# Cost Minimization of $f(x) = min(x_1,x_2) + x_3$

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?

• What problem you´re trying to solve? You have a production function and what else? Jan 25 at 16:37
• Ok - I wasnt expecting to get any comments or possible answers here. This is literally the only information that I have! Jan 25 at 17:05
• What is the word-by-word text of your exercise? It cannot be the two lines only. Jan 25 at 17:09
• 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! Jan 25 at 17:14

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.

• 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 $p_y < p_3$ Jan 25 at 17:12
• 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: economics.stackexchange.com/questions/41963/… Jan 25 at 17:46
• 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? Jan 25 at 17:51
• "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). Jan 25 at 18:12