# Finding production given total cost (shephard's lemma)

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 $$q(k,l)= \frac{4k^{1/4} l^{3/4}}{27^{1/4}}$$