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Exercício Resolvido Maximização de Lucro

Exercício Resolvido Maximização de Lucro

A função lucro de uma firma é dada pela maximização de seu lucro:


Π(p,v,w)⇔max⁡π\Pi(p,v,w) \Leftrightarrow \max \pi


π=pq−vKc−wLc\pi=pq-vK^c-wL^c


Se esta firma é tomadora de preços, tanto de seus insumos como de seu produto, econtre a função lucro para a função de produção Cobb-Douglas q=f(K,L)=KαLβq=f(K,L)=K^\alpha L^\beta.

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Este exercício é difícil, por isso eu destaquei o foco de cada passagem em vermelho.


q=KαLβq=K^\alpha L^\beta


π=pq−vKc−wLc=pq−CT\pi=pq{\color{#c90000}-vK^c-wL^c}=pq-CT


Onde usamos a função custo total de uma Cobb-Douglas genérica.


Primeiro, calculando as quantidades contingentes de K e L e depois colocando-as na função custo total:


CT=q1α+β(α+β)(vαwβααββ)1α+βCT=q^{\frac{1}{\alpha+\beta}} (\alpha+\beta) \left ( \dfrac{v^\alpha w^\beta}{\alpha^\alpha \beta^\beta} \right)^{\frac{1}{\alpha+\beta}}


π=pq−q1α+β(α+β)(vαwβααββ)1α+β\pi=pq-{\color{#c90000} q^{\frac{1}{\alpha+\beta}} (\alpha+\beta) \left ( \dfrac{v^\alpha w^\beta}{\alpha^\alpha \beta^\beta} \right)^{\frac{1}{\alpha+\beta}}}


Agora, derivamos π\pi com relação a qq e igualamos a zero, para encontrar a quantidade que maximiza o lucro:


∂π∂q=p−1α+βq1α+β−1(α+β)(vαwβααββ)1α+β{\color{#c90000} \dfrac{\partial \pi}{\partial q}}=p-\dfrac{1}{\alpha+\beta}q^{{\color{#c90000} \frac{1}{\alpha+\beta}-1 } }(\alpha+\beta) \left ( \dfrac{v^\alpha w^\beta}{\alpha^\alpha \beta^\beta} \right)^{\frac{1}{\alpha+\beta}}


1α+β−1=1−α−βα+β\dfrac{1}{\alpha+\beta}-1=\dfrac{1-\alpha-\beta}{\alpha+\beta}


p−1α+βq1−α−βα+β(α+β)(vαwβααββ)1α+β=0p-\dfrac{1}{\alpha+\beta}q^{{\color{#c90000} \frac{1-\alpha-\beta}{\alpha+\beta} } }(\alpha+\beta) \left ( \dfrac{v^\alpha w^\beta}{\alpha^\alpha \beta^\beta} \right)^{\frac{1}{\alpha+\beta}}={\color{#c90000} 0}


p=q1−α−βα+β(vαwβααββ)1α+βp=q^{{\frac{1-\alpha-\beta}{\alpha+\beta} } } {\color{#c90000} \left ( \dfrac{v^\alpha w^\beta}{\alpha^\alpha \beta^\beta} \right)^{\frac{1}{\alpha+\beta}}}


q1−α−βα+β=p(ααββvαwβ)1α+βq^{{\color{#c90000} \frac{1-\alpha-\beta}{\alpha+\beta} } }=p \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{\alpha+\beta}


q∗=pα+β1−α−β(ααββvαwβ)11−α−β{\color{#c90000} q^*}=p^\frac{\alpha+\beta}{1-\alpha-\beta} \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta}


Isolamos qq em função de p,v e wp,v \ e \ w, otendo a quantidade ótima de produção.


Agora, precisamos substituir q∗q^* de volta na função π\pi para encontrar a função lucro máximo Π\Pi.


Π=pq∗−q∗1α+β(α+β)(vαwβααββ)1α+β\Pi=p{\color{#c90000} q^*}-{ {\color{#c90000} q^*}^{\frac{1}{\alpha+\beta}} (\alpha+\beta) \left ( \dfrac{v^\alpha w^\beta}{\alpha^\alpha \beta^\beta} \right)^{\frac{1}{\alpha+\beta}}}


Π=p⋅pα+β1−α−β(ααββvαwβ)11−α−β−[pα+β1−α−β(ααββvαwβ)11−α−β]1α+β(α+β)(vαwβααββ)1α+β\Pi=p \cdot {\color{#c90000}{ p^\frac{\alpha+\beta}{1-\alpha-\beta} \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta}}}-\left[ {\color{#c90000}{ p^\frac{\alpha+\beta}{1-\alpha-\beta} \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta}}} \right]^{\frac{1}{\alpha+\beta}} (\alpha+\beta) \left ( \dfrac{v^\alpha w^\beta}{\alpha^\alpha \beta^\beta} \right)^{\frac{1}{\alpha+\beta}}


