Inflation and Consumer Basket

Formation of consumer basket is investigated by means of probability theory. Inflation on the consumer market can be managed by reducing its rate and lowering inflation-related risks. The approach is based on the treating the inflation risks of particular ingredients of the consumer goods basket as component of the whole complex rather than separate units. The proposed management strategy is focused on the degree of correlation between the rates of price increase of the items in the basket. Portfolio theories of Markowitz and Tobin are used.


I. INTRODUCTION
It is difficult to govern consumer basket under inflation. And it is especially difficult to make it than future prices are unknown. The risk of unpredictable price rises can significantly distract our plan of necessary purchases. Inflationary processes should necessarily be taken into account in business, state and household expenses. But this account as a rule use only predictions of inflation level and do not consider a volatile character interconnections of prices for various items in consumer basket.
There is similarity between compiling of consumer basket portfolio of securities compiling. Investor wants to maximize expected profit and minimize risk while buying securities. Ordinary customer wants not only to satisfy his necessities but to minimize expected prices and to minimize inflationary risk of jumping prices beyond some level.
Similar to portfolio of securities it is possible to govern consumer basket in order to lower expected cost and risk. In other words it is possible to reduce inflation effect an inflation risks by compiling consumer basket in proper way.
We use Markowitz and Tobin portfolio theories and consider inflationary risks of consumer basket ingredients in unity. Prices of various goods are interconnected. So there is correlation between rates of price rise of various goods. Prices on consumer market rise? But change in price of one commodity lead to change of in price of other commodities. So the risk of unexpected price jump of a whole consumer basket may be not so large even if risks of its isolated items price jump are large enough. One has to define *Address correspondence to this author at the Financial University, Moscow, Russia; Tel: +79265335744; Fax: +74992772321; E-mail: vlapopov@gmail.com proportions of ingredients in proper way. The process of diversification of consumer is realized in mathematical model. It is possible to calculate exact proportions of ingredients using mathematical formulas.

II. RANDOM CHARACTER OF PRICES
Statistical analysis of securities market made by Cowles and Working showed random character of price changing. If P k is price of asset at end the period numbered k then r k = ln P k P k !1 and R k = r 1 + r 2 +!+ r k are independent random values (Cowles, 1939), (Working, 1934). It contradicts to the contemporary opinion that prices have some rhythms, cycles, trends. The idea of random pricing was confirmed and developed in works of Kendal, Roberts and Osborn in which there are explained some statistically observed effects such as for example clustering (Kendal, 1953), (Roberts, 1959), (Osborn, 1959). Samuelson formulated hypothesis of effective market. He showed that expected price P k +1 of an asset at the end of a period k +1 equals conditional expectation of previous prices P 1 , P 2 ,…P k ; E(P k +1 / P 1 , P 2 ,…P k ) (Samuelson, 1965).
The famous Markowitz work (Markowitz, 1952) laid the foundation of the theory of securities. The theory dials with investment optimization problem under condition of uncertainty and risk. Analysis from the point of view of probability theory shows very important value of correlation between profitableness of assets. The most fruitful idea of Markowitz theory is an idea of diversification while complying securities portfolio. It explains securities market behavior and gives practical recommendation to compiling optimal portfolio. We will use securities portfolio theory to investigation of consumer market.
We consider price changes of different goods in S in ! P in ! P in P io in n periods as normally distributed random values with expectation E in = E(S in ) , standard deviation ! in = D(S in ) and correlation matrix P ijn = !(s in ,; s jn ). Price change of consumer of consumer basket is calculated by formula S n = x 1 s 1n + x 2 s 2n +!+ x k s kn , where x i is the share of item number i of a consumer basket, x 1 + x 2 +!+ x k = 1 . Expected inflation and dispersion of value S n are calculated by formulas Here ! j ,! j are standard deviations of values s in , s jn .
Let us consider mean inflation h = 1 n S n of consumer basket and mean inflation inflation of whole consumer basket in one period, and ! be standard deviation of whole consumer basket in one period. Let E(q i ) = E(s in ) = µ i be expected mean inflation of item number i . Let ! (q i ) = ! i be standard deviation inflation of item number i and !(q i , q j ) = ! ij be correlation coefficient of two random values q i and q j . Than formulas (1) can written as follows.
It is possible change level of inflation µ risk of higher inflation ! by changing price shares x i . We can find confidence interval of values h and S n with given probability. Now we let us investigate the situation than price changes of every item in consumer basket is unregulated. So for every item level of inflation is unpredicted and ! i " 0.
First of all let us consider the consumer basket which consists only of two items. In this case there is only one correlation coefficient ! = ! 12 . Les us denote µ 1 = t than µ 2 = 1 ! t . System of equations (2) will be as follows.
System (3) can be considered as definition of a function µ = f (! ) . If µ ! ±1 the curve is hyperbola with branches directed along ! . Coordinates of its vertex (µ 0 ;! min ) can be calculated by standard mathematical methods [3]. The risk ! min of corresponding consumer basket is minimal risk among risks of all possible consumer baskets. As in portfolio theory we call consumer basket with standard deviation ! = ! min minimal consumer basket. Price shares of this basket can be calculated by standard mathematical methods too [3]. On Figure 1 horizontal axe is axe of risk and vertical axe is axe of level of inflation. Point M is point of minimal risk ! min . One can solve system of equation (3) and calculate price shares x 1 and x 2 .
For given values µ 1 , µ 2 ,! 1 ,! 2 , " we can change price shares x 1 , x 2 and lessen expected level of inflation µ of the consumer basket and its risk ! . Let us consider the following model example.
Monthly forecasting of price rising for two items equal to µ 1 = 0.65% and µ 2 = 0.99% respectively. Risks of deviation from this expected level of inflation equal to ! 1 = 3.23% and ! 2 = 4.86% for first and second items respectively. Correlation forecast is equal to ! = 0.39 . Initial plan of consumption is as follows. Price share for the first ingredient of consumer basket equals x 1 = 0.7 and price share for the second ingredient of consumer basket equals x 2 = 0.3 . It easy to calculate by formulas (3) that µ = 0.75%, ! = 3.12% .
Annual inflation rate is equal to µ y = 12µ = 9% . Risk of annual rate deviation is equal to ! y = ! 12 = 3.12 " 12 = 10.81 . So standard annual deviation is equal to ! y = 9 "10.81 100 = 0.97% . According to (3) H CMy ! (6.08%;11.92%) . σ rule" we have confidence interval with confidence probability 0.9973 Structure of consumer would be better (cheaper and less risky) if parameters of consumer basket correspond to point Q min which lies on arc NLQ min KF (this arc is determined by values of µ and ! ). But to point Q min corresponds to another structure of consumption. From formulas (4)  H CMy ! (5.86%;11.42%).
It would even better to choose point L where with the same risk as in point K inflation is less than in point Q min . But proportions of ingredients corresponding to that point are such that consumption of second ingredient almost disappears, and that is no good for buyer. If we change ingredients proportions in order to go along the way FTM min DN (this is more visually demonstrative in comparison with the way FKQ min lM ) from point M min to D than we will reduce inflation but increase risk. One has to calculate inflations and risks for various structures of consumer basket in order to choose optimal proportion of ingredients.

