Offer and Demand

In the previous examples (Grazing Fields, Food, Prey and Predator and Wildlife Reserve) we have used fuzzy cognitive maps to model ecosystems. In this example we are going to use a fuzzy cognitive map to model an economic system.

1. Description of the problem

The arrival of the world wide web in the mid nineties was an enormous boost for sales of personal computers aimed at the home user market. The high level of sales lasted for a few years, but then it came to an halt. Analysts of the computer industry have detected that the sales of computers for home users have been steady for a while. In the computer vendors' headquarters the alarms have gone off. The business managers of a main computer vendor have got together to analyse the situation and to devise a way forward. The question is: How can they provoke a raise in the demand of home PCs? We could try and see whether a fuzzy cognitive map can offer some answers.

2. Factors involved

The first step in the construction of the fuzzy cognitive map is to identify the factors related to the problem at hand. We can assume that there are (at least) six factors that may play a role:

Demand

The focus of this whole exercise is on the demand of PCs from home users. We want to provoke a raise in the demand. Therefore we should include this parameter in the fuzzy cognitive map.

Price

Usually, the demand of a product or service is somehow related to the price the customer has to pay for it. We should add the price factor to the picture as well.

Publicity

Publicity and marketing affect the level of demand of a product (or at least that's their purpose!).

Production

The level of production may affect, or be affected by, the demand.

Market saturation

The market saturation is the percentage of potential customers that have already bought the product. The demand is related to the saturation of the market.

Need of replacement

Every now and then things break down, and we have to replace or fix them. This affects the demand.

3. Relationships between factors

Once we have identified the factors that will be present in the fuzzy cognitive map, we have to establish how they are related to one another.

The price of the product affects the demand in a negative way: the higher the price, the lower the demand.

The level of production affects the price. If the production is very high, the final price per item will be lower. This is one of the benefits of mass production. The contrary is true as well: the lower the production, the higher the price per unit. Therefore there is a negative relationship of causality between production and price.

Publicity and marketing affect the level of demand in a positive way: more publicity, more demand.

The level of demand has an effect over three other factors. The demand affects the price positively: high demand makes prices raise. The demand leads to more market saturation as customers purchase the product (positive relationship). Finally, the demand affects the level of production. If the demand is high, the vendors are encouraged to raise production (positive relationship).

The degree of market saturation affects the demand. Once a prospect customer has bought the product, he is not any more a prospect customer because he is not going to buy the product again. Market saturation causes a drop in the demand (negative relationship).

The statement above is only half true: a customer that has bought a product is not going to buy another product... as long as the product does not break down. Things break down, or wear out, or get obsolete, and we have to replace them. A home PC user eventually has to upgrade his system to keep pace with technology. More market saturation generates more need of replacements (positive relationship) and those who need their product replaced do not belong to the saturated part of the market any more (negative relationship).

4. The fuzzy cognitive map

5. Levels of factors

In the description of the problem we said that the level of sales (demand) of computers is not raising any more. It is worth highlighting here the difference between the amount of sales per unit of time (month or year) and the increment in the amount of sales per unit of time. The former is analogous to speed, and the latter is analogous to acceleration or change of speed. The problem of the computer vendor is not that they are not able to sell computers any more. They do sell computers; the amount of sales per unit of time has not dropped (deacceleration). The problem is that the sales are at a constant level, they are not increasing (accelerating), which is what the computer vendors want in order to earn more money from their investments.

Therefore, we are dealing with a steady state of the system. The level of demand is constant, and certainly constant are the market saturation and the need of replacement. We can assume that the levels of the other factors (price, production and publicity) are constant as well.

The levels of the factors of the system are measured using different units: price is measured in $ per item, market saturation is measured in percentage, production is measured in items/month, publicity is measured in $/month spent, etc. Every factor of the system has its own level measured in its own units. Instead, in the fuzzy cognitive map we measure the levels of the factors with a dimensionless parameter that ranges from 0 to 100. We can arbitrarily establish a correspondence between the level of each factor in the actual system in the initial state and a level of 50 out of 100 in the fuzzy cognitive map. For example, if the average price of a PC in real life is $1200 then that is represented by level 50 in our fuzzy cognitive map, and if the market saturation is 30% then that is represented by level 50 in our map.

6. Intensities of effect

All the intensities of effect are, for the moment, left at value 50.

7. Effect of price on demand

Computer vendors can only directly modify the levels of three factors in the system: price, production and publicity. They cannot change the level of demand, market saturation nor need of replacement directly, because those are factors that are not under their control, directly at least. Therefore, their objective is to find a combination of levels of price, production and publicity that triggers a raise in the level of demand.

First, we modify the level of price only, leaving the levels of production and publicity at 50, in order to see what the effect of price is on the system.

We can run simulations for the price leves of 0, 20, 40, 60, 80 and 100, for example. All these simulations are convergent. The final state of the system for each of them is shown in the table below:

We can notice two kinds of final states. One of them is reached in the simulations for initial prices of 0 and 20. In this final state the levels of production, demand market saturation and need of replacement are quite high (this is precisely what the computers vendors want to achieve). The price of computers, on the other hand, is rather low. The other kind of final state is reached in the simulations for initial prices of 40, 60, 80 and 100. In this final state the production, demand, need of replacement and market saturation are rather low, and the price is a bit high. This is certainly not what the computer vendors are interested in.

