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I work for a start-up that has encountered an economics problem no one on our team has been able to crack.

A story form and a general form of the problem are below. I believe they are equivalent, but if there is a difference, please defer to the general form.

We understand there is no precise answer to this question and that the value of any currency considers prices in other factors that this question ignores. Still, we figured, or hoped, that there was some general way of approximating this, even if it's under idealized conditions. Thanks!

Story Form

The sultan of Slickcrudistan has gone insane. He has decreed that his country will only accept pre-2015 sand dollars in exchange for oil1. Barrels of oil will be sold to the bidder offering the most sand dollars per barrel. Additionally, the sultan is willing to sell sand dollars back to the public at approximately the current US dollar to sand dollar exchange rate.

His country produces 20,000,000 barrels of oil at a market value of $1,000,000,000 USD/year. Additionally, 100,000,000 pre-2015 sand dollars exist. Assuming liquid and efficient markets, what is the equilibrium price of a sand dollar? How would this change if there were a stable inflation in the quantity of sand dollars?

1 Suppose an extremely cheap process exists for verifying this.

General Form

Currency X has a total circulation of a units and a stable inflation rate of currency units of m percent.

Currency Y has a total circulation of b units and a stable inflation rate of currency units of n percent.

Currency Y is completely debased and of no value in comparison to currency X (at market exchange rates, b << a ).

A market with a value of c units per year in currency X can now only be transacted in a currency Y. c is growing at a stable rate of p percent.

Assuming liquid and efficient market, what is the new equilibrium of exchange?

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    $\begingroup$ I'm voting to close this question as off-topic because it is the functional equivalent of a homework question with no effort shown. What did you try? Where did you get stuck? $\endgroup$ – BKay Jul 9 '15 at 12:04
  • $\begingroup$ Why has currency Y got inflation? From what you've described, it should have deflation of ~p percent. What's the use of the story form if the general form contains all available information? What are urchins? What's the velocity of circulation? It's all rather unclear. $\endgroup$ – EnergyNumbers Jul 9 '15 at 14:44
  • $\begingroup$ Are there complete markets, such that there's an equilibrium USD-to-urchin exchange rate? (I am assuming that there are three currencies here, sand dollars, USD, and urchins.) $\endgroup$ – dismalscience Jul 9 '15 at 14:47
  • $\begingroup$ This is not a homework question. I'm a physicist/programmer and far outside my domain knowledge. $\endgroup$ – Jeremy Kauffman Jul 9 '15 at 17:13
  • $\begingroup$ Urchins and sand dollar are equivalent (a sand dollar is a dead urchin), I will edit the question to clarify this. $\endgroup$ – Jeremy Kauffman Jul 9 '15 at 17:14
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Using a simple textbook-approach, I will provide one possible answer here, showing that in this approach, the one thing that needs to be specified is "velocity of money" as regards sand dollars.

I will use also the numerical values provided in the question, after cutting out six zeros. I assume that the Law of One Price holds, so

$$P_s\cdot S_{USD/s} = P_{USD} \tag{1}$$

$P_s$ is "the price level" related to sand dollars, $P_{USD}$ is the price level related to USD, and $S_{USD/s}$ is the exchange rate we wish to find ("how many USD per one sand dollar").

While the price levels are usually price-indexes, here we are looking at a single good, oil, so we can take the $P's$ to mean "price of oil".

The Sultan's oil is the only output that can be bought with sand dollars. And we do know the value of this output in USD

$$P_{USD} \cdot Q_s = P_{USD} \cdot 20 = 1,000 \;\text{USD} \tag{2} $$

Using the Law of One Price, we get

$$P_{USD} \cdot Q_s = P_s\cdot S_{USD/s}\cdot Q_s \implies S_{USD/s} = \frac {P_{USD} \cdot Q_s}{P_s\cdot Q_s} \tag{3}$$

(don't simplify).

Assuming the Quantity Theory of Money, we have

$$P_s\cdot Q_s = V_s \cdot \bar M_s \tag{4}$$

where $\bar M_s$ is the fixed amount of sand dollars available, and $V_s$ is the "velocity of money" in relation to sand-dollars (we will return to $V_s$). Inserting $(4)$ into $(3)$ and using also $(2)$ and the fixed value of $\bar M_s$ we have

$$(4),(3),(2) \implies S_{USD/s} = \frac {P_{USD} \cdot Q_s}{V_s\bar M_s} = \frac {1,000}{V_s\cdot 100} = \frac {10}{V_s}$$

So, if we can find a value for $V_s$, we have found the exchange rate we are after. The velocity of money can be thought as "how many times that same unit of currency, say the same paper bill, will be used in transactions during a production cycle".

The M1 velocity of money for USD is approximately $V_{USD} \approx 6$, using this value for the sand-dollars also, we get

$$S_{USD/s} = \frac {10}{6} \approx 1.67$$

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