<p>If the technology is CRS, and the parameters are such that 0 profit is both maximal and attainable with nonzero production, then the optimal solution is not unique, any multiple of it will also be optimal.</p>

CC BY-SA 4.0
2023-03-10T16:46:22.527Z
2023-03-10T16:46:22.527Z
Giskard
<p>Assume that we have a profit function of the form:
<span class="math-container">$$
\pi=pf(Q)-wl-rk
$$</span>
where <span class="math-container">$Q$</span> represents total volume of output, <span class="math-container">$w$</span> represents wages
and <span class="math-container">$r$</span> represents rental rate on capital <span class="math-container">$k$</span>. One way to choose
optimal quantities of inputs is to set up the Lagrangean, where we
obtain optimal demand functions for <span class="math-container">$l^{*}$</span> and <span class="math-container">$k^{*}.$</span> Therefafter,
one obtains for each value of <span class="math-container">$Q,$</span> the value of <span class="math-container">$l$</span> and <span class="math-container">$k$</span> such
that profits are maximized. In that case, we have two equations in
two unknowns, and we can solve for <span class="math-container">$l^{*}$</span> and <span class="math-container">$k^{*}$</span>. Now, let
us assume that <span class="math-container">$f(.)$</span> exhibits constant returns to scale. In that
case, we know that:
<span class="math-container">$$
\pi=pf(Q)-wl^{*}-rk^{*}=0
$$</span>
owing to Euler's theorem. With which other equation can the profit
function above be combined to yield the unique equilibrium values
of <span class="math-container">$l^{*}$</span> and <span class="math-container">$k^{*}?$</span> Will it be with the tangency condition between
<span class="math-container">$l^{*}$</span> and <span class="math-container">$k^{*}?$</span></p>

<optimization>
Use of zero profit condition in determining unique solution
Kwame Brown
# Use of zero profit condition in determining unique solution

(Asked by: username_0 on 2023-03-10)

Assume that we have a profit function of the form:
$$
\pi=pf(Q)-wl-rk
$$
where $Q$ represents total volume of output, $w$ represents wages
and $r$ represents rental rate on capital $k$. One way to choose
optimal quantities of inputs is to set up the Lagrangean, where we
obtain optimal demand functions for $l^{\*}$ and $k^{\*}.$ Therefafter,
one obtains for each value of $Q,$ the value of $l$ and $k$ such
that profits are maximized. In that case, we have two equations in
two unknowns, and we can solve for $l^{\*}$ and $k^{\*}$. Now, let
us assume that $f(.)$ exhibits constant returns to scale. In that
case, we know that:
$$
\pi=pf(Q)-wl^{\*}-rk^{\*}=0
$$
owing to Euler's theorem. With which other equation can the profit
function above be combined to yield the unique equilibrium values
of $l^{\*}$ and $k^{\*}?$ Will it be with the tangency condition between
$l^{\*}$ and $k^{\*}?$

**Tags:** optimization

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## 1 answers

### Answer by username_1 on 2023-03-10. Score: 4 | Accepted answer

If the technology is CRS, and the parameters are such that 0 profit is both maximal and attainable with nonzero production, then the optimal solution is not unique, any multiple of it will also be optimal.


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