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Mediterranean Journal of Social Sciences
ISSN 2039-2117 (online)
ISSN 2039-9340 (print)
MCSER Publishing, Rome-Italy
Vol 5 No 24
November 2014
Income Distribution and Market Demand: The Case of Heterogeneous Preferences
Ibragimov M.
Tufetulov A.M.
Kazan Federal University, Institute of Management, Economics and Finance, Kazan, 420008, Russia
Doi:10.5901/mjss.2014.v5n24p185
Abstract
In this note, we obtain sufficient conditions under which changes in income inequality lead to an increase or decrease in the
market demand elasticities in the case of heterogeneous preferences among the consumers. In this paper, we applied
majorization theory to study dependence of market demand elasticity on the inequality in income distribution among the
consumers. In this note, we extend the results obtained to the case where consumers’ preferences are heterogeneous and the
condition on equality of individual demand functions does not necessarily hold. This case is more realistic because consumers’
preferences are affected by a variety of different factors.
Keywords: Income inequality, market demand, elasticity, heterogeneous preferences
In recent years, a number of studies have focused on modeling income inequality using majorization relation (see, e.g.,
Marshall and Olkin [6]) and applications of the latter concept to the problems in economics. The approach to the analysis
of income inequality based on majorization which dates back to Lorenz [5] has been used, among others, by Atkinson [1],
Dasgupta, Sen and Starrett [2], Shorrocks [8] and, more recently, Saposnik [7]. Using related concepts and methods,
Lambert and Pfahler [4] presented an analysis of the effects of income (re-)distribution on the market demand for a good
or service.
In [3], the authors applied majorization theory to study dependence of market demand elasticity on the inequality in
income distribution among the consumers. However, in [3] it is assumed that consumers’ preferences are the same for
given prices on goods independently of their income levels. In this note, we extend the results obtained in [3] to the case
where consumers’ preferences are heterogeneous and the condition on equality of individual demand functions does not
necessarily hold. This case is more realistic because consumers’ preferences are affected by a variety of different factors.
Let there be Ʉ consumers and M goods in an economy. Denote by ߮௠௞ ሺܲǡ ‫ܫ‬௞ ሻ the function of the kth consumer’s
demand on the mth good, by ࡵ ൌ ሺ‫ܫ‬ଵ ǡ ǥ ǡ ‫ܫ‬௄ ሻ the vector of incomes of the consumers and by ࡼ ൌ ሺ‫݌‬ଵ ǡ ǥ ǡ ‫݌‬ெ ሻ the
vector of prices on goods.
‫)ݐ݁ܮ‬௠ ሺࡼǡ ࡵሻ ൌ σ௄௞ୀଵ ߮௠௞ ሺܲǡ ‫ܫ‬௞ ሻ be the function of market (aggregate) demand on good m and ݈݁‫݁ݐ‬௠ ሺࡵሻ ൌ
߲ ݈‫) ݃݋‬௠ ሺࡼǡ ࡵሻȀ߲ ݈‫݌ ݃݋‬௠ stand for its own-price elasticity. Denote by ܵ௠௞ ‫ ܀ ؿ‬௠ାଵ the domain of definition of the
function ߮௠௞ ሺܲǡ ‫ܫ‬௞ ሻ and by ܵ௠ ൌ ሼሺࡼǡ ࡵሻ ൌ ሺࡼǡ ‫ܫ‬ଵ ǡ Ǥ Ǥ Ǥ ǡ ‫ܫ‬௄ ሻ ‫ࡾ א‬ெା௄ ǡ ሺࡼǡ ‫ܫ‬௞ ሻ ‫ܵ א‬௠௞ ǡ ݇ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ‫ܭ‬ሽ the domain of definition
of the function )௠ ሺࡼǡ ࡵሻǡ ݉ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ‫ܯ‬.
