<p>Your assumption 3 is compatible with corner solutions of the kind <span class="math-container">$y_i=0$</span> for some <span class="math-container">$i$</span>, and is not sufficient to avoid corner solutions for some prices small enough.</p>
<p>With production functions it is common to assume Inada's conditions to avoid corner solutions. With cost functions, such conditions are quite naturally expressed as
<span class="math-container">$$ \lim_{y_i\rightarrow 0} \frac{\partial c}{\partial y_i}(y_i,y_{-i})=0 $$</span>
<span class="math-container">$$ \lim_{y_i\rightarrow +\infty} \frac{\partial c}{\partial y_i}(y_i,y_{-i})=+\infty. $$</span></p>
<p>In your discussion, you proposed the cost function <span class="math-container">$$c(y_1,y_2)=y_1^2+y_2^2+2y_1$$</span> to illustrate a corner solution. This function does not satisfy the first Inada conditions wrt <span class="math-container">$y_1$</span>, hence the corner solution.</p>
<p>A general comment to conclude: it is not fully natural to forbid firm's exit from a given output market. In IO there is a quite dynamic literature on entry and exit. And no firm is producing all products available in the economy.</p>

CC BY-SA 4.0
2021-04-03T13:34:56.783Z
2021-04-03T13:34:56.783Z
Bertrand
<p>A function <span class="math-container">$c: \mathbb{R}^K_+ \xrightarrow{} \mathbb{R}_+$</span> is is said to be a <em>cost function</em> if</p>
<ol>
<li>The value of function <span class="math-container">$c$</span> at <span class="math-container">$y = \textbf{0}$</span> is <span class="math-container">$0$</span>: <span class="math-container">$c(\textbf{0}) = 0$</span></li>
<li><span class="math-container">$c$</span> is continuous on the domain <span class="math-container">$\mathbb{R}^K_+$</span>, strictly increasing and strictly convex on <span class="math-container">$\mathbb{R}^K_{+}$</span></li>
<li>For any <span class="math-container">$p \in \mathbb{R}^K_{++}$</span>, the set <span class="math-container">$B(p) = \{y \in \mathbb{R}^K_+| c(y) \leq p \cdot y\}$</span> is compact, the interior of <span class="math-container">$B(p)$</span> is nonempty.</li>
</ol>
<p>Suppose a firm has cost function <span class="math-container">$c(\cdot)$</span> and <span class="math-container">$p\in \mathbb{R}^K_{++}$</span> is the market price. What sort of topological assumption I need to add to guarantee the existence of an <strong>interior solution</strong> to the profit maximisation problem?
<span class="math-container">$$
\pi (p) = \max_{y\in\mathbb{R}^K_+} \left\{ p\cdot y - c(y) \right\}
$$</span></p>
<p><strong>Remark 1</strong>: From assumption (2) and (3), the existence of solution is guaranteed. Due to strict convexity, uniqueness is also guaranteed. However, the solution is probably not inside the interior of <span class="math-container">$\mathbb{R}^K_{+}$</span>. I could not find an example for this at the moment.</p>
<p><strong>Remark 2</strong>: My prototype cost function to look at is <span class="math-container">$c(y) = y_1^2+y_2^2$</span>. However, the cost is not necessarily to be additive separable.</p>

