Percentages, God Damn Percentages

Percentages, god damn percentages; is there anything more abundantly abused in scientific literature? Gross misrepresentations of this notation haunt me back to my high school days, when science was first inserted in my curriculum. In this blog post, I shall expose exactly what a percentage is as I feel many persons need a rehearsal on the notion and then explain what I think is an unacceptable and gross abuse of the notion of what a percentage is. This, I hope, will shed light on the subject in a decisive manner.

Here is a formal definition of the percentage definition using the mathematical equality.
Which means in laymen’s terms that a number, followed by the percentage sign, is equivalent to this number divided by a hundred. With such a straightforward and easy to grasp definition, one may wonder how this notation can be so outrageously abused. Here is a blatant example. One wants to evaluate the percentage of variation between two real numbers x and y. The following formula is how one would accomplish such a calculation “the wrong way”.Can you spot the logical flaw in this? If not, then refer to the definition. Try the calculation with two numbers, which we expigeously choose to be x = 100 and y = 101. Intuitively, the variation of x is 1%. Is this logical phrase true or false? In other words, is the equation mathematically right ?
Well, it makes sense, so it must be right ! No shit, Sherlock? Well, let me remind you that:
Hence, the multiplication by 100 is not only useless; it makes the equation mathematically wrong. Some manuals use the following notation, which I still find absurd but I guess is somewhat of a compromise.
Much like the hexadecimal, the octal or the decimal system, the percentage notation is just a way to present the numbers in a more convenient manner; one does not have to expose the calculation he makes to switch a number from the decimal notation to the hexadecimal and the same is true for percentages. A number presentation system is a number presentation system, period. The multiplication by 100, probably used to make it easier for whoever calculates (well, you do see 1 instead of 0.01 on the calculator’s screen, makes a whole lot more sense doesn’t it ?), is harmful and simply not right. Stop using it. Now.

Teachers should take note of this and take the time to explain properly the notation instead of taking damaging pedagogical shortcuts.

Of Ice Storms and Candles

In 1998, the province of Quebec was affected by the most important ice storm in its recent history, cutting electricity supply to over 1 million customers for a period of time varying between a few days and a whole month. During that time, I was living in a small town in the middle of the most affected area, sometimes referred to in the media as “The Triangle of Darkness”. Being only a decade old at the time, it is of very profound interest to reanalyze these events and put them under the scrutiny of my acquired sagacity. In this blog entry, I shall explain a particular event that caught my eye eleven years ago and that I ended up reanalyzing in a completely different manner after understanding the basic principles of economics. By extension, my conclusion also serves as a specific example of how capitalism works and why it is an efficient system.

One of the most important things to have in an electricity outage is lighting, and candles provide a cheap and reliable alternative to flashlights or other alternatives to lamps. After a while, it seemed apparent the outage provoked by the ice storm would be very long. In my hometown, the suppliers of candles could not anticipate such a high demand and the stores were quickly depleted of their candles by eager customers. Furthermore, distribution was paralyzed by the outage, putting an ever greater pressure on the supply. All suppliers ended up running out of candles except one who made a sound business decision whilst saving the town of a complete penury: the grocery store.

It appeared that the owner of the store, a very clever and opportunist businessman purchased the remaining stock of candles in all the stores in the area and sold them for a much higher price. This, at the time, appeared to my relatives and myself as a disgusting act of greed. In fact, on a strictly personal and motivational basis, it was probably an act of greed. Seeing the stocks of candles dropping, the manager quickly understood that the rarity of the good was skyrocketing, in a time where its main substitution competitor, electricity, was unavailable. Although one can question the ethical implications of such a move, the actions of the manager prevented, most certainly involuntarily, an even worst problem: a penury of candles at the time when they were the most needed.

In the case we study, the supply of candles from suppliers is constant since the usual supply routes are disrupted and the studied amount of time is very short, whilst the demand varies according to price, as it usually does. The situation is shown in the graphic below.
The two blue curves are the demand curves. The higher the price is, the lower the demanded quantity is. The curve with the steeper slope is the one representing the ice-storm situation. In this situation and compared to the normal situation, customers will always buy more candles. The initial price is P_i and the supply is S. If the price is not adjusted following the new situation, the demanded quantity will be Q. Since the supply is fixed and Q > S, there will rapidly be a penury, meaning that S candles will be sold at P_i but could have been sold at P_e, preventing a penury.