According to the idea going back to Lorenz [5] (see Marshall and Olkin [6]), a vector ࡵሺଵሻ ൌ ሺ‫ܫ‬ଵሺଵሻ ǡ ǥ ǡ ‫ܫ‬௄ሺଵሻ ሻ
represents a more uniform distribution of the total income Y among Ʉ consumers than a vector ࡵሺଶሻ ൌ ሺ‫ܫ‬ଵሺଶሻ ǡ ǥ ǡ ‫ܫ‬௄ሺଶሻ ሻ if
ሺ௝ሻ
ሺଵሻ
ሺଶሻ
ሺଵሻ
ሺଶሻ
σ௟௜ୀଵ ‫ܫ‬ሾ௜ሿ
൑ σ௟௜ୀଵ ‫ܫ‬ሾ௜ሿ , l=1,…,K-1, and σ௄௜ୀଵ ‫ܫ‬ሾ௜ሿ ൌ σ௄௜ୀଵ ‫ܫ‬ሾ௜ሿ ൌ ܻ , where ‫ܫ‬௜ , j = 1, 2, are the income levels of the ith
ሺ௝ሻ
ሺ௝ሻ
ሺ௝ሻ
consumer and ‫ܫ‬ሾଵሿ ൒ ‫ܫ‬ሾଶሿ ൒ ‫ ڮ‬൒ ‫ܫ‬ሾ௄ሿ denote the components of the vectors I(j), j = 1,2, in decreasing order (if the
above conditions hold, it is said that the vector ࡵሺଵሻ ൌ ሺ‫ܫ‬ଵሺଵሻ ǡ ǥ ǡ ‫ܫ‬௄ሺଵሻ ሻis majorized by ࡵሺଶሻ ൌ ሺ‫ܫ‬ଵሺଶሻ ǡ ǥ ǡ ‫ܫ‬௄ሺଶሻ ሻǡ ‫ݎݓ‬itten ࡵሺଵሻ ‫ط‬
ࡵሺଶሻ Ǥ
A function f (I) is called Schur-convex (resp., Schur-concave) in I if ሺࡵሺଵሻ ‫ࡵ ط‬ሺଶሻ ሻ ֜ ቀ݂൫ࡵሺଵሻ ൯ ൑ ݂൫ࡵሺଶሻ ൯ቁ (resp.
ሺࡵሺଵሻ ‫ࡵ ط‬ሺଶሻ ሻ ֜ ቀ݂൫ࡵሺଵሻ ൯ ൒ ݂൫ࡵሺଶሻ ൯ቁሻǤ
Theorem 1. (i) Let the individual demand functions ߮௠௞ ሺܲǡ ‫ܫ‬௞ ሻ be twice continuously differentiable and let, for all
ሺࡼǡ ࡵሻ ‫ܵ א‬௠ such that ‫ܫ‬௥ ൑ ‫ܫ‬௦ , the following conditions hold:
డఝ೘ೝ ሺ௉ǡூೝ ሻ
డூೝ
൑
డఝ೘ೝ ሺ௉ǡூೞ ሻ
డூೞ
ǡ ሺͳ)
185
Mediterranean Journal of Social Sciences
ISSN 2039-2117 (online)
ISSN 2039-9340 (print)
డమ ఝ೘ೝ ሺ௉ǡூೝ ሻ
൑
డ௣೘ డூೝ
Vol 5 No 24
November 2014
MCSER Publishing, Rome-Italy
డమ ఝ೘ೝ ሺ௉ǡூೞ ሻ
, (2)
డ௣೘ డூೞ
where pm is the price of the mth good in consideration. Then the absolute value of the elasticity |em(I)| is Schur –
concave in I on the set Sm. That is, the more non-uniform is the distribution of the total income among consumers in the
economy, the smaller is the elasticity of the aggregate demand on the considered good by the absolute value.
(ii) If in conditions (1) and (2) the inequality sign ≤ is replaced by ≥, then the absolute value of the elasticity |em(I)|
is Schur-convex in I on Sm. That is, the more non-uniform is the distribution of the total income among the consumers,
the larger is the elasticity of the aggregate demand on the considered good by the absolute value.
