Procedimiento

\[ Q (K,L) = AL^{\alpha} K^{\beta} \] \[ CT =p_L \cdot L + p_K \cdot K \]

Antes de plantear el problema de optimización, notar que podemos obtener la productividad marginal del trabajo \(L\) y capital \(K\) y reescribirlas de las siguientes formas útiles:

\[ \frac{\partial Q}{\partial L} = A \alpha L^{\alpha-1}K^{\beta} = \alpha A L^{\alpha}L^{-1}K^{\beta} = \frac{ \alpha A L^{\alpha}K^{\beta}}{L} = \alpha \frac{Q}{L} \]

\[ \frac{\partial Q}{\partial K} = A \beta L^{\alpha}K^{\beta-1} = \beta A L^{\alpha}K^{\beta}K^{-1}= \frac{ \beta A L^{\alpha}K^{\beta}}{K} = \beta \frac{Q}{K} \]

Notar que si despejamos para \(\alpha\) y \(\beta\) obtenemos las elasticidades insumo de la producción: \(\alpha= \frac{\partial Q}{\partial L} \frac{L}{Y}\) y \(\beta= \frac{\partial Q}{\partial K} \frac{K}{Y}\) Si \(\alpha = 2\), entonces un aumento del \(10\%\) en la mano de obra (\(L\)) dará como resultado un aumento del \(20\%\) en la producción (\(Y\)).

Planteamos ahora el problema de minimización restringida:

Resolución por método de Multiplicadores de Lagrange

\[ \underset{K,L}{\text{Min }}CT =p_L \cdot L + p_K \cdot K \\ \] \[ \text{sujeto a } Q=AL^{\alpha} K^{\beta} \]

\[ \textbf{L} = p_L \cdot L + p_K \cdot K + \lambda [Q-AL^{\alpha} K^{\beta}] \]

Obtenemos las condiciones de primer orden

\[ \frac{\partial \textbf{L}}{\partial L}=0 \Rightarrow p_L = \lambda \alpha A L^{\alpha-1}K^{\beta} \] \[ \frac{\partial \textbf{L}}{\partial K} =0 \Rightarrow p_K = \lambda \beta A L^{\alpha}K^{\beta-1} \] \[ \frac{\partial \textbf{L}}{\partial \lambda}=0 \Rightarrow Q = L^{\alpha} K^{\beta} \]

Dividimos las primeras dos CPO para simplificar el problema:

\[ \frac{\frac{\partial \textbf{L}}{\partial K}}{\frac{\partial \textbf{L}}{\partial L}}= \frac{p_K}{p_L} = \frac{\lambda \beta A L^{\alpha}K^{\beta-1}}{\lambda \alpha A L^{\alpha-1}K^{\beta}} \] Simplificamos y resolvemos para cualquiera de los dos insumos. (Aquí resolvemos para \(L\))

\[ \frac{p_K}{p_L} = \frac{\beta}{\alpha}\frac{L}{K} \] \[ L= \frac{p_K}{p_L} \frac{\alpha}{\beta} K \] Sustituímos este valor en nuestra tercer CPO (que es nuestra función de producción)

\[ Q - A L^{\alpha} K^{\beta} =0 \]

\[ Q - A \left( \frac{p_K}{p_L} \frac{\alpha}{\beta} K \right) ^{\alpha} K^{\beta} =0 \]

\[ Q = A \left( \frac{p_K}{p_L} \frac{\alpha}{\beta}\right) ^{\alpha} K^{\alpha +\beta} \] \[ \frac{Q}{A} \left( \frac{p_K}{p_L} \frac{\alpha}{\beta}\right) ^{-\alpha} = K^{\alpha +\beta} \]

\[ K = \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}} \left( \frac{p_K}{p_L} \frac{\alpha}{\beta}\right) ^{\frac{-\alpha}{\alpha +\beta}} \] \[ K^* = \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}} \left( \frac{p_L}{p_K} \frac{\beta}{\alpha}\right) ^{\frac{\alpha}{\alpha +\beta}} \] Ahora podemos resolver para \(L= \frac{p_K}{p_L} \frac{\alpha}{\beta} K\)

