# Goodfriend Memorial Lecture: Knowledge Creation and Diffusion

**Economic Brief**

Introduced in 2023, the Goodfriend Memorial Lecture series honors the legacy of Marvin Goodfriend, long-time Richmond Fed economist, research director and senior policy advisor. The lecture was delivered as part of the Richmond Fed's Collaboration of Research Economists (CORE) Week model, which brings together Richmond Fed economists and visiting economists from a range of disciplines for seminars, conferences, networking and collaboration.

*On May 11, 2023, Hugo Hopenhayn delivered the inaugural Marvin Goodfriend Memorial Lecture with a presentation of his paper "Knowledge Creation and Diffusion with Limited Appropriation," which was co-authored with Liyan Shi. The paper highlights how intellectual property rights must trade off incentives for innovation and knowledge diffusion and considers the optimal assignment of property rights as a Ramsey problem in a dynamic model where knowledge diffusion takes place under random matching. Hopenhayn is a professor of economics at UCLA, a Guggenheim Fellow and a long-term Richmond Fed research consultant.*

*The following is a lightly edited transcript of the lecture.*

This is the Goodfriend Memorial Lecture.

Many of you probably do not know Marvin Goodfriend, so I wanted to give a couple of words about him. In order to come up with this these words, I did use ChatGPT. I put in Marvin Goodfriend and knowledge creation and diffusion to see what comment this might have. I got the following: Marvin Goodfriend is an economist who has made significant contributions to the field of monetary economics, particularly in the areas of macroeconomic policy and central banking. He has conducted extensive research on topics such as the role of monetary policy in stabilizing the economy, the effectiveness of unconventional monetary policies, and the design of central bank operating procedures. Then, in terms of knowledge creation, Goodfriend has contributed to the development of new economic models and theories through his research, which has been published in a number of academic journals. He has also co-authored several books, etc. In terms of knowledge diffusion, Goodfriend has been an influential figure in the economics community. He has taught at several universities, including Carnegie-Mellon, and has mentored numerous students. He has also served as a visiting scholar at central banks around the world, including the Federal Reserve Bank of Richmond, European Central Bank and others. So that's my summary of the connection to knowledge creation and diffusion. I couldn't find any papers.

So with that said, this is a paper joint with Liyan Shi that some of you might have seen; she came to a couple of these CORE Weeks. It's a step back in some ways relative to the previous paper that sort of examined channels of diffusion through knowledge spillovers across people and tried to measure those, in the sense that I'm going back to the sort of models where knowledge blowers are modeled a little bit more in abstract without really relating to the links. But I'm going to try to bring in a consideration that has not been brought into those models, which is the idea of, we do have some intellectual protection systems. And so trying to see whether through the lens of these models, we can say something about the design of these intellectual protections and its implications for knowledge diffusion and innovation.

So that question is, how do intellectual property rights affect growth? That's going to be one of the questions that I'm going to answer within the context of these models. And second, how strong should they be? As is usual, when you write a model, the model has a lot of specifics you model in a certain way. And at the end, you might come up with answers to the questions that might be more or less convincing, but along the way you learn things. And I think that's the stage that we're in right now. I'm going to convey a little bit more about what we learned rather than put my finger on the results that we have right now.

Basically, we're going to be looking at a model where there's going to be firms that are going to be innovating, firms that are going to be trying to learn from other firms, and these are going to be exclusive activities. And then we're going to follow the literature in that sense. What we're going to be looking at is how these property rights might affect both the incentives to innovate and the incentives to sort of copy or learn from others.

And so there's two margins. On the one margin, if you consider innovation versus imitation, free riding. Another's innovation reduces the incentive to innovate. So if I can free ride on others, why should I innovate? And the other is when learning has a cost. So if it's costly to go and learn, there might be a holdup in teaching somebody or having this knowledge transmission might entail a very high tax to imitators or learners. So the model will have a little bit of those two features.

I'm going to take a standard sort of model of diffusion. I'm going to build on Perla and Tonetti's model. Basically what we're going to add is two things. Those models, when you learn you basically just copy what others are doing. There's no transfer; that surplus that is generated for you, it's kept fully by the person that is learning. The innovator does not appropriate any part of the surplus. One of the things that we're going to do is we're going to introduce a sort of Nash bargaining, or somehow a notion of patent strength, that will affect the surplus division.

