<p>While not said explicitly in the question I am guessing from the Langrangian function you set up that the problem you intend to solve is</p>
<p><span class="math-container">$$\min_{x_1,x_2,x_3} p_1x_1 + p_2x_2 + p_3x_3 \\[8pt]
h(x_1,x_2,x_3) = \min\{x_1,x_2\} + x_3\geq z$$</span></p>
<p>This assignment combines perfect complements in production <span class="math-container">$f(x_1,x_2) = \min \{x_1,x_2\}$</span> with perfect substitues <span class="math-container">$g(y_1,y_2) = y_1 + y_2$</span>.</p>
<p>One approach is sometimes to solve problems in two stages. Intuitively, with <span class="math-container">$y_1:=\min\{x_1,x_2\}$</span> firm will produce <span class="math-container">$y_1$</span> at the unit cost <span class="math-container">$p_{y_1} = p_1+p_2$</span> for whatever level of <span class="math-container">$y_1$</span> is produced it can never be cost minimizing to use more of <span class="math-container">$x_1$</span> than <span class="math-container">$x_2$</span> or other way around. Hence <span class="math-container">$x_1=x_2 = \min\{x_1,x_2\} = y_1$</span> and the seperate cost for each unit of <span class="math-container">$y_1$</span> is therefore <span class="math-container">$p_1+p_2$</span>.</p>
<p>Next, solve cost minimizing with production</p>
<p><span class="math-container">$$g(y_1,y_2) = y_1 + y_2 = y_1 + x_3 = z$$</span></p>
<p>with input prices <span class="math-container">$p_{y_1}$</span> and <span class="math-container">$p_{3}$</span>. Since factors <span class="math-container">$y_1$</span> and <span class="math-container">$x_3$</span> are equally productive (constant equal marginal productivity) production will happen with cheapest factor so cost must be <span class="math-container">$C(z,p) =\min\{p_{y_1},p_3\}z = \min\{p_1+p_2,p_3\}z$</span>.</p>
<p>I avoid using Lagrangian multiplier method because the production function is non-differentiable.</p>

CC BY-SA 4.0
2021-01-25T17:12:35.323Z
You are correct in you assumption. This is exactly the problem that I am trying to solve. Your approach is rather simple, intuitive and makes sense.

It would make sense, to put restrictions on the relationship between $p_3$ and $p_y$ I'd guess; such that Id be able to make a statement on the line of $\frac{m}{p_y}$ if $p_y < p_3$
TiredStudent
CC BY-SA 4.0
2021-01-25T17:46:21.780Z
Yes, that is true. But that is the case of perfect substitues as I said. You can find a solution of that part of the problem here: https://economics.stackexchange.com/questions/41963/cost-function-from-an-easy-ces-production-function/41971#41971
Jesper Hybel
CC BY-SA 4.0
2021-01-25T17:51:43.357Z
You have my sincere gratitude. And I am certain that ALOT of students will thank you. I will not pick it as answer yet.

I haven't tried your solution out yet, but Id guess the same logic applies if I were to do a profit maximization problem - right?
TiredStudent
CC BY-SA 4.0
2021-01-25T18:12:11.903Z
"Id guess the same logic applies if I were to do a profit maximization problem" - I wouldn't say so, but offcourse profit max implies cost minimization so any argument that is solid for cost minimization is also valid by profit max. Notice however, that you have constant marginal costs that are therefore unit costs, so profit max does not have unique solution for quantity produced when revenue is $pz$ as under perfect competition on output market. If demand is for example $z^d := A/p^\epsilon$ and producer is monopolist the story is a little different (monopolist with constant marginal costs).
Jesper Hybel
CC BY-SA 4.0
2021-01-25T16:51:34.963Z
2021-01-25T16:51:34.963Z
Jesper Hybel
<p>The following production function is given,</p>
<p><span class="math-container">$f(X) = min\{x_1,x_2\} + x_3$</span></p>
<p>There is a solution here <a href="https://math.stackexchange.com/questions/605925/constrained-maximization-of-leontif-utility-function-minx-1-x-2">https://math.stackexchange.com/questions/605925/constrained-maximization-of-leontif-utility-function-minx-1-x-2</a>, which is similar to this function. However, it does not include an additive <span class="math-container">$x_3$</span>.</p>
<p><strong>My initial solution attempt</strong></p>
<p>Redefine the production function, such that,</p>
<p><span class="math-container">$f(X) = x_1 + x_3$</span> if <span class="math-container">$x_2 &lt; x_1$</span> and <span class="math-container">$f(x) = x_2 + x_3$</span> otherwise. This gives the following Lagrangian;</p>
<p><span class="math-container">$L = p_1 x_1 + p_2 x_2 + p_3 x_3 - \lambda_1(x_1 +x_3 - Q) - \lambda_2(x_2 + x_3 - Q)$</span></p>
<p>With each <span class="math-container">$\lambda_i &gt; 0$</span> both constraints are binding, such that,</p>
<p><span class="math-container">$Q=x_1+x_3$</span> and <span class="math-container">$Q=x_2 + x_3$</span>; and then the constraints reduces to</p>
<p><span class="math-container">$x_1 - x_2 = 0$</span> implying that <span class="math-container">$x_1 = x_2$</span>. However, this contradicts the initial conditions that I made on the relationship between <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> and, thereby, the arising production functions.</p>
<p>I'm caught in a bit of confusion here. How should I proceed with this problem?</p>

CC BY-SA 4.0
2021-01-25T16:37:44.570Z
What problem you´re trying to solve? You have a production function and what else?
callculus42
CC BY-SA 4.0
2021-01-25T17:05:37.707Z
Ok - I wasnt expecting to get any comments or possible answers here. This is literally the only information that I have!
TiredStudent
CC BY-SA 4.0
2021-01-25T17:09:47.253Z
What is the word-by-word text of your exercise? It cannot be the two lines only.
callculus42
CC BY-SA 4.0
2021-01-25T17:14:36.303Z
Ill post it just for clarity;

A firm uses 3 inputs to produce 1 output. The production function is given by: $f(x_1,x_2,x_3) = min(x_1,x_2) + x_3$, let $(w_1,w_2,w_3)$ denote the price vector of the three inputs. 

