Bob revisits capital and interest theory to show why the textbook result “interest = MPK” only holds in a one-good world, and why …
Transcript
This is the Human Action podcast where we debunk the economic, political, and even cultural myths of the days. Here's your host, Dr. Bob Murphy. Hey everybody, welcome back to Human Action Podcast. In this episode, I am going to do a deep dive into the Austrian approach to capital and interest theory. And in particular, I'm going to relate it to the mainstream neocclassical approach. Now, for those of you who have followed my work for a while, you might say, "Bob, didn't you already do this?" Yes and no. So, I have definitely hit these areas before in previous podcast episodes and things I've written in various outlets. But what prompted me to come back and hit this again is number one, we've got a different audience now than perhaps was the case three years ago. So, it's fine to have repetition on these important matters. But also, I was recently interviewing um one of my old professors from NYU, uh great guy, Alberto Bazine, u very open-minded. He was mathematical economist trained in Chicago. And we were just talking about this the controversy that's raging lately on Twitter at least uh of over the use of mathematics and economics. Okay? And I was, you know, I wasn't arguing. I was just letting him explain his perspective. And I made the point that on the one hand I had sort of acknowledged to him that yes I noticed in my own work when I was doing my dissertation at NYU and I was working on what's called the Austrian pure time preference theory of interest that one of the points I was trying to make to the Austrian crowd was that the term time preference is used to mean different economic things in the literature and it's the same words and that it's important just to be clear about what do you mean by that term. term and I said to Alberto now it was very easy you know I can explain it to you as a mainstream mathematical economist the distinction and and I went ahead and said it in math terms and he understood immediately oh yeah there's there's that thing or it's this thing both could plausibly be called time preference but you got to stick with one meaning and you know be consistent with it you can't just flip back and forth mid argument okay but then I said now having you know congratulated you guys Alberto for precision and formalism. On the other hand, I have to say what also became apparent to me while I'm doing my dissertation at NYU is that um the mainstream economists are walking around thinking that interest is due to the marginal product of capital just like they think that um real wages are due to the marginal product of labor or are equal to it in you know monetary terms and and at least with competitive compitive markets and so on, right? and and I said, and that is a fundamental confusion in the Austrian tradition, and yet um it it seems like it just pops out of a standard neocclassical model like the kind that we learned as first year grad students at NYU where, you know, you have a production function that takes K and L like capital and labor as inputs and out pops output and then the representative household can divide the output into consumption or saving. and you know carry the saved goods forward one period and then they get thrown in the production function again and so on rinse and repeat. And that in that kind of a model if you ask assuming competitive markets blah blah blah what is the real return to labor it's oh you take the production function you take the first derivative with respect to the labor input variable and that's what the wage rate is in real terms and the idea is oh yeah if output goes up a little bit because you put in a little bit more labor into the mach into the factory or whatever the production function And you know, you put in one more labor hour and output goes up by 10 apples. Well, then in equilibrium in a competitive market, that worker needs to be paid 10 apples for that hour of labor. And so he gets his marginal product. Okay? Now, the Austrians agree with that and so do the mainstream economists in their mathematical model. Of course, the Austrians would have all kinds of quibbles and deeper objections to that model, but the economic intuition is the same, but that correspondence does not carry over to the determination of interest. And so again in a simple model like that of the kind that first year grad students at least back when I went through and I think probably still to this day learn is like the you know the introduction get you warned up how to work with these models is you say okay well what's the real interest rate in equilibrium? Well capital has to earn its marginal product and so put in a little bit dose more of capital into the production function. output physically goes up a little bit and then that increment that's the return to capital and so in equilibrium the real weight uh sorry the real interest return to capital they have to earn that increment and output and hence in equilibrium that's what the real wage or sorry the real interest rate equals okay and I want to say there the Austrians are going to fall out of their chair and say whoa that's not right and it's we're not just arguing about you can't collapse the economy into a simple mathematical model that takes two inputs and you know that that's crazy but beyond all that just the economics of it is wrong even the dimensions are wrong okay and that's what I want to unpack and so and I was telling Alberto this and I and I said but now the issue the thing that was weird and what I struggled with when I was in grad school is when we were learning those models you know we're going doing problem sets and taking the first derivative and blah blah blah and doing all that the math was correct. Right? So in the world of the model, I understood why that's what the textbook said the return to the capitalist was going to be. And so it wasn't wrong. And yet that seemed to be directly at odds with what Ogan von Bomba a second generation Austrian after you know Manger was the founder of the school in 1871 and then Bomba was in the next wave and you know was a huge pioneer everybody agrees in capital interest theory and one of the things one of his contributions was he wrote this critique of what he called the naive productivity theory of interest and in that bomb shows when people are trying to understand what is the source of the interest income that capitalists earn in a market economy. A natural hypothesis is to say, "Oh, well, because capital is productive. You can make more stuff with capital goods than without. So, duh, that's how the capitalists earn an interest return." And Maverick showed that no, no, that's a fun there's a fundamental confusion going on there. You're getting mixed up. That isn't a good explanation. Okay. So, I'm going to come back, folks, and spell that out. Don't worry. In this episode, by the time you're done, if you invest your time listening to this, I will explain that clearly and you you'll totally get it. Okay? But take my word for it the moment. But Bob is definitely right. when he when he you understand what he's saying and he walks you through and shows the fact that a farmer can have a bigger harvest physically that can get more bushels of wheat or ears of corn or whatever using a tractor than if he didn't have the tractor available. that the physical productivity of the tractor, its ability technologically speaking to augment the yield of his operations. That does not explain why a farmer who has a sum of money and buys a tractor then can use it and over time when he does the accounting or has his accountant do it that he ends up with a bigger sum of money acrewing from the purchase of the tractor, right? That in a sense he earned a return on the financial capital that he poured into the tractor if you want to think of it like that. And Bombber showed to explain that phenomenon. Why is it that maybe he earned 5% a year correctly considered on his investment in the tractor in money terms and you can make it real interest like 5% real return if you account for you know changing money prices over time for price inflation. You can do that too. But the point is when you're trying to explain that phenomenon, why is it that in general in the market economy, capitalists who have a sum of money at the outset can buy things with them and invest their proceeds such that over time they tend to earn a net return on that. So they end up with more financial capital than they started out with even accounting for rising prices. How do you explain that? And Bomb said to say that capital goods are physically productive is completely missing the point that that's not an explanation at all. Right? Again, we'll come back and explain that. All right? But just take my word for it. That's what he did. And so there I was at NYU, a grad student working on this dissertation focusing on Bobky and capital interest theory, but then you know trying to relate it to what I was learning in the mathematical mainstream curriculum. And I was like, "What the heck's going on here?" That the math of these, you know, that I'm learning on the blackboard or whatever in class is correct. And yet that seemed to enshrine the naive productivity theory into the equations. They were clearly saying in equilibrium in our models, the capitalists who start out and have a bunch of capital and contribute it to the production process. How do we explain what the real return is percentage-wise? You know, you know that like what what fraction of their capital, how much does it grow measured in terms of itself. So you know that there wasn't some just basic confusion about inflation or something that no the real rate of interest in that model was directly equal to just looking at the physical production function and saying if you do a slight increment you add epsilon more capital physically into the production process that boosts the physical output by a certain amount and then that relationship the magnitude of that of the one increment to the other then pins down what has to be the real rate of interest in equilibrium and and I was just staring at that like I don't remember how long it took me might be a better story if I did but I don't remember but then I had the Eureka moment and oh I see what's going on it's because in those simple models that you learn as a first year grad student and I think probably still do everybody would acknowledge oh yeah yeah the Um, we know that in the real world things are way more complicated, but for simplicity sake, for tractability, just to get your, you know, get warmed up in how these things work. We'll start with a real simple case where assume there's just one production function, one type of labor, and we'll just call it L to like just measure or, you know, L with a subscript of T if you want to make it index by time, and K with a subscript of, you know, T and T plus1, T+2, whatever. And so, yeah, we know in the real world there's all kinds of different types of capital goods and whatever, but here we're just going to be real simple. And I realized that apparently innocuous assumption for convenience was driving the whole result. Okay? And all the like Bombav's critique of the naive productivity theory falls away. It's a moot point. He's worrying over something that in practice can't ever be or can't have an implication, any significance. If you were in a world where there was just one type of good and that the capital good was the same thing as the consumption good, they were physically the same thing. Um, for example, Irving Fischer, who was one of the giants in neocclassical interest theory, when he was in his original pioneering works trying to lay out this framework and get people to think through the the, you know, the issues, he would use simplistic examples and one of them was like, oh, imagine the only good are sheep. And so they are like the capital and the consumption good at the same time. that you got like a flock of a hundred sheep and you want to decide, you know, how many do I want to consume? And if you don't, it's like, oh, you're saving it, but you're also kind of investing it because if you don't, you know, touch them, you just let them graze and whatever, the hundred sheep, let's say, will physically turn into 110 sheep next year, right? you know, some will die and some will be born, but on net, imagine there's an internal 10% biological growth rate. And so in that world, if sheep are the only good that exists in this economy, it would have to be the case that the real rate of interest is 10%. Right? If you were going to, you know, do you say, "What do you mean real rate of interest?" Well, the real part means we're not worried about money prices. We're worried about like actual goods. And so what it