# The question

A firm’s production function is given by

$$q=F(L,K)=L^{1\over{4}}K^{1\over{4}}$$

find the firm's cost function $$C(w,r,q)$$,

# What I know so far

I'm aware that the Technical rate of subsitution (TRS) is:

$${MP_L\over{MP_K}}={{\partial{F}\over{\partial{L}}}\over{\partial{F}\over{\partial{K}}}}={K\over{L}}$$

but I'm not sure how this can help me find the cost function

## 1 Answer

A firm's production function is a relationship between inputs and outputs (thus what are the combinations of input and out that can be obtained given the available technology). y-axis is Output and x-axis is for inputs. A firm's cost function is a relationship between costs (expressed in monetary terms) and output. Thus, for every level of output, and market prices for inputs, what is the cheapest way to produce it. Output on the x-axis and Cost (monetary units) on the y-axis.

This relationship isn't as straight forward if prices aren't linear. However, most models will assume linear prices. If that is the case, you can kind of see the cost function as the inverse of the production function (essentially flipping the axes), because more inputs for an unit of output imply more costs.

As far as actually getting the cost function from the production function, you have to start by finding the argmin for this cost minimization problem: $$\min (w \cdot k + r \cdot k)$$ s.t: $$q=L^{1/4}K^{1/4}$$ F.O.C.

$w={\partial{F}\over{\partial{L}}}=f_{L}(L,K)$

$r={\partial{F}\over{\partial{K}}}=f_{K}(L,K)$

So wages will be equal to marginal productivity of Labor and interest rate will be equal to marginal productivity of capital in equilibrium. Let me know if you don't know how to keep going from here.

• This is fine. My issue is what is the form of the cost function being that I know this. Do the FOC solutions for $w$ and $r$ become $g(w,r)$? – FreakconFrank Oct 31 '16 at 15:11
• Which is $HOD$ $({1\over{2}})$ correct? – FreakconFrank Oct 31 '16 at 15:21
• I'm not 100% sure how calculate that also – FreakconFrank Oct 31 '16 at 15:40
• Once you have the marginal products equal to each of the prices (wages and interest rate), you can divide the two equalities, you will get K for example in terms of L, w and k. – Jéssica Dutra Nov 2 '16 at 1:29
• Once you have the marginal products equal to each of the prices (wages and interest rate), you can divide the two equalities, you will get K for example in terms of L, w and r. Then you substitute back into the constraint (which will then become K in terms of q, w and r. Since you have the K to L relationship you derived you can also get L in terms of q, w and r – Jéssica Dutra Nov 2 '16 at 1:31