Neoclassical Growth In An Interdependent World

>> Stephen Redding: Thanks very much for the invitation to present today. So this is a joint paper with Benny Kleinman in the Economics Department here, Ernest Liu and Motohiro Yogo, and it's called Neoclassical Growth in an Interdependent World. So what are we trying to do in this paper? Well, if you think about understanding country growth rates, one of the kind of standard canonical frameworks remains the so called closed economy neoclassical growth model, going back to Ramsey, Solow and Swan.

And obviously a key prediction of that model in the closed economy, so countries are in both trade and financial autarky in the simplest version of that model. The key prediction, as we all know, is that each country should converge to its own steady state level of income per capita.

The so called conditional convergence prediction. If you think about open economy versions of that model, they often make very strong assumptions about substitutability in goods and capital markets and or about frictions in goods and capital markets. So for example, in some open economy versions of that model, goods seem to be homogeneous across countries and traders assume to be perfectly costless.

Or as we all know, in the real world, trade costs are potentially important in shipping goods between countries. Similarly, if you think about open economy versions of that model, in capital markets, capital is assumed to be homogenous, which implies potentially very large and net capital flows to arbitrage away differences in rates of return.

And again, that doesn't seem to be a feature that we observe in the data. So what we're going to try to do in this model is take that standard canonical Ramsey neoclassical growth model and then try to open it up in a realistic way where we allow for the fact that there can be frictions in both goods and capital markets to allocating goods and capital across countries.

And there can also be imperfect stability in both goods and capital markets, which is going to help us match a key feature of the data, which is a so called gravity equation for both trading goods and capital markets, so what do I mean by that? Well, if you look at both goods flows and capital flows between countries, you see that they decline very sharply with distance between countries, but they're also proportional to some measure of origin country size and destination country size.

And that's going to emerge very naturally from our framework when we make these extensions to allow for frictions in goods and capital markets and imperfect suitability. So in particular what we're going to try to do is develop a tractable quantitative framework which we can take directly to many country world.

We've got many countries trading and goods markets having bilateral capital holdings with one another and we're going to try to simultaneously keep track of trading goods at a point in time. So how do countries source their trade flows from one another? We're also going to try to keep track of capital holdings at a point in time.

So countries are going to have portfolios of capital holdings across one another and at the same time we want to allow for intertemporal consumption saving decisions. So we're going to have endogenous current accounts as so we want to be able to keep track of these three things and we want to do so in a multi country world where the model stays tractable, we can take it to the data and we can think about policy questions.

So what we'll show is there's actually a relatively tractful way of doing all of that, that's going to capture key features of the data on trade flows and capital holdings. Yeah, John.
>> John: Neoclassical meaning each country is stuck with its own productivity and we don't trade in ideas.

>> Stephen Redding: That's going to be the main, so countries have a given stock of technology, we'll take it as being exogenous and then it'll be neoclassical in the sense you have capital accumulation to some steady state level of capital per worker.
>> John: Puzzle of course is why don't they just do it our way and triple their quintuple their GDP per capita, but that's not today.

>> Stephen Redding: Well, great question, we have something to say about that. So related to your question there is a so called Lucas puzzle which is sort of widely capital flow from capital rich countries to capital scarce countries will have something to say about that. There's actually a force in our model that partially explains why we don't see those flows to arbitrage rate of return differences.

But you're absolutely right, we will take the stock of technology as exogenous. You could add on top of this sort of endogenous innovation and technology diffusion and so on. The reason we've chosen to focus on exogenous technology is we want to really understand what happens to the standard Ramsey model once you open up the capital holdings.

And so we try to keep the other features as simple as possible.
>> John: Get that part first.
>> Stephen Redding: Exactly, you're absolutely right. You could build on a model of endogenous technological change as well. So as I mentioned, our framework is going to hit some key features of the data.

I've already mentioned one of them which is the so called gravity equation relationship for trading goods and capital holdings that'll emerge very naturally in our framework. We'll also get some other key features of the data that OBS and many other people have emphasized. So in particular, the model will naturally explain home bias in both trade and capital holdings.

So countries disproportionately invest in themselves relative to investing in other countries. The model will also generate determinant predictions for gross and net capital holdings. In particular, gross capital holdings will be much larger than net capital holdings, which is again something we see in the data. And our model will give determinant predictions for those gross holdings.

As we were just discussing with John, the model also have something to say about why we don't see larger capital flows from capital rich countries to capital scarce countries. We have a natural way of thinking about that in our framework, and it'll also explain the so called Felstein-Horioka Puzzle.

Why is there a very strong correlation between domestic investment rates and domestic savings rates even though we're in an open economy world? Again, we'll have something natural to say about that. And so we're going to try to build the model to match those features of the data and to keep it tractable to sort of think about various policy implications.

So what are the two sets of policy implications that sort of come out from our framework? Well, firstly, we're going to show how you can do counterfactuals for policy changes in our environment in a way that's very easy and very tractable. So suppose you want to know what happens if the US and China decouple, if there's an increase in trade frictions between the US and China, as we've been seeing in recent weeks or months.

Well, there's also an increase in capital market frictions in the US and China if the restrictions on US firms investing in China. So we'll show inside our model you can do counterfactuals for those types of policy experiments. In particular, you can use what are sometimes called dynamic exact-hat algebra counterfactuals, what does that mean?

Well, it means that if you want to understand something like a shot to trade and capital frictions in our framework, you can answer that counterfactual question just using the observed data. So if you see observed trade flows in the data in an initial period, observe capital holdings in the data, things like gdp, population, wealth, stocks in the data, then you can do your counterfactuals just using the observed endogenous variables.

You don't need to know what are the unobserved frictions, the trade frictions, the capital frictions, the unobserved productivities things that are difficult to measure. You can do the counterfactuals just using the observed endogenous variables. That's a technique that's been very popular in trade in recent years, and we show how you can apply open economy framework to both trade and capital holdings.

Our model will generate what we think are more realistic predictions for income convergence across countries than the standard Ramsey model. So one puzzle about the standard closed economy near classical growth model is that the empirical transitions you see in the data, the rates of convergence you see in the data, typically tend to be much slower than you'd expect with a plausible intertemporal Illassistia substitution.

The standard Ramsay model tends to give too fast convergence with a plausible intertemporal Illassistia substitution. Once we open that model up. And allow for capital holdings in other countries. The model is naturally going to provide an explanation for why rates of convergence to steady state might be slower than you'd observe.

And that's something that will naturally emerge in our framework. And in fact, our model is gonna give really subtle predictions for how openness in goods markets interacts with openness in capital markets to shape rates of convergence to steady state. And depending on time, I'll try to give you a sense of why that is and what those new predictions are.

>> John: Are when you say capital, do you mean human capital too?
>> Stephen Redding: So, great question. So today we're going to be mainly thinking about physical capital. So think about machinery and equipment used in production. And I'll tell you a little bit about how we measure capital in the data which will connect with that sort of interpretation in the model.

As you're highlighting, we are abstracting from human capital investment in skills. And again, that's something that could be added into the model. The reason we've chosen to do it like this is we want to sort of take the standard Ramsey model and say what happens when we open that up?

How does it change things? And so we try to keep all the other related to John's question, all the other features of the model as close as possible to the standard Ramsey framework. But it's a great question. And then as I was highlighting a moment ago, one of the sort of key centerpieces about the framework is we can use it to think about things like what will be the effect of a decoupling between the US and China.

And one of the key predictions that comes out of our model is if you want to understand the answer to that question, it's really important whether you're in a world where both goods and capital markets are open or you're in financial autarky and trade autarky. So for example, we'll show the decoupling between the US and China in goods markets has really different predictions in our framework where capital markets are open to the Ramsey model when capital markets are completely closed.

