1) [tex]z=6 x^7 y^2+3 x^3 y^3+y^{16}[/tex]
2) [tex]Z=5 x^4 y^8[/tex]
4) [tex]z=8 x^2 y-11 x y^3[/tex]
f) [tex]z=3 x^2+12 x y+5 y^2[/tex]
Verify all the first and second order derivatives and prove Young's theorem in Economics
Answer (2)
We computed the first and second partial derivatives for each of the four functions and confirmed that the order of differentiation yields the same result, validating Young's theorem for all cases. Each function's analysis shows that z x y = z y x holds true. Thus, Young's theorem is verified across all provided functions.
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Calculate the first partial derivatives z x and z y for each function.
Calculate the second partial derivatives z xx , z yy , z x y , and z y x for each function.
Verify Young's theorem by checking if z x y = z y x for each function.
Young's theorem is verified for all four functions.
Young’s Theorem Verified for all functions
Explanation
Problem Analysis We are given four functions and asked to verify the first and second order derivatives and to prove Young's theorem for each function. Young's theorem states that if the second partial derivatives are continuous, then the order of differentiation does not matter, i.e., z x y = z y x . We will compute the first and second partial derