Balanced growth in theory

Posted by Dietrich Vollrath on September 07, 2016 · 13 mins read

The last post was on the empirics behind balanced growth paths (BGP), the key organizing principle of economic growth models. I was skeptical of these empirics, and it looked like only weak evidence was there to support the idea of BGP’s.

As I said in that post, essentially all the models of growth written down in the last half-century work very hard to ensure they have a BGP. Here, I want to show that this generates a set of pretty tight restrictions about how you can model the world. And hence it generates a set of pretty tight restrictions on how you think the world actually works. So tight that perhaps it seems implausible. Like I said in the prior post, I’m trying to be highly skeptical of these claims, in order to test my own beliefs.

Uzawa and BGP’s

Leave aside the questions regarding the Kaldor facts from the last post entirely. Let’s assume that you do want your model of growth to have a BGP. What does this imply about how growth works?

The key paper here is by Uzawa (1961), but I highly suggest you not read it. It is baffling. I suggest you take a look at Jones and Scrimgeour who have written a much more accessible explanation.

Let me try to give you a flavor of Uzawa’s theorem. I’m going to try a different tactic than Jones and Scrimgeour, because I think I can keep it more accessible to a blog audience. There is math, but I hope nothing too terribly awful.

To start, we’re going to assume that we have some kind of constant returns to scale production function. That CRS is crucial is necessary for BGP’s to exist. Let y be output per worker, and k be capital per worker. Let B denote the capital-augmenting technology - think of this as an index of how useful capital each unit of capital is for production. Let A denote labor-augmenting technology - again an index of how useful labor is for production. Okay, then let’s write the growth of output per worker as

\[\hat{y} = \epsilon_K \hat{B} + \epsilon_K \hat{k} + \epsilon_L \hat{A}\]

where the “hats” on a variable denote its growth rate. The values $\epsilon_K$ and $\epsilon_L$ are the elasticity of output with respect to that specific input. Because of constant returns, it must be that $\epsilon_K + \epsilon_L = 1$, and that will prove useful in a moment.

What the equation says is that growth in output per worker is a weighted sum of capital-augmenting technological progress, labor-augmenting technological progress, and growth in capital per worker. The weights are the elasticities, which tell us how important capital and labor are to production.

Now, one of the conditions for a BGP is that the rate of return on capital is constant. If you remember the prior post, we said that the rate of return could be calculated as

\[r = s_K \frac{y}{k}\]

or capital’s share in output, $s_K$, times the output/capital ratio. The BGP also requires that the capital share is constant over time. So for the return, r, to be constant it must be that the ratio y/k is also constant. Which implies that $\hat{y} = \hat{k}$.

Go back to the growth rate equation knowing this. Plugging in for $\hat{k} = \hat{y}$, rearranging, and using the assumption that $\epsilon_K + \epsilon_L = 1$, we have that

\[\hat{y} = \frac{\epsilon_K}{1-\epsilon_K} \hat{B} + \hat{A}.\]

The growth rate of output per worker depends only on the two types of technological progress. I didn’t have to say anything about exactly how capital is accumulated, but if we are insisting that the return on capital is constant, then it has to be that capital and output grow at the same rate, which means the growth rate of output per worker is dependent on only the growth rate of the technology.

Ok, what does this leave us with? Along a BGP, the growth rate of output per worker is supposed to be constant. There are two ways to make this happen:

  1. Cobb-Douglas production, constant labor- and capital-augmenting progress. $\hat{B}$ can be non-zero, but then we have to ensure that $\epsilon_K$ is constant as well. This restricts the production function to be a Cobb-Douglas, where the capital and labor elasticities never change. In this case, the fact that we call it “capital-augmenting” or “labor-augmenting” progress is irrelevant, as mathematically they are equivalent.

  2. Non-Cobb-Douglas production, but $\hat{B} = 0$. When production isn’t Cobb-Douglas, the elasticity $\epsilon_K$ will change if $\hat{B}$ is non-zero. So the only way for this to work without a Cobb-Douglas production function is if capital-augmenting progress is exactly zero. Here, there really is a difference between “capital-augmenting” and “labor-augmenting” progress, and we have to have zero capital-augmenting progress.