And so it's really important to model these two dimensions of globalization, integration in goods markets and integration in capital markets simultaneously. It really matters. The policy predictions come out of the framework. Obviously what we're doing stands on a large body of work to which many people here have contributed.

We'll obviously build closely on standard neoclassical growth models. Ramsey Solo Swan we also connect with some of the recent quantitative trade literature. We now have these multi country quantitative models of trade. You can take to the data but so far it's proved relatively more challenging to develop sort of quantitative multi country models where you have open capital markets.

And so we're trying to bring the trade and capital holdings models together in a framework where things stay tractable. Obviously what we're doing also relates to a large literature in international finance and macroeconomics, again to which many people here have contributed. One thing that's going to be distinctive about our model relative to some of the other work in international finance is we're going to have a model where there's no aggregate uncertainty.

So we're going to be abstracting from aggregate uncertainty. So obviously we're abstracting from issues of international risk diversification which, you know, many, many people have considered. Nonetheless, we'll show that our model can match a number of features of the data on capital holdings. And we sort of built the model in this way to sort of keep it tractable to enable us to do multi country quantitative work.

And that's why we're kind of abstracting from international risk diversification. And you'll sort of see how that works as we go through. I'll explain that in a little bit more detail. So let me get into the meat of what we're going to do. The paper's quite technical. I'm going to try to sort of get into the economics during my talk here and try not to go too much into the technical details.

But I'm really happy to sort of delve further into those details as we go through. So I'll outline the theory, tell you a bit about the data, how we take the model to the data and then try to tease out the key new predictions we get for sort of convergence to steady state in this model compared to the Ramsey model.

And then the new predictions for thinking about policy interventions like a decoupling between the US and China. So first, give you a sense of the model setup in intuitive terms and then I'll kind of go into some little bit more detail to highlight some of the features that are important for the analysis.

So as I said, we want to be in a multi country world. We can take to the data. So we're going to think about a world economy consisting of many countries indexed by nni. Time is going to be discrete, indexed by tau. We're going to try to keep the trade bit of the model as standard as possible.

So we're going to think about each country supplying a differentiated good. So that's the so called Armington model of trade goods are differentiated by country of origin. However, having Said that the trade module of this model is completely standard and holds in any trade model. The entire kind of class of trade models where you have a concept gravity.

So today I'm gonna focus on the Armington model. But if you want to consider an Eaton quarter model of Ricardo and technology differences in trade, if you want to consider a Krugman model of love of variety and trade, if you want to consider a Mallett's model with heterogeneous farms and a Pareto distribution of productivity, all of those models would be completely isomorphic.

All you really need is a constant elasticity gravity equation for trading goods. And so we just pick the Armington trade model because it's the simplest to develop our framework, the simplest to understand, but all the results go through throughout that class of model.
>> Ken Judd: The Armington assumption makes every good in every country an essential part of everybody's consumption.

So how if you cut off trade, utility is minus infinity.
>> Stephen Redding: Good question. So a couple of clarifications there. So you're right that the Armington model will have constant illicit substitution preferences across countries with a single illicit substitution. But we'll actually look at the case where countries goods and substitutes.

So that elasticity is bigger than one. And so if you don't trade with one country, utility is still well defined and utility is still finite. So and that's the empirically realistic case. Typically trade elasticities people think are around four is a sort of standard number.
>> Ken Judd: So your utility function is defined if one of the goods is zero.

>> Stephen Redding: Correct. Exactly right.
>> Ken Judd: That's a major departure from Armington.
>> Stephen Redding: Well, it's major departure from Armington with complements. It's standard if you have Armington and substitutes.
>> Ken Judd: Yeah.
>> Stephen Redding: So, and it will work with any trade model where goods are substitutes. So not just Armington, also you can call-

>> Ken Judd: A much better model this way.
>> Stephen Redding: Yeah, it's more.
>> Ken Judd: You had to have some way out of this.
>> Stephen Redding: Yeah, no, and it's crucial to matching the data that we have that future, yeah. And so then, you know, whichever one of these trade models you pick, goods are going to be produced with labor and capital under constant returns of scale.

So that's the standard neoclassical assumption. Markets are going to be competitive. Goods can be traded between countries subject to bilateral trade costs. So the usual iceberg trade costs, each country is going to have an inelastic supply of labor. Ln so endowments, exogenous endowment of labor in the setup as in many trade models.

And then here's going to be how we're going to start thinking about capital holdings in the model at the beginning of period t, the representative agent in each country is going to inherit an existing stock of wealth ant and then she's then going to decide where to hold that wealth.

And so she can choose to hold her wealth in capital in any country around the world. So that's the key departure from the standard Ramsey or Solo Swan setting. You've got wealth in your country, but you can actually choose to hold it in any country around the world as physical capital used in production in that country.

And we'll see how we're going to model that process in a moment. Some key features of it is that they're going to be frictions in capital markets. If I allocate my wealth abroad, that's going to be more costly to manage than if I hold it at home. That's how we're going to match the home bias in investment allocations.

And then when I allocate my wealth in countries abroad, I'm going to be subject to some idiosyncratic shocks to the returns on those holdings. And again, I'll explain in more detail how that works. But those idiosyncratic shocks are going to integrate out. So as I mentioned earlier, there'll be no aggregate uncertainty.

And again, I'll discuss that in more detail in a moment.
>> John: Adjustment cost. Or can I take my capital and simply move it to Canada tomorrow?
>> Stephen Redding: So the way we build the model today is going to be like the kind of classic Ramsey model. No adjustment costs.

So I can allocate my capital anywhere around the world. I can shuffle it costlessly and the model will give a prediction for exactly how I allocate that capital. It'd be pretty easy to kind of add on some adjustment costs into this setting. But for the moment we abstract from.

>> John: That means physical capital as well. So I can just move a factory from country to country B, no problem.
>> Stephen Redding: Yep, So, and you'll see how that works in a second. So we'll make some strong assumptions there and you'll see exactly how we model that. And. Yeah, and that's a great question, but if you wanted to, you could add on some adjustment costs.

So I can say more about that later.
>> John: Asked cuz there's often a confusion of hot money, but you can't hot the money unless you can actually move this stuff.
>> Stephen Redding: Right, right.
>> John: You're not making that confusion, so, good.
>> Stephen Redding: Beginning of period T, you'll choose your wealth allocation across countries.

You also make your intertemporal consumption saving decision so you can accumulate your wealth over time and again. We'll see how that works in a second. And then at the beginning of period T, the investment returns will be T +1. The investment returns will be realized, depreciation will occur, and then the whole process will start again.

You'll allocate your wealth again, make your consumption saving decision again. As I mentioned, no aggregate uncertainty. So you'll solve everything under perfect foresight. Okay, so let's go into some of the details. So the intertemporal consumption saving decision is gonna be a completely standard macro specification. So plain vanilla, off the shelf.

With one slight modification, which I'll explain now. So we're going to think of our representative agent. She's maximizing her intertemporal utility. So just adding up utility every period into the future. Beta is the discount rate and we're going to make the standard assumption constant relative risk aversion preference.

>> Ken Judd: Subscript notation is confusing. I don't mean. I suspect you don't mean NL plus S.
>> Stephen Redding: And I'll pass that. Okay, so let me talk through that notation. So C is going to be the flow of consumption of the representative agent in country n at time t +s.

>> Ken Judd: Yeah, yeah. Should be a comma between.
>> Stephen Redding: Okay, cool. Yeah, quite confusing otherwise. So she'll maximize her intertemporal utility subject to her period by period budget constraint, which takes the standard form. So each period she gets some income. What's her income? Well, that's her labor income from her endowment of labor plus the income she gets from her existing stock of wealth.

