<p>There isn't any good way to rehabilitate the quantity theory when there are other currencies that are <em>perfect</em> 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, <a href="http://research.stlouisfed.org/fred2/graph/?g=1aMa" rel="nofollow">as reserves become a perfect substitute for other short-term nominal assets once the nominal interest rate hits zero</a>.</p>

<p>That said, I'd make two points.</p>

<p><strong>1. Anything less than perfect substitutability rescues the quantity theory.</strong></p>

<p>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) $XB$. </p>

<p>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.</p>

<p>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.</p>

<p>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 <em>almost</em> 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 <em>perfect</em> substitutes in absolutely every application.</p>

<p>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 <a href="http://fmwww.bc.edu/ec-p/wp772.pdf" rel="nofollow">Ireland (2011)</a>.</p>

<p><strong>2. The central bank can pin down the price level in other ways, even without the quantity theory.</strong></p>

<p>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, <a href="http://rads.stackoverflow.com/amzn/click/0691010498" rel="nofollow">Woodford's canonical text</a> 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.)</p>

<p>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.</p>

<p>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.</p>

CC BY-SA 3.0
2015-05-09T01:46:26.043Z
2015-05-09T02:14:24.010Z
2015-05-09T02:14:24.010Z
nominally rigid
<blockquote>
  <p><em>That leaves me with two variables ($X$ and $P$) but only one equation. What pins down the values of these variables?</em></p>
</blockquote>

<p>If Bitcoins are accepted as payment medium to <em>all transactions where USD is also accepted</em> (always speaking for one national economy), as Landsburg's scenario asks us to imagine, then, it <em>must</em> 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.  </p>

<p>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) </p>

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

<p>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".</p>

CC BY-SA 3.0
2015-05-08T00:23:22.877Z
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
CC BY-SA 3.0
2015-05-08T01:11:16.487Z
@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
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2015-05-08T01:31:19.837Z
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
CC BY-SA 3.0
2015-05-08T03:14:55.983Z
@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
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2015-05-08T03:34:05.630Z
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
CC BY-SA 3.0
2015-05-13T16:23:21.030Z
Absence of monetary illusion is my argument against this. What makes you believe people in bitcoin would take 1 bitcoin as their unit of currency, Alecos?
VicAche
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2015-05-13T18:03:34.747Z
@Vicache _Some_ unit  would have to be declared, don't you think? In your experience, do people "follow" in this the issuer of the currency and what he -the issuer- declares as the unit, or do they do each whatever each wants? Remember, we _start_ by the assumption that "bitcoin" is equally credible as the USD.
Alecos Papadopoulos
CC BY-SA 3.0
2015-05-13T18:22:27.960Z
@AlecosPapadopoulos Credible is one thing (and your answer isn't stupid, it don't think it deserves a downvote), but other things are to be accounted for, for example avaibility (dollars are used for trade in Asia, Africa) and divisibility (Bitcoin is easily divided, and people tend to speak in µ฿ rather than in ฿ actually)
VicAche
CC BY-SA 3.0
2015-05-13T18:25:11.317Z
@AlecosPapadopoulos in fact I believe cents would align with Satoshies, as until they do, people that need to trade very precise amounts would prefer one currency to the other, which violates the terms of the question
VicAche
CC BY-SA 3.0
2015-05-08T00:16:23.140Z
2015-05-08T00:16:23.140Z
Alecos Papadopoulos
<p>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.</p>

CC BY-SA 3.0
2015-05-13T18:33:24.723Z
2015-05-13T18:33:24.723Z
VicAche
<p>In <a href="http://www.thebigquestions.com/2013/05/13/whats-a-bitcoin-worth-wonkish/" rel="nofollow noreferrer">this blog post</a>, economist Steve Landsburg posed a question about the value of Bitcoin which he didn't know the answer to.</p>
<blockquote>
<p>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) <span class="math-container">$X$</span> dollars per Bitcoin.</p>
<p>Then if there are <span class="math-container">$M$</span> dollars and <span class="math-container">$B$</span> bitcoins in circulation, the money
supply (measured in dollars) is effectively <span class="math-container">$M + X B$</span> .</p>
<p>Money demand is presumably <span class="math-container">$P D$</span>, where <span class="math-container">$P$</span> is the general price level and
<span class="math-container">$D$</span> depends on things like the volume of transactions and the payment
habits of the community. (If it helps, we can write <span class="math-container">$D = T/V$</span> where <span class="math-container">$T$</span> is
the volume of transactions and <span class="math-container">$V$</span> is the velocity of money.)</p>
<p>Equilibrium in the money market requires that supply equals demand, so</p>
<p><span class="math-container">$M + X B = P D$</span></p>
<p>Now <span class="math-container">$M$</span> is determined by the monetary authorities; <span class="math-container">$B$</span> is
determined by the Bitcoin algorithm, and <span class="math-container">$D$</span>, as noted above, is
determined outside the money market.</p>
<p>That leaves me with two variables (<span class="math-container">$X$</span> and <span class="math-container">$P$</span>) but only one equation.
What pins down the values of these variables?</p>
</blockquote>
<p>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.</p>
<p>So does anyone know the answer to Landsburg's question?  Are there any models that shed light on what determines the exchange rate <span class="math-container">$X$</span> and the price level <span class="math-container">$P$</span> in a situation like this?</p>

