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 
A reswitching post – awesome. I worked through it when I was writing my thesis, and I even taught it once or twice. I remember almost nothing of it. I wonder if I have my lecture notes somewhere.
Stephen: Thanks! It made my brain hurt. I am very impressed you once worked through it. And taught it!!! How come? How did you discover it? (I don’t remember it being on the course outlines at UWO in my day, but that might be just my memory.
My thesis was on investment, so I ended up reading a lot on how capital stocks are measured. A nasty, thorny thicket that literature turned out to be.
Aha! Yep. It sure is.
I think I missed the punch line.
Sorry, that was a joke. I missed the whole article. So the capital controversy could have been an anything controversy.
You are referencing Descartes Rule of Signs for real polynomial roots.
Edeast: Yep. Capital, food, swallows.
Lord: Aha! Good find!!
This was the clearest description of re-switching I’ve ever seen.
God, the amount of time we spent on the reswitching controversy in advanced macro back in the early 1970s. I don’t remember very much of it (but I think I still have a book about the whole thing–Some Cambridge Controversies in the Thoery of Capital, by Geoffrey Harcourt, which you, too, can experience (http://www.amazon.com/Some-Cambridge-Controversies-Theory-Capital/dp/0521096723/ref=sr_1_fkmr0_1?ie=UTF8&qid=1345257808&sr=8-1-fkmr0&keywords=come+cambridge+controversies+in+the+theory+of+capital).
Indeed; I suspect 1 and 2 are identical and both ‘wrong’, at least by the lights of the reswitching controversy. Namely, 1 is plausible but does not follow from core assumptions on consumer behavior. It has to be imposed as an aggregate assumption.
There is a whole rhetorical dust-up as to whether Giffen goods and other failures of the law of demand (aka where 1 is false) is an appropriate analogy to reswitching – in both there is a perverse equilibration that messes with ‘nice’ behavior, but Giffen goods are largely easy to detect, have straightforwardly unlikely theoretical requirements, and don’t seem to exist, but reswitching is hard to detect, has utterly confusing theory, and who knows about existence…? The Sraffians straightforwardly exist that it is prevalent and the neoclassicals say it is rare at best, in the absence of empirics, and the burden of argument is obviously on the other side, etc. etc.
David, the Sonnenschein-Mantel-Debreu theorem is often cited as an “anything goes” result, showing that violations of the law of market demand are always possible. Yet such violations are unlikely to occur in the real world, since they depend on pathological interactions between substitution and distribution effects. Informally, suppose that folks with highly income-elastic food demand happened to hold a long position on fertilizer, as part of their endowment. (They might have no motive to hedge that position, because there is no uncertainty inside the model: we are doing comparative statics). Then as the price of fertilizer falls, these folks’ endowment becomes less valuable and their demand for food falls, so food production falls in turn and so does fertilizer use. This might count as a violation of the law of market demand, but it shows how contrived such violations can be.
I definitely don’t understand the roots of the debate, but I’m somewhat surprised to learn that it all just hinged on an absurd misconception about the term structure of interest rates. My understanding – and maybe I’m totally wrong – is that neoclassical theory held that the profit rate was meaningfully explained as the “marginal product of capital.”
Leftists (Cambridge UK), while taking offense to the fact that this was easily conflated with the marginal product of capitalists, further objected that the theory was circular and meaningless. I have no idea how these ideas were presented in the ’60s, but my modern/finance perspective is that returns (ie profits from capital) are determined by the risk tolerances of investors, rather than their marginal contribution to output. I.e. investors make money because they are scared, nothing to do with their fearless advancement of “technology.” (Maybe we should revisit the fallacy of the Cruesonia plant here). I thought that was what was at the core of the debate, but I’ll admit that I hadn’t fully understood the reswitching dispute, nor is it clear to me the extent to which it was critical in the Cambridge UK critique of the neoclassical view of capital.
