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Grat problem for the basket brigade model.
In this model, poor people have baskets mostly empty, rich people mostly full. The poor people will pay an interest charge, soon. The reason theypay a charge is because, with finite probability, there will occasionally be a poor person who gets his basket full, so the S&L machine can infer the basket size, and it knows, therefore, what 'too empty' means, (conditional probability of this purchase fitting into that basket).

What to the two groups do? They change basket sizes such that consumption optimally fills the basket and no interest payouts pass.
Why would th S&L machine do such a thing? We ask it to. We ell the S&L machine to keep these basket sizes so our personal basket is efficiently filled. So the total goods coming to our door arrives with the minimum number of transactions.

Multiplier?
Let the S&L machine suddenly widen its bounds for price discovery. What happens? If it is done well, the S&L gets multiplied up in market share, at the expense of some other S&L machine. The catch, the basket brigade model assumes no third party trades, their is no multiplicand. It is simply an artifact of the mathematical model being the simplest one for basket brigade. Make it complex; but at its base, the multiplier concept is gone, replaced by a random, generally zero mean, currency uncertainty.

I would like to follow your logic but need a mechanical model. I'll try to describe (in a way you can follow) the model you seem to be using.

Grab the back of an envelope and draw a circle on it. Label the circle the "Entire Economy" (Assumed to be confined to the part we are considering).

Within that circle, draw two smaller circles; label one circle "Hand-to-Mouth" and the second circle "Autonomous".

Outside the circle and to the right side, draw two boxes. Label one box "savings" and one box "consumption".

Next, we link the circles to money flows:

We begin with an external (one time) injection of money. Draw this as an arrow entering the left side of the Entire Economy circle.

This injection of money is divided between two players represented by the Hand-to-Mouth and Autonomous circles. Split the initial injection as desired. You use a 50-50 distribution initially.

Each player makes a money use decision. The Hand-to-Mouth player sends his share to Consumption; The Autonomous player sends his share to Savings. Draw arrows from each player to the correct destination boxes.

This completes the model until we consider that you would like to describe rounds two and more. Round two (and more) will require additional injections of money into the economy. Following your logic, this additional money would come from the Consumption box. [The additional money could come from savings or a new external injection but that would be a different scenario.]

We could mechanically describe rounds two and more with additional arrows indicating money flows from the consumption box back to the (input side of) entire economy circle (we could call it a feed-back loop).

By using this mechanical model and making some assumptions about how to apply the 1% consumption change, I can reproduce most of your numbers.

Does this mechanical model work for you?

Despite trying very hard to avoid errors, I commonly find them.

"Outside the circle and to the right side, draw two boxes. Label one box "savings" and one box "consumption". should read "Outside the Entire Economy circle and to the right side, draw two boxes. Label one box "savings" and one box "consumption".

Roger: if you want a "hyraulic" model: imagine the Autonomous agents having a stock of money, so they can choose their flow out of spending, and the HTM agents instantly spending a flow of money equal to their incoming flow of income. The Autonomous agents determine the flow between the agents. There is no flow to or from the outside of the system, so one agent's flow out is another agent's flow in.

But it doesn't clarify anything.

Using your comment as a foundation, we could build a "continuous chain" model. We can think of a continuous chain having buckets mounted at intervals. Each bucket can be loaded with money.

The Autonomous agents decide how much money to put into each bucket each time a bucket passes them. The HTM agents receive the bucket full of money and immediately return it, using the same bucket (instantaneously).

When the bucket returns to the Autonomous agents (from the HTM agents), they can decide whether money should be added (or removed) to the bucket.

I don't see how we can get a multiplier from this model. The HTM agents seem to be completely dependent upon the decisions of the Autonomous agents.

Roger: this is very basic Old Keynesian macro. If the Autonomous agents are half the economy, then if Autonomous agents increase their spending by $100 per year, HTM increase theirs by $100 too, so total spending rises by $200. Which is a multiplier of 2.

Thanks Nick.

For me, it's all about mechanical models. It helps to see how you (and other economist) relate mechanical models to mathematical models.

