(a) Explanation Given Information: The provided production formula is P ( x , y ) = K x a y b ; ( a + b = 1 ) . Formula Used: The partial derivatives of f with respect to x is the derivative of f with respect to x , when all other variables are treated as constant and is denoted as ∂ f ∂ x . A particular value of ∂ f ∂ x at any point, say ( x , y ) is denoted as: ∂ f ∂ x | ( x , y ) Calculation: It is provided a production formula of the form P ( x , y ) = K x a y b ; ( a + b = 1 ) . First, find the partial derivative of the provided function with respect to x , considering y as constant: ∂ P ∂ x = K y b ( a x a − 1 ) = K a x a − 1 y b Now, since a + b = 1 , then a = 1 − b . Substitute 1 − b for a : ∂ P ∂ x = K a x a − 1 y b = K a x 1 − b − 1 y b = K a x − b y b = K a ( y x ) b Now, find the partial derivative of the provided function with respect to y , considering x as constant: ∂ P ∂ y = K x a ( b y b − 1 ) = K b x a y b − 1 Now, since a + b = 1 , then b = 1 − a

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ISBN: 9781337514309

Author: Waner

Publisher: Cengage

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Chapter 8.2, Problem 65E

(a)

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To calculate: The formulas for ∂P∂x and ∂P∂y, and show that ∂P∂x=∂P∂y precisely when x/y=a/b assuming that it is given a production formula of the form P(x,y)=Kxayb ; (a+b=1).

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Chapter 8.2, Problem 64E

Chapter 8.2, Problem 66E

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The formulas for ∂ P ∂ x and ∂ P ∂ y , and show that ∂ P ∂ x = ∂ P ∂ y precisely when x / y = a / b assuming that it is given a production formula of the form P ( x , y ) = K x a y b ; ( a + b = 1 ) .

Solution Summary: The author explains the formulas for partial x and

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To calculate: The formulas for ∂P∂x and ∂P∂y, and show that ∂P∂x=∂P∂y precisely when x/y=a/b assuming that it is given a production formula of the form P(x,y)=Kxayb ; (a+b=1).

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