The fact that we, as customers or producers, are all “greedy”, or said more gently, “economically sound”, is what permits a distribution of resources that is optimal. There is a finite set of resources on Earth and infinite needs; we thus need rationing methods. The simple fact is that there was an upcoming penury of candles in the town, aggravated by the fact that the electricity outage was anticipated to last for a very long time. People were buying candles like crazy, probably even irrationally so. This is a perfect example of customer greed. After all, how is it possible for an atomic customer to have the whole picture? There is not an electronic board in the store detailing how many candles are left and how many each person should buy so nobody is left without candles. And even if there was one, how could anyone assess, without a prohibitively costly system, the metrics associated with such an information system ? Furthermore, everybody values candles in a different manner. If I already have a flashlight home, a candle is of very little use. I might be ready to pay $1 for a candle, but certainly not $3. On the other hand, if I have no alternatives, I would certainly be happy to pay $1, but I am still ready to pay $3.

In a sense, the grocery’s manager’s greed, combined with the greed of customers, is what prevented an even less enviable situation, i.e. unused or incorrectly used resources and customers left with pressing needs. By setting the price to P_e, the demanded quantity was adjusted appropriately. Sooner or later, there would have been no candles left and many people really needing them would not have been able to get them, simply because they did not anticipate the candle-run or were not fast enough to evaluate their need in candles. On the other hand, some would have had too many and they would not have felt it was necessary to use them with parsimony, even if there was a critical situation of penury in the community.

Supposing no market adjustments would have been made, the story would have been very different. On the one hand, you would have a few customers with a lot of useless candles, most of them wasted or not used in a time where they were needed by others . On the other hand, you would have a store owner who has not maximized his profit and, even worst, customers left with no candles at all. The increase in price caused a situation where a few people do not lose a lot (the value of the excedentary candles is lesser than the first ones, as utility decreases rapidly with these kinds of goods; having 5 or 100 candles yields a far less than proportional augmentation in utility) and where most of the community gains. In other words, the price adjustment did not change much except the distribution of candles amongst the population.

All this seems beautiful but I will temperate my enthusiasm with two very important remarks. First, the market system works at its best when there is a lot of suppliers, making it impossible for any supplier to set the price of the goods it sells. However, in this situation, since supply is fixed it has an infinite price-elasticity, meaning that even if there is only one supplier, the market still dictates the price, even if the supplier is a price-taker. If the merchant would have set the price too high or too low, profit would not have been maximized. As such, price is a variable the store owner could play with but that had to be maximized.

My second remark is one that is often made in cases similar to this one. On paper, the system works very well and prevents waste or badly allocated resources. Unfortunately, the price increase might mean some people cannot afford candles anymore, meaning they are unable to obtain a good that is essential. When this happens, government subsidies are the best solution. Wealth redistribution is, in theory, not efficient in a strictly economical sense (i.e. it is Pareto-inefficient), but it is simply the right thing to do. Collectively, we cannot accept that a certain portion of the population is deprived from what is essential, be it water, food, health care or candles, in this particular case. Without directly subsiding the purchase of candles in a specific town (which would have required a degree of flexibility way above any bureaucratic organization’s means), the Gouvernement du Québec provided an emergency financial aid during the crisis. Although I do not remember and have not researched the specifics of this plan, it is basically an aid aimed at the poorest to surmount the hardships associated with such a prolonged lack of electricity, including sudden hikes in essential goods.

In the end, one could say the store’s manager was greedy. One could say he was a heartless person profiting from other people’s misery. What is often forgotten is that customers are also very greedy, intentionally or not. This would never have been mentioned as such if there would have been a penury of candles; the stores would probably have been blamed for a lack of supply. After all, how is it possible to blame micro-actions when it is their aggregated effect that causes macro-problems?

Optimizing the Cobb-Douglas Production Function under Budget Constraint

In this blog entry, I wish to explore the theoretical optimization of the production function according to a budget constraint, i.e. producing the most with the least possible.

