Treat this as a rough draft, on a topic I haven't thought much about for over three decades. The punch line is at the end. I may be wrong (of course).
A farmer considers whether to switch to a new technique for growing food. Compared to the old technique, the new technique would produce less food in the first year, but more food in the second year and all subsequent years.
How does the rate of interest affect his decision? (But watch out for that "the", because it hides a massive implicit assumption.)
There will be one rate of interest r* at which the Net Present Value of making the switch will be zero, and the farmer will be indifferent between the old and new techniques. (r* might be negative). At any rate of interest above r* he will prefer the old technique and at any rate of interest below r* he will prefer the new technique.
Forget about interest rates. Suppose the farmer can neither borrow nor lend, because he is Robinson Crusoe growing food for his own consumption. Which technique will be choose? Well, it depends on his preferences for consumption in the present vs consumption in the future. (But watch out for that "the", because it hides a massive implicit assumption.) The more "patient" he is, the more likely he is to prefer the new technique.
We might say that the new technique is more "capital-intensive" than the old technique. It has costs in the present and benefits in the future. Switching to the new technique requires current saving/investment that will produce returns in the future. The lower the rate of interest, or the greater the preference for consumption in the future relative to the present, the more "capital-intensive" will be the technique chosen.
Now lets take a more complicated example.
Compared to the old technique, the new technique will produce less food in the first year, more food in the second year, less food in the third year, more food in the fourth…..and some complicated but specified pattern after that.
How does the rate of interest affect his decision? Let's calculate the Net Present Value of the switch, defined as:
NPV(new minus old) = A + B/(1+r) + C/(1+r)^2 + D/(1+r)^3 + … + N/(1+r)^n
This assumes that the switch in techniques affects the output of food for n future years. Some of the parameters A, B, C, D,…, N are positive and others are negative. (The techniques are maybe different systems of crop rotation.)
We can solve for the rate of interest r* at which NPV(new-old)=0, so the farmer will be indifferent between the old and new techniques.
Mathematicians will immediately notice a problem. We are solving an n-degree polynomial equation, which, in general, may have up to n different solutions for r*. (And I think the number of solutions for r* , where r* is a real number and greater than minus 1, will be less than or maybe it will be equal to the number of times the sequence of parameters {A, B, C, D,…N} switches sign, but I'm not at all sure about that.) [Update: Lord in comments points us to Descartes' Rule of Signs].
That's what the "reswitching controversy" was all about. If there are (say) 7 different solutions for r*, then as the rate of interest starts out high and slowly falls, the farmer will switch back and forth between the two techniques 7 times. We cannot say that a fall in the rate of interest causes a switch to more "capital-intensive" techniques. We cannot even say which technique is more "capital-intensive".
But first notice something important. When I said "as the rate of interest starts out high and slowly falls" I am not talking about a process that is happening over time. I am not saying "suppose r is 100% in the first year, 99% in the second year, 98% in the third year…". I can't be saying that, because In doing the NPV calculation I have assumed that r stays exactly the same in all years. I have assumed a perfectly flat term structure of interest rates. It's that assumption which lets us talk about "the" rate of interest. Rather, I am imagining different possible worlds, and asking what happens as we slowly traverse from the first possible world, where r is and always will be 100%, to a second possible world where r is and always will be 99%, etc. And I am looking at what technique a farmer would choose in each of those many possible worlds.
Forget about interest rates. Suppose the farmer can neither borrow nor lend, because he is Robinson Crusoe growing food for his own consumption. Which technique will be choose? Well, it depends on his preferences for consumption in the present vs consumption in the future. But there isn't just one future period; there are many future periods. And the switch in techniques reduces the farmer's consumption in the present, increases his consumption in some future periods, and decreases his consumption in other future periods. If the farmer became more patient, and cared more about his consumption in the future relative to the present, would that make him tend to prefer the new technique to the old, or vice versa? We cannot say. There isn't just one future period; there are many future periods. And just as the farmer is not indifferent between consumption this year and consumption next year, so he isn't indifferent between consumption next year and consumption the year after next. You can't measure his patience in a single parameter. If the first year is a lean year, the second is a fat year, and the third is lean again, he will hate the new technique, which makes the lean years even leaner and the fat years even fatter. If it were the other way around, he would love the new technique, because it would make the fat years leaner and the lean years fatter. His marginal preferences for extra food in different years won't always be a simple geometric series depending on how far into the future he gets the extra food.
Now let's forget about time and talk about "food".