means to say what's the real rate of interest in that world, it's well, if you had 10 sheep right now and you sold them to somebody and he was going to give you a claim for, you know, you could redeem it one year from now to get a certain number of sheep in equilibrium, his claim would have to say, I'll give you 11 sheep next year for an implicit 10% net return given that you know the sheep multiply the way I said. Okay? And if you just think through the logic of why would that be? Because he he wouldn't he wouldn't be able to promise you more and he wouldn't promise you less because you could just hold the sheep and they would physically multiply, right? So that's kind of how it gets pinned down. All right. And so so I made that point to Alberto more succinctly. I didn't take looking at the clock there. Took me 13 minutes to get here. I made the point much more quickly than that. And I said, "So you see, Alberto, you know, and I did that in my dissertation. I did a model. I did exactly what you guys said that the defenders of mathematical economics will say, "Yeah, yeah, we know there's unrealistic assumptions." But you know what? If you think that one of our unrealistic assumptions is driving the result and that it's not just an innocuous thing done for convenience and tractability, but if you think the question we're trying to answer that that the answer popping out of our model is dependent upon an unrealistic simplifying assumption we made, then you should just go ahead and take our model and tweak that. Make the assumption one that you think is more realistic and conforming with the real world. And then when you get a different answer popping out, you show us that. And if we agree with your logic and so forth, then yes, we will we will agree that you won. You showed us that our earlier model, which everybody agrees is wrong in a sense. It's not really simulating the world. There's all sorts of simplifying assumptions made. But if the conclusion we were drawing on the question of interest to us as analysts, as economists, if the answer popping out of our simple model and you get one that's more realistic and you get the different answer and we, you know, we agree with your argument and whatnot, then yep, we'll agree you're right and we should stop saying the first thing. And so I said to Elbero, that's what I did in my dissertation. you know after you know the verbal quoting from Mombav and going through stuff in my critiques that I I gave of Messian approach to the pure time preference theory that was in the earlier chapters and I gave my own version of what I thought could replace it and be a a more robust Austrian theory but then I had a mathematical appendix where I showed I'm going to just add one good I'm going to have a mathematical model just like they taught us at NYU and I'm going to have a capital good and a consumption Good. And then I'm going to in that world I derived what the equilibrium wage rate was and it was as it normally is that the you take the derivative of the production function with respect to labor. Boom. Real wage rate. But in that world because the capital and consumption goods were distinct things physically the real interest rate was a more complicated expression that what you needed to include in the you know the equation to say what it equals in equilibrium was not just oh you take the derivative of the production function with respect to the capital but you also had to worry about the market price of the capital good measured in the consumption Good, right? So, there's not money in this. Is all real just to keep things simple and to make sure we're not getting misled by nominal inflation. Okay? And then when I wrote that equation out, I'm sitting there looking at it and I realized, oh, if we just assume that the capital and consumption good are the same thing, and so that the market price of the capital good is always one, right? Because if if they're if they're the same thing, the the price of something measured in terms of itself is always one for all time periods. And if you plug that in, my more complicated expression reduced to saying the real interest rate equals the marginal product of capital. And I was like, oh, so I thought, you know, it's kind of like analogous to special relativity, right? Einstein showing that oh Newtonian mechanics is basically right at slow speeds because I'm writing out as Einstein the more correct general formulation of things and but in my equations that are more complicated than the ones Newton did. If you assume that the velocities of all the items in question are relatively low compared to the speed of light, my system of equations approximates the old simpler Newtonian ones. And so that's the sense in which Newton was basically right in the special scenario where things weren't traveling very fast relative to light. Okay. And so what I was showing is I came up with a more general expression of what the real interest rate equals in equilibrium. And I showed oh in the special case where there's only one good where the capital and consumption good are the same thing physically my more complicated expression reduces down to interest equals the marginal product of capital. And so I thought, isn't this what I'm supposed to do that you guys told us? If you think what we're saying is wrong with our real simple math models, then go do one that's more robust, that has a more realistic assumption. I said, I think we can all agree it's more realistic to assume there are two goods rather than one good, right? And yet I'm saying that little shift. So I said that to him, again, I said it much more succinctly. And Albero's response was, if I understood what he was getting at, it's like, oh yeah, we we know that. And but I mean look, it's the same kind of thing. There's more than one type of labor, right? But in those simple workhorse models that you learn as a first year grad student, we assume one type of labor, but everyone knows in the real world there's different skill levels and wage rates aren't it's not just the And so again, I I think he's missing the point there. And and I'm just picking on him just because I had the conversation with him. The whole reason I was talking to him is he's was very open-minded. He would go to the Austrian colloquium and so forth back in at NYU days. And so that's why I was I was trying to talk to him because I thought, you know, this guy might at least give us a shot whereas the other people would just say, "Get out of here, Austrians. Give me a break." Right? So what I want to unpack in the remainder of this episode is first review Bumbers's critique of the naive productivity theory, but then just try to drive home why um there still is a fundamental qualitative difference between those two situations. So yes, it is true that in a very simplistic mainstream math model where they say you know like y equals f of k comm, l that you know to say output is is y is a function of the capital input and the labor input. there. It's assuming all the different types of labor in the economy are like one homogeneous number that it makes sense to say how many how much labor went into the production function this period that that's just some number when in reality if you were going to make it more realistic. Of course, you'd say, "Well, how many, you know, manh hours of skilled brain surgery went in? And how many man-h hours of unskilled construction work went in? And how many manh hours of uh people, you know, harvesting fruit went in? And how many hours of people laying bricks went, you know, that there's all sorts of different skill sets. And that even if you're doing it like physically to just try to make it a mechanical engineering problem in terms of inputs mapping into outputs still the thing going into the production function is not just some number measuring how much total labor like you'd have to you know there's multiplicity of of types of labor but still the basic economic intuition is still correct to say how do you know at least in a competitive market, how much the owner of labor hours who sells them on the marketplace, how much will he get paid for that contribution to output? It's like, oh, at least in equilibrium, in a competitive market, you would think there'd be a tendency that he would get paid according to him putting in his labor hour, how much does total output go up? He gets wages that could go by that in the marketplace, right? So he doesn't literally get paid in terms of apples, but he gets money. But the idea is he goes into the marketplace and then the money he gets as wages could buy the amount of apples that his labor physically produces. Okay? And that's the idea. And so yes, it's a simplification to keep the model tractable to assume all let's say there's one type of labor and that we can summarize all of the labor input by just some number each period. Okay. But by making it more realistic and breaking up into different types of labor, the economic logic of how you would explain each type of wage rate to say how much do brain surgeons get paid per hour or per year, how much do baseball players get paid? How much do brick layers get paid? It would still be the same logic. You would just say each consider isolation. How much do they contribute to the increment in output? And that's what they get paid. Okay? Whereas with capital, no, that's not what's going on. Right? If you're trying to say, well, right now the real interest rate's 8%. Why? It's it's not correct to say, oh, because there was, you know, 8,000 units of capital that went in the production function and that raised output by a certain amount. And then if you take the d, you know, one divided by and you say, oh, but there's more than one type of capital good. Okay. And then it's it's no, it's not that. For one thing, notice it's not that the interest rate is different on each capital good, right? That's not what's going on. Okay? And so very quickly you realize, oh, no, no. The analog of the real wage rate for capital goods, physical things, you know, machinery, tools, whatnot. The analog of that is not the interest rate. It's the rental price or the rental rate of those particular goods. Right? So, you could say, "Oh, yes. Using um you know, uh like if if your carpet's all just wrecked, like somebody spills something really bad on your carpet and you you tried spraying with cleaner, that's a work." You can go to like a Home Depot or something and rent like a heavyduty carpet cleaner, you know, not something that would ever make sense for you to have in your house that you own, but you can rent these really, you know, heavy duty things that you come and, you know, they really scrub and shampoo your carpet or whatever. Then you bring it back. And so maybe you, you know, maybe it's a daily rate and you got to pay whatever, $200 to rent the thing for a day. Okay? And so there when you say what determines how come that thing rents for $200 a day, whereas if you needed whatever a hammer from somebody and you wanted to rent a hammer and that would only be $5 a day. Okay? Why you say, "Oh, because the carpet cleaner is more productive." that using that for a day boosts revenues or it, you know, spares me something, I'm I'm willing to pay $200 for that thing whereas the hammer doesn't boost output as much. Okay? And so that explains the rental prices or like a farmer who just like he wants to hire some extra hands to pick some more crops during the day and he he pays them accordingly. Likewise, if somebody is renting whatever a tractor, how much would the farmer pay per day to rent the tractor? He can run those numbers. But again, that's not explaining what the percentage interest rate is. That's just showing what is the per unit time period of the monetary rental rate of that item. Okay? And so you realize, wait, something's screwy. There's something much deeper going on. Okay. So that's the issue and that's what I want to resolve in the balance of time of this episode. So let me first just do a quick review. Well for many of you it'll be review some be fresh of critique of the naive productivity theory. All right. So again, what we're trying to explain when we explain the phenomenon of interest is why is it that capitalists who have some purchasing power, some financial capital, some financial wherewithal, why is it that if they invest in various concrete items that over time they seem to earn a net return. They not only get back their original investment but a yield on top of that. You know, how do we explain that as economists? Okay. And so, um, one obvious explanation is to say, well, because the capital goods are physically productive, you produce more with capital than without. And so, duh. And so, Bombber said, no, that's not right. And specifically he said because the capital goods contribution to output that explains