The Growth Equation Derived

From Jodi Beggs' website (ec10help.blogspot.com) - Thanks Jodi

In our discussion, we are assuming a particular functional form for the f function (remember, Y=A*f(K,L). 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

Hopes this makes sense for the math inclined, otherwise, you dont really need to know this.