Google this

2 sqrt(-abs(abs(x)-1)*abs(3-abs(x))/((abs(x)-1)*(3-abs(x))))(1+abs(abs(x)-3)/(abs(x)-3))sqrt(1-(x/7)^2)+(5+0.97(abs(x-.5)+abs(x+.5))-3(abs(x-.75)+abs(x+.75)))(1+abs(1-abs(x))/(1-abs(x))),-3sqrt(1-(x/7)^2)sqrt(abs(abs(x)-4)/(abs(x)-4)),abs(x/2)-0.0913722(x^2)-3+sqrt(1-(abs(abs(x)-2)-1)^2),(2.71052+(1.5-.5abs(x))-1.35526sqrt(4-(abs(x)-1)^2))sqrt(abs(abs(x)-1)/(abs(x)-1))+0.9

 And that's how you map complex functions my friends!

Two quick links - Krugman edition

Paul Krugman is the ultimate polarising force. People either love him or loathe him. I suppose that's what comes with the territory when you are, arguably, the world's most high profile economist.

Anyway, seeing as I referred to him in my previous posts on climate change and food prices, here are two links that seem kinda relevant.

1) Roger Pielke Jr has a post criticizing Paul Krugman's citiation of extreme weather events in which he questions any causation between long-run carbon levels and food prices. I left a comment saying, among other things, that I regard his invocation of the historical food record as tempting the gods of false equivalence. I also think that he's being uncharitable in his basic reading of Krugman's argument. Still, it's an interesting debate with thought-provoking points raised on both sides... I'd especially like to see whether more people pick up on my comments about Norman Borlaug and the possibilities of another "Green Revolution".

2) Back to PK himself, who, in commenting on the American Economic Review's "Top 20" papers of the last 100 years, had this to say:

It’s also worth noting that a number of the papers on the list involve very simple, intuitive models — that includes Friedman on the natural rate, Krueger on rent-seeking, Mundell on currency areas; Shiller’s great piece on stock volatility was also a remarkably simple concept yielding a powerful insight. My own paper was actually pretty math-heavy; uncharacteristically, it’s also a paper in which doing the math fundamentally changed my mind about things (I didn’t believe the home market effect was real until it popped out of the equations; only then did I realize it was obvious.)

Hmmm. Sounds kinda familiar...

Further, and I would say more importantly, [maths] enables us to fully work through the consequences and dynamics of [economic] relationships. Our intuition may suggest certain outcomes, but intuition can be pretty limited in scope. Working through a set of equations enables you to arrive at a final result with a cool clarity, which may then show up unexpected results that only become “intuitive” upon further reflection.

;)

o-[-<

Karl Smith on the importance of maths-free papers

... And simple allegorical stories to help our intuitive understanding of economic concepts. Here is an excerpt:

I posted a link to Bryan Caplan’s paper on Behavioral Economics and the Welfare State. Many of the comments I got from economists were predictable:  

   1. Where is the formal model and existence proofs?

   2. Where is the data analysis?

   3. How is this a paper?

   4. Do you mean to tell me this is publishable? 

I too was shocked initially by these features or lack thereof. However, that’s part of what made the paper compelling.

Some papers get a wonderful data set, perform magnificent identification and get a result that really changes your mind about something you care about. Most don’t.

Most are cases that are of very narrow interest or do a 90% good job at the ID but leave enough doors open that you are not really sure if  the result is meaningful or not.

On the other hand, one could as Bryan and his co-author did, attack an important question, string together some non-obvious points and in my case leave the reader thinking about whether he or she should reexamine an import view.

The profession should rightly celebrate the first kind of paper. However, what about the relative worth of the second and the third?

I submit that bringing up arguments that use the economic way of thinking matter. This is true even if the argument is not definitive, has no mathematical proof behind it and marshals no data.

Now, clearly I've written in support of using maths in economics on this blog before. However, in case anyone missed it, my point was not to say that we have to have put forward a mathematical equation for each and every economic argument or study. Indeed, I took some pains to make it explicit that this wasn't the case... Shoehorning a maths proof into an economics paper does not (cannot) constitute analytical rigour, let alone intellectual honesty.

I wholeheartedly agree with the excerpt above. We should embrace any economic paper and/or argument that aids our conceptual understanding, or forces us to re-examine our views on a particular subject. However, I continue to submit that theory alone is insufficient... 

Smith goes on to describe the baby-sitting coop allegory, which he says did more to make him a "confident Keynesian" than any other analysis. The important caveat here is that - as pointed out in the comments section - we have competing parables such as Bastiat's "Broken Window" that appear to offer completely different policy prescriptions. Again, this reinforces my belief that we ultimately need to seek empirical validation/refutation of our theories, whether they come in the form of a complex mathematical proof or a simple children's story. As I wrote here:

Knowing when to use your tools is just as important as knowing how to use them. Also, the ultimate test for any model - mathematical or otherwise - should always be how well it describes real-life data.

THOUGHT FOR THE DAY: It's always good to have a explanation at hand that everyone can understand.

PS - The Bryan Caplan paper that Smith is making reference to here is well worth a read. There are also some interesting critiques in the comments section of Smith's initial post about it.

Physics Envy (again)

Seeing as I've covered the topic a few times already, here is an interesting - and surprisingly amusing - presentation by Andrew Lo of MIT, based on a paper he has co-authored with (physicist) Mark T. Mueller: "WARNING: Physics Envy May be Hazardous To Your Wealth!"

Lo's discussion is somewhat contextualised towards financial economics and markets, but he certainly builds his case by first discussing economics in general:

(HT: Bill Easterly. The actual presentation begins around the 03:10 mark)

As I see it, Lo and Mueller's general propositions dovetail nicely with the arguments that I (and many others for that matter) have put forward in the past. For instance (+/-07:05), focusing on when to use your tools rather than simply throwing them to one side:

So what we thought we would do in this paper is to try to trace the origins of physics envy and then see whether or not it really is the cause of crisis, or when it is and when it isn't, and how we might deal with it in a somewhat more productive manner than simply saying that quant is broken and we should forget all about all of this fancy mathematics.

Because, frankly, when I first came across this very popular sentiment that quant is at the root of financial evils, I have to say that it struck me a little bit odd. It's a bit like blaming arithmetic and the real number system for accounting fraud. It's true that they're involved, but you're sort of missing a piece!

Among the many interesting points Lo makes, it's also intriguing to see how reverent he is to the supposed father of physics envy, Paul Samuelson. For example, at (+/-) 12:45 he defends Samuelson against a charge of blind physics envy by quoting a passage from the latter's famous dissertation:

[O]nly the smallest fraction of economic writings, theoretical and applied, has been concerned with the derivation of operationally meaningful theorems. In part at least this has been the result of the bad methodological preconceptions that economic laws deduced from a priori assumptions possessed rigor and validity independently of any empirical human behavior. But only a very few economists have gone so far as this. The majority would have been glad to enunciate meaningful theorems if any had occurred to them. In fact, the literature abounds with false generalization. (Samuelson, 1947, p. 3) 

[Editors note: Samuelson goes on to add (p. 4): "By a meaningful theorem I simply mean a hypothesis about empirical data which could conceivably be refuted, if only under ideal conditions." Karl Popper anyone? I would also note that Lo makes repeated reference to the need for empirical testing as a means of validating economic theory, at the same time as he is wide-eyed about the problems we may encounter in doing so.]

Why we need maths in economics 2(b) - Schelling's segregation model

Slightly later than planned... Here is the second example of a little maths helping to improve our analysis of a real economic problem. Seeing as I received some complaints about the calculus in my first example being too much for a blog post, I've tried to go for something much more basic this time. As such, the only maths that we'll only be using here is the arithmetic that everyone was taught in school. Again, however, those that are only interested in the final outcome can skip to the concluding "THOUGHT FOR THE DAY" and accompanying video at the bottom of the post!

