# What determines the exchange rate of two fiat currencies if the supply of each is known?

In this blog post, economist Steve Landsburg posed a question about the value of Bitcoin which he didn't know the answer to.

Imagine a future in which Bitcoins (or some other non-governmental currency) are widely accepted and easily substitutable for dollars, at an exchange rate of (say) $$X$$ dollars per Bitcoin.

Then if there are $$M$$ dollars and $$B$$ bitcoins in circulation, the money supply (measured in dollars) is effectively $$M + X B$$ .

Money demand is presumably $$P D$$, where $$P$$ is the general price level and $$D$$ depends on things like the volume of transactions and the payment habits of the community. (If it helps, we can write $$D = T/V$$ where $$T$$ is the volume of transactions and $$V$$ is the velocity of money.)

Equilibrium in the money market requires that supply equals demand, so

$$M + X B = P D$$

Now $$M$$ is determined by the monetary authorities; $$B$$ is determined by the Bitcoin algorithm, and $$D$$, as noted above, is determined outside the money market.

That leaves me with two variables ($$X$$ and $$P$$) but only one equation. What pins down the values of these variables?

As he suggests in parenthesis, this isn't a problem specific to Bitcoin, it's a general problem for non-governmental currencies that are perfect substitutes for dollars.

So does anyone know the answer to Landsburg's question? Are there any models that shed light on what determines the exchange rate $$X$$ and the price level $$P$$ in a situation like this?

• The current title is a little confusing; a cryptocurrency isn't really a fiat currency. I would suggest that the answer has everything to do with frictional holdings of the cryptocurrency— the equilibrium value of the cryptocurrency will be very low, and the exchange brokers will profit off their holdings through the spread on the exchange rate. The cryptocurrency will, in effect, be a transaction technology, not full-fledged money in and of itself. – dismalscience May 7 '15 at 2:20
• @dismalscience Well, Landsburg's question isn't something specific to cryptocurrencies. It's just a general question about any currency that is a perfect substitute for dollars. – Keshav Srinivasan May 7 '15 at 2:24
• @ Wouldn't the nominal exchange rate just "move around" the real rate like any other currency? Meaning that since real rates are determined by relative prices, the nominal rate would reflect changes in these relative prices as well as other factors. – Hessian May 7 '15 at 2:35
• In addition, I'm not sure that applying quantity theory of money is the best way of thinking about the demand for money. Usually there are two ways of generating "demand" for currencies in economics models, and neither uses quantity theory: cash in advance (agents have to "visit the ATM" before consumption) and money in utility models (where you derive satisfaction from holding currency by utility being a function of both consumption and the stock of money being held.) – Hessian May 7 '15 at 2:39
• The premise is a bit absurd, then. No currency is a perfect substitute for dollars (this is generally true of fiat currencies). Try paying US taxes in another currency, or settling any USD contract in another currency. Both are impossible. To do business in a currency other than the one in which one's obligations are owed is to accept foreign exchange risk. – dismalscience May 7 '15 at 2:54

There isn't any good way to rehabilitate the quantity theory when there are other currencies that are perfect substitutes for dollars - so in that sense, there isn't any answer to Landsburg's question. Indeed, the irrelevance of the quantity theory under perfect substitutability - which has always been theoretically clear - has become a practical reality recently, as reserves become a perfect substitute for other short-term nominal assets once the nominal interest rate hits zero.

That said, I'd make two points.

1. Anything less than perfect substitutability rescues the quantity theory.

Suppose we replace the left side of Landsburg's quantity theory equation with the more general form $$F(M,XB) = PD$$ where $F$ is a homogenous production function that produces aggregate "monetary services" using money $M$ and bitcoins (with value stated in terms of money) $XB$.

Landsburg's equation corresponds to the case of perfect substitutes, $F(M,XB) = M+XB$. In fact, there's another assumption that's implicit in Landsburg's formulation, which is that either $X$ is constant over time or, if $X$ varies, the risk-adjusted expected return on bitcoins relative to money is zero: otherwise, you'd strictly prefer to hold one or the other, whichever gives the highest return, given that they have equal transactional value. I'm going to continue assuming that this assumption holds for simplicity - but keep in mind that in a full-fledged dynamic model, allowing returns to differ and individuals to substitute on this basis might endogenously produce the extra equation that Landsburg seeks.

If money and bitcoins have the same returns, then anyone holding them will want to equate the marginal transactional value of the two, setting $F_M=F_{XB}$. This holds for any relative quantities of $M$ and $XB$ in Landsburg's perfect substitutes formulation, which is why he's struggling, but for general (homogenous) $F$ it will only hold for one ratio $M/XB$ of the two. This will pin down relative demand.

For instance, if $F$ is Cobb-Douglas, with $F(M,XB) = M^\alpha (XB)^{1-\alpha}$, then $F_M = \alpha F/M$ and $F_{XB} = (1-\alpha)F/XB$, and equating the two gives us $M/XB = \alpha/(1-\alpha)$. Suppose $\alpha=1/3$. Then we have $XB = 2M$, and it is trivial to solve for $P$ from $M$ and $D$: $$F(M,XB)=PD\Longleftrightarrow M^{1/3} (2M)^{2/3} = PD\Longleftrightarrow P = 2^{2/3}\frac{M}{D}$$ Cobb-Douglas is just one parameterization I'm using for illustrative purposes, but we'll similarly be able to solve as long as $F$ has a declining marginal rate of substitution between $M$ and $XB$ - which would be true, for instance, if $M$ and $XB$ were almost perfect substitutes, but not quite. Landsburg's case of perfect substitutes is very much non-generic in this sense: it's probably not true that fiat currency and bitcoins will ever be perfect substitutes in absolutely every application.

By the way, the idea that two forms of currency combine in an imperfectly substitutable way to provide overall monetary services isn't just something I made up - you can see assumptions like this in the literature in a number of places, like equation (3) in Ireland (2011).

2. The central bank can pin down the price level in other ways, even without the quantity theory.

The modern view on monetary policy is that what really matters is the central bank's ability to set the short-term interest rate. Traditionally, this has been done by changing the supply of money through open-market operations, but that doesn't need to be the case. Indeed, Woodford's canonical text shows how it is possible to implement monetary policy even in a "cashless" world where there is no demand for money: the central bank simply pays interest on money. (By the way, this result is hard to escape when you try to microfound the "quantity theory" equation by writing down a dynamic, internally consistent model: you realize that the quantity theory operates in general equilibrium via the response of interest rates to money, and that by manipulating interest rates directly we can get the same outcome.)

Indeed, we're moving closer to Woodford's hypothetical world all the time: for instance, one of the options for the Fed when it decides to raise interest rates in the coming months will be to push up the rate of interest on reserves, while keeping its expanded balance sheet intact.

From this viewpoint, Landsburg's observation just isn't very relevant. The central bank is dedicated to price stability, and it'll enforce this by adjusting interest rates in response to deviations of inflation from trend. If it can adjust interest rates through the traditional method of adjusting $M$ via open-market operations, great. But if it can't do this (because we live in Landsburg's world of perfect substitutability), then the central bank will just adjust nominal interest rates by changing the interest it pays on reserves, and ultimately accomplish exactly the same thing.

My guess would be that $\Delta B$, the amount of Bitcoins introduced in the economy in a given year (which you can get from the value of Bitcoin and characteristics of Bitcoin algorithm) and $\Delta S$ (same for dollars, supposingly given by the Fed) should verify $B/S = \Delta B / \Delta S$, or people would decide to keep their liquidities in the currency that is less eased to avoid depreciation.

That leaves me with two variables ($X$ and $P$) but only one equation. What pins down the values of these variables?

If Bitcoins are accepted as payment medium to all transactions where USD is also accepted (always speaking for one national economy), as Landsburg's scenario asks us to imagine, then, it must be the case that Bitcoin enjoys the exact same credibility like USD. If this is so, then their exchange rate cannot be anything else than unity.

If I feel equally safe in accepting Bitcoins and USD (safe as regards what can I do with Bitcoins), why should I ever accept an exchange rate different than unity? The moment I demand a different exchange rate than unity, it implies that the risk profiles of the two currencies are no longer perceived as being the same (risk profiles in the very broad sense, taking into considerations everything)

It's all about expectations then, what we project in the future for these two currencies.

This in turn requires modelling: on what foundations does the credibility of USD relies, and on what foundations does Bitcoin credibility relies, and "what we have in mind" for these foundations. This will eventually provide "the missing equation".

• That seems like a fallacious argument to me. Couldn't that same argument be used to argue that the exchange rate between Bitcoins and pennies must unity? (A penny is a coin worth one-hundredth of a dollar.) So I think that just because two currencies are completely credible doesn't mean that their exchange rate must be unity. – Keshav Srinivasan May 8 '15 at 0:23
• @KeshavSrinivasan I presented an argument as to why their exchange rate should be unity. What is your argument that it could be something else? – Alecos Papadopoulos May 8 '15 at 1:11
• My argument is simple: apply your argument, mutatis mutandis, to show that the exchange rate between Bitcoins and pennies must be unity (since pennies and Bitcoins are equally credible). That implies that the exchange rate between Bitcoins and dollars must be 100 Bitcoins per dollar, since 100 pennies is equal to 1 dollar. – Keshav Srinivasan May 8 '15 at 1:31
• @KeshavSrinivasan The unit of currency is declared as one "dollar", and a "penny" is declared as only 1/100 of the unit of currency. Of course these are totally arbitrary -but that's the essence of fiat money: A source enjoying credibility in the eyes of citizens declares something -and the citizens accept it as is. Under USD credibility, no one would give you "one dollar" if you gave them "one penny of a dollar". Enter another fiat money: if it has credibility, people will compare and assess "unit of currency" with "unit of currency" -apples with apples, not apples with oranges. – Alecos Papadopoulos May 8 '15 at 3:14
• Well, let's take an example of something that doesn't have an obvious "unit", like gold. (Suppose for simplicity that gold had no intrinsic value.) Then assuming gold had equal credibility as the dollar, then couldn't your argument be used to show that the exchange rate between pounds of gold and dollars is unity, and also that the exchange rate between ounces of gold and dollars is unity? – Keshav Srinivasan May 8 '15 at 3:34