yunjia0810
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Production function

Economists have found that the output of a finished product, f(x,y), is sometimes described by the function

f(x,y) = ax^by^1-b

where x stands for the amount of money expended for labor, y stands for the amount expended on capital, and a and b are positive constants with 0<b<1.

a. If p is a positive number, show that f(px,py) = pf(x,y).

b. Use the result of part (a) to show that if the amount of money expended for labor and capital are both increased by r%, then the output is also increased by r%.
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mqb2766
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axby1-b

Can you improve the formatting of this? Also, what question are you asking, what have you done so far?
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Y12_FurtherMaths
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I don’t see anything
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mqb2766
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Just substitute rx and ry into f(x,y) and use the laws of exponents
(rx)^p = r^p * x^p
and similarly for (ry)^(1-p).

What problem do you have?
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yunjia0810
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(Original post by mqb2766)
Just substitute rx and ry into f(x,y) and use the laws of exponents
(rx)^p = r^p * x^p
and similarly for (ry)^(1-p).

What problem do you have?
i do not how to answer the question b.
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yunjia0810
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Production function

Economists have found that the output of a finished product, f(x,y), is sometimes described by the function

f(x,y) = ax^by^1-b

where x stands for the amount of money expended for labor, y stands for the amount expended on capital, and a and b are positive constants with 0<b<1.

a. If p is a positive number, show that f(px,py) = pf(x,y).

b. Use the result of part (a) to show that if the amount of money expended for labor and capital are both increased by r%, then the output is also increased by r%.
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yunjia0810
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(Original post by mqb2766)
axby1-b

Can you improve the formatting of this? Also, what question are you asking, what have you done so far?
Production function

Economists have found that the output of a finished product, f(x,y), is sometimes described by the function

f(x,y) = ax^by^1-b

where x stands for the amount of money expended for labor, y stands for the amount expended on capital, and a and b are positive constants with 0<b<1.

a. If p is a positive number, show that f(px,py) = pf(x,y).

b. Use the result of part (a) to show that if the amount of money expended for labor and capital are both increased by r%, then the output is also increased by r%.
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mqb2766
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part b) follows directly from part a) if you increase x by r% then x becomes
(1+r/100)x
so
p = (1+r/100)
Then part a) shows that f increases by the same amount p so the percentage increase is the same?
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yunjia0810
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(Original post by Y11_Maths)
I don’t see anything
Production function

Economists have found that the output of a finished product, f(x,y), is sometimes described by the function

f(x,y) = ax^by^1-b

where x stands for the amount of money expended for labor, y stands for the amount expended on capital, and a and b are positive constants with 0<b<1.

a. If p is a positive number, show that f(px,py) = pf(x,y).

b. Use the result of part (a) to show that if the amount of money expended for labor and capital are both increased by r%, then the output is also increased by r%.
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