<p>The &quot;min&quot;-form of the production function requires that
<span class="math-container">$$
Q_f = \beta^M M_f,
$$</span>
and
<span class="math-container">$$
Q_f = \Omega_f H(L_f, K_f, \beta).
$$</span>
As such, we can write the cost minimization problem as:
<span class="math-container">$$
C(Q_f) = \min_{M_f, L_f} P^M_f M_f + P^L_f L_f \text{ s.t. } Q_f = \beta^M M_f \text{ and } Q_F = \Omega_f H(L_f, K_f, \beta).
$$</span>
The Lagrangian is given by:
<span class="math-container">$$
P^M_f M_f + P^L_f L_f - \lambda_1(\beta^M M_f - Q_f) - \lambda_2(\Omega_f H(L_f, K_f, \beta).
$$</span>
Here <span class="math-container">$\lambda_1$</span> and <span class="math-container">$\lambda_2$</span> are the two Lagrange multipliers associated with the two constraints. The first order conditions give:
<span class="math-container">$$
\begin{align*}
&amp;P^M_f \psi^M_f = \lambda_1 \beta^M,\\
&amp;P^L_f \psi^L_f = \lambda_1 \Omega_f \frac{\partial H}{\partial L_f}.
\end{align*}
$$</span></p>
<p>Next, the envelope theorem theorem gives that the marginal cost is given by:
<span class="math-container">$$
\frac{\partial C(Q_f)}{\partial Q_f} = \lambda_1 + \lambda_2.
$$</span>
Substituting out the first order conditions gives:
<span class="math-container">$$
\frac{\partial C(Q_f)}{\partial Q_f} = P^M_f \psi^M_f \frac{1}{\beta^M} + P^L_f \psi^L_f \frac{1}{\Omega_f \frac{\partial H}{\partial L_f}}.
$$</span></p>
<p>Next, differentiating the two constraints with respect to <span class="math-container">$Q_f$</span> (at the optimal solution) gives:
<span class="math-container">$$
1 = \beta^M \frac{\partial M_f}{\partial Q_f}.
$$</span>
and
<span class="math-container">$$
1 = \Omega_f \frac{\partial H}{\partial L_f} \frac{\partial L_f}{\partial Q_f}.
$$</span></p>
<p>As such,
<span class="math-container">$$
\frac{\partial C(Q_f)}{\partial Q_f} = P^M_f \psi^M_f \frac{\partial M_f}{\partial Q_f} + P^L_f \psi^L_f \frac{\partial L_f}{\partial Q_f}.
$$</span></p>

CC BY-SA 4.0
2024-02-21T21:20:23.340Z
That's great, thank you very much!
nini
CC BY-SA 4.0
2024-02-19T06:56:50.253Z
2024-02-19T06:56:50.253Z
tdm
<p>Seems like one would make use of the fact that <span class="math-container">$M_f$</span> and <span class="math-container">$L_f$</span> are functions of <span class="math-container">$Q_f$</span>, while the input prices are functions of the respective inputs, then take the first derivative of the goal function
<span class="math-container">$$
P_{f}^M M_{f}+P_{f}^L L_{f}-\lambda_{f}\left(Q\left(M_{f}, L_{f}, K_{f } ; \beta^M, \boldsymbol{\beta}\right)-Q_{f }\right)
$$</span>
w.r.t. <span class="math-container">$Q_f$</span>.</p>
<p>E.g.:
<span class="math-container">$$
P_{f}^M \psi_{f}^M \frac{\partial M_{f}}{\partial Q_{f}}
$$</span>
is
<span class="math-container">$$
\begin{align*}
P_{f}^M \psi_{f}^M \frac{\partial M_{f}}{\partial Q_{f}}&amp; = P_{f}^M\left(\frac{\partial P_{f}^M}{\partial M_f}\frac{M_f}{P_{f}^M}+1\right)\frac{\partial M_{f}}{\partial Q_{f}}
\\
&amp; =
\left(\frac{\partial P_{f}^M}{\partial M_f}M_f+P_{f}^M\right)\frac{\partial M_{f}}{\partial Q_{f}} \\
&amp; =
\frac{\text{d} P_{f}^MM_f}{\text{d} M_{f}}\frac{\partial M_{f}}{\partial Q_{f}} \\
&amp; = \frac{\text{d} P_{f}^MM_f}{\text{d} Q_{f}}.
\end{align*}
$$</span></p>

