Abstract: Following a brief description of demand function in econometrics, the fuzzy consumption function theory is introduced that characterizes the dynamically behaviors of market demand and fluctuation. Two real-world examples are included to show the efficacy of this new theory and the forecast module built into DataX™.

I.    Brief Background on Demand Theory

• Consumption Demand - it is the amount of consumption on goods (purchase amount). In general, it is related to the objective factors of commodities (such as physical, chemical and artistic characters) and the subjective value of the consumer (preference, personal habits, health conditions, etc.).  Demand is affected by the total spending capability and population of a customer group, as well as the consumer prices.

• Consumption Demand Function - The behavioral characteristics of financial market can be represented by the concept of consumption demand, and the consumption demand function is the mathematical description of this concept.  In addition,  consumption demand can be determined by the following types of factors:

  • objective factors - the physical, chemical, biological and artistic properties of goods;

  • subjective factors - consumer's interest, preference and psychological state;

  • Group factors - population and wealth of the consumers (consumer group);

  • Comparative factors - the ratio of prices of different goods, ratio of different preference, and ratio of subjective values on different goods (comparisons of different consumption can directly affect the consumption demand);

  • Fluctuation factors - wealth, population and price fluctuations

II.    Fuzzy Consumption Demand Functions

• In studying advanced methodology for the analysis of consumption demand, AI researchers at Zaptron Systems have developed technology and software tool based on the so called fuzzy demand functions. They can model and describe the market demand, or consumption demand, on various commodities or services, based on consumption data available.  The fuzzy demand functions discussed here are developed based on the fuzzy consumption utility function theory developed by Zaptron scientists.

• In fact, consumption demand is a fuzzy logic concept. Mathematically, the fuzzy demand function is a more accurate measure on the consumption demand, compared against a traditional (non-fuzzy) demand function.  It can describe relationships among wealth, price, consumption composition (decomposition), preference and subjective measure on commodity or service values.

• Mathematical model of fuzzy consumption demand functions -  Given prices P1, P2, ..., for different goods and total spending amount C of consumer group, the fuzzy demand  function for the i-th commodity can be expressed as Di = Di(P1, P2, ..., C). The form of Di(.) is complex and will not be given here. However, such a Di(.)  must satisfy the following conditions:

  • Non-negative - Di > 0.

  • Additive - the sum of demands on all goods is equal to the total spending amount.

  • Linear - Di(kP1, kP2, ..., kC) = Di(P1, P2, ..., C), k is a constant. This means that when all prices Pi increase by k times and the total spending increase by k times, the each individual demand Di remains unchanged.

  • Monotonic - Di is monotonic w.r.t. total spending C, total population P, and price Pi

  • Substitutive - substitution by i-th commodity due to price change in j-th commodity is equal to the substitution by j-th commodity due to price change in i-th commodity. (People tend to substitute one commodity by another when they are not satisfied)

• Computation of Fuzzy Demand Functions and Parameters - Based on the Maximum Utility Principle, they can be computed by solving a set of complex mathematical equations.  Zaptron's DataX™ software suite has special modules to compute them.

III.    Market Demand Forecast -- A Case Study using DataX™

• Problem background - Available (but not given in this article) are the annual data, from one rural area in Asia, of (1) 27 consumer prices, (2) total spending amount of the consumer group and (3) the spending amount on each individual commodity for 8 years (1977-1984).  The objectives are

  • Use the data from 1977 to 1984 to forecast the market demand on these 37 commodities for 1985 to 2000.

  • To perform demand fluctuation analysis by changing the price of one commodity.

• Computation of Demand Functions and Parameters Estimation - All computations of this example were conducted by using Zaptron's DataX™.  In computation, it was assumed that the annual spending capability rises by 8% and annual population increases by 0.9%.

• Results -  see Table1- and Table-2 below.

  • Table-1 lists the demand forecast for the 27 commodities for the years 1985, 1990, 1995 and 2000.

  • Table-2 shows the impact of fluctuation of one commodity (cereals - the basic food) price on the market demand on all 27 commodities.

• Observations from Table-1

  • The demand on high-priced (luxury) commodities will continuously increase from 1985 to 2000 . For example, demand on TVs increases by 5.4 times, refrigerators by 5.8 times, washing machines by 5.9.  Look at the demand on foods, beef increases by 4.1 times, lamb by 4.2 times.  Demand on clothing increase by 4 times. 

  • However, the demand on basic consumer goods will experience only slight changes - cereals by 0.486 times and coal by 0.958 times. On the other hand, demand on FM radios will decrease by 67.3% from 1985 to 2000. This means, FM radios will be substituted by recorders and TV.

• Observations from Table-2

  • In this case, the price of cereals (the basic food of the consumer group under study) has been increased by 5%, while the prices of other 26 commodities, population and spending capability remain unchanged. Table-2 shows the new demand on these 27 commodities caused by this fluctuation in cereal price. 

  • Except for AM radios, which will see an increased demand by 0.1527% (explained below), the demand on all other commodities will be reduced by 0.0134% (salt) to 1.8628% (cereals).  The demand on cereals will see largest reduction (1.8628%), while demand on luxury goods (TV, refrigerators, washing machines, fans, recorders) will have larger reduction (1.18% to 1.34%) than on other basic consumer goods (clothing, food).

  • Explanation - Raise in commodity price is equivalent to a reduction in the spending capability of a consumer group.  Thus in case of price raise, people tend to cut down spending on luxury goods (TV, washing machines, etc.) more than on basic goods (salt, food, clothing).

  • Why is there an increase in the demand on AM radios while demand on all other goods is reduced? - This is in contradictory with common sense. This is so because AM radios belong to the category of obsolete goods, to which a different analytical methodology (Fuzzy Demand Function for Obsolete Goods) applies.