===

Topic: The formation of segregated neighbourhoods.

Aim: To show how seemingly innocuous discriminatory preferences among neighbours (in terms of race, sex, etc) can lead to completely segregated outcomes at the aggregate level.

When research has been done in racially segregated neighbourhoods, one interesting (and consistent) finding is that respondents from these areas generally claim to desire more integration. “We don’t want to live in segregated neighbourhoods!” they implore. However, there are usually some important caveats thrown in and a typical response might be, “I want to live in an integrated neighbourhood... just as long as I am not in too much of a minority”. Nevertheless, what if such “very minor” discriminatory preferences are still enough to lead us to completely segregated outcomes?

This is the question that nobel laureate Thomas Schelling first explored in a seminal paper back in 1969. He later extended his analysis in a series of subsequent books and articles (e.g. here). Alongside his contributions to game theory and conflict strategy, the brilliance of Schelling's work was to examine not only the underlying motivations characterising individual behaviour, but also the implications of individuals acting on each other in the aggregate.

I say “in the aggregate”, though this is a potentially misleading phrase. We have become used to interpreting “aggregates” as something like the average behaviour of individuals. However, much of Schelling’s research has been aimed at proving the exact opposite; i.e. that aggregate results in society are not necessarily simple extrapolations from the individual. Instead, aggregate outcomes are often much more complex since they result from a system of interactions between individuals and their environment. In other words, we impact others and our environment by our actions, while they impact us in turn. These complex interactions can lead to the emergence of surprising and even undesirable outcomes when considered at the societal level. This led Schelling to make the famous distinction between "Micromotives and Macrobehaviour".[*]

Right, so let’s establish the stylised “facts” for the particular model that we’ll be using here. The most important assumptions are as follows:

• People live in different neighbourhoods. If someone is unhappy with their current neighbourhood, then they can costlessly move to a new one. In this model, the only thing that makes people (un)happy about where they live is the racial profile of their neighbours.
• For simplicity we specify a population that consists of only two ethnic groups: greens and reds. In this example, we’ll assume that there are 50 greens in the population and 100 reds. (I briefly consider a different scenario at the end of this post.)
• Both green and red individuals have a variety of “tolerance levels”, reflecting the maximum ratio of race mixing that each person is prepared to accept in his or her neighbourhood. If the colour ratio exceeds a person’s particular tolerance ratio, then they will move to another neighbourhood where they are satisfied. (Those with the highest intolerance will move first.)
• Finally, we assume that tolerance levels among greens and reds can be ordered sequentially from high to low.[**] For both groups, let’s say that the most tolerant individual will accept a ratio of 2:1... In other words, be willing live in a neighbourhood as a one-third minority. The median individual will tolerate a ratio of 1:1, while the least tolerant will accept no person of opposite colour in their neighbourhood.

From the above, it should be obvious that there are a number of greens and reds who would be happy to live together in some combination. However, in order to analyse which combinations are most likely to occur, as well as the processes that cause people to leave or join a neighbourhood, we must turn to a little maths...

The first thing to do is create tolerance schedules for our two groups. Recalling our tolerance ratios from earlier (most = 2:1, median = 1:1, least = 0:1, etc), we depict these as follows:

Next, we translate these tolerance schedules into “absolute-numbers” curves by simply multiplying the population level by the corresponding tolerance ratio. In other words, we're finding out how many greens each red is prepared to tolerate, and vice versa. The parabolic shape of the resulting curves reflect the diminishing level of tolerance among each population, as you move from the most "tolerant" individual of the group to the most “racist”:

Note the placement of the green population on the vertical axis, and how this allows us to easily compare the interaction with the red population. There are three areas: 1) Any point within the overlap area (bottom left-hand corner) represents a combination of reds and greens that can coexist happily in the same neighbourhood. 2) Points beneath the red curve, but to the right of the green curve, represent a mixture where all the reds will be satisfied, but not all the greens. 3) In contrast, any point within the green curve and above the red curve corresponds to a combination where all the greens in the neighbourhood are content, but not all the reds.