Π=p1+α+β1−α−β(ααββvαwβ)11−α−β−p11−α−β[(ααββvαwβ)11−α−β]1α+β(α+β)(vαwβααββ)1α+β\Pi= {\color{#c90000}{ p^{1+\frac{\alpha+\beta}{1-\alpha-\beta}} \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta}}}-p^\frac{1}{1-\alpha-\beta}\left[ {\color{#c90000}{ \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta}}} \right]^{\frac{1}{\alpha+\beta}} (\alpha+\beta) \left ( \dfrac{v^\alpha w^\beta}{\alpha^\alpha \beta^\beta} \right)^{\frac{1}{\alpha+\beta}}


1+α+β1−α−β=11−α−β1+\dfrac{\alpha+\beta}{1-\alpha-\beta}=\dfrac{1}{1-\alpha-\beta}


Π=p11−α−β(ααββvαwβ)11−α−β−p11−α−β[(ααββvαwβ)11−α−β]1α+β(α+β)(vαwβααββ)1α+β\Pi= {\color{#c90000}{ p^{\frac{1}{1-\alpha-\beta}} \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta}}}-p^\frac{1}{1-\alpha-\beta}\left[ {{ \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta}}} \right]^{\frac{1}{\alpha+\beta}} (\alpha+\beta) \left ( \dfrac{v^\alpha w^\beta}{\alpha^\alpha \beta^\beta} \right)^{\frac{1}{\alpha+\beta}}


Π=p11−α−β(ααββvαwβ)11−α−β[1−(α+β)((ααββvαwβ)11−α−β)1α+β−1(vαwβααββ)1α+β]\Pi= {\color{#c90000}{ p^{\frac{1}{1-\alpha-\beta}} \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta}}} \left[1-(\alpha+\beta) \left( \left(\dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta} \right) ^{\color{#c90000}{\frac{1}{\alpha+\beta} -1}} \left(\dfrac{v^\alpha w^\beta}{\alpha^\alpha \beta^\beta } \right) ^\frac{1}{\alpha+\beta} \right]


Π=p11−α−β(ααββvαwβ)11−α−β[1−(α+β)((ααββvαwβ)11−α−β)1−α−βα+β(vαwβααββ)1α+β]\Pi= {\color{#c90000}{ p^{\frac{1}{1-\alpha-\beta}} \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta}}} \left[1-(\alpha+\beta) \left( \left(\dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta} \right) ^{\color{#c90000}{\frac{1-\alpha-\beta}{\alpha+\beta}}} \left(\dfrac{v^\alpha w^\beta}{\alpha^\alpha \beta^\beta } \right) ^\frac{1}{\alpha+\beta} \right]


Π=p11−α−β(ααββvαwβ)11−α−β[1−(α+β)((ααββvαwβ)11−α−β⋅1−α−βα+β)(vαwβααββ)1α+β]\Pi= {\color{#c90000}{ p^{\frac{1}{1-\alpha-\beta}} \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta}}} \left[1-(\alpha+\beta) \left( \left(\dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^{\frac{1}{1-\alpha-\beta}\cdot {\color{#c90000}{\frac{1-\alpha-\beta}{\alpha+\beta}}}} \right) \left(\dfrac{v^\alpha w^\beta}{\alpha^\alpha \beta^\beta } \right) ^\frac{1}{\alpha+\beta} \right]


Π=p11−α−β(ααββvαwβ)11−α−β[1−(α+β)(ααββvαwβ)1α+β(vαwβααββ)1α+β]\Pi= {\color{#c90000}{ p^{\frac{1}{1-\alpha-\beta}} \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta}}} \left[1-(\alpha+\beta) \left(\dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^{ {\color{#c90000}{\frac{1}{\alpha+\beta}}}} \left(\dfrac{v^\alpha w^\beta}{\alpha^\alpha \beta^\beta } \right) ^\frac{1}{\alpha+\beta} \right]


Π=p11−α−β(ααββvαwβ)11−α−β[1−(α+β)]\Pi= {\color{#c90000}{ p^{\frac{1}{1-\alpha-\beta}} \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta}}} \left[1-(\alpha+\beta) \right]


Π=(1−α−β)\Pi=(1-\alpha-\beta) p11−α−β(ααββvαwβ)11−α−β{{ p^{\frac{1}{1-\alpha-\beta}} \left( \dfrac{\alpha^\alpha \beta^\beta}{v^\alpha w^\beta} \right)^\frac{1}{1-\alpha-\beta}}}

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