<p>Yes, the problem is solvable.</p>
<p><strong>Hint:</strong></p>
<ol>
<li><p>To find the cost function you need to solve the following:
<span class="math-container">$$\begin{align}
\min_{K,L\geq 0} \quad &amp; wL+rK \\
\textrm{s.t.} \quad &amp; f(L,K) \geq Q
\end{align}$$</span></p>
<p>The solution to the above gives the conditional input demand functions:<br />
<span class="math-container">$L^{c}(Q,w,r)$</span>, <span class="math-container">$K^c(Q,w,r)$</span></p>
<p>substituting the input demands into the objective of the above problem gives the cost function:  <span class="math-container">$\boxed{C(Q,w,r)=wL^{c}(Q,w,r)+rK^c(Q,w,r)}$</span></p>
</li>
<li><p>Using the cost function <span class="math-container">$C(Q,w,r)$</span> derived in <em>1.</em> we can solve the following problem to find the supply function for a competitive firm:
<span class="math-container">$$\begin{align}
\max_{Q\geq 0} \quad &amp; p\cdot Q-C(Q,w,r) \\ \\
\textrm{FOC:} \quad &amp; p-C'(Q,w,r)\overset{set}{=}0\\
\textrm{SOC:} \quad &amp; -C''(Q,w,r)&lt;0 \text{ or } \\ &amp;C''(Q,w,r)&gt;0
\end{align}$$</span></p>
<p>Assuming <span class="math-container">$\text{SOC}$</span> holds the supply function is given by: <span class="math-container">$\boxed{Q^s(p,w,r) \;\text{ s.t. }\; p=C'(Q^s(p,w,r),w,r)}$</span></p>
</li>
</ol>

CC BY-SA 4.0
2024-12-12T14:35:49.813Z
the conceptual solution is comprehansive. But when in comes to the particular functional form from OP, it doesn't provide conveniet $Q$ from FOC of the (1) part of the problem. In CES or loglinear settings the algebra seems to be less cumbersome. Is there a workaround?
garej
CC BY-SA 4.0
2024-12-26T13:39:33.117Z
@garej I'm not sure what you mean by does mean by 'convenient $Q$ from FOC'. If you can better rephrase what you mean perhaps I can edit the solution
mynameparv
CC BY-SA 4.0
2024-12-26T16:17:08.920Z
I'm not saying that you should correct your answer. I mean, $L^{1/3} + ln K$ has both logarithmic and power functions in one objective. Not very friendly for algebra manipulations.
garej
CC BY-SA 4.0
2023-06-09T19:22:36.990Z
2023-06-09T19:22:36.990Z
mynameparv
<p>Find the cost function and supply function for <span class="math-container">$Q = L^\frac{1}{3} + \ln K$</span>. Note that the price of <span class="math-container">$Q$</span> is <span class="math-container">$P$</span>, the price of <span class="math-container">$L$</span> is <span class="math-container">$w$</span> and the price of <span class="math-container">$K$</span> is <span class="math-container">$r$</span>. <span class="math-container">$Q$</span>, <span class="math-container">$L$</span>, and <span class="math-container">$K$</span> are all traded in a perfectly competitive market. Is this a solvable problem?</p>

CC BY-SA 4.0
2024-09-30T20:57:39.073Z
Where does this exercise come from?
Bertrand
<microeconomics>
Derive cost function and supply function from production function
user44341
# Derive cost function and supply function from production function

(Asked by: username_0 on 2023-06-09)

Find the cost function and supply function for $Q = L^\frac{1}{3} + \ln K$. Note that the price of $Q$ is $P$, the price of $L$ is $w$ and the price of $K$ is $r$. $Q$, $L$, and $K$ are all traded in a perfectly competitive market. Is this a solvable problem?

**Tags:** microeconomics
> **Comments:**
> **username_1**: Where does this exercise come from?  

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## 1 answers

### Answer by username_2 on 2023-06-09. Score: 2 | Accepted answer

Yes, the problem is solvable.

**Hint:**

1. To find the cost function you need to solve the following:
   $$\begin{align}
   \min\_{K,L\geq 0} \quad & wL+rK \\
   \textrm{s.t.} \quad & f(L,K) \geq Q
   \end{align}$$

   The solution to the above gives the conditional input demand functions:  
   $L^{c}(Q,w,r)$, $K^c(Q,w,r)$

   substituting the input demands into the objective of the above problem gives the cost function: $\boxed{C(Q,w,r)=wL^{c}(Q,w,r)+rK^c(Q,w,r)}$
2. Using the cost function $C(Q,w,r)$ derived in *1.* we can solve the following problem to find the supply function for a competitive firm:
   $$\begin{align}
   \max\_{Q\geq 0} \quad & p\cdot Q-C(Q,w,r) \\ \\
   \textrm{FOC:} \quad & p-C'(Q,w,r)\overset{set}{=}0\\
   \textrm{SOC:} \quad & -C''(Q,w,r)<0 \text{ or } \\ &C''(Q,w,r)>0
   \end{align}$$

   Assuming $\text{SOC}$ holds the supply function is given by: $\boxed{Q^s(p,w,r) \;\text{ s.t. }\; p=C'(Q^s(p,w,r),w,r)}$

> **Comments:**
> **username_3**: the conceptual solution is comprehansive. But when in comes to the particular functional form from OP, it doesn't provide conveniet $Q$ from FOC of the (1) part of the problem. In CES or loglinear settings the algebra seems to be less cumbersome. Is there a workaround?  
> **username_2**: @garej I'm not sure what you mean by does mean by 'convenient $Q$ from FOC'. If you can better rephrase what you mean perhaps I can edit the solution  
> **username_3**: I'm not saying that you should correct your answer. I mean, $L^{1/3} + ln K$ has both logarithmic and power functions in one objective. Not very friendly for algebra manipulations.  

---