The Cobb-Douglas production function for k non-null inputs is of the following form (please see this post for an introduction to the Cobb-Douglas function and this post for an introduction to its parameters and the effect theirs settings have on production).

Where Y is the production (output), A a circumstantial constant global efficiency constant, I_i the i^th production input and α_iits output elasticity. The function is often linearized using logarithms.
We shall also introduce a typical isocost function, which is of the following general form.
Where C is the total cost of the production associated to the input vector (I₁,I₂,…I_k) and c_i is the cost of one unit of the i^th production factor. For a given level of production N, there exists an infinite number of input possibilities. Only a subset of these possibilities, which is the most interesting of all possible subsets, minimizes the total cost. This is the subset we shall find.

Finding this subset is feasible using the Lagrange multiplier method. We have the following conditions.

And the general solution for partial derivatives of the first order:
Which yields the following solution subset:
As a solution, we get the following equality, which is in fact the equation of a line in R^k.
Which gives us the following production function, for a given level of production N, where j is the index of an arbitrary production input.
For the sakes of readability, we should pose:
It then follows that:
Now, since the right part of the equation is a constant specific to output elasticities (i.e. the technological advancement), we shall group all the terms in a constant which we shall call the inputs’ cost/output-elasticities ratio constant and note with χ.
We can thus derive the optimal production input vector for a production N. Also, expressing the cost of a production, provided it is optimized, will yield the following equation.
Deriving this function will give us the marginal cost function, which is the price at which the N^th item is produced.
This is, in fact, the offer curve of the firm represented in the cost and output-elasticities data. If the sum of the output-elasticities is less than one, as was previously proven, the function monotonously increasing. Equalizing it with the demand curve, one could derive the quantity sold by the company to maximize its profits given it optimizes its production process. To illustrate this, one company will never produce the Nth item if its production price is higher than the market’s price, simply because doing so will mean it loses money. Equalizing the marginal cost function with the market’s price means all the previous produced units will be profitable and the last one will be produced at null profit, hence maximizing profits. We shall also introduce the average item cost function, which is simply:
Here is a typical optimized offer curve (marginal cost curve with diminishing returns to scale in blue) and a constant demand curve, as well as the average cost curve (in grey).


The Lagrange multiplier method finds critical points (i.e., points where the scalar product of the two gradients is null), but one could ask if the critical point is really a minimum. The test of second-order derivatives using a bordered Hessian matrix would prove this, but in our case the test is inconclusive, as the second-order partial derivatives of the cost functions are always 0, meaning the Hessian determinant is always null.

Supposing the solution we found is a maximum, there exists no input vector satisfying the production function such that, for a given production level N and using the optimal cost function derived earlier:

Prooving that there exists such a vector will therefore prove the Lagrange multiplier’s solution is a minimum. Both conditions can be linked using the average cost function.
Intuitively, there exists an infinity of solutions to that equation. Hence, the critical point we found is the minimum.

I shall use the general solution in a future blog entry to introduce a model for customer satisfaction production in perfect and imperfect markets. This will end the Cobb-Douglas blog entries series. I shall satisfy my most mathematically uninclined readers with day to day life subjects in my next posts.

Parameters of the Cobb-Douglas Function

Now that the notion of elasticity has been presented, I shall introduce a more rigorous study of the parameters of the Cobb-Douglas and the effect they have on output. Again, here is the equation of the Cobb-Douglas function:
Where Y is the output, I_i is the used input of the i^th production factor, α_i the output elasticity of the i^th production factor and A is total factor productivity, a constant that accounts for circumstantial variations in output not directly related to inputs.

We shall start our study with the total factor productivity. I often introduce this concept to non-economists by talking about a very simple example: a farm. Let’s say last year was neither a good nor a bad year in terms of weather and the farmer harvested 10 tons of wheat. Since it is a non-exceptional situation, this production level yields the total factor productivity A = 1. In other words, only the inputs influenced the outputs, as one would expect.