why it has a purchase price. But the interest return if it's positive the interest rate is positive. What you're trying to explain is why is it that the original purchase price of that item is actually less than the long run flow of extra revenue you're going to get from owning it because that really explains the net return when you invest in something. Right? If you have $100,000 and you buy a tractor and and what you know what do you do? you make forecasts like oh yes with the tractor my harvest will be higher by this amount you know over time and then at some point that you know the tractor's breaking down or whatever and at some point I just scrap it okay so over the lifetime if I'm just looking ahead let's say I make accurate forecasts oh yeah the amount of bushels I bring to market will be this much higher every year because I have that tractor assisting me and then I can sell the extra bushels make extra money so you can you know at time zero look forward and compute and estimate how much more money you'll be bringing in over time. And so Maverick's point was the only way you taking your $100,000 and buying the tractor and earning more over time is if the long run extra revenues you receive from owning that tractor are higher than $100,000. That's how you earn a return. And so another way of putting it is that original purchase price is actually less than the contribution you expect that thing to make to your production process over time. That's what makes a net return possible. And so Baba said citing the productivity of the of the good is going the wrong way. We're trying to explain its apparent undervaluation. We're not trying to explain how come people value capital goods. We're trying to explain how come they don't seem to value them enough. If you know that I have the if I have this tractor in my possession over 10 years, it's going to bring in an extra, you know, at the end of it $120,000, how come I'm not willing to pay $120,000 for it right now? I'm only been willing to pay 100, let's say. And that's what explains the interest return. And so Babar said clearly when we're trying to explain the apparent discount on capital goods why are you able to buy them for less than it seems like they're worth in some sense clearly to say oh because capital goods are productive that that doesn't make any sense right okay so the resolution that BAV came up with is he said oh it's because we're ignoring the time element it's if we naively think that the fact that the tractor is going to yield extra bushels of wheat over time. That um we can't treat, you know, a bush extra bushels of wheat harvested 10 years from now as the same thing economically is extra bushels of wheat today. And so like the the financial capital that I'm using to buy the tractor today, I could go buy present bushels of wheat, but what the tractor is is it's like a technological claim ticket on a flow of future bushels of wheat, some of which aren't going to acrue until many years in the future. And so if you just accept for the moment that present wheat is more valuable, should have a higher market price than bushels of wheat that won't be delivered until 10 years from now, well then it all makes sense that the present capitalized value of that future flow of bushels in today's prices does not equal the same as the same physical number of bushels of wheat available right now would. Okay? And so that's why in a sense uh you could like getting a hundred units of of wheat today like the purchasing power could entitle you to more than 100 bushels of wheat over time measured in their money terms all along the way. Okay, so that's his solution. All right, and then he went on to explain, well, why should present goods be more valuable than future goods? And he went through all that. Okay. So, let me now that I've you know walked you through that so you can see I hope that I did a good job persuading you. Yes. To say why is it that oh the interest rate is 8%. Why to say because people can produce more with capital goods than without that it's it's not just that that's wrong like that's missing it's like the wrong dimension. Okay to you know one thing what's where's the percentage coming into play right so um you know and and tractors it's not that tractors make more tractors right so you see when I was going back with that Irving Fiser example of a 100 sheep if we don't if we don't consume them they just multiply biologically into 110 sheep next period and oh yeah see and so that pins down the real rate of interest in that world has to 10%. Do you see how seductive that is and slippery and misleading it is? Because there there's a lot packed into that that the consumption good we assume is cheap. And we also Oh yeah, and it's also the capital good to keep things simple. And that the capital good creates more of itself, but it's also the measure of real purchasing power in this example where it's the only good like you know a basket of consumption goods. What could it be? Just sheep. because that's the only thing there is, right? So, you see how all these different concepts all get squashed into the same thing. And it's like, oh yeah, 100 sheep turn into 110 sheep next year. Real rate of interest is 10%. If a capitalist has financial capital and he, you know, has the the ability right now to buy 100 sheep, he could lend that out at interest. And it's got to be the case that next period he gets paid back enough money to go buy 110 sheep. And that's all pinned down by the technological productivity of sheep production, right? You take a 100 sheep, you put them in the production function, out pops 110 sheep next period. What could be simpler than that? But I'm saying that is extremely misleading because in general, capital goods don't make copies of themselves. They make other goods. And if if we're picking the unit of measure in terms of just like what do we mean by the real rate of interest? You got to have some like basket of consumption goods to say if you make a loan over time, how much you know how how many units of that basket of consumption goods can you buy over time? That's to give operational meaning to the real rate of interest. That's an obvious candidate. Okay. And so like if we want to say it's bushels of wheat and the tractor makes you know contributes to them the point is what Mavver showed the issue is not having you know adding more tractors to my farming operation does that boost how many bushels of wheat I harvest each year. The issue is what is the initial purchase price of the tractors measured in bushels of wheat and how does that relate to the increment? That's what would explain the real rate of interest. If we're picking bushels of wheat as like the numer now just flip and explain what I did in my appendix. Oh, and I forgot to say the reason I realized I need to do this episode is because I said to Alberto, um, you know what? I'll I'll try to send you something like to, you know, where I I spelled all this out and I I said to the, you know, the listeners of that podcast when I did the the intro and the outro and I said, I'll I'll go find something for those of you who want to go, you know, read up on this. And I realized I this isn't written anywhere in an accessible form. It's in the appendex to my dissertation. I had done a standalone article for the a history of economic thought journal. But when I got to this, you know, part that was my my personal favorite where I had this equation that shows the general expression, but then if you plug in that there's one good, it reduces to what the, you know, mainstream guys were saying. The the a referee said, "No, this is too much. Just forget You know, I I had done earlier stuff about Bomba and then this was like the, you know, PA songs and he and the referee said, "No, you got too much here. Cut this part." And so I did, you know, get the thing published. Got to do what you got to do. But the point is I realized, oh, I really don't have this written anywhere except in the my at the end of this, you know, big dissertation. Then no one's going to slog through that or some of you do, but I can't expect people to do that. So that's why I'm partly doing here is I want to have it in one accessible place for people to see what I'm talking about. And again, this is partly for Austrian grad students. So in case you're losing your mind like I was trying to reconcile two things that seem contradictory and yet you can't point out the specific fallacies. So there's that element. Also, if you are talking to a mainstream colleague and they say, "Well, you know, this verbal logic, I don't even you're walking around in a circle. You can give them this and show them what you're talking about." Like, I think I made it crystal clear. Okay, so um let's just work with a hypothetical example. Suppose we're talking about a net, right? somebody Robinson Crusoe on his island, he builds a net that allows him to catch more fish per unit of time than he could with his bare hands. All right? And let's just suppose that the net augments his productivity by 100 fish per unit of time, right? It could be days, months, whatever, you know, whatever. Okay? And suppose that the net is such that physically it's good for 10 uses but then after you use it the 10th time it just frays and you got to throw it out. Okay. So there is physical depreciation in that sense. It it it's equally serviceable for the first 10 times you use it. It gives you 100 extra fish than what you would have gotten with your bare hands. But then after you use it the 10th time it tears and Crusoe just throws it out and it's it's useless. Okay. So now the question is you know what if we're going to model this in a mainstream fashion and try to talk about uh the real interest rate how do we do it? So let's just assume that the consumption good in this world are the fish and that the only type of capital good is this net. Okay so there's only two goods in his labor. All right if you want to call that a good you can but he can't consume that or anything. Okay. And so let me now show you you can go ahead and write out a bunch of equations and things, but the the money line is we flashed up on the screen. This was the equation for that I end up deriving. And so the I there stands for the real interest rate. You might have expected it to be an R, but I didn't do that because earlier I had used that to be the rental rate of the capital. And again, that's part of the problem in this literature is people use R to mean interest rate, but also it could mean rental rate. In some models, it's it's better like they do distinguish these things, but I'm saying in these workhorse introductory models where there's just one type of capital good and it's the same thing as the consumption good, they tend to be, you know, they tend to use R and that it's it embeds the confusion. So here I called it I but it's this is the real rate of interest. Okay. So I'm saying it at time t the real interest rate is the derivative of the production function with respect to that first argument. That's what the f1 means. And that k is the capital input at time t. And then l is how much labor goes in at time t. And then but that's not it. That's in the numerator. And then you subtract. And then what that pi stands for, the pi is saying that's the price of the capital good measured in consumption good. The first term there is at time t. And then you subtract away its value, its price at time t plus one. And then you divide all of that by the original purchase price of the capital good. Okay? Now don't be intimidated here. Let me just give you a simple example. And so let's say that let me just do the example. Let me let me just tell you let's make it so that the interest rate the real interest rate is 5%. Okay. So that means in fish if you if Crusoe has 100 fish and Friday comes along and Crusoe says hey Friday here I'll lend you 100 fish in equilibrium Friday has to give him 105 fish next period. Okay let's just say that that's what we want the answer to be. what ends up happening here given the physical the brute physical facts that I've outlined in this story right with the net and how it works. So if effectively when you're want like what's the what's the initial purchase price of that net let's assume the way that it works is you use the net over the course of a period of time and then at the end of that period you get that yield of the hundred extra fish. Right? All right. So, that's the way we're going to handle the the time element here. So, we're going to say like you you buy the net at time zero. And that what that's doing is entitling you, you know, as as t turns into one, you all of a sudden have that extra hundred fish now that you would have gotten over and above what you could catch with your bare hands. And um like so if you could catch eight fish and in