The switch to the new technique uses more ammonium nitrate fertiliser but grows more food. There will be a relative price of food to fertiliser R* at which the farmer will be indifferent between the old and new techniques. If R is above R* he will use the new technique and if R is below R* he will use the old technique. A rise in R causes a switch to a more food-intensive technique.
Now let's take a more complicated example.
The new technique uses more fertiliser, and produces more bananas, but less carrots, more dates, less eggs, more figs…..etc. (Think of it as chalking the soil, where the increased PH is good for some crops and bad for others, or figure out your own example). The Net Present Value of the switch in technique, (using fertiliser as the numeraire so all prices are measured in tons of Ammonium nitrate) is:
NPV(new minus old) = A + Rb.B + Rc.C + Rd.D + … + Rn.N
Where A is negative, because it's the cost of the extra tons of fertiliser, Rb is the price of bananas in terms of fertiliser, B is the extra bananas produced, etc.
Which of the two techniques produces more "food"? We cannot say. The new technique produces more of some foods and less of other foods than the old. We can't say which one produces more "food" in total without specifying some way to add apples and oranges.
Will an increase in the price of "food" cause the farmer to be more or less likely to switch from the old to the new technique? That depends, on which particular food prices rise, and by how much. It could go either way.
Suppose we assumed that food prices increased geometrically, [update: I meant "decreased", since R is less than 1 in the analogy] as we move alphabetically along the list of foods. So that Rc=Rb^2, and Rd=Rb^3, etc. Then we could restate the NPV equation as a polynomial:
NPV(new minus old) = A + R.B + R^2.C + R^3.D + … + R^n.N
And we could then solve for R* at which NPV=0 so the farmer would be indifferent between the two techniques. And there would be up to n different solutions for R*. So if the price of food started out really low, and slowly increased, we might see the farmer switch back and forth between the two techniques up to n times.
Why should we assume that food prices increase geometrically as we move from the lowest valued to the highest valued food types? Why should we assume that the term structure of interest rates is flat? R is equivalent to 1/(1+r). The assumption of geometrically increasing food prices as we move along the alphabet is formally equivalent to a flat term structure of interest rates where the 1-period rate of interest is constant into the distant future. Even if we relax the assumption of geometric progression as we move across food types and time periods, we will need to replace it with some assumption about how individual food prices increase when the price of food increases.
The NPV calculation for n different types of food in one period of time is formally identical to the NPV calculation for 1 type of food over n future periods. There can be reswitching that is exactly the same way in both examples.
If reswitching means we can't talk about "capital" being scarce, then reswitching means we can't talk about "food" being scarce.
What do we mean by "more capital"? What do we mean by "more food"?
Here's one way to think about it: Let F be the vector of foods. Either different foods at the same time, or the same food at different future times. Let R be the vector of prices of those foods, relative to current food or fertiliser. The value marginal product of investment is then R.dF/dA. (Note that is not the same as d(R.F)/dA, because at the macroeconomic level, more investment will change relative prices). If R.dF/dA > 1 the investment is profitable.
There's another, much much simpler way to think about it. Switch numeraires.
1. Suppose the price of fertiliser goes down, holding the prices of all the different types of food constant. Will the farmer use more fertiliser? Maybe. Probably yes. He certainly won't use less fertiliser.
(Update: Oops! david notes in comments that I had forgotten about the possibility that fertiliser might be a Giffen good!)
2. Suppose the current 1-period rate of interest goes down, holding constant all future 1-period rates of interest. (So the term-structure twists). Will the farmer invest more this period? Maybe. Probably yes. He certainly won't invest less.
1 and 2 are formally identical.
The investment demand curve slopes down as a function of the current 1-period rate of interest, holding expected future 1-period rates of interest constant. (Or rather, it does not slope up.) Reswitching is irrelevant to that question.
What happens to investment today if the whole term structure shifts down equiproportionately is a totally different question. "The future" is not a single period, just as "food" is not a single good. Current investment may be a substitute or a complement for future investment.
(Thanks to Determinant and K for helping me on the math. Errors are mine.)
Worthwhile Canadian Initiative 
Nick I provided a lengthy reply here [link here NR]. Thanks for answering my questions. JakeS, note that I said that lower rates of interest might lead to higher demand for consumption goods, but not necessarily. And you are quite right that in the current conditions with debt deleveraging processes going on the effects of more expansionary monetary policy are almost nil.
“1. Suppose the price of fertiliser goes down, holding the prices of all the different types of food constant. Will the farmer use more fertiliser? Maybe. Probably yes. He certainly won’t use less fertiliser.”
