Question

A “Cobb–Douglas” production function relates production (Q) to factors of production,
capital (K), labor (L), and raw materials (M), and an error term u using the equation:
? = ???1??2M?3? ?, where ?, ?1, ?2, and ?3 are production parameters.

a) Suppose that you have data on production and the factors of production from a
random sample of firms with the same Cobb–Douglas production function. How
would you propose to use OLS regression analysis to estimate the above production
parameters, which is a nonlinear function?

b) Based on your regression model, how to interpret ?, ?1, ?2, and ?3? Suppose that
you believe that the marginal return of labor is not a constant but rather increases
when K increases. How could you modify your model to capture this effect?

Answer 1

(a) The production function es given as : Q=AK L Me, which is nonlinear. Hawever, this can be linearize with ligarithmic treansferz- mation and then OLS regreRsion analysis can be used to estimate the production funetion, given data on Q,K, L and M .Let ui kog Logacithms Log Q - log (Ax"'2*mse") > Log a = log a t+ Bi log k+ Pz bog L + Balogm + u This representation is linear dn parcametegy Hence, te applying Logarcithmic treanstoemation on given data Cire., Q, K, L and M), we can estimate te production function. on both sides as follows : (b) In thi above model, p.'s can be dnterRpzeted as elasticeties. For example, B, is thi autput elasticity of capital, sinee Sèmitarhy, B2 and Bg are the output elasti city af labor ond raw materials cespectively. Considercing that marginal ceturen of labot increaxs when k in creases, Geven Q= 1 kP'Lm* we have aeginal retuen of increases if kinerea) labore - da AKPLPM3 [multiplying and dividing by 2] = B2 to show that P2u thi autput elesticity This is an alternative way of labore. On ehe other hand 1 s lhi production level cith 1 curit of each of lhe inpunt, which can be venitied from beh speciticatims. Using ariginal specification Q= 1C1)P"1)*c1n -. A ltennatively, when K=L=M=1, log k-logL= logm= 0 The original specification au well as the dogarcithmic specitication ensures that marginal reetuen of laborc inereases as K inereases asshown above.