The Solow Model and the Steady State

Welcome back. Let's continue our exploration of the Solow Growth Model. In our last video, we covered how physical capital faces the iron logic of diminishing returns.

Now let's turn to another unfortunate aspect of physical capital: capital rusts. Roads get potholes and need to be repaired, tools wear out, trucks break down. In short, we say that capital depreciates. Now let's put the amount of capital on the horizontal axis and the amount of depreciation on the vertical axis. We can then model the relationship like this. Depreciation increases at a constant rate as the capital stock increases. The more capital you have, the more capital depreciation you have.

Now let's add a new aspect to our model. Where does the money for capital accumulation come from? From savings and investment. When we create economic output, we can either consume it or save it. What we don't consume can be saved and invested in new capital. So, suppose we invest a constant fraction of our output. Let's say we devote 3 of every 10 units of output or 30% of output to investment. We can now add an investment curve to our graph. It'll mimic the shape of the output line since investment is just a constant fraction of output. Notice that our first units of capital -- they're very productive and so they create a lot of output and thus also a lot of investment. But as we add more and more units of capital, we get less output and also less investment. That's the iron logic of diminishing returns once again.

Now let's put investment and depreciation on the same graph. Depreciation is growing at the same rate as the capital stock grows. Each new unit of capital creates an equal amount of depreciation. Now notice that when investment is greater than depreciation, that means the capital stock must be growing. We're adding more units of capital than are depreciating. But as the capital stock grows – investment and depreciation -- they're on a crash course to intersect. When this happens, we've reached what is called the Steady-State Level of Capital. The steady-state is the key to understanding the Solow Model. At the steady-state, an investment is equal to depreciation. That means that all of investment is being used just to repair and replace the existing capital stock. No new capital is being created.

Now remember, we've assumed that all the other variables in the model -- they're not changing. So, if the capital stock isn't growing, nothing is growing. In other words, when we reach the Steady-State Level of Capital we've also reached the Steady-State Level of Output. Now suppose you ended up on the other side of the steady-state point -- over here. You'd find that depreciation is greater than investment. That means some of the capital stock needs repair, but there isn't enough investment to do all of the needed repairs, so the capital stock shrinks, pushing you back towards the steady-state. So, to the left of the steady-state we have investment greater than depreciation and the capital stock is growing. To the right of the steady-state we have the opposite -- depreciation is greater than investment, and the capital stock -- it's shrinking. Either way, we always end up moving towards the steady-state.

Now let's go back to our earlier example of Germany after the end of World War II. Since the capital stock is low, it's also very productive and we get a lot of output from the first new roads and factories after the war. We've already mentioned that point. But in addition, we now see that when the capital stock is very productive and producing a lot of output, we will also be producing a lot of investment. So, in the next period the capital stock will be even bigger than before and we'll get even more output. Plus, since the capital stock is low, we don't have much depreciation to take care of. So, with the investment, it will mostly be generating new capital, not replacing old capital.

Now over time, however, both of these forces -- they weaken. The returns to capital diminish and depreciation eats up more and more of investment. A country with a lot of roads, and bridges and factories -- it's doing well, but it also has to invest a lot just to maintain all those roads and bridges and factories. And this is exactly what we saw in Germany and Japan after World War II. Growth rates started out very high, but as those countries caught up, growth rates declined. Now perhaps our friend K still has one more trick up his sleeve to get the economy growing.

What if we started to save more of our output? A higher savings rate shifts the investment curve up like this. Now investment is higher than depreciation, so we're adding to the capital stock and the economy is back to growing. However, you can see that the same dynamic exists as before. The iron logic of diminishing returns means that we'll again end up at a new steady-state level of capital. The higher savings rate -- it spurs growth for a time and it does increase the steady-state level of output. But, at the new steady-state, investment once again equals depreciation and we get zero economic growth. Accumulation of physical capital can only generate temporary growth.

In our next video, we'll take a look at how human capital influences growth.