<p>Let <span class="math-container">$q= f(x,y)$</span> be the prodution function. the equivalence of utility maximization for production is the output maximisation problem:
<span class="math-container">$$
q(p_x, p_y, c) = \max f(x,y) \text{ s.t. } p_x x + p_y y = c.
$$</span>
This maximizes the output given the total expenditure on inputs. The function <span class="math-container">$q(p_x, p_y, c)$</span> is similar to the indirect utility function.</p>
<p>Given the solutions <span class="math-container">$x = a c/p_x$</span> and <span class="math-container">$y = (1-a) c/p_y$</span> we easily arrive at the expression:
<span class="math-container">$$
q(p_x, p_y, c) = c \left(\left(\frac{a}{p_x}\right)^a \left(\frac{1-a}{p_y}\right)^{1-a}\right).
$$</span></p>
<p>Now, the cost minimization problem is given by:
<span class="math-container">$$
c(p_x, p_y, q) = \min p_x x + p_y y \text{ s.t. } f(x,y) \ge q.
$$</span>
The function <span class="math-container">$c(p_x, p_y, q)$</span> gives the minimal expenditure necessary to produce an output level of <span class="math-container">$o$</span>. It is similar to the expenditure minimization problem.</p>
<p>It is known that inverting <span class="math-container">$q(p_x, p_y, c)$</span> with respect to <span class="math-container">$c$</span> gives the function <span class="math-container">$c(p_x, p_y, q)$</span>. So in this case, we immediately get that:
<span class="math-container">$$
c(p_x, p_y, q) = q\left(\left(\frac{p_x}{a}\right)^{a}\left(\frac{p_y}{(1-a)}\right)^{1 - a}\right).
$$</span>
This shows that marginal costs are constant and equal to <span class="math-container">$\left(\left(\frac{p_x}{a}\right)^{a}\left(\frac{p_y}{(1-a)}\right)^{1 - a}\right)$</span></p>
<p>Here, trying to get an expression for the profit function does not make much sense. As the Cobb-Douglass production function is CRS, profits (under price taking) are either zero or unbounded.</p>

CC BY-SA 4.0
2024-06-14T14:10:47.547Z
"*Here profit maximization does not make sense.*" This seems strangely phrased, since many GE models assume CRS + profit maximization. (You yourself mention zero profit as a possible solution.) Or are you arguing that GE models do not make sense?
Giskard
CC BY-SA 4.0
2024-06-14T14:42:27.487Z
@Giskard I mean that it does not make sense to try to have an expression for the profit function (as a function of output and input prices).
tdm
CC BY-SA 4.0
2024-06-14T21:37:38.700Z
I see and like your rephrasing.
Giskard
CC BY-SA 4.0
2024-06-14T12:08:12.700Z
2024-06-14T14:41:56.087Z
2024-06-14T14:41:56.087Z
tdm
<p>for cobb-douglas utility functions,
aI/px and (a-1)I/py gives the marshallian demand for x and y.</p>
<p>Are there any parallels for cobb-douglas production functions,
q = x^a * y^(1-a) ?</p>
<p>Thank you.</p>

CC BY-SA 4.0
2024-06-10T06:21:47.907Z
Are you doing cost minimization or profit maximization?
Giskard
CC BY-SA 4.0
2024-06-10T06:41:03.683Z
I wasn't looking for a specific one, just wondering if there's a shortcut for either cost minimization or profit maximization.
Mark An
CC BY-SA 4.0
2024-06-10T07:13:09.623Z
Yes, there are, but demand does not exist independently of the objective.
Giskard
<production-function>
any contingent labor and capital demand functions shortcuts for cobb-douglas functions?
Mark An
# any contingent labor and capital demand functions shortcuts for cobb-douglas functions?

(Asked by: username_0 on 2024-06-10)

for cobb-douglas utility functions,
aI/px and (a-1)I/py gives the marshallian demand for x and y.

Are there any parallels for cobb-douglas production functions,
q = x^a \* y^(1-a) ?

Thank you.

**Tags:** production-function
> **Comments:**
> **username_1**: Are you doing cost minimization or profit maximization?  
> **username_0**: I wasn't looking for a specific one, just wondering if there's a shortcut for either cost minimization or profit maximization.  
> **username_1**: Yes, there are, but demand does not exist independently of the objective.  

---

## 1 answers

### Answer by username_2 on 2024-06-14. Score: 2

Let $q= f(x,y)$ be the prodution function. the equivalence of utility maximization for production is the output maximisation problem:
$$
q(p\_x, p\_y, c) = \max f(x,y) \text{ s.t. } p\_x x + p\_y y = c.
$$
This maximizes the output given the total expenditure on inputs. The function $q(p\_x, p\_y, c)$ is similar to the indirect utility function.

Given the solutions $x = a c/p\_x$ and $y = (1-a) c/p\_y$ we easily arrive at the expression:
$$
q(p\_x, p\_y, c) = c \left(\left(\frac{a}{p\_x}\right)^a \left(\frac{1-a}{p\_y}\right)^{1-a}\right).
$$

Now, the cost minimization problem is given by:
$$
c(p\_x, p\_y, q) = \min p\_x x + p\_y y \text{ s.t. } f(x,y) \ge q.
$$
The function $c(p\_x, p\_y, q)$ gives the minimal expenditure necessary to produce an output level of $o$. It is similar to the expenditure minimization problem.

It is known that inverting $q(p\_x, p\_y, c)$ with respect to $c$ gives the function $c(p\_x, p\_y, q)$. So in this case, we immediately get that:
$$
c(p\_x, p\_y, q) = q\left(\left(\frac{p\_x}{a}\right)^{a}\left(\frac{p\_y}{(1-a)}\right)^{1 - a}\right).
$$
This shows that marginal costs are constant and equal to $\left(\left(\frac{p\_x}{a}\right)^{a}\left(\frac{p\_y}{(1-a)}\right)^{1 - a}\right)$

Here, trying to get an expression for the profit function does not make much sense. As the Cobb-Douglass production function is CRS, profits (under price taking) are either zero or unbounded.

> **Comments:**
> **username_1**: "*Here profit maximization does not make sense.*" This seems strangely phrased, since many GE models assume CRS + profit maximization. (You yourself mention zero profit as a possible solution.) Or are you arguing that GE models do not make sense?  
> **username_2**: @Giskard I mean that it does not make sense to try to have an expression for the profit function (as a function of output and input prices).  
> **username_1**: I see and like your rephrasing.  

---

