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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!

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  • $\begingroup$ There is a typo in your goal function, $Q_f$ should be inside the bracket: $$ P_{f}^M M_{f}+P_{f}^L L_{f}-\lambda_{f}\left(Q\left(M_{f}, L_{f}, K_{f } ; \beta^M, \boldsymbol{\beta}\right)-Q_{f }\right) $$ $\endgroup$
    – Giskard
    Commented Feb 15 at 11:05

2 Answers 2

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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, $$ \frac{\partial C(Q_f)}{\partial Q_f} = P^M_f \psi^M_f \frac{\partial M_f}{\partial Q_f} + P^L_f \psi^L_f \frac{\partial L_f}{\partial Q_f}. $$

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  • $\begingroup$ That's great, thank you very much! $\endgroup$
    – nini
    Commented Feb 21 at 21:20
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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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