Epsilon example: additional notes

Unorganized!


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 Additional Notes (not essential reading):


The CB initially purchases Ms0 worth of assets on the open market (Ms0 = 8.33 in Figure 1.). Thus the bank starts off with a fixed reserve balance balance of Ms0. Assume the CB does not conduct further open market operations (OMOs) until the average price level, P, reaches steady state. Let P0 represent this initial steady state value of P. Since there's no cash, base money (and thus the medium of account (MOA) in monetarist theory) consists solely of the bank's reserve balance, while broad money consists solely of the total deposit liability of the bank. The exchange equation relates four values:

M*V = P*Y

M = quantity of money
V = velocity of money
P = average price level
Y = real output

According to the theory of the long term neutrality of money, should the quantity of money (M) change, then the price level will change too and eventually reach a new steady state value, such that the new P will have changed in proportion to M (assuming all else is equal). Note that sticky wages or prices in conjunction with a change in M may cause real changes in the near term to V or Y, but all else equal, in the long term V and Y should return to their original steady state values, leaving only M and P changed. This is also called the quantity theory of money (QTM). Thus suppose M0 is changed to M1, then we can describe the new steady state value of P as

P1 = P0*M1/M0

Now let's assume that the CB sells $M0*(1-epsilon) of its assets so that M1 = M0*epsilon, where epsilon lies on the interval (0,1]. Assume the CB's assets are the same price when sold as when purchased. Using the above formula, we can then see that the new steady state value for P will be:

P1 = P0*epsilon

That is the Scott Sumner/Mark A. Sadowski interpretation, and in fact both Scott and Mark agreed that as epsilon approaches 0, P1 also approaches 0. However, Nick Rowe pointed out that Scott and Mark have implicitly assumed that the demand for MOA would not change with epsilon. Nick gave a different answer: he claimed that due to the nature of the hypothetical, the demand for MOA would decrease along with the quantity of MOA (and thus with the supply of MOA). Nick is very particular in using "supply" and "demand" to mean functions or curves rather than scaler quantities. He uses "quantity of" or "stock of" to refer to scaler quantities. For supply and demand for MOA to move towards zero together in proportion to the quantity of MOA supplied, must mean that both the supply and demand curves for MOA are also somehow scaled by epsilon as well. Nick gave some clues as to what this meant, and those in conjunction with this article resulted in the interpretation that is depicted in Figure 1. Both Nick's and Scott's solutions are shown there, and if you click on Figure 1 or this link you can see an animation of what happens to both solutions as epsilon approaches zero.

Nick informed me that the demand for money (in this case reserves) is plotted in an unusual way. First of all, the independent variable is plotted on the y-axis, and it is represented by 1/P, where again P is the average price level. The dependent variable, demand, is on the x-axis. This is an oddity of the economics profession, according to Nick. The article fills in some of the gaps here and specifies that it's the nominal demand (Mdn) which is plotted in this way. There is also a real demand (Mdr) and the relation between the two is:

Mdr = Mdn/P

What made sense was to consider that epsilon must be scaling the real demand (Mdr) in Nick's case. Meanwhile, the (nominal?) supply curve (Ms) is depicted as a vertical line crossing the x-axis at the (nominal?) quantity of MOA supplied. This is true for both Nick's and Scott's cases. In other words, the supply curve is

Ms = Ms0*epsilon

for both Nick's and Scott's cases with Ms = quantity of MOA supplied. This is plotted on the same x-axis as is Mdn. Ms0 is the initial value for Ms. Equating Ms0 = M0 from my previous description, then in my particular example in Figure 1, we see that Ms0 = 8.33. Also, in my example,

Mdr = Mdr0*epsilon

in Nick's solution, with Mdr0 = 25 for both Nick's and Scott's (Scott of course keeps this value fixed throughout). With epsilon = 1, there's no change in the solution. I'm assuming Nick's and Scott's solution start off the same at epsilon = 1, with the solution for the initial steady state average price level being P0 = 1/3. But as epsilon is decreased towards zero, Nick's solution and Scott's solution for P begin to diverge. Since Scott assumes that Mdr does not change with epsilon, then his Mdn curve does not change either, and as epsilon goes to 0, the 1/P crossing point between Ms and his Mdn goes to infinity: in other words P goes to 0. Nick's Mdn curve, however, is changed by the scaling of Mdr. This keeps the 1/P crossing point between Ms and Mdn fixed at 3, or at P = 1/3, in Nick's solution.

Why do I use an epsilon which only approaches zero, rather than just stating that Ms1 = 0 as my second case? This is because I tried the direct approach, and got contradictory and somewhat uninteresting answers from both Nick and Scott. I still get contradictory answers from them, but at least now they are interesting! It turns out I had to sneak up on the problem in more ways than one to accomplish that and introducing epsilon helped. I explain more about the iterative way in which this question was developed in the notes below.

Nick's solution is more intuitively satisfying in that it's hard to believe that prices would drop to zero just because the CB got rid of all its assets. Recall the commercial bank is still available to make loans. However Nick's solution seems oddly in conflict with normal MM thought about this, since it implies that demand for MOA will scale with open market purchases to cancel out  any effect. Granted it's a special case, but I still find this surprising.

My question to Nick at this point, is what is the minimum set of changes that need to be made to this hypothetical for his solution and Scott's to agree? Can we simply introduce a second commercial bank? Do we need cash (Scott implies this is not the case in his "cashless society" case: see the additional notes below). What do we need to add, other than cash, to move Nick towards Scott on  this?

