<ol>
<li><span class="math-container">$C(q)$</span>: given fixed input prices, this is the cost of producing <span class="math-container">$q$</span> units, assuming the use of inputs is optimal given the technology and the fixed input prices.</li>
<li><span class="math-container">$C(w_1, w_2 ,q)$</span>: same as above, but the function is more general because the cost is given for all <span class="math-container">$w_1, w_2$</span> input prices, not just some specific fixed ones.</li>
<li><span class="math-container">$C(\mathbf{w}, \mathbf{x})$</span>: I have never seen this notation. <span class="math-container">$C(\mathbf{w},q)$</span> is the solution to the minimalization problem
<span class="math-container">$$
\min_{\mathbf{x}} \mathbf{w} \cdot \mathbf{x}
$$</span>
<span class="math-container">$$\text{s.t.:} \ \ f(\mathbf{x}) = q.$$</span></li>
</ol>

CC BY-SA 4.0
2022-10-13T14:39:03.050Z
What do we use for something like $C = wr + kl$? I was denoting that with the third notation.
Patricia
CC BY-SA 4.0
2022-10-13T14:39:41.557Z
What do mean "*What do we use for*"? Your purpose is not clear to me. $\mathbf{w} \cdot \mathbf{x}$ is good for some uses, $C(\mathbf{x},q)$ is good for others.
Giskard
CC BY-SA 4.0
2022-10-13T14:40:58.390Z
Like, what notation do we use for $C = wr + kl$? Don't we use $C(w, r, k, l)$? I was denoting this as $C(\mathbf{w},\mathbf{x})$ where $\mathbf{w}$ is the price vector and $\mathbf{x}$ is the input vector.
Patricia
CC BY-SA 4.0
2022-10-13T14:43:03.380Z
What notation do we use for $x_1\cdot \alpha^y - \gamma$...? I don't see why every formula would have its function. Is there something wrong with writing $wr+kl$?
Giskard
CC BY-SA 4.0
2022-10-13T14:51:07.173Z
Nothing wrong with it. But without a function, how can we perform various operations on it? To add, differentiate or even plug in values, it has to be a function of some sort, right? Perhaps not necessarily with a notation. And that's what you're trying to convey I think.
Patricia
CC BY-SA 4.0
2022-10-13T14:58:49.413Z
$wl+rk + 1$, $\frac{\text{d} (wl+rk)}{\text{d} w}$, $5l+9k$
Giskard
CC BY-SA 4.0
2022-10-13T14:36:57.053Z
2022-10-13T14:46:48.973Z
2022-10-13T14:46:48.973Z
Giskard
<p>I see three types of cost functions:</p>
<ol>
<li><span class="math-container">$C(q) = 4q$</span> where <span class="math-container">$q$</span> is the quantity</li>
<li><span class="math-container">$C(w_1, w_2 ,q) = w_1w_2q$</span> where <span class="math-container">$w_1, w_2$</span> are the rental rates and <span class="math-container">$q$</span> is the quantity of the product</li>
<li><span class="math-container">$C(\mathbf{w}, \mathbf{x}) = \mathbf{w} \cdot \mathbf{x}$</span> where <span class="math-container">$\mathbf{w}$</span> is the price vector and <span class="math-container">$\mathbf{x}$</span> denotes the input factors</li>
</ol>
<p>Can anyone tell me what each of these mean? The second one is probably called the conditional cost function and the third one comes from the definition of cost. What's the first one? I have seen it being used in perfect competition and monopoly for instance, but what's the difference between 1 and 2 then? We can also use (2) in perfect competition and monopoly I think, or can we not?
I am confused.</p>
<p><strong>Edit:</strong> I think (1) and (2) are the same. They both are probably conditional cost functions. (2) probably considers the prices as values (with the assumption that prices aren't changing). (3) is the generic definition or the unconditional cost function. When optimized to produce quantity <span class="math-container">$q$</span>, (3) produces (1) and (2).</p>

<microeconomics><cost><cost-functions>
Different types of cost functions (curves)
Patricia
# Different types of cost functions (curves)

(Asked by: username_0 on 2022-10-13)

I see three types of cost functions:

1. $C(q) = 4q$ where $q$ is the quantity
2. $C(w\_1, w\_2 ,q) = w\_1w\_2q$ where $w\_1, w\_2$ are the rental rates and $q$ is the quantity of the product
3. $C(\mathbf{w}, \mathbf{x}) = \mathbf{w} \cdot \mathbf{x}$ where $\mathbf{w}$ is the price vector and $\mathbf{x}$ denotes the input factors

Can anyone tell me what each of these mean? The second one is probably called the conditional cost function and the third one comes from the definition of cost. What's the first one? I have seen it being used in perfect competition and monopoly for instance, but what's the difference between 1 and 2 then? We can also use (2) in perfect competition and monopoly I think, or can we not?
I am confused.

**Edit:** I think (1) and (2) are the same. They both are probably conditional cost functions. (2) probably considers the prices as values (with the assumption that prices aren't changing). (3) is the generic definition or the unconditional cost function. When optimized to produce quantity $q$, (3) produces (1) and (2).

**Tags:** microeconomics, cost, cost-functions


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## 1 answers

### Answer by username_1 on 2022-10-13. Score: 0 | Accepted answer

1. $C(q)$: given fixed input prices, this is the cost of producing $q$ units, assuming the use of inputs is optimal given the technology and the fixed input prices.
2. $C(w\_1, w\_2 ,q)$: same as above, but the function is more general because the cost is given for all $w\_1, w\_2$ input prices, not just some specific fixed ones.
3. $C(\mathbf{w}, \mathbf{x})$: I have never seen this notation. $C(\mathbf{w},q)$ is the solution to the minimalization problem
   $$
   \min\_{\mathbf{x}} \mathbf{w} \cdot \mathbf{x}
   $$
   $$\text{s.t.:} \ \ f(\mathbf{x}) = q.$$

> **Comments:**
> **username_0**: What do we use for something like $C = wr + kl$? I was denoting that with the third notation.  
> **username_1**: What do mean "*What do we use for*"? Your purpose is not clear to me. $\mathbf{w} \cdot \mathbf{x}$ is good for some uses, $C(\mathbf{x},q)$ is good for others.  
> **username_0**: Like, what notation do we use for $C = wr + kl$? Don't we use $C(w, r, k, l)$? I was denoting this as $C(\mathbf{w},\mathbf{x})$ where $\mathbf{w}$ is the price vector and $\mathbf{x}$ is the input vector.  
> **username_1**: What notation do we use for $x_1\cdot \alpha^y - \gamma$...? I don't see why every formula would have its function. Is there something wrong with writing $wr+kl$?  
> **username_0**: Nothing wrong with it. But without a function, how can we perform various operations on it? To add, differentiate or even plug in values, it has to be a function of some sort, right? Perhaps not necessarily with a notation. And that's what you're trying to convey I think.  
> **username_1**: $wl+rk + 1$, $\frac{\text{d} (wl+rk)}{\text{d} w}$, $5l+9k$  

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