the time interval with your bare hands, with the net, you can catch 108 fish. Okay? So, the net is adding 100. It's the net, no, no pun intended, it's the net yield physically of that um during that time period, the flow, so it's over and above what you get with your bare hands. Okay? And then that happens for 10 periods in a row and then the net is frayed and useless and you throw it out. So there's a sense in which it's like the net is a bond that just pays a 100 fish starting at the end of the first period and then at the end of the second D for 10 periods in a row. And so if we're going to discount that flow of future fish at a 5% discount rate, the present value of that I'm rounding is like 772. Okay. So that's the initial purchase price. To buy that net right now, his brand new net, Crusoe would have to hand over 772 fish that, you know, he could have accumulated over. He's catching fish with his bare hands, salting them and saving them, building up this big stockpile. And then, you know, some entrepreneur is going out and getting vines and making a net and selling them. And how much would Crusoe value that? Well, in equilibrium, you know, its price is going to be 772 fish, the spot price of that net when it's brand new. Now, how does this work? So, he, you know, Crusoe goes and gets a, you know, harvests 100 extra fish compared to what he would have gotten with his bare hands. And so in that expression there that equation four that first term is 100 right it's saying what is the increment that f1 means you just taking the the derivative with respect to the first argument so because you put in one net output went up 100 fish okay so that's what that part is but then you're subtracting the change in the price of the capital good over the course of that period so to to fill that in we need to know, oh, during the course of the period, the market value of that net goes down, right? That when I when I buy the net, it's brand new, I use it once. Now, at the start of the next period, that net is no longer brand new at that point. Now, it only is going to entitle me to nine future bursts of 100 fish. And so if you then just view that as a bond that's going to give you a 100 fish, you know, starting at the end of this period and then for nine total periods, discounting it at a 5% rate, I'm rounding. It's about a 711 fish. Okay? So in this equation four, you would say, oh, the top left one is 100 fish minus and then in parenthesis there, it would be 772 minus 711, right? because the original purchase price of the capital good was 772, it drops to 711. Okay? And so if you go ahead and do that, you end up with 39 is the numerator, right? Because it dropped by 61, right? So the market price of the net dropped by 61 fish. So the yield, the net yield, no pun intended, of owning that capital good for that first period is not the full 100 fish because that's partially offset by the drop in the market value of your capital good. So the 100 fish is what you got. Like if you had just rented out the net, you could have rented it out for 100 fish. But then the value of the net itself, the market value measured in terms of fish dropped by 69 or sorry 61. And so the net yield for that period was only the 39 fish, right? And they say, "Oh, so the so the interest rate's 39. What? 39% 39 what? 100%." know, you got to divide by the original purchase price, right? Because it's that the top thing is 39 fish. And so if the interest rate is a percentage, it's a rate, you need to divide by some number of fish, right? To get the units of fish to cancel out because interest rates aren't fish. You know, you say, "Oh man, the Fed cut interest rates today down to two fish." What the heck are you talking about? Right? Because I'm saying like a lot of the discourse over this, the units aren't even right. The dimensions aren't right. Okay? So if the top of the of this numerator or if the numerator in this expression is 39 fish then you divide by that original purchase price which in this example we already said is 772 fish and so then oh 39 fish divided by 772 fish equals approximately 5%. It's like 5.05 but that's because I would been rounding. If you if I gave you the exact numbers it would have been exactly 5%. Okay so that's how that works in an example like this. So you can I'm just showing you even though that equation that I flashed up there might initially have been intimidating say I'm not going to get it. I'm saying when you see what it's saying it's it's not hard and it makes intuitive sense. It's saying yes of course the extra physical yield in the consumption good that the capital good provides you matters but there's other things going on. It's more complicated if the capital good is a distinct thing and hence you have to worry about its value. measured in terms of that consumption good. So again, this is Bombick's critique of the naive productivity theory and I'm putting it here in mathematical form in a you know good little neocclassical model with all the eyes dotted and te's crossed. Okay. Now let me just show you something before I move on to the punchline. Let me show you why the physical productivity of capital is not sufficient for a positive interest rate. I can use the same framework and it's consistent with the real interest rate measured in fish being 0%. And what would that look like? You say, "Oh, if the nets couldn't harvest more fish, well, yeah, that would work. But you know what? It doesn't you don't need to invoke that. Even if we still stipulate that physically speaking, a guy can go make a net and sell it to Crusoe and then Crusoe can get a 100 extra fish period for 10 periods and then the net is useless physically. It, you know, wears away and he just discards it. That is consistent with a world on this island, an economy on this island where the real interest rate measured in fish is 0%. And how could that be? Well, when Crusoe goes to originally buy the net, its market price is 1,000 fish. And so, he buys it. Okay. He uses it. He gets a 100 fish by the end of that period, over and above what he would have got with his bare hands. But then what is the market price now of the net going into the next period? Oh, it's dropped down to 900. And then in the next period, it's 800 and 700 and so on. Okay? And such by the end of it, he's just recovered his fish back. And so for each period to use that equation he would say oh in the numerator you've got the 100 fish that he gets you know due to the productivity of the net physically but then you subtract the change in the market price from one period to the next which is also 100. And so that numerator is zero in the first period it's you know 0 divided by a th00and. So the the net interest rate is 0% even though the net is still just as physically productive entitles you to an extra flow of real fish as it was before. And again you see that the trick if you will or how does that what's going on it's that I can just just determine I can tell you I can make that work if the rate of interest is 5% I can make it worth the rate of interest 0%. And so what's pinning down the valuation of the net is knowing what is the intertemporal exchange rate of fish that if I know that oh yeah people prefer present fish to future fish then there's going to be a positive discount rate measured in fish and that will make the original purchase price of the net measured in fish lower than what the you know a thousand and that's why originally In that first scenario when Crusoe puts in 772 fish to buy that net and we we determined after that first period he had earned a net return of 39 fish. Okay. So like that's his free and clear. In other words, he still got his 772 fish of his original capital intact and he could eat that extra 39 and not impair his capital. That's another way of putting it, right? That he's he starts out with the 772 fish that he needs to buy the net and then in that first period he would be allowed to eat 39. He couldn't eat 100, by the way. If he ate the full 100 extra that that net yielded, he would be implicitly eating some of his capital. He'd be consuming capital. But he can eat 39 and then he still has the 772 like rolling over such that at the very end when he does it and that net wears away, he's still got 772 fish that he can roll over and then buy a brand new net. and he can just keep doing that and just gets that constant flow. Okay. And so again, it it would be tempting to say, geez, if he just scrims and saves and in the beginning when he doesn't have a net and has no savings and just lives below his means, you know, he's he's catching eight fish period with his bare hands, but let's say he only consumes seven and every period he socks away a fish and just lets that accumulate. And then eventually he gets up to 772. He's got to have good storage. I don't know how he's doing that in the island, but go with me on this one, folks. And then once he gets up to seven to 72, he can go buy the net. And now forever on, he can get uh what? 39 plus 8. 47 fish period that he can consume forever and maintain that 772 of his capital intact. that he's getting the eight implicitly from his labor and the 39 is like interest earnings on his saved capital would be the way you would decompose it like as an accountant or a macroeconomist looking at what are the returns to the factors in this economy. Okay. So it would be tempting to say geez how's he getting at the flow of 39 and like there there is a sense in which yes the physical productivity of the net allows for that but you can see how when you want to understand why is the interest rate 5% and not 10 or not two or not zero clearly the physical productivity of the net is not the explanation because I can make the interest rate any of those numbers I just said and be consistent with the net having the same physical productivity that we stipulated. Okay, now finally the great reveal. Put that equation back up. What if we don't have an economy where the net and the fish are different physical things and hence you have to have a price of the net measured and quoted in fish which allows for all these complications. What if instead we're back in Fischer's world where it's just sheep and that's it? So in that world, we can still apply this, you know, this equation is still true. It's just, oh, here the capital good and the consumption good are the same physical thing. Instead of it being a net and a and fish, it's just sheep. Instead of being a net gives you more fish, it's sheep give you more sheep. And so, oh, at any given time, what is the market price of a sheep measured in sheep? Oh, it's one. Not because of any deep economic insight, just because it's the same thing. So, of course, one sheep has a value of one sheep. Duh. As opposed to a brand new net, you know, being 772 or being a thousand or whatever. Okay. And so in this equation, what happens if pi of t is always one for all t? Well, the denominator is just a one. So that goes away. And then pi of t is one. And pi t + 1 is one. And so that top right, the pi t minus p t + 1, that's one minus one. That's zero. And that goes away. Oh, and so that equation four reduces to if the capital good and the consumption good are the same physical thing so that the price of each is just one all the time. Oh, look at that. That more complicated expression reduces down to the interest rate, the real rate of interest equals the marginal product of capital, right? The derivative of the production function with respect to that first argument, which is K. Okay, so that's what I'm saying. I think a lot of mainstream economists are walking around believing that interest income is due to the physical productivity of capital in the same way that labor income w you know wage income is due to the physical productivity of labor and as the Austrians have been saying since bomb no that's a fundamental confusion that's not what's going on it's way more complicated than that. The productivity of physical capital goods explains their rental rate. It does not explain the ratio of their unit rental rate to their spot purchase price, which is kind of like what the at least uh gross rate of interest is. Okay, I will wrap up there. I'll put a link if you really are an overachiever and want to check out the dissertation. I'll put a link in the show notes page. Um, I'll also link to my other conversation with Alberto Bazine, which is, you know, was very interesting in many other respects, but I did want to flesh this particular thing out. And again, if you're an Austrian in a grad program somewhere and your head is swimming, I hope now I've shown you what's going on. Thanks for your attention, everybody. See you next time. Check back next week for a new episode of the Human Action podcast. In the meantime, you can find more content like this on mises.org. Heat.
Interest Is Not the Marginal Product of Capital
