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If, hypothetically, money velocity would reach an infinite speed, what would happen to that economy? If it would be infinite then it would mean that there would be infinite transactions and therefore Y would be infinite aswell? But infinite transactions would also be a sign of infinite demand and therefore the price would become infinite? Furthermore, if the money velocity is infinite, then it also would mean that every economic agent of the economy is in possession of the same amount of money and thus no wealth inequality would exist? I am curious to if there is a more solid theoretical answer to this and or what others think about this question.

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No $Y$ would not become infinity. $Y$ represents the real output of an economy. A real output is limited by the economy's production possibility frontier. For example, if we assume simple production function $F(L)=10 \sqrt(L)$ with assuming the economy's stock of labor being $100$ such economy would not be able to produce more than $100$ units of output whether velocity is infinite or not.

In the short run an economy might not be using its resources to the full extent and in that case increase in the velocity could have some effect - although causality would go through prices not directly from velocity (see Blanchard et al. Macroeconomics an European Perspective). Furthermore, to see the effect of velocity we could use a simplified model for money market equilibrium given by the equation of exchange:

$$MV=PY$$

Where $M$ is money supply, $V$ velocity, $P$ price level and $Y$ output. As explained above $Y$ is limited by the production capability of an economy. Hence $Y$ can be treated as fixed. Next $M$ is under a control of a government through central bank or other institutions. Since in your problem you dont specify any government action regarding $M$ we can also treat it fixed. Then the only other variable that remains is $P$. Hence if $V$ wold increase to infinity $P$ would also increase ceteris paribus.

However, as alluded to above in a short run economy might not be operating at its full potential. In such scenario increase in $P$ in the short run would also stimulate firms to produce more according to standard macro AS-AD model. Hence in short run some of the increase in velocity could also be soaked by $Y$ (through increase in $P$ induced by increase in $V$) but certainly not anywhere near to reach infinity. $Y$ must in the end respect the country's PPF which in a world of scarcity will be finite.

Furthermore, if the money velocity is infinite, then it also would mean that every economic agent of the economy is in possession of the same amount of money and thus no wealth inequality would exist?

No, a velocity of money is defined as: "the frequency at which one unit of currency is used to purchase domestically- produced goods and services within a given time period" (see FRED). High velocity does not imply more equal distribution of money. For example, imagine an economy with two people A and B, with A having one 1 $\\\$1$ banknote and B having 2 $\\\$1$ banknotes - and lets for simplicity assume anytime they spend money they instantly get it back as an income. If A uses his banknote only twice to purchase goods and services (he purchases something - immediately gets it back as an income and purchases something again) but B uses both of his two banknotes 10 times the velocity will be $22/3 \approx 7.33$. If B decides to use his banknotes 50 times the average velocity will be 34 and so on we could go on. Point is that higher or even infinite velocity does not necessary tell us by itself much about distribution of money in economy.

Furthermore, in public economics wealth is usually defined as a value of net assets. The value of net assets a person has does not correspond to the amount of money one has. For example, imagine person with house worth $\\\$ 100,000$ and no debt but absolutely no money. An economist would say that person's wealth is $\\\$ 100,000$. Now imagine person B with $\\\$1,000,000$ in a bank account while having also $\\\$1,000,000$ debt. Such person would be recorded as having $0$ wealth. Different scholars might disagree about whether wealth should include things like social security and on other details but I know of no economist that would equate wealth to money holdings. Hence there is no simple relationship between distribution of money and wealth distribution.

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    $\begingroup$ Just a comment on a point that is implied by this answer, but not obvious to people new to the field. Not all transactions count for GDP. For example, financial market transactions do not directly add to GDP. Therefore, there can be an extremely high velocity with respect to transactions, but standard definitions of velocity apply to the transactions in goods and services that add up to GDP. $\endgroup$ – Brian Romanchuk Aug 25 '20 at 0:39
  • $\begingroup$ But isn't it that you cannot have infinite money velocity when resources are limited and therefore the PPF would be limited as well? So if money velocity would be infinite then it would mean that there is an infinite amount of resources and therefore an infinitely large PPF? Or what am I missing here? $\endgroup$ – Kroko Aug 25 '20 at 15:11
  • $\begingroup$ @Kroko well in the end it’s physically impossible to use a unit of currency infinite times but it is not equal to the size of GDP. For example, if A sells apple for 10e to B the velocity will be 1 as each unit of currency was used only once while GDP will be 10e. I think it’s also possible to construct example where we transact one apple back and forth where a resale won’t be counted in but the unit of currency is still used in transactions for final goods and services. Not that any such situation would be realistic. Also, I think that rest of your Q is addressed by Brian’s +1 comment. $\endgroup$ – 1muflon1 Aug 25 '20 at 15:32

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