Hi Nick,

I don't think the Kalecki analogy works unless the Autonomous are also capitalists. When Kalecki said, "the capitalists get what they spend," he was making the same point Keynes made in the Treatise, viz. that, if the output of consumption goods is fixed, at least in the short run, then additional consumption spending raises prices and profits. In other words, what the capitalists spend, they get back in profits. So, for the analogy to go through, I think your Autonomous group must be capitalists (or residual claimants at least).

Suppose something causes the Autonomous agents to increase their consumption by 1%. The first round effect is to increase aggregate income by 0.5% (because they are only half the economy). In the second round the Hand-To-Mouth agents increase their consumption by 0.5%, which increases aggregate income by an additional 0.25%. And so on, until the increase in aggregate income converges to 1% in the limit of the multiplier process.

Oh! How do you know that?

Two "autonomous" agents: I offer to mow her lawn for money and she offers to fix my car for the same money. We both increase our consumption by 1%, so what's it to you? Why would this have any effect on anyone else in the system?

What I'm saying is that an economy must necessarily have a network topology, not an homogeneous goo topology so therefore you have no conclusion about what will happen with the Hand-To-Mouth agents. They might take up some proportion of the additional spending, or they might not.

You can see why "Forward Guidance" matters a lot in New Keynesian models. Anything that causes expected future consumption to increase by 1% causes current consumption to increase by the same 1%, regardless of how far in the future agents expect that thing to happen.
Give or take a modicum of Fed Credibility.

"Take a 2-period example to see how it works. The Autonomous agents have a 0.5 Marginal Propensity to Consume out of current income."

Where did this come from? You started out by stating that autonomous agents have a 0 MPC.

I do not know if I misunderstand your argument, but

"..The first round effect is to increase aggregate income by 0.5% (because they are only half the economy). In the second round the Hand-To-Mouth agents increase their consumption by 0.5%, which increases aggregate income by an additional 0.25%. And so on, until the increase in aggregate income converges to 1% in the limit of the multiplier process."

Doesn't this depend on the following ratio assumption for each round (/ refers to division):

spending of hand-to-mouth agents going to hand-to-mouth agents (SHGH)/spending of hand-to-mouth agents going to autonomous agents (SHGA) = spending of first-round autonomous agents going to autonomous agents (SAGA)/spending of first-round autonomous agents going to hand-to-mouth agents (SAGH)

or simply, SHGH:SHGA = SAGA:SAGH, where : refers to ratio.

Though I am not sure if this is the artifact of the (Old Keynesian multiplier) simplification, not main arguments. I am yet to read the Farhi-Werning paper.

@JKH:

The 0.5 MPC comes from it being a two-period model. Autonomous agents are consumption smoothers, so an increase in one period's income is spent half in the first period, half in the second period. This spending increase occurs regardless of whether the income increase happens in the first or second period, since autonomous agents are not liquidity-constrained.

My previous comment assumed the single-period analysis, plus no action after first round on spending by autonomous agents. Now I am relaxing that assumption, just as in the quote below:

"..the Autonomous agents have a 0.5 Marginal Propensity to Consume out of current income. So if they get news from the central bank that leads them to expect that future income will be 1% higher than they had previously expected, their first round effect is to increase their consumption by 0.5% in both the current and future periods, which means (since they are half the economy) that aggregate income will increase by 0.25% in both the current and future periods. In the second round of their figuring, they figure the Hand-To-Mouth agents will respond by increasing their consumption by 0.25% in both periods, which means aggregate income increases by an additional 0.125% in both periods. But then the Autonomous agents decide to increase their consumption by an additional 0.125% in both periods. And so on. And (0.25+0.125+0.125)+(0.0625+0.03125+0.3125)+......etc. = 0.5+0.25+.....etc. = one. So in the limit of their figurings, their expectation of a 1% increase in future income becomes a self-fulfilling rational expectation, and also causes current income to increase by 1% too."

But just like my previous comment, this seems to depend on where spending goes to. So start with aggregate income shared equally between autonomous and hand-to-mouth agents, but with spending outflow destination ratio not 1:1. Suppose that spending mostly always goes to hand-to-mouth agents for all rounds. Then multiplier process indicates higher-than-expected income increase. Then I guess different non-self-fulfilling expectation equilibria are possible, though maybe by equilibrium process forces (first-order conditions based on utility functions?) the agents converge on self-fulfilling expectation equilibria. Though again, I am yet to read the paper mentioned, so I do not know whether this is relevant.

