Thursday, June 30, 2011

Thinking through Chapter 3 of the General Theory

Bob Murphy is seeking some insights on Chapter 3 of the General Theory, and I thought I'd give it a shot. I haven't read the book carefully through for several years now, although certain chapters I've given multiple careful reads since. Chapter 3 is not one of them. It's worth noting - before jumping into this - that Keynes himself says that his exposition of the theory of effective demand at this stage, before introducing subsequent theory, "may not be fully intelligible".

Question 1: What is the unit of aggregate supply price and aggregate demand? In money?

As far as I can tell it's in money. The question, of course, is whether it's in current money or some constant value of money. For Keynes, money is not some numeraire. There is a reason for its value at any given point in time, and the value of money at any point in time, and the change in the value of money relative to expectations has consequences. So I would guess (and Keynes seems to give no reason not to suppose) that it is current money. He doesn't seem to make it explicit either way, but that's how I read it. Later in the chapter he talks about how the analysis of wage units can be done either in "money wages" or real wages.

Question 2: Is the aggregate supply price referring to the unit price, or the total proceeds?

This he does make explicit. The aggregate supply price is the total proceeds of the output associated with a given employment level, N. This is, of course, confusing. That's not a "price". But Keynes lays out his definition, and it is what it is. He says in the text that "the aggregate supply price of the output of a given amount of employment is the expectation of proceeds which will just make it worth the while of the entrepreneurs to give that employment." Murphy quotes this portion, and it's true it's not entirely clear from this sentence whether he's talking about total proceeds or unit price (i.e. - he could be refering to the unit price consistent with a total outpu that will make it with the while of the entrepreneurs). But the footnote to this sentence clarifies: "Not to be confused (vida infra) with the supply price of a unit of output in the ordinary sense of this term". This is still odd. We don't talk about the total proceeds of output much. However, I think it becomes clearer later, and it makes more sense when we think in terms of the quantity theory of money. The other clue is that the demand function - which equilibrates with supply and therefore must be measured in the same units for both variables - is clearly the entrepreneurial proceeds (not profits - which are proceeds minus factor costs). In the first paragraph he also calls the "proceeds" the "aggregate income (i.e. factor cost plus profit)". So I think it's quite definitely the total rather than unit proceeds.

Question 3: Related to this, are the functions Z=φ(N) and D=f(N) upward sloping? (And what is the Y axis here–money?) So Keynes is saying that in the general case, D starts above Z, but has a lower slope, so that when N is really low, D is above Z, but eventually they intersect as N increases?

Yes, I think so. This stands to reason. If Z and D are the total proceeds rather than unit price, then Z=φ(N)=pQs, and D=f(N)=pQd. We know that Qs is increasing in p and Qd is decreasing in p by the basic law of supply and demand, so at Z=D (or, equivalently, Qs = Qd) Z is steeper than D. Is that a unique equilibrium? We don't know. We take local equilibria and don't usually ask any more questions, unless circumstances or research interests compel us to ask questions. Multiple equilibria aren't relevant to the issue at hand, though.

The vertical axis, Z and D, is measured in money - presumably current money for the reasons I gave in response to the first question.

Consider the following data from a simple AD-AS graph which follows. After the AD-AS graph is the associated Z-D graph:

There's one major difference between my last graph here and the version that Keynes presents: my horizontal axis is in terms of the aggregate price level (i.e., the CPI) and Keynes's horizontal axis is in terms of N - employment.

Strange, isn't it? How did I get that on the horizontal axis? It's because we typically think of quantity as a function of price. So we talk about "quantity demanded" and "quantity supplied", but we never talk about "price demanded" or "price supplied". To get output (which we assume to be increasing with employment up to a level of diminishing returns at least) on the horizontal axis, all we need to do is sub Q for Qd and Qs in my data chart and sub pd and ps for p. You can see in this interchangability the origins of the Phillip's Curve*.

Question 4: Keynes says that when N is below the equilibrium point, then D is above Z, and so entrepreneurs have an incentive to hire more workers. But why? It sounds intuitive at first, but I’m not so sure it is. In particular, Keynes says that when N is such that D and Z intersect, the entrepreneurs profit has been maximized. But it seems to me the profit is zero at that point? (In a standard micro model, it’s fine for the producers to maximize profit at the point of zero-profit, because they’d earn negative profits at different levels of output. But that’s not what happens here. If N went below the intersection point, then wouldn’t aggregate profits go up?).