And as you can see here, this is the one modification to the standard setup that the representative agent in country N is holding her wealth in each producing country I as capital in that producing country I. So to get a total wealth, I have to add up her holdings across each of the destination countries where she's using her wealth, and she gets some return on those wealth holdings.

And in equilibrium, that return will be the same obviously across all the countries where she uses her wealth. And so that's why I've taken the return V outside the summation sign, because equilibrium will be the same across each of the countries where she holds her wealth. And then she'll also have some depreciation on her existing stock of wealth.

And as you can see here, this is a standard sort of macro setup where I can either eat my my own good, I can consume it, or I can accumulate it as wealth I can save for the future. And so my income in each period has to equal my consumption of that single final good and my holdings and wealth going forward into the next period.

>> John: You mentioned the Lucas puzzle. Capital doesn't seem to flow to equalizer returns, but here we have in capital flowing instantly equalizing all rates of returns.
>> Stephen Redding: Hold on one second. So there's. That won't in fact be true. There's gonna be a caveat to that. There's going to be a sense in which what you said is true and a sense in which it's not.

It's more subtle than that, and I'll explain that in a second. Great question. So, but you know, just before I get to that, because this return in equilibrium is across all the countries where you're using your capital, I can obviously just collapse those summations down into an existing stock of wealth.

And this is then exactly a standard macro problem maximizing my intertemporal utility subject to a standard period by period budget constraint. And so with constant relative risk aversion preferences, we get exactly the standard solution, which is that equilibrium consumption is a linear function of the agent's current period of wealth.

And that's the net present value of her future labor income and her wealth from her accumulation of capital held in each country around the world. And so with log utility, this linear function would just be a constant var sigma here would just be 1 minus beta, the discount rate more generally with constant relative risk aversion preferences.

There's a linear consumption rule. Okay, standard macro result. And so then, related to John's question a second ago, let's now delve into the part of the model that's new, that's this kind of really novel element here. So let me explain how that works. And so that's the capital allocation decision.

We're opening up the Ramsey model, allowing you to allocate your wealth not just at home, but in every country around the world. So let me explain how we think about that. Well, when we think about this stock of wealth as consisting of a continuous measure of units of wealth, each of those units representative agent is going to decide where to hold that unit around the world.

She can allocate it and use it as physical capital in any country around the world. And so let's think about what's her return if she uses a unit of her wealth as capital in any destination country I well, her return is going to be the rental rate on capital in that destination country I so we denote that by R here.

She's then going to incur some management costs from allocating that capital from origin country N to destination country I. So kappa are gonna be capital management costs and those are gonna think about those as frictions. It's more costly to manage your capital abroad. And that's going to help us match this gravity equation for capital holdings I mentioned a moment ago.

So she allocates her capital in a destination country I, she gets this rental rate per effective unit of capital r, the net of management costs, the cost of managing those investments. But then she's also going to get an idiosyncratic shock to how productive her unit of capital is in that destination country I and so think of this as an idea for an investment opportunity in destination country I.

And for each of her units of wealth, she gets one of these idiosyncratic shocks for every possible destination country. So she's got some ideas, some projects, some investment projects. Each of those has got an idiosyncratic shock to effective units of capital. So she takes her one unit of capital and puts it in each destination country.

She gets this random shock to her number of effective units of capital in that destination country I and then she gets the rental rate per effective unit adjusted by these management costs. We're going to assume that those idiosyncratic shocks are drawn from an extreme value distribution as shown over here.

And we'll let the mean of that extreme value distribution be different for every possible destination country. So some countries, like the United States, may have good protection of property rights, may be quite good countries have desirable assets that many countries want to allocate their wealth as capital in the United States.

And so this parameter eta will allow for those differences in the productivity of investments.
>> Ken Judd: Now, since you have those kinds of transaction costs, aren't there going to be corner solutions? Why decide zero for some countries?
>> Stephen Redding: It's a great question, so not quite. And in particular, why not?

Let's understand that because we've got an extreme value distribution for these idiosyncratic shocks, we get a very nice closed form solution for what is the probability that the representative agent in origin country N allocates her wealth to destination country I. In fact, that probability will take the constant elasticity of substitution or logit form.

That it's just gonna depend upon relative rental rates in each of the countries you can invest in. Relative management frictions and then the quality of property rights, the efficiency of capital markets, the productivity of investments in destination country I. And so to answer both John's question and Ken's question, rental rates are not necessarily going to be equalized across countries in this model.

Why? Because of this elasticity here at Psylon. Because there are these idiosyncratic shocks. If I want to attract more capital into my destination country, I have to pay a higher rental rate. Why? Because when I pay that higher rental rate, I attract units of capital with worse idiosyncratic drawers.

And so that's why a small difference in rental rates is not immediately arbitraged away as it would be in an open economy. Ramsey model is because we put some curvature into the models from these idiosyncratic shocks to effective units of capital. And that's going to help us match the gravity equation in the data.

That's why we've built that feature into the model. And so, rental rates can differ because each country effectively faces an upward sloping supply function for capital. Now, to answer Ken's question, just answer your first question before taking another one. As long as these management costs are finite and rental rates are finite in general, each country will have a diversified portfolio.

The representative agent will allocate some of her wealth to to every country around the world. We won't have any corner solutions in the data. We'll sometimes see zero bilateral investment flows. And how will we think about that in the model? Well, that will be the case where these management costs are infinite.

There's just something about it that makes it really difficult for me to invest in that country. And that's why the capital holding goes towards zero.
>> Ken Judd: Corner solutions are with finite capital, very natural. But also your use of extreme value distribution. But it's a very useful trick in IO.

But there definitely you explicitly take into account the fact that, yes, I'm at a corner or not. I mean, so your claim is that you're never at a corner.
>> Stephen Redding: So let me explain why that doesn't occur. So we allow for the possibility of corner solutions. The reason it doesn't occur is because the extreme value distribution has this property that the support of the distribution is unbounded from above.

And so there's always a unit of capital for which you just get a really great idiosyncratic draw for each country that makes you want to invest there. Unless these management costs are prohibitive, unless kappa goes towards Infinity continually. So, yeah, but great question. So one way to think about your question would be if you didn't have an extreme value distribution, if you had a distribution with bounded support, then you would get corner solutions.

>> Ken Judd: Yeah.
>> Stephen Redding: And we picked this distribution, so we don't have to worry about that.
>> Ken Judd: Well, everybody picks this one.
>> Stephen Redding: Yeah, exactly. Right.
>> Ken Judd: Avoid some problems.
>> Stephen Redding: So great, great, great question. So that this gives us sort of one feature of the model that helps things tractable.

We've got these constant illicit substitution expressions for each country's capital holdings. And countries generally generically have this diversified portfolio where they allocate their capital across countries according to a gravity equation that we see fitting the data. The extreme value assumption also gives us something else that's really useful and keeps the model tractable.

And again, it relates to one of John's questions, which what's the other feature of the extreme value distribution? Well, it turns out that the expected return to allocating a unit of capital to each destination country conditional on allocating the capital there is actually equalized across all destination countries.

And that's this expression here at the bottom of the slide there. So what does that say? Well, although the rental rates differ across countries, as we just discussed, the expected return to these investments, once you take account the idiosyncratic draws is equalized across all the destination countries. And that means that each origin country n has the same rate of return across all the destination countries where it's allocating its capital.

That's crucial to keeping the model tractable. Why? Because it means we have this common rate of return to investment that goes into the intertemporal problem we just looked at in the previous slide. What's the intuition for what's going on here? Well, imagine one country pays a higher rental rate.

That raises the return to investing in that country. So what's that going to mean? Well, that means that the investor is going to allocate units of capital with worse idiosyncratic draws there. And the extreme value distribution has this property that that composition effect, that batting average effect, exactly offsets the impact of the higher rates of return.