Thus concludes this post. I've supplied some additional (and mostly VERY redundant) notes below, including some additional links to my original questions and answers:



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Additional Notes on Getting My Question Right and Initial Responses; Set #1:

This is an an example that I presented to Nick Rowe, Scott Sumner and Mark Sadowski in March of 2014. It's about the long term neutrality of money and the hot potato effect (HPE) in a hypothetical situation. I have to thank Nick especially for giving me just enough information to go on to come up with the above chart (Figure 1):

Figure 1 illustrates two of the solutions to my "Epsilon" hypothetical monetary problem as put forward by Nick Rowe and Scott Sumner (and Mark A. Sadowski, agreeing with Scott's solution). I have links to the final form of the question as it appears on Scott's blog, but I'll reprint it here with some further explanation:
Scott,
OK, let’s start with your own example #7 from this post:

“7. Now let’s assume a cashless economy where the MOA is 100% reserves.”

You’ve clearly identified the MOA there: reserves. Banking doesn’t matter, so why not assume a single commercial bank? And my other assumptions: [reserve requirements = 0%,]* no taxes, gov spending, foreign trade, etc. So if initially the CB buys $X in assets, this gives us an initial $X in reserves, which you’ve identified as MOA.

Since we have an MOA, we will reach a steady state price level, P, right? Now what if the CB sells 1-epsilon of its assets? The new eventual price level should be: 

new steady state price level = P*(epsilon*X/X) = P*epsilon

So as epsilon approaches zero, the new steady state price level should approach zero too??
As can be seen in Figure 1, Scott's and Mark's solutions to the problem is that the steady state price level (P) would indeed go to zero as epsilon goes to zero (this BTW is completely counter-intuitive to me, but is what I'd expect based on what I know of Scott's theory). Nick, on the other hand, pointed out that Scott probably thought that the demand curve for reserves (which I diagram as Mdn, meaning the nominal demand curve for MOA, as reserves are the only component of MOA in this example) wouldn't change, so as the supply curve for MOA (determined solely by the CB) moved to the left, the crossing point with the demand curve would go to infinity, which corresponds to P = 0. Nick, on the other hand, said that both supply and demand would go to zero, which when translated into this type of diagram I think means that the real demand for MOA, expressed as

real demand for MOA = Mdr = Mdn/P

would also go to zero along with Ms shifting to the left. Thus I took that to mean that epsilon would multiply Mdr as well as Ms. If that's the case, then the crossing point of supply and demand does not change as epsilon goes to zero, and thus there is no change in the steady state average price level P (which is P = 0.33, the starting steady state value for P at epsilon = 1 in Figure 1). Nick's answer is more intuitive to me, but still doesn't quite sit right with me. I'll explain why in a future update.

Since I suspect it's best to keep blog questions to popular blogs like themoneyillusion.com (Scott's) and Worthwhile Canadian Initiative (Nick's) as short and sweet and too the point as possible (a big challenge for me, especially since I rarely get my question right the first time), I didn't want to elaborate much more than that on Scott's blog: both Scott & Nick answer a lot of questions from the public, so I can't be too greedy with their time (or they'll just ignore me, Lol). I struck upon the idea of using Scott's own example as a starting point because my previous iterations with Scott didn't get very far; he briefly responded each time that without a medium of account (MOA) there could be no average price level (P). Scott's own example demonstrated that there indeed was an MOA. Nick's initial response was similar in that it "escaped" the point of my question, but it was different than Scott's: Nick's responses to my early versions were that there would be no demand for bank reserves, and thus the commercial bank would be taking on the role of the central bank (CB).

When answering early versions of this question, Nick had the advantage of me explaining why I set the hypothetical up the way I did (single commercial bank, no taxes, foreign trade, etc.): I was really intending to make the reserves (the MOA in Scott's Case 7) as useless as possible: they wouldn't be needed to intermediate transactions with the government, or in foreign trade or with foreign governments or other CB deposit holding institutions: the single commercial bank would essentially be the sole holder of a CB deposit. Also, since there's only one commercial bank, the reserves would not be needed to settle payments, and since there's no cash, they would not be needed to buy cash (paper reserve notes and coins) from the CB either. The only value of the reserves here is to be an offsetting asset for the commercial bank should the CB purchase assets from either the commercial bank itself or from the non-bank public (this is where a balance sheet might be helpful to add). I'm not sure Scott and Mark fully understood that that was what I was after in setting the example up the way I did.

At this point I'm also not sure I'm happy with any of the answers I received, but I am happy that they all took the time to answer. I certainly learned something! I will draw out some balance sheets at some point to illustrate why I'm not totally happy with any of the answers, and I might also embed an interactive version of the spreadsheet I used to create Figure 1.


Additional Notes on Getting My Question Right and Links to Originals; Set #2:

It took me several iterations to get the question the way I wanted it and in the process I probably confused Nick and Scott a little, mostly Nick. Both Nick and Scott gave me initial answers to early versions of the question. Their early answers were different from each other. I did eventually get the question boiled down to one self-contained comment* which I gave to Scott. At that point I showed it again to Nick. And finally to Mark. Here's the interesting thing: Both Nick and Scott changed their answers for the new self-contained "epsilon" question, but still their answers didn't match. Mark's answer agreed with Scott's. I wish I'd just asked all three of them the same question once, at the same time: it would have been less confusing for everyone!

Here's the best version of the question as I presented it to Scott (and then to Nick and Mark later on):


For completeness here's the follow up comment specifying that reserve requirements = 0% a couple of comments later in the stream:


After discussing Nick's solution with him here:


and here:


And reading this article:


 *Note: I had to put this asterisk, because I still managed to forget something in the "self-contained" epsilon question (that the required reserve ratio was 0%), but that was a one liner in a second comment I put up immediately after the first. 

[Update 01, 2014.3.18] Nick Rowe says that my Figure 1 looks roughly right.
[Update 02, 2014.3.18] I also link to this post on Cullen Roche's pragcap.com blog Forum to see if any discussion gets started there.  

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