One can easily imagine a situation in which the constant is different than one. If A < 1, certain phenomena affected the harvest which were not related to the inputs used, for example excessively bad weather, an insect invasion or quasi-absent rain; ceteris paribus, the output is thus reduced. The contrary is also possible: excessively good weather, no insects destroying the crops and a perfect quantity of rainfall; in this situation, A > 1 and ceteris paribus, the output is augmented. The constant can only account for variations that affect the overall productivity. Otherwise, the output-elasticities need to be adjusted.

In the graph below, there are three Cobb-Douglas productivity function with exactly the same output elasticties. The blue has A > 1, the purple A = 1 and the green A < 1. The functions act exactly in the same manner, except exactly the same inputs yield more or less output (in other words, the planes are parallel).
One should not confuse total factor productivity with technological advances, which are accounted for by the output elasticities. In fact, using an overall multiplier to account for technological advances would be very unpractical since technological improvements, more often than not, are very precise and limited to a single production factor.

In the last post, elasticity was discussed and I derived a very interesting result concerning the elasticity of the Cobb-Douglas function. Let E be the elasticity operator as defined in the previous post. We conclude that:
Which means that the point elasticity of substitution of any variable in the Cobb-Douglas function is constant. It also means that the increase of one percent of input i yields an increase of α_i percent in output, at all times. The higher the output elasticity, the higher the return-rate on the output is. Here is a graphical illustration with three different classic capital/labor Cobb-Douglas functions of the form:

For each of the functions, the output elasticity of capital is the same but the output elasticity of labor varies (α = 0.1 for the blue plane, α = 0.4 for the purple plane and α = 0.6 for the green plane; β = 0.3 ; A = 1).

The output elasticities can also be studied in a holistic manner. In fact, it can be observed, for a Cobb-Douglas production function with n production factors, that:
Then, for all values of ϕ:
Which means, respectively, that there is a decreasing return on scale, a constant return on scale or an increasing return on scale.

Now that we have explored all the different parameters of the Cobb-Douglas function, we are finally ready to explore the optimization of the function, which shall be the subject of the next blog post.

Of the Point-Elasticity of a function – Addendum

One of my readers sent me a message asking me what the general rules for the point-elasticity of an addition, subtraction and the multiplication by a constant would be. As I already had the solution on a sheet of paper, I decided to create an addendum to my previous blog entry and post them there. For the record, since the general solution yields a very non-elegant result, I did not feel like posting it, but it will help illustrate the statement that “point-elasticity is not a linear operator”, which I have not proved in my previous post.

Let f and g be two continuous functions of n variables. Then, the rule for addition/subtraction is:

And, for a real constant c:
In other words, constants have no incidence on the elasticity. Now, let me remind you of the conditions that makes an operator linear. Let O be any operator, c a constant of any vector space and x, y members of any vector space. O is a linear operator if:

These two conditions are obviously not respected with the elasticity operator and, as such, elasticity is not a linear operator.

Of the Point-Elasticity of a Function

Reading again my last blog post, I realized that I talked a lot about elasticity without properly defining what it is. I wish to introduce rigorously (not as much as a mathematician would hope) and conclusively the notion of elasticity in mathematics and, more particularly, in economics before stepping into much more challenging subjects. Of course, I promised in my last post that I would introduce the optimization of the Cobb-Douglas function but all things shall come at the right time.

Elasticity is the measure of the effect the increment of one variable has on another variable. In mathematical terms, the elasticity of y in regards to x is the ratio of the change in variable x and the change in variable y, which can be written as follows.
Where ∆x is the increment of the variable x and ∆y the increment on variable y. This is the x arc elasticity of y. For a function y = f(x):
The increments are divided by two to get the average value. This formula is useful in discrete cases but is not the best solution in continuous cases.

Considering the studied function is continuous, it is possible to assign any value to both increments. We can thus obtain the x point-elasticity of y, which is in fact the arc-elasticity when the increment is made very close to 0.