Summary
Bob revisits capital and interest theory to show why the textbook result “interest = MPK” only holds in a one-good world, and why …Transcript
This is the Human Action podcast where we debunk the economic, political, and even cultural myths of the days. Here's your host, Dr. Bob Murphy. Hey everybody, welcome back to Human Action Podcast. In this episode, I am going to do a deep dive into the Austrian approach to capital and interest theory. And in particular, I'm going to relate it to the mainstream neocclassical approach. Now, for those of you who have followed my work for a while, you might say, "Bob, didn't you already do this?" Yes and no. So, I have definitely hit these areas before in previous podcast episodes and things I've written in various outlets. But what prompted me to come back and hit this again is number one, we've got a different audience now than perhaps was the case three years ago. So, it's fine to have repetition on these important matters. But also, I was recently interviewing um one of my old professors from NYU, uh great guy, Alberto Bazine, u very open-minded. He was mathematical economist trained in Chicago. And we were just talking about this the controversy that's raging lately on Twitter at least uh of over the use of mathematics and economics. Okay? And I was, you know, I wasn't arguing. I was just letting him explain his perspective. And I made the point that on the one hand I had sort of acknowledged to him that yes I noticed in my own work when I was doing my dissertation at NYU and I was working on what's called the Austrian pure time preference theory of interest that one of the points I was trying to make to the Austrian crowd was that the term time preference is used to mean different economic things in the literature and it's the same words and that it's important just to be clear about what do you mean by that term. term and I said to Alberto now it was very easy you know I can explain it to you as a mainstream mathematical economist the distinction and and I went ahead and said it in math terms and he understood immediately oh yeah there's there's that thing or it's this thing both could plausibly be called time preference but you got to stick with one meaning and you know be consistent with it you can't just flip back and forth mid argument okay but then I said now having you know congratulated you guys Alberto for precision and formalism. On the other hand, I have to say what also became apparent to me while I'm doing my dissertation at NYU is that um the mainstream economists are walking around thinking that interest is due to the marginal product of capital just like they think that um real wages are due to the marginal product of labor or are equal to it in you know monetary terms and and at least with competitive compitive markets and so on, right? and and I said, and that is a fundamental confusion in the Austrian tradition, and yet um it it seems like it just pops out of a standard neocclassical model like the kind that we learned as first year grad students at NYU where, you know, you have a production function that takes K and L like capital and labor as inputs and out pops output and then the representative household can divide the output into consumption or saving. and you know carry the saved goods forward one period and then they get thrown in the production function again and so on rinse and repeat. And that in that kind of a model if you ask assuming competitive markets blah blah blah what is the real return to labor it's oh you take the production function you take the first derivative with respect to the labor input variable and that's what the wage rate is in real terms and the idea is oh yeah if output goes up a little bit because you put in a little bit more labor into the mach into the factory or whatever the production function And you know, you put in one more labor hour and output goes up by 10 apples. Well, then in equilibrium in a competitive market, that worker needs to be paid 10 apples for that hour of labor. And so he gets his marginal product. Okay? Now, the Austrians agree with that and so do the mainstream economists in their mathematical model. Of course, the Austrians would have all kinds of quibbles and deeper objections to that model, but the economic intuition is the same, but that correspondence does not carry over to the determination of interest. And so again in a simple model like that of the kind that first year grad students at least back when I went through and I think probably still to this day learn is like the you know the introduction get you warned up how to work with these models is you say okay well what's the real interest rate in equilibrium? Well capital has to earn its marginal product and so put in a little bit dose more of capital into the production function. output physically goes up a little bit and then that increment that's the return to capital and so in equilibrium the real weight uh sorry the real interest return to capital they have to earn that increment and output and hence in equilibrium that's what the real wage or sorry the real interest rate equals okay and I want to say there the Austrians are going to fall out of their chair and say whoa that's not right and it's we're not just arguing about you can't collapse the economy into a simple mathematical model that takes two inputs and you know that that's crazy but beyond all that just the economics of it is wrong even the dimensions are wrong okay and that's what I want to unpack and so and I was telling Alberto this and I and I said but now the issue the thing that was weird and what I struggled with when I was in grad school is when we were learning those models you know we're going doing problem sets and taking the first derivative and blah blah blah and doing all that the math was correct. Right? So in the world of the model, I understood why that's what the textbook said the return to the capitalist was going to be. And so it wasn't wrong. And yet that seemed to be directly at odds with what Ogan von Bomba a second generation Austrian after you know Manger was the founder of the school in 1871 and then Bomba was in the next wave and you know was a huge pioneer everybody agrees in capital interest theory and one of the things one of his contributions was he wrote this critique of what he called the naive productivity theory of interest and in that bomb shows when people are trying to understand what is the source of the interest income that capitalists earn in a market economy. A natural hypothesis is to say, "Oh, well, because capital is productive. You can make more stuff with capital goods than without. So, duh, that's how the capitalists earn an interest return." And Maverick showed that no, no, that's a fun there's a fundamental confusion going on there. You're getting mixed up. That isn't a good explanation. Okay. So, I'm going to come back, folks, and spell that out. Don't worry. In this episode, by the time you're done, if you invest your time listening to this, I will explain that clearly and you you'll totally get it. Okay? But take my word for it the moment. But Bob is definitely right. when he when he you understand what he's saying and he walks you through and shows the fact that a farmer can have a bigger harvest physically that can get more bushels of wheat or ears of corn or whatever using a tractor than if he didn't have the tractor available. that the physical productivity of the tractor, its ability technologically speaking to augment the yield of his operations. That does not explain why a farmer who has a sum of money and buys a tractor then can use it and over time when he does the accounting or has his accountant do it that he ends up with a bigger sum of money acrewing from the purchase of the tractor, right? That in a sense he earned a return on the financial capital that he poured into the tractor if you want to think of it like that. And Bombber showed to explain that phenomenon. Why is it that maybe he earned 5% a year correctly considered on his investment in the tractor in money terms and you can make it real interest like 5% real return if you account for you know changing money prices over time for price inflation. You can do that too. But the point is when you're trying to explain that phenomenon, why is it that in general in the market economy, capitalists who have a sum of money at the outset can buy things with them and invest their proceeds such that over time they tend to earn a net return on that. So they end up with more financial capital than they started out with even accounting for rising prices. How do you explain that? And Bomb said to say that capital goods are physically productive is completely missing the point that that's not an explanation at all. Right? Again, we'll come back and explain that. All right? But just take my word for it. That's what he did. And so there I was at NYU, a grad student working on this dissertation focusing on Bobky and capital interest theory, but then you know trying to relate it to what I was learning in the mathematical mainstream curriculum. And I was like, "What the heck's going on here?" That the math of these, you know, that I'm learning on the blackboard or whatever in class is correct. And yet that seemed to enshrine the naive productivity theory into the equations. They were clearly saying in equilibrium in our models, the capitalists who start out and have a bunch of capital and contribute it to the production process. How do we explain what the real return is percentage-wise? You know, you know that like what what fraction of their capital, how much does it grow measured in terms of itself. So you know that there wasn't some just basic confusion about inflation or something that no the real rate of interest in that model was directly equal to just looking at the physical production function and saying if you do a slight increment you add epsilon more capital physically into the production process that boosts the physical output by a certain amount and then that relationship the magnitude of that of the one increment to the other then pins down what has to be the real rate of interest in equilibrium and and I was just staring at that like I don't remember how long it took me might be a better story if I did but I don't remember but then I had the Eureka moment and oh I see what's going on it's because in those simple models that you learn as a first year grad student and I think probably still do everybody would acknowledge oh yeah yeah the Um, we know that in the real world things are way more complicated, but for simplicity sake, for tractability, just to get your, you know, get warmed up in how these things work. We'll start with a real simple case where assume there's just one production function, one type of labor, and we'll just call it L to like just measure or, you know, L with a subscript of T if you want to make it index by time, and K with a subscript of, you know, T and T plus1, T+2, whatever. And so, yeah, we know in the real world there's all kinds of different types of capital goods and whatever, but here we're just going to be real simple. And I realized that apparently innocuous assumption for convenience was driving the whole result. Okay? And all the like Bombav's critique of the naive productivity theory falls away. It's a moot point. He's worrying over something that in practice can't ever be or can't have an implication, any significance. If you were in a world where there was just one type of good and that the capital good was the same thing as the consumption good, they were physically the same thing. Um, for example, Irving Fischer, who was one of the giants in neocclassical interest theory, when he was in his original pioneering works trying to lay out this framework and get people to think through the the, you know, the issues, he would use simplistic examples and one of them was like, oh, imagine the only good are sheep. And so they are like the capital and the consumption good at the same time. that you got like a flock of a hundred sheep and you want to decide, you know, how many do I want to consume? And if you don't, it's like, oh, you're saving it, but you're also kind of investing it because if you don't, you know, touch them, you just let them graze and whatever, the hundred sheep, let's say, will physically turn into 110 sheep next year, right? you know, some will die and some will be born, but on net, imagine there's an internal 10% biological growth rate. And so in that world, if sheep are the only good that exists in this economy, it would have to be the case that the real rate of interest is 10%. Right? If you were going to, you know, do you say, "What do you mean real rate of interest?" Well, the real part means we're not worried about money prices. We're worried about like