Is it really the case that Solow and Samuelson were made to issue contrite retractions when all that would have been required is a statement that “obviously profits/rates aren’t the same in all periods, and to the extent that we ever wrote that, it was just a convenient…”
Lord: Decartes’ rule of signs”
Cool! Never seen that. Was going to pipe in with my “discovery” of a proof that you were right, Nick (see last comments in previous post), that oscillation of the sign of terms is in fact a necessary condition for having n positive real roots. Oh,well…
but I’m somewhat surprised to learn that it all just hinged on an absurd misconception about the term structure of interest rates.
IIRC, the issue was capital aggregation. Due to re-switching, you cannot aggregate capital based on market price, because demand for capital A and capital B changes when interest rates change. But interest rates are themselves determined from the market price of capital. So there is no well-defined way of aggregating heterogenous quantities of capital, or there might be multiple ways of doing this that give different answers, each equally valid. But then you cannot say “as the quantity of capital increases, the return on capital falls” because you are unable to define “quantity of capital”.
See, e.g. http://en.wikipedia.org/wiki/Cambridge_capital_controversy
As someone who studied reswitching with Don Coffin (above) and later published on it in JET (1983), I think there are some things missing from this post, interesting as it is. The most important is what is key to reswitching, namely a time pattern of returns to a technique that is complicated over time in the sense of having some non-monotonicity. These can arise in environmental examples where there may be costs up front then positive returns but then delayed costs. Or at least in comparing two techniques there are relative changes over time the pattern of net benefits. These are what bring about the phenomenon.
The immediate relevance here is that I do not see any of that going on in these agricultural examples. I doubt they exhibit reswitching or even more garden variety capital-theoretic paradoxes.
Which brings me to a second point: what about the term structure of interest rates? Maybe I missed it, but I think what is potentially interesting there is when we get inversions of that structure, or better yet, non-monotonicities such as we are currently seeing in many countries, with minimum interest rates (in some cases nominally negative) for bonds of about two years in maturity. I suspect that such shifts in patterns can bring about some curious and apparently paradoxical shifts of relative rankings of techniques that have complicated time patterns of relative net returns.
Oh, and the bottom line from my old paper was that when reswitching can occur, discontinuities can occur in optimal time paths of growth for capital-labor ratios. This result was inherent in some of the models put forward during the original debates by such figures as Garegani and Pasinetti, although they did not pursue the full implications of it, although Pasinetti in particular seemed more aware of this possibility.
but my modern/finance perspective is that returns (ie profits from capital) are determined by the risk tolerances of investors, rather than their marginal contribution to output.
OK, but orthodoxy says that returns are a function of both preferences and real physical returns — where the preference curve is tangent to the production frontier curve. It is not one or other. And re-switching calls into question the physical side of things — what is this aggregate production function, and how do you define the “K” that goes into it in a way that is independent of preferences, so that you have two independent curves that can jointly determine the equilibrium profit rate and quantity of capital employed. If the production side is itself a function of preferences, then you have an under-determined system and you are back to Sraffa’s view that the profit rate is exogenous.
If you throw all that overboard and say “it’s only preferences”, then you are not taking the neo-classical side of the argument, which is basically “we don’t care about logical problems in passing from micro to macro such as re-switching”. It is possible, at least conceptually, to aggregate consumption goods, but it is not conceptually possible to aggregate capital goods.
I think “we don’t care” is a legitimate view to take, but it is not compatible with a demand for microfoundations.
I may be completely wrong, but I think that once you conceive of capital as time, of time as uncertain, of uncertainty and risk-bearing being related with returns, and of there being no meaningful difference between quantity of capital and value of capital, all this fades away?
I thought Hicks had solved this for us for the certain view of the world, and Fischer Black solved this for the uncertain view of the world?
I am full on board with K’s finance view of the world. Too many people pay lip service to ‘capital is just the PV of expected future returns’ and then try to ground those expected returns into some function of the current physical stock of capital, which pushes them back into the capital is buildings and factories view of the world. Expectations and uncertainty is all that matters, and the ‘quantity’ of capital is stochastic.