Tel: sure. Take a large economy, where the Autonomous/HTM distinction is orthogonal to the pattern of demand among agents. That's the benchmark case. With correlation, the multiplier could be either smaller or larger.

JKH: With consumption-smoothers in an n-period model, their mpc out of current income is 1/n. In the standard New Keynesian model, which is infinite period, the mpc out of this instant's income becomes zero, so we can call them "Autonomous". I should probably have named them "Smoothers", because they are only strictly "Autonomous" in the limit. As Majro says.

TGV: yes, I *think* that's right (see my reply to Tel above). But that is the standard (often implicit) Keynesian (Old and New) assumption. Think of every agent's consumption spending distributed equally across all other agents, so they only differ in what determines their total spending.

Thanks Nick.

I suspect my exposure to the use of the terminologies “autonomous” and “MPC” is implicitly old Keynesian, obtained in roughly the same era as my experience with drawdown/redeposit.

JKH: Yep. I called them "Autonomous" in the first bit precisely to make people think in Old Keynesian terms. But then the name stuck, even though it didn't fit so well when I was writing the last bit! I was thinking about giving them a new name throughout, but decided against.

[The following description follows the "back-of-envelope" mechanical model previously described in these comments.]

We have two agents, "Autonomous" and HTM. Each agent represents populations; populations that can be unequal in size while having equal income on a per head basis.

The HTM will spend 100%; the Autonomous 0%. That does not mean that the Autonomous do nothing with their income. In every case, the Autonomous delay their spending until n rounds have completed. (This means that Autonomous spending is ALWAYS considered as an initial driving event.)

Nick begins with a money block passing through the system one time. With populations equal and per-head income equal, each agent group had equal income.

In the next scenario, Nick allows a 1% deviation between group spending. In the mechanical description, HTM would now spend 50% + 1% = 51% and Autonomous 50% - 1% = 49%, so we would consider that the population of HTM is 51% and Autonomous 49%. This 1% change will increase total economy consumptive spending on every round, beginning a multiplier effect.

[If you try to set this up on a computer, remember that the population (as individuals) has equal income. The two agents will have (usually) unequal income.]

We should consider the multiplier effect further. The initial money injection was a fixed amount. The portion received by HTM can be re-spent, thereby re-entering the system and enlarging the potential consumption (and delayed spending) by agents. This re-entry of money drives the "multiplier effect".

Nick begins a "multiplier effect" description by a inserting a second round, then a adding the income from the initial event and the second event. The initial event is driven from money sourced outside of the system. The second event is driven by money re-spent by the HTM consumers (as described above). A portion of the re-spent money is captured by the Autonomous group in a fashion identical to the Autonomous capture of the initial money injection. (We can have additional rounds. Each round will "recycle" (only) money from the HTM agent and will be smaller due to the money delayed by the Autonomous agent.) Eventually, a limit is approached where-in all (where "all" is considered as a meaningful measurement) the externally injected money is in the hands of the Autonomous agents.

[Money sourced outside of this system can actually come from Autonomous' delayed spending (as previously mentioned), or it could come from QE.]

Does my mechanical model make more sense now? It seems like a Keynesian description but I don't know the flavor.

That's the benchmark case. With correlation, the multiplier could be either smaller or larger.
So I'm going to give you a model and it delivers an answer... but if you want a bigger answer we can tweak the model to do that; and by the way if you ever feel the need for a smaller answer, we can deliver that too. We deliver everything! No customer goes home unsatisfied here.
Think of every agent's consumption spending distributed equally across all other agents, so they only differ in what determines their total spending.
In other words, you have boiled the economy down to green goo and removed any structural concept. By doing that you also cannot possibly have structural problems, nor can any policy ever create structural problems.

My recap interpretation:

The post uses the term MPC in two different ways (which I think Nick has acknowledged in a comment above).

The first is as it applies in the old K way, characterizing the expected consumption response to an “autonomous” injection of new spending and income. Such an old K autonomous injection is typically in the form of an assumed amount of investment, government, or export expenditure. But the post transforms/simplifies this to the case of autonomous consumption (in a non-consumption economy for example), to the exclusion of these usual old K categories above (“it's got bugger all to do with inventories (think haircuts)”).