It will take me a couple of reads to process this post as a whole, but this stood out. If the price of fertilizer goes down it creates an income effect for the farmer’s expenditure as a whole. This could change the composition of what he buys – no need to invoke giffen goods.
Nick: “I disagree. Suppose (for example) we define “the” rate of interest as the one set by the central bank. Now suppose the central bank targets a very high inflation rate. Say 100% inflation. Then the nominal rate of interest set by the central bank will be (say) 105%, while (almost) all real interest rates will be much lower than this.”
Okay. Grand. Now, say that the CB holds the rate of interest at 105% and causes the 100% inflation rate — leaving aside for a moment if they can do this. The the real rate of interest is 5% and, while all the other rates of interest will not be equal to this (I never said that), they would tend toward it.
What you’re postulating is an EXOGENOUS shock imposed upon the economy by the CB. The immediate result will be one, in neoclassical terms, of disequilibrium. But in the long-run it is assumed that the economy will return to equilibrium (in your example, at a new price level, so far as I can see).
“But even in that special case of a stationary equilibrium, you still can’t say what determines that equilibrium (including the interest rates)…”
I don’t recall that I ever did say what “determined” the equilibrium. That is irrelevant to the capital debates, so far as I understand them. What everyone was talking about was an economy in a static state. Because neoclassical economics views the economy as moving toward equilibrium, the Cambridge crowd focused on this equilibrium state in order to show that the equilibrium state itself was incoherent.
@TheIllusionist:
I’m not arguing that permanently low interest rates (ZIRP in the (non-Gesellian) limit) eliminates the business cycle. I’m arguing that for a given regime of macrostabilization, lower interest rates will give a higher baseline of economic activity (and greater growth during periods of growth). Because low interest rates (a) reduce the required rate of return for doing stuff, meaning that more stuff will be done (entire sectors of economic activity can become uneconomical at higher or more volatile interest rates) and (b) take less income from net debtors, who are typically the people who spend money (otherwise they wouldn’t be net debtors) and give less income to net creditors, who are not inclined to spend money (or they wouldn’t be net creditors).
Now, you can argue that (a) will not contribute to aggregate demand, because this investment will simply (for a given state of aggregate demand) crowd out consumption. However, it does contribute to long-run growth, for precisely that reason – at least in economies with structurally insufficient domestic investment (such as the Western(TM) economies of the last three or four decades).
And the point of (b) is that not all income is created equal. Interest income to creditors, in particular, goes into a black hole. Unless you either believe in loanable funds or believe that the creditors have any appreciable inclination to spend their interest income, rather than stash it in bonds or engage in speculative games with it. Whereas interest costs to debtors directly impacts the income of people who are inclined to spend that income.
@Nick Rowe,
causality runs both ways between the quality premium and the total value of sheep and the composition of the total volume of sheep.
Does this create practical problems for agricultural economists doing empirical work? Yes, it probably does. […] But would anybody argue from this that neoclassical economics is incoherent, or that the price premium for meat that is perceived to be of higher quality is somehow illegitimate, and not reflective of relative scarcity? I don’t think so.
No, because agricultural economists (usually) don’t do silly things like writing down a production function that takes the total capitalization of sheep farmers and attempts to impute a long-run equilibrium value for their revenue or their share of the value added from sheep farming.
But a lot of central bankers do precisely that for the capital plant and profit share of value added.
The problem with capital aggregation in far too many neoclassical models isn’t the aggregation per se, it is that (a) the implied causal relationship in the neoclassical model is the wrong way around or (b) there’s no correspondence between the aggregates reported by statistical services and the aggregates going by the same name in the models. That’s an inclusive ‘or’ by the way.
Also, as a more general pet peeve, I really think it obscures more than it enlightens to use the term interest rate to describe something that is not a credit instrument. Return to capital, equity or investment, as appropriate, avoids the confusion between access to capital which, in a capitalist economy, is rationed by price and availability of capital, and access to credit which, in a monetary economy, is rationed by political fiat.
@Greg Ransom
If you want to do the microeconomics right, you have do to pure logic of choice of the Robinson Crusoe type WITHOUT categories and goods that only exist in a real, money and credit using economy.
Only if you want to do microeconomics for a species of sentient creatures who in some respects resemble, but are not actually, homo sapiens sapiens.
If you want to do microeconomics properly, you need to start with a good grounding in organization theory, psychology and political science.
