00:01
So here we're given a production function and we're asked a whole bunch of stuff about it.
00:04
The first thing we're asked is the elasticity of substitution or what economists would call sigma kl, right? that's capturing the elasticity of substitution, which is the percentage change in k over l with respect to the percentage change in r over w, right? so this is saying at a very basic level, or, you know, if there is a percentage change in the ratio of factor prices, i .e.
00:36
If capital gets twice as expensive relative to labor, what happens to the firm's use of capital relative to labor? if capital gets more expensive relative to labor, does the firm use more capital or less capital, right? that's what's going on.
00:51
So this in the language of calculus is the derivative.
00:56
Of log k over l with respect to the derivative of the log of r over w.
01:03
And you may say this is too many arguments.
01:07
I don't understand what's going on here.
01:09
But we know that in a competitive factor market, we can pin down some things.
01:16
We know that in particular, that the rental rate on capital should be equal to the marginal product of capital.
01:25
And the wage should be equal to the marginal product of labor, right? those are pretty common definitions, and finding those out is pretty straightforward, right? so we have the marginal product of capital is equal.
01:40
We're going to use our chain rule here to the 0 .5 plus l to the 0 .5.
01:47
So that's the outside part.
01:49
Now the inside part is 0 .5k to the minus 0 .5 .5.
01:55
And similarly, the marginal product of labor is chain rule, bring the two down, k to the 0 .5 plus l to the 0 .5, 0 .5l to the minus 0 .5.
02:11
Right? so this gives us expressions for relating the real interest rate to capital and labor and the marginal product of labor to capital and labor.
02:26
Okay, so now let's try to relate the two quantities in the derivative, right? so r over w here is going to equal to the marginal product of capital over the marginal product of labor.
02:41
And when we divide those things, you see how much cancels out, right? you get the two k to the 0 .5 plus l to the 0 .5 on the top and bottom canceling out.
02:50
So you get 0 .5k to the minus 0 .5 over 0 .5 l.
02:56
To the minus 0 .5, which is equal to k over, sorry, the 0 .5s cancel out, and we get the square root of l over the square root of k, right? so this tells me, right, if we continue manipulating things just a little bit, i'm going to start off by squaring both sides, and i want to keep r over w as a variable, right? so i am going to try to do that.
03:26
And then this tells me that i get k over l is equal to r over w to the minus two, right? i flip both sides, basically speaking.
03:41
So now i can take the lawn of both sides, the log of k over l is therefore equal to minus two times the log of r over w.
03:51
That's just my basic properties of logarithms.
03:54
And when i complete this now, well, i've got the function that i've needed...