And so when I come in and want to learn something from Grey [Gordon], he can say "No way, I'm not going to teach you." He can basically allow me not to learn or not to participate, though that leads to a bargaining problem. And so the kind of question that we're looking at is, how does the split of the surplus in that bargaining problem affect these two margins, the margin of diffusion and the margin of innovation.

The second part that we're going to add to the literature is that we're going to have congestion in knowledge diffusion. I'm going to step back right now and explain a little bit better later why we want congestion and why it plays a key role in this setting.

So we're going to look at two things: I'm going to present you the model, and then we're going to ask two questions of the model. One is long-run growth, so what are implications of different strengths if you want protection, intellectual protection, on longer growth. And the second one will be steps that we have done in solving for a Ramsey problem, which was sort of the optimal property rights in a calibrated model.

So there's a big literature that we build on. Perla and Tonetti is one of the key papers that we're building on. Boyan [Jovanovic] and Zhu [Wang] have a paper that is sort of related also to ours, and they're looking at the role of licensing in a repeated innovation and diffusion model. And Ufuk [Akcigit] has done papers thinking about how knowledge is diffused in the economy through the sales of patents and so on. And then I'm going to rely on this paper by [Daniel] Garcia-Macia, [Chang-Tai] Hsieh, and [Peter] Klenow, which is a paper that sort of tried to decompose the different sources of growth, and I'm going to use that for the purpose of calibration. And there are other papers that I didn't mention, but they're really important too.

So we're going to start with an illustrative model. So a very simple model, like a one-period model, if you want, not even two periods, to illustrate the mechanisms that are behind it. Then I'm going to the dynamic model with sort of this endogenous growth framework, Perla and Tonetti. Talk about calibration, long-run growth, and the optimal policies.

Here's the illustrative model. So basically, think about this as an endowment economy, there's a unit mass of firms that are endowed with this, call it productivity *z*, distributed with this distribution *F*. And they can improve the productive capacity through innovation or imitation. So firms can either choose to be an innovator or what we're going to call an imitator. So if they decide to be in innovation, firms will decide whether to innovate and how much to innovate. Innovators and then imitators later. So we're going to have two stages, even in one period. First, the innovators will decide how to innovate. And then the second stage will be a matching between those who did not innovate, the imitators if you want to call it, and the innovators, in which the learning process will be mediated.

So innovation will create basically *µ*, it's going to be the parameter that defines the level of innovation, the cost of that innovation. We're going to scale everything up by the initial knowledge that the firm comes with it. So that at the end of the day, we're going to get things being invariant to this productivity, all firms are going to choose the same level of innovation. So think about here a fraction α will innovate in the equilibrium. This fraction α obviously will be selected in the sense that the firms that have higher values of productivity, higher *z*, will be innovating and the other ones will not.

So diffusion takes place in the following way. There's a matching function. The matching function is going to be a standard constant returns-to-scale matching function. For example, this could be a Cobb-Douglas matching function where α is the fraction of innovators, right? And 1 − α is the remaining fraction, which are the ones that are imitating. So there's congestion in this matching process, and I'll describe a little bit later why it is sort of critical for the questions that we're asking. So λ(α) is going to be the contact rate for innovators. That's going to play an important role in calculation. ι(α) is going to be that for imitators; ι for imitators. We didn't have more than one I, so we kept λ for innovators.

So the surplus to be divided between the firms is basically — the assumption here is that the imitator gives up on the possibility of producing unless he learns from somebody. So the surplus for the imitator is basically he learns completely and is able to replicate what the innovator does. And so the flow of payoffs that the imitator will get is *μ *times *z*. And that's going to be the total surplus, because otherwise to get zero in the innovator bargaining weight will be β, which is the fraction of that surplus that the innovator would get in this Nash bargaining arrangement.

So let me do a short deviation because this is going to be important later on. I'm going to give you a very simple micro foundation for that bargaining weight and link it to what you might think of the strength of IP protection. So here's this very simple enforcement game that occurs once the two meet. Let's say the innovator will make a take-or-leave-it offer of a license that the imitator needs to pay in order to be able to do it legally. The imitator either rejects to do that and just copies and takes a risk. And with probability β, it's going to be caught and basically not let produce. So β here now would represent the probability of this sort of patent being enforced and the innovator being kicked out if he didn't pay the license. And if he's caught, cannot use the knowledge and product output is zero. I'm going to make the story short; I'm not going to get you through this. What this implies is a transfer which is essentially β times *s*. And so this justifies this parameter β here translates into the Nash bargaining power of the innovator. And 1 − β is what the imitator gets to keep.