Nick:
Let me create an aggregate investment demand schedule that slopes up. There are two kinds of farmers in the world. Those who have cash and those who don’t. They’re competing for the same market of final output. When the price of fertilizer goes down, the cash-rich farmer does not buy more fertilizer, because even the cash-deficient farmer can now buy fertilizer and the rich farmer will have to fight harder in the marketplace. But when the price of fertilizer goes up, the cash-deficient farmer is priced out, and the cash-rich farmer loads up, creating sustainable advantage. The actions of the cash-rich farmer dominate those of the cash-deficient farmer. The investment demand schedule slopes up.
When firms finance the bulk of investment out of retained earnings and large incumbents dominate new/small busineses, this can easily happen.
rsj: thanks! I was trying to be as simple and clear as possible. That’s the only way I can understand things.
david: Ooops! Yep. I forgot about Giffen goods!
rsj: “IIRC, the issue was capital aggregation. Due to re-switching, you cannot aggregate capital based on market price, because demand for capital A and capital B changes when interest rates change. But interest rates are themselves determined from the market price of capital. So there is no well-defined way of aggregating heterogenous quantities of capital, or there might be multiple ways of doing this that give different answers, each equally valid. But then you cannot say “as the quantity of capital increases, the return on capital falls” because you are unable to define “quantity of capital”.”
That’s basically my “IIRC” too. And I’m saying it seems to me you can’t aggregate “food” either, for exactly the same reason. So when farmers charge us for what they call “food”, it’s really all just mystification and exploitation and them grabbing part of the surplus.
Barkley: JET! Wow. I can’t even read that journal. Thanks for saying the post is interesting. But I don’t think it’s missing some of the bits you say it’s missing.
“Or at least in comparing two techniques there are relative changes over time the pattern of net benefits. These are what bring about the phenomenon.”
I had got that bit.
” Maybe I missed it, but I think what is potentially interesting there is when we get inversions of that structure, or better yet, non-monotonicities such as we are currently seeing in many countries, with minimum interest rates (in some cases nominally negative) for bonds of about two years in maturity. I suspect that such shifts in patterns can bring about some curious and apparently paradoxical shifts of relative rankings of techniques that have complicated time patterns of relative net returns.”
Yep. That’s something it would be interesting to explore, but I didn’t explore it here. You didn’t miss it.
“Oh, and the bottom line from my old paper was that when reswitching can occur, discontinuities can occur in optimal time paths of growth for capital-labor ratios.”
That’s interesting. I think a closely related thing is that you might get weird fluctuations in the optimal time path for output. I was sort of getting at that in my Robinson Crusoe example. But then I wonder if those fluctuations couldn’t be smoothed out by using a mix of the two techniques, and slowly changing the mix over time. (You farm half the land using the old technique and the other half with the new technique.) That is the question that I (or someone else) really should explore next (though it’s maybe already been done somewhere). Take a reswitching example, then explore the optimal mix of the two techniques over time in a GE example. For example, if the switch causes ultra low output in the third year after the new technique is introduced, then just introduce the new technique slowly over time to more and more of the economy, to smooth out that low spot.
I wish I had spent more time talking to Luigi Pasinetti on this stuff when I had the chance. But I was a new prof, trying to teach 3 new courses, and get my thesis done. Stuff happens.
“But then I wonder if those fluctuations couldn’t be smoothed out by using a mix of the two techniques”
I wonder if those discontinuities don’t go away once you realize that that the present production decisions are made based on expectations of uncertain future paths, and that those expectations always change continuously unless the underlying stochastic output processes themselves have discontinuities.
I am intrigued by the implications of re-switching for the simple productivity/policy discussions that suggest that current low interest rates should solve our investment problem and hence productivity problem. If the issue is expectations about future harvests and markets as well as interest rate evolution, it seems to me that some of the simplistic policy pronouncements of various central bankers, finance ministers and other talking heads get problematic.
I just realised: you can always make food prices look like a geometric series as you move along the alphabet. Just choose different units for different foods to make it happen.
Im not sure this matters but in combinatorics they don’t even bother solving for x, or r of your polynomials. Just use them as a clothes line for generating sequences of numbers.. Aka generating functions. Anyway changing the units seems valid, it’s the coefficients that matter.