So the expected return conditional on investing in a country taking into account idiosyncratic returns is actually equalized in equilibrium. And that's another feature of the model that keeps everything tractable. And so we're almost at the position we can sort of think about the model's properties and thinking about taking it to the data.

Before I do that, I have to tell you a little bit more about the trade side of the model. Again, Related to some of the earlier questions, we've tried to keep the trade side as tractable as possible. We've got countries goods are differentiated by country of origin. So the consumption index in importing country N depends upon the good produced by each exporting country I.

And we've got a constant illicit substitution preferences across countries goods here. So sigma is the elasticity of substitution. As we're discussing earlier, we're gonna look at the case where countries goods are substitutes. Sigma is bigger than one and theta is going to be a key object. It's called the trade elasticity.

It will be the elasticity of trade flows with respect to trade costs. And in the Armington model it's just equal to sigma-1. So as you will know, that trade model gives you a very nice feature that sort of connects with the data which is that trade flows also satisfy a gravity equation.

So SNI is the expenditure share of importing country N on the goods produced by exporting country I. It just follows a standard gravity equation prediction. It depends upon relative prices at the point of production PI and then iceberg variable trade costs, tau and I, because the cost of shipping a good from producing country I to consuming country air.

How about the production side of the model? Well, I already mentioned, we keep that very plain vanilla. Like in the Ramsey model, goods are produced with labor and capital. So what is the marginal cost of producing a good? Well, that's just the wage and the rental rate weighted by the Cobb Douglas exponents.

We'll assume a Cobb Douglas technology for simplicity. And we'll also let countries differ in terms of their productivity and production. Obviously, when you go to the data related to John's question earlier, some countries are much more productive than others and we need to be allowed able to allow for that in the model.

So z will be productivity, which can differ across producing countries. And because we've got perfect competition, the price of sourcing a good and importing country N from producing country I just equals the marginal cost at the point of production adjusted by variable trade costs. So each country has got its own consumption index, which we saw at the top of the slide here.

And this is just going to be the dual price index for that country's consumption good. Again, using the standard CES functional form, I've nearly developed all the components of the model. The final thing we need to be careful about and taking the model to the data is thinking about the market clearing conditions.

And so let me explain that now. So one thing we need to keep track of is that payments to capital at the point of production. So that's just going to be the rental rate times the capital stock. Taking into account effective units of capital, they're going to be have to be equal to the income received by all the investors that are allocating their wealth into producing country I.

And that's summing across all investing countries n the wealth that each investing country n holds in producing country I multiplied by that expected return. That's just the kind of income accounting equation of the model payments to capital that point to production have to be for the income of the investors allocating their capital to that producing country.

So that's good. And all fits together. And obviously, with Cobb Douglas, payments to capital, including effective units of capital, will just be a constant multiple of the wage bill.
>> Speaker 4: So I have a question about. How does the good that is produced in country I get converted into capital that's useful in some other country at a different point in time?

>> Stephen Redding: Yep. So, great question. I'll say more about that in a second. Just before answering that question, let me just finish one detail about this income accounting relationship, just to keep our message. When we sort of apply this income accounting relationship, we obviously need to take account of what the effective units of capital are.

We have to calculate what's the average idiosyncratic draw to the productivity of these investments conditional on allocation your capital to a producing country. And it turns out that with this extreme value distribution, again, we actually have a closed form expression for those average effective units of capital conditional on investing in a destination country.

And so when we apply this income accounting relationship, we need to take account of that relationship between capital, including the average efficiency draw and the wealth that's allocated to a country. And with an extreme value distribution, that average productivity just depends upon the share of your portfolio you allocate to a destination country.

So that just finishes that thought. And so then to come to answer your question, key feature of the model that keeps things really simple is that each country accumulates its wealth in terms of its own consumption index. So like in macro, I produce a good, I can either consume it or I can use it to accumulate wealth.

Once I convert my good into wealth, I can then invest it in different destination countries according to this investment technology here. So I've got some wealth. So you think about, I can put that wealth into a suitcase, I can take it, I can invest in the destination country, and I can convert it into capital in my destination country.

And that's the way we're going to think about those investments that each country saves in terms of its own consumption good. We're just working on a revision of the paper at the moment, and one of the things the referees asked about, related to your question is how sensitive are the models predictions that that assumption that every country accumulates wealth in terms of its own consumption are good.

So we now have another version of the model where instead of accumulating wealth in terms of your own consumption good, and then having this investment technology where you can invest directly in capital in every producing country, we make a different assumption. So what's that alternative assumption? We say that what I can do is I can issue claims in capital markets, so I can sell a claim to my consumption index and buy a claim in capital markets to your consumption index.

So then there's no investment technology that's moving units of Consumption between countries. I'm just issuing claims in capital markets on the consumption index in different countries. It turns out that version of the model is slightly more complicated to analyze because state variables in the model actually become a matrix.

You have to keep track of a matrix of wealth holdings. But it turns out we can solve that version of the model using very similar techniques to this sort of simpler version I'm going to present here.
>> Speaker 4: So that version keeps the physical capital, or the physical good that gets produced stays in the country and then the ownership of the capital is different across countries.

>> Stephen Redding: Exactly. So that's a little bit more empirically plausible. It captures the idea that there could be a factory in France that's owned by a US investor, sold to Japanese investor. The factory doesn't move, it's just the claim that moves. And so that version of the model is maps to the data even more cleanly.

Turns out to be more complicated to analyze. It took us a while to figure out how to do it. But you can use the same techniques.
>> Speaker 5: And the market clearing prices of those claims doesn't change all the other assumptions you've made?
>> Stephen Redding: No, because inside the model, capital is in this alternative version of the model, capital is constructed out of producing countries consumption index.

And so the price of each claim is the price of that consumption index. And when I trade these claims in capital markets, the relative price is the consumption in relative price of the consumption indices. And that's part of the reason it's more complicated to analyze is because you get valuation effects.

So, right, if I've got some of my capital, my wealth, and this version of the model means you don't have to worry about those valuation effects. Whereas in the alternate version you have to take account of them, you can keep track of them, it's just more complicated to keep track of them.

>> Speaker 4: But in this version, like, you know, something gets produced in the country, you know, part of it gets eaten, part of it gets put into wealth, and.
>> Stephen Redding: Then you've got a technology directly invest that wealth.
>> Speaker 4: You can invest that wealth by putting it into the suitcase.

It becomes capital.
>> Stephen Redding: Exactly. And it's converted into capital in this way you get these idiosyncratic drawers as to how productive each unit of that wealth is. You use that wealth as effective units of capital with that efficiency draw and you get a rental rate and incur some management costs.

So you've got this sort of direct investment technology that converts your wealth into physical capital.
>> Ken Judd: Steve, apologies if you covered this, I'm missing something I taught since so. In your layout you're Basically assuming that's inconsistent with balanced growth. So anything about that out of conditions.
>> Stephen Redding: Right, so great-
>> Stephen Redding: So great question.

So one simplifying assumption we've made here is Cobb Douglas production technology. And so, as you're pointing out, one feature of Cobb Douglas is it means that whether technical changes hicks neutral labor augmenting or capital augmenting doesn't really matter. They're sort of isomorphic up to a power transformation. But as you're pointing out, more generally with CS technology or another general production technology, it would matter whether technical change was labor augmenting or capital augmenting.

We just chosen to try to keep that part of the model as simple as possible because we want to open up the Ramsey model. So we take the simplest version of the Ramsey model with Cobb Douglas technology, open it up to allow for these holdings of capital in other countries.

There's nothing inside of the model that would prevent us doing this with CES technology. We take efficiency as exogenous here. So we could specify any sort of exogenous process for labor augmented technical change, any convergence sequence of expected techn change into the future. Many of the results would go through.

