It's good to think about weird worlds, which you know are different from the real world. It helps you understand the real world better. Here is a weird world where commercial banks' wanting to expand loans and deposits would reduce aggregate demand and cause deflation. But the only weird assumption is that currency and deposits are complements, rather than substitutes. If you want to understand the macroeconomics of banks, you need to think about this assumption.
I think that currency and demand deposits are substitutes. Not perfect substitutes, but substitutes nevertheless. But here I want to make the extreme opposite assumption, and assume that currency and deposits are perfect complements. People always want to hold the same quantities of currency and deposits. Just like right shoes and left shoes, where one is no good without the other, and we talk about how many pairs of shoes people want to own. The desired currency/deposit ratio is always exactly one, just like the desired left/right ratio for shoes is always exactly one.
To keep it simple, let's ignore reserves. The monetary base is all currency, and banks choose to hold no currency reserves at all.
Starting in equilibrium, if the central bank created an additional one dollar of currency, people would deposit 50 cents in the banks, which would expand loans and deposits, just like in the textbook story, until that 50 cents was back in public hands, alongside one extra dollar of deposits. If we define the quantity of money as currency plus deposits, as we normally do, then with a desired currency/deposit ratio bolted down at one, the money multiplier would be two. Every extra dollar created by the central bank would create an additional dollar of deposits.
(But in this world we would most usefully define the quantity of money the same way we define the quantity of shoes. We would define the quantity of pairs of shoes as Q=min{quantity of right shoes; quantity of left shoes}. Because a right shoe without a left shoe, or a left shoe without a right shoe, is useless. A dollar in currency together with a dollar in deposits is a composite commodity, just like a pair of shoes. We would not define the quantity of money as M=currency+deposits. We would define it as M=min{currency; deposits}. Which would make the money multiplier one.)
Start again in equilibrium, but now suppose the shock comes from the commercial banks and not from the central bank. Suppose there is some change in banking technology, so that every individual bank finds it is no longer maximising profits at the initial equilibrium, and wants to expand its loans and deposits. What happens? That depends on how the central bank responds.
Suppose the central bank accommodates the desire by banks to create more deposits by increasing the stock of currency by an equal amount. The money supply (however defined) has increased, and with no change in the public's desire to hold more money, there would be an excess supply of money, which people would try to get rid of, forcing up prices in the process. The change in banking technology would be inflationary. (An inflation targeting central bank would not fully accommodate banks' desire to create more deposits.)
(If the central bank fully accommodates banks' desire to create more deposits, we get the same results whether the initial shock comes from the commercial banks or from the central bank itself. The only thing that matters is what the central bank does.)
Suppose instead the central bank refuses to accommodate the desire by banks to create more deposits. It holds the stock of currency constant. The banking system as a whole cannot create more deposits, because people would want to convert those deposits into currency. But that doesn't stop any individual bank from creating more deposits, if it can persuade people to hold more of its deposits and fewer deposits at some other bank. Each individual bank will compete against other banks by raising the rate of interest it pays on deposits (or doing something else to make holding its deposits more attractive than holding other banks' deposits). Competition between individual banks for deposits will cause the interest rate paid on deposits to increase until no individual bank wants to increase its loans and deposits. The banking system is again in equilibrium, with the stock of currency and deposits unchanged.
But people are not in equilibrium. They hold the same quantities of currency and deposits as before, but they are earning a higher interest rate on their stocks of the composite commodity. It is as if they hold the same number of pairs of shoes as before, but right shoes now pay interest. They would want to hold more pairs of shoes. They would want to hold more deposits and more currency, with equal amounts of both. There would be an excess demand for the composite commodity of money. Each individual would try to get more money by selling more other goods and buying less other goods. That would create an excess supply of other goods. The result would be deflation.
(An inflation targeting central bank would therefore need to increase the supply of currency to offset that deflation. It would partially accommodate banks' desire to create more deposits. The degree of accommodation would depend on the interest-elasticity of the demand for the composite money.)
I don't need to assume currency and deposits are strict complements to get this result; just complements will do. An increase in the supply (curve) of deposits by commercial banks, with no change in the demand (curve) for deposits by the public, that is not accommodated by the central bank, will be deflationary if currency and deposits are complements.
Now do you guys get it?
If you have an (implicit or explicit) model of the effect of banks on aggregate demand, and if your model does not recognise the importance of the degree of substitutibility between currency and deposits, then your model is wrong. And if deposits are convertible on demand into currency at a fixed exchange rate, if currency is in excess demand (supply) against other goods, then so are deposits.
But then, as I have heard so many times, us Market Monetarists don't understand banks at all. So we just ignore them.
Worthwhile Canadian Initiative 
Sorry, that should read:
“…if the partial derivative of Hb wrt Rd is greater than the negative partial derivative of Hn wrt Rd then PY and Rd will be negatively correlated.”
In words: if the base money stock increases with increasing Rd, given PY is held constant, then PY decreases with increasing Rd given base money stock held constant.
Tom Brown
I think Nick is saying there is no reserve requirement in this specific example. In this case you don’t need to hold currency to back up your deposits. It took me a few reads to understand and I think he is correct in his conclusion so long as there is no RR. Every bit of currency a banks receives it can just lend out.
Of course the current banking system is different to this example though.
dannyb2b, I agree that’s what Nick R is supposing in the world he’s created for this post, but his comment here:
http://worthwhile.typepad.com/worthwhile_canadian_initi/2014/04/deflationary-banks.html?cid=6a00d83451688169e201a3fceb29fc970b#comment-6a00d83451688169e201a3fceb29fc970b
is more general as it applies to Nick Edmonds’ model, which is not restricted to the rules of Nick Rowe’s weird world. Nick Edmonds is saying you don’t need such a weird world for increases in the interest rate on deposits to be deflationary: it can work in some circumstances when currency in circulation and deposits are imperfect substitutes and when reserves indeed are > 0. Reserves > 0 are an explicit component of Nick E’s model: that’s what his parameter “b” is all about.
