<p>Consider a profit-maximizing (or cost-minimizing) firm with a production function <span class="math-container">$f(x,y)$</span>. If I am given the inputs <span class="math-container">$x,y$</span>, the input costs <span class="math-container">$w,r$</span> and the cost function <span class="math-container">$c(w,r,q) = w^{0.4}r^{0.6}q$</span>, does that mean it is related to <span class="math-container">$wx+ry$</span>, possibly in the following way:</p>
<p><span class="math-container">$w^*x_q + r^*y_q = w^{0.4}r^{0.6}q$</span> where <span class="math-container">$(w^*, r^*) := \underset{w,r}{\text{argmin}}(wx + ry) \text{ subject to } q=f(x,y)$</span> and <span class="math-container">$(x_q,y_q)$</span> are the inputs that produce output <span class="math-container">$q$</span>?</p>

CC BY-SA 4.0
2022-10-10T16:16:24.643Z
Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
Giskard
CC BY-SA 4.0
2022-10-10T16:52:58.563Z
@Giskard I am given the cost function $c(w,r,q) = w^{0.4}r^{0.6}q$. I read that cost functions are of the form $C = w \cdot x$ (where $w$ is the input price vector and $x$ is the demand vector).  I am confused if they are the same thing or not. As in, if $c$ is the optimal $C$ (cost) of producing an output of $q$?
bountyhunter
CC BY-SA 4.0
2022-10-10T20:51:44.057Z
Yes, $c(w,r,q)$ is the minimal cost of producing $q$ units given the input prices $w,r$.
Giskard
CC BY-SA 4.0
2022-10-12T08:25:27.453Z
If the firm could choose the prices $w,r$ without constraint, it would set their level to $0$ (or even $-\infty$).
Bertrand
<microeconomics><cost-functions>
Is the cost function always of the form $w \cdot x$?
bountyhunter
# Is the cost function always of the form $w \cdot x$?

(Asked by: username_0 on 2022-10-10)

Consider a profit-maximizing (or cost-minimizing) firm with a production function $f(x,y)$. If I am given the inputs $x,y$, the input costs $w,r$ and the cost function $c(w,r,q) = w^{0.4}r^{0.6}q$, does that mean it is related to $wx+ry$, possibly in the following way:

$w^\*x\_q + r^\*y\_q = w^{0.4}r^{0.6}q$ where $(w^\*, r^\*) := \underset{w,r}{\text{argmin}}(wx + ry) \text{ subject to } q=f(x,y)$ and $(x\_q,y\_q)$ are the inputs that produce output $q$?

**Tags:** microeconomics, cost-functions
> **Comments:**
> **username_1**: Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.  
> **username_0**: @Giskard I am given the cost function $c(w,r,q) = w^{0.4}r^{0.6}q$. I read that cost functions are of the form $C = w \cdot x$ (where $w$ is the input price vector and $x$ is the demand vector).  I am confused if they are the same thing or not. As in, if $c$ is the optimal $C$ (cost) of producing an output of $q$?  
> **username_1**: Yes, $c(w,r,q)$ is the minimal cost of producing $q$ units given the input prices $w,r$.  
> **username_2**: If the firm could choose the prices $w,r$ without constraint, it would set their level to $0$ (or even $-\infty$).  

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