The second use of the MPC terminology refers to autonomous expenditure itself responding in such a way that “something causes the autonomous agents to increase …” This seems like a conflation of application between the old and new K (which I think Nick has acknowledged in comment above), and a commingling of the usual difference in old K meaning between autonomous and induced (HTM). In addition, this MPC works as new K “smoothed” consumption spending over multiple time periods, where the so-called MPC factor works uniformly (additively in effect) over time, rather than multiplicatively as in the old K fashion.

So in the old K way, a 50 per cent MPC effectively multiplies a given autonomous expenditure/income injection into successive income/spending rounds, where in each round 50 per cent of newly created income is spent on consumption.

And in the new K way as applied in the post, a 50 per cent MPC (for example) refers to autonomous expenditure undertaken in each of 2 periods at that 50 per cent rate (according to new K consumption “smoothing”).

The post then splits the characterization of old and new K multipliers as cross sectional and times series phenomena respectively.

This is confusing, because the old K multiplier is inherently and logically a time series in itself. It works in effect through iteration of compounding of MPC on MPC – over time, step by step. Whereas the time series characterization of the New K distribution of “autonomous” spending refers to the distribution of “planned” installments of such autonomous spending over time.

Thus, the cross sectional characterization of the Old K induced (HTM) spending has to do with the correspondence of the initiation of an old K multiplier response to each new K spending installment, where each old K multiplier response considered in its entirety is in fact a time series as well.

Moreover, given the apparent feedback in the example between the responses of autonomous and HTM to each other’s income effect, the cross sectional characteristic would seem to apply to both equally.

JKH "This is confusing, because the old K multiplier is inherently and logically a time series in itself. It works in effect through iteration of compounding of MPC on MPC – over time, step by step."

Yes and no. If we look at the math (and accounting) of the Old Keynesian multiplier, and take it literally as written, it all happens instantly. Given a consumption function C(t)=a + bY(t), and Y(t) = C(t)+ A(t) (where A(t) is autonomous expenditure), the multiplier effect of A(t) on Y(t) happens instantly. There is no time-series process. But when the Old Keynesian profs taught the story, in words or diagrams, it was always a process of successive rounds that seemed to happen over time.

The simplest way to reconcile the apparent contradiction is to say that when people decide on their spending for the current period they do not know their current period income (which is determined by others' spending decisions), so we change the consumption function too C(t)= a + bYe(t) (it depends on *expected* current income), then assume adaptive expectations so Ye(t)=Y(t-1).

A second way to reconcile the apparent contradiction is to think of the successive rounds of the multiplier process as happening in "virtual time" inside people's heads, as they try to solve mentally for the new equilibrium before deciding how much to spend. "Let's see, the govt has announced its is increasing spending by $100 this period, so Y goes up by $100, so everyone will respond by increasing their C by $50. But hang on, that means Y goes up by $150, so they increase by $75. But hang on...etc." They are trying to solve for the rational expectations equilibrium (the Nash Equilibrium) by iteration, rather like students first learning algebra who can't solve simultaneous equations the proper way. And this is exactly the assumption made by the Farhi & Werning paper. And it's what I'm doing in this post too (because my post is about their paper). It's sort of like Rational Expectations, but stops short of fully rational, because the agents only do a finite number of iterations before they get bored doing the math.

And a third way to reconcile the apparent contradiction, and one the old profs often used, is to talk about inventories, and firms finding that their inventories had fallen and increasing output with a one-period lag to desired expenditure (there is negative undesired inventory investment when the multiplier starts rolling). This is really a more complicated version of my first story above, only it's firms who get surprised by their sales, rather than households being surprised by their incomes. And if firms have a desired *stock* (rather than flow) of inventories this is actually a much more complicated story, because the desired flow (change in stock) depends on past surprises. And it doesn't work for services.

Nick,

Maybe I should represent myself as a Keynesian big bang concrete steppes guy.

I had quite an extended discussion with David Glasner over the course of several of his posts regarding my own interpretation of the old K multiplier. In fact, he concluded it with a post exclusively dedicated to a response in part. Perhaps I’ll dig it up later.

With regard to the algebra and the corresponding accounting, I would beg to differ in that there is nothing there that indicates it is not a time series process. Logically, if people are in fact spending from income that is created algebraically and accounting step-wise, that represents a time series process.

