Posted in History & Literature

Demand And Supply

How does the economy function? On the surface, economics seems extremely complex and intricate, changing dynamically due to what appear to be insignificant factors. For example, one can predict a recession from the increasing attractiveness of waitresses. But economics relies largely on two simple laws: the law of demand and the law of supply.

The law of demand states that as the price of a good goes up, people demand less quantities of that good. This makes logical sense as (rational) consumers want to spend the least amount of money possible for something. When plotted on a graph with price (P) as the y-axis and quantity (Q) as the x-asis, we can show that demand (D) is a downward-sloping curve.

The law of supply states that as the price of a good goes up, people supply more quantities of that good. This makes logical sense as (rational) producers want to sell something for as much money as they can. When plotted on a graph with price (P) as the y-axis and quantity (Q) as the x-asis, we can show that demand (S) is a upward-sloping curve.

When we superimpose these two laws on the same graph, we get a nice X-shape as the two lines cross over. The point where they cross is called the market equilibrium and this is where the consumers demand exactly the right amount of goods that the producers are willing to supply at a given price. If a good is being sold at a price higher than the equilibrium price, consumers demand less of the good and producers are left with an excess. If the price is lower, then there is a shortage as demand exceeds supply. Over time, the price is pushed towards the market equilibrium. Thus, thanks to the laws of supply and demand, the market automatically adjusts to the price to accurately reflect the value of the good.

Adam Smith, the father of economics, called this the invisible hand – a force driven by the individual ambitions of consumers and producers to balance the market. This force is not found in centrally planned economies of communist states, as the price and quantity supplied is fixed by the government. A pure free market is only driven by the invisible hand. Most modern countries’ economies are mixed economies, usually in the form of free markets with some government intervention.

Although economics appears complicated, it essentially boils down to the laws of supply and demand. By understanding the principles of demand and supply, one can begin to understand more complicated economic theories such as aggregate demand and supply, elasticity, foreign currency exchange and trade. Real economic situations such as oil cartels, trade embargos and taxation can be broken down and modelled using PQ-diagrams (depicting the demand and supply curves).

The laws of supply and demand are two crucial laws of economics that everyone should have some understanding of, as it can be extremely useful in everyday life. Not only do they apply in obvious situations such as running a store or a business, or understanding how the economy works, but it can be applied to negotiating too. One of the fundamental principles of negotiating is finding the balance between what one person wants (demand) and what the other person is willing to do (supply). It is amazing how useful knowing that simply being slightly flexible is in negotiations.

Posted in Science & Nature

Seven Bridges Of Konigsberg

The city of Königsberg (capital of Prussia, now Kaliningrad, Russia) has the Pregel River running through the middle, with islands at the centre of the river connected by seven bridges. Is it possible to cross all of these bridges while only crossing them only once each?

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If you try to solve this problem, you soon discover that it is incredibly difficult not to cross the same bridge twice. But it is difficult to tackle this problem in a brute force manner. To calculate all of the permutations in the order of bridges, you use 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040, meaning that there are 5040 possible arrangements of bridges. Then how can you prove if the problem is solvable or not?

The great mathematician Leonhard Euler, upon being asked to solve the problem, is reported to have said that the problem is impossible to solve on the spot. In 1735, he proved his answer by modelling the seven bridges of Königsberg in a diagram of four dots connected by lines (representing the bridges).

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By using this model, the problem is converted into a “draw in one stroke” problem, which is also called a Euler walk to honour Euler’s contributions. Euler discovered many properties and laws regarding such problems. If a certain point is the starting point, then the line must first leave the point, then even if it comes back to the point, it must leave again. Ergo, the starting point must have an odd number of lines connected to it. The opposite applies to the ending point, where a line must enter the point, and if it leaves the point it must come back to it. Ergo, the ending point must also have an odd number of lines connected to it. In the case of a Euler walk, the starting and ending points are identical, so the number of lines is the sum of two odd numbers, making it an even number. Thus, to find out whether a picture can be drawn using one line, use the following laws:

  1. If there are no points of odd degree (odd number of lines), the starting and ending points are identical.
  2. If there are two points of odd degree, the starting and ending points are different.
  3. If there are one of more than two points of odd degree, it is impossible to draw using one stroke.

Thus, a Euler walk is only possible if there are 0 or 2 points of odd degree. Looking at the seven bridges of Königsberg problem, we can see that A is connected to 5 lines and B, C and D are connected to 3 lines each. As there are four points of odd degree, we have thus proved that it is impossible to draw a path that crosses all the bridges while not crossing any bridge more than once.

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