CC BY-SA 4.0
2021-03-31T06:52:49.423Z
To clarify the question mathematically: by guarantee, do you mean a sufficient condition? Then specifying a particular cost function will work, but is not very general. More general might be something like: cost increases slower than linearly near 0, eventually faster than linearly. 
How do you define a *topological* assumption? What makes it topological?
Sander Heinsalu
CC BY-SA 4.0
2021-03-31T06:55:43.417Z
@SanderHeinsalu Yes, by guarantee I mean sufficient condition. Topological assumption, from my viewpoint, may include assertions about openness, closedness and even convexity of certain sets but not differentiability assumption on the cost function. The prototype function is just an example to look at and learn about the minimal set of conditions that ensures interior solution.
Khánh Toàn
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2021-03-31T07:04:06.073Z
Differentiability is not needed. Something about the growth rate of the cost probably is, vaguely along the lines: there exist k, K such that c(1)-c(0)<k<c(K+1)-c(K). 
One way to approach it is from the other end: find counterexamples of costs that satisfy your assumptions but that generate corner solutions. Your assumption 3. may already be sufficient.
Sander Heinsalu
CC BY-SA 4.0
2021-03-31T08:48:56.357Z
@SanderHeinsalu I just learn that convex function always have right (and left) partial derivatives. Let $\partial^+_i c(y_i,y_{-i})$ be the right partial derivative in the $y_i$ direction. A sufficient condition could be $\partial^+_i c(0,y_{-i}) = 0$ for all $i$. However, as you suspected, (1) (2) and (3) are perhaps sufficient already and adding extra assumption just complicates the problem. I spend days finding a counter-example but I could not. Could you find one?
Khánh Toàn
CC BY-SA 4.0
2021-03-31T08:55:41.540Z
@SanderHeinsalu P/S I just want you to know you was my game theory teacher at the ANU. Good to see you here.
Khánh Toàn
CC BY-SA 4.0
2021-03-31T09:18:50.767Z
@SanderHeinsalu I just found one: $c(y_1,y_2) = y_1^2 + y_2^2+2y_1$ satisfies (1) (2) (3), but at price $p=(2,2)$, the optimal production plan is $(y_1^*,y_2^*) = (0,1)$ which lies on the boundary
Khánh Toàn
CC BY-SA 4.0
2021-03-31T11:21:21.523Z
Could you include a source for your definition of cost function?  The cost function of a firm is often (normally?) understood to be the function determining the minimum cost of producing any level of output given the costs of factor inputs.  Such a function could be either concave or convex depending on whether there are economies or diseconomies of scale in production.
Adam Bailey
CC BY-SA 4.0
2021-03-31T14:37:18.027Z
It seems `c(y_1,y_2)=y_1^2+y_2^2+2y_1` does not satisfy the nonempty interior assumption in 3 but I may be wrong. 
A convex function may be discontinuous on its domain boundary: `f(1)=1`, `f(x)=0` otherwise on the domain `[0,1]`. The partial derivatives may need special definition in this case.
Nice to virtually meet you again, @Khánh Toàn!
Sander Heinsalu
<microeconomics><consumer-theory><profit-maximization><producer-theory>
Interior Solution for profit maximisation problem
Khánh Toàn
# Interior Solution for profit maximisation problem

(Asked by: username_0 on 2021-03-31)

A function $c: \mathbb{R}^K\_+ \xrightarrow{} \mathbb{R}\_+$ is is said to be a *cost function* if

1. The value of function $c$ at $y = \textbf{0}$ is $0$: $c(\textbf{0}) = 0$
2. $c$ is continuous on the domain $\mathbb{R}^K\_+$, strictly increasing and strictly convex on $\mathbb{R}^K\_{+}$
3. For any $p \in \mathbb{R}^K\_{++}$, the set $B(p) = \{y \in \mathbb{R}^K\_+| c(y) \leq p \cdot y\}$ is compact, the interior of $B(p)$ is nonempty.

Suppose a firm has cost function $c(\cdot)$ and $p\in \mathbb{R}^K\_{++}$ is the market price. What sort of topological assumption I need to add to guarantee the existence of an **interior solution** to the profit maximisation problem?
$$
\pi (p) = \max\_{y\in\mathbb{R}^K\_+} \left\{ p\cdot y - c(y) \right\}
$$

**Remark 1**: From assumption (2) and (3), the existence of solution is guaranteed. Due to strict convexity, uniqueness is also guaranteed. However, the solution is probably not inside the interior of $\mathbb{R}^K\_{+}$. I could not find an example for this at the moment.