Take a real and continuously derivable function of n variables. Its point-elasticity for a chosen variable x_i is:It should be noted that as an operator, elasticity is non-linear. As it is the case with any operator, some basic rules can be derived from the definition. Let f and g be two continuous functions of n variables. The rule for multiplication is:

And the rule for division is:
As an example, we can study the point-elasticity of the generalized form of the Cobb-Douglas function. Here is its equation:
The general form of the first-order partial derivate for this function is:
Applying the elasticity operator for the variable I_i, we get:

This means that elasticity is constant for any value. In fact, the Cobb-Douglas function is a special case of the constant elasticity of substitution (CES) production/utility function, introduced in 1961 by Arrow, Chenery, Minhas, and Solow in their landmark article Capital-Labor Substitution and Economic Efficiency.

It is interesting to note that many kinds of elasticity are studied in economics, such as the price elasticity of demand (the effect the price of a good has on its consumption), the price elasticity of offer (the effect the price of a good has on its production) and the income elasticity of demand (the effect the income of consumers has on the purchase of goods). To learn more about them, see this Wikipedia article.

Now that we have demystified what elasticity is, the next blog entry will introduce in a more rigorous way the parameters of the Cobb-Douglas production function, which will lead our way into its optimization under constraints. We have a very interesting journey ahead of us !

The Cobb-Douglas Function

The Cobb-Douglas function was introduced in 1928 by the American economist Paul Douglas and the American mathematician Richard Cobb. Its form makes it possible to take into account such phenomena as the law of diminishing returns, scale economies and personal preferences. It is applied to production as well as utility and was tested microeconometrically in numerous studies. The Cobb-Douglas production function for k non-null production inputs (production factors) is of the following form:Where Y is the total number of units produced (outputs), I_i is the amount of the i^th production factor that has been used, α_i the output elasticity of the i^th production factor and A is total factor productivity, which is in fact a constant that accounts for variations in output not related to inputs (e.g. bad weather, inventory losses, accounting errors, etc.).

The concept of elasticity is very central in economics. We can view output elasticity as the effect an additional unit of input has on output, considering the level of input already used. Let’s say we have a very simple company which uses two inputs to produce an output, namely labor and capital. We shall write the production function as follows.
Now, let’s suppose different values for the output elasticities to analyze the effect they have on the output. First, let’s consider the case where α=β=0.5. Since both output elasticities have a value smaller than one, their output return diminishes. Since they are equal, it also means that an increase of one worker and one capital unit will make output grow of one. Here is an example graphic (A = 1).

This situation could exist in a small farm, where one unit of capital is one acre of exploitable agrarian surface. Suppose a situation where ten persons are hired to work on one unit of surface. This would be extremely inefficient, as one could imagine workers stepping on each other’s foot or doing nearly nothing. This is an illustration of the law of diminishing returns. Imagine, on the other hand, a situation where one worker has to work with ten units of capital. This is also an inefficient combination, as one worker is clearly insufficient for ten acres.

Let’s imagine a situation where α=0.4 and β=0.1. In this situation, labor is clearly more important to output than capital is. The graphic (A = 1) representing this situation is below.


Comparing the precedent graphic and this one, we can see that the overall production will be lower and the optimization of production will require a completely different labor/capital ratio. We could see this situation happening in a shop where one machine needs multiple operators to be efficient, for example.

Now that we have given many examples of how the production function is used, we shall look into utility. It can be written exactly like production, except the inputs are replaced by the consumed quantities of goods. For k goods with non-null consumption:

The output elasticities are the preferences of the consumer for which the utility function applies. Usually, goods have an output elasticity that makes their returns diminishing. Think of yourself sitting outside on a terrace on a warm summer afternoon. You order your first soda and drink it; your satisfaction grows rapidly. Then, you order your second soda, which is still refreshing but not as much as the first one. By the third one, your satisfaction plateaus; you do not deem it useful to order a fourth one because you can consume other goods, for the same price, which will increase your satisfaction more efficiently. Rarely (and almost certainly never) does a good become more satisfactory the more it is consumed, with maybe the exception of certain drugs, which, it should be noted, create utility but not necessarily well-being.

Optimizing utility our utility is something we do every day, every time we purchase goods, although it rarely comes to our mind to try to optimize our Cobb-Douglas utility function. But, for the benefit of my readers and for the pleasure of the most mathematically inclined, I will do just that in my next blog entry.