Proof. (i) Let ݃௠ ሺܲǡ ࡵሻ ൌ ߲Ȱ௠ ሺʟǡ ࡵሻȀ߲˓௠ ൌ σ௄
௞ୀଵ ߲߶௠௞ ሺȬǡ ࢩ௞ ሻȀ߲˓௠ be the deriva tive of the function of
aggregate demand on the mth good with respect to its price. If conditions (1) and (2) are satisfied, then the following
inequalities hold:
െ
డ஍೘ ሺ௉ǡࡵሻ
డ σ಼
ೖసభ ம೘ೖ ሺ௉ǡூೖ ሻ
డூೞ
డூೝ
െ
డ௚೘ ሺ௉ǡࢩሻ
డమ ஍೘ೝ ሺ௉ǡࢩሻ
డூೞ
డ௣೘ డூೝ
డ஍೘ ሺ௉ǡࢩሻ
ሺ‫ܫ‬௥ െ ‫ܫ‬௦ ሻ ቀ
డூೝ
and
డ௚೘ ሺ௉ǡࢩሻ
ሺ‫ܫ‬௥ െ ‫ܫ‬௦ ሻ ቀ
డூೝ
ቁ ൌ ሺ‫ܫ‬௥ െ ‫ܫ‬௦ ሻ ቀ
ቁ ൌ ሺ‫ܫ‬௥ െ ‫ܫ‬௦ ሻ ቀ
െ
െ
డ σ಼
ೖసభ ம೘ೖ ሺ௉ǡூೖ ሻ
డ஍೘ೝ ሺ௉ǡ௶ೝ ሻ
డூೞ
డூೝ
ቁ ൌ ሺ‫ܫ‬௥ െ ‫ܫ‬௦ ሻ ቀ
െ
డ஍೘ೞ ሺ௉ǡூೞ ሻ
డூೞ
ቁ
డమ ஍೘ೞ ሺ௉ǡࢩሻ
డ௣೘ డூೞ
ቁ൒Ͳ
In addition, from the definition of the functions Ȱ௠ ሺܲǡ ࢩሻ and ݃௠ ሺܲǡ ࢩሻit follows that they are symmetric on the
set Sm, that is,
ሺଵሻ
ሺଵሻ
ሺଵሻ
ሺଵሻ
ʣ௠ ሺܲǡ ‫ܫ‬ଵ ǡ Ǥ Ǥ Ǥ ‫ܫ‬௄ ሻ ൌ ʣ௠ ൫ܲǡ ‫ܫ‬గሺଵሻ ǡ Ǥ Ǥ Ǥ ‫ܫ‬గሺ௄ሻ ൯
ሺଵሻ
ሺଵሻ
ሺଵሻ
ሺଵሻ
݃௠ ሺܲǡ ‫ܫ‬ଵ ǡ Ǥ Ǥ Ǥ ‫ܫ‬௄ ሻ ൌ ݃௠ ൫ܲǡ ‫ܫ‬గሺଵሻ ǡ Ǥ Ǥ Ǥ ‫ܫ‬గሺ௄ሻ ൯
for all permutations ߨ ‫ ׷‬ሼͳǡ ǥ ǡ ‫ܭ‬ሽ ՜ ሼͳǡ ǥ ‫ܭ‬ሽ of the set ሼͳǡ ǥ ‫ܭ‬ሽ
Consequently, according to Theorem 3.A.4 in [6], the functions ߔ௠ ሺܲǡ ࢩሻand ݃௠ ሺܲǡ ࢩሻܽ‫ݎ‬e Schur-convex in ࢩ, that
is, ࢩሺଵሻ ‫ࢩ ط‬ሺଶሻ implies ߔ௠ ൫ܲǡ ࢩሺଵሻ ൯ ൑ ߔ௠ ൫ܲǡ ࢩሺଶሻ ൯ and ݃௠ ൫ܲǡ ࢩሺଵሻ ൯ ൑ ݃௠ ൫ܲǡ ࢩሺଶሻ ൯Ǥ
Since the function ݃௠ ሺܲǡ ࢩሻ is non-positive, from ࢩሺଵሻ ‫ࢩ ط‬ሺଶሻ it thus follows that
௚೘ ൫௉ǡࢩሺభሻ ൯
஍೘
൫௉ǡࢩሺభሻ ൯
௚೘ ൫௉ǡࢩሺమሻ ൯
൑
஍೘ ൫௉ǡࢩሺమሻ ൯
or, equivalents
݁௠ ሺࢩሺଵሻ ሻ ൌ
డ஍೘ ൫௉ǡࢩሺభሻ ൯
ή
డ௣೘
ሺଶሻ
௣೘
൫௉ǡࢩሺభሻ ൯
஍೘
൑
డ஍೘ ൫௉ǡࢩሺమሻ ൯
డ௣೘
ή
௣೘
஍೘ ൫௉ǡࢩሺమሻ ൯
ሺଵሻ
ൌ ݁௠ ሺࢩሺଶሻ ሻ
That is, ࢩሺଵሻ ‫݅ ࢩ ط‬mplies ห݁௠ ሺࢩሺଶሻ ሻห ൑ ห݁௠ ሺࢩ ሻห, as claimed.
(ii) If in conditions (1) and (2) the inequality sign ൑ is replaced by ൒, then the functions Ȱ௠ ሺܲǡ ࢩሻ and ݃௠ ሺܲǡ ࢩሻ
