Monday, November 16, 2009

Caplan on Velocity and Money Demand

Bryan Caplan gave an nice discussion of velocity on Econlog.

He explains that velocity is defined to be V = Y/M, where Y is nominal income and M is the quantity of money.

He then argues that it is better understood as the reciprocal of the amount of money people keep relative to their nominal incomes.

He states:

Velocity is therefore essentially a measure of income-adjusted money demanded.
The only thing I would add is that Alfred Marshall came up with this idea, and called it "k" and in monetary economics we call it the "Cambridge k."

Caplan makes the interesting point that if there were no real output and income, nominal income would be zero and so would velocity. He says that money would still be spent multiple times on whatever assets or remaining goods existed.

Caplan makes a great point that the demand for money (adjusted for income or not) is something about which individuals can make a choice. And it is possible to add up individual money demands to get an aggregate demand to hold money. It isn't obvious how this can be managed for velocity understood as number of times a dollar is spent.

Finally, Caplan brings up the tautology canard:
Economists occasionally dismiss MV=PY as a mere tautology. Whenever I've
taught macroeconomics, however, I've found that it's an immensely useful

Caplan is saying that it is useful even if it is a tautology, but I think that it isn't a tautology, and that this can be understood using Caplan's approach.

The equation of exchange follows from the equilibrium condition that the quantity of money is equal to the demand to hold money: Ms = Md.

As Caplan notes,

Md = kY

Where k is the income-adjusted demand for money.

Add the equilibrium condition Ms = Md

And solve for: Ms = kY

k = 1/V (definition)

MsV = Y

Y = Py (definition, nominal income equals the price level multiplied by real income)

M = Ms (convention to drop the "s" on the quantity of money)

And the result is the equation of exchange, MV = Py

So what's up with this notion that it is a tautology? Well, Ms = Md can be understood in two ways.

It is a tautology because all existing money is held by someone. In this sense, to hold money is to "demand" it.

What is the equilibrium condition?

It is that actual money balances must be equal to desired money balances. That is, people hold the money and they want to hold it.

The equilibrium condition is Ms = Md because people will adjust their spending until desired balances equal actual balances.

As Caplan notes, this only holds in the aggregate if nominal income adjusts, and it is certainly plausible that changes in aggregate expenditure will impact both the levels of production and prices, and so y and P.

But don't forget, there is this "liquidity effect" by which attempts to spend away excess money can impact the difference between the nominal yields on other assets and money, causing "k" to adjust. (Think about how a given increase in demand for a good raises its relative price and reduces its quantity demanded.)

In fact, I find it doubtful whether k or V are terribly useful.

Still, these ideas are the fundamental ideas of monetary theory.

It would be nice if all introductory macroeconomics students were as well grounded in these relationships as they are in basic supply and demand.


  1. You are right.
    You could even go further and analyse the different macro theories:
    - Neo-classical: V doesn't change, y given (labor market, technology), increase in M leads proportional increase in P.
    -Keynes: when M increases, P given, so y increases even though V must decrease a bit.
    -Monetarism: short run: y increase, P constant or increases a little, V decreases a litlle too (as i decreases). Long run: y constant, P increases more than M (bce V increases as i increases).
    -Same more or less with new keynesianism.
    -Same with new classical: long run monetarism.
    Eh voila!
    Andre Fourcans

  2. I've been wondering lately why the emphasis is usually placed on the MV=PY formulation of the equation of exchange rather than on the more general MV=PT version. It seems that the former version implicitly assumes that excess money balances will be spent exclusively on components of current GDP (or at least assumes that velocity can be meaningfully defined in that way), rather than potentially also on existing assets.

    The whole issue as to whether to target asset prices in addition to prices of the components of GDP (or, alternatively, growth/level of NGDP) seems to me to really be a debate about the demand for money and what people do with excess money balances. Another way to frame the discussion would be to ask “what terms do we put on the right hand side of the equation of exchange”?

  3. I was never taught the equation of exchange in a classroom, but instead discovered it when trying to formalise my intuitive reasoning. When doing so, it seemed sensible to include total transactions on the right side of the equation.

    It seems to me that economic growth is always just the rearrangement of existing stuff. In the only sense that matters, resale of goods increases the productivity of an economy; however, convention has it that at some point we stop calling the rearranging of stuff "production."

  4. The process of production, earning income, and the expenditure of incomes on the goods produced is central to macroeconomics.

    The use of income to purchase various consumer goods is an important aspect of microeconomics.

    The demand for money is how much money people want to hold. It isn't how much money people want to spend.

    I don't think "T" fits into this approach very well.

    I think that the yields on assets other than money can impact the demand to hold money.

  5. Hi Bill. Forgive my obsession with demand for money (I was a student of David Laidler).

    You said:

    "The demand for money is how much money people want to hold. It isn't how much money people want to spend."

    True(ish) but:

    a) I'm not sure that allows one to distinguish between Y and T since they both involve the use of money balances for spending;

    b) at least in the case of the transactions demand for money, it is assumed that the average amount of money held is the result of income receipt at discrete intervals and spending at a roughly continuous rate over the course of each such interval so that money can in fact be held for transactions ("spending") purposes, albeit future transactions purposes;

    c) regardless of what determines their desired level of money balances, an excess supply of money assumes people will try to get rid of it by spending it on something and whether they spend that excess exclusively on components of current NGDP or existing assets as well presumably has implications for monetary policy.

  6. David:

    Thanks for the comment.

    I don't use the equation of exchange at all.

    So, to me, you are saying:

    Md = P md(Rb-Rd, T)

    Where T is the number of transactions you make. Well, I never think about the number of transactions I need to make. And so that doesn't impact my demand for money.

    I don't think the demand for money is proportional to the amount people plan to spend.

    While people do hold money in order to be able to spend it, that doesn't mean they plan to spend it at any particular time.

    The transactions demand for money is useful, in my view, mostly to think about how changes in payment patterns, trade credit, or the use of credit cards might cause changes in the demand for money.

    Following Mises, I take the precautionary demand for money as central, though I grant that holding money between the time it is received and spent is always involved.

    It provides benefits by reducing inconvenience in an uncertain world. But like all good things, having more of it requires the sacrfice of something else. And so, income creates a budget constraint. I will grant that wealth might be better, but changes in asset prices shades over into the opportuntity cost arguments.

    As for disequilibrium, I don't assume that excess supplies or demands for money soley cause changes in expenditure on current components for GDP.

    What is interesting about the income elasticity of the demand for money is that decreases in real income can reduce the demand for money, but if this is due to demand-contrained production, there remains monetary disequilibrium. Given the poverty created by the monetary disequilibrium people are satisfied holding the existing quantity of money. But really, we are seeing one of the worst problems created by monetary disequilibrium.

    That is why I focus on real income.