We could, we could handle the CES case. It would be slightly more complicated, but I don't think it would present a problem for anything we're doing. Good question.
>> Ken Judd: Another way to think about it is there is a global frontier and the countries approach that frontier at different rates for different reasons.

Mapping of your conditional convergence.
>> Stephen Redding: Right, so great question. And so here, the way we think about that convergence to the frontier is each country's got these Higgs neutral technology shifters, and then they're accumulating capital, which is taking them to their steady state capital labor ratio. We take the technology as exogenous.

We really focus on the capital accumulation. That's where we're going to get all our new predictions. You're right. You could do several extra things with technology. You could make the production technology more general, like CES, and have augmenting capital augmenting. You can also make technology endogenous. Those are all great extensions.

Really interesting extensions.
>> Ken Judd: Make it more complicated.
>> Stephen Redding: Yeah. You're going to show even though we kind of tie our hands by taking this very plain vanilla model of the production technology, just by opening up the capital holdings, we get really nuanced predictions that come out of the model.

It gets really, really rich relative to the sort of standard financial or taki version of the Ramsey model. Yeah, it's a great question. Yeah, no, I'd love to work more on that. It's great ideas, yeah.
>> Ken Judd: In what sense? This neoclassical saying? Neoclassical means that there's an alternative universe.

Is there anything that comes very close? Flexible classification. Is it fair that you. What do you exclude?
>> Stephen Redding: Yeah, so great question. No, it is a really good question. So what do we mean by neoclassical is the core of the question? So what we mean two things? So the first thing we mean is we want to contrast this model, which is essentially a Solo Swan Ramsey style model, with a model of endogenous growth.

So what we don't do is we don't endogenize productivity. Related to Michael's question as well. And so that's part of what we mean by neoclassical is you want to contrast the earlier growth models of Solo Swan, Lucas Ozawa with the later kind of Chad Jones, Paul Romer, Aguyon Howitt models of endogenous technical change.

And so that might have been a misnomer the way using neoclassical to do that, but that's what we were trying to do, was to try to signal that difference. The we technology is exogenous. That's a simplifying assumption. We're making the other sense of neoclassical, as you pointed out, the other thing we're doing is we're going to work with competitive markets, complete wage and rental rate flexibility.

So we don't have any nominal rigidities. We don't have any real rigidities in the model. And so we don't nest new Keynesian models which would feature nominal or real rigidities. We haven't built those into the model. We abstract from them, that's something we haven't thought about. I need to kind of think more about what the implication of putting in some nominal real rigidities would be.

Something we haven't thought about. It's really interesting idea, but that's what we mean by neoclassical. It's capital accumulation without endogenous technology and its competitive markets, wage and rental rate flexibility. Answer the question.
>> Ken Judd: I think so, yeah.
>> Stephen Redding: Yeah.
>> Ken Judd: I guess I'm somewhat concerned you have this kind of variable that phases out heterogeneity.

>> Stephen Redding: So there is actually a deep sense in which there's another sense in which this model is actually Keynesian, which relates to the idiosyncratic draw. So what do I mean by that? This model of capital holdings actually gives you a downward sloping marginal efficiency of capital schedule in each country because I've got these measure of units of wealth and I get these idiosyncratic drawers.

Each unit I can allocate to each destination country. So if exactly for every unit I've got a downward sloping marginal efficiency of capital schedule. So all the countries I can allocate this capital to. And so in some sense the way we model these idiosyncratic drawers is actually very related to Keynes notion of the marginal efficiency of investment.

It's that you've got one of these schedules in each country you can invest in. So that's another layer in which you can connect it with Keynes if you wanted to. So there we-
>> Ken Judd: Excluding the creative model. For every member of that model.
>> Stephen Redding: We do exclude some things, right.

So we're excluding nominal and real rigidities. We're excluding a kind of label Azure choice, but we're gonna work within a market clearing setting.
>> Ken Judd: You can add another, move into that part of the space. But seems to me you're getting very close to advocating reversal model.
>> Stephen Redding: So I guess my take on that would be I'm going to try to convince you that making this one change to the Ramsey model opening up, we get a lot of bang for our buck.

We get a lot of extra insights even if we don't add all these other features. And so that's why we focused on this one feature because we think we get a lot of juice out of the model, a lot of insight. And so that's why we justified our focus on it.

We thought about some other extensions, we haven't thought about others such as some of the ones you mentioned. But the reason we focus on this extension as you'll see it, and hopefully I'm gonna have enough time to show you just how much juice we get out of making this one change to the model.

>> Speaker 4: To let you get to it, obviously, I think it's great. I mean we can think of these as just to be determined later how many of these conclusions are orthogonal to.
>> Stephen Redding: Right, so we can come back later and think, well, how would, how would this new insight change if we added these extra features in?

And it would be great to talk more about that later. I've got. We finish at 1:15, right? Is that right, John? Yeah, so let me just briefly mention a couple of features of the model, tell you about the data and then try to get to the punchline what we get out of this framework.

So the models just like Solo, Swan and Ramsey, there's convergence to a steady state capital labor ratio in every country. But because you've got openness in goods and capital markets. Now that steady state capital labor ratio depends upon things like goods and capital market frictions between countries. It doesn't just depend upon your own productivity because we built the model so we have constant elasticities.

It's a constant elasticity specification. As I mentioned earlier, we can use those tricks from trade for asking counterfactual questions. How would the world change if US China decouple? We can undertake those counterfactuals just using things we see in the data. Initial trade shares, initial capital holdings, initial stocks of wealth.

So we don't need to know what are trade costs. That's very hard to estimate and measure. Just give us trade shares and we can do our counterfactuals. It's a very useful property of the model. And so when we solve one of these counterfactuals, we can consider any expected convergence sequence of future changes in trade costs, capital frictions, and we can solve for the entire transition path of each country along the convergence path towards its steady state in the full sort of nonlinear model.

>> John: This is because you can back out all of the unobservable primitives from observable stuff.
>> Stephen Redding: It's related to it. That's the intuition for why this works, is there's actually kind of an invertibility in the model that up to some choices of units, once you see these observed endogenous variables, you could actually infer the unobserved productivities.

>> John: What you're observing is, is just trade flows between countries. That's enough. Or what do you actually observe?
>> Stephen Redding: So the key thing we need to observe is we need to observe an expenditure share matrix S ni so the expenditure share of importer N on exporter I. Okay.

And then we have to observe the mirror image of that, the T matrix. So what's T? It's the income share matrix. What is the share of the income of exporter I that comes from market N under balance trade? They would be very closely related, but we don't have to.

We allow for imbalanced trade in the model. So not necessarily they can differ in more interesting ways. The other things we have to see are capital holdings. So the matrix of portfolio shares, so the share of investor ends wealth that's used as capital in producing country I. And then again a mirror image of that.

The share of capital payments in destination country I that are paid out to investors in origin country end. So those are the key things we need to see. And then two initial stocks of wealth. The vector of wealth.
>> John: Yes, I just didn't understand the notation.
>> Stephen Redding: Yeah, I was going quickly.

Yeah, yeah, but that and those are, once you see those things, that's all you need to solve for counterfactuals.
>> John: So somehow I can tell the differ. For example, you and I might not trade because there's. You have low productivity or because there's big iceberg costs, but somehow you're able to.

There's a matrix there that lets you tell the difference between those.
>> Stephen Redding: But great question. That invertibility as you pointing out holds up to some normalization. So as you're highlighting tra, trade costs are not separately identified from an increase in productivity is the same as a reduction in your trade cost with all your partners, including yourself.

Yeah, right. And so that inversion holds up to some choices of units. An advantage of just doing the counterfactuals based on the indigenous variables is you don't need to take a stand about those choices of units. If you see the shares, that's good enough to do the counterfactuals.