– Jake
@ Jake
“Because low interest rates (a) reduce the required rate of return for doing stuff, meaning that more stuff will be done (entire sectors of economic activity can become uneconomical at higher or more volatile interest rates)…”
Totally disagree. “More activity” or “stuff” requires an amount of aggregate demand to facilitate. This is obvious today. If there are not enough buyers — i.e. too many unemployed or underemployed — you will not sell your produce and depression will result. No matter of interest rates, they are secondary. Their effectivity falls to nil at a certain point. We’re there!
“…and (b) take less income from net debtors, who are typically the people who spend money (otherwise they wouldn’t be net debtors) and give less income to net creditors, who are not inclined to spend money (or they wouldn’t be net creditors).”
This is true… in theory. But not in practice. See: the Credit Suisse evaluation:
https://doc.research-and-analytics.csfb.com/docView?language=ENG&source=ulg&format=PDF&document_id=930221251&serialid=WCjh4HraBteNnwZn29w46PybhgK%2BBIXDfe0rAQogpwQ%3D
“The side-effect of the Fed’s near-zero interest medicine – the collapse in personal interest income over the last few years. The decline in interest income actually dwarfs estimates of debt service savings. Exhibit 2 compares the evolution of household debt service costs and personal interest income. Both aggregates peaked around $1.4 trn at roughly the same time – the middle of 2008. According to our analysis of Federal Reserve figures, total debt service – which includes mortgage and consumer servicing costs – is down $206bn from the peak. The contraction in interest income amounts to roughly $407bn from its peak, more than double the windfall from lower debt service.”
My economics — and my investment decisions — involve taking account of actual market conditions. Everything else is nonsense.
A couple of years of following the public debate on economics gives one a healthy skepticism for any analysis coming out of a bank – in particular a Swiss bank.
But even taking their numbers at face value, I’m not prepared to bet money that the marginal propensity to consume or invest (as opposed to gamble on securities) is less than twice as high for net debtors as it is for net creditors. And you shouldn’t take their numbers at face value, because bid-ask spreads and bank markups over the policy rate would have increased whether the policy rate was lowered or remained constant. So the spread between interest savings and interest income would still have been there – it might have been smaller, but it’s hard to guess how much.
Of course in the current environment, where everybody is deleveraging, a reduction in debt servicing costs doesn’t boost demand so much – it does, however, contribute to reducing the duration of the crisis, assuming appropriate fiscal macrostabilization is performed.
– Jake
FWIW: Financial Times commenting
http://blogs.ft.com/money-supply/2012/08/23/qe-bono/#axzz24NeLw7WZ
on Bank of England report on who benefits from QE
Click to access nr073.pdf
HT Mark Thoma Economist’s View
Frankly, Jake, I think that its VERY believable. Much money made in the US today is financial money. This is because the “spreads” between profits and wages have gotten very “wide”. More and more money is being made in the financial sphere and less and less is being made in the productive sphere:
Yes, we can question the “evil” banks so-called “agenda”… but I think that’s bullshit. The above is a left-winger and it seems obvious what is going on. Those that deny it are pursuing an ideological agenda and frankly, I have no time for them.
Believe what you want, but don’t believe for a moment that it overlaps with reality….
“Illusionist” have you read Bohm-Bawerk?
Hicks learned about his stuff from Hayek, and he admits as a British trained exonomist he never really “got” what Hayek and Bohm-Bawerk doing, couldn’t fully escape his training — Hicks tells us that Hayek was making him think of inputs. Kling before outputs in a process talk place across time. Hicks sought to capture this using British/Marshall and Walras/Pareto methods.
Back at you, “Illusionist”, I don’t think you understand what the deep issues are here or what a consistent marginalist looks like, or what explanatory role marginalist logic can play, or what the real & deep differences between the Ricardo/classical and the consistent marginalist position really looks like in the domains of valuatational relatikns across time in production goods, and the relation of this pure logic to the explanation of money interest rates in the very different context of the real world.
The Illusionist: “The decline in interest income actually dwarfs estimates of debt service savings. ”
That’s because
1) Consumers borrow long (30 yrs) and lend short (ignoring retirement accounts)
2) we have a consumer debt crisis and under water mortgage borrowers are unable to renegotiate or move.
Since banks no longer run ALM mismatches, the consumer duration imbalance is actually balanced by retirement accounts, which are in significant surplus (the bond part). But because consumers can’t sell those assets, and can’t borrow against them, they find themselves massively liquidity constrained. It’s a nice trick whereby the financial industry has arranged thing such that consumers are unable to obtain funding by repoing their liquid securities (in registered accounts) and instead are forced to use expensive long term mortgage finance (and policy rate insensitive credit cards).
