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by Valentino Piana (2012)



1. Overview of the model
2. The differences in the cumulative bundle after a large number of periods
  3. The price of the goods and the relative market share of the rich and the poor  
4. Vulnerability to negative shocks
5. Competition between the rich and the poor
6. Policy implication

7. Conclusions



1. Overview of the model

This short paper explores the differences between a rich and a poor in consumption and need satisfaction. It's a fairly abstract analysis because it is oriented by a formal model expressed in an MS Excel spreadsheet and it leverages a new key concept we introduced: the cumulative bundle.

This paper considers two people with widely different income per period (e.g. month), facing a flow of decision to select whether to buy or not a certain good. The properties of the goods are explicited in a matrix in which each good has a price, lifetime (0 to infinity), percentage of need satisfaction on a range of needs (negative values possible), a revenue it provides and the costs for operation and maintenance it requires.

In the present version, the rich has an income of 100 and the poor of 20 but you can change the routine you find in the button labelled "Action!" to change these parameters - and others, such as the number of goods (currently 20), the number of periods (currently: 50) and the rule followed by the poor and the rich to select purchases [1].

For the moment the rule is a random extraction from the full range of goods, with acceptance being automatic if the available income is higher than its price. Available income is the period inflow of income (as said 100 or 20) minus the purchases already made during the period plus past savings. It also includes the revenues gained thanks to the elements of the previous-period cumulative bundle that bear interests / revenue. It is reduced by the (mandatory) maintenance and operation costs of the previous-period cumulative bundle.

You might want to change the rule, including other suggested here, such as the "equalization" rule, the "specialization" rule - or further ones, such as specific coherence with previously cumulated bundle and a time-linked synusoidal need satisfaction generated in response to graph similars to biorythms and Circadian cycles, possibly drawing on more general cronobiological phenomena. You might also introduce some sort of comparison about a certain set of proposed goods, especially to test substitutes.

You can explore and change the "properties of the goods" in the homonymous sheet. As default, prices have two decimals. No good fully satisfies a need (100%), whereas typically it satisfies a few, not all, in varying degrees.

The capability of goods to satisfy material, cognitive, emotional, symbolic and social needs can be due to their use, ownership, any activity taking advantage of the good and social interaction based or making use of the good. One could introduce purchasers' skills in using the good and performing activities in which it enters as input, especially if this is difficult.

"Lifetime" is the "longevity" of the good: 1 means just 1 period (the period in which the purchase takes place), so it reflects a non-durable good, with the cumulative bundle being each period equal to current purchases. 2 means that the good remains in the cumulative bundle also in the following period.

2. The differences in the cumulative bundle after a large number of periods

The rich has a much larger cumulative bundle, in terms of the number of goods and a higher percentage of need satisfaction (even higher than 100% and with duplicative items, which could be elements for the definition of "overconsumption").

The poor is dissatisfied on many need axes, as his small cumulative bundle doed not provide enough satisfaction. If needs were to be divided into "basic needs" and "other needs", it is fully possible that even basic needs are not covered.

The attempt to reach a high level of satisfaction on one need is paid with even a more pronounced dissatisfaction in another.

Unsatisfied needs cannot be compensated by the fulfillment (or overfulfillment) of other needs (you can die for lack of water even if you eat a lot). However, it's interesting to look at averages in need satisfaction to have a single number for comparison purposes, when changing goods properties and other parametres. You'll find these averages in cells AN2 and AO2 of the "Synthesis of results" sheet.

Try imposing different longevity (lifetime duration) of the goods. You'll discover that there is a strong positive relationship between product durability and need satisfaction, both for the rich and the poor, with a reduction in absolute poverty in terms of needs not satisfied (material deprivation - a measure of which is to be introduced by the European Union to measure the achievement of Europa 2020 poverty reduction goal).

3. The price of the goods and the relative market share of the rich and the poor

Even with the very basic and apparently neutral random rule that we mentioned above, a very strong correlation can be detected between the price of the good and the unit market share in sales of the rich. The higher the price, the higher the market share of the rich, i.e. the percentage of sales (measured by the number of items) that are made by the rich. Since the price is not assumed to change over time, unit market share is directly mirrored in market shares over the monetary value of sales. Conversely, the lower the absolute price, the larger the market share due to the purchases of the poor.

In particular, when computing the total sales across all periods for each good, in the columns BA-BA of the "Synthesis of results" sheet, you will see the sales of each good to the rich and to the poor. The rich buys systematically more than the poor, to the effect that firms that might need to differentiate the distribution channels, advertising and special features of the good would enter in synch with the rich and not with the poor.

