Math Formula

Tuesday, September 25, 2012

Should I Eat That Apple Today Or Tomorrow? - Part I

Disclaimer: The study conducted is not at all scientific, and this is by purpose. I know it has several flaws.
Apart from that, there might be several other studies on that topic already. Still, here are my thoughts. You have been warned!

Imagine a basket of apples, which is magically re-filled every day with fresh apples. You have free access to that basket, you are not starving, and you are generally fond of apples.
Every day you can either take one apple, eat and enjoy it; or reject to do so.
Should you eat an apple every day, just because it is free, or can it make sense to constrain yourself on certain days, in order to enjoy it even more the next day?

Not too surprising, the answer is, similar to all questions that are worth being asked at all: It depends!
According to the first model I will present today, it depends on one factor only: On your individual saturation factor. If eating an apple today reduces your level of satisfaction tomorrow by more than a half, skip the apple today; otherwise, eat it.

I will try to explain my chain of thought, as easy and clearly as possible. That's for three reasons:
  • I want to show you that a micro-economic model is nothing that's for university graduates and PhD's only (alright I'll admit, that idea is credited to Karl Popper)
  • I want to show myself that this micro-economic model is nothing that's for university graduates and PhD's only
  • I know this theory is still very weak and probably has fundamental flaws. Probably even similar theories already exist, which I do not know about. Still, I want to fail fast, in order to improve it.
Basic idea
We are talking about the consumption of a freely available good.
This could be an apple, a glass of water, watching TV, or having sex (whereas arguably some of the assumptions made below do not hold).
Each period (say, a day), the consumer faces the decision whether to consume that good or not. Consumption provides some form of satisfaction, which I will further on refer to as utility. The good is assumed to be saturating. If it was consumed in previous periods, it keeps providing utility, but less than in previous periods - because the consumer gets saturated (the technical term is marginal utility).
The consumer wants to maximize her gross utility - the sum of the utility provided every day. How should she decide every day in order to maximize gross utility?

As this is the first version of the model, assumptions are very restrictive. I'll try to weaken some of these in the upcoming weeks; of course also your input and ideas are warmly welcome!

Assumption 1:
There is only one free good, which saturates the consumer.

Assumption 2:
The good provides some form of utility, but is not necessary to survive.

Assumption 3:
The consumption decision is to be made every period (e.g., a day). The amount consumed cannot be chosen, only whether to consume or not at all. Goods cannot be taken and donated, which might provide some utility, too. They can only be consumed or not.

Assumption 4:
If the good was already consumed in the directly precedent period, utility is reduced linearly by the saturation factor. Imagine watching TV every day. Clearly, the level of enjoyment in each period is not that high each time, as if you would watch only once a month.

Assumption 5:
Contrary to a typical spending function, the consumer is indifferent between consuming the apple today or tomorrow. The apple today provides exactly the same utility as the apple tomorrow (in accordance with A4, given no apple was consumed the day prior to that).

Assumption 6:
Consumption of the good in one period does NOT provide utility in the following periods. However, according to A4, if consumed again, utility is reduced.

Assumption 7:
The consumer seeks to optimize the overall utility gained; that is, the sum of all utilities (resulting from A5 and A6).

Alright, being well prepared with these assumptions, it gets slightly mathematical now. Don't worry, I'll try to keep it as short and easy as possible. I hear the critics among you already shouting about homo economicus misusage - and rightly so. Let me respond to this critique at the end.

The variables we'll use are as follows:
  • \(c(t)\) refers to the consumption of the good in the period t. In each period, it can be either 0 (no consumption) or 1 (consumption).
  • \(u(t)\) refers to the utility gained in period t. According to A2 and A4, this depends on the consumption in t, and also the previous period. That is: \[u(t) = f(c(t), c(t-1))\]
  • \(g\) is the gross utility. According to A5, this is the sum of all u(t) over all periods. This is the value to be maximized.
  • The utility factor \(a\). According to A2 and A5, this is the level of utility one single unit of the good (e.g., one apple) would provide if none was consumed in the previous period.
  • The saturation factor \(s\): How much the utility is reduced in period t if the good was consumed in the previous period already, and therefore the consumer is saturated a little bit already. This results from A4. This factor can be any value between 0 and 1. The higher it is, the less the additional utility gained in the period after consumption.
  • The utility function \(f\): How all of the aforementioned variables are related. Resulting from A4, A5 and A6, we can say: \[u(t) = a c(t) - a c(t) c(t-1) s = a c(t) (1 - c(t-1) s)\]
That was a little bit theoretical now, so let me give you an example.