CC BY-SA 4.0
2024-02-15T11:05:10.943Z
2024-02-15T11:17:22.440Z
2024-02-15T11:17:22.440Z
Giskard
<p>I'm trying to derive the first-order condition of cost minimization when inputs are non-substitutable. The question is based on the problem raised by Rubens (AER 2021).</p>
<p>Suppose firms <span class="math-container">$f$</span> produce <span class="math-container">$Q_{f}$</span> units  of output using intermediate goods <span class="math-container">$M_{f}$</span>, labor <span class="math-container">$L_{f}$</span>, and fixed assets <span class="math-container">$K_{f}$</span>. Intermediate goods cannot be substituted with either labor or capital. The amount of intermediate goods needed to produce one unit of output is a scalar <span class="math-container">$\beta^M$</span>. The production function be given by the following equation:
<span class="math-container">$$
Q_{f}=\min \left\{ \beta^M M_{f}, \Omega_{f} H\left(L_{f}, K_{f}, \boldsymbol{\beta}\right)\right\}
$$</span></p>
<p>where manufacturers differ in terms of their productivity level <span class="math-container">$\Omega_{f}$</span>. Firms use a production technology <span class="math-container">$H(\cdot)$</span> in labor and capital with common parametrization <span class="math-container">$\boldsymbol{\beta}$</span>. We assume <span class="math-container">$H(\cdot)$</span> is twice differentiable in both labor and capital.</p>
<p>Manufacturers exert oligopsony power over both labor and intermediate inputs. The extent of oligopsony power of a manufacturer <span class="math-container">$f$</span> over inputs is equal to one plus the inverse elasticity of input supply and is denoted by:</p>
<p><span class="math-container">$$
\psi^M_{f}= \frac{\partial P^M}{\partial M}\frac{M}{P^M}+1,\qquad
\psi^L_{f}= \frac{\partial P^L}{\partial L}\frac{L}{P^L}+1$$</span></p>
<p>where <span class="math-container">$P^M$</span> and <span class="math-container">$P^L$</span> are the input prices of intermediate inputs and labor.</p>
<p>Assume firms choose labor to minimize their variable costs. Given that labor and intermediate inputs are not substitutable, choosing labor implies choosing a quantity of intermediate inputs, and hence also output. Through their input quantity choices, firms also set input prices if input supply is upward-sloping. The cost minimization is thus:
<span class="math-container">$$
\min _{L_{f }}\left(P_{f}^M M_{f}+P_{f}^L L_{f}-\lambda_{f}\left(Q\left(M_{f}, L_{f}, K_{f } ; \beta^M, \boldsymbol{\beta}\right)\right)-Q_{f }\right)
$$</span></p>
<p>Using the cost minimization problem, the marginal costs <span class="math-container">$\lambda_{f}$</span> can be written as:
<span class="math-container">$$
\lambda_{f}=P_{f}^L \psi_{f}^L \frac{\partial L_{f}}{\partial Q_{f}}+P_{f}^M \psi_{f}^M \frac{\partial M_{f }}{\partial Q_{f }}
$$</span></p>
<p>How would one derive this FOC? Thank you very much in advance!</p>

CC BY-SA 4.0
2024-02-15T11:05:50.677Z
There is a typo in your goal function, $Q_f$ should be inside the bracket: $$
P_{f}^M M_{f}+P_{f}^L L_{f}-\lambda_{f}\left(Q\left(M_{f}, L_{f}, K_{f } ; \beta^M, \boldsymbol{\beta}\right)-Q_{f }\right)
$$
Giskard
<profit-maximization><lagrangian><perfect-complements>
Deriving FOC with non-substitable goods
nini
# Deriving FOC with non-substitable goods

(Asked by: username_0 on 2024-02-15)

I'm trying to derive the first-order condition of cost minimization when inputs are non-substitutable. The question is based on the problem raised by Rubens (AER 2021).

Suppose firms $f$ produce $Q\_{f}$ units of output using intermediate goods $M\_{f}$, labor $L\_{f}$, and fixed assets $K\_{f}$. Intermediate goods cannot be substituted with either labor or capital. The amount of intermediate goods needed to produce one unit of output is a scalar $\beta^M$. The production function be given by the following equation:
$$
Q\_{f}=\min \left\{ \beta^M M\_{f}, \Omega\_{f} H\left(L\_{f}, K\_{f}, \boldsymbol{\beta}\right)\right\}
$$

where manufacturers differ in terms of their productivity level $\Omega\_{f}$. Firms use a production technology $H(\cdot)$ in labor and capital with common parametrization $\boldsymbol{\beta}$. We assume $H(\cdot)$ is twice differentiable in both labor and capital.

Manufacturers exert oligopsony power over both labor and intermediate inputs. The extent of oligopsony power of a manufacturer $f$ over inputs is equal to one plus the inverse elasticity of input supply and is denoted by:

$$
\psi^M\_{f}= \frac{\partial P^M}{\partial M}\frac{M}{P^M}+1,\qquad
\psi^L\_{f}= \frac{\partial P^L}{\partial L}\frac{L}{P^L}+1$$

where $P^M$ and $P^L$ are the input prices of intermediate inputs and labor.