Alright, so what's the innovator's problem now? The innovator will basically get an output *μ* times *z*, that will be the output *μ* times *z*, it gets that for itself for sure. And with probability λ(α), which is the contact rate for innovators, will meet an imitator and will get β of that surplus, which is *μ* times *z* also. What is the cost of innovation will be *c*(*μ*), also scaled by *z*. So this gives us the total innovator's surplus and so the problem, the innovator's maximizing *μ* as a simple first order condition, you know, it's kind of marginal cost is equal to marginal return.

And then we have the other problem defining which is the indifference firm, the ones that are above *Z* will be innovators, the ones below are going to be imitators. For the innovators, the surplus is the one that I just gave you, which is the total surplus that they get from both using their own knowledge and also licensing it. On the side of imitators, their contact rate, that is the probability of meeting an innovator, is going to be ι(α). And in that case, they get the 1 − β of a surplus. And since it's a random meeting, random matching, the expected surplus is the [inaudible] of all *z*'s that are above this threshold. Okay, so that's basically the equilibrium. And α here is going to be the share of firms that decide to innovate to begin with. That's all the model, that's all the story. I was going to say that I liked your remark saying, I have a model that fits in one page. I wanted to say, I have a model that fits in all the pages, so you're going to see equations throughout.

Alright, so what's the planner's problem here? So what the planner will do is the planner will choose — I'm going to write it this way — there's one parameter only the planner controls, which is β, it's the strength of this patent protection. We're going to write it this way indirectly through the constraints, α and *μ* are going to be tied to β. But it's just more convenient to think of it this way. And so there's the output from the incumbents on the one side. There is then the output, the surplus for matching which is just the total number of matches that take place multiplied by the expected return of these matches, the expected surplus of each of those matches. So the planner maximizes this, subject to conditions 1 and 2, and implementing this allocation with a choice of a particular β. So the alphas and mus are not independent. Because at the end, they're implemented only by one policy, which is β.

Alright, so this is a key slide. I just rewrote this here. The first-order conditions from this problem — and I don't want you to get to think through the first-order conditions — but the first-order conditions have three components. The first component, which I call congestion, is a component that appears in all matching models, in all search random matching models. I'm going to call it the Hosios term for those of you that are familiar with that. And basically, if that were the only term which is it captures the congestion. Basically, there's an externality, if I am an innovator, part of the returns I'm getting is displacing other innovators, and I'm giving a positive return to more imitators. And so there are these externalities are balanced out in the Hosios condition, when β is equal to ω. ω was the elasticity of the matching function with respect to innovators. So basically, this will tell you if there were no other effects you would want, you know, the Hosios condition to hold. I will get into that later in our quantitative section.

The second effect is what we call the displacement externality, which is basically the following idea. If I'm a marginal innovator and I am there, right? If I join the set of innovators, I'm thinking, you know, do I join or not, I join a set of innovators. I am, in some ways, making the distribution of potential matches worse for the imitators. And so what am I doing is essentially I'm displacing to some extent, with me being there, I'm displacing an average innovator. So the calculation of this distortion is the mean versus the marginal. And this is really important, right? Instead of me meeting a randomly chosen innovator, you're going to meet me that I'm the marginal one. And so that's going to provide a loss for the point of imitators and a social loss, which will be measured by mean versus margin. Okay, so, of course this is going to be related in some ways to the dispersion of the distribution. And that's going to play an important role in determining this.

And the last one is that by changing the share of innovators, that's going to — indirectly, because that's going to imply that you need to change the β in order to induce that — that's going to have an impact on innovation. So that's going to have an effect on innovation. And this is what we call the innovation spillover. So the effect of putting more people on one side, it will cost you potentially or give you a benefit of more or less innovation. And that's there. Don't worry about this term yet. I mean, it's essentially, in some ways, part of that is internalized when a firm decides to innovate, it's doing it to maximize the returns it gets. So the real externality is not what that firm gets, but what the other party will get, which is, in this case, what the innovators that face that firm will get.

**Audience Member**: Hugo, can I ask a quick question? So when I tried to imitate and I get caught, or how was it happening? So if I get caught, I pay punishment or I'm just not allowed to copy? What was the β?