**Remark 2**: My prototype cost function to look at is $c(y) = y\_1^2+y\_2^2$. However, the cost is not necessarily to be additive separable.

**Tags:** microeconomics, consumer-theory, profit-maximization, producer-theory
> **Comments:**
> **username_1**: To clarify the question mathematically: by guarantee, do you mean a sufficient condition? Then specifying a particular cost function will work, but is not very general. More general might be something like: cost increases slower than linearly near 0, eventually faster than linearly.  How do you define a *topological* assumption? What makes it topological?  
> **username_0**: @SanderHeinsalu Yes, by guarantee I mean sufficient condition. Topological assumption, from my viewpoint, may include assertions about openness, closedness and even convexity of certain sets but not differentiability assumption on the cost function. The prototype function is just an example to look at and learn about the minimal set of conditions that ensures interior solution.  
> **username_1**: Differentiability is not needed. Something about the growth rate of the cost probably is, vaguely along the lines: there exist k, K such that c(1)-c(0)<k<c(K+1)-c(K).  One way to approach it is from the other end: find counterexamples of costs that satisfy your assumptions but that generate corner solutions. Your assumption 3. may already be sufficient.  
> **username_0**: @SanderHeinsalu I just learn that convex function always have right (and left) partial derivatives. Let $\partial^+_i c(y_i,y_{-i})$ be the right partial derivative in the $y_i$ direction. A sufficient condition could be $\partial^+_i c(0,y_{-i}) = 0$ for all $i$. However, as you suspected, (1) (2) and (3) are perhaps sufficient already and adding extra assumption just complicates the problem. I spend days finding a counter-example but I could not. Could you find one?  
> **username_0**: @SanderHeinsalu P/S I just want you to know you was my game theory teacher at the ANU. Good to see you here.  
> **username_0**: @SanderHeinsalu I just found one: $c(y_1,y_2) = y_1^2 + y_2^2+2y_1$ satisfies (1) (2) (3), but at price $p=(2,2)$, the optimal production plan is $(y_1^*,y_2^*) = (0,1)$ which lies on the boundary  
> **username_2**: Could you include a source for your definition of cost function?  The cost function of a firm is often (normally?) understood to be the function determining the minimum cost of producing any level of output given the costs of factor inputs.  Such a function could be either concave or convex depending on whether there are economies or diseconomies of scale in production.  
> **username_1**: It seems `c(y_1,y_2)=y_1^2+y_2^2+2y_1` does not satisfy the nonempty interior assumption in 3 but I may be wrong.  A convex function may be discontinuous on its domain boundary: `f(1)=1`, `f(x)=0` otherwise on the domain `[0,1]`. The partial derivatives may need special definition in this case. Nice to virtually meet you again, @Khánh Toàn!  

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## 1 answers

### Answer by username_3 on 2021-04-03. Score: 2 | Accepted answer

Your assumption 3 is compatible with corner solutions of the kind $y\_i=0$ for some $i$, and is not sufficient to avoid corner solutions for some prices small enough.

With production functions it is common to assume Inada's conditions to avoid corner solutions. With cost functions, such conditions are quite naturally expressed as
$$ \lim\_{y\_i\rightarrow 0} \frac{\partial c}{\partial y\_i}(y\_i,y\_{-i})=0 $$
$$ \lim\_{y\_i\rightarrow +\infty} \frac{\partial c}{\partial y\_i}(y\_i,y\_{-i})=+\infty. $$

In your discussion, you proposed the cost function $$c(y\_1,y\_2)=y\_1^2+y\_2^2+2y\_1$$ to illustrate a corner solution. This function does not satisfy the first Inada conditions wrt $y\_1$, hence the corner solution.

A general comment to conclude: it is not fully natural to forbid firm's exit from a given output market. In IO there is a quite dynamic literature on entry and exit. And no firm is producing all products available in the economy.


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