The number of iterations that it takes to reach convergece is not the same for the six initial conditions that we have tried. From price 0 to price 100, it takes 27, 28, 82, 55, 28 and 27 iterations to reach convergence, respectively. We may suspect that, although the final states are the same, they are reached following different paths of evolution. It would be interesting to plot the levels of each factor as a function of the iteration number and see how the system evolves depending on the initial level of price.

8. Effect of publicity on demand

Another factor that is in computer vendors' hands to modify is the effort and money they spend (or invest) in publicity. To determine the effect of the level of publicity on the other factors we run simulations for initial levels of publicity of 0, 20, 40, 60, 80 and 100. The levels of all the other factors are left at 50.

In this case all the simulations run to convergence as well, but the final states are different for different initial levels of publicity. The results are shown in the table:

In the sets of final states it seems that the levels of all factors are directly proportional to the level of publicity: more publicity, more everything. There is however something that is worth highlighting. In the final state of the simulation for publicity level at 60, the levels of production, demand, need of replacement and market saturation are significantly lower that in the initial state, and the price is higher. The fuzzy cognitive map suggest that if the vendors raise the level of publicity moderately, they are shooting themselves on the foot. According to our fuzzy cognitive map, if they want to raise the level of publicity they have to do it significantly, not moderately, if they want to improve the state of their industry.

9. Effect of production on demand

The third factor that computer vendors can modify is the level of production. The table below shows the final states of simulations for initial production levels of 0, 20, 40, 60, 80 and 100.

The simulations converge to two different final states. These two states are roughly the same as those we obtained when we varied the price (see Section 7). In this case, there is no meaningful relationship between the initial production level and the final state reached. We could argue that these two final states are particularly stable ones and the system tends to converge to any of them independently from the initial state. The number of iterations needed to reach convergence are 26, 121, 79, 40, 121 and 26, respectively. There seems that there is not relationship between the number of iterations and the kind of final state reached either. As in Section 7, it would be interesting to plot the path followed by the evolution of each simulation.

10. Combined effect of price, production and publicity on demand

In the previous examples we have investigated the effect of the levels of production, price and publicity on the system separately. However, these are no very realistic situations. In a real system it seldom happens that only one of the factors is changed.

Computer vendors cannot decide, for example, to raise production while leaving all the other factors at the same level. To raise production it is necessary to buy more infrastructure, to hire more workers and to expand the distribution networks, and this does not come free. To raise production computer vendors need more capital for their companies. The capital can be obtained from new investors, but it is difficult to attract new investors when the level of business is steady. More likely, what computer vendors have to obtain more money for production by raising the price of the product or by cutting expenses somewhere else, for example in publicity.

Computer vendors cannot reduce the price of computers only, because otherwise they would not get enough revenue to maintain their companies. If they reduce the price of computers they have to cut expenses by reducing the production or by spending less in publicity. On the other hand, if they decide to raise the price of computers they may have more cash available to raise production or publicity.

Computer vendors do not wish to reduce the production, or at least, not if they are not forced by the circumstances. If they reduce the production they have to close factories, sell infrastructure and make workers redundant. If later on they decide to raise production, they need to acquire the resources again, which is difficult, slow and expensive.

We can explore three different scenarios. The first one is that computer vendors reduce the price of computers and cut down the expenses in publicity to make up for the reduction in revenue. The level of production is left at 50 initially. The second scenario is that they leave the price of computers as it is (level 50) but they raise the production, obtaining the necessary capital by a cut in expenses in publicity. The third scenario is that they raise the price of computers and invest the additional revenue in publicity.

The alternative of raising the price of computers and invest the additional revenue in raising the production is not feasible. A raise in the price of computers would trigger a drop in sales, at least initially, and it does not make sense to raise the production and commit resources when the demand is going to diminish.

Therefore we try the following initial states:
Reduction of price and publicity: price/publicity levels of 40/40 and 20/20.
Increase of production and reduction of publicity: production/publicity levels of 60/40 and 80/20.
Increase of price and publicity: price/publicity levels of 60/60 and 80/80.

The results of the simulations are shown in the table below:

The only combination of levels with which we achieve a raise in demand is with production = 80 / publicity = 20. The demand raises to level 66 (a very moderate increase) but the price of computers drops to 22 (a significant decrease). Therefore is not clear whether the final situation is any more profitable at all than the initial situation.

The other final states are worse or much worse than the initial situation. In none of the final states there is a clear increase of revenue for the computer vendors. For the cases where there is more income (higher price of computers at the same level of demand) from sales there is also an extra expense in publicity; therefore there is no net gain.

11. Conclusions

In our simulations on the fuzzy cognitive map we have failed to find a way to increase the demand of computers, and therefore the revenue of computer vendors. The reason for this may be that it is not possible to raise the demand (very unlikely, but possible) or that the fuzzy cognitive map is not good (this is more likely). While constructing the fuzzy cognitive map we may have left out some important factor(s) or we may have set the relationships wrongly. In addition, the intensities of the relationships have been all set at value 50, and this may not reflect the reality.

The only conclusion that we can draw from the results of our simulations is that the fuzzy cognitive map does not seem to be good at all. More work is needed to refine the fuzzy map or perhaps to come up with an alternative, better fuzzy map.