Assume firms choose labor to minimize their variable costs. Given that labor and intermediate inputs are not substitutable, choosing labor implies choosing a quantity of intermediate inputs, and hence also output. Through their input quantity choices, firms also set input prices if input supply is upward-sloping. The cost minimization is thus:
$$
\min \_{L\_{f }}\left(P\_{f}^M M\_{f}+P\_{f}^L L\_{f}-\lambda\_{f}\left(Q\left(M\_{f}, L\_{f}, K\_{f } ; \beta^M, \boldsymbol{\beta}\right)\right)-Q\_{f }\right)
$$

Using the cost minimization problem, the marginal costs $\lambda\_{f}$ can be written as:
$$
\lambda\_{f}=P\_{f}^L \psi\_{f}^L \frac{\partial L\_{f}}{\partial Q\_{f}}+P\_{f}^M \psi\_{f}^M \frac{\partial M\_{f }}{\partial Q\_{f }}
$$

How would one derive this FOC? Thank you very much in advance!

**Tags:** profit-maximization, lagrangian, perfect-complements
> **Comments:**
> **username_1**: There is a typo in your goal function, $Q_f$ should be inside the bracket: $$ P_{f}^M M_{f}+P_{f}^L L_{f}-\lambda_{f}\left(Q\left(M_{f}, L_{f}, K_{f } ; \beta^M, \boldsymbol{\beta}\right)-Q_{f }\right) $$  

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## 2 answers

### Answer by username_2 on 2024-02-19. Score: 0 | Accepted answer

The "min"-form of the production function requires that
$$
Q\_f = \beta^M M\_f,
$$
and
$$
Q\_f = \Omega\_f H(L\_f, K\_f, \beta).
$$
As such, we can write the cost minimization problem as:
$$
C(Q\_f) = \min\_{M\_f, L\_f} P^M\_f M\_f + P^L\_f L\_f \text{ s.t. } Q\_f = \beta^M M\_f \text{ and } Q\_F = \Omega\_f H(L\_f, K\_f, \beta).
$$
The Lagrangian is given by:
$$
P^M\_f M\_f + P^L\_f L\_f - \lambda\_1(\beta^M M\_f - Q\_f) - \lambda\_2(\Omega\_f H(L\_f, K\_f, \beta).
$$
Here $\lambda\_1$ and $\lambda\_2$ are the two Lagrange multipliers associated with the two constraints. The first order conditions give:
$$
\begin{align\*}
&P^M\_f \psi^M\_f = \lambda\_1 \beta^M,\\
&P^L\_f \psi^L\_f = \lambda\_1 \Omega\_f \frac{\partial H}{\partial L\_f}.
\end{align\*}
$$

Next, the envelope theorem theorem gives that the marginal cost is given by:
$$
\frac{\partial C(Q\_f)}{\partial Q\_f} = \lambda\_1 + \lambda\_2.
$$
Substituting out the first order conditions gives:
$$
\frac{\partial C(Q\_f)}{\partial Q\_f} = P^M\_f \psi^M\_f \frac{1}{\beta^M} + P^L\_f \psi^L\_f \frac{1}{\Omega\_f \frac{\partial H}{\partial L\_f}}.
$$

Next, differentiating the two constraints with respect to $Q\_f$ (at the optimal solution) gives:
$$
1 = \beta^M \frac{\partial M\_f}{\partial Q\_f}.
$$
and
$$
1 = \Omega\_f \frac{\partial H}{\partial L\_f} \frac{\partial L\_f}{\partial Q\_f}.
$$

As such,
$$
\frac{\partial C(Q\_f)}{\partial Q\_f} = P^M\_f \psi^M\_f \frac{\partial M\_f}{\partial Q\_f} + P^L\_f \psi^L\_f \frac{\partial L\_f}{\partial Q\_f}.
$$

> **Comments:**
> **username_0**: That's great, thank you very much!  

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### Answer by username_1 on 2024-02-15. Score: 1

Seems like one would make use of the fact that $M\_f$ and $L\_f$ are functions of $Q\_f$, while the input prices are functions of the respective inputs, then take the first derivative of the goal function
$$
P\_{f}^M M\_{f}+P\_{f}^L L\_{f}-\lambda\_{f}\left(Q\left(M\_{f}, L\_{f}, K\_{f } ; \beta^M, \boldsymbol{\beta}\right)-Q\_{f }\right)
$$
w.r.t. $Q\_f$.

E.g.:
$$
P\_{f}^M \psi\_{f}^M \frac{\partial M\_{f}}{\partial Q\_{f}}
$$
is
$$
\begin{align\*}
P\_{f}^M \psi\_{f}^M \frac{\partial M\_{f}}{\partial Q\_{f}}& = P\_{f}^M\left(\frac{\partial P\_{f}^M}{\partial M\_f}\frac{M\_f}{P\_{f}^M}+1\right)\frac{\partial M\_{f}}{\partial Q\_{f}}
\\
& =
\left(\frac{\partial P\_{f}^M}{\partial M\_f}M\_f+P\_{f}^M\right)\frac{\partial M\_{f}}{\partial Q\_{f}} \\
& =
\frac{\text{d} P\_{f}^MM\_f}{\text{d} M\_{f}}\frac{\partial M\_{f}}{\partial Q\_{f}} \\
& = \frac{\text{d} P\_{f}^MM\_f}{\text{d} Q\_{f}}.
\end{align\*}
$$


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