Math Formula

Wednesday, October 17, 2012

The Tragedy Of Public Spending In Six Sentences

Political leaders are supposed to give precise spending forecasts, even when the details of the projects to be approached are not yet known.

If a project budget is not fully used up, the politician is said to be a poor forecaster, and less money than actually needed will be allocated to her future projects. Thus, a project budget is always fully used.

If the forecasts are exceeded, the politician is said to be a poor forecaster as well, plus he has to struggle to get the additional funds needed. Thus, there is an incentive to make the initial forecast bigger than actually expected.

As there is no incentive to counter these increased costs, always more money is spent than actually needed.

Monday, October 15, 2012

Updated Blog Style

After more than a year of writing this blog and a extremely lame layout and overall look & feel, I felt it was about time to make a small redesign and several improvments for things that always bothered me.

First of all, you will note a new theme - instead of the blue font on what background you will now find a combination of dark colors (black, grey) and orange.

Next, I tried to get rid all the unnecessary clutter on the page - label overview, archive, about me section are all now cut away from the main page. Instead, I created dedicated pages for archive and a quite rudimentary "about" page as well.

Furthermore, I found that I did not offer any possibilites to get in touch with me - so here they are!

Finally, I moved to feed burner for feed generation, which also allows subscribing via email now.

And what do you, dear reader, think about the new design? Each kind of feedback is truly welcome!

Tuesday, October 2, 2012

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

In last week's post, I discussed the question of a whether a free good should always be consumed, or if it might make sense to abstain from consumption under certain circumstances.

We went through four examples of consumption.
In example 1 and 2, the saturation factor was low (\(s = 0.2\)).
In example 1, the consumer always consumed, and gross utility \(g = 26\).
In example 2, the consumer abstained in period 2, and gross utility \(g = 20\).

In example 3 and 4, the saturation factor was high (\(s = 0.6\)).
In example 3, the consumer always consumed, and gross utility \(g = 18\).
In example 4, the consumer abstained in period 2, and gross utility \(g = 20\).

So, here is the answer to our initial question: You should eat or not eat the apple depending on the saturation factor \(s\) ! For a big \(s\) , not consuming in period 2 was better; for a small \(s\), consuming was better.

However, at which threshold does the decision change?

Let's make a step back first. As you might have noticed, I only varied the decision for period 2, and here's why.

In period 1, consuming is always superior to not consuming. There was no prior period from which the consumer might still be saturated, so he always consumes. Thus, we can assume \(c_1 = 1\), and can disregard the consumption decision from now on.

As we have seen, changing the decision in period 2 has an effect on the gross utility, so \(c_2\) remains a variable to be considered.

However, period 3 can be disregarded again. Why? If \(c_2\) was 0, there is no saturation, and similar to period 1, consumption is better. If \(c_3\) was 1, there is saturation, but there is no 4th period to save consumption for, so again, consumption is better. Thus, \(c_3 = 1\) , and can be disregarded further on.
The only variable to maximize against is \(c_2\).

In the light of the above, let's reconsider our gross utility function:
\(c_1 = 1\)
\(c_2 =\) to be seen
\(c_3 = 1\)
\(u_1 = a c_1  = a\)
\(u_2 = a c_2 (1 - c_1 s) = a c_2 (1 - s)\)
\(u_3 = a c_3 (1 - c_2 s) = a (1 - c_2) s) = a - a c_2 s\)

\(g = u_1 + u_2 + u_3 = a + a _2 (1 - s) +  a - a c_2 s = \)
\(2 a + a c_2 - a c_2 s - a _2 s = \)
\(2 a + a _2 - 2 a _2 s = 2 a + a _2 (1 - 2s)\)

Now we can directly compare the two outcomes with each other; the gross utility in case of consumption in period 2 (which is , and the gross utility in case of no consumption in period 2.

So, for \(c_2 = 1, g\) would be: \(2 a + a (1 - 2s)\)
And for \(c_2 = 0, g\) would be: \(2 a + 0 = 2a\)

So, this question can be formulated as inequation:

\(2 a + a (1 - 2s) > 2 a\)
\(a (1 - 2s) > 0\)
\(1 - 2s > 0\)
\(1 > 2 s\)
\(1/2 > s\)
\(s < 1/2\)

So, if the saturation factor \(s < 1/2\), gross utility is bigger with \(c_2 = 1\).
For \(s > 1/2\), gross utility is bigger with \(c_2 = 0\).
For \(s = 1/2\), the consumer is indifferent, so the gross utility is equal.

Of course, your real saturation factor \(s\) is hardly known. However, I find it quite interesting to keep in mind that for a big saturation factor, I should rather consider not consuming. The bigger the impact on the reduction of the satisfaction of tomorrow's consumption, the more I should be inclined to defer consumption.

As I mentioned above, I'm well aware of the fact that this model is still very weak.

First of all, the approach of trying to quantify utility of consumption, especially of non-tangible goods might be quite inappropriate. After all, that's the major weakness of the homo economicus altogher, right?
As a defense, I'd like to see the approach chosen not as a purely numerical, but rather as a concept as whole. You can imagine and include whatever you want into this utility function.

Second, the assumptions and constraints are very restrictive. Consequently, the results might not only be inaccurate, but even wrong and misleading.
The assumptions should incrementally be loosened in further research. I intent to do so in upcoming weeks.

Third, some empirical studies should be conducted, until the theory can eventually be rejected (Karl Popper again).

I hope that I managed to make my point, and am looking forward to all kind of additional critique and feedback.

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