I think I hear what you’re saying in terms of rational expectations and instantaneous response, but again there is nothing in the multiplier algebra or accounting that implies that an instantaneous response is what is being depicted. The instantaneous response in my mind would be the result of some sort of rational expectations implementation, as I think you suggest. But it would be hard to demonstrate empirically that this is the way that the world works. The world actually works in discrete time, and that encourages the expectation of a time series process regarding this issue, if anything.

I’m obviously not denying your knowledge and understanding of the relevant academic theory, but I think that theory is not the only theory when it comes to fitting with the facts of economic development. I can’t imagine that a multiplier from infrastructure spending occurs without a time series element. Workers tend to want to see at least some income from their employment before they start putting in their orders for new houses, cars and smartphones.

But above all else, the algebra and the accounting alone does not imply the absence of a time series requirement, and I’d debate that point anytime (and I did, among other things, with David Glasner). An instantaneous algebraic calculation does not mean that the mathematics depicts an instantaneous event.

"An instantaneous algebraic calculation does not mean that the mathematics depicts an instantaneous event."

I think an "instantaneous algebraic calculation" of a time-series-event automatically implies an assumption that the time series reached it's limit of expansion within the time period under consideration .

JKH: the math is unequivocal: the only solution is Y(t)= [(1/(1-c)]x[a+ A(t)] . So a change in A(t) changes Y(t) instantly. And the period (t) can be arbitrarily short, right into continuous time. (This is similar to the argument between Ramamanan and Steve Keen a couple of years back). But our intuition is to think of $ flowing first into our pockets, and only *subsequently* being spent. Which is like C(t)=a+bY(t-1). And with that consumption function, you do indeed get a slow multiplier process, which only converges slowly towards the same answer.

But in the real world, does it take time for the multiplier process to work itself out through successive rounds? Yes. And I've done a couple of post on this, way back, because the same applies to the Old Monetarist Hot Potato process too.

But if people anticipate the increase in G, and consume based on their expected future income (in part), then the multiplier works (or starts working) *even before* G increases.

Nick,

“the math is unequivocal … changes Y(t) instantly”

Of course it is. But that’s got nothing to do with what I’m saying.

Of course the mathematics is instantaneous through the use of a computer or somebody with a good capacity for mental arithmetic. So is the Keynesian cross intersection with the 45 degree line. I can see it right away - it's all there instantaneously.

So what? That doesn’t mean that the *basic* underlying process represented by the *basic* math (algebra or geometry) is instantaneous in time or logic.

The basic algebra series that you’ve used yourself is something like ½ + ¼ + …

This comes about as the result of an MPC acting on previously generated new income.

If you want to modify that into an accelerated response due to rational expectations, then that’s fine. But you’re no longer using that series. You’ve increased the early stage MPC when you do.

Similarly, if you want to modify the start time to have an MPC as a function of expected income in the future, before the injection of “autonomous” income even occurs, that’s fine too. But you’re no longer using that basic series. You’ve accelerated the timing of the early stage MPC so that it is no longer a function of actual new income but of expected income.

Those modifications don’t contradict my point that the basic series represents successive marginal responses to newly generated income at each step, which can only be representative of a time series in operations and logic. Otherwise everything would have to occur instantaneously, which once again requires a revision to the definition of the upfront MPC, because in that case there is only one MPC, and the argument of its function is very complicated.

So I acknowledge whatever modifications you want to make to the basic series, but what I said about the basic series being a time series is correct from the perspective of operations, accounting AND logic – in my view.

“And the period (t) can be arbitrarily short, right into continuous time”

Sure, but that’s got nothing to do with it either. That produces a derivative rate of change that gets integrated over time. The integration still requires a continuous time series of what it is that’s being integrated – unless everything that will happen happens in an instant of that continuous time. Continuous time is a red herring.

And I think we should remember that the calculus was created by taking the limit from a starting point of discrete (concrete) steps.

“But in the real world, does it take time for the multiplier process to work itself out through successive rounds? Yes.”

That’s good enough for me. Otherwise, everything gets done at once, and there is no relevance to time.

That, which allows for your quite reasonable and understandable modifications to the basic time series, is what I’m saying.

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