Transcription
Let me give you an application, an economic application of this. Recall the Cobb-Douglas production function is this, where capital A is some positive parameter, little a is some number between zero and 1, and little b is between zero and 1. This is a non-linear function. However, it’s possible to linearize this function if you take the logarithms of both sides. For example, the logarithm of Q must be equal to the logarithm of this, A times K to the a times L to the b. You just take the logarithms of both sides.
And, if we apply this rule #1, and actually, we’re applying an extended rule #1, if you have three terms – let me just write this. If you have the logarithm of u times v times w, what is that equal to? What would you guess? If you have three terms in there, u, v, and w, u, v, and w are variables. [Student comment] You just add them, each of them. So, it’s the log of u plus the log of v plus the log of w.
Well, let’s apply this extended rule #1 to this right-hand side expression here. So, this is the logarithm of A plus the logarithm of K to the a plus the logarithm of L to the b. Because you have three terms: A, K to the a, L to the b. So, you can do that. And you can break these two terms down further. So, this is the logarithm of A. I’ll just repeat that first term. But this second term, you can bring this little a down. Plus little a times the logarithm of K. Plus b times the logarithm of L. And what’s nice about this expression is this is a nice linear expression in terms of the logarithms of the variables. This is log Q equals some constant, the log of A, plus little a times the log of K plus little b times the log of L.
Are some of you taking 351? So, you know about linear regression? And what you can do is, if you have, for example, a time series expression. You have quantity for several different time periods. You have measures of the capital stock for several different time periods. You have measures of the labor stock over time. Then, what you could do is measure output, not in its natural values, but in its logarithmic values. And regress that on the logarithm of capital and the logarithm of labor. You could do this linear regression of log Q on log K and log L. Then, the coefficients you get, little a and little b, will just be the coefficients from doing a linear regression. Only you’re doing your regression analysis in the logarithms of the variables instead of their natural values. And the constant you get in your regression will just be interpreted as the log of A. So, this has great practical significance. Sometimes, you could take non-linear functions and you could use the logarithmic transformation to linearize the equations. And then estimate them in a straightforward way using linear regression.