The higher the price, the stronger this relation, to the final effect that for very expensive goods, the rich is the only to buy (for the prices of the goods, see column B in the sheet "Properties of the goods"). To the extent that certain needs are satisfied only by expensive goods, possibly because the price is positively related to the number of needs fulfilled, the poor remains systematically dissatisfied with them.

The smaller total purchases of the poor over time mean that he buys less frequently than the rich. He faces more inventory breaks and longer periods in which he suffers from the unavailability of the good in his cumulative bundle.

Depending on repurchasing rules and the lifetime duration of the good (possibly made endogenous, with repurchasing provoking substitution of the older good), one could demonstrate that the poor has - on average - older goods than the rich (which might lead to decay in their performance because of tearing-and-wearing as well as to unfashionable or less technologically advanced versions). In the current default, this effect does not take place (the objective lifetime duration is the main determinant of the age of the cumulative bundle for both the rich and the poor).

The result of the high market share of the rich in total sales is dependent on the number of rich and of poor in the population (1 and 1, in our model); since in reality the poor outnumber the rich, there would be a certain re-balance. However, the poor need to be many times more than the rich just to reach an equal market shares. In another vein, our numerical example exposes a polarised society, whereas one could have a society with a large middle class: try changing the parametres for different shape of income distribution and see the difference!

For a comparison to real world data see here and with microdata from another model see here.

4. Vulnerability to negative shocks

Experiment with the current version of the model a 10% fall in income and a 10% inflation reflected in a systematic increase of all prices by 10%. Is the impact on need satisfaction of the poor more than proportional than on the rich?

In order to take into account stochastic negative shocks, you can explore the income-related inssues of insurance. For any given poverty line, the lower the starting income, the higher probability of going below the poverty line because of substantial negative shocks. Insurance could provide a remedy but many poor tend to be uninsured, because of premiums amount, as you can experiment with this formal model.

5. Competition between the rich and the poor

The current version simply juxtaposes the rich and the poor. But it may be worth exploring what happens when the rich actively compete with the poor, for instance in an auction for the same good or in matching with prospective partners.

In the case of business partner and family partner - the rich might be preferred because of the larger cumulative bundle to share and pool with the partner. He will receive more proposals of partnering - if he must choose just one, then he will have the advantage of choosing a richer partner, if he can accept all he will have a much wider pooled bundle.

If the partner is looking for a specific good in the cumulative bundle, he will more likely ask the rich (who has a larger probability of owning it and earlier).

In another vein, there are at least eleven good reasons for the poor to pay more than the rich to get the same good or the same level of satisfaction of needs, as we explain here.

6. Policy implication

The model allows for simulating income and wealth redistribution, taxes and subsidies (both in income and prices, as it would be the case with VAT), as well as the possibility of taking a non-collateralised loan to purchase revenue-generating assets (microcredit).

The strong positive relationship between the durability of goods and the level of needs' satisfaction for both the rich and the poor suggests new policy approaches. In particular, product mandatory and voluntary standards that emphasise longer lifetime, resistance to tear-and-wear, reusability and recyclability would be important in the fight to poverty and for better livelihoods. It's an example how a "green" policy can help social inclusion, boosting at the same time several dimension of sustainability.

Further policies can be devised by tackling the cases where the poor pay more than the rich.

7. Conclusions

This short paper opens a reflection on structural differences between the rich and the poor in a formal model of consumption, leveraging the new key concept of cumulative bundle. At EWI, we are eager to listen to your consideration on the matter.


[1] The practical method to modify the routine is to click on the MS Excel Menu "View" and prompt the Visual Basic (VBA) Toolbar to be visible.

In a picture, click on the items in red:

Click on this icon:. You shall be in the "Project" mode that allows to edit the content of the "Action!" button, by double clicking on it.

You shall see the following routine, operating on the sheets (Foglio1, Foglio2,...), with comments preceded by a ' sign:

Private Sub CommandButton1_Click()


nperiods = 50
nproducts = 20
richincome = 100
poorincome = 20

'clean tables

For t = 1 To nperiods
richavailableincome = richincome
pooravailableincome = poorincome

If t > 1 Then
richavailableincome = richavailableincome + (richincome - Foglio4.Cells(t, 2)) + Foglio4.Cells(t, 15) - Foglio4.Cells(t, 16) ' It includes what was saved in the previous period, the revenues and the cost of the cumulative bundle of the previous period
pooravailableincome = pooravailableincome + (poorincome - Foglio4.Cells(t, 3)) + Foglio4.Cells(t, 17) - Foglio4.Cells(t, 18)
End If

richngoods = 0
poorngoods = 0
richcumulativeitem = 0
poorcumulativeitem = 0
richaverageage = 0
pooraverageage = 0