And then if you have n terms in your polynomial,where n is infinite, you switch to Having a power series. Hey is that the problem with aggregation, the fact that we don’t subdivide a category to infinity. Because apparently you need power series to overcome some of zeno’s paradoxes. Maybe its the same problem.
For a real world example of what is involved, Raymond Prince and I published a paper on this in 1985 in Growth and Change. We looked at numbers available then to compare cattle grazing and strip mining of coal in the US Southwest. We determined that switch points occurred at 2% and 7% roughly for real discount rates. In between coal mining was more profitable while in the higher and lower ranges it was cattle grazing. Driving this was high upfront costs for coal strip miniing that made it unprofitable at the higher rates, and then assuming somebody made them pay for the delayed cleanup costs, those delayed cleanup costs made it relatively unprofitable at the much lower rates.
A warning from this, identified by others prior to us, such as Herfindahl and Porter, is that one cannot necessarily and unambiguously say that lower interest/discount rates are necessarily “more conservationist.”
As I understood it, the capital values depend on the interest rate structure which the creation of capital itself changes.
Nick,
And I’m saying it seems to me you can’t aggregate “food” either, for exactly the same reason. So when farmers charge us for what they call “food”, it’s really all just mystification and exploitation and them grabbing part of the surplus.
I’m sorry, I missed this part. I thought you were saying that the supply curve of different types of food dependent on the interest rate. We can still aggregate consumption, at least conceptually, by the market price, say using a laspeyeres index. We would not be able to do this if the interest rate was a function of the price of food because then we would have a loop a la capital. But just saying that the interest rate influences the relative supplies of foods (say like the weather) does not in and of itself create a self-referential loop.
Lord: “As I understood it, the capital values depend on the interest rate structure which the creation of capital itself changes.”
True (for the economy as a whole). But it will also be true for food. The value of chalking the fields depends on the food price structure. And chalking the fields changes the relative supplies of different foods, and will change relative food prices.
rsj: if we can aggregate apples today and bananas today, we can aggregate apples this year and apples next year.
My disaggregated foods example was supposed to ignore time. Fertiliser affects output today only, so interest rates don’t matter. (Yep, not a great example to illustrate the idea).
Barkley: I think those are good examples. If there are costs at the beginning, and none at the end, it’s simple investment. With costs at the end, and none at the beginning, it’s like negative investment. And with both, it’s some sort of mixture.
Implicit in this though, is the question: what is the alternative? Is the alternative not mining at all? Or is it waiting till next year to mine? I expect the answer is: whatever happens to be the next best alternative.
Which makes me think (though I don’t think it’s an entirely original thought): what’s underlying all this stuff is the concept of “joint production”. Sheep: wool and meat. Investment: output next year and the year after. Car: new car and 1-year old car and 2-year old car…etc. And cost of production theories cannot handle joint products. I should probably do a post on this.
Actually, a better example than fertiliser and food would be sheep. Suppose the farmer switches to a different breed of sheep, that produces more meat, less wool, more fertiliser, etc. You cannot define the benefit of the new breed of sheep independently of the relative prices of meat, wool, and fertiliser. Therefore sheep are an illusion.
if we can aggregate apples today and bananas today, we can aggregate apples this year and apples next year.
Let me try again.
Re-switching is important because it is the non-trivial portion of a two part argument. The first (trivial) part says that the price of a capital good is the interest rate (or one over it). The second (non-trivial) part part is re-switching, which says that the relative prices of capital goods are also a function of the interest rate.
The attack against aggregation requires both parts.
So what is your equivalent attack against consumption aggregation? What is trivially true for capital goods is not true for consumption goods. You need more than just the re-switching part of the argument.
rsj: “The first (trivial) part says that the price of a capital good is the interest rate (or one over it). The second (non-trivial) part part is re-switching, which says that the relative prices of capital goods are also a function of the interest rate.”
First. That should be interest rates, plural.
Second: let’s try it with sheep. The first (trivial) part says that the price of a sheep is the sum of the prices of the various joints of meat and wool. The second (non-trivial) part part is re-switching, which says that the relative prices of different breeds of sheep are also a function of the relative prices of the various joints of meat and wool. (Merinos produce lots of wool but not so good on the meat.)