Importantly, the above figure also depicts the dynamics of motion of the system. This is what the arrows are showing us: We can see how the populations of the two groups will be changing at any particular point.[***] For example, the bottom-left arrow (pointing up and to the right) indicates that the numbers of reds and greens will be increasing together at low population levels. However, if reds begin to settle in the neighbourhood at a faster rate than greens, then some greens will be motivated to leave. This in turn exacerbates the problem, since the ratio of greens to reds now becomes even worse which prompts more greens to leave! And so... we inexorably move towards neighbourhood comprised entirely of reds. (This is depicted by the bottom-middle arrow that is pointing down to the right.) In this way, the dynamics of motion show us that we can approach three possible equilibria. However, only two of these -- the completely segregated outcomes -- are stable. The mixed equilibrium combination is unstable since any disturbance, i.e. the departure or arrival of a new neighbour, has the potential to set off a chain reaction that will ultimately lead to one colour completely dominating the neighbourhood!

Now, of course, this was a rather specific example used for illustration. You might ask whether the segregated outcome depends for instance, on the relative sizes of our green and red populations? The answer to that question, however, is “not really”. A one-colour equilibrium is still the inevitable result even when we have reds and greens in equal numbers. Having said that, having equal numbers together with steeper tolerance schedules will tend to produce a stable equilibrium. For example, if there are 100 reds and greens and they both have a tolerance schedule where the median individual can tolerate being in a 2.5:1 minority, then we end up with the following:

In this alternative schedule, we see that all three equlibria are stable. Importantly, this includes the interior solution (a mix of 80 reds and 80 greens), which is robust to fairly large perturbations. (However, if we are already at a segregated outcome, then a move towards a stable mixed result will require the concerted entry of more than 25 percent of the other colour.)

THOUGHT FOR THE DAY: It’s easy to assume that people living in segregated neighbourhoods are relatively racist. Similarly, we might also assume that people who express a desire to live in integrated neighbourhoods will automatically arrive at such an outcome through conventional market processes. However, the basic Schelling model shows that even small preferences for a degree of homogeneity – for just a few of our neighbours to be “like us” – is enough to cause segregated outcomes at the aggregate. (In other words, completely segregated neighbourhoods may be inevitable even when the majority of people are fine with being in a minority!) The model depicted here is very simplified and hardly perfect, but still provides extremely valuable insights into the emergent dynamics of segregation. It also illustrates how relatively simple maths can be used to deal with complex and counter-intuitive phenomena.

PS - The above model is part of a broader literature called “agent-based modelling”, which is used for analysing anything from traffic flows to health epidemics. While maths was crucial to proving that his results held generally, Schelling initially used coins and a chessboard to illustrate his point. For a pretty cool illustration of agent-based modelling at work, try one of these two online versions of the segregation model. Or just watch the below video!

[Note: Those of you paying attention might notice that this video is slightly different in terms of set-up to the model that I have discussed above, in that it each person (/egg's) neighbourhood is limited to the spaces immediately alongside them. Schelling called this a "spatial proximity model". In contrast, the model presented above analyses the make-up of the neighbourhood as a whole and is called the "bounded neighbourhood model".]

---

[*]  "Micromotives and Macrobehaviour" is the title of Schelling's brilliant book, which forms the basis for today's post. If the type of emergent economic outcomes that I discuss here is of interest to you, then M&M is a must-read.
[**] In technical language: we assume that the cumulative frequency distribution of the “tolerances” of individuals is represented by a straight line. This is done for ease of illustration, although it is not overly problematic to experiment with different distributions (as Schelling does here). 

[***] What we have here is a simple phase diagram, which is very useful for analysing the stability of any dynamic system where you have to worry about things like multiple equlibria, tipping points, saddle paths, etc. You usually determine the direction of your arrows in a phase diagram by taking the first derivatives of your equations, but since I'm trying to keep things simple here I won’t bother with that now.