**Hugo Hopenhayn**: You're out. You're out. Later on, in the dynamic model, you're not going to be allowed to use that idea. And so you can sort of keep searching for ideas.

**Audience Member**: You will just prevent me from using it, there's no additional cost?

**Hugo Hopenhayn**: There's no additional cost. There could be. I agree. I mean, there could be. I'm looking at a very limited set of instruments here. I mean, I have only one instrument and in some ways, I want to implement more than one thing, because I would want to implement innovation levels, and I want to implement the right fraction of innovators and imitators. That's a game we're playing. We're playing, you know, this is not about optimal policies unrestricted but it's in terms of this particular instrument, which we call patent policy and enforcement policy and thinking about what's the optimal in that.

**Audience Member**: The imitators have no productive value, right? They only act to steal, but you still want them around. Why?

**Hugo Hopenhayn**: They're valuable. They're valuable because they contribute to learning. They will contribute eventually once they learn. Once they learn, they have productive value.

**Audience Member**: Why is a planner not choosing *μ*, set by set? So I arranged a productive guy, I can be assigning higher *μ*, innovate more for the —

**Hugo Hopenhayn**: Because the planner has, as I just said, only one instrument, which is the patent strength. That's it. Not selectively different patent strengths for different firms.

**Audience Member**: Would the implication be different if my monopoly right over my idea expires with some probability?

**Hugo Hopenhayn**: Yeah, that could be. That could be easily introduced. That's going to reduce the claim over value that the innovator will have, the claim over its innovation value. So, it might to some extent be — I think in some ways, it's going to be similar to the effect of changing β. I mean, if you think about β representing some of the expected discounted time, the duration the patent will go. This is just from the back of my head, don't put a lot of weight on that.

Okay, so here are some functional forms. Let's say you have a Pareto distribution for this productivity matching function, which we're going to use also in the dynamic model. You know the Cobb-Douglas ω, here's the elasticity, that relevant electricity. And then a cost function, which is, I think we've seen this before, like homogeneous cost function, where basically the term ε dictates the elasticity of the cost with respect to *μ*. It's kind of an important parameter there. This is just an example, but later I'm going to use this in the quantitative model.

This Pareto parameter goes from high to low here. Twenty means essentially all the firms are the same, so there's no heterogeneity. Even if there's no heterogeneity, a fraction of the firms will innovate, a fraction will imitate and that's efficient. You don't want everybody to innovate, because there is this alternative of being able to learn from others too. And so as we go to one, one would be more dispersed. And so if you recall from what we said before, the mean versus marginal is going to change as we go in that direction. So you can see here, the mean versus marginal goes from one to 11 in these examples. So the Hosios condition, because we're going to assume ω is equal to 0.5, so this elasticity would say β should be equal to 0.5. If we go back here and take these two conditions and set those equal to zero, so the Hosios and the displacement, that would lower β to 0.28. So you want to discourage crappy innovators from being there, right? So it's a selection effect. And then, once you bring in the spillover effect, and you put the three effects together to induce more innovation, then you increase a little bit β. So you want property rights to be a little bit stronger to induce more innovation.

This is a result that we're going to see consistently when we look at the planner's problem and the Ramsey problem. In this particular example, if you were looking only at spillover, you want to maximize the innovation rate. It actually happens that it's the same as the Hosios condition would be, 0.5. In the absence of heterogeneity, it's 0.5. And that, you know, magically coincide there.

Alright, so, let me now go to the dynamic model. So I lay out the basics, the basic ingredients. We have this sort of innovation, we have learning from others through matching. Oh, I should have said, I'm going to say this in words, because I should have said this — why do we care about congestion? And why do we have congestion in this exercise? A lot of the literature doesn't have congestion. Basically, if I want to learn, I'm not affected by how many people are out there to teach me. So basically, the probability of meeting someone in all these diffusion models, it's exogenous, it's independent of how many people are on the other side. Well, that would make the result very trivial. What you want is just the most productive firm to be the only one to innovate and all the rest to learn from that one. Right. So that trivializes a problem, so we think congestion is a reasonable thing and that's why we brought it in the picture.

