After regressing historical data, I got an equation c*ln A+ a*ln K+b* ln L+error, where A = technological change, K=kapital and L=labour. How do I change this equation to a general Cobb-Douglas function ?
I am worry that converting "c*ln A+ a*ln K+b* ln L+error" is not simple as to "A K^a L^b".
in cobb-douglous function, intercept denotes technological change. Why do you have another intercept?
what is c? If by C, you mean constant/intercept, then your constant = LnA.
A K^a L^b". transforms into lnA + aln K+ bln L + u
transform K and L and Output into log forms, regress log output on log K and Log L, the regression must have a constant term.
Consider the Cobb douglas production function as :-
Q=ALaKb
Where Q=output,K=capital i/p L=Labor i/p and A-technology change
Taking natural log on both sides ,we get,
lnQ=ln A+a lnL +b lnK where ln denotes natural log
So you will regress ln Q on ln L and ln K
This is kind of log-log or double log model where your intercept term or constant term is equal to lnA and a,b denotes the labor and capital elasticities of output...
Hope this helps...
Thanks for all answer given here.
Supposed that a production function is
Y = A*K^a*L^b,
then,
MPL = (A*b)* K^a*L^(b-1) = b (Y/L)
MPK = (A*a ) *K^ ( a-1 ) *L^b= a (Y/L)
My questions are :
1). How do I find an optimal labour and optimal capital ?
2). if L dan K are in monetary terms, What are equations for MPK and MPL ?
Apply Ln (Natural Logarithm) to the given equations on both sides. Estimate the parameters by OLS or GLS (if required). Get the predicted equations for Ln(MPL^) and Ln(MPK^). Exponentiate these equation and get the optimal values for MPL and MPK.
Exp(Ln(MPL^))=MPL^ & Exp(Ln(MPK^))=MPK^. I hope this will help.
I hope one way is you can use the form of returns: Suppose that b1 + b2 = 1, model shows constant returns to scale. If b1 + b2 > 1, increasing returns to scale and if b1 + b2 < 1, diminishing returns to scale. So the b are elasticity, that can indicate optimal combination of resources returns to what ever variable under inquiry.
The generalized transformation of the C-D production function can help to noise elimination, easy translation and can ease correlation problems. The logarithmic transformation of the production function provides a log-linear form which is convenient and commonly used in econometric analyses using linear regression techniques. Employing a more general form of the function can allow for estimation of the coefficients value and statistically testing hypotheses about return to scale:
Taking the derivative of the logarithm form which can be interpreted as a percentage change of the parameters you are using.
Here you can further investigate on:
MC= ∆VC/∆Y ( output you assumed)
∆VC= W∆L
∆L/∆Y = which can state the change in quantity of labor you study assumed to affect a single (one) unit change in your inquired (dependent) which is 1/MPL
MC= W/MPL
MPL will rise then MCL will fall and vice-versa if the assumption is constant wage rate
On the other way APL indicates labor productivity but yet it arise which labor categories (specialization and division of labors also appears for scientific justification)
On the other side the diminishing returns of the parameters on your investigation related by the position of MPL APL
Maximizing the average product of your study L of your variable can be calculated on the slop of the curves with the deviation of dependent with L (Slope= dY/dL)
The two important answers have been addressed by
Niharika Joshi Bhatt
Basheer Ahmad