\[ L= \frac{p_K}{p_L} \frac{\alpha}{\beta} \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}} \left( \frac{p_L}{p_K} \frac{\beta}{\alpha}\right) ^{\frac{\alpha}{\alpha +\beta}} \] \[ L= \frac{p_K}{p_L} \frac{\alpha}{\beta} \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}} \left( \frac{p_L}{p_K} \frac{\beta}{\alpha}\right) ^{\frac{\alpha}{\alpha +\beta}} \] \[ L= \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}} \frac{p_K}{p_L} \left( \frac{p_L}{p_K} \right) ^{\frac{\alpha}{\alpha +\beta}} \frac{\alpha}{\beta} \left( \frac{\beta}{\alpha} \right) ^{\frac{\alpha}{\alpha +\beta}} \] Notar que podemos expresar \(\frac{p_K}{p_L}\) como \(\left( \frac{p_L}{p_K} \right) ^{-1} = \left( \frac{p_L}{p_K} \right) ^{\frac{{-\alpha +\beta}}{{\alpha +\beta}}}\)

\[ L= \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}} \left( \frac{p_K}{p_L} \right) ^{\frac{\beta}{\alpha +\beta}} \left( \frac{\alpha}{\beta} \right) ^{\frac{\beta}{\alpha +\beta}} \] \[ L^*= \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}} \left( \frac{p_K}{p_L} \frac{\alpha}{\beta} \right) ^{\frac{\beta}{\alpha +\beta}} \]

Con los resultados en parámetros de \(K^*= \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}} \left( \frac{p_L}{p_K} \frac{\beta}{\alpha}\right) ^{\frac{\alpha}{\alpha +\beta}}\) y \(L^* = \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}}\left( \frac{p_K}{p_L} \frac{\alpha}{\beta} \right) ^{\frac{\beta}{\alpha +\beta}}\) podemos obtener la función de costos mínimos o función valor del problema de optimización:

\[ C_{min}= p_K \cdot K^* + p_L \cdot L^* \] \[ C_{min}= p_K \cdot \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}} \left( \frac{p_L}{p_K} \frac{\beta}{\alpha}\right) ^{\frac{\alpha}{\alpha +\beta}} + p_L \cdot \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}}\left( \frac{p_K}{p_L} \frac{\alpha}{\beta} \right) ^{\frac{\beta}{\alpha +\beta}} \] Factorizamos \(\left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}}\)

\[ C_{min}= \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}} \left( p_K\left( \frac{p_L}{p_K} \frac{\beta}{\alpha}\right) ^{\frac{\alpha}{\alpha +\beta}} + p_L\left( \frac{p_K}{p_L} \frac{\alpha}{\beta} \right) ^{\frac{\beta}{\alpha +\beta}} \right) \] Simplificamos los precios de los factores y factorizamos

\[ C_{min}= \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}} p_K^{\frac{\beta}{\alpha+\beta}} p_L^{\frac{\alpha}{\alpha+\beta}} \left( \left(\frac{\beta}{\alpha}\right) ^{\frac{\alpha}{\alpha +\beta}} + \left(\frac{\alpha}{\beta} \right) ^{\frac{\beta}{\alpha +\beta}} \right) \] ## Caso particular con valores arbitrarios

Utilizando los resultados para nuestro problema \(\alpha=\frac{3}{4}\) \(\beta=\frac{1}{4}\) \(Q=800\) \(p_K=4\) \(p_L=2\)

\[ K^* = \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}} \left( \frac{p_L}{p_K} \frac{\beta}{\alpha}\right) ^{\frac{\alpha}{\alpha +\beta}} = \left( \frac{800}{1} \right)^{\frac{1}{(3/4) +(1/4)}} \left( \frac{2}{4} \frac{(1/4)}{(3/4)}\right) ^{\frac{(3/4)}{(3/4) +(1/4)}} \]

\[ L^*= \left( \frac{Q}{A} \right)^{\frac{1}{\alpha +\beta}} \left( \frac{p_K}{p_L} \frac{\alpha}{\beta} \right) ^{\frac{\beta}{\alpha +\beta}} = \left( \frac{800}{1} \right)^{\frac{1}{(3/4) +(1/4)}} \left( \frac{4}{2} \frac{(3/4)}{(1/4)} \right) ^{\frac{(1/4)}{(3/4) +(1/4)}} \]

\[ C_{min}= \left( \frac{800}{1} \right)^{\frac{1}{(3/4) +(1/4)}} 4^{\frac{(1/4)}{(3/4)+(1/4)}} 2^{\frac{(3/4)}{(3/4)+(1/4)}} \left( \left(\frac{(1/4)}{(3/4)}\right) ^{\frac{(3/4)}{(3/4) +(1/4)}} + \left(\frac{(3/4)}{(1/4)} \right) ^{\frac{(1/4)}{(3/4) +(1/4)}} \right) \]

Cálculo mediante paquetería R

##                         [,1]
## Capital (K) óptimo  208.6779
## Trabajo (L) óptimo 1252.0677
## Costo mínimo       3338.8471

  1. El Colegio de México, ↩