The general model borrows a lot of features of some of the models that you've seen before. Think about this as a monopolistic, this is an aggregator, you can think of a monopolistic competition setting in the log utility intertemporally. And then, in terms of production, there were going to be what we call active firms and inactive. The active firms are going to be the ones that are doing two things at the same time, they're going to be producing and innovating. The inactive firms, think about those of entrants. They're out of the industry; they're just looking for ideas. And once they find one, they're going to enter and produce and also innovate as they were there.

So the production function for the incumbent firms is linear, again, *L* is labor. Then this is just the way we're phrasing this is think about this as being a manager, c(*μ*) is the time the manager devotes to innovating, the rest of the time devotes to producing. So one minus that is the time allocated to production in that production will be proportional to that human capital input, the managerial input. So basically the costs, this is convenient, are scaled to be in terms of your final good; they're proportionate also in that way to your productivity. Productivity is *z*, we're going to assume *z* is log normal. That's we have* e ^{z}* in here.

Alright, so the inactive firms try to find ideas, α(*t*) again denotes the mass of active firms, and we're going to assume there's a mass one of firms. And so α(*t*) is the rate of those firms, the ones that are producing and innovating; 1 − α(*t*) are the set of potential entrants that are looking for ideas.

**Audience Member**: Maybe I should have asked this before; it is related to this α. Is there going to be a threshold, like a threshold for who innovates? And then in the previous table that you showed for the other model, was that there too?

**Hugo Hopenhayn**: It was identical in that sense. α was the fraction of the firms above that *Z*.

**Audience Member**: So, implicit in that α, there is a *z* threshold —

**Hugo Hopenhayn**: Implicit in the α is exactly — well, it is a *z* conditional on the existing distribution of productivities, which evolves over time. So it's endogenous, in this dynamic model will be endogenous.

So in terms of the productivity process, it's going to be a Brownian motion.* **μ* is the drift of the Brownian motion, so that what firms are choosing are the drift of the Brownian motion. And there's going to be volatility σ. By the way, this is related to work that Sam [Kortum] has done in terms of when I talk about diffusion and early work from Sam [Kortum]. Volatility will play an important role, because it's kind of going to tell you something about randomness, and it's going to be very relevant in determining the overall dispersion in the distribution of existing firms' productivity. So we're going to see that that is going to play an important role here.

When the imitator meets an innovator, it draws a random quality. We're going to allow for imperfect imitation, right? So it's going to run a random quality and actually can be imperfect, up or down. Our calibration will tend to move it down in the average of an imitator will be like 30 percent of the existing innovator. We're going to introduce something that you know as creative destruction. Since I knew Ufuk was going to be here, I had to have creative destruction, right? But we're going to put it under the rug a little bit in the following way. So when an imitator learns something, we can define two possible states. It could be a completely new product that is not competing with the product that was offered by the existing firm that they learned from. Or with probability δ, if it's better than the product of the existing firm — remember, because there is this incomplete learning potential — if it's better, then it's going to displace that firm. Okay, so that's kind of our way of putting in creative destruction.

**Audience Member**: Yeah. So I have a question. So here imitators, or inactive firms, are learning from innovators. Usually we think about the spillovers of innovation between innovators. What about spillover from innovation, which would affect in a way that logic, which I really like, this fact that from marginal versus average? Then you may think that now this marginal guy's entering and really approaches the problem's solution in a very different way. So there is a value for other innovators in learning something.

**Hugo Hopenhayn**: Yeah, I mean, we have a one-dimensional thing here, which is just the level of knowledge. Following those models, we assume that learning something that you already know, it's not useless. It's useless for a firm. This is a little bit literature driven, I should say, in that in Perla and Tonetti there were these, in some ways the cost of being an imitator is that you give up the possibilities of producing and innovating. I think if I would have called this exit, nobody would have had a question about that. There are a lot of these models, [inaudible] has a model where firms, when they enter, their initial shock is kind of a random draw from the distribution of. Yes, I mean, I agree there is that margin. It's important; it's not here.

Alright, so now when will the creative destruction take place? It will take place not only when, you know, this is a potential creative destruction, it has to be sufficiently good to be better than the previous one. That, remember, will occur with this thing we call delta hat. And remember, delta hat is the probability then that when a match between an innovator and an imitator takes place, that imitator will creatively destroy. Rather than add a new product with the value that it gets, it's going to add that new product but at the same time it will displace the existing one.