For i = 1 To 100

' the rich tries to buy
goodtobuy = Int(Rnd * nproducts)
If richavailableincome > Foglio1.Cells(goodtobuy + 1, 2) Then
richngoods = richngoods + 1
Foglio2.Cells(t + 1, richngoods + 1) = goodtobuy
richavailableincome = richavailableincome - Foglio1.Cells(goodtobuy + 1, 2)
Foglio4.Cells(t + 1, 2) = Foglio4.Cells(t + 1, 2) + Foglio1.Cells(goodtobuy + 1, 2)
End If

'the poor tries to buy
goodtobuy = Int(Rnd * nproducts)
If pooravailableincome > Foglio1.Cells(goodtobuy + 1, 2) Then
poorngoods = poorngoods + 1
Foglio3.Cells(t + 1, poorngoods + 1) = goodtobuy
pooravailableincome = pooravailableincome - Foglio1.Cells(goodtobuy + 1, 2)
Foglio4.Cells(t + 1, 3) = Foglio4.Cells(t + 1, 3) + Foglio1.Cells(goodtobuy + 1, 2)
End If

Next i

'width of bundle of purchases
Foglio4.Cells(t + 1, 4) = richngoods
Foglio4.Cells(t + 1, 5) = poorngoods

' cumulative bundle at time t

For tt = 1 To t
' look at purchases in each previous period and their lifetime
' max 255 goods
richagecumulativeitem = 0
pooragecumulativeitem = 0

For i = 1 To 255
If Foglio2.Cells(tt + 1, i + 1) <> 0 And Foglio1.Cells(Foglio2.Cells(tt + 1, i + 1) + 1, 5) > t - tt Then
richcumulativeitem = richcumulativeitem + 1
richagecumulativeitem = richagecumulativeitem + 1
Foglio5.Cells(t + 1, richcumulativeitem + 1) = Foglio2.Cells(tt + 1, i + 1)
Foglio4.Cells(t + 1, 15) = Foglio4.Cells(t + 1, 15) + Foglio1.Cells(Foglio2.Cells(tt + 1, i + 1) + 1, 3)
Foglio4.Cells(t + 1, 16) = Foglio4.Cells(t + 1, 16) + Foglio1.Cells(Foglio2.Cells(tt + 1, i + 1) + 1, 4)

Exit For
End If
Next i
richaverageage = richaverageage + richagecumulativeitem * (t - tt)

For i = 1 To 255
If Foglio3.Cells(tt + 1, i + 1) <> 0 And Foglio1.Cells(Foglio3.Cells(tt + 1, i + 1) + 1, 5) > t - tt Then
poorcumulativeitem = poorcumulativeitem + 1
pooragecumulativeitem = pooragecumulativeitem + 1
Foglio6.Cells(t + 1, poorcumulativeitem + 1) = Foglio2.Cells(tt + 1, i + 1)
Foglio4.Cells(t + 1, 17) = Foglio4.Cells(t + 1, 17) + Foglio1.Cells(Foglio3.Cells(tt + 1, i + 1) + 1, 3)
Foglio4.Cells(t + 1, 18) = Foglio4.Cells(t + 1, 18) + Foglio1.Cells(Foglio3.Cells(tt + 1, i + 1) + 1, 4)

Exit For
End If
Next i

pooraverageage = pooraverageage + pooragecumulativeitem * (t - tt)

Next tt

'width of cumulative bundle
Foglio4.Cells(t + 1, 8) = richcumulativeitem
Foglio4.Cells(t + 1, 9) = poorcumulativeitem

' average age of cumulative bundle
If richcumulativeitem <> 0 Then Foglio4.Cells(t + 1, 12) = richaverageage / richcumulativeitem
If poorcumulativeitem <> 0 Then Foglio4.Cells(t + 1, 13) = pooraverageage / poorcumulativeitem

' need satisfaction
For s = 1 To 250
For need = 1 To 10
If Foglio5.Cells(t + 1, s + 1) <> 0 Then Foglio7.Cells(t + 1, need + 1) = Foglio7.Cells(t + 1, need + 1) + Foglio1.Cells(Foglio5.Cells(t + 1, s + 1) + 1, need + 5)
If Foglio6.Cells(t + 1, s + 1) <> 0 Then Foglio8.Cells(t + 1, need + 1) = Foglio8.Cells(t + 1, need + 1) + Foglio1.Cells(Foglio6.Cells(t + 1, s + 1) + 1, need + 5)

Next need
Next s

Next t
End Sub






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