Since the banks are profiting enormously from the current situation, the only (quick) way to realize the benefit of low rates would be via massive legislated mortgage restructuring. Which doesn’t seem likely either at this point.
K
I agree about the banks not having any ALM mismatches, but I fail to see how there’s a consumer duration imbalance. The savings are locked up in retirement accounts, but the 30 year mortgage is not financing the financial assets minus these retirement accounts. It’s financing the house, which will presumably last more than 30 years. Where’s the imbalance?
In general, if you have a 30 year mortgage at a fixed rate and you have provisioned for mortgage payments out of your income, the decline in house prices is of little concern to you. You care about house prices and mortgages only if you’re net short or net long housing, i.e. you have more or less than you need.
@ Ransom
I looked into it. Yes, the Austrians have a typically idiosyncratic (and I would say: woolly) approach to this based on some sort of subjectivism. If this is the theory of interest rates similar to that put forward by Hayek then it is well out of date by now. Sraffa destroyed it in the 1920s and Keynes finished it off in the 1930s.
Ritwik,
The house is a real asset. Forget it. I’m talking about fixed rate nominal assets, which are in zero net supply in the economy. For every 30 year fixed rate borrower there’s a 30 year fixed rate lender. Since the financial sector is roughly flat, and the corporate and government sector are short, that puts the consumer long fixed rate debt. But there are two factors that prevent most consumers from realizing their MTM gains on their net bond positions:
1) Most of them are actually just short (i.e. they are young or poor and they are debtors). These people actually took mark to market losses on their fixed rate debt exposures.
2) Of the rest, the vast majority hold their bond positions in registered retirement accounts, company pensions or social security. So they have no way of accessing that wealth. You can’t take it out, and you can’t borrow against it. Though a drop in the term structure of interest rates results in net wealth gains for the consumer sector, they have no way to use it to relieve their liquidity constraints if they are unable to roll their mortgage at reasonable terms.
The problem is partly a balance sheet crisis but also consumer cash flow crisis aggravated by regulatory features of our system of savings which is designed to prevent consumers from efficiently borrowing against their most liquid assets, and which makes the balance sheet problems look even more serious than they actually are.
Why did borrowers have to renew mortgages at outrageous spreads in ’08/’09 when the government bonds in “their” social security account could be repoed below 0%? It’s a very elaborate financial Rube Goldberg. But it’s still a scam.
K:
In Canada, you can’t access your employer-sponsored retirement account or your CPP-QPP account but your RRSP ( registered retirement savings plan)can be accessed for home-buying and higher-education retraining. Was the decision based on the realization that the problem you raised is very real? I don’t know but sure thre was a great media and public opinion pressure. Though the practice is sometimes discouraged on the fear that it won’t be paid back and that you are stopping the accumulation of assets.
Unfortunately, for most people, the only moment when they access their RRSP is at bankrupcy proceedings , when it’s too late to do any good.
Jacques: “Though the practice is sometimes discouraged on the fear that it won’t be paid back and that you are stopping the accumulation of assets”
I think that’s largely “concern trolling” by the financial industry. If we had wanted to address that issue, we could have given people an easy way to access a mortgage from their RRSP at the same terms as (and pari passu to) their bank mortgage. That way it can only be a substitute for, rather than an addition to, other debt. This also precludes any argument for the $25K limit. I would have loved to give a 30-year mortgage to myself (with outrageous prepayment penalties!) in my RRSP 15 years ago. For some reason, that solution wasn’t proposed by the (deeply concerned) financial industry.
Not that I’m mainly advocating that, though a bank mortgage provides an excellent return and lending to yourself is a really excellent risk. But beyond that, we need to be able to access all of our superannuation assets as collateral. Securities are way better collateral than houses.
@theIllusionist
The Godley/Lavoie models indicate that lowering interest rates provides a short-term boost to demand, but creates a long-term drag due to lost interest income.
id need to see that carefully (any references?)
for one, there are distributinal issues involved (welath inequality is higher than income inequaltiy). only a very limited fraction of the population would see their income increased this way, but financial costs would increase for everybody. I cant see why the income increase effect would dominate the cost increase effect.
Anyhoo, the way you expressed it, it would contradict the very logic of inflation targeting,. what would stop it from generating an inflatinary process like so
higher interests -> higher income -> higher demand -> higher prices -> higher interests
“TheIllusionist” — beg the question much?