Consider the following true (CobbDouglas) production function: ln Y i = 0 + 1 ln L 1i + 2 ln L 2i

Consider the following “true” (Cobb–Douglas) production function:

ln Y_i = α₀ + α₁ ln L_1i + α₂ ln L_2i + α₃ ln K_i + u_i

where

Y = output

L₁ = production labor

L₂ = nonproduction labor

K = capital

But suppose the regression actually used in empirical investigation is

ln Y_i = β₀ + β₁ ln L_1i + β₂ ln K_i + u_i

On the assumption that you have cross-sectional data on the relevant variables,

a. Will E(β̂₁) = α₁ and E(β̂₂) = α₃?

b. Will the answer in (a) hold if it is known that L2 is an irrelevant input in the production function? Show the necessary derivations.