are Schur-concave in I, that is, ࢩሺଵሻ ‫ࢩ ط‬ሺଶሻ implies Ȱ௠ ൫ܲǡ ࢩሺଵሻ ൯ ൒ Ȱ௠ ൫ܲǡ ࢩሺଶሻ ൯ and ݃௠ ൫ܲǡ ࢩሺଵሻ ൯ ൒ ݃௠ ൫ܲǡ ࢩሺଶሻ ൯ The
rest of the arguments is completely similar to part (i).
Example 1. Suppose that the function of market demand for good m has the CES form: Ȱ௠ ሺߏǡ ࢩሻ ൌ
‫ܭ‬
σ݅ൌͳ ߶൫ܲǡ ߙሾ݅ሿ ǡ ‫ܫ‬ሾ݅ሿ ൯, where ‫ܫ‬ሾͳሿ ൒ ‫ܫ‬ሾʹሿ ൒Ǥ Ǥ Ǥ ൒ ‫ܫ‬ሾ௄ሿ ǡ ͳ ൒ ߙሾଵሿ ൒ ߙሾଶሿ ൒Ǥ Ǥ Ǥ ൒ ߙሾ௄ሿ ൐ ͳȀʹǡ ߶ሺܲǡ ߙǡ ‫ܫ‬ሻ ൌ ߰ሺܲǡ ߙǡ ሻ‫ ܫ‬and
߰ሺܲǡ ߙǡ ሻ ൌ
షభȀሺభషഀሻ
௣೘
షഀȀሺభషഀሻ
ಾ
σೕసభ ௣ೕ
are the factors at the individual CES utility functions (that is, the consumers with a higher income I have a higher
elasticity of substitution െͳȀሺͳ െ ߙሻ). We have
డథሺ௉ǡఈೝ ǡூೝ ሻ
డூೝ
డథሺ௉ǡఈೝ ǡூೝ ሻ
డ௣೘ డூೝ
ൌ
ଵ
షഀೝ Ȁሺభషഀೝ ሻ
௣೘ ቀσಾ
ೕసభ൫௣ೕ Τ௣೘ ൯
ቁ
షഀೝ Ȁሺభషഀೝ ሻ
ൌ
ቀఈೝ ିσಾ
ೕసభ൫௣ೕ Ȁ௣೘ ൯
ቁ
మ ሺଵିఈ ሻቀσಾ ൫௣ Ȁ௣ ൯షഀೝ Ȁሺభషഀೝ ሻ ቁ
௣೘
ೝ
ೕసభ ೕ ೘
ିఈ Ȁሺଵିఈ ሻ ିଵ
ೝ
ೝ
Simce the function ቀσெ
ቁ is increasing in ߙ ‫ א‬ሺͲǡͳሻ for ‫݌‬௝ ൒ ‫݌‬௠ , ݆ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ‫ܯ‬, ݆ ് ݉ , we
௝ୀଵ൫‫݌‬௝ Ȁ‫݌‬௠ ൯
have that ʣ௠ ሺʟǡ ࡵሻ satisfies conditions (1) if ‫݌‬௝ ൒ ‫݌‬௠ , ݆ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ‫ܯ‬, ݆ ് ݉. Further, since the function ݄ሺ‫ݔ‬ሻ ൌ
ߙ‫ ݔ‬ଶ െ ‫ ݔ‬is increasing in x for ‫ ݔ‬൒ ͳȀሺʹߙሻ , we get that ʣ௠ ሺʟǡ ࡵሻ satisfies conditions (2) if
‫݌‬௝ Ȁ‫݌‬௠ ൒ ݉ܽ‫ݔ‬௜ୀଵǡǥǡ௄ ቀ
ெିଵ
ሺଵିఈ೔ ሻΤఈ೔
ቁ
ଶఈ೔ ିଵ
for ݆ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ‫ ܯ‬, ݆ ് ݉. From part (i) of Theorem 1 we obtain that, in this
domain, an increase in income inequality leads to a decrease in the absolute value of the market demand elasticity.