Ritwik,
Capital is not time. When an entrepreneur decides to create a firm that sells, say, on line ticketing services, then he needs many ingredients. He needs to hire and train people, acquire land, buy servers, routers, write software, and create relationships with the relevant gatekeepers that control access to the payment system or to local vendors.
You can say that this process requires time just as it requires the ability to bear risk, the ability to hire talented people, the ability to see profit opportunities and access to credit relationships.
But the resulting capital of the firm — the collection of software, hardware, land, specialized skills, and the supporting infrastructure — is not equal to “time” anymore than it is equal to market insight or the ability to bear risk.
That is a category error. And the whole point of the re-switching debate is that these issues do not go away once you aggregate by market price.
let’s try it with sheep. The first (trivial) part says that the price of a sheep is the sum of the prices of the various joints of meat and wool. The second (non-trivial) part part is re-switching, which says that the relative prices of different breeds of sheep are also a function of the relative prices of the various joints of meat and wool. (Merinos produce lots of wool but not so good on the meat.)
No problem, you can still aggregate based on price in that example.
The thing is that with consumption goods you can always argue that the ratio of market prices of two consumption goods reflect some sort of indifference between the two goods as a result of consumer preferences. And this allows you to, at least conceptually, aggregate based on price by letting one of the goods be a numeraire.
So the consumer is the aggregator of consumption.
Then to prevent something like re-switching, you only need downward sloping marginal utility of each good + non-satiation. Perhaps you need some other things — you know better than I.
You do not care about non-convexities across time because consumption goods are not aggregated across time. They are assumed to be consumed in each period. If a good provides future consumption but is sold today, then it is a capital good by definition even if it is just a piece of paper that says “bananas tomorrow”. The “tomorrow” turns the paper into a bond rather than a consumption good. If it is possible to purchase the good but not consume it in the same period as it is purchased, then it’s not a consumption good.
But with capital, you can have these non-convexities that prevent the investor from being as effective an aggregator as the consumer. The investor can still try to aggregate, but there may not be a unique ratio of prices, there may be multiple ratios, leading to multiple definitions of quantity.
rsj
Ok, forget about time. Let’s just talk about aggregation.
When the stock market says that Apple is worth $600 billion (assume no debt), it is aggregating by market price. If I now say that the value of the firm’s assets, and hence capital is $600 billion, at this moment, what mistake am I making?
Ritwik, the mistake would be putting 600 billion, divided by some aggregate deflator as the value of K into a production function that determines the quantity of output produced by Apple. We do need some production function in order to make things work, and we try to relate interest rates to the rental rate on this K and whatever preferences we have.
The whole problem of the valuation of multiple production goods across time in the “Robinson Crusoe” case with NO money just if the “problem of interest” as posed by Bohm-Bawerk. (Not one good, not two goods — multiple production goods).
Here’s the key issue — no one will extend the length of a production process unless it promised superior output, and most production goods can be used in various ways taking various lengths of time to produce alternative outputs.
The pricing of borrowed money in the real world, ie interest rates in the actual world, is related to the theoretical problem of “interest” as first clearly articulated by Bohm-Bawerk — but anyone who knows their history of economic thought knows that these are two different questions.
@anon
I am curious as to whether “unlikely to occur” has been rigorously quantified anywhere in the literature. It is easy to find neoclassicals asserting that the interaction is rare and heterodoxist asserting that it is common.
@Greg Ransom
I dare you to define “the length of a production process”!
Greg: “The pricing of borrowed money in the real world, ie interest rates in the actual world, is related to the theoretical problem of “interest” as first clearly articulated by Bohm-Bawerk — but anyone who knows their history of economic thought knows that these are two different questions.”
The difference is: in the theoretical Robinson Crusoe case, there is one person, who only has to “coordinate” his actions with Nature. In the real world there are two or more people, who also have to coordinate their actions with each other, as well as nature. Right?
“Here’s the key issue — no one will extend the length of a production process unless it promised superior output,…”
That’s not quite right. We store apples for later consumption, even though we know the apples won’t multiply in storage, and some will go bad. The real 6-month rate of interest, deflated by apple inflation, is negative.
david: A farmer has the choice of different crop rotation techniques. There are (at least) three different crops in each technique, and maybe fallows too, where there is no crop.