Solve the cost minimization problem and find the cost function.

This is literally the only text!
TiredStudent
<leontief>
Cost Minimization of $f(x) = min(x_1,x_2) + x_3$
TiredStudent
# Cost Minimization of $f(x) = min(x_1,x_2) + x_3$

(Asked by: username_0 on 2021-01-25)

The following production function is given,

$f(X) = min\{x\_1,x\_2\} + x\_3$

There is a solution here <https://math.stackexchange.com/questions/605925/constrained-maximization-of-leontif-utility-function-minx-1-x-2>, which is similar to this function. However, it does not include an additive $x\_3$.

**My initial solution attempt**

Redefine the production function, such that,

$f(X) = x\_1 + x\_3$ if $x\_2 < x\_1$ and $f(x) = x\_2 + x\_3$ otherwise. This gives the following Lagrangian;

$L = p\_1 x\_1 + p\_2 x\_2 + p\_3 x\_3 - \lambda\_1(x\_1 +x\_3 - Q) - \lambda\_2(x\_2 + x\_3 - Q)$

With each $\lambda\_i > 0$ both constraints are binding, such that,

$Q=x\_1+x\_3$ and $Q=x\_2 + x\_3$; and then the constraints reduces to

$x\_1 - x\_2 = 0$ implying that $x\_1 = x\_2$. However, this contradicts the initial conditions that I made on the relationship between $x\_1$ and $x\_2$ and, thereby, the arising production functions.

I'm caught in a bit of confusion here. How should I proceed with this problem?

**Tags:** leontief

> **Comments:**
> **username_1**: What problem you´re trying to solve? You have a production function and what else?  
> **username_0**: Ok - I wasnt expecting to get any comments or possible answers here. This is literally the only information that I have!  
> **username_1**: What is the word-by-word text of your exercise? It cannot be the two lines only.  
> **username_0**: Ill post it just for clarity;  A firm uses 3 inputs to produce 1 output. The production function is given by: $f(x_1,x_2,x_3) = min(x_1,x_2) + x_3$, let $(w_1,w_2,w_3)$ denote the price vector of the three inputs.   Solve the cost minimization problem and find the cost function.  This is literally the only text!  

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## 1 answers

### Answer by username_2 on 2021-01-25. Score: 4 | Accepted answer

While not said explicitly in the question I am guessing from the Langrangian function you set up that the problem you intend to solve is

$$\min\_{x\_1,x\_2,x\_3} p\_1x\_1 + p\_2x\_2 + p\_3x\_3 \\[8pt]
h(x\_1,x\_2,x\_3) = \min\{x\_1,x\_2\} + x\_3\geq z$$

This assignment combines perfect complements in production $f(x\_1,x\_2) = \min \{x\_1,x\_2\}$ with perfect substitues $g(y\_1,y\_2) = y\_1 + y\_2$.

One approach is sometimes to solve problems in two stages. Intuitively, with $y\_1:=\min\{x\_1,x\_2\}$ firm will produce $y\_1$ at the unit cost $p\_{y\_1} = p\_1+p\_2$ for whatever level of $y\_1$ is produced it can never be cost minimizing to use more of $x\_1$ than $x\_2$ or other way around. Hence $x\_1=x\_2 = \min\{x\_1,x\_2\} = y\_1$ and the seperate cost for each unit of $y\_1$ is therefore $p\_1+p\_2$.

Next, solve cost minimizing with production

$$g(y\_1,y\_2) = y\_1 + y\_2 = y\_1 + x\_3 = z$$

with input prices $p\_{y\_1}$ and $p\_{3}$. Since factors $y\_1$ and $x\_3$ are equally productive (constant equal marginal productivity) production will happen with cheapest factor so cost must be $C(z,p) =\min\{p\_{y\_1},p\_3\}z = \min\{p\_1+p\_2,p\_3\}z$.

I avoid using Lagrangian multiplier method because the production function is non-differentiable.

> **Comments:**
> **username_0**: You are correct in you assumption. This is exactly the problem that I am trying to solve. Your approach is rather simple, intuitive and makes sense.  It would make sense, to put restrictions on the relationship between $p_3$ and $p_y$ I'd guess; such that Id be able to make a statement on the line of $\frac{m}{p_y}$ if $p_y < p_3$  
> **username_2**: Yes, that is true. But that is the case of perfect substitues as I said. You can find a solution of that part of the problem here: https://economics.stackexchange.com/questions/41963/cost-function-from-an-easy-ces-production-function/41971#41971  
> **username_0**: You have my sincere gratitude. And I am certain that ALOT of students will thank you. I will not pick it as answer yet.  I haven't tried your solution out yet, but Id guess the same logic applies if I were to do a profit maximization problem - right?  
> **username_2**: "Id guess the same logic applies if I were to do a profit maximization problem" - I wouldn't say so, but offcourse profit max implies cost minimization so any argument that is solid for cost minimization is also valid by profit max. Notice however, that you have constant marginal costs that are therefore unit costs, so profit max does not have unique solution for quantity produced when revenue is $pz$ as under perfect competition on output market. If demand is for example $z^d := A/p^\epsilon$ and producer is monopolist the story is a little different (monopolist with constant marginal costs).  

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