Exercício Resolvido Teoria do Produtor

A função lucro é dada por:

$\Pi={{ p^{\frac{1}{1-\alpha-\beta}} (1-\alpha-\beta) \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta}}}$

Para encontrar o grau de homogeneidade desta função, devemos multiplicar todos os argumentos dela por uma constante positiva $t$:

$\Pi(p,v,w)\Rightarrow \Pi(tp,tv,tw)=\Pi'$

$\Pi'={{ (tp)^{\frac{1}{1-\alpha-\beta}} (1-\alpha-\beta) \left( \dfrac{\alpha^\alpha \beta^\beta}{(tv)^\alpha (tw)^\beta} \right)^\frac{1}{1-\alpha-\beta}}}$

Agora, devemos escrever, se possível, a função com a constante $t$ em evidência:

$\Pi'={{ t^{\frac{1}{1-\alpha-\beta}} \cdot p^{\frac{1}{1-\alpha-\beta}} (1-\alpha-\beta) \left( \dfrac{\alpha^\alpha \beta^\beta}{t^\alpha v^\alpha t^\beta v^\beta} \right)^\frac{1}{1-\alpha-\beta}}}$

$\Pi'={{ t^{\frac{1}{1-\alpha-\beta}} \cdot p^{\frac{1}{1-\alpha-\beta}} (1-\alpha-\beta) \left( \dfrac{\alpha^\alpha \beta^\beta}{t^{\alpha+\beta} v^\alpha v^\beta} \right)^\frac{1}{1-\alpha-\beta}}}$

$\Pi'={{ t^{\frac{1}{1-\alpha-\beta}} \cdot p^{\frac{1}{1-\alpha-\beta}} (1-\alpha-\beta) \cdot t^{-(\alpha+\beta) \cdot \frac{1}{1-\alpha-\beta}}}} \cdot \left( \dfrac{\alpha^\alpha \beta^\beta}{ v^\alpha v^\beta} \right)^\frac{1}{1-\alpha-\beta}$

$\Pi'={{ t^{\frac{1}{1-\alpha-\beta}-(\alpha+\beta) \cdot \frac{1}{1-\alpha-\beta}}}} \cdot p^{\frac{1}{1-\alpha-\beta}} (1-\alpha-\beta) \left( \dfrac{\alpha^\alpha \beta^\beta}{ v^\alpha v^\beta} \right)^\frac{1}{1-\alpha-\beta}$

$\Pi'={{ t^{\frac{1-\alpha-\beta}{1-\alpha-\beta}}}} \cdot p^{\frac{1}{1-\alpha-\beta}} (1-\alpha-\beta) \left( \dfrac{\alpha^\alpha \beta^\beta}{ v^\alpha v^\beta} \right)^\frac{1}{1-\alpha-\beta}$

$\Pi'=t \cdot p^{\frac{1}{1-\alpha-\beta}} (1-\alpha-\beta) \left( \dfrac{\alpha^\alpha \beta^\beta}{ v^\alpha v^\beta} \right)^\frac{1}{1-\alpha-\beta}$

$\Pi'=t \cdot \Pi$

$\Pi(tp,tv,tw)=t \cdot \Pi(p,v,w)$

A função é homogênea de grau 1.