January | 2014 | The Everyday Economist

David Andolfatto has written a very good post on Bitcoin and why it might have positive value. In particular, he provides an excellent overview of what Bitcoin actually is (an electronic record of transactions) and how this relates to the insight that “money is memory.” (On this point, see also, William Luther’s paper, “Bitcoin is Memory.”) Nevertheless, I have some questions about the post regarding David’s discussion of the volatility of Bitcoin and how this impacts the choice of what to use as money. In this post, I hope to address this point and perhaps add some additional substance to the issue.

David ends his post talking about whether Bitcoin would make for a good form of money. This is an interesting question and one that often isn’t given sufficient thought. In David’s own research, however, he has emphasized that the characteristic that determines whether an asset is useful as money is whether that asset is information-sensitive (this claim is often prevalent in Gary Gorton’s work as well). The basic idea is that if the value of an asset is sensitive in the short-run to news that has private value, but no social value, then short run fluctuations in the price of the asset will preclude its use as a medium of exchange. Thus, David suggests looking at how the value of Bitcoin has changed over time. In his graph, he plots the price of bitcoins in terms of dollars. As you can see if you read his post (or if you know anything about Bitcoin), the price of bitcoins relative to dollars is quite volatile — especially over the last year.

However, I wonder whether looking at the volatility of the exchange rate between Bitcoin and the dollar is the best way to determine whether Bitcoin would be a good form of money. There are two reasons why I say this.

First, as David points out, this volatility could be the result of the fact that people view the supply of Bitcoin as being fixed (the supply of Bitcoin will eventually be fixed), but the demand for Bitcoin is fluctuating. David notes that this is consistent with the type of behavior we observe under commodity standards. When there is a change in the demand for gold, the purchasing power of gold varies (at times considerably) even though the long run purchasing power is constant.

I have heard others make this argument as well and this seems very plausible. Nevertheless, it is useful to recall the way in which free banking systems operated. For example, in a free banking system in which banks issued notes that were backed by gold, the supply of bank notes fluctuated with demand. Increases in the demand for money caused an increase in the supply of notes. These changes in the supply of notes, however, needn’t imply any change in the purchasing power of gold. Issuing bank notes redeemable in gold was thereby capable of reducing the volatility of the purchasing power of gold. Similarly, a financial intermediary today could issue bank notes redeemable in bitcoins and reduce the volatility of the purchasing power of bitcoins.

[A quick note: It is perhaps true that the U.S. government might decide that they don’t want to allow financial intermediaries to issue bank notes, in which case my proposed solution to this source of volatility would not be operable. I would add though that it is not operable because of a legal restriction and not the volatility of the asset price.]

Second, and perhaps much more importantly, in models with competing money supplies the exchange rate does not factor in to the choice of allocation or welfare analysis. This is true even of David’s own research. For example, in quoting the price of bitcoins in terms of dollars, David is citing an exchange rate. However, in his research, the price volatility that matters for an asset is the own rate of return volatility. I think that this distinction matters.

To illustrate why I think that this distinction matters, let’s consider a simple overlapping generations model. There are two types of agents, young and old. Each lives for two periods. At any point in time, there is a generation of young and a generation of old. The population is assumed to be constant. There is one good to trade and it is non-storable. The young receive an endowment, y, of the consumption good. The old do not receive an endowment of goods. Thus, money is essential. There are two assets that can be used as a possible medium of exchange. The first is fiat currency. The second is bitcoins. The initial old carry both currency and bitcoins into the first period. The aggregate supply of bitcoins is fixed. The aggregate supply of currency, $N_t$ , is assumed to grow at the gross rate $x$ (i.e. $N_{t + 1} = x N_t$ ).

Let’s consider the first and second period budget constraints for future generations in our model (i.e. everybody except the initial old). In the first period, future generations can use their endowment for consumption or they can sell some of this endowment for money and/or bitcoins. Thus, the first-period budget constraint is:

$c_{1,t} + m_t + b_t \leq y$

where $c_{1,t}$ denotes the consumption when young in period t, $m$ is real currency balances, and $b$ denotes real balances of bitcoins.

Denote $v_t$ as the price of currency in terms of goods at time t. Similarly, denote the price of bitcoins in terms of goods as $u_t$ . Thus, the rate of return on currency is $v_{t + 1} / v_t$ . Now let’s assume that there is some cost, $\tau$ that individuals have to pay when they use bitcoin to make a purchase. The rate of return on bitcoins is then given as $(1-\tau){{u_{t+1}}\over{u_t}}$ . Thus, the second-period budget constraint can be written as

$c_{2,t+1} = {{v_{t+1}}\over{v_t}} m_t + (1 - \tau){{u_{t+1}}\over{u_t}} b_t$

But we can derive a precise definition of the rate of return on money. It follows from our first period budget constraint that we have:

$m_t = v_t n_t = (y - b_t - c_{1,t})$

where $n_t$ denotes nominal currency balances. Define the total nominal currency stock as $N_t$ and the size of the population, which as assumed to be constant as $P$ . This implies an aggregate demand function for currency:

$v_t N_t = P(y - b_t - c_{1,t})$

Thus, the rate of return on money is

${{v_{t+1}}\over{v_t}} = {{P(y - b_{t+1} - c_{1,t+1})}\over{P(y - b_t - c_{1,t})}}{{N_t}\over{N_{t + 1}}}$

From above, we know that the currency supply grows at a gross rate $x$ . This implies that in a stationary allocation (i.e. where consumption paths a constant across generations), the rate of return on currency is

${{v_{t+1}}\over{v_t}} = {{1}\over{x}}$

By similar logic, it is straightforward to show that in a stationary allocation ${{u_{t+1}}\over{u_t}} = 1$ because the supply of bitcoins was assumed to be fixed.

Thus, our stationary allocation budget constraints are:

$c_1 + m + b \leq y$

$c_2 = {{1}\over{x}}m + (1 - \tau) b$

In the present model, money and bitcoins are perfect substitutes (i.e. there only purpose is to serve as proof of a previous transaction when trading with future young generations). Thus, the real rates of return on money and bitcoins must be equal for both to exist in equilibrium. In other words, it must be true that ${{1}\over{x}} = (1 - \tau)$ . We can re-write the second-period constraint as

$c_2 = {{1}\over{x}}(m + b)$

Combining these budget constraints, we have a lifetime budget constraint:

$c_1 + x c_2 \leq y$

Now let’s consider the basic implications of the model. First, the conditions under which both currency and bitcoins would be held in equilibrium is dependent on their relative rates of return. If these rates of return are equal, then both assets are held. This condition is independent of the exchange rate. Second, lifetime budget constraint outlines the feasible set of allocations available given the agents’ budget. Assume that utility is a function of consumption in both periods. The allocation decision in this case is dependent on the rate of return on currency, which is the same as the rate of return on bitcoins. The allocation decision is therefore contingent on the equilibrium rate of return. The exchange rate between currency and bitcoins plays no role in the allocation decision. In addition, one can show that this is the identical lifetime budget constraint that would exist in a currency economy (i.e. one in which Bitcoin doesn’t circulate). This last characteristic implies that neither the existence of bitcoins nor the exchange rate between bitcoins and currency have any effect on welfare.

So what does this mean? Basically what all of this means is that the exchange rate between currency and bitcoins is irrelevant to the decision to hold bitcoins, to the allocation decision, and to welfare. [Note: This isn’t new, Neil Wallace taught us this type of thing over 30 years ago.]

The analysis above is a bit unfair to David for two reasons. First, the framework above skirts David’s main point, which is that Bitcoin is information sensitive whereas currency is not. Second, David doesn’t necessarily mean that the exchange rate between the dollar and bitcoins is relevant for the type of analysis above. Rather, what he means is that since the dollar is relatively stable, the exchange rate serves as a proxy for the own price of bitcoins in terms of goods.

What I would like to do now is to amend the framework above to make bitcoins information sensitive. The results regarding the exchange rate of bitcoins into dollars remain. However, an interesting result emerges. In particular, one can show that it is the risk premium associated with bitcoins that can help us to understand the choice of whether to hold bitcoins or dollar-denominated currency as well as factor into the allocation decision. I will then speculate as to whether this risk premium is sufficient to explain the differences in rates of return between bitcoins and dollar-denominated currency.

For simplicity, let’s now denote the rate of return on bitcoins as $r$ . In addition, we will assume that $r$ is stochastic. In particular the assumption is that the rate of return is entirely determined by random news events. Thus, we can re-write our first- and second-period budget constraints, respectively, as

$c_{1,t} + m_t + b_t \leq y$

$c_{2,t+1} = {{1}\over{x}} m_t + r_{t+1} b_t$

The objective of future generations is to maximize $u(c_{1,t}) + v(c_{2,t+1})$ subject to the two constraints above. Combining the first-order conditions with respect to $m$ and $b$ and using the definition of covariance, we have the following equilibrium condition:

$E_t r_{t+1} = {{1}\over{x}} - {{cov[r_{t+1},v'(c_{2,t+1})]}\over{v'(c_{2,t+1})}}$

where the second term on the right-hand side measures the risk premium associated with bitcoins (note that this is in fact a risk premium since the covariance between the rate of return on bitcoins and the marginal utility of consumption is negative). If both assets are to be held in equilibrium, then the equibrium condition must hold. If the risk premium is too high, it is possible that nobody would hold bitcoins and they would only hold currency. This confirms David’s view that information sensitivity could affect the decision to hold bitcoins. However, this does not show up in the exchange rate, but rather in the relative rates of return. The risk premium similarly affect the allocation decision. Consider, for example, that the lifetime budget constraint can now be written as

$c_{1,t} + x c_{2,t+1} + x \omega \leq y$

where $\omega$ is used to simplify notation and denote the risk premium and the aggregate supply of bitcoins has been normalized to one. It is straightforward to see that when the risk premium is zero (i.e. bitcoins are not information sensitive) then the lifetime budget constraint is the same as that outlined above. The existence of a positive risk premium alters the budget set.

So what does all of this mean?

Essentially what it means is that looking at the exchange rate between bitcoins and the dollar is not a useful indicator about whether or not bitcoins would actually make for a good money. Even if we view the exchange rate between bitcoins and dollars as a useful proxy of the price of bitcoins in terms of goods, the exchange rate is not the correct measure for analysis. Rather, to evaluate whether bitcoins are a viable alternative/substitute for dollars, we need to know the relative rates of return on bitcoins and dollars and the risk premium associated with the fact that bitcoins are information sensitive.

This might all seem like semantics, after all, if we think the exchange rate is a good proxy of the price of bitcoins in terms of goods, then the rate of return could just be measured as the rate of change in the exchange rate. Nonetheless, this distinction seems especially important given the nature of the exchange rate between bitcoins and dollars. In particular, just looking at David’s graph of the exchange rate, it is plausible that the time series follows a random walk with a drift (I had trouble acquire actual data rather than ready-made graphs on this so if anybody has the data please send it along). This is important because if this is correct, the variance of the exchange rate is time-dependent. However, in terms of rates of change, the data would be stationary and therefore have some constant, finite variance. Thus, in this hypothetical example, looking at the exchange rate using David’s criteria about information sensitivity would indicate that bitcoin is a very bad money because the variance of the exchange rate is time dependent. In contrast, if the rate of return on bitcoin is stationary, then it is not immediately clear from the data whether or not bitcoin is a good money. This is why we need the model as it helps us to understand what properties the rate of return must possess to make a good money.

Monthly Archives: January 2014

Exchange Rate Volatility and Alternative Money: The Case of Bitcoin

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