actual goods. And so what it means to say what's the real rate of interest in that world, it's well, if you had 10 sheep right now and you sold them to somebody and he was going to give you a claim for, you know, you could redeem it one year from now to get a certain number of sheep in equilibrium, his claim would have to say, I'll give you 11 sheep next year for an implicit 10% net return given that you know the sheep multiply the way I said. Okay? And if you just think through the logic of why would that be? Because he he wouldn't he wouldn't be able to promise you more and he wouldn't promise you less because you could just hold the sheep and they would physically multiply, right? So that's kind of how it gets pinned down. All right. And so so I made that point to Alberto more succinctly. I didn't take looking at the clock there. Took me 13 minutes to get here. I made the point much more quickly than that. And I said, "So you see, Alberto, you know, and I did that in my dissertation. I did a model. I did exactly what you guys said that the defenders of mathematical economics will say, "Yeah, yeah, we know there's unrealistic assumptions." But you know what? If you think that one of our unrealistic assumptions is driving the result and that it's not just an innocuous thing done for convenience and tractability, but if you think the question we're trying to answer that that the answer popping out of our model is dependent upon an unrealistic simplifying assumption we made, then you should just go ahead and take our model and tweak that. Make the assumption one that you think is more realistic and conforming with the real world. And then when you get a different answer popping out, you show us that. And if we agree with your logic and so forth, then yes, we will we will agree that you won. You showed us that our earlier model, which everybody agrees is wrong in a sense. It's not really simulating the world. There's all sorts of simplifying assumptions made. But if the conclusion we were drawing on the question of interest to us as analysts, as economists, if the answer popping out of our simple model and you get one that's more realistic and you get the different answer and we, you know, we agree with your argument and whatnot, then yep, we'll agree you're right and we should stop saying the first thing. And so I said to Elbero, that's what I did in my dissertation. you know after you know the verbal quoting from Mombav and going through stuff in my critiques that I I gave of Messian approach to the pure time preference theory that was in the earlier chapters and I gave my own version of what I thought could replace it and be a a more robust Austrian theory but then I had a mathematical appendix where I showed I'm going to just add one good I'm going to have a mathematical model just like they taught us at NYU and I'm going to have a capital good and a consumption Good. And then I'm going to in that world I derived what the equilibrium wage rate was and it was as it normally is that the you take the derivative of the production function with respect to labor. Boom. Real wage rate. But in that world because the capital and consumption goods were distinct things physically the real interest rate was a more complicated expression that what you needed to include in the you know the equation to say what it equals in equilibrium was not just oh you take the derivative of the production function with respect to the capital but you also had to worry about the market price of the capital good measured in the consumption Good, right? So, there's not money in this. Is all real just to keep things simple and to make sure we're not getting misled by nominal inflation. Okay? And then when I wrote that equation out, I'm sitting there looking at it and I realized, oh, if we just assume that the capital and consumption good are the same thing, and so that the market price of the capital good is always one, right? Because if if they're if they're the same thing, the the price of something measured in terms of itself is always one for all time periods. And if you plug that in, my more complicated expression reduced to saying the real interest rate equals the marginal product of capital. And I was like, oh, so I thought, you know, it's kind of like analogous to special relativity, right? Einstein showing that oh Newtonian mechanics is basically right at slow speeds because I'm writing out as Einstein the more correct general formulation of things and but in my equations that are more complicated than the ones Newton did. If you assume that the velocities of all the items in question are relatively low compared to the speed of light, my system of equations approximates the old simpler Newtonian ones. And so that's the sense in which Newton was basically right in the special scenario where things weren't traveling very fast relative to light. Okay. And so what I was showing is I came up with a more general expression of what the real interest rate equals in equilibrium. And I showed oh in the special case where there's only one good where the capital and consumption good are the same thing physically my more complicated expression reduces down to interest equals the marginal product of capital. And so I thought, isn't this what I'm supposed to do that you guys told us? If you think what we're saying is wrong with our real simple math models, then go do one that's more robust, that has a more realistic assumption. I said, I think we can all agree it's more realistic to assume there are two goods rather than one good, right? And yet I'm saying that little shift. So I said that to him, again, I said it much more succinctly. And Albero's response was, if I understood what he was getting at, it's like, oh yeah, we we know that. And but I mean look, it's the same kind of thing. There's more than one type of labor, right? But in those simple workhorse models that you learn as a first year grad student, we assume one type of labor, but everyone knows in the real world there's different skill levels and wage rates aren't it's not just the And so again, I I think he's missing the point there. And and I'm just picking on him just because I had the conversation with him. The whole reason I was talking to him is he's was very open-minded. He would go to the Austrian colloquium and so forth back in at NYU days. And so that's why I was I was trying to talk to him because I thought, you know, this guy might at least give us a shot whereas the other people would just say, "Get out of here, Austrians. Give me a break." Right? So what I want to unpack in the remainder of this episode is first review Bumbers's critique of the naive productivity theory, but then just try to drive home why um there still is a fundamental qualitative difference between those two situations. So yes, it is true that in a very simplistic mainstream math model where they say you know like y equals f of k comm, l that you know to say output is is y is a function of the capital input and the labor input. there. It's assuming all the different types of labor in the economy are like one homogeneous number that it makes sense to say how many how much labor went into the production function this period that that's just some number when in reality if you were going to make it more realistic. Of course, you'd say, "Well, how many, you know, manh hours of skilled brain surgery went in? And how many man-h hours of unskilled construction work went in? And how many manh hours of uh people, you know, harvesting fruit went in? And how many hours of people laying bricks went, you know, that there's all sorts of different skill sets. And that even if you're doing it like physically to just try to make it a mechanical engineering problem in terms of inputs mapping into outputs still the thing going into the production function is not just some number measuring how much total labor like you'd have to you know there's multiplicity of of types of labor but still the basic economic intuition is still correct to say how do you know at least in a competitive market, how much the owner of labor hours who sells them on the marketplace, how much will he get paid for that contribution to output? It's like, oh, at least in equilibrium, in a competitive market, you would think there'd be a tendency that he would get paid according to him putting in his labor hour, how much does total output go up? He gets wages that could go by that in the marketplace, right? So he doesn't literally get paid in terms of apples, but he gets money. But the idea is he goes into the marketplace and then the money he gets as wages could buy the amount of apples that his labor physically produces. Okay? And that's the idea. And so yes, it's a simplification to keep the model tractable to assume all let's say there's one type of labor and that we can summarize all of the labor input by just some number each period. Okay. But by making it more realistic and breaking up into different types of labor, the economic logic of how you would explain each type of wage rate to say how much do brain surgeons get paid per hour or per year, how much do baseball players get paid? How much do brick layers get paid? It would still be the same logic. You would just say each consider isolation. How much do they contribute to the increment in output? And that's what they get paid. Okay? Whereas with capital, no, that's not what's going on. Right? If you're trying to say, well, right now the real interest rate's 8%. Why? It's it's not correct to say, oh, because there was, you know, 8,000 units of capital that went in the production function and that raised output by a certain amount. And then if you take the d, you know, one divided by and you say, oh, but there's more than one type of capital good. Okay. And then it's it's no, it's not that. For one thing, notice it's not that the interest rate is different on each capital good, right? That's not what's going on. Okay? And so very quickly you realize, oh, no, no. The analog of the real wage rate for capital goods, physical things, you know, machinery, tools, whatnot. The analog of that is not the interest rate. It's the rental price or the rental rate of those particular goods. Right? So, you could say, "Oh, yes. Using um you know, uh like if if your carpet's all just wrecked, like somebody spills something really bad on your carpet and you you tried spraying with cleaner, that's a work." You can go to like a Home Depot or something and rent like a heavyduty carpet cleaner, you know, not something that would ever make sense for you to have in your house that you own, but you can rent these really, you know, heavy duty things that you come and, you know, they really scrub and shampoo your carpet or whatever. Then you bring it back. And so maybe you, you know, maybe it's a daily rate and you got to pay whatever, $200 to rent the thing for a day. Okay? And so there when you say what determines how come that thing rents for $200 a day, whereas if you needed whatever a hammer from somebody and you wanted to rent a hammer and that would only be $5 a day. Okay? Why you say, "Oh, because the carpet cleaner is more productive." that using that for a day boosts revenues or it, you know, spares me something, I'm I'm willing to pay $200 for that thing whereas the hammer doesn't boost output as much. Okay? And so that explains the rental prices or like a farmer who just like he wants to hire some extra hands to pick some more crops during the day and he he pays them accordingly. Likewise, if somebody is renting whatever a tractor, how much would the farmer pay per day to rent the tractor? He can run those numbers. But again, that's not explaining what the percentage interest rate is. That's just showing what is the per unit time period of the monetary rental rate of that item. Okay? And so you realize, wait, something's screwy. There's something much deeper going on. Okay. So that's the issue and that's what I want to resolve in the balance of time of this episode. So let me first just do a quick review. Well for many of you it'll be review some be fresh of critique of the naive productivity theory. All right. So again, what we're trying to explain when we explain the phenomenon of interest is why is it that capitalists who have some purchasing power, some financial capital, some financial wherewithal, why is it that if they invest in various concrete items that over time they seem to earn a net return. They not only get back their original investment but a yield on top of that. You know, how do we explain that as economists? Okay. And so, um, one obvious explanation is to say, well, because the capital goods are physically productive, you produce more with capital than without. And so, duh. And so, Bombber said, no, that's not right. And specifically he said because the capital goods contribution