Cost minimizing choice of inputs | Microeconomics | Khan Academy

We are now going to continue our discussion of factor markets, and we're going to go beyond just thinking about labor as a factor. In fact, in this video, we're going to start thinking about capital as well, which we know is another one of the factors of production.

But just as a little bit of review, we've already thought about it from a firm's perspective on what is the rational amount of labor to hire based on the marginal revenue product of labor and based on the marginal factor cost of labor. So in the horizontal axis, we have the quantity of labor hired by the firm, and in the vertical axis, you have the wage rate, which you could view as the price of labor. We've seen this multiple times; you're likely to have a downward sloping marginal revenue product curve (MRP).

I'm going to be very specific that this one is the marginal revenue product of labor, and then we have the marginal factor cost curve. If we're assuming that this firm is in a competitive, perfectly competitive labor market, well, they're just going to have to pay whatever the wage is in the market, and so that's why we have a horizontal line there; so that's the marginal factor cost of labor. We've talked about multiple times that it's rational for the firm to keep hiring as long as the marginal revenue product of labor, as long as the incremental revenue that the firm gets for each of those people or each of those units of labor that they hire, is higher than the incremental cost of each of those units of labor.

It will keep hiring until these two lines intersect, and so it would be rational for it to hire that quantity of labor. I'll do this as the labor for the firm, and I'll put a little star over here so that quantity of labor. We can draw an analogous thing for capital, so this is how a firm thinks about that input, how it thinks about labor. But we could also do something similar for capital, or we could do it for land as well.

Hopefully, this is going to be the firm; the firm, as they think about capital, will see that they have analogous axes. The horizontal axis, right over here, is going to be the quantity not of labor but the quantity of capital. Then the vertical axis will be the price of capital; you could view that as the rent rate if you're thinking about maybe you're renting some type of machinery.

So you will have your marginal revenue product of capital; we could still imagine that you have diminishing returns, so that's why it's downward sloping. The marginal revenue product, and we typically use a K for capital, just as we don't get the C confused with other things. Then we have our marginal factor cost, which is really just... and we'll assume once again that this is a perfectly competitive capital market.

So you just have to pay whatever the market rate for renting that capital is, and so that would be the marginal factor cost of the capital. Once again, it makes sense to keep bringing on more and more capital as long as the incremental revenue that you get from each of those extra units of capital is higher than the cost of each of those extra units of capital.

Here it would be rational for the firm, if we're just looking at the dimension of capital, to produce this much. So this would be... actually, let me... this would be the capital; the quantity of capital for the firm to employ.

Now, an interesting question that might have already crossed your minds is that firms have a certain amount of resources that they are going to think about: well, how much do I put in labor versus how much do I put into capital? They don't just think about these dimensions of how many inputs of these factors they want; they have to think about them relative to each other.

To help us think through this, let's say that we are at a certain level of output. So let's say that our output right now... I don't know: our current output, our current output is... I'm just going to make up something: 1000 units per day. At our current output, we know what the marginal product of labor and the marginal product of capital is. Let's say that we know that our marginal product of labor at this output, remember it changes as we have very different output and we bring on more labor or more capital, so our marginal product of labor at that level is 90 units.

So another way to think about it: for every incremental unit of labor we bring on, we're going to be able to produce 90 more units of output. Let's say that the price of labor, which is the wage rate, is equal to ten dollars. Ten dollars per unit of labor. So let me call this output units... output units.

Let's say that the marginal product of capital, I'm just in a different color, the marginal product of capital right now is 80 output units. Output units. So every unit of this factor of capital, we are able to produce an incremental 80 output units. Let's say that the price of capital, which would be the rent, is equal to five dollars. Five dollars per input unit of the factor.

So at right at this moment, if I have an incremental dollar, would it be more rational for me to add more labor, or would it be more rational for me to add more capital? Pause this video and see if you can figure that out.

Well, to think about which one is more rational, you just think about which one do I get more of a bang for my buck. So per dollar, how many output units do I get when I put a dollar into labor versus per dollar how many output units do I get when I put that dollar into capital?

So let's do it first for labor. If you want your bang for the buck, so to speak, you would just take your marginal... let me do this in a different color. If you want your bang for a buck, you would just take your marginal product of labor, so your output, and divide it by the price. This is going to tell you output per dollar.

So in this situation, it's 90 output units—could we could say widgets for a general term for output units—output units over ten dollars. Over ten dollars. And so this is going to be equal to nine output units per dollar. So this is equal to nine output units per dollar, so that's our measure of our bang per bang for our buck when we put an incremental buck into labor.

Now, what about for capital? Well, our marginal product of capital divided by the price of capital right at this moment—remember it changes depending on our output level and different combinations—is going to be equal to 80 output units divided by five dollars, which is equal to 16 output units per dollar.

So which one would I get a better bang for my buck? Well, right at this moment, I'm getting a better bang for my buck from investing in capital. Every extra dollar I put, I get 16 output units. So it would be rational for this firm that wants to maximize its profit and reduce its cost, if it has an extra dollar to invest, it would put it into capital.

So maybe it puts it into capital, and then it gets a little bit more output, and then the marginal product of capital is likely to go down. You could imagine at some point these things might be equal, and then the firm might be indifferent between the two. Then maybe at some point, if they kept adding capital, then maybe you get a better bang for your buck from the labor.

In general, a firm would want to keep investing in one or the other until these two things are equal to each other. So, big picture, you would look at the marginal product of the factor divided by the price of the factor, and then you compare that to the marginal product of the other factors divided by the price of those other factors.

Whichever one has the best bang for the buck, that's where it would be rational to invest in. In some ways, one way is a very efficient combination is if you get to that point that you're indifferent when the marginal product divided by the prices of the various factors are equal to each other.

For example, if I were to tell you that we're at a different point of production, let me cordon this off. If we're at a different level of production where our marginal product of labor is equal to... I'll call it 10 widgets... saves time. Let's say that the price of labor is equal to five dollars, and let's say that the price of capital is equal to ten dollars.

What would have to be the marginal product of capital for me to be indifferent between labor and capital? Pause this video and try to figure that out.

Well, in order for me to be indifferent right over here, that means my marginal product of labor divided by price of labor needs to be equal to my marginal product of capital divided by my price of capital. And so I would have 10 over 5 would have to be equal to my marginal product of capital over 10. So 10 over 5; this is 2 widgets per dollar.

So if I want two widgets per dollar over here, this has got to be equal to 20. So at this point, I'm indifferent between producing between bringing on more capital versus labor because in either case, every dollar I bring on, if the marginal product of capital is 20, then I'm able to get two widgets per dollar of investing in either factor.