Математика. Прикладная математика

Доктор физико-математических наук, профессор Лавриненко Н.М.

Томашевский П.В.

Донецкий Национальный Университет Экономики и Торговли

имени Михаила Туган-Барановского

Demand function. The coefficient of elasticity.

As a result of solving problems of optimal choice, it is possible to trace the relationship between changes in systems of price and income groups of consumers, on the one hand, and the demand of this group of goods and services on the other, as well as construct a function of optimal demand.
In a sufficiently general form, the optimal demand is expressed by functions of the form - , where

In some cases, the optimal demand functions are particularly simple form. Thus, if the utility function has a logarithmic form, then the optimal demand expressed by the formula , where .
The exact form of the demand function is determined by statistical processing of results of special observations for revenues and expenditures for various social groups. A study of the demand functions are typically installed some classification features of goods.

If for some goods, the condition , they are called normal goods, as the demand for it decreases with the increase of its price. However, there are goods for which demand increases, despite the price increase. This paradoxical situation arises when an increase in price an ineffective product (e.g., potatoes), a group of consumers with low incomes simply cannot acquire more high-calorie products (meat) and is forced to compensate for the lack of calories enhanced buying potatoes. Goods for which the inequality , they are called anomalous or goods Giffin. With a fixed income, and for practical purposes for normal products are used as a rule, the demand function of two types:

1) A linear function of demand , where are statistically estimated parameters of the model;

2) A power function of demand .

In many applied research significant role played by the coefficient of elasticity - a measure of response of the endogenous variable to change the exogenous variable. However, this definition is too general. Specifically elasticity can be defined as the limit of the relationship between the relative increment of the function y: (dependent variable) and the relative increment of the independent variable x.

Thus, the elasticity can be expressed in the formula ,
or in the continuous case .

For practical reasons, elasticity attributed to the percentage increase in the independent variable. In this case, the elasticity shows how percent increase or decrease endogenous variable y, if the independent variable x changes by 1%.
There are arc elasticity, the average at some part of the curve, and scatter the elasticity value of the derivative at a given point. For the practical calculation of the elasticity formula is used for English mathematician and economist Robert Allen (19061983). He proposed using the midpoint of the interval on which the change occurs as the denominator. Then calculate the relative change in the endogenous variable and the relative change in the exogenous variable . And then calculated the ratio of first to second. It must be remembered that the formula of Allen, though popular, but not the only possible. However, it should not be applied to a very wide range, as in this case it may be misleading. To determine the elasticity of demand on the price, you can use the formula , or .

The coefficient of price elasticity of demand indicates the percentage decrease (increase) the demand if the price of goods will increase (decrease) by 1%. For a linear demand function is assumed that, , where the average price, the average demand for the used sample. Obviously, the power function of demand .

If the coefficient of elasticity is close to zero, the demand for a product does not depend on its price. In this case we say that demand is inelastic to price. This applies mainly to the basic necessities. Demand is normally elastic, if , which holds for durable goods. For luxury items are usually , Super-elastic demand is.

At constant prices of products differ in changes in demand, depending on the income . Product is called a valuable (or higher number of goods), if , demand for it increases as we move from less profitable consumer groups to be more profitable. For low-value goods takes place opposite inequality means that the displacement of the product from a range of consumer groups of consumers to increase its revenue categories.

On the basis of the known classification of commodities into three groups (essential items, durables, luxury) can be modified depending on the demand increase of income through the graph (Figure 1.1)

Fig.1.1. Changes in demand based on income
Here on the horizontal axis () plotted the relative values of income, and the vertical share of costs for the three groups of products.
It is easy to see that the proportion of demand for essential goods fell from 70% (low income) to 35% (with an income 10 times greater line AA); relatively stable (ranging from 20% to 27%) share of the cost of goods of the second group (line BB) and significantly increased spending on luxury goods (from 10% to 43% of the line CC). To study the changes in demand depending on the income of different consumer groups are mainly used in the model of two types:

1. Models of power type (function Engel): .

Here the index g has a sense of elasticity, as it shows the percentage will increase demand for goods, if income increases by 1%. The coefficient of elasticity of demand on income is as .

For essentials index , an increase in income the additional cost of these products in this category are all decreasing share. For durable goods, the elasticity approximately equal to 1, which means approximately constant share of expenditure on these items in additional income. For luxury items, the elasticity . This means that when a significant increase in income increasing proportion of its growth is spent on goods in this group.
2. The idea of separation of goods and services consumed by a number of different groups developed further in the design of the so-called functions of Tornqvist. For essential commodities, this function is defined as ,
where , parameters of the model.

With a very large income, conventionally represented as (), the value of demand and, that expresses the fact that the asymptotic saturation of the consumer essentials.

The demand function for the Tornqvist durables reflects the fact that the demand for these goods arises only from a certain (fairly high) level of income. The corresponding expression is:, where , parameters of the model, if .

As can be seen, the demand for products in this group also has an asymptotic tendency to saturate, because. For luxury formula is used, which has no tendency to saturation, and demand starts to even higher levels of income: , if .

It is easy to see that for sufficiently large values of the income .

This means that in this situation, virtually the entire increase in income is spent on luxuries.

Let's consider on an example application of functions of demand of Tornkvist.

Let's assume that income level is. At the analysis of the statistical data, such parameters for essential commodities have been established: and .

Then using the formula of Tornkvist for essential commodities: , we will receive .

For the goods of long using have been received such parameters: , , and .

Using the formula of Tornkvist for the goods of long using: , we will receive: .

For the goods of a category of luxury, have been received such parameters: , , , .

Using the formula of Tornkvist for the goods of a category of luxury: , we will receive .