<p>Are you assuming a perfectly competitive market? If so, then a profit-maximizing firm with constant returns to scale (CRS) production can only feasibly earn zero economic profits as <span class="math-container">$P = MC$</span> in the long-run. If not, then either <span class="math-container">$P &gt; MC$</span> and the firm continues to produce indefinitely until it hits capacity constraints or <span class="math-container">$P &lt; MC$</span> and the firm ceases to operate since producing any units would be unprofitable (in which case doubling inputs would be even more unprofitable).</p>

<p>To see this, it's easiest to consider the firm's cost function <span class="math-container">$C(w, r, y)$</span>. Since the firm's production is CRS, the firm's <em>average cost</em> of producing <span class="math-container">$y$</span> is constant and thus <span class="math-container">$C(w, r, y) = C(w, r, 1)y$</span>. The firm's unconstrained profit maximization problem is thus</p>

<p><span class="math-container">\begin{align*}
\max_{y} \prod = Py - C(w,r,1)y.
\end{align*}</span></p>

<p>Assuming an interior solution, the first-order condition is</p>

<p><span class="math-container">\begin{gather*}
\frac{\partial \Pi}{\partial y} = P - C(w,r,1) = 0 \\
\implies P = C(w,r,1).
\end{gather*}</span></p>

<p>So at <span class="math-container">$P = C(w, r, 1)$</span>, profits are zero and output <span class="math-container">$y$</span> is indeterminate. If instead <span class="math-container">$P &gt; C(w,r,1)$</span>, for instance, then we are no longer at an interior solution and it is clear that the firm will continue to produce and use more inputs up until capacity constraints bind.</p>

CC BY-SA 4.0
2018-10-14T13:39:48.927Z
2018-10-15T00:00:03.680Z
2018-10-15T00:00:03.680Z
Kenneth Rios
<p>Suppose we have 2 inputs a and b , output is <strong>y=f(a,b)</strong>.
In the long run, let us suppose profits are maximized at a* and b*.
Profit is <strong>py-wa-kb</strong>[p is price and w and k are constants]. Now for max profit, profit equals <strong>py-wa*-kb*</strong>.
Now for constant/increasing returns to scale firms, doubling input clearly doubles or more the profit which means there can be <strong>no finite</strong> a* and b* for such firms which does not make logical sense to me. 
Where am i wrong?</p>

CC BY-SA 4.0
2018-10-14T14:21:06.917Z
You are "wrong" when assume that double profits are larger then the normal profit. This is not always true, e.g. $2 \cdot (-2) < -2$.
Giskard
<microeconomics><profit-maximization>
Profit maximization and returns to scale relation
Polario
# Profit maximization and returns to scale relation

(Asked by: username_0 on 2018-10-14)

Suppose we have 2 inputs a and b , output is **y=f(a,b)**.
In the long run, let us suppose profits are maximized at a\* and b\*.
Profit is **py-wa-kb**[p is price and w and k are constants]. Now for max profit, profit equals **py-wa\*-kb\***.
Now for constant/increasing returns to scale firms, doubling input clearly doubles or more the profit which means there can be **no finite** a\* and b\* for such firms which does not make logical sense to me.
Where am i wrong?

**Tags:** microeconomics, profit-maximization

> **Comments:**
> **username_1**: You are "wrong" when assume that double profits are larger then the normal profit. This is not always true, e.g. $2 \cdot (-2) < -2$.  

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## 1 answers

### Answer by username_2 on 2018-10-14. Score: 2

Are you assuming a perfectly competitive market? If so, then a profit-maximizing firm with constant returns to scale (CRS) production can only feasibly earn zero economic profits as $P = MC$ in the long-run. If not, then either $P > MC$ and the firm continues to produce indefinitely until it hits capacity constraints or $P < MC$ and the firm ceases to operate since producing any units would be unprofitable (in which case doubling inputs would be even more unprofitable).

To see this, it's easiest to consider the firm's cost function $C(w, r, y)$. Since the firm's production is CRS, the firm's *average cost* of producing $y$ is constant and thus $C(w, r, y) = C(w, r, 1)y$. The firm's unconstrained profit maximization problem is thus

\begin{align\*}
\max\_{y} \prod = Py - C(w,r,1)y.
\end{align\*}

Assuming an interior solution, the first-order condition is

\begin{gather\*}
\frac{\partial \Pi}{\partial y} = P - C(w,r,1) = 0 \\
\implies P = C(w,r,1).
\end{gather\*}

So at $P = C(w, r, 1)$, profits are zero and output $y$ is indeterminate. If instead $P > C(w,r,1)$, for instance, then we are no longer at an interior solution and it is clear that the firm will continue to produce and use more inputs up until capacity constraints bind.


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