# Office Hours: The Solow Model: Investments vs. Ideas

Ideas are a major factor in economic growth . But so are saving and investing. If you were given the choice between living in an inventive (more ideas) or a thrifty

Ideas are a major factor in economic growth. But so are saving and investing. If you were given the choice between living in an inventive (more ideas) or a thrifty (more savings) country, which would you choose?

The Solow model of economic growth, which we recently covered in Principles of Macroeconomics, can help you make the choice. In this Office Hours video, Mary Clare Peate will use our simplified version of the Solow model to show you an easy way to work out each country’s economic prospects, and then compare them to see where you’d rather be.

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## Transcript

Today, we're going to take a closer look at the Solow Model by evaluating how different inputs affect a country's economy. Consider the following two Countries: Inventive and Thrifty. In Inventive, the country's economy grows according to the following production function: gross domestic product equals two times the square root of K, and it devotes 25% of GDP to making new investment goods. Thrifty's production function is given by GDP equals the square root of K, and it devotes 50% of its GDP to making new investment goods. Both countries begin with \$100 worth of capital, and both countries have the same capital depreciation rates and the same population.

If you had to choose, in which country would you prefer to live? As always, check out our recent videos on the Solow Model, and then try to solve this problem by yourself. If you're stuck, then come back and we'll work through it together. Ready? I really like this question. To get a better idea of what this question is actually asking, let's compare the two countries side by side to understand similarities and differences.

First, we'll compare the two countries' production functions, and we see that they differ by a multiple of two, which loosely translates to the country's ideas or productivity. So Inventive, as its name suggests, is more productive with its factor of production, capital, than Thrifty is. So, what does Thrifty have going for it? Not surprisingly, Thrifty has that higher savings rate. It's saving 50% of everything it produces GDP-wise each year, versus Inventive's 25%. And everything else is the same: capital stock, depreciation rates, and population.

So, what this question is really asking is, is it more important for a country to have a high savings rate like Thrifty, or have more ideas and therefore be more productive, like Inventive? Where would you prefer to live? The trickiest part here is translating what an ordinary citizen cares about into something the Solow Model actually tracks. Solow doesn't measure faster Wi-Fi, even though we all care about that. I mean, sure, we can and we will look at how much GDP each country has, how much it's investing in its capital stock, the usual Solow suspects. But the real key here is not so much GDP per se, but rather the GDP that's left over once we're done investing: consumption. Consumption is that neglected variable in the Solow Model, but it's arguably what citizens will care most about given the Simple Solow Model framework.

So, to outline our steps for solving the problem, we'll first track Thrifty's economic prospects on those three dimensions: GDP, investment, and consumption. We'll then do the exact same thing for Inventive, and finally we'll compare the two to decide where we'd rather live. The first step is to find Thrifty's economic prospects. Thrifty's production function is GDP equals the square root of K. Its initial capital stock is 100, so the square root of 100 is 10. This country is producing 10. And, if this country is saving 50% of its GDP each year, then the country is saving 5 of that 10.

More formally, we can graph its investment function as I equals 0.5 times the square root of K. If it's producing 10 and investing 5, what's left over for consumption? 10 minus 5 is 5. Now on to step two, which is to do the exact same thing for Inventive. Its production function is GDP equals 2 times the square root of K. And, given that it has the same initial capital stock as Thrifty, 100, its GDP this year is the square root of 100 times 2, or 20. If it's investing 25% of GDP per year, 25% of 20 is 5.

More generally, its investment curve is 0.5 times the square root of K. And again, consumption is just the leftover GDP after investment. So, 20 minus 5, or 15. A quick aside here, notice that the two countries' investment curves are the same. We'll revisit this later. So, we now move on to step three, which is to compare the two. Inventive seems like the clear winner here. Not only does it have a much higher GDP than Thrifty, but more importantly for the citizen, the amount of GDP available for consumption is much higher: Inventive's 15 compared to Thrifty's 5.

Two things to note here. First, you may think the difference between consuming something like 5 and 15 is really boring. Like, who cares? Those numbers are really small. So, let's try to put it in real-world terms. Inventive citizens consume three times as much as Thrifty citizens. This means that if Thrifty citizens consumed, say, \$30,000 worth of stuff this year, Inventive citizens would be consuming \$90,000 worth of stuff this year. Suddenly, 5 versus 15 seems like a much bigger deal. And second, even though population doesn't factor directly into our Super Simple Solow Model, it's important that the populations of these two countries are equal, as the problem originally states. Given equal populations, we know that GDP and consumption per person, or per capita, will also be higher in Inventive than in Thrifty.

Now, if we were in a normal classroom right now, this is probably the time when you would raise your hand and say something like, "This looks great. But, what about these two countries in their steady states? What if Thrifty, because of all of their saving, will be far better off than Inventive in another, I don't know, say 10 years?" This is exactly the question you should be asking. It means that you understand the whole point of the Solow Model. It turns out that our answer will hold in the steady state. Inventive will produce and consume more GDP in the long run.

If you want to better understand why and how it holds, check out our practice problems at the end of the video. In summary, Inventive citizens get to consume more not only today, but also, tomorrow, making it a more desirable country to live in. What does this tell us? It is incredibly important for a country to have new ideas and become more productive. Saving is great, and will do a lot to further a country's economic growth and prosperity, but it can only get us so far.

Let's focus on the first practice problem and put together what we know from the previous videos about steady state. We know that the steady state level of capital stock is that level of capital stock at which capital neither grows, nor shrinks, but stays the same. It would grow if the quantity taken away from GDP to be invested as new capital (call this "investment") was greater than the quantity of capital stock which is eroded or destroyed at that time (call this "depreciation"). And it would decrease, if depreciation was greater than investment. So the level of capital at which capital is constant is the level at which investment equals depreciation.
Now let's look at the graph to see that level. There is only one level of capital (on the horizontal axis) at which investment equals depreciation: the point of intersection between the investment line (function) and the depreciation line (function). To the left of that point, investment is greater than depreciation, so capital still grows. To the right of it, investment is lower than depreciation, so capital, on the net, erodes, or shrinks.
Let's see if that point is 100, the value of capital with which we start.
We are looking at country Inventive.
The video shows how to calculate GDP; investment, and depreciation.
GDP = 2 times square root of K = 2 times square root of 100 = 2 times 10 = 20
investment = investment rate times GDP at that point = .25 times 20 = 5
depreciation = depreciation rate times capital at that point = .03 times 100 = 3
So, can 100 be the steady state level of capital. No, because investment is greater than depreciation at that point. Capital is on the net growing. On the graph, this means we are still at the left of the intersection. So the steady state level of capital must be greater than 100.
Perhaps it is 111?
If we calculate we see that even when capital is 111, investment will still be greater than depreciation, so we shall still be at the left of the steady state level.
Can it be 278?
Yes! At that point, if we perform the calculations, we discover that investment is approximately equal to depreciation.
So it seems that what we know from the videos was enough to solve this problem.
If we search for youtube videos about "calculating the steady state level of capital in the Solow model" we see how to find the general solution, by equating the investment function with the depreciation function. But we did not need to do that here. We just tried possible values of capital to see whether, at any of these possible values, it so happened that investment was equal to depreciation.