<p>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.</p>

<p>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.</p>

<p>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:
$$
q=L^{1/4}K^{1/4}
$$
<strong>F.O.C.</strong></p>

<p>$w={\partial{F}\over{\partial{L}}}=f_{L}(L,K)$</p>

<p>$r={\partial{F}\over{\partial{K}}}=f_{K}(L,K)$</p>

<p>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.</p>

CC BY-SA 3.0
2016-10-31T15:11:51.663Z
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 $g(w,r)$?
Bensstats
CC BY-SA 3.0
2016-10-31T15:21:52.617Z
Which is $HOD$ $({1\over{2}})$ correct?
Bensstats
CC BY-SA 3.0
2016-10-31T15:40:39.903Z
I'm not 100% sure how calculate that also
Bensstats
CC BY-SA 3.0
2016-11-02T01:29:52.343Z
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
CC BY-SA 3.0
2016-11-02T01:31:28.293Z
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
CC BY-SA 3.0
2016-10-31T03:36:50.100Z
2016-11-02T01:28:00.277Z
2016-11-02T01:28:00.277Z
Jéssica Dutra
<h1>The question</h1>
<p>A firm’s production function is given by</p>
<p><span class="math-container">$$q=F(L,K)=L^{1\over{4}}K^{1\over{4}}$$</span></p>
<p>find the firm's cost function <span class="math-container">$C(w,r,q)$</span>,</p>
<h1>What I know so far</h1>
<p>I'm aware that the <strong>Technical rate of subsitution (TRS)</strong> is:</p>
<p><span class="math-container">$${MP_L\over{MP_K}}={{\partial{F}\over{\partial{L}}}\over{\partial{F}\over{\partial{K}}}}={K\over{L}}$$</span></p>
<p>but I'm not sure how this can help me find the cost function</p>

<microeconomics><production-function>
How to find a firm's cost function based on its production function
Bensstats
# How to find a firm's cost function based on its production function

(Asked by: username_0 on 2016-10-30)

# 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

**Tags:** microeconomics, production-function

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## 1 answers

### Answer by username_1 on 2016-10-31. Score: 1

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.

> **Comments:**
> **username_0**: 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 $g(w,r)$?  
> **username_0**: Which is $HOD$ $({1\over{2}})$ correct?  
> **username_0**: I'm not 100% sure how calculate that also  
> **username_1**: 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.  
> **username_1**: 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  

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