Example 1
Let's look at the consumption of apples across three days.

On the first day, the consumer is totally into apples, so he'll eat the apple. For the sake of that example, let's also eat it on day 2 and 3. So: \[c_1 = 1, c_2 = 1, c_3 = 1\]
Choosing of factor \(a\) is arbitrary, so why not make it 10 (so \(a = 10\))?
Saturation factor \(s\) would be subject to empirical studies; however, I'll just as arbitrarily set it to 0.2 (\(s = 0.2\)).
Question: What is the utility u in each of the 3 periods? What is the gross utility?

Let's start with the utilities in each period.
Utility in first period \(u_1\) is straightforward, because there was no previous period and therefore no saturation to take into account. Thus \[u_1 = a c_1 (1 - c_0 s) = 10 ⋅ 1 (1 - 0 ⋅ 0.2) = 10\]

Utility in second period we get by simply filling into the utility function f:
\[u_2 = a c_2 (1 - c_1 s) = 10 ⋅  1 ⋅ (1 - 1 ⋅ 0.2) = 10 (1 - 0.2) = 8\]

The same is true for period 3:
\[u_3 = a c_3 (1 - c_2 s) = 10 ⋅ 1 ⋅ (1 - 1 ⋅ 0.2) = 10 (1 - 0.2) = 8\]

The gross utility g is the sum of those three:
\[g = u_1 + u_2  + u_3 = 10 + 8 + 8 = 26\]

We can sum up this example as follows:

Period 1 2 3
c(t) 1 1 1
u(t) 108 8
g 10 18 26

Table 1

Wasn't too hard, was it? So let's directly dive into a second example, which empirically comes close to our final answer already.

Example 2
Now, let's assume that the consumer consumed the good in period 1 (similar to example 1, \(c_1 = 1\)). There is no previous period, so the consumer is not at all saturated, so abstinence does not make sense. In period 2 however, the consumer abstains from consumption (\(c_2 = 0\)).
In period 3, he consumes again (\(c_3 = 1)\).
Question: What is the utility u in each of the 3 periods? What is the gross utility?

Similar to the method from above,
\(u_1 = a c_1 = 10\)
\(u_2 = a c_2 (1 - c_1 s) = 10 ⋅ 0 ⋅ (1 - 0.2) = 0\)
\(u_3 = a c_3 (1 - c_2 s) = 10 ⋅ 1 ⋅ (1 - 0 ⋅ 0.2) = 10\)
\(g = 20\)

... or in table representation:
Period 1 2 3
c(t) 1 0 1
u(t) 100 10
g 10 10 20

Table 2

\(g = 20\) is less that in example 1, where \(g = 26\)! That's an interesting finding, because now we know that given an utility factor \(a = 10\) and saturation factor \(s = 0.2\), it does NOT make sense to abstain from consumption. On the contrary, the good should always be consumed!
Let's repeat those two examples, but with another saturation factor s instead; say, \(s = 0.6\).

Example 3
Same setup as in example 1, the consumer always consumes, but \(s = 0.6\).
\(c_1 = 1\)
\(c_2 = 1\)
\(c_3 = 1\)


\(u_1 = a c _1 = 10\)
\(u_2 = a c_2(1 - c_1 s) = 10 ⋅ 1 ⋅ (1 - 0.6) = 4\)
\(u_3 = a c_3 (1 - c_2 s) = 10 ⋅ 1 ⋅ (1 - 1 ⋅ 0.6) = 4\)
\(g = 18\)

Period 1 2 3
c(t) 1 1 1
u(t) 104 4
g 10 14 18

Table 3

Example 4
Same setup as in example 2, the consumer abstains in period 2, but \(s = 0.6\)

\(c_1 = 1\)
\(c_2 = 0\)
\(c_3 = 1\)

\(u_1 = a c _1 = 10\)
\(u_2 = a c_2(1 - c_1 s) = 10 ⋅ 0 ⋅ (1 - 0.6) = 0\)
\(u_3 = a c_3 (1 - c_2 s) = 10 ⋅ 1 ⋅ (1 - 0 ⋅ 0.6) = 10\)
\(g = 20\)

Period 1 2 3
c(t) 1 0 1
u(t) 100 10
g 10 10 20

Table 4

Now the results are the other way around - it pays off now to abstain in period 2!
A modification in the utility factor would not make any difference, because it would simply result in a higher or lower overall utility.

Alright, we're almost there. I want to leave it up to the reader to draw his own conclusions, until I will offer mine during next week.

Should I Eat That Apple Today Or Tomorrow? - Part II can be found here

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