So the reason we have this is also to have a close mapping of what we're going to use for calibration, which is this paper by Garcia-Macia, Hsieh, and Klenow, where they try to sort of — we might question their identification criteria, we're using this for now; we don't have another, maybe you could suggest a better alternative, but we're using this — that will separate the sources of growth into the growth that is from innovators, from innovation of incumbents, or from incumbents, and the growth from entrants that could come from two parts, from creative destruction or new product variety. So basically, you know, that's the idea.

Okay, so I think I'm going to have to skip a little bit these value functions. But the value functions sort of have a key term as an analogue to what we had before in terms of the surplus created. This is a surplus of a match, which is basically conditional on the match being successful in the sense that you get a draw that is worthwhile. It's the value of the draw for the new firm minus *W* here represents the outside value that it has, so *W* is the value you have if you continue being an imitator, you're outside, you're not producing. The net gain is the surplus that is generated by this match. So, there is going to be a split with β of this net gain with a game I told you before will be obtained by the incumbent firm; 1 − β will be retained by the imitator, that would be the licensing. But on top of that, when you look at the value function, so if you think about the value function of the innovator, it has all the terms that you would expect here from purely the profit flow of the firm. And then the terms that are determined by the random Brownian motion. Then the new terms are: upon meeting in a match, it gets β of that surplus that we explained before, but with probably delta hat, it gets destroyed. And so it loses the value it has now, it's thrown back to the pool of imitators and it's out of the industry and gets *W(t).*

**Audience Member**: Could you think of intellectual property protection as also something that allows the firms to not get creatively destroyed? Maybe I don't want to share; if my β is high, I choose not to share? I don't care how much of the surplus you give me.

**Hugo Hopenhayn**: So given our choices, given this enforcement problem, if you'd interpreted this enforcement game, you would never want not to be kicked out. Because the transfer that you receive is actually — no, I'm sorry, you would always accept the license as opposed to leaving the person with the risk of kicking you out. And if you could have different, that policies would be the strength of bad — maybe this is what you're saying. This strength of protection being more directed toward being — yeah, I mean, that would make sense. I mean, a lot of the way this works is, you prosecute, you go to the prosecuting, and maybe the incentives for prosecution are bigger if you are being destroyed than not, and so enforcement could be bigger then. We don't have that here. We didn't have this δ before, but we thought it would be better to match the data with creative destruction.

So we're going to be looking at a balanced growth path. So this slide is explaining the different sources of growth, and this is what's going to allow us to go in and match and sort of calibrate the model to the data. So the first source is own innovation. Obviously, there is the drift *μ* of the first. There's also this unfortunate term for those of you familiar, "it enters," we don't kill the variants in the drift, and it appears sort of unfounded here. But this is own innovation.

Creative destruction, in some ways this reflects mean versus marginal, so the level of creative destruction. And then there's new varieties minus exit. Creative destruction basically generates a gain, which is the draw, the improvement over the existing product, that would be this draw *q*. So this would be the δ in the models of like [inaudible]. Minus one because you're basically displacing this firm. And this one doesn't have that minus one. This is a contribution of new varieties. This is a contribution of exit, the firms that are at the threshold exit. The relative importance of those firms depends on what that threshold is relative to the mean. So that's a negative effect on exit.

So this is going to generate an endogenous distribution in a steady state. The only thing I want to highlight is the tail of the distribution; it's going to have a Pareto tail, it's going to be affected by volatility, as I mentioned before. So higher volatility will lead to longer tails. Diffusion is a source of mean reversion. So when other firms learn, they're kind of catching up. And so that's mean reversion. Mean reversion acts in the opposite way and sort of tends to thin the Pareto tail. So when we see mean reversion here, more mean reversion will mean thinner tails; more volatility in innovation will mean thicker tails. That's all what I want to say there.

So calibration, I've kind of described the first two before. These are parametric forms we pick. *G* is going to be an exponential distribution. We calibrate the existing level of β, it's going to be 0.5. This reflects the fact that licensing, your royalty rates, tend to be about 10 percent of revenues. And profits are 20 percent of revenues. And so that makes it like 50 percent of profits. So that's what we use for calibrating these parameters. Own innovation is 1.15 percent. We still need to separate this between these two. The own variants, this is from again this paper by Garcia-Macia, Hsieh, and Klenow, is calibrated to σ = 0.11, and so on.