Maths and Economics - The Interlude

Those of you paying attention may have noticed that I yet to deliver on my promise to provide a second economic-maths example. Apologies for the delay, but I've had a workload from hell this passed week. Honestly, I'll try and get around to it during the next few days when the storm clears. (The good news is that I at least know what I am going to post about...) 

Anyway, I'd thought I'd share a little statistic in the interim:

As of the present moment, my original post on why we need maths in economics has been viewed more than five times as much as the follow-up oil-tank model.

I found this pretty interesting. The first post was meant to act as a general defence of the subject. However, realising that general defences can be a bit vague I was motivated to offer specific examples to illustrat my points. I really thought that this was where the "value" would lie, since readers would be able to see the actual mechanics at work. Not the case apparently...

Perhaps the calculus in the “oil-tank” example put a few people off. Fair enough, I don't expect everyone to care for going through the maths... But I at least expected them to look over the post and skim to the model's conclusions. However, the click-through rates* indicate that even this hasn't been happening. Rather, it seems that people simply aren't as interested in a specific example of maths being useful to shed light on an economic problem, as they are in having a general discussion about it. In their own little way, these posts offer some insight into behavioural attitudes towards maths in economics – a sort of informal experiment if you will, which shows people leaning towards what I would regard as an important, but less precise approach.

THOUGHT FOR THE DAY: It's easy to take a position on things without getting to specifics. Providing concrete examples to back-up or demonstrate your point can be harder, but at least you give people something to work with. However, whether everyone will be interested in the evidence at hand is still open to question.** 

---

* Clearly, Stickman's Corral isn't competing with any blogs worth mentioning in terms of numbers. Still, I think that my first maths post in particular has drawn a pretty respectable audience for a lowly grad student blogging in a sea of anonymity :)

** For the dramatic version of what I’m talking about, we have some classic Hollywood.

Why we need maths in economics 2(a) - The Oil Tank Model

As promised, here is the first example of a highly stylised model, which -- despite its unrealistic simplicity -- still produces valid (and unexpected) results for real-life decisions. 

Topic: The optimal depletion of an oil resource.

Aim: To show that changes in the interest rate don't always have clear implications for our rate of withdrawal.

[NOTE: Those of you with no desire to go through all the equations (although that kinda defeats the purpose) can just scroll down to "THOUGHT FOR THE DAY".]

The model that I am going to refer to here is commonly known as the "oil tank" model. As the name suggests, imagine we have an oil resource that can be exploited exactly as if it were in a tank or drum. A key assumption here is that rate of production is proportional to the number of wells; you drill two holes, the oil comes out twice as fast. This is grossly unrealistic. As any geologist would probably tell you: drilling holes affects well pressure around the borehole, geological structure and reserves are not uniform, etc, etc. There are also other economic considerations that we ignore here, such as the fact that we shall assume a constant oil price[*], uniform costs of production, and an oil field that has been unitized. (Unitization essentially means that the resource is controlled by one supplier. This is not completely correct, but just go with it...) Bottom line: lots of simplifications. 

So, let's specify the model and then solve to see what we get.

$Q =$ Amount of oil that can be recovered
$q_t = q = $ The (constant) rate of production
$T = Q/q =$ Extraction time (remember that extraction is proportional to the number holes you drill)
$c =$ The cost to develop each well
$p =$ Price of oil
$r =$ Interest rate

The present value of the well to us as producers can thus be written as:

\begin{equation}
V = -cq + \int_0^T \! pqe^{-rt} \, \mathrm{d}t
\end{equation}

Where the first term on the RHS equals the cost of our investment. (Note: We assume upfront payments for simplicity[**], and the second term under the integral represents the total revenue flow from production until we exhaust the resource at time, $T$.)

