# Difficult quantity theory of money question

Suppose that you live on an island with 100 units of currency. The cultural beliefs of the islanders discourage an excess focus on commerce, which has created two important rules of commerce. First, you are legally permitted to do 10 transactions per year. Second, the maximum permitted value of each transaction is $10, expressed in USD. With these assumptions, can you solve for the maximum legal FX rate of the islander's currency expressed in US dollars? I feel like you should be able to do this with some kind of simple, recursive calculation, but for some reason I just can't figure it out right now. GDP is obviously$100 USD per islander and I feel like you can know velocity by knowing the number of transactions per year, but I am not sure how.

• I find this comment disrespectful towards the OP, in assuming that the OP doesn't know what s/he wants to ask. Opinions about the OP's participation in other forums are not relevant to answer the question. – JoaoBotelho Apr 27 '18 at 7:32

Even in simplest of models I don't think it is enough info to constrain the fx rate. Even if velocity can be worked out we can't get m0.

In the simplest model all islanders are equally capable or some equilibrium arises where everyone gets the same proportions of resource. Then if resource x has $a$ units per person and total of all resources or sum of all units of the different resources is $s$ the price of each resource $x$ is simply $\frac{100a}{s}$. If the price of a resource is known in a foreign currency then one can theoretically obtain the foreign currency by selling goods in the foreign currency. In this simple theory any difference in resources would be arbitrage opportunities seeking to balance or distribute resources till they have the same abundance.

This is very simple theory. If one resource is short one place and another resource short some where's else there may be no arbitrage and thus do not cancel out.

So we assume eventually or in long run of existence of island nation, absent of any kind of trade barriers, there is no more arbitrage. Now we have $$\frac{m_a a}{s_a}=\frac{m_b b}{s_b}$$ Assuming resources eventually are evenly distributed, or even that the ratios of such resources are same, the then we see that one unit of currency equals $1_{unit \ a} =\frac{1_{unit \ b} m_b}{m_a}$.

So in this model if islanders have $m_a=100$ and u.s. have $m_b=30000$ then one island currency would be worth $\frac{30000}{100}=300$ USD.

So we see a lot of assumptions are made here. Also maybe this is too granular. Probably more needs to be considered about resource pricing. In reality there are different levels of abundance of resource and that is the primary motivating factor here.

The velocity should just be the amount of times the average unit of currency exchanges hands per year. That is $10$ if we take the highest possible number.

Money supply is $100$ units of island currency ($IC$), so that the left side of

$$MV=PY$$

turns into $100*10$ Nominal GDP, so $YP$, is \$$100 per Islander, or$$\$100/XR*n$$in$IC$, where$XR$is the exchange rate of$USD/IC$and$n$is the amount of people. Then we get$XR=1/10*n$. However, I'm a bit lost for what$n\$ would be in this case. I'm guessing you got this far yourself?