They say that he who treats what also misses. This is certainly not the case in the financial sector, for perhaps no other economic sector has as much money moving around as here. Investors who manage portfolios, banking executives, or scientists who make up the nation's economic models treat unimaginable financial flows. They have such a big responsibility, and that's why they can't shield themselves from differential calculus. Indeed, it provides powerful tools for estimating the future or maximizing profit. So let's try to uncover some of the tricks used in dealing with money.
In physics, the laws of preservation apply, but in finance? Regular banks that store people's money and then lend it to other people may not seem to follow suit. So before we go straight to the true differential, let's stick with the banks and look at the methods they use.
This is a common practice, e.g. in the Czech Republic, the smallest reserve allowed by law is $p$ two percent.
So let's have a bank that takes money from people to save. Naturally, people will want to keep money in the bank for a long time, so the bank will want to invest with it for that time. So he sets a rule that he keeps only $p=10\,\%$ of his money as a reserve if someone comes to collect.
Let us now look at the total number of money that exists. Let's say at the beginning the bank had some amount of $C$. Subsequently, it loaned out an amount of $(1-p)C$, as it had to keep $pC$ as financial reserves. But suppose that people spent the money they borrowed and the new owners of the money hid it, of course, in a bank. This is a pretty realistic assumption, because people put money in the bank a lot, and even if there was another bank, let's assume that all banks have the same $p$.
So the bank has now got a new $(1-p)C$, but of course it won't keep all of it, but it will go on loan again. That means it will lend $(1-p)(1-p)C=(1-p)^2C$ of money. These will come back to her again and lend her $(1-p)^3C$ money etc. So the question is how much of the total $S_N$ money so overall will be in circulation just because of the original $C$. Let's write this down using the equation: $$\begin{align*} S_N &= C + (1-p) C + (1-p)^2 C + (1-p)^3 C + \dots + (1-p)^N C \\ S_N &= \sum_{i=0}^N (1-p)^i C \,. \end{align*}$$
One possible reason for a geometric name is that each additional member is a multiple of the previous one. Multiplication is a very important concept for geometry, as we can use it to describe volume.
In this equation, we added a total of $N+1$ of members. We also wrote the equation using notation with the sum for convenience. Our example is actually the sum of a sequence that we call a geometric sequence. We've done a similar task before, trying to figure out a formula for the sum of the first $n$ of natural numbers in Alive Geometry. The best method turned out to be telescopic, the kind where a lot of members were subtracted. For this geometric sequence, we can achieve telescopicity by multiplying the entire sum by $(1-p)$: $$\begin{align*} S_N &= C + (1-p) C + (1-p)^2 C + (1-p)^3 C + \dots + (1-p)^N C \\ (1-p)S_N &= (1-p) C + (1-p)^2 C + (1-p)^3 C + \dots + (1-p)^N C + (1-p)^{N+1} C \\ \end{align*}$$
We see a lot of similar members emerging, so let's try to subtract the first and second lines from each other: $$\begin{align*} S_N - (1-p) S_N &= C + (1-p) C - (1-p) C + \dots + (1-p)^{N+1} C \\ S_N(1 - (1-p)) &= C + (1-p)^{N+1} C \\ S_N &= \frac{C + (1-p)^{N+1} C}{1 - (1-p)} \,. \end{align*}$$
This gave us a generally valid formula for the sum of $N+1$ members of a geometric sequence. Naturally, we're interested in what happens if $N$ is very large, that is, if it grows beyond all limits. To do this, we do a limit transition: $$\begin{align*} S \equiv \lim_{N \to \infty} S_N = \lim_{N \to \infty} \frac{C + (1-p)^{N+1} C}{1 - (1-p)} = \frac{C}{p} \,. \end{align*}$$
We took advantage of the fact that $(1-p)$ is less than $1$, this way $(1-p)^N$ is getting smaller and smaller and so it goes to zero. But we got a very surprising result: in some mysterious way, we put $C$ of money into the bank at the beginning, and thanks to the bank, $C/p = 10 C$ of money is now in circulation. So the bank was able to increase the total amount of money tenfold just by using some financial strategy.
This is far from being a theoretical example, for example, in the US there is theoretically no lower limit to $p$. However, the lower the $p$, the more money the bank can just create. Of course, if $p$ is very low, the bank has almost no reserves and no one will believe it, however how many people actually know how the banking system works? In fact, noted businessman Henry Ford once said that if people knew how the money system worked, there would be a revolution.
The example of reserve banking may not have been so mathematically interesting, but it did show us one thing: in economics, we try to maximize our profit at all costs. Fortunately, the differential number is highly tailored to this, as we can simply look for the highs and lows of function to help. So all we have to do is write our gain as a function of some variable, and we can find the highest point using the derivative.
Let's imagine, for example, the function $f(x) = - (x+1)^2 + 4$. We ask for which $x$ is peaking. Fortunately, the derivative can help with that. Derivation expresses the tendency of the tangent to function. And if the function gets maximum, the tangent slope is zero, see the $f(x)$ diagram below:
$$\begin{align*} f'(x) &= 0 \\ (-x^2 + 2x - 1 +4)' &= 0 \\ -2x + 2 + 0 &= 0 \\ x_{\mathrm{max}} &= -1 \,. \end{align*}$$
Indeed, you can see from the sketch that the function will peak at $x=-1$. The function value at maximum is simply given as $f(x_{\mathrm{max}})= 4$.
But we withheld a little thing: we can't tell by derivative if it's the maximum of a function or the minimum: the derivative is zero in both the maximum and the minimum. Collectively, we indicate highs and lows by extremes. At the same time, this is just a local maximum, that is, a maximum function only in its immediate vicinity.
We can check everything for $f(x) = x^2$. This one has a derivative of zero for $x=0$. But at the same time, there is minimal function. We can't find the maximum by derivative, because the function doesn't even have a maximum: it grows beyond all limits for $x\to \infty$. So using the derivative, you can find the extremes of the function, but then you have to figure out if it's the maximum or the minimum.
The extreme search method works for almost any $f(x)$ function. That may not sound very interesting, but we should pause and appreciate this fact. We can substitute $f(x)$ for the changing temperature over time, the density of the Earth depending on depth, the popularity of the president changing over time or according to geography, or perhaps the average number of people's money depending on their age. Differential calculus thus provides us with an entirely general method that finds applications in many fields that we would not have expected: and we can always rely on a familiar mathematical basis.
But for now, let us limit ourselves to the field of economics. Here, for example, firms looking for an optimal strategy for selling their products can use the search for a maximum function. Suppose, as a company, you sell, for example, jars of honey. Everyone knows that the law of supply and demand applies, so the more glasses you want to sell, the less each state will be. The question is, what number of glasses is most optimal for sale?
In the $P(Q)$ function and the entire calculation, we omitted units.
Let's say the number of glasses sold is $Q$. At the same time, the price of one glass of honey is a function of $P(Q) = 100 - Q$: the more I sell, the less one glass will cost. So the revenues can be described as $P(Q)\cdot Q$. Furthermore, we have some expenditure on honey production, which may be described by the function $C(Q)=20 + Q$: this function represents the fixed costs of production and the cost of producing each additional slider. Total earnings will be $$\begin{align*} V(Q) &= P(Q) - P_-(Q) \\ V(Q) &= 100Q - Q^2 -20 - Q = -Q^2 + 99 Q -20\,. \end{align*}$$
What $Q$ maximizes this feature? We have to derive by $Q$: $$\begin{align*} V'(Q) = -2Q + 99 &= 0\\ 2Q &= 99\\ Q &= 49{,}5\,. \end{align*}$$
According to this forecast, I should make $45{,}5$ jars of honey. If I made more, the price of honey would fall, along with the cost of production, resulting in less earnings. In the same way, with less production, I wouldn't sell enough goods and I wouldn't achieve full earnings.
Using the linear function for $P(Q)$ is a neglect that applies to small scales (such as our honey business), the actual market is driven more by the function $P(Q) = a/(b+Q^2)$. Furthermore, production costs per unit in large production tend to decrease with quantity as we are able to make production more efficient. Then it would make sense to choose a cost function such as $C(Q)=c + d\sqrt{Q}$. However, the calculation with these parameters is already too complicated for us, but only computationally! In principle, the same method could still be used to find the optimal number of goods.
The so-called invisible hand of the market is often referred to in economics. This is the idea that market prices and the quantity of goods sold are not driven by some central planning or by the will of the people, but are organized spontaneously, organized by that invisible hand.
Indeed, this analysis makes quite sense, since there are typically many more firms than one in a market environment. A company that did not respect the optimal prices and quantities of its goods would be lost, therefore, in a process similar to Darwinian evolution, there would always be one firm left with the price and quantity of products in the optimal configuration.
The two terms that come with such ideas are supply and demand. The offer is synonymous with the number of goods sold: it is the quantity of goods the trader offers. Demand means how much goods people actually want and it comes down to price: the more demand, the more price a trader can afford to put on a product. Therefore, we can call the $P(Q)$ curve the demand curve and the $Q(P)$ curve the supply curve. It is the work of the invisible hand of the market that these two curves intersect and the goods are sold at a single price. For if it were sold below cost, a profitable man would immediately notice, buy it, and sell it for more.
Let's stick with our simple supply-demand model for a moment. What would happen if one more firm selling honey went to the honey market? Suppose our company produces meadow honey and the incoming company produces mountain honey. We can write two equations again for the price of two firms' glasses of honey: $$\begin{align*} P_l(Q_l, Q_h) &= 100 - 2Q_l - Q_h \,, \\ P_h(Q_l, Q_h) &= 100 - 2Q_h - Q_l \,. \end{align*}$$
In these curves, we see how demand for mountain honey is slightly influenced by meadow honey. But since they are not exactly identical products, they do not affect demand in the same way as mountain honey. We see similar behaviour in demand for meadow honey. Now we still need to know what the output prices will be for both firms, assuming a similar dependency to the previous case: $$\begin{align*} C_l (Q_l) &=20 + Q_l \,, \\ C_h (Q_h) &=20 + Q_h \,. \end{align*}$$
Now let's see what both companies will earn: $$\begin{align*} V_l(Q_l,Q_h) &= (100-2Q_l-Q_h)Q_l -20 -Q_l = P_l Q_l - C_l \,,\\ V_l(Q_l,Q_h) &= (100-2Q_h-Q_l)Q_h -20 -Q_h \,. \end{align*}$$
We can simplify this: $$\begin{align*} V_l(Q_l,Q_h) &= -2Q_l^2+ 99 Q_l - 20 - Q_hQ_l = P_l Q_l - C_l \,,\\ V_h(Q_l,Q_h) &= -2Q_h^2+ 99 Q_h - 20 - Q_hQ_l \,. \end{align*}$$
But to what balance will this two-seller market settle? That will depend on the strategy the companies adopt.
One possibility is that companies would conspire together in a so-called collusion. This means they would offer enough jars of honey to maximise the sum of earnings. So we're trying to find an extreme function $$\begin{align*} V(Q_l,Q_h) \equiv V_l(Q_l,Q_h) + V_h(Q_l,Q_h) &= -2Q_l^2+ 99 Q_l - 20 - Q_hQ_l -2Q_h^2+ 99 Q_h - 20 - Q_hQ_l\\ &= -2Q_l^2-2Q_h^2 - 2Q_hQ_l + 99 Q_h + 99 Q_l - 40 \,. \end{align*}$$
To find the maximum, we would like to use a similar method to last time: looking for a zero derivative. However, there is a slight problem: this is a function of two variables and not just one. However, we can assume that e.g. $Q_h$ is constant (view $Q_h$ as a parameter), and then we get the function of one variable. We can already apply our method to that without any problem and we get one minimum condition. Then we reverse the $Q_h$ and $Q_l$ roles and do the same operation. From two conditions, we may be able to find what values $Q_h$ and $Q_l$ should have.
The derivative of the function of two variables according to one variable is called the partial derivative. Unfortunately, we do not have enough room to adequately explain why the procedure described above is working in detail. Next, we can't prove if the $Q_l$ and $Q_h$ we found are really highs. So just think of this example as a rough introduction to the issue.
So now we can go derivative: $$\begin{align*} \frac{\mathrm{d} V(Q_l) }{\mathrm{d} Q_l} = - 4 Q_l - 2 Q_h + 99 &= 0\,,\\ \frac{\mathrm{d} V(Q_h) }{\mathrm{d} Q_h} = - 4 Q_h - 2 Q_l + 99 &= 0\,. \end{align*}$$
Subtracting the first equation from the second gives us $Q_h = Q_l$. This result makes sense, since the equations were symmetrical from the beginning. Let's put this in the first equation: $$\begin{align*} - 4 Q_l - 2 Q_l + 99 &= 0 \Rightarrow Q_l = 16{,}5 \,. \end{align*}$$
The total earnings will therefore be: $$\begin{align*} V(16{,}5,\,16{,}5) &= 1\,593{,}5 \,. \end{align*}$$
For verification, we can try other values, like $20$ and $10$: $$\begin{align*} V(20,\,10) &= 1\,530 \,, \end{align*}$$
which is less than the equilibrium found.
Now let's assume that two companies aren't working together. This presumption is mostly fulfilled as collusion in competition is mostly prohibited by law. So both firms are trying to maximize their earnings independently. That's what they call the Cournot Competition. We get then: $$\begin{align*} \frac{\mathrm{d} V_l (Q_l)}{\mathrm{d} Q_l } = (-2Q_l^2+ 99 Q_l - 20 - Q_hQ_l)' = -4Q_l + 99 - Q_h &= 0 \,,\\ \frac{\mathrm{d} V_h (Q_h)}{\mathrm{d} Q_h } = -4Q_h + 99 - Q_l &= 0 \,. \end{align*}$$
Again, we solve equations by subtracting the first from the second, giving us $Q_l=Q_h$. Then we put in the first: $$\begin{align*} -4Q_l - Q_l + 99 &=0 \\ Q_l &= 99/5 = 19{,}8 \,. \end{align*}$$
We get the following total earnings (each company's earnings are half): $$\begin{align*} V(19{,}8,\,19{,}8) &\doteq 1530 \,. \end{align*}$$
So we are back to a position that is less advantageous than if the two firms worked together.
Overall, a lot of other conditions for interaction between the two firms could still be devised, e.g. a situation where one firm is already selling and the other enters the market. We could certainly easily solve such equations and calculate the price at which the market would settle. At the same time, it would be straightforward to add more players to the market, only once again the outcome would be more complicated.
To make the models more realistic, we would also have to use functions other than linear, and we would entrust the calculation to the computer for simplicity. However, the spirit of our method would remain and we could thus analyse different market positions. The use of these methods seems to be on offer: in the financial departments of a variety of companies.
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