Physical Capital and Diminishing Returns

In our last video, we introduced the variables in our Super Simple Solow Model. We have physical capital, represented by "K," human capital, represented by "e" times "L," and ideas, represented by "A." In this video, we're going to hold human capital and ideas constant. That will let us focus in on K so we can show what happens to output when the amount of physical capital changes.

Since capital is the only input, output is a function just of the quantity of capital. Let's write output with the letter "Y." Then we can say that Y is a function of K. Output is a function of the quantity of capital. What properties should our production function have? First, it makes sense that more K increases output. Recall from our earlier video, our farmer. A farmer with a tractor can produce a lot more output than a farmer with just a shovel. Similarly, a farmer with two tractors can produce more output than a farmer with just one tractor. If we graph capital on the horizontal axis and output on the vertical axis, we're going to see a positive relationship. As capital goes up, output goes up. That seems pretty straightforward. The second property our production function should have is that while more capital produces more output, it should do so at a diminishing rate.

What do I mean by that? Let's go back to our farmer. The first tractor he gets is the most productive. It helps him grow a lot more wheat. The second tractor he might use if the first tractor -- it breaks down. So, the second tractor is less productive than the first. The third tractor is maybe just a spare in case both break down. So, the third tractor will boost his output even less than did the second. Said another way, the farmer will allocate his tractors so that the first tractor, he's going to allocate to the most important, the most productive task. Meaning that subsequent tractors -- the farmer will allocate them to less and less productive tasks. We call this the Iron Logic of Diminishing Returns.

To represent both of these properties, we can use a simple production function, one which we're already familiar with: the square root function. Output equals the square root of the capital inputs. So, if we input 1 unit of capital, output is 1. If we input 4 units of capital, output is 2. If we input 9 units of capital, output is… 3. The marginal product of capital describes how much additional output is produced with each additional unit of capital. Notice that the marginal product of the first unit of capital is really high. But as the capital stock grows, the marginal product of capital is less and less and less.

Already, we can explain one of our puzzles. Recall that growth was fast in Germany and Japan after World War II. That makes sense, because after the war, those countries -- they didn't have a lot of capital. So that meant that the first units of capital had a very high marginal product. The first road between two cities or the first tractor on a farm, or the first new steel factory -- that gets you a lot of additional output. Capital's very productive when you don't have a lot of it. But don't forget that Germany and Japan were growing from a low base. You can grow fast when you don't have a lot, but all else being the same, you'd rather have more and grow slower. So, capital can drive growth, but because of the iron logic of diminishing returns, the same additions to the capital stock may get you less and less output. Unfortunately for K, in the next video we'll show that capital has another problem to deal with.