III. PRICE CHANGES OF SOME ITEMS IN CONSUMER BASKET IS PREDERTENENED
Above we discussed situation where the theory of Markowitz portfolio can be applied. I. e. we considered positively determined correlation matrixes (! ij ) . This supposition often occurs in securities portfolio and can be applied for certain types of consumer basket.
Generally correlation matrix is degenerated. For many goods inflation is absent or strictly regulated i. e. have zero deviation (σ = 0). Let us unite this goods and services as one ingredient and define it as component of consumer basket without risk. In portfolio theory a model with component without risk is well studied and called Tobin portfolio. (Tobin, 1965).
Let us divide consumer basket in two parts. First part with share x 0 has no risk and its expected inflation equals µ 0 . Second part which is called "marketing part" consists of n components with shares x 1 , x 2 ,…, x n inflationary expectations µ 1 , µ 2 ,…, µ n risks ! 1 ,! 2 ,…,! n and correlation matrix (! ij ) . Mean inflation and risk of a consumer basket as a whole are calculated by following formulas.
We want to maximize profit µ while investigating securities portfolio. But we are minimizing inflation µ while investigating consumer basket. We use the equation of minimal boundary from Tobin's portfolio The securities portfolio profit of risky security is greater than profit security without risk, µ i ! µ 0 . For consumer basket it is often vice versa and what's more we can guess that µ i ! µ 0 . So we have an equation of minimal boundary ! = µ 0 " µ d .
The risky part of securities basket X = (x 1 ; share of ingredient without risk equals x 0 = 1 ! ! I T ! X.

IV. CONCLUSIONS
A Markowitz and Tobin theory allows choosing the consumer basket of minimal inflation risk. But such basket has purely theoretical application as it demands the serious change customer's habitual consumption structure. So given above formulas are aimed to show the possibility of sufficient changing of consumer's consumption structure by substituting items of arbitrary chosen basket for the higher quality analog by reducing expected price and risk of price jumps.
But the serious change in consumer's consumption structure can be possible only by state regulation. Such conclusion does not contradict to classical Keynes's evaluations of state regulations in economy. State should implement tax, expenditures, and monetary regulations to eliminate any economic instability. So as for Keynes state should act in the situations where private business fails. And the regulations of inflation risks and price increases are one of the state priorities too.
We hope that idea of using the theory of securities portfolio to investigate consumer basket formation is fruitful and deserves further development. It is possible to take into consideration utility function in order to compile consumer basket precisely. On the other hand changing of quantity of goods in consumer market should be restricted. But choosing of utility function and restrictions on amount of items should be made differently for different consumer baskets. And analysis of special consumer baskets is the subject of other investigations.