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Given a total cost function, for example, $$ C = q {w}^{3/4}{v}^{1/4} $$ and Shephard's Lemma, how do you find the underlying production function $q(k,l)$?
For this example, Shephard's Lemma provides the constant output demand functions: $$ {l}_{c} = \frac{3}{4}q({\frac{v}{w}})^{1/4} $$ $$ {k}_{c} = \frac{1}{4}q({\frac{w}{v}})^{3/4} $$
How do we use this information to find $q(k,l)$?
To find the production function, you can solve for $\frac{v}{w}$ in $ {l}_{c}$ and ${k}_{c}$ and set $\frac{v}{w}$ = $\frac{v}{w}$ then solve for $q$.
This will yield
$$ \frac{v}{w} = (\frac{4{l}_{c}}{3q})^{4} $$ $$ \frac{v}{w} = (\frac{4{k}_{c}}{q})^\frac{-4}{3} $$
Set both equations equal
$$ (\frac{4{l}_{c}}{3q})^{4} = (\frac{4{k}_{c}}{q})^\frac{-4}{3} $$ Take both sides to the power of $1/4$ this eliminates the exponent in the left expression $$(\frac{4{l}_{c}}{3q}) = (\frac{4{k}_{c}}{q})^{-1/3} $$
Then solve for $q$
$$ (q^{-4/3}) \frac{4l_c}{3}= 4^{-1/3} k_c^{-1/3}$$ take both sides to the power of $-3$ $$ (q^4) \frac{3^3}{4^3 l_c^{3}}= 4 k_c $$ $$ q^4= \frac{256k_c l_c^{3}}{27} $$ This the production function $$ q(k,l)= \frac{4k^{1/4} l^{3/4}}{27^{1/4}} $$