Let's see. Creative destruction. This again comes from the contribution of creative destruction. The incidence of creative destruction for entrants is 5.4 percent, that determines this term here. The average quality of creative destruction — again, this comes from the contribution to growth in this paper — is 1.26. So those give us this full term of creative destruction, that was a contribution of 0.46 percent growth. And finally, new varieties, again, taking a similar approach, looking at the entrants and new varieties circulating by entrants, in the average qualities give us a contribution — oh by the way, exit is almost no, quantitatively, it doesn't have any effect. The calibration is meant to match then. Again, this growth decomposition in Garcia-Macia, Hsieh, and Klenow, so GHK.

So here's some of the results. First, I'm going to talk to you about the results of growth and long-run growth and how it changes with β. And then I'm going to talk about the exercise of solving the Ramsay problem. So this is sort of total growth. And what you can see is that it increases with β. So long-run growth is increasing in β.

There are two effects. One is this is the own innovation of firms, that's increasing in β. And so the more property rights they have, the more innovation they get. And total growth is increasing in β. There's very little effect on diffusion. So it doesn't affect too negatively diffusion. You might think, and this is true in the extreme, if β is equal to one so that the imitators get nothing, you would have no imitators and no diffusion. And that's true. But there's a discontinuity at β equal to one. And what happens is that there's these forces, well, maybe I'm going to tell it later. I mean, there is a reason in the model. If you're interested in these nerdy things, as I am, I'll tell you what they are.

This is important. It traces a little bit the connects to what we saw in the first place, which is as β decreases, what we get is that the innovator contact rate decreases in the mean. As β increases, the mean-to-margin ratio increases. I'm not going to go through this slide, because this is part of that nerdy stuff.

Optimal policy. So what we're solving is the Ramsey problem. We're maximizing the present discounted value of the utility. By picking a process for β subject to, given those sequence of β, you are generating an equilibrium that evolves over time. And what we're characterizing in this is similar to what they do in an optimal taxation, when they're looking at limiting tax on capital, we're looking at the tail of that β distribution and it converges. And so the process converges to a balanced growth path and the limit. So that's what we're characterizing here. But even though we're characterizing here, that doesn't mean that we're putting all the weight on long-run growth. We're really thinking about how the planner wants to choose his β. And also thinking at that point, putting up a discounting future. So it's different from long-growth maximizing.

So the optimal β set has the three effects that we had before. It has that congestion effect, it has the displacement externality effect, and it has the growth effect. They're obviously more complicated through the lens of this model. But the three effects are there, the same three effects.

And here are the results. So this is the final slide. This is the calibration that we have. The optimal β is actually — the enforcement property is actually zero, so it's at the corner. The intuitive reason for this is we're trying to match the data on the distribution of firms' productivities, which is sort of the distribution of firm sizes. What do we know about that? That the tail of the distribution, it looks like a Pareto distribution, it's a very, very long tail. What does that mean? That mean to marginal is fairly high. And so mean to marginal is bad here. The planner wants to discourage the participation of very marginal firms, and so if mean marginal is very large, it will try to compress and have less firms participating. So that pushes the claims of the incumbent firms down to zero. And in fact, the optimal is let firms just get the returns to innovation that they get for their own sake, no returns from innovation that they get from transferring knowledge to others.

That's basically what we have, so let me just conclude with this slide. Is it ok if I go through a conclusion slide? You know, I thought, it's my conference, I have to present, I shouldn't have invited such good people. You make me look bad.

So consider this, you know, heterogeneity of growth through knowledge diffusion, incentives for imitation, and imitation innovation. I talked about the Hoisos condition as sort of a benchmark to think about the congestion. But in the model, the externalities that you recall, displacement externalities kind of play a really important role. In fact, you get this optimal β, optimal share of a surplus that innovators get, it's very little. Oh, by the way, I should say something that's important. In the Ramsey problem that we solve, we have not been able to solve the problem with endogenous innovation. So what we're solving is exogenous innovation. So it's basically dictated by, rather than all the effects, by the two first effects here. We expect that once we are able to solve with innovation, that's going to bring up a little bit the β but we're not there yet.

Hugo Hopenhayn is a professor of economics at UCLA, a Guggenheim Fellow and a long-term Richmond Fed research consultant.

*To cite this Economic Brief, please use the following format:* Hopenhayn, Hugo. (June 2023) "Goodfriend Memorial Lecture: Knowledge Creation and Diffusion with Limited Appropriation." *Federal Reserve Bank of Richmond Economic Brief*, No. 23-18.

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