I think my model is consistent with Nick’s. Doing away with reserves means setting b = 0, which means the condition is simply that ah1 < 0. This implies that demand for currency is positively correlated with the deposit rate, making it act as a complement with deposits.
Whilst my version may be less weird, I wouldn’t want to give the impression that I think it is representative of the real world. But I agree with Nick that this sort of exercise is useful for building a sense of how elasticities matter.
Nick Edmonds, yes, setting b=0 and taking ah1 < 0 does make your model more like Nick R’s. To make it exactly like Nick R’s we’d need to set ah0 = ad0 and ah1 = -ad1 as well. Then Hn = D = H, reserves = 0, and dPY/dRd < 0. But I’m still not sure how that relates to this statement of Nick R’s:
“So the relative sizes of reserves and currency will matter for whether base is a substitute or complement.”
Since, even in Nick R’s model, the quantity of reserves is always 0, so currency [in circulation] (M0 in the US) will always exceed it, but as your model shows (in my exercise above) we can get dPY/dRd < 0 with M0 < reserves or M0 > reserves. And although I didn’t show it, M0 = reserves also works.
Maybe I’m misunderstanding what Nick R meant by that sentence, since I’m just referring to dPy/dRd < 0, not whether or not the base as a whole is a substitute or a complement for deposits. I’m not sure dPY/dRd < 0 is a test for that. When Nick R wrote:
“I think what matters is whether base as a whole is a substitute or complement for deposits.”
I don’t know what it matters for. In light of the model you’d just presented, which he seemed to be commenting on, I’d supposed he could have meant “matters for whether or not raising Rd is deflationary” (i.e. dPY/dRd < 0), but since that doesn’t seem to be true, I’m not sure. Any ideas what the complement-ability or substitutability of the base as a whole (as NR says, determined by their relative sizes) matters for?
Nick Edomonds,
“Whilst my version may be less weird, I wouldn’t want to give the impression that I think it is representative of the real world.”
As we’ve shown your model CAN be just as weird (exactly the same as NR’s world), but it has a range from that extreme to less weird, showing when dPY/dRd < 0 in all of them, so that’s why I think it’s so interesting! You took a single example and created a set of examples from it, and showed that dPY/dRd < 0 works across the boundary of complementary to substitutable relationships between M0 (your Hn) and D. Well done!
Tom: “You took a single example and created a set of examples from it, and showed that dPY/dRd < 0 works across the boundary of complementary to substitutable relationships between M0 (your Hn) and D.”
M0 is not Nick E’s Hn. “Base as a whole” is currency (Nick E’s Hn) PLUS reserves. If currency is a substitute for deposits, but reserves are a complement for deposits, then whether base is a substitute or complement for deposits (and whether an increase in interest on deposits will cause an increase or decrease in PY, for a given stock of base) will depend on BOTH the degree to which currency is a substitute and reserves are a complement AND on the relative magnitudes of currency and reserves.
Nick R, Nick E writes:
“total base money H = Hn + Hb constant.”
So when he wrote “total base” is H, then I took “H” to be “base as a whole” and Hn to be the currency in circulation part of it.
He’s holding total base (H) constant, not Hn!
“Assume non-bank demand for base money is Hn” … assuming demand equals supply, the base money held by non-banks is what M0 is in the US. It’s different in the UK.
That’s what the 1st table in this article says about M0
http://en.wikipedia.org/wiki/Money_supply
“Notes and coins in circulation (outside Federal Reserve Banks and the vaults of depository institutions) (currency)”
However, later it says this:
“M0: The total of all physical currency including coinage. M0 = Federal Reserve Notes + US Notes + Coins. It is not relevant whether the currency is held inside or outside of the private banking system as reserves.’
So that seems to be a discrepancy. First it says “outside of depository institutions” and then it says it’s not relevant.
Tom: in Nick E’s formulation, reserves are a strict complement to deposits, and reserves are part of the base. So base could be either a substitute or a complement to deposits, depending on the size of b (the reserve ratio).
Nick E and I are on the same page.
Over and out.
Nick R: I never said you weren’t on the same page… his model is the same as yours if you set b=0, etc (all the parameter settings we discuss above). Completely on the same page, I agree. But what his model shows is that dPY/dRd < 0 does not depend on the relationship of the stock of Hb to Hn, (my numerical example shows dPY/dRd < 0 can happen for Hb > Hn or Hb < Hn), it depends only on the terms multiplying Rd (i.e. -ah1 and b*ad1).
So it’s this part of your statement I question:
“…AND on the relative magnitudes of currency and reserves.”
If by that you mean how Hn compares to Hb. Again, take a look at my numerical example above: I demonstrate that dPY/dRd < 0 can happen for both Hn < Hb and for Hb < Hn
Tom: “Again, take a look at my numerical example above: I demonstrate that dPY/dRd < 0 can happen for both Hn < Hb and for Hb < Hn”
OF COURSE IT CAN! That does NOT contradict what I said!
“But what his model shows is that dPY/dRd < 0 does not depend on the relationship of the stock of Hb to Hn,..”
He shows it depends on b, and Hb depends on b. (And elasticity is not the same as slope.)
Stop now. This is annoying me.
Nick R,
“OF COURSE IT CAN!” … OK, good to know. It wasn’t clear to me you agreed with that. Sorry for annoying you.
Nick,
market monetarism post:
[link here strongly recommended NR]