Marginal and Total Revenue: A Mathematical Link

For many, the concept of the marginal is a tough one. I've always enjoyed economics with a mathematical edge, and this is quite a quaint way of tying the two incredibly linked economic ideas together.
We know the relationship that when MR=0, TR is maximised (as once MR becomes negative TR must fall). But this relationship can be explained using first differentials and turning points.
We know that MR is the gradient function of TR, and as this value is always falling (negative bell shape) we know that MR must have a negative gradient. Also, the turning point of the TR curve is where the gradient function has a value of zero, or essentially:
QTRmax = QMR0.


If you look at a total revenue curve, its upside-down bell curve shape resembles that of a negative quadratic, and that is essentially what it is. Therefore, we can give this TR curve a general mathematical formula of y=ax-x^2 (where a is a positive integer).
Because of the nature of the MR curve, we can differentiate the TR curve to find a generic formula for this curve as well:
y = ax - x^2
dy/dx = a - 2x
This formula matches the negative linear function of the MR curve, hence when MR=0:
a - 2x = 0
a = 2x
a/2 = x
Therefore revenue maximisation is achieved where x (or quantity) is equal to half the distance between the two x axis intercepts of the curve.
Now we didn't necessarily need to differentiate to show that a/2 is the output where revenue is maximised, as we can see it by symmetry of the TR curve, but the use of the maths just proves it nicely.
This esssentially proves that the value of a/2 is the output where revenue is maximised, so for those economists who struggle with the wordy explanations of the relationship, this nice link may be helpful.
From this we can find the TR at any value of x, merely by setting the value of a and hence x for a specific firm.

Economic History: John Nash

Economics is an area of study which has played host to a series of incredibly influential individuals, and many 'big name' conflicts; from Hayak to Keynes in the world of the macroeconomy and fiscal policy, to Nash and Smith in the nuances of the microeconomy, there have always been two sides to every answer. In this post I will be taking a look at the inroads into game theory and behavioural economics carved by John Forbes Nash.
Nash was a highly skilled American mathematician and was heralded for much of his work. The feature film 'A Beautiful Mind' was based on him and portrayed his life as a mentally-troubled economic genius. In his mid-life, he experienced extreme paranoia, and believed that there was a Soviet plot trying to overturn America. He was later diagnosed to be a sufferer of Schizophrenia, which is still nowadays confused with dual personality disorder, a very different ailment. Essentially Nash feared that the Soviet nations were trying to take over America, and using his great mind tried to find links and messages everywhere, always finding nothing, to his dismay, as there was nothing to find.
Nash is considered to be one of the forerunners in the world of game theory, a highly mathematical and intricate area of microeconomics. One of Nash's most famous 'games' is discussed an explained in the film in the layman's terms of 'pulling', it is known as the Nash Equilibrium and forms part of his concept of Governing Dynamics.
The Nash Equilibrium is the scenario where a group of two or more people interact, and assumes all parties involved know the strategies of all other parties. Nash himself stated that this is one way to reach an efficient outcome, or 'pay-off'', which contrasted with the long-held and respected theory of Adam Smith coined nowadays as the 'Invisible Hand'. In essence, Smith first used the Invisible Hand to describe 'cooperation without coercion', by which consumers and producers would act to benefit themselves, which would in turn benefit the society. Since then the metaphor has been extended to fit any scenario when one self-interested action generates a positive pay-off in society. Nash's theory, however, was one of cooperation with communication. He argued that if everyone acted to benefit society primarily, and themselves after putting the needs of everyone else first, this would see the best results for society and as a consequence, the individuals. If you consider it at its most basic level, Smith said acting selfishly will benefit everyone best, ensuring a maximum pay-off for all individuals, whilst Nash said acting selflessly will benefit everyone best, and therefore as a consequence, the individual will receive an optimal pay-off.
This was a massive leap forward in economic theory, as it was so widely applicable. The theory could be used in the macroeconomy (and provides a justification for left-wing government policy and a command economy), as well as in the field of games, where it could refer to groups as small as two individuals acting in the interest of society. His theory has sometimes been called ‘the greater good’, as it refers to collaboration of separate groups to reach a goal beyond that of which can be achieved through self-interest.
So with this theory, Nash achieved recognition, something which he always sought. I do not necessarily advocated that Nash had the right ideas, on the contrary, I would personally put my faith in Smith’s theory in terms of the wider economy, but I will tackle this debate in a later post.
The theory proposed by Nash is excellent in smaller markets and games, where it is possible for everyone to know everyone else’s strategy (a world of perfect information), as this is the fundamental variable he specifies. It is true that it can produce the best outcome for society under certain circumstances and provides a viable alternative to Smith’s notions; if anything their ideas represent right- and left-ism in a basic form, an age-old contrast.
I suppose one of the ironies from Nash’s life was that his governing dynamics if anything offered good arguments for a leftist, equalitarian government as he promoted acting the interest of everyone for the benefit of the individual, whilst at the height of his illness; he feared that very system’s propagators were trying to take over the USA.