to output that explains why it has a purchase price. But the interest return if it's positive the interest rate is positive. What you're trying to explain is why is it that the original purchase price of that item is actually less than the long run flow of extra revenue you're going to get from owning it because that really explains the net return when you invest in something. Right? If you have $100,000 and you buy a tractor and and what you know what do you do? you make forecasts like oh yes with the tractor my harvest will be higher by this amount you know over time and then at some point that you know the tractor's breaking down or whatever and at some point I just scrap it okay so over the lifetime if I'm just looking ahead let's say I make accurate forecasts oh yeah the amount of bushels I bring to market will be this much higher every year because I have that tractor assisting me and then I can sell the extra bushels make extra money so you can you know at time zero look forward and compute and estimate how much more money you'll be bringing in over time. And so Maverick's point was the only way you taking your $100,000 and buying the tractor and earning more over time is if the long run extra revenues you receive from owning that tractor are higher than $100,000. That's how you earn a return. And so another way of putting it is that original purchase price is actually less than the contribution you expect that thing to make to your production process over time. That's what makes a net return possible. And so Baba said citing the productivity of the of the good is going the wrong way. We're trying to explain its apparent undervaluation. We're not trying to explain how come people value capital goods. We're trying to explain how come they don't seem to value them enough. If you know that I have the if I have this tractor in my possession over 10 years, it's going to bring in an extra, you know, at the end of it $120,000, how come I'm not willing to pay $120,000 for it right now? I'm only been willing to pay 100, let's say. And that's what explains the interest return. And so Babar said clearly when we're trying to explain the apparent discount on capital goods why are you able to buy them for less than it seems like they're worth in some sense clearly to say oh because capital goods are productive that that doesn't make any sense right okay so the resolution that BAV came up with is he said oh it's because we're ignoring the time element it's if we naively think that the fact that the tractor is going to yield extra bushels of wheat over time. That um we can't treat, you know, a bush extra bushels of wheat harvested 10 years from now as the same thing economically is extra bushels of wheat today. And so like the the financial capital that I'm using to buy the tractor today, I could go buy present bushels of wheat, but what the tractor is is it's like a technological claim ticket on a flow of future bushels of wheat, some of which aren't going to acrue until many years in the future. And so if you just accept for the moment that present wheat is more valuable, should have a higher market price than bushels of wheat that won't be delivered until 10 years from now, well then it all makes sense that the present capitalized value of that future flow of bushels in today's prices does not equal the same as the same physical number of bushels of wheat available right now would. Okay? And so that's why in a sense uh you could like getting a hundred units of of wheat today like the purchasing power could entitle you to more than 100 bushels of wheat over time measured in their money terms all along the way. Okay, so that's his solution. All right, and then he went on to explain, well, why should present goods be more valuable than future goods? And he went through all that. Okay. So, let me now that I've you know walked you through that so you can see I hope that I did a good job persuading you. Yes. To say why is it that oh the interest rate is 8%. Why to say because people can produce more with capital goods than without that it's it's not just that that's wrong like that's missing it's like the wrong dimension. Okay to you know one thing what's where's the percentage coming into play right so um you know and and tractors it's not that tractors make more tractors right so you see when I was going back with that Irving Fiser example of a 100 sheep if we don't if we don't consume them they just multiply biologically into 110 sheep next period and oh yeah see and so that pins down the real rate of interest in that world has to 10%. Do you see how seductive that is and slippery and misleading it is? Because there there's a lot packed into that that the consumption good we assume is cheap. And we also Oh yeah, and it's also the capital good to keep things simple. And that the capital good creates more of itself, but it's also the measure of real purchasing power in this example where it's the only good like you know a basket of consumption goods. What could it be? Just sheep. because that's the only thing there is, right? So, you see how all these different concepts all get squashed into the same thing. And it's like, oh yeah, 100 sheep turn into 110 sheep next year. Real rate of interest is 10%. If a capitalist has financial capital and he, you know, has the the ability right now to buy 100 sheep, he could lend that out at interest. And it's got to be the case that next period he gets paid back enough money to go buy 110 sheep. And that's all pinned down by the technological productivity of sheep production, right? You take a 100 sheep, you put them in the production function, out pops 110 sheep next period. What could be simpler than that? But I'm saying that is extremely misleading because in general, capital goods don't make copies of themselves. They make other goods. And if if we're picking the unit of measure in terms of just like what do we mean by the real rate of interest? You got to have some like basket of consumption goods to say if you make a loan over time, how much you know how how many units of that basket of consumption goods can you buy over time? That's to give operational meaning to the real rate of interest. That's an obvious candidate. Okay. And so like if we want to say it's bushels of wheat and the tractor makes you know contributes to them the point is what Mavver showed the issue is not having you know adding more tractors to my farming operation does that boost how many bushels of wheat I harvest each year. The issue is what is the initial purchase price of the tractors measured in bushels of wheat and how does that relate to the increment? That's what would explain the real rate of interest. If we're picking bushels of wheat as like the numer now just flip and explain what I did in my appendix. Oh, and I forgot to say the reason I realized I need to do this episode is because I said to Alberto, um, you know what? I'll I'll try to send you something like to, you know, where I I spelled all this out and I I said to the, you know, the listeners of that podcast when I did the the intro and the outro and I said, I'll I'll go find something for those of you who want to go, you know, read up on this. And I realized I this isn't written anywhere in an accessible form. It's in the appendex to my dissertation. I had done a standalone article for the a history of economic thought journal. But when I got to this, you know, part that was my my personal favorite where I had this equation that shows the general expression, but then if you plug in that there's one good, it reduces to what the, you know, mainstream guys were saying. The the a referee said, "No, this is too much. Just forget You know, I I had done earlier stuff about Bomba and then this was like the, you know, PA songs and he and the referee said, "No, you got too much here. Cut this part." And so I did, you know, get the thing published. Got to do what you got to do. But the point is I realized, oh, I really don't have this written anywhere except in the my at the end of this, you know, big dissertation. Then no one's going to slog through that or some of you do, but I can't expect people to do that. So that's why I'm partly doing here is I want to have it in one accessible place for people to see what I'm talking about. And again, this is partly for Austrian grad students. So in case you're losing your mind like I was trying to reconcile two things that seem contradictory and yet you can't point out the specific fallacies. So there's that element. Also, if you are talking to a mainstream colleague and they say, "Well, you know, this verbal logic, I don't even you're walking around in a circle. You can give them this and show them what you're talking about." Like, I think I made it crystal clear. Okay, so um let's just work with a hypothetical example. Suppose we're talking about a net, right? somebody Robinson Crusoe on his island, he builds a net that allows him to catch more fish per unit of time than he could with his bare hands. All right? And let's just suppose that the net augments his productivity by 100 fish per unit of time, right? It could be days, months, whatever, you know, whatever. Okay? And suppose that the net is such that physically it's good for 10 uses but then after you use it the 10th time it just frays and you got to throw it out. Okay. So there is physical depreciation in that sense. It it it's equally serviceable for the first 10 times you use it. It gives you 100 extra fish than what you would have gotten with your bare hands. But then after you use it the 10th time it tears and Crusoe just throws it out and it's it's useless. Okay. So now the question is you know what if we're going to model this in a mainstream fashion and try to talk about uh the real interest rate how do we do it? So let's just assume that the consumption good in this world are the fish and that the only type of capital good is this net. Okay so there's only two goods in his labor. All right if you want to call that a good you can but he can't consume that or anything. Okay. And so let me now show you you can go ahead and write out a bunch of equations and things, but the the money line is we flashed up on the screen. This was the equation for that I end up deriving. And so the I there stands for the real interest rate. You might have expected it to be an R, but I didn't do that because earlier I had used that to be the rental rate of the capital. And again, that's part of the problem in this literature is people use R to mean interest rate, but also it could mean rental rate. In some models, it's it's better like they do distinguish these things, but I'm saying in these workhorse introductory models where there's just one type of capital good and it's the same thing as the consumption good, they tend to be, you know, they tend to use R and that it's it embeds the confusion. So here I called it I but it's this is the real rate of interest. Okay. So I'm saying it at time t the real interest rate is the derivative of the production function with respect to that first argument. That's what the f1 means. And that k is the capital input at time t. And then l is how much labor goes in at time t. And then but that's not it. That's in the numerator. And then you subtract. And then what that pi stands for, the pi is saying that's the price of the capital good measured in consumption good. The first term there is at time t. And then you subtract away its value, its price at time t plus one. And then you divide all of that by the original purchase price of the capital good. Okay? Now don't be intimidated here. Let me just give you a simple example. And so let's say that let me just do the example. Let me let me just tell you let's make it so that the interest rate the real interest rate is 5%. Okay. So that means in fish if you if Crusoe has 100 fish and Friday comes along and Crusoe says hey Friday here I'll lend you 100 fish in equilibrium Friday has to give him 105 fish next period. Okay let's just say that that's what we want the answer to be. what ends up happening here given the physical the brute physical facts that I've outlined in this story right with the net and how it works. So if effectively when you're want like what's the what's the initial purchase price of that net let's assume the way that it works is you use the net over the course of a period of time and then at the end of that period you get that yield of the hundred extra fish. Right? All right. So, that's the way we're going to handle the the time element here. So, we're going to say like you you buy the net at time zero. And that what that's doing is entitling you, you know, as as t turns into one, you all of a sudden have that extra hundred fish now that you would have gotten over and above what you could catch with your bare hands. And um like so if you