>> John: Okay.
>> Speaker 4: But this is in a model that's very abstract that abstracts from political risk, abstracts from expropriation, abstracts from all sorts of things. And Neil, that exist in the real world that are going to explain why I don't want to invest in. In some country like, you know, China, you know, people thought, it was a great place to invest.

It's going capitalist now. It's not, and that's not perfect foresight.
>> Stephen Redding: Great question. So there are some things we can capture with the framework and some things we can't. So let me explain what we can and then flag what we can't. So we can capture something close to what you mentioned.

It's not exactly what you mentioned, but something close to it. So it could be true that China has a really bad environment for doing business, for investing in, and that's captured in these management costs. And then the sort of average draw for your efficiency units are capital. So the model can allow for that and in fact it can match the portfolio share we see in the data related to John's questions.

Because we've got these management costs and, and so we can rationalize as an equilibrium the entire matrix of bilateral capital holdings. One thing we don't have, as you pointed out, is we don't have aggregate uncertainty. And so there's no aggregate uncertainty for, for the US investing in, in, in China.

We solve the model under perfect foresight. And that's correct. That's a simplification we're making and I'll try to convince you we get some interesting insights. Despite this simplification, the big thing it lets us do is go multi country quantitative.
>> Ken Judd: You seem to be asserting a very strong inversion principle that somehow from you say you don't need to know the fundamentals.

Well, I gather that what you're telling us is that from the endogenous variables you can pin down the, can identify.
>> Stephen Redding: Up to some normalizations.
>> Ken Judd: Yeah, units.
>> Stephen Redding: Yeah.
>> Ken Judd: So, but then when you solve this, what you have equations that only use the, they're only in the endogenous variables that don't involve the.

>> Stephen Redding: That's the beauty of it. You can solve for the entire transaction path just using the observer. So think about it like this. You could write it in terms of the fundamentals and then you just rewrite it in terms of indulgence variables. And if everything's a constant-
>> Ken Judd: That's because of all the special functional form assumptions.

>> Stephen Redding: Constant elasticity in particular.
>> Ken Judd: Yes, okay.
>> Stephen Redding: Yeah, okay.
>> Ken Judd: Yeah, yeah, yeah, yeah.
>> Stephen Redding: Okay.
>> Ken Judd: So if you want this is, is. This just a local analysis or this global.
>> Stephen Redding: So the cool thing about this is global.
>> Ken Judd: Okay.
>> Stephen Redding: Even for large changes.

So that's what's really powerful about it is I can do a counterfactual in the full nonlinear model and these techniques go forward related to two questions you just asked that we can also linearize the model. In that case we can actually generalize functional forms. And we also as well as.

Yeah, and I'll say a little bit more about that later. We actually get even more results from the linearized version of the model. But given my constraints on time, let me just tell you a bit about the data. So how we take the model to data. So we're going to take the data going to be national income accounts.

So Penworld tables, the model will exactly match gdp, labor income compensation, population. From the Penworld tables, we'll also exactly match bilateral trade flows between countries. From UNCOM trade, we'll also match bilateral capital holdings between countries. So there the data are more limited as you all know, we try to take the state of the art data we have on bilateral capital holdings.

So that comes from the CPIS data from the International Monetary Fund as restated by Matteo Maggiore's Global Capital Allocation Project here at Stanford. So we take their state of the art matrices for dealing with the difference between residents and nationality to try to deal as best as possible with issues such as tax havens and related issues.

And because we want to hew as close as possible to the Ramsey model, we're going to treat all types of capital together. We can aggregate them all together just like we do in the Ramsey model. So we've got direct investment portfolio, investment debt and equity. We aggregate everything together and think of there being an aggregate capital stock in the model.

To illustrate the extra sort of insights we get from opening up the Ramsey model, we'll try to keep the parameter values as standard as possible. So nothing is driven by exotic parameter values. So standard discount rate, intertemporal elasticity of substitution, standard trade elasticity, we use sort of five, four or five is standard in the trade literature.

The one parameter where there's a bit less evidence is this sort of capital elasticity. So how up it's sloping is your capital supply function when you raise your rental rate, what is the elasticity of capital holdings with respect to that rental rate? We take a parameter estimated by our co-author Moto Yogo in a paper with Ralph Kojian, we take their estimate of four as a kind of central value and then the model will exactly match the Penwell table.

So we let the labor share differ across countries.
>> Ken Judd: That's less than infinity.
>> Stephen Redding: Yes. Yeah, yeah, yeah. We've got a relatively small number of countries in the quantitative application, about 40 countries.
>> John: So that's from a very different experiment. Right. Thousands of microstructure of markets with upward sloping demand curves and limited agents.

And so-
>> Stephen Redding: So they report a couple of different estimates. We tried to pick the estimate. This is close to what we're doing. We definitely don't have fun level data. So we tried to pick the thing that's as close as possible to what we're doing. But yeah, it's a good point.

Cool. So let me just convince you that building this model there's a lot of empirical motivation for trying to have a model that matches gravity and trade and capital holdings. So let me just try to illustrate that to you. So what am I doing here? We all know that the trade flows are characterized by A gravity equation.

And we just show that in the first two columns here we're going to regress the simplest specification, the log of bilateral trade flows on destination fixed effects, origin fixed effects and bilateral distance. Log of bilateral distance with a constant elasticity and then a regression error. We all know that if you estimate that by OLS log linear extremely good fit to the data, R squared well above 80%, distance elasticity of around minus 1.

As we were discussing earlier, there are some zeros in the trade data. So you might want to allow for that sort of state of the art way of doing that is to use the Poisson pseudo maximum likelihood estimator. Again you get a very sort of similar pattern of results for trade flows.

Gravity is a very good approximation to the data. Maybe less well known, but also very strong feature of the data is that gravity holds for capital holdings. So in column three we show that in a log linear specification, again R squared of over 80%. And then again similar results.

If you estimate it with plus on pseudo maximum likelihood, you might worry, well, is this sort of good fit of a gravity equation here just driven by the fixed effects? Maybe it's all the origin destination fixed effects. And so in the paper we show even if you use Frischer Laval Waugh and you partial out the origin destination fixed effects and you look at the conditional correlations, distance is around as important for capital holdings as it is for bilateral trade flows.

And so, this is the kind of key first order feature of the data that we wanna hit with the model. And that's why we've built the model to try to match it.
>> Ken Judd: I don't think that's certainly not why.
>> Stephen Redding: Yes, what's-
>> Ken Judd: That's gonna make a lot of sense and monitor your factory, etc.

>> Stephen Redding: It's actually really interesting. It's actually really interesting. So here we aggregate them, but we've also run it, we report in the paper what happens if you split it out. And the elasticity is actually really similar for portfolio and direct investment. One interpretation might be information frictions, that maybe it's partly capturing information frictions and that's why gravity holds for the capital holdings.

And we think of that as part of our management costs. Those kind of information frictions, it's costly to find out about investments abroad.
>> Ken Judd: Our regional free trade agreements talk about how free trade they are.
>> Stephen Redding: Right.
>> Ken Judd: Partly reflected in the data looking like this.
>> Stephen Redding: Right.

Interesting question. So you could enrich this with free trade agreements and other measures of frictions between countries. My guess is whatever you throw at it, you're going to see that distance matters. Obviously, if you put enough proxies for sort of information frictions, social frictions, these are other variables gonna start to explain away the physical effect.

>> Ken Judd: Of this regional trade agreements, meaning you're more likely to capital flow across borders within the region.
>> Stephen Redding: And so, actually, you're going to see that's going to be a key prediction of the model. And so let me get onto that. That's one of the things we have a lot to say about.

So I'll try to get onto that. So it's just to kind of motivate that it makes sense to develop a model with these features. So now, in my remaining 15 minutes, let me try to kind of tease out the two kind of two main sets of insights we get from opening up the Rams Model.