Solving for the above gives:
\begin{equation}
V = q \left[ -c + \frac{p \left(1-e^{-rT}\right)}{r} \right]
\end{equation}
Now that we have a PV equation for our oil field, we can used the standard optimization procedure to find the optimal production rate... simply derive w.r.t. $q$ and set equal to zero. Recalling that $T = Q/q$:

\begin{align} \frac{\mathrm{d}V}{\mathrm{d}q} = 0 &= - c + \frac{p \left(1-e^{-rT}\right)}{r} + pq \frac{\mathrm{d}T}{\mathrm{d}q}e^{-rT} \\ &= - c + \frac{p \left(1-e^{-rQ/q}\right)}{r} - \frac{pQ}{dq}e^{-rQ/q} \end{align}
Since it usually makes sense to think in terms of normalised costs (i.e. how much it costs to develop a field relative to the price of oil), we can just divide through by p and solve to get:

\begin{equation}
\frac{c}{p} = \frac{1-e^{-rQ/q}}{r} - \frac{Qe^{-rQ/q}}{q}
\end{equation}
Okay, we're getting there. The above equation is important because we can use it to analyse how changes in two "given" variables -- investment cost per well $\left( \frac{c}{p} \right)$ and interest rate $\left( r \right)$ -- affect our "choice" variable, namely the production rate $\left( q \right)$. I want to focus on the latter in this blog post. So let's think about what our intuition would tell us about the relationship between production rates and interest.

If an oil producer asks, "What should I do if the interest rate goes up?", the instinctive response is "produce faster". After all, this seems perfectly intuitive; higher rates of interest signal increasing impatience and we can put the money that we receive from production into the bank at the higher interest. So, let's see if this is how things work out.

For this last part, it will make things easier to follow if we define the previous equation as being equivalent to a function, which we'll call $f$:

\begin{equation}
f \equiv \frac{c}{p} = \frac{1-e^{-rQ/q}}{r} - \frac{Qe^{-rQ/q}}{q}
\end{equation}

We can then use a general derivation rule,

\begin{equation}
\mathrm{d}f = \frac{\partial f}{\partial r} \mathrm{d}r + \frac{\partial f}{\partial q} \mathrm{d}q,
\end{equation}
to get our second last equation (with a little help from the chain, quotient and exponent rules):

\begin{align}
\mathrm{d}f = &\left[ \frac{ \left(\frac{Q}{q}e^{-rQ/q} \right)r - \left( 1 - e^{-rQ/q} \right)}{r^2} - \frac{-Q}{q}\frac{Qe^{-rQ/q}}{q} \right] \mathrm{d}r \\ &+ \left[ \frac{-r \frac{Q}{q^2}e^{-rQ/q} }{r} - \frac{\left( r \frac{Q}{q^2}Qe^{-rQ/q}\right)q - Qe^{-rQ/q} }{q^2} \right] \mathrm{d}q .
\end{align}

Simplifying and setting equal to zero again for optimization yields,

\begin{equation}
\mathrm{d}f = 0 = \left[ \frac{ \left( r \frac{Q}{q} \right) e^{-rQ/q} - \left( 1 - e^{-rQ/q} \right)}{r^2} + \frac{Q^2}{{q}^2}e^{-rQ/q} \right] \mathrm{d}r - \left[ \frac{rQ^2}{q^3}e^{-rQ/q} \right] \mathrm{d}q.
\end{equation}
And now -- if you've made it this far! -- we are at last ready to produce our final equation... i.e. The one that describes the relationship between the production rate $(q)$ and the interest rate $(r)$. All we need to do is rearrange the above equation and then simplify this bad boy as follows,

\begin{align}
\frac{\mathrm{d}q}{\mathrm{d}r} &= \frac{\frac{ \left( r \frac{Q}{q}\right)e^{-rQ/q} - \left( 1 - e^{-rQ / q} \right) }{r^2} + \frac{Q^2}{q^2}e^{-rQ/q}}{\frac{rQ^2}{q^3} e^{-rQ/q}} \\ &= \frac{ -\frac{1}{r} \left[ \frac{ \left( 1 - e^{-rQ / q} \right) }{r} - \frac{Q}{q}e^{-rQ/q} \right] + \frac{Q^2}{q^2}e^{-rQ/q}}{\frac{rQ^2}{q^3} e^{-rQ/q}} \\ &= \frac{ -\frac{1}{r} \frac{c}{p} + \frac{Q^2}{q^2}e^{-rQ/q} }{\frac{rQ^2}{q^3} e^{-rQ/q}} .
\end{align}

(For this last step, just recall our equation for $\frac{c}{p}$ above.)

