The classic reference on the equation of exchange approach is Irving Fisher (1911) The Purchasing Power of Money, though the approach is tied to the quantity theory of money, which goes back centuries in various forms.
This awkward situation underlines the chasm between "micro" and "macro" economics and shows that economists do not agree on the actual role that money plays in the real-world economy. There are two main approaches to dealing with money in economic analysis, each having strengths and weaknesses.
Probably all economists, and even many non-economists, are familiar with the time-honored "equation of exchange" approach to money, which takes a macro perspective by essentially overlaying the entire stockpile of money on top of the underlying "real" micro economy.1
There are several uses of the term "equation of exchange" in economics. The most common modern usage refers to money demand and supply, or Irving Fisher's MV = PT (or MV = PY where Y represents income). A more general, earlier use of the term was introduced by William Stanley Jevons. See Inflation, by Lawrence H. White in the Concise Encyclopedia of Economics for more on the modern monetary use of the term.
The original and more intuitive version of the equation is MV = PT, where M stands for the total quantity of money in the economy, V is the "velocity of circulation" (meaning how many times, on average, a dollar bill changes hands for the time period in question), P is the average price of a transaction ("price level"), and T is the total number of transactions. In this form, one can see that the equation is not a statement of economic theory. Instead, it is an accounting identity. That is, the statement must be true. The left side, MV, measures how much is spent in the aggregate during the period in question. The right side, PT, is the exact same thing, from a different perspective. Since both sides measure how much total money is spent in the economy, the two expressions necessarily equal each other.
The problem with stating the equation of exchange in this form is that the term T includes all transactions, including the sale of previously-produced goods and even of assets. For example, if Bill buys 100 shares of stock from Sally at $5 per share, then this transaction is included in T, the price of the shares influences the "average price" P, and the turnover of the $500 from Bill to Sally is included in the "velocity of circulation" V. However, economists typically don't welcome this result because they want to exclude financial transactions from the analysis in order to focus on production.
Consequently, it is more common nowadays to see the equation expressed in the form MV = PQ, where Q stands for newly-produced goods and services ("real output") during the period in question, and where both V and P are restricted to transactions involving newly-produced goods and services. In this updated form, the equation of exchange can be very useful to illustrate certain relationships.
For example, the equation shows that if the quantity of money in circulation doubles while velocity and real output remain constant, then the price level doubles, too. It can also show that if the economy's production of "stuff" grows faster than the stock of money, and if people don't change how quickly they spend the average dollar bill in their possession, then prices will have to fall over time.
As with the transactions version of the equation, this version is necessarily true, and economists of all stripes can use it to illustrate their rival theories. On the one hand, a Keynesian might use it to show that during a liquidity trap, monetary expansions by the central bank will not lead to rising prices because they will be perfectly offset by drops in velocity ("increased hoarding"). On the other hand, a Chicago School proponent of efficient markets might argue that monetary policies (especially in the long run) can't affect the "real economy" Q and, instead, merely change the growth rate of P, that is, the inflation rate.
The great virtue of the equation of exchange is that it provides a simple and intuitive framework with which to keep track of the various constraints in the economy. Whatever story an economist wants to tell, the equation of exchange forces him to "stay honest" with the listener. For example, if an economist wants to warn that a growing money stock will cause prices to rise too rapidly, the equation reminds him that a full explanation requires him to show that velocity and real output will not react in ways that contradict his prediction.
The great weakness of the equation of exchange it that it is artificial, providing no real insight into the function of money in the first place. With this approach, economists typically will "solve" for an economy in terms of barter, and then, as an afterthought, overlay the total stock of money on these relative price ratios to come up with absolute prices (quoted in money). Thus, the economist might use the fundamental logic of utility maximization and resource supplies to conclude that, in equilibrium, one apple will trade for two oranges. Then, he might say that with a certain stock of money M, an apple will have a price of $1 while an orange will have a price of 50 cents. Now, if we double the stock of money to 2M, other things equal, apples will be priced at $2 and oranges at $1.
To reiterate, the problem with this (standard) exercise is that there is no explanation for why people in the real world use money. After all, if the economist is able to "solve" for the relative price of apples and oranges without the use of money, it's not clear why people in the real world use intrinsically useless pieces of paper in their dealings. Why not consult their preferences and swap fruits directly against each other, as the economist assumes they can do?
Note that the criticism here is not the familiar one that a given model is "unrealistic"; just about every economic analysis assumes away real-world complications to isolate particular relationships. Rather, the objection here is that if we are going to use a model to talk about money, then there should be a good reason in the model for people to use money. The standard equation of exchange approach does not obey this simple principle.
Rather than taking a bird's eye view of the entire stock of money in the economy, an alternative tradition focuses on the typical individual and his holdings of cash balances.2 The central insight here is that, strictly speaking, money is never "in circulation"; at any given moment, every single dollar is in someone's possession. Rather than focusing on dollar bills as they flow around the economy, the cash-balance approach focuses on individuals and their optimizing decisions.
In the cash-balance approach, the economist recognizes that in equilibrium—though it may be a fleeting condition—it has to be true that every individual is satisfied with the amount of money he holds. After all, if the individual thought he'd be happier by trading away 50 cents to receive just one more gumball, then it would not be equilibrium. Having thought through the necessary and sufficient conditions for a typical individual to be in equilibrium with respect to his cash holdings, the economist finishes the analysis by saying that the sum of all desired cash balances must equal the total quantity of money. In equilibrium, it must be the case that the community collectively desires to hold the exact amount of money in existence at that moment—no more, no less.
The great virtue of the cash-balance approach is that it focuses on the interesting fact that people want to hold money—even though qua money it's intrinsically useless—but not too much money.3 Going down this path forces the analyst to think about the different things one might buy with money, including not just consumable goods and services, but also assets. Thus, we see that cash can "burn a hole in one's pocket" not simply because of a desire to buy pizza, but also because of a desire to diversify wealth out of currency and into corporate stock, bonds, precious metals, etc.
As with the equation of exchange, the cash-balance approach, too, is very flexible, allowing economists to implement their preferred theories explaining monetary activity. A Keynesian can argue that at very low nominal interest rates, there is little reason to prefer relatively illiquid bonds over money; thus, the demand to hold money can rise substantially, perhaps frustrating the central bank's efforts to boost aggregate demand. In contrast, an inflation hawk might use the approach to argue that an expected burst of money printing in the future would lead people to reduce their desire to hold money in the present. Since the nominal quantity of money is fixed (for the moment), the only way to restore equilibrium is if prices generally rise, so that the "real" amount of purchasing power everyone holds can decline.
The downside of the cash-balance approach is intimately connected to its strength. It's all well and good to praise the method for focusing on the optimizing behavior of the individual, thus treating the commodity "money" with the same conceptual tools that economists use on every other good and service. The problem is that it's very difficult to explain the demand to hold money. (In contrast, explaining the demand for apples, or even the demand for unskilled labor, is relatively trivial.) In explaining why people choose to hold money, economists are ultimately forced to take positions on the role of institutions, how individuals deal with uncertainty, and even the mechanics of commercial transactions. Because the cash-balance approach carries with it so much baggage, economists cannot use it to quickly settle an argument over, say, whether the Fed should drop the dual mandate to care about both unemployment and inflation and, instead, adopt an explicit inflation target.
As we have seen, both of the approaches to money can be flexible, allowing the economist of a particular school of thought to tell a more specific story involving behavioral assumptions he favors. It is also true that economists can ultimately handle a given scenario with either approach, and the decision about which approach to use boils down to which is more convenient or intuitive for the scenario at hand. In practice, the two approaches may often blend into one unified analysis, drawing on elements from both perspectives.
For example, suppose that an economist wants to discuss David Hume's thought experiment of a magical doubling of the quantity of money during the night. Using the equation of exchange, the economist might say:
Assume that we had full employment originally, so there is little scope for increasing real output. In this case, if velocity is constant, doubling M will cause P to double. However, if the overnight magic causes people to worry that their money is not a good store of value, then velocity will rise, meaning that P ends up more than doubling.
In contrast, the economist could handle the stipulated scenario with the cash-balance approach, perhaps saying:
The typical individual was originally in equilibrium with a certain nominal quantity of dollars, and with respect to a certain level of prices. The next morning, everyone wakes up to discover that he is holding twice as much money as he thinks is optimal. Each person will try to get rid of the excess cash holdings. However, collectively, the community can't get rid of money; one person's spending is another person's income. If we assume that what people really care about is the amount of goods and services that can be purchased at a moment's notice with cash balances, then equilibrium can be restored once prices rise. If the magical doubling wouldn't change this desired amount of "real" cash balances, then prices must double. However, if the episode has soured the typical person on storing wealth in the form of currency, then he will re-optimize and reduce his desired level of real cash balances. This means that prices must more than double in order to induce the community to hold the doubled quantity of nominal dollars.
This example shows that, in practice, the distinct approaches may merge into a unified explanation. Even when an economist uses the equation of exchange, he will probably motivate discussions of "what happens to V" by reference to individuals' preferences and expectations about future purchasing power. On the other hand, even the economist who writes strictly in terms of the cash-balance approach may often reach the qualitative answer in his mind by first thinking in a macro framework and then giving the reasoning a "micro foundation" by incorporating it into the perspective of a typical individual's demand for money.
Although money is essential to modern civilization, it occupies an awkward position in economic theory. The equation of exchange provides an intuitive framework for quickly analyzing various scenarios, but it can lead to faulty policy conclusions because it doesn't really explain why individuals hold money in the first place. In contrast, the cash-balance approach seems like a natural extension of standard microeconomic analysis, applied to the special case of money. But this special case is qualitatively more complicated and controversial than standard applications.
In practice, when assessing the impact of a policy or preference change, economists often draw on elements of both approaches. Despite the typical non-economist's belief to the contrary, economists do not "study money," and, in fact, most economists are very uncomfortable doing so.