Input-output relationships for the Cobb-Douglas production function

The production formula: Q = Bo·L^BL ·K^BK, where Q=output, L=labor, K=capital, and Bo, BL and BK are positive parameters.

1. Elasticity of output for a Cobb-Douglas production function: the exponent on each factor of production = the elasticity of output (Q) with respect to that factor.
So for the function here, B_L = %DQ/%DL = e_Q,L and B_K = %DQ/%DK = e_Q,K.

Proof: Consider e_Q,L, and recall the formula for a point elasticity estimate, which in this case would be:

              dQ L
   e_Q,L = --- · --.
              dL   Q

Calculating dQ/dL:

    dQ                                       BL· Bo·L^BL·K^BK     BL· Q
    --- = BL· Bo·L^BL-1·K^BK = --------------------- = -------- (since Q = Bo·L^BL·K^BK).
    dL                                                   L                       L

Cutting out the middle terms to summarize results, we have:

    dQ     BL· Q
    --- = ---------
    dL         L

We can then multiply both sides by L/Q to isolate the BL term:

             dQ    L
    BL = --- · --- which = e_Q,L.
             dL    Q

2. Returns to scale for a Cobb-Douglas production function: the sum of the exponents on the factors of production = the elasticity of output (Q) with respect to the scale of production.

Proof: Since each exponent is the elasticity of output with respect to the related factor of production, we have:
B_L = %DQ/%DL which => %DQ = B_L·%DL, and
B_K = %DQ/%DK which => %DQ = B_K·%DK.

So when both L and K change,

%DQ = B_L·%DL + B_K·%DK.

For a change in scale, %DL = %DK = %DScale, so

%DQ = B_L·%DScale + B_K·%DScale =>
%DQ = (B_L + B_K)·%DScale.