could catch eight fish and in the time interval with your bare hands, with the net, you can catch 108 fish. Okay? So, the net is adding 100. It's the net, no, no pun intended, it's the net yield physically of that um during that time period, the flow, so it's over and above what you get with your bare hands. Okay? And then that happens for 10 periods in a row and then the net is frayed and useless and you throw it out. So there's a sense in which it's like the net is a bond that just pays a 100 fish starting at the end of the first period and then at the end of the second D for 10 periods in a row. And so if we're going to discount that flow of future fish at a 5% discount rate, the present value of that I'm rounding is like 772. Okay. So that's the initial purchase price. To buy that net right now, his brand new net, Crusoe would have to hand over 772 fish that, you know, he could have accumulated over. He's catching fish with his bare hands, salting them and saving them, building up this big stockpile. And then, you know, some entrepreneur is going out and getting vines and making a net and selling them. And how much would Crusoe value that? Well, in equilibrium, you know, its price is going to be 772 fish, the spot price of that net when it's brand new. Now, how does this work? So, he, you know, Crusoe goes and gets a, you know, harvests 100 extra fish compared to what he would have gotten with his bare hands. And so in that expression there that equation four that first term is 100 right it's saying what is the increment that f1 means you just taking the the derivative with respect to the first argument so because you put in one net output went up 100 fish okay so that's what that part is but then you're subtracting the change in the price of the capital good over the course of that period so to to fill that in we need to know, oh, during the course of the period, the market value of that net goes down, right? That when I when I buy the net, it's brand new, I use it once. Now, at the start of the next period, that net is no longer brand new at that point. Now, it only is going to entitle me to nine future bursts of 100 fish. And so if you then just view that as a bond that's going to give you a 100 fish, you know, starting at the end of this period and then for nine total periods, discounting it at a 5% rate, I'm rounding. It's about a 711 fish. Okay? So in this equation four, you would say, oh, the top left one is 100 fish minus and then in parenthesis there, it would be 772 minus 711, right? because the original purchase price of the capital good was 772, it drops to 711. Okay? And so if you go ahead and do that, you end up with 39 is the numerator, right? Because it dropped by 61, right? So the market price of the net dropped by 61 fish. So the yield, the net yield, no pun intended, of owning that capital good for that first period is not the full 100 fish because that's partially offset by the drop in the market value of your capital good. So the 100 fish is what you got. Like if you had just rented out the net, you could have rented it out for 100 fish. But then the value of the net itself, the market value measured in terms of fish dropped by 69 or sorry 61. And so the net yield for that period was only the 39 fish, right? And they say, "Oh, so the so the interest rate's 39. What? 39% 39 what? 100%." know, you got to divide by the original purchase price, right? Because it's that the top thing is 39 fish. And so if the interest rate is a percentage, it's a rate, you need to divide by some number of fish, right? To get the units of fish to cancel out because interest rates aren't fish. You know, you say, "Oh man, the Fed cut interest rates today down to two fish." What the heck are you talking about? Right? Because I'm saying like a lot of the discourse over this, the units aren't even right. The dimensions aren't right. Okay? So if the top of the of this numerator or if the numerator in this expression is 39 fish then you divide by that original purchase price which in this example we already said is 772 fish and so then oh 39 fish divided by 772 fish equals approximately 5%. It's like 5.05 but that's because I would been rounding. If you if I gave you the exact numbers it would have been exactly 5%. Okay so that's how that works in an example like this. So you can I'm just showing you even though that equation that I flashed up there might initially have been intimidating say I'm not going to get it. I'm saying when you see what it's saying it's it's not hard and it makes intuitive sense. It's saying yes of course the extra physical yield in the consumption good that the capital good provides you matters but there's other things going on. It's more complicated if the capital good is a distinct thing and hence you have to worry about its value. measured in terms of that consumption good. So again, this is Bombick's critique of the naive productivity theory and I'm putting it here in mathematical form in a you know good little neocclassical model with all the eyes dotted and te's crossed. Okay. Now let me just show you something before I move on to the punchline. Let me show you why the physical productivity of capital is not sufficient for a positive interest rate. I can use the same framework and it's consistent with the real interest rate measured in fish being 0%. And what would that look like? You say, "Oh, if the nets couldn't harvest more fish, well, yeah, that would work. But you know what? It doesn't you don't need to invoke that. Even if we still stipulate that physically speaking, a guy can go make a net and sell it to Crusoe and then Crusoe can get a 100 extra fish period for 10 periods and then the net is useless physically. It, you know, wears away and he just discards it. That is consistent with a world on this island, an economy on this island where the real interest rate measured in fish is 0%. And how could that be? Well, when Crusoe goes to originally buy the net, its market price is 1,000 fish. And so, he buys it. Okay. He uses it. He gets a 100 fish by the end of that period, over and above what he would have got with his bare hands. But then what is the market price now of the net going into the next period? Oh, it's dropped down to 900. And then in the next period, it's 800 and 700 and so on. Okay? And such by the end of it, he's just recovered his fish back. And so for each period to use that equation he would say oh in the numerator you've got the 100 fish that he gets you know due to the productivity of the net physically but then you subtract the change in the market price from one period to the next which is also 100. And so that numerator is zero in the first period it's you know 0 divided by a th00and. So the the net interest rate is 0% even though the net is still just as physically productive entitles you to an extra flow of real fish as it was before. And again you see that the trick if you will or how does that what's going on it's that I can just just determine I can tell you I can make that work if the rate of interest is 5% I can make it worth the rate of interest 0%. And so what's pinning down the valuation of the net is knowing what is the intertemporal exchange rate of fish that if I know that oh yeah people prefer present fish to future fish then there's going to be a positive discount rate measured in fish and that will make the original purchase price of the net measured in fish lower than what the you know a thousand and that's why originally In that first scenario when Crusoe puts in 772 fish to buy that net and we we determined after that first period he had earned a net return of 39 fish. Okay. So like that's his free and clear. In other words, he still got his 772 fish of his original capital intact and he could eat that extra 39 and not impair his capital. That's another way of putting it, right? That he's he starts out with the 772 fish that he needs to buy the net and then in that first period he would be allowed to eat 39. He couldn't eat 100, by the way. If he ate the full 100 extra that that net yielded, he would be implicitly eating some of his capital. He'd be consuming capital. But he can eat 39 and then he still has the 772 like rolling over such that at the very end when he does it and that net wears away, he's still got 772 fish that he can roll over and then buy a brand new net. and he can just keep doing that and just gets that constant flow. Okay. And so again, it it would be tempting to say, geez, if he just scrims and saves and in the beginning when he doesn't have a net and has no savings and just lives below his means, you know, he's he's catching eight fish period with his bare hands, but let's say he only consumes seven and every period he socks away a fish and just lets that accumulate. And then eventually he gets up to 772. He's got to have good storage. I don't know how he's doing that in the island, but go with me on this one, folks. And then once he gets up to seven to 72, he can go buy the net. And now forever on, he can get uh what? 39 plus 8. 47 fish period that he can consume forever and maintain that 772 of his capital intact. that he's getting the eight implicitly from his labor and the 39 is like interest earnings on his saved capital would be the way you would decompose it like as an accountant or a macroeconomist looking at what are the returns to the factors in this economy. Okay. So it would be tempting to say geez how's he getting at the flow of 39 and like there there is a sense in which yes the physical productivity of the net allows for that but you can see how when you want to understand why is the interest rate 5% and not 10 or not two or not zero clearly the physical productivity of the net is not the explanation because I can make the interest rate any of those numbers I just said and be consistent with the net having the same physical productivity that we stipulated. Okay, now finally the great reveal. Put that equation back up. What if we don't have an economy where the net and the fish are different physical things and hence you have to have a price of the net measured and quoted in fish which allows for all these complications. What if instead we're back in Fischer's world where it's just sheep and that's it? So in that world, we can still apply this, you know, this equation is still true. It's just, oh, here the capital good and the consumption good are the same physical thing. Instead of it being a net and a and fish, it's just sheep. Instead of being a net gives you more fish, it's sheep give you more sheep. And so, oh, at any given time, what is the market price of a sheep measured in sheep? Oh, it's one. Not because of any deep economic insight, just because it's the same thing. So, of course, one sheep has a value of one sheep. Duh. As opposed to a brand new net, you know, being 772 or being a thousand or whatever. Okay. And so in this equation, what happens if pi of t is always one for all t? Well, the denominator is just a one. So that goes away. And then pi of t is one. And pi t + 1 is one. And so that top right, the pi t minus p t + 1, that's one minus one. That's zero. And that goes away. Oh, and so that equation four reduces to if the capital good and the consumption good are the same physical thing so that the price of each is just one all the time. Oh, look at that. That more complicated expression reduces down to the interest rate, the real rate of interest equals the marginal product of capital, right? The derivative of the production function with respect to that first argument, which is K. Okay, so that's what I'm saying. I think a lot of mainstream economists are walking around believing that interest income is due to the physical productivity of capital in the same way that labor income w you know wage income is due to the physical productivity of labor and as the Austrians have been saying since bomb no that's a fundamental confusion that's not what's going on it's way more complicated than that. The productivity of physical capital goods explains their rental rate. It does not explain the ratio of their unit rental rate to their spot purchase price, which is kind of like what the at least uh gross rate of interest is. Okay, I will wrap up there. I'll put a link if you really are an overachiever and want to check out the dissertation. I'll put a link in the show notes page. Um, I'll also link to my other conversation with Alberto Bazine, which is, you know, was very interesting in many other respects, but I did want to flesh this particular thing out. And again, if you're an Austrian in a grad program somewhere and your head is swimming, I hope now I've shown you what's going on. Thanks for your attention, everybody. See you next time. Check back next week for a new episode of the Human Action podcast. In the meantime, you can find more content like this on mises.org. Heat.