Marginal cost given (Cobb-Douglas) production

The Cobb Douglas production function with constants returns to scale

$$y = prod_i x_i^{alpha_i} = A prod_i left(frac{x_i}{alpha_i}right)^{alpha_i} ,$$

where $A:= prod_i alpha_i^{alpha_i}$ annoying constant.

Cost minimization with perfect competition

$$min_x p^top x lvert y = prod left(frac{x_i}{alpha_i}right)^{alpha_i},$$

implies FOC

$$p_j - lambda left[Aprod_i left(frac{x_i}{alpha_i}right)^{alpha_i}right] left(frac{x_j}{alpha_j}right)^{-1}=0 Leftrightarrow frac{lambda y}{p_j} =frac{x_j}{alpha_j},$$

insert in production function to get

$$y = Aprod_i left( frac{lambda y}{p_i}right)^{alpha_i} Leftrightarrow lambda = frac{bar p}{A},$$

where $bar p := prod_i p_i^{alpha_i}$ which is a price index for production factors. Reinsert in FOC to get

$$p_j - frac{bar p}{A}left[Aprod_i left(frac{x_i}{alpha_i}right)^{alpha_i}right] left(frac{x_j}{alpha_j}right)^{-1}=p_j - frac{bar p}{A}y left(frac{x_j}{alpha_j}right)^{-1}=0,$$

from which you find

$$(1) p_j x_j = frac{alpha_j bar p}{A} cdot y,$$

using this formula several implications follow. First take sum over $j$ and use $sum_j alpha_j = 1$ to get cost function

$$(2) C(p,y)= frac{bar p}{A} y.$$

Set $y=1$ to get unit cost, diffrentiate with $y$ to get marginal cost, divide with y to get average cost and it follows that

$$(3) C(p,1) = frac{partial C(p,y)}{partial y} = frac{C(p,y)}{y} = frac{bar p}{A}.$$ Finally from (1) divide with $x_j$ to get

$$(4) x_j^star(p,y) = frac{alpha_j C(p,y)}{p_j} = frac{alpha_j}{p_j} frac{bar p}{A} y,$$ stating that the share $alpha_j$ of the costs are used on production factor $x_j$ and under perfect competition there is zero profit so revenue equals costs, hence $alpha_j$ of the revenue must be used (price of $y$ must be unit cost, marginal cost, average cost = $C(p,1)$).

With only two production factors the cost function can be written as

$$C(p,y) = C(p_1,p_2,y) = frac{bar p}{A} y = frac{p_1^{alpha} p_2^{1-alpha}}{alpha^alpha (1-alpha)^{(1-alpha)}} y,$$ hence marginal cost are easily seen to be

$$frac{partial C(p,y)}{partial y} = frac{p_1^{alpha} p_2^{1-alpha}}{alpha^alpha (1-alpha)^{(1-alpha)}}.$$