Entrepreneurs have an incentive to hire more workers for the same reason that they have an incentive to hire (i.e. - produce more output) when Q is below equilibrium in the supply and demand model: some entrepreneur is earning a marginal revenue that exceeds his marginal cost. What happens when N is below equilibrium? At that point Z, which is the minimum aggregate proceeds required to engage the employment of N units of labor, is lower than the expected proceeds of that labor. Profits are positive, as Bob notes. What do entrepreneurs do when they see positive profits? They enter the market. As Keynes writes "if for a given value of N the expected proceeds are greater than the aggregate supply price, i.e. if D is greater than Z, there will be an incentive to entrepreneurs to increase employment beyond N and, if necessary, to raise costs by competing with one another for the factors of production, up to the value of N for which Z has become equal to D". This is the basic market equilibrium of setting marginal revenue equal to marginal cost. Keynes does something you usually don't see, which is to present it in terms of the total proceeds from the market (i.e. price times quantity), because this is the relevant functional relationship in macroeconoimcs, where we look into the aggregate properties of human action.

Question 5: Later on, when discussing the implications of the classical view, where D=Z at all levels of N, Keynes says “the forces of competition between entrepreneurs may be expected to push [N] to this maximum value.” But why? If Z and D overlap each other for all N, and Keynes has earlier argued that at the intersection point, aggregate profits are maximized, then why would entrepreneurs have an incentive to move N one way or the other, if Say’s Law holds?

This part is confusing to me. If you read it all together, I'm not sure he's saying that the functions are identical - I think he's saying that for any given N, the D=f(N) curve shifts to be equal to Z=φ(N). Paying factors of production Z total proceeds creates the same amount of demand in the market, so demand increases in response. An increase in demand is a shift of the effective demand curve itself, not a shift along it. This seems to make the most sense to me after reading Keynes talk about demand "accomodating itself to the aggregate supply price", and "the proceeds D assume a value equal to aggregate supply price Z". That implies to me a change in demand, not an identical function.

If all values of N are admissable equilibrium, then Bob rightly asks on what basis would entrepreneurs move to a high value of N under Keynes's rendition of classical economics? We know entrepreneurs produce output until profits are driven to zero. What would require them to maximize N? If profits are zero, then income is made up of two components: factor costs of employment and what Keynes calls the "user cost". If we maximize the factor cost of employment (i.e., the wage bill) we have to minimize this "user cost". What is "user cost"? Keynes defines it as "the amounts which he [the entrepreneur] pays out to other entrepreneurs for what has has to purchase from them together with the sacrifice which he incurs by employing the equipment instead of leaving it idle" (emphasis mine).

People find private value in leaving things idle - in keeping capital or money ready at hand. This, more than anything else, is the point of the General Theory. The most valuable thing to keep idle, of course, is money. We only keep anything else idle because we think it may serve some purpose similar to what we turn to money for: either because we think it will store value, or because we think it can act as a medium of exchange.

Keynes argues that no classical economist really thought through the implications of leaving things idle. By ignoring it, they implicitly assumed that user cost would be minimized and that's what drives N up to the limits imposed on it by an inelastic supply curve and the marginal disutility of labor. Many classics touched on what would become known as "hydraulic Keynesianism". If you have a leak in the circular flow, your income level would go down. The mercantilists and Malthus got that far but were later pushed off the stage by classical economists who somewhat unfairly identified mercantilism with the protectionist ideas of kings and merchants. But this isn't really sufficient. This is what I've called the "whack-a-mole" theory of general gluts. Saying "demanding idle cash lowers income" doesn't do the trick because of the real balance effect. As the price level increases, the cash becomes more valuable and everything else becomes less valuable. As long as prices eventually adjust everything is fine. There's no reason for anything to change other than the price of money (i.e. the inverse of the price level). There's no reason for any relative price adjustments, in other words, and therefore no reason for a change in the employment rate.

However, if variations in the value of money (presumably driven by the demand for money) can influence Keynes's "user cost" of capital - if leaving capital idle can earn a return - then there is no reason to expect a rebalancing. We have the raw ingredients for a theory of a stable underemployment equilibrium.

So Bob's fifth question is by far the toughest (I wrote most of this last night and then had to sleep on my response to the fifth question), but I think the clearest way to think about it is (1.) Keynes thinks the classics ignore user cost or the cost of not leaving things idle, (2.) the implication of this is entrepreneurial competition maximizing N, (3.) In Z-D terms, this means that increasing N provides the additional demand to employ that N, and the D schedule is driven up the Z schedule to the point where Z is inelastic.

That's my reaction - any thoughts? Later in the chapter Keynes provides a great synopsis of the General Theory, which I think helps clarify what is going on early in his discussion of the classics and "user cost". I haven't read Chapter 6 again, yet, but that also goes into more detail on user cost.

* This was actually an accident. In writing the post, I realized "that shouldn't be on the horizontal axis!" and had to think through why it came out that way. So it's a diversion, but an interesting diversion I thought. The point is in (1.) deriving a Z and D function where the D function isn't as steep at equilibrium as the Z function, and (2.) demonstrating why it follows necessarily from the definition of Z and D as the total proceeds of the output.

1 comment:

  1. Wow thanks for putting so much effort into this. I'll be sure to point my students at this post.


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