difference between optimal changes in a consumer price
in relation of maximizing the gross margin and turnover.
Hereafter,
it is possible to allocate two cases depending on the value of the purchasing
price:
With
increasing the trade margin 
the value of the difference
coefficient decreases. This means it is easier to find a compromise between
often conflicting requirements of increasing the turnover on the one hand and raising
the marginal profit on the other in determining consumer prices of high-margin
products. The reason lies in the fact that, under other equal conditions,
increase of the trade margin on the product leads to convergence of optimal
prices in terms of the turnover and marginal profit with each other.
The
limit of the function 
is needed to be calculated:
.
So, one
could argue that the considerable growth of the trading margin leads to deleting
difference between the two types of optimal prices.
The
range of trade margin values 
generates a field of values
of the function of the coefficient of difference between optimal price changes
on the gross margin and the turnover 
.
As the
value 
is positive, if the purchasing
price is not equal to zero (this condition is always done in the trading practice),
. It means that the price, at which the maximum value of the gross
margin is reached, is always higher than the price, at which the maximum value
of the turnover is achieved, by the amount 
:
.
Hence
it follows:
,
where 
– price, at which the maximum
value of the gross margin is achieved, $; 
– price, at which the maximum
value of the turnover is reached, $; 
– current purchasing price, $.
Thus, the determinants of the optimal consumer price are received and
the correlation between the two types of the optimal price (in the terms of the
turnover and the gross margin) is defined as a result of the research.
References:
1. Èçâåêîâ Í.Þ., Êàëàøíèêîâà Ò.Â. Èñïîëüçîâàíèå öåíîâîé ýëàñòè÷íîñòè â ïðîöåäóðàõ îïòèìèçàöèè ôèíàíñîâûõ ïîêàçàòåëåé ñåòè ðîçíè÷íîé òîðãîâëè // Àíàëèòè÷åñêèé æóðíàë «ÐÈÑÊ: ðåñóðñû, èíôîðìàöèÿ, ñíàáæåíèå, êîíêóðåíöèÿ». – 2011. – ¹3. – Ñ. 384-389.
2. Catherine Cazals,
Frederique Feve, Patrick Feve, Jean-Pierre Florens. Simple structural
econometrics of price elasticity // Economics Letters. – Elsevier, 2005. – Vol.
86(1). – P. 1-6.
3. Joseph G. Eisenhauer,
Kristine E. Principe. Price knowledge and elasticity // Journal of empirical
generalisations in marketing science. – 2009. – Vol 12, ¹2. – P. 31-52.
4. Martijn Brons,
Peter Nijkamp, Eric Pels, Piet Rietveld. A meta-analysis of the price
elasticity. A system of equations approach // Tinbergen Institute Discussion
Paper. – 2006. – ¹106/3.