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It is not generally true that cost minimization for a given output levelIf $Y$ results in overall profit maximization. The reason is quite simple: Out of the many output levels generally not all are optimal. So it is not true that you can always get all necessary conditions for profit maximization from cost minimization. A notable exception is if the production function$F(K,L)$ is a homogeneous function of degree one, meaning that then so is $$ \Pi(K,L) = F(K,L) - R \cdot K - w \cdot L. $$ This follows straight from the economiesdefinition of scale are constanthomogeneity. In this case all output levels(A definition of homogeneous function can be optimalfound (if an optimum existshere.), hence cost minimization is the same as This means that if a maximal profit maximizationexists it is zero.

On the other hand, because profit maximization does imply cost minimization Otherwise you could get the necessary conditions of cost minimization

\begin{align} R + \lambda F_K & = 0\\ \lambda & = \frac{R}{F_K}\\ w + \lambda F_L & = 0\\ \lambda & = \frac{w}{F_L}\\ \frac{R}{F_K} & = \frac{w}{F_L} \end{align}increase all inputs by say 100%, thereby increasing both revenues and costs and thus profits by 100%. So $\Pi(K^*,L^*) = 0$.

from the necessary conditions of profit maximizationBy Euler's Homogeneous Function Theorem we have $\forall K,L$:

\begin{align} R &= F_k\\ w &= F_L \end{align}\begin{align} \Pi(K,L) &= \Pi_K(K,L) \cdot K + \Pi_L(K,L) \cdot L \\ \Pi(K,L) &= (F_K(K,L) - R) \cdot K + (F_L(K,L) - w) \cdot L. \end{align}

by simply dividing the first condition bySince $\Pi(K^*,L^*) = 0$, we have $$ -(F_K(K^*,L^*) - R) \cdot K^* = (F_L(K^*,L^*) - w) \cdot L^* $$ We know that $K^*,L^* \geq 0$, so if we can show that the secondsigns of $(F_K(K^*,L^*) - R)$ and $(F_L(K^*,L^*) - w)$ match we will have proven them to be equal to zero. Otherwise one side of the equation would be negative and the other positive. From cost minimization you have \begin{align} R + \lambda F_K & = 0\\ w + \lambda F_L & = 0. \end{align} If (provided$\lambda >1$ then \begin{align} F_K(K^*,L^*) - R & < 0\\ F_L(K^*,L^*) - w & < 0, \end{align} if $w \neq 0$)$\lambda =1$ then \begin{align} F_K(K^*,L^*) - R & = 0\\ F_L(K^*,L^*) - w & = 0. \end{align} and if $\lambda <1$ then \begin{align} F_K(K^*,L^*) - R & > 0\\ F_L(K^*,L^*) - w & > 0, \end{align}

\begin{align} \frac{R}{w} &= \frac{F_k}{F_L} \\ \frac{R}{F_K} & = \frac{w}{F_L}. \end{align} so the signs do indeed match, hence \begin{align} F_K(K^*,L^*) - R & = 0\\ F_L(K^*,L^*) - w & = 0. \end{align}

It is not generally true that cost minimization for a given output level $Y$ results in overall profit maximization. The reason is quite simple: Out of the many output levels generally not all are optimal. So it is not true that you can always get all necessary conditions for profit maximization from cost minimization. A notable exception is if the production function is a homogeneous function of degree one, meaning that the economies of scale are constant. In this case all output levels can be optimal (if an optimum exists), hence cost minimization is the same as profit maximization.

On the other hand, because profit maximization does imply cost minimization you could get the necessary conditions of cost minimization

\begin{align} R + \lambda F_K & = 0\\ \lambda & = \frac{R}{F_K}\\ w + \lambda F_L & = 0\\ \lambda & = \frac{w}{F_L}\\ \frac{R}{F_K} & = \frac{w}{F_L} \end{align}

from the necessary conditions of profit maximization

\begin{align} R &= F_k\\ w &= F_L \end{align}

by simply dividing the first condition by the second one (provided $w \neq 0$)

\begin{align} \frac{R}{w} &= \frac{F_k}{F_L} \\ \frac{R}{F_K} & = \frac{w}{F_L}. \end{align}

If $F(K,L)$ is a homogeneous function of degree one then so is $$ \Pi(K,L) = F(K,L) - R \cdot K - w \cdot L. $$ This follows straight from the definition of homogeneity. (A definition of homogeneous function can be found here.) This means that if a maximal profit exists it is zero. Otherwise you could increase all inputs by say 100%, thereby increasing both revenues and costs and thus profits by 100%. So $\Pi(K^*,L^*) = 0$.

By Euler's Homogeneous Function Theorem we have $\forall K,L$:

\begin{align} \Pi(K,L) &= \Pi_K(K,L) \cdot K + \Pi_L(K,L) \cdot L \\ \Pi(K,L) &= (F_K(K,L) - R) \cdot K + (F_L(K,L) - w) \cdot L. \end{align}

Since $\Pi(K^*,L^*) = 0$, we have $$ -(F_K(K^*,L^*) - R) \cdot K^* = (F_L(K^*,L^*) - w) \cdot L^* $$ We know that $K^*,L^* \geq 0$, so if we can show that the signs of $(F_K(K^*,L^*) - R)$ and $(F_L(K^*,L^*) - w)$ match we will have proven them to be equal to zero. Otherwise one side of the equation would be negative and the other positive. From cost minimization you have \begin{align} R + \lambda F_K & = 0\\ w + \lambda F_L & = 0. \end{align} If $\lambda >1$ then \begin{align} F_K(K^*,L^*) - R & < 0\\ F_L(K^*,L^*) - w & < 0, \end{align} if $\lambda =1$ then \begin{align} F_K(K^*,L^*) - R & = 0\\ F_L(K^*,L^*) - w & = 0. \end{align} and if $\lambda <1$ then \begin{align} F_K(K^*,L^*) - R & > 0\\ F_L(K^*,L^*) - w & > 0, \end{align}

so the signs do indeed match, hence \begin{align} F_K(K^*,L^*) - R & = 0\\ F_L(K^*,L^*) - w & = 0. \end{align}

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Giskard
  • 29.7k
  • 11
  • 47
  • 81

It is not generally true that cost minimization for a given output level $Y$ results in overall profit maximization. The reason is quite simple: Out of the many output levels generally not all are optimal. So it is not true that you can always get all necessary conditions for profit maximization from cost minimization. A notable exception is if the production function is a homogeneous function of degree one, meaning that the economies of scale are constant. In this case all output levels can be optimal (if an optimum exists), hence cost minimization is the same as profit maximization.

On the other hand, because profit maximization does imply cost minimization you could get the necessary conditions of cost minimization

\begin{align} R + \lambda F_K & = 0\\ \lambda & = \frac{R}{F_K}\\ w + \lambda F_L & = 0\\ \lambda & = \frac{w}{F_L}\\ \frac{R}{F_K} & = \frac{w}{F_L} \end{align}

from the necessary conditions of profit maximization

\begin{align} R &= F_k\\ w &= F_L \end{align}

by simply dividing the first condition by the second one (provided $w \neq 0$)

\begin{align} \frac{R}{w} &= \frac{F_k}{F_L} \\ \frac{R}{F_K} & = \frac{w}{F_L}. \end{align}