Eurozone Turmoil

Since the Euro became the legal currency of over a dozen nations in the EU, it has been seen as a genuine challenge to the English Pound, and a currency that the British parliament has frequently considered joining. However, since the start of the recession in 2008, the strength of the Euro has come into dispute.

In the news of late have been figures of macroeconomic collapse in the Eurozone nations of Portugal, Ireland, Spain, Greece and Italy, where several of the aforementioned countries have experienced a drop in National Output, high unemployment rates (in 2011 Spanish unemployment hit a rate of 21.1%), and reports of shocking national debt (Italy's national debt is now about 115% of its GDP). Greece, Portugal and Ireland have all required sizeable bailouts by the ECB due to their low growth and inflationary pressures due to cost-push scenarios of more expensive raw materials and issues with employment.

With all this going on, it seems a wonder that the Eurozone as a whole hasn't seen more trouble, and largely, this has been due to the high growth rates seen in Germany and France. While the weaker Eurozone nations were faltering, Germany saw a sharp upturn in exports and growth in 2010 and early 2011, with growth rates for 2010 hitting around 3.6%.

This highlights one of the first issues with the Euro as a currency for several nations with different economic strengths and compositions, namely; high German growth requires a raising of the ECB's rate of interest, yet the lower growth and problems with credit in nations like Spain require a lower interest rate. So here lies the issue, do the ECB promote growth in the struggling nations by keeping interest rates low, whilst harming the French and German economies which will suffer from higher inflation due to demand-pull pressures, or do they raise interest rates, helping Germany and France avoid high inflation whilst potentially stagnating the economies of Spain and Greece.

In the end, the ECB chose to raise the interest rates to help out the nations saving the Eurozone, but now, in August 2011, that Germany has seen near static growth of just 0.1% in the most recent quarter the issues in the rest of the Eurozone may become a lot worse.

Germany has been the largest contributor to the Greek bailout packages, yet back in June the Greeks took to the streets, calling Angela Merkel (German Chancellor) a Nazi and carrying banners of the EU stars rearranged in the form of a Swastika. Now that Germany is seemingly struggling too, the Greeks may now realise that their previous outrage was totally misplaced.

It seems that it is predominantly the nations of the Eurozone which are typically fuelled by tourism as a key export have suffered greatly due to the tighter availability of credit for the holiday-makers from the UK and Northern Europe. The UK chancellor George Osbourne has discussed an 'export-lead recovery' from its own economic issues, it seems that for the nations of Spain and Portugal, more time and investment needs to go into making the economy more self-sufficient and less dependent on the outside world.

Though possibly the most worrying development of the Eurozone turmoil is the massive bad-debt in Italy. The buying of government bonds in Italy has acted like a loan to the government, but you have to ask if this is truly sustainable. It is fair to say the Eurozone could survive the collapse of the Greek economy and maybe even a flailing Irish and a floundering Portuguese economy, but if Italy goes down, as the third largest Eurozone economy it will surely have sufficient repercussions to put the Eurozone beyond repair, especially if this is coupled with the low growth seen now in Germany.

The debt of Italy has to, in part, be accredited to some level of corruption within the government, or issues of the principal-agent problem, by which the government act in a way undesired by the people and not in the best interests of the nation as a whole. The severe austerity measures now seen in Italy reflect the fear that people have that if the national debt gets any worse, it could enter a second recession, of far greater proportions.

To try and counter this, the Italian government has entered into near draconian spending cuts and austerity measures, but I have to question if it can recover whilst being dragged down by other economies.

This is why the Euro is such a contentious issue; without the group interest of the Eurozone nations in Italy and Greece's welfare, nations like France and The Netherlands would not worry so much about bailing them out of their issues by buying up government bonds, but due to the way all 17 economies are now so dependent on each other now, it is hard to see them all recover. If the stronger nations could in some way just release responsibility for keeping Greece and co. afloat, they could probably all make their way back to full strength eventually.

This is where I'd like to propose a theory; the Euro is a bad idea. Without such a large group of nations holding onto the same currency, Germany could regain its powers as a key exporter, fuelling its own economic growth, which even without being connected to other Eurozone nations by the same currency would still provide a solid foundation for a multiplier effect to kick in. All the nations could set interest rates appropriate for their own levels of growth, not those of the most important economy, and the feeling of some countries that because they are all under one currency the stronger countries can support the weaker ones would diminish, leaving a greater onus on recovering from the debt crisis.