First insight is for convergence to steady state. Rates of convergence to steady state. And, and I'm going to illustrate that by looking at impulse responses for a productivity shock in the model. So I'm going to take a country that's small in the model in the sense that it's small relative to global portfolios.

We also report results for a large country in the paper. And you'll see in a moment why I'm just doing this as a small country. Nothing I say is really going to. The main insight I want to tease out is not sensitive to being small. Just considering a small country helps to make the point as concisely and insightfully as possible.

So I'm going to do an impulse response for a productivity shock in four different models to sort of illustrate how our model generates new predictions for convergence along the transition path to steady state. So the first model is this second column here. So this is the closed economy Ramsey model.

I'm considering a productivity shock in a small country. The third column here is going to be. Second column is a Ramsey model. First column is our model. So our baseline model with open goods and capital markets. Third column is a version of our baseline model with no trade or capital market frictions.

So that's setting the taus and the kappas to be equal to 1. There's no frictions when you trade or invest in other countries. And then the fourth column has no frictions in our model and takes the capital elasticity to infinity, which is obviously the open economy Ramsey model in that fourth column.

So second column is closed economy Ramsey model. Fourth column is open economy Ramsey model. And then the first column is our model. So let's think about effect of a small productivity shock in all of these models. Let's start with the simplest, the one we're very familiar with, the Ramsey model.

If I shock productivity in a small country, what happens to capital and wealth in that small country? Well, because the country is small, relative, and it's in financial autarky. So there's going to be no direct effect of the shock immediately on wealth. Because when I get more productive, what happens in the Ramsey model?

Well, it means I want to accumulate capital to go to the new steady state. The only way I can accumulate capital is by accumulating wealth. And that requires me to save and gradually accumulate wealth. So with this small productivity shock in a small country, you're gonna have gradual accumulation of both capital and wealth.

And obviously in this model, capital and wealth are the same thing. And so you just get gradual convergence to new higher level of capital and wealth as a result of this higher productivity. So very simple, very familiar, what happens in the open economy Ramsey models. Let's consider the fourth column here.

So no frictions and an infinite capital elasticity. Well, when I shock productivity in the small country, what happens to wealth in that country? Well, absolutely nothing. Why? Because there are no frictions here and there's infinite capital elasticity. So in this open economy Ramsey model, there's a single global portfolio that every country holds.

And when I shot productivity in one country, that's measure zero relative to that global portfolio, it has no effect on the return to the global portfolio and no effect on the accumulation of wealth. So wealth just stays completely flat. But because I've made my small country more productive, what's going to happen?

Well, obviously there's going to be reallocation of capital, instantaneous flow of capital from the rest of the world to this small country, reflecting the fact that it's become more productive. And so the closed economy Ramsey model and the open economy Ramsey model have these polar opposite predictions to how you respond to a productivity shock.

In the closed economy Ramsey model, the only way you can respond is with perspiration. You have to say gradually accumulate things domestically. In the open economy Ramsey model, you can instantaneously adjust by getting capital flows from the board. What happens in our model? Well, you'll see that our model is going to generate impulse responses in between those two extremes.

So in particular, let's now think what happens, well, when there's this productivity shock, because we now have open capital markets, my small countries become more productive. Some of the adjustments going to occur through reallocation of capital. Some capital will move from abroad to the domestic economy, but the adjustment will be less than in the open economy Ramsey model.

Why? Because there are these capital frictions and capital is imperfectly substitutable. Every country has this upward sloping capital function. To attract more capital, I have to pay a higher rental rate. As the rental rate goes up, some capital reallocates, but not enough to completely arbitrage away the differences in the rental rates.

So you get some initial capital reallocation. But at that point onwards, rental rates still differ across countries. The rate of return to investment still different across countries. And so there's some perspiration, there's some additional domestic accumulation. And so in other words, in response to this productivity shock, we get the sort of more plausible prediction.

You get adjustment both through the reallocation of capital internationally and through domestic accumulation. Why does that matter?
>> John: Do you actually end up wealthier in, you know, relative to the.
>> Stephen Redding: It's a great question. It's a great question. I'm going to say something more about that in a second.

Come on to that. Let's now overlay the dynamics in our model on the top of. And I'm going to come around to answering the question in a second on top of the closed economy Ramsey model. So the purple dashed line is our dynamics just put on top of the Ramsey dynamics.

And it relates to the question you just asked a moment ago. What do we see? Well, you'll see in our model there's this initial reallocation which helps to dampen the difference in rental rates. In the Ramsey model, that can't occur because you're in the closed economy, you're in financial autarky.

So because there's this initial reallocation in our model, it dampens the differences in rental rates. What does that mean? It means the slower convergence from that point onwards, the transition path cuts from above the transition path in the Ramsey model. And so, that's the intuition for why our model generates slower conversions to steady state and then the closed economy Ramsey model.

Why? Because once you're open, some of the differences in rental rates gets arbitraged away. And so from that point onwards, a slower convergence to steady state. But then related to your question, does that mean you're better off in this world relative to. Is slow convergence good or bad in terms of welfare?

Slow convergence might be actually good in terms of welfare. Why? Because you get the initial reallocation that takes you closer to steady state. So we're getting slower convergence than the Ramsey model. But you know, the welfare properties are going to be very different from that because our model allows for this initial reallocation of capital.

So very good question.
>> Ken Judd: We have to see consumption to know that.
>> Stephen Redding: Yeah. And I'll show you consumption in a second when we do our counterfactuals. Yeah, so that's essentially going to be one of the. That's now that first key insight we get out of our model.

Generically, our model delivers slower convergence to steady state and the closed economy Ramsey model. Why? Because capital markets are open and that can dampen away some of the variation in rental rates. And I'm short on time, so I can't explain how we do this, but it turns out we can actually, in the linearized version of our model, if we linearize it, we can actually solve in closed form for the speed of convergence, steady state and we can actually calculate it.

So let me try to give you an intuition for where this is gonna happen, how this works. What are the state variables in the model? The state variables in the model are the vector of wealth in every country around the world. And so those are my state variables.

And in general, in the data, countries start off not necessarily being in steady state. Each country can have some deviations of its wealth from its steady state level. We can represent any shock, such as the productivity or capital frictions, trade frictions, anything that affects the state variables. We can represent it using the eigenvectors of a particular matrix in the linearized model.

And so what that means is that for any vector of empirical shocks to the fundamentals in the model, we can represent that vector of empirical shocks as a linear combination of the eigenvectors of something called a transition matrix in the model. Let me just give you a more sense of where that comes from.

Once we linearize the model, we actually get a autoregressive representation for the state variables in the linear model here, where the vector of wealth at time t is just the linear function of the vector of wealth at time T minus 1 and then A vector of shocks. And so the dynamics of the state variables just depend upon something we call an impact matrix, which is the initial effect of the shocks in the first period they occur, and then a transition matrix p which governs the updating of the state variables one period to the next.

Where we get this equation from is essentially solving a second order difference in the state variables. Solving a second order difference equation in the state variables. The solution has this autoregressive representation where this transition matrix can be eigen decomposed into its eigenvalues and its eigenvectors. And we can basically use that property so that any shock to the state variables can be represented as a linear combination of the eigenvectors of the transition matrix matrix.

Each of those eigenvectors has an eigenvalue, and the speed of convergence just depends upon that eigenvalue. And so here I'm showing you the speed of convergence to steady state as measured by the half life of convergence to steady state for the entire spectrum of eigenvalues. So there's as many eigenvalues here as there are state variables in our model.

In other words, as many countries as there are. We've got around 48 countries in the quantitative model. The blue dash line are these speeds of convergence in our model. So we have half lives ranging from around 20, well, I guess around 10 years to more than 60 years.

The red line is the speed of convergence as measured by the half life in the closed economy Ramsey model. It differs across countries because countries have different labor shares. And as you can see here, half lives in our model are generically longer than in the Ramsey model. We have slower convergence.

And we already walked through the kind of intuition for why that is. And so our model is going to provide a kind of a simple explanation for this apparent puzzle that the empirical transitions in the slower than would be implied by the closed economy Ramsey model for plausible intertemporal illnesses as a substitution.

Why? Because open capital markets help to dampen away some of the variation in rental rates.
>> John: We've seen large decreases in labor share after all the Caldor alleged constants disappeared. And so using current labor shares. And so another question later, maybe, there seems to have been either a shifting of what's showing up with labor and human capital and capital or there's a lot of movement across countries in this.

That's another.
>> Stephen Redding: Yeah, great question. So in order to get the maximum country coverage to get up to 48 countries, the capital data are only available for a relatively limited time period. So we've got 2013 to 2017. So we're in a relatively small time period. You're right that if we wanted to, if we had data over a longer time period, that change in that labor share would be really important.

And I think it would be super interesting to think about that in the model and maybe also relates to your question about labor augmented capital augmenting and so on, where labor share wouldn't be a parameter, it would evolve endogenously. And I'd love to talk more about that. Let me in my last five minutes, just try to do my policy counterfactual, yep.

>> John: Some opening capital markets means the interest rate has go up as much, but also means more capital can come in.
>> Stephen Redding: Yep.
>> John: So it's got to be some race between this elasticity of supply and your intertemporal elasticity willingness to save up. It's not as easy as just open stuff up cuz then it will come in faster.

>> Stephen Redding: Absolutely, and so the way I'd interpret the point you're making there is that this curve here is gonna depend upon a couple of things. It's going to. Well, several things. It's going to depend upon the matrix capital frictions across countries. It's going to depend upon the capital elasticity, a whole bunch of stuff.

This is drawn with the standard parameters I mentioned to you in the observed capital matrices. So you're actually absolutely right. This is giving you the intuition for what's going on. But the size of this capital reallocation here, as you're pointing out, depends upon lots of things like the capital elasticity, matrix of capital frictions, all of that.

Yeah, absolutely.
>> Ken Judd: I need that, one comment that I've been itching to make. One is that your notion of tractability is quaint and primitive, that people have been for 15 years looking at methods that can solve more general versions of this model than this. So that's just, I don't have the time to go into it.

But this is primitive stuff relative to what can be done.
>> Stephen Redding: Good question. I'd actually push back against that. If you look at the work in international macroeconomics.
>> Ken Judd: I'm not talking about international macroeconomics. I'm talking about Computational methods in general.
>> Stephen Redding: No, I agree that completely.
>> Ken Judd: Okay, that's relevant.

That's relevant.
>> Stephen Redding: Computation methods in general. We're not trying to innovate here. We're actually using tools from the dynamics, stochastic, general equilibrium literature.
>> Ken Judd: Old ones.
>> Stephen Redding: But the really big contribution, and it's fundamental as a massive contribution, is relative to the international macro literature, which typically uses two country models.

>> Ken Judd: Yes.
>> Stephen Redding: Limit to three.
>> Ken Judd: Comparing yourself to an easy target is. That's standard procedure. Many of these issues that people have raised could be analyzed if you moved away from these quaint primitive methods.
>> Stephen Redding: We're not aware of anybody else who's done this. We're not aware of anybody else.

>> John: Understand what comes out of the big black box.
>> Stephen Redding: Yeah, yeah.
>> John: To do something very nice on that direction.
>> Stephen Redding: Yeah. We get analytical results. We can characterize all this stuff analytically. So I agree. We're using computational methods elsewhere. We never would wanna claim, we're innovating that.

And no one's applied those in international finance. We're going beyond a two country model to work in a world, you know, we can handle 100 countries, 150 countries, as many countries as we have data. But I've only got two minutes left. So let me try to show you what else we get out of this framework.

So the big policy thing is we want to think about what happens if the US and China decouple. And the other big takeaway from our model is that it really matters answering that question, whether you take account that capital markets are open. So again, to connect with Ken's point here, going from a two country open economy macro model to a model where we can go to the data, we can really model capital market linkages.

We get major new insights for the impact of this policy counterfactual. Let me illustrate that for you. In the first column here we've got a trade cost shock in a model of capital autarky. In the second column we've got a capital friction shock in a model of trade autarky.

In the third column we've got a tracost shock in our model of open capital markets. And in the fourth column we've got a shock to capital frictions in our model with open trading capital markets.
>> Ken Judd: It's just the US and China couple?
>> Stephen Redding: Yeah, just bilateral.
>> Ken Judd: Okay, I just want to-

>> Stephen Redding: That's a particular experiment here. And as you're hinting that's fine, I just want to know whether it does, whether other countries change as well. And you can also look at that with a model. That's what's so great about it is you can play with all the different experiments.

So let me show you what what happens when you're in capital Otaki in the first column here. Well, firstly let's focus on an increase in trade costs between the US and China. Well, you get static welfare losses in both the US and China. Those are the standard foregone static gains from trade.

When you put trade frictions and the China loses more than the US, why? Because the US market is much more important to China than the Chinese market is for the US Then already in our model you get something different. In this version of our model with capital autochea you get something different from a standard trade model which is when you increase trade frictions between the US and China, that increases the consumption price index because you accumulate capital in terms of the consumption price index that makes investment more costly.

And so we actually get dynamic losses from these trade frictions. So the cost to these decoupling are now bigger for the US and China because I've now raised trade frictions. That doesn't just lead to static welfare losses, it also leads to a decumulation of capital and wealth over time.

And you see that here those initial losses are magnified as I transition to the new steady state. Now let me show you that you get really different predictions if you now do the same shock when bilateral trade cost frictions and you're in a world of open capital markets.

So let's look here. Well, you're going to have the same static welfare losses and again you find that the welfare losses are bigger for China than they are for the United States. But you now see that the transition path is very different here. And in fact in this open economy world we see that in the dynamics it's actually the US that loses more than China.

So what's going on there? Well, let's think through what happens when I'm in this world with open capital markets when bilateral trade cost frictions go up. Both the US and China can now reallocate the capital around the world. You get these third country effects. But once trade frictions are more higher between the US and China, US Investors, for example, might want to reallocate capital to Singapore and served the Chinese market from Singapore.

So you get this kind of new endogenous generally reallocation of capital in capital markets, which actually there's quite a bit of evidence that happened after the first Trump tariffs that a lot of investment went to Vietnam. So that's kind of a new channel that you get here. What's driving the fact that the US Accumulates whereas China doesn't.

China starts to accumulate here. Well, when these bilateral trade cost frictions go up between the US And China, as we said, the US Market is much more important for China than the Chinese market is for the US So a lot of the goods China's shipping to the US Go back home to China.

That bids down goods prices and factor prices in China and that reduces the consumption price index and induces China to accumulate more wealth and accumulate investment over time. Whereas in the United States, because it's sourcing a lot of goods from China, when trade costs go up, the consumption price index goes up a lot in the United States and that leads to, that makes investment more costly and leads to this process of accumulation.

And so whether you have open capital markets in our model or closed capital markets really matters for counterfactual predictions. But given that I'm just about 30 seconds over, let me try to finish. What did we try to do? We tried to take a familiar Ramsey model, generalize it, open it up.

So we studied trade at a point in time, capital holdings at a point in time, endogenous consumption saving decisions. We hit a bunch of things in the data. We develop a tractable multi country framework you can use for policy experiments goes considerably beyond the two country settings considered in the open economy macro literature.

And we saw how you can get new insights for this question of convergence and decoupling by bringing these things together in this multi country tractable framework. But let me stop there to stay on time and thanks for all the great comments and questions very much.