Looking at our final equation above, it should be clear that we cannot directly see how the rate of production ($q$) should change. That's because we are left with an equation of ambiguous sign:

\begin{equation}
\frac{\mathrm{d}q}{\mathrm{d}r} =\frac{ \text{something negative} + \text{something positive} }{\text{something positive} }.
\end{equation}
In other words, it is not certain whether $\frac{\mathrm{d}q}{\mathrm{d}r}$ will be negative (i.e. we should decrease our production rate) or positive (i.e. we should increase production). It all depends on how high our costs of investment $(\frac{c}{p})$ are! If investment costs are "high", then we should produce less, since $\frac{\mathrm{d}q}{\mathrm{d}r} < 0$. On the other hand, if they are "low", then we should produce more, since $\frac{\mathrm{d}q}{\mathrm{d}r} > 0$.

THOUGHT FOR THE DAY: The "oil tank" model is crude and dramatically simplified. Yet, it still provides very useful insights. We have isolated key variables to see how they might affect each other on a ceteris paribus basis. What we focused on here is how changing the rate of interest actually has an ambiguous effect on the optimal depletion of an oil resource. Accordingly, we can say that a higher rate of interest (r) plays two roles:

1) "Impatience" - We want our money more quickly (and capital costs are high)

2) "Opportunity cost of capital" - To get more money, we need to drill more wells. However, this is expensive in the face of high interest rates.

The first of these is the one that comes completely naturally to us. As such, we're almost always predisposed to think that we should just do something "quicker" (in this case producing oil) when faced with a higher rate of interest. However, having worked through the analysis, we see that the interest rate has another important impact that is easily glossed over; namely it represents the opportunity cost of capital. If we relied solely on our intuition, then it's easy to make the error of ignoring this latter effect. Indeed, as I argued in my previous post, working through the maths has helped us to make an intuitive argument that only becomes "intuitive" upon reflection!

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[*] Although, we can compare the fact that oil companies typically make forecasts based on "set" prices to analyse profitability thresholds and so forth.
[**] We could take PVs for investment costs over time if we really wanted...

Why we need maths in economics

Secondary subject line: Why words and intuition aren’t always good enough.

A lot has been written about the use of mathematics in economics. Much of it negative and particularly so in the wake of the financial crisis (e.g. here, here, here and here). The root cause of these troubles apparently being that economists suffer from a major case of penis envy physics envy. Now, I’m no John Nash – or Paul Samuelson for that matter – but I feel the need to weigh in with my meagre two cents all the same.

First, let me say that I can wholly agree that a fixation with mathematical formulation has been detrimental to economics in a number of instances. By any reasonable account, many an economist has become enamoured of their elegant theoretical models and abstract equilibria points, all the while ignoring the implausibility of their founding assumptions and the unpredictability of real-life phenomena.  Point made, let’s correct for these imbalances and move on.

Unfortunately, not quite that simple. I’ve noticed an increasing tendency – especially among those in the blogosphere – to decry the use of virtually any maths in economics as at best worthless, and at worst downright harmful. Again, there are certain groups preaching that economics has no need for maths (or empirical validation for that matter), or that all useful economic knowledge can be derived through a set of self-evident axioms anchored in logic. “Real life is too complicated to be laid out in a set of abstract functions!” “The use of simplified models and assumptions merely distorts our understanding of the facts!” To me, this hypoallergenic position is to throw out the baby screaming with the bathwater and, moreover, it largely misses the point.

Some comments: