Thursday, February 09, 2006

Growth Rate Form of Production Function

Recall that in class I gave you the production function Y=Af(K,L), and then a couple of minutes later I gave you an equivalent equation in growth rates form, given by:

%chgY = %chgA + a%chgK + (1-a)%chgL

(I apologize for the abuse of notation, I'm somewhat limited in the characters I can use!) Some of you are likely wondering how on earth we can go from one of these to the other. So I will try to explain. Last semester I explained that a change in a variable (change in total cost for example) could be represented by the derivative of the variable (eg. marginal cost is the derivative of total cost). Now, a question for you: what is the derivative of ln(x)? It's dx/x, right? Or alternatively, the change in x divided by x. This is the definition of percent change, is it not? Thus, (instantaneous) percent change in a variable is the derivative of the natural log of the variable. Now, why is this useful?

In our discussion, we are assuming a particular functional form for the f function given above. Specifically, we are assuming what is called a Cobb-Douglas production function. This function is f(K,L)=(K^a)*(L^(1-a)) for some a between 0 and 1. So now you have Y=A*(K^a)*(L^(1-a)). To get the percent changes, take the natual log of both sides and then the derivative:

lnY = ln(A*(K^a)*(L^(1-a)))

lnY = lnA + ln(K^a) + ln(L^(1-a))

lnY = lnA + alnK + (1-a)lnL

dy/y = dA/A + adK/K +(1-a)dL/L (this is the total derivative since the terms are separable)

%chgY = %chgA + a%chgK + (1-a)%chgL

I hope that makes sense!


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