These are pretty restrictive. If you think an economy has a BGP, then what Uzawa indicates is that you have to think it either has a very specific production function (Cobb-Douglas) or a very specific rate of progress in capital-augmenting technology (zero).

It feels implausible that the real world actually would have the exact right conditions for a BGP to happen. Taking a step back, maybe production isn’t even constant returns to scale in capital and labor in the first place. Are the empirics behind the BGP so strong that they should compel us to buy the theoretical restrictions they imply?

Relaxing Uzawa

Well, maybe Uzawa’s restrictions aren’t necessary or aren’t as knife-edged as they seem. Here’s a few examples. Grossman, Helpman, Oberfield, Sampson, who got me going on this whole topic, propose one solution. What GHOS suggest is that human capital growth may be sufficient to break the strict nature of Uzawa’s conditions. That is, they can write down a model of economic growth that doesn’t require Cobb-Douglas production, has a positive growth rate of capital-augmenting technological change, and yet still has a BGP.

Chad Jones has a different approach, and he establishes conditions by which the aggregate production function may be Cobb-Douglas, even though the underlying technical production functions of firms are not. This makes a BGP a reasonable outcome.

But this all presumes that we want to rescue the BGP - that economic growth necessarily involves a constant long-run growth rate, a stable capital share, and a constant return to capital. Given the information in the prior post, the question remains whether in fact the BGP is worth rescuing.

Structural Change

Let me give an example of where the dependence on BGP’s is possibly a problem. A lot of effort has been expended writing down models of economic growth that explicitly allow for multiple sectors (agriculture, manufacturing, services, for example) but that also display a BGP. You can think of there being an ancillary Kaldor fact that these papers want to explain, which is that there has been significant shifts of labor between sectors (generally, ag to manufacturing to services) over time in most economies.

Acemoglu and Guerrieri, Kongsamut, Rebelo, and Xie, Ngai and Pissarides, among others, are built to deliver both structural change and a BGP. To do so is not trivial, so these models are quite clever in how they set things up to make this work out. In some way, they are writing down new versions of Uzawa that also allow them to account for structural change. They establish conditions for preferences or production functions that allow for economic activity to wax and wane in different sectors, but in aggregate continues to be consistent with Kaldor’s facts.

The danger here is presuming the original Kaldor facts are required. They work very hard to show that structural change can happen while the growth rate of output per capita remains constant in the long run. But what if the growth rate of output per capita isn’t constant in the long run? Or, if you don’t want to abandon the concept of a BGP altogether, what if there are different BGP’s depending on what stage of structural change you are in?

Remember the French figure from the prior post. You could argue that France had both a pre-WWII BGP (with relatively low levels of GDP per capita and a low growth rate) and a post-WWII BGP (with a higher level and growth rate). Who is to say that this kind of shift to a different BGP, perhaps with a lower growth rate, isn’t possible in a country like the US today? Who is to say that level shifts in the BGP, perhaps up or down, are not possible in the future based on structural change?

Are we missing the point?

Existing models that assume a BGP cannot, by design, deliver the kinds of jumps seen in the French example, or in cases like South Korea or Japan as they developed. Yes, those countries appeared to go from one BGP to another, so the existing models are good at capturing the dynamics within each BGP. But they have no way of explaining why it was that South Korea took off for a different BGP in about 1960, or why France’s growth rate shot up after World War II. These shifts are completely exogenous in these models.

Let me ask it this way. If I could write down a model that could predict the conditions under which a country would accelerate like South Korea from poverty to rich-country living standards, would I discard it just because it failed to predict a strict BGP in the long run? I hope I wouldn’t, because a model that could identify those conditions would be incredibly valuable. But there is a tendency for growth economists to be skeptical of anything that doesn’t have a BGP in the end.

To take it from the other side, it may be that even if BGP’s are not strictly true in the data, that models with BGP’s are effective in capturing most of the interesting dynamics. The strict assumptions behind them need not be exactly true in real life, but are useful for simplifying our analysis. But the danger here is that we go too far, assert these strict assumptions are true because they deliver a BGP, and that would appear to be going in the wrong direction.

I’ve been operating in full skepticism mode, and so I should probably stop now before I manage to insult everyone who has done growth work in the last few decades. That, and I don’t want to discourage everyone from buying my textbook with Chad Jones that talks very explicitly about BGP’s and models that deliver them.