Similarly, in the above domain, the market demand function ʣ௠ ሺʟǡ ࡵሻ ൌ σ௄௜ୀଵ ߶൫ܲǡ ߙሺ௜ሻ ǡ ‫ܫ‬ሾ௝ሿ ൯ , where ߙሺଵሻ ൑ ߙሺଶሻ ൑
‫ ڮ‬൑ ߙሺ௄ሻ and ‫ܫ‬ሾଵሿ ൒ ‫ܫ‬ሾଶሿ ൒ ‫ ڮ‬൒ ‫ܫ‬ሾ௄ሿ are ordered in the opposite ways, satisfies conditions (1) and (2) with the inequality
signs ൑ replaced by ൒. From part (ii) of Theorem 1 we conclude that, in this case, an increase in income inequality leads
to an increase in the absolute value of the market demand elasticity.
Example 2. Suppose that the function of market demand for good m has the form
186
Mediterranean Journal of Social Sciences
ISSN 2039-2117 (online)
ISSN 2039-9340 (print)
MCSER Publishing, Rome-Italy
Ȱ௠ ሺ‫݌‬ǡ ࢩሻ ൌ σ௄௜ୀଵ ߶൫‫݌‬ǡ ߙ௜ ǡ ߚ௜ ǡ ‫ܫ‬ሾ௜ሿ ൯ , where ߶ሺ‫݌‬ǡ ߙǡ ߚǡ ‫ܫ‬ሻ ൌ
ఈூ
൫ூାఉ೛ ൯
Vol 5 No 24
November 2014
is a typical function on goods of first
necessity,ߙ௜ ǡ ߚ௜ ൐ Ͳ, ݅ ൌ ͳǡ Ǥ Ǥ Ǥ ǡ ‫ܭ‬, are some constants and, as in Example 1, ‫ܫ‬ሾଵሿ ൒ ‫ܫ‬ሾଶଵሿ ൒Ǥ Ǥ Ǥ ൒ ‫ܫ‬ሾ௄ଵሿ . It is not difficult
to check that conditions (1) and (2) of part (i) of Theorem 1 are satisfied if and only if, for ‫ ݎ‬൒ ‫ݏ‬,
ఈೝ ఉೝ
ఈೞ ఉೞ
൑
మ
మ
(3)
൫ூሾೝሿ ାఉೝ ௣൯
൫ூሾೞሿ ାఉೞ ௣൯
ఈೝ ఉೝ ൫ூሾೝሿ ିఉೝ ௣൯
ఈೞ ఉೞ ൫ூሾೞሿ ିఉೞ ௣൯
య
൫ூሾೝሿ ାఉೝ ௣൯
൑
య
൫ூሾೞሿ ାఉೞ ௣൯
(4)
Let ‫ ݎ‬൒ ‫ݏ‬. Assume that the vector ࡵ ൌ ሺͲǡ Ͳǡ Ǥ Ǥ Ǥ ǡ Ͳሻ belongs to the domain of definition of Ȱ௠ ሺ‫݌‬ǡ ࢩሻ. Suppose that
conditions (1) and (2) of Theorem 1 are satisfied. Then from inequalities (3) and (4) for ࡵ ൌ ሺͲǡ Ͳǡ Ǥ Ǥ Ǥ ǡ Ͳሻ it follows that
ߙ௥ Τߙ௦ ൌ ߚ௥ Τߚ௦ (5)