We could say:
1. “No farmer will switch to a technique that produces lower quality of food unless it produces higher quantity of food”.
2. “No farmer will switch to a technique that produces food later unless it produces a higher quantity of food”.
ISTM you need a price/interest rate index in both cases, to define “quality”, “later”, “quantity”. And you need to talk about preferences.
(You maybe don’t disagree).
rsj: assume sheep produce 3 different cuts of meat: rack is best and dearest, leg is second best, shoulder is worst and cheapest. Different breeds of sheep produce different quantities of the three cuts. Let r represent the discount rate as you move to the next lower quality of cut. So the relative price of leg to rack is 1/(1+r), and the relative price of shoulder to rack is 1/(1+r)^2.
Then as r changes, you can get re-switching of breed of sheep.
BTW: TypePad seems to be playing up. If your comment doesn’t immediately appear, go backpage, forward page, and re-post.
Nick,
” So the relative price of leg to rack is 1/(1+r), and the relative price of shoulder to rack is 1/(1+r)^2.”
I don’t get it. So all of this depends on some arbitrary exogenously imposed set of price constraints? So why do we care?
Same goes for capital. Different capital has different rental rates, and those rates vary feom period to period. We know that from MPT. So why the obsession over the interest rate? Also it’s really confusing to use the word “interest” for expected profit. Let’s use “interest” for debt, and profit for capital in general.
@Nick Rowe
I don’t disagree. But your modus ponens is another’s modus tollens, and it is hard to intuit “later” and “quantity” in a straightforward way. And of course the price vector isn’t independent here.
K: “I don’t get it. So all of this depends on some arbitrary exogenously imposed set of price constraints? So why do we care?”
You get it.
david: “But your modus ponens is another’s modus tollens,…”
Took me a while to figure that one out, but I think I get it. But nobody ever says “you can’t define “sheep” independently of the price vector”, and takes that as a starting point to talk about the distribution of income between shepherds and everyone else.
“You get it.”
:-)))
Ok, but then which part didn’t Samuelson, Solow et al get at first? Did they really write multi-period models with flat term structures, and think that would work? What was Samuelson’s (flawed) “non-reswitching” theorem all about, and what was he trying to prove? Why the big fight over nothing? (These guys weren’t stupid.)
K: Dunno. I don’t get it.
Maybe they thought that under constant returns to scale it ought to be possible to determine relative prices independently of preferences based on technology alone???
I blame Marshall. Not 100% sure why, but I still blame Marshall. He tried to downplay 1871. If Jevons had been a better swimmer, and hadn’t died young from drowning (1882) this would never have happened.
If you produce sheep under constant returns to scale, using labour only (assume land is free), then you can pin down the value of sheep from technology only, but you need preferences to determine the relative prices of wool and meat, which are joint products. If the shepherds need to wear woolen cloaks to look after the sheep (so the shepherds don’t freeze), you get production of sheep by means of sheep. You can’t talk about the cost of producing a sheep independently of the relative prices of wool and meat, which depend on preferences. With two different breeds of sheep, you can get reswitching of breeds of sheep.
@Nick Rowe
You need to poke the hornet’s-nest of the heterodox more often! The post-Keynesians will not generalize the Cambridge aggregation problem too much – they have their own aggregates to think about, after all. But the econophysicist guys will talk about instability of price tâtonnement and anything-goes and Sonnenschein-Mantel-Debreu until your ears fall out.
I regret to inform you that people who say “you can’t define ‘sheep’ independently of the price vector” do exist. Usually it’s ‘deserved’ labour wages or somesuch rather than ruminants, though. Grandiose charges that the entire price system fails to reflect individual productivity and contribution in any way are limited to the more frothy types. But they do exist.
Wasn’t the great dream of the Sameulson era an easily-computable, stable, and unique general equilibrium?
I’m a little reminded of Nick’s old fish-don’t-see-the-water-they-swim-in post. The first and second welfare theorems didn’t even exist back then, guys.