CC BY-SA 3.0
2015-05-07T02:20:33.870Z
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
CC BY-SA 3.0
2015-05-07T02:24:39.617Z
@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
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2015-05-07T02:35:53.037Z
@ 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
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2015-05-07T02:39:18.037Z
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
CC BY-SA 3.0
2015-05-07T02:54:52.310Z
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
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2015-05-07T03:05:22.100Z
@Hessian Well, if we assume the quantity theory of money, then what would the answer to Landsburg's problem be?
Keshav Srinivasan
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2015-05-07T03:05:38.897Z
@dismalscience Consider it as a theoretical question then, rather than a question about whether existing currencies are perfect substitutes for dollars in the US,
Keshav Srinivasan
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2015-05-07T03:22:36.060Z
@KeshavSrinivasan Well first off, I wouldn't start off by "assuming" QTM. It's just straight up not the way to think about exchange rates. IMHO, it's best to think about exchange rates from a PPP/LOP perspective. Of course his setup doesn't determine the price level, there are no supply side economics involved, so I'm not sure why he expects it to be determined in the first place. He could easily determine the exchange rate assuming everything else is exogenously given -> 1 unknown in 1 equation. Again, QTM is not the best way to approach it, whether it's assumed or not.
Hessian
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2015-05-07T03:32:52.547Z
@KeshavSrinivasan I emphasize that it's not the best approach since it's an  accounting identity, and certainly wasn't developed with exchange rate determination in mind. Probably best to go with (a) something micro-founded/structural and (b) something with a sense of a supply side for determining prices.
Hessian
CC BY-SA 3.0
2015-05-07T03:40:42.033Z
@Hessian Well, if we eliminated the second currency, wouldn't the price level just equal $M/D$?  So it's the introduction of the second currency that's preventing us from determining the price level.
Keshav Srinivasan
CC BY-SA 3.0
2015-05-07T03:43:35.433Z
@KeshavSrinivasan Why would this be how prices are determined? So you can grasp why this is so absurd, consider that determining prices in this manner would COMPLETELY ignore the cost of producing the goods in the index, the market structure, etc.You really CAN'T do these kinds of things with what is only an identity.
Hessian
CC BY-SA 3.0
2015-05-07T04:12:29.053Z
Whichever has the most frictions would determine the other, no? One of these, in a practical sense would be more resistant to change on the aggregate. There would then be regional exceptions where that direction would be reversed because the frictions likely have non-uniform distributions.
RegressForward
CC BY-SA 3.0
2015-05-07T04:18:10.463Z
@Hessian Well, doesn't the stuff you're talking about just constitute the determinants of relative prices?  Why does that affect the price level?
Keshav Srinivasan
CC BY-SA 3.0
2015-05-07T04:21:11.577Z
@KeshavSrinivasan I don't really have the time to explain producer theory but I assure you that all of what I mentioned is relevant for price levels as well as relative prices. If you want to learn more id refer you to Nicholson or Varian's intermediate micro books.
Hessian
CC BY-SA 3.0
2015-05-07T04:23:21.447Z
@Hessian Well, is $P = M/D$ a true or false statement if we assume the quantity theory of money?
Keshav Srinivasan
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2015-05-07T12:57:23.193Z
I think you guys should have moved this discussion to chat quite some time ago ;)
FooBar
<exchange-rates><money-supply><price-level><cryptocurrency><bitcoin>
What determines the exchange rate of two fiat currencies if the supply of each is known?
Keshav Srinivasan
# What determines the exchange rate of two fiat currencies if the supply of each is known?

(Asked by: username_0 on 2015-05-07)

In [this blog post](http://www.thebigquestions.com/2013/05/13/whats-a-bitcoin-worth-wonkish/), 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?

**Tags:** exchange-rates, money-supply, price-level, cryptocurrency, bitcoin

> **Comments:**
> **username_1**: 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.  
> **username_0**: @username_1 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.  
> **username_2**: @ 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.  
> **username_2**: 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.)  
> **username_1**: 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.  
> **username_0**: @username_2 Well, if we assume the quantity theory of money, then what would the answer to Landsburg's problem be?  
> **username_0**: @username_1 Consider it as a theoretical question then, rather than a question about whether existing currencies are perfect substitutes for dollars in the US,  
> **username_2**: @KeshavSrinivasan Well first off, I wouldn't start off by "assuming" QTM. It's just straight up not the way to think about exchange rates. IMHO, it's best to think about exchange rates from a PPP/LOP perspective. Of course his setup doesn't determine the price level, there are no supply side economics involved, so I'm not sure why he expects it to be determined in the first place. He could easily determine the exchange rate assuming everything else is exogenously given -> 1 unknown in 1 equation. Again, QTM is not the best way to approach it, whether it's assumed or not.  
> **username_2**: @KeshavSrinivasan I emphasize that it's not the best approach since it's an  accounting identity, and certainly wasn't developed with exchange rate determination in mind. Probably best to go with (a) something micro-founded/structural and (b) something with a sense of a supply side for determining prices.  
> **username_0**: @username_2 Well, if we eliminated the second currency, wouldn't the price level just equal $M/D$?  So it's the introduction of the second currency that's preventing us from determining the price level.  
> **username_2**: @KeshavSrinivasan Why would this be how prices are determined? So you can grasp why this is so absurd, consider that determining prices in this manner would COMPLETELY ignore the cost of producing the goods in the index, the market structure, etc.You really CAN'T do these kinds of things with what is only an identity.  
> **username_3**: Whichever has the most frictions would determine the other, no? One of these, in a practical sense would be more resistant to change on the aggregate. There would then be regional exceptions where that direction would be reversed because the frictions likely have non-uniform distributions.  
> **username_0**: @username_2 Well, doesn't the stuff you're talking about just constitute the determinants of relative prices?  Why does that affect the price level?  
> **username_2**: @KeshavSrinivasan I don't really have the time to explain producer theory but I assure you that all of what I mentioned is relevant for price levels as well as relative prices. If you want to learn more id refer you to Nicholson or Varian's intermediate micro books.  
> **username_0**: @username_2 Well, is $P = M/D$ a true or false statement if we assume the quantity theory of money?  
> **username_4**: I think you guys should have moved this discussion to chat quite some time ago ;)  

---

## 3 answers

### Answer by username_5 on 2015-05-09. Score: 2

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](http://research.stlouisfed.org/fred2/graph/?g=1aMa).

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)](http://fmwww.bc.edu/ec-p/wp772.pdf).

**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](http://rads.stackoverflow.com/amzn/click/0691010498) 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.


---

### Answer by username_7 on 2015-05-13. Score: 0

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.


---

### Answer by username_6 on 2015-05-08. Score: -4

> *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".

> **Comments:**
> **username_0**: 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.  
> **username_6**: @KeshavSrinivasan I presented an argument as to why their exchange rate should be unity. What is your argument that it could be something else?  
> **username_0**: 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.  
> **username_6**: @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.  
> **username_0**: 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?  
> **username_7**: Absence of monetary illusion is my argument against this. What makes you believe people in bitcoin would take 1 bitcoin as their unit of currency, Alecos?  
> **username_6**: @Vicache _Some_ unit  would have to be declared, don't you think? In your experience, do people "follow" in this the issuer of the currency and what he -the issuer- declares as the unit, or do they do each whatever each wants? Remember, we _start_ by the assumption that "bitcoin" is equally credible as the USD.  
> **username_7**: @AlecosPapadopoulos Credible is one thing (and your answer isn't stupid, it don't think it deserves a downvote), but other things are to be accounted for, for example avaibility (dollars are used for trade in Asia, Africa) and divisibility (Bitcoin is easily divided, and people tend to speak in µ฿ rather than in ฿ actually)  
> **username_7**: @AlecosPapadopoulos in fact I believe cents would align with Satoshies, as until they do, people that need to trade very precise amounts would prefer one currency to the other, which violates the terms of the question  

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