It is easy to see that condition (3) is thus equivalent to ߚ௥ Ȁ൫‫ܫ‬ሾ௥ሿ ൅ ߚ௥ ‫݌‬൯ ൑ ߚ௦ Ȁ൫‫ܫ‬ሾ௦ሿ ൅ ߚ௦ ‫݌‬൯ or ‫ܫ‬ሾ௥ሿ Ȁߚ௥ ൒ ‫ܫ‬ሾ௦ሿ ൅ ߚ௦ Ȁߚ௦ Ǥ
Since ‫ܫ‬ሾ௥ሿ ൑ ‫ܫ‬ሾ௦ሿ we conclude that, for conditions (3) and (4) to be satisfied it is necessary that (5) holds for all ‫ ݎ‬൒ ‫ݏ‬
and, in addition, for all ‫ ݎ‬൒ ‫ݏ‬,
ߚ௥ ൑ ߚ௦ ǡ ߙ௥ ൑ ߙ௦ (6)
Suppose that the satiation level for good m is the same for all the consumers, that is, for ‫ ݌‬ൌ Ͳ and all r, s,
߶ሺͲǡ ߙ௥ ǡ ߚ௥ ǡ ‫ܫ‬௥ ሻ ൌ ߶ሺͲǡ ߙ௦ ǡ ߚ௦ ǡ ‫ܫ‬௦ ሻ. Then from the definition of the individual demand functions ߶ and (5) it follows that
ߙ௥ ൌ ߙ௦ and ߚ௥ ൌ ߚ௦ for all r, s. Since, as is easy to see, from the above analysis it follows that inequalities (6) are
strict for I[r] < I[s] if conditions (1) and (2) are satisfied, we conclude that part (i) of Theorem 1 cannot hold.
As above, we get that part (ii) of Theorem 1 holds if and only if (3) and (4) are satisfied with the inequality sign ൑
replaced by ൒. For I[r] = I[s] = 0 this implies conditions (5). Assuming that the satiation level for good m is the same for all
the consumers, we get that, as above, ߙ௥ ൌ ߙ௦ and ߚ௥ ൌ ߚ௦ for all r, s. Thus, it is easy to see that part (i) of Theorem 1
holds if and only if, for all ‫ ݎ‬൒ ‫ݏ‬,
ଷ
ଷ
൫‫ܫ‬ሾ௥ሿ െ ߚ‫݌‬൯ൗ൫‫ܫ‬ሾ௥ሿ ൅ ߚ‫݌‬൯ ൒ ൫‫ܫ‬ሾ௦ሿ െ ߚ‫݌‬൯ൗ൫‫ܫ‬ሾ௦ሿ ൅ ߚ‫݌‬൯ (7)
where ߚ ൌ ߚ௥ ൌ ߚ௦ . Similar to Example 1 in [3], it is not difficult to check that conditions (7) are satisfied if ൫‫ܫ‬ሾ௄ሿ ൒
ʹߚ‫݌‬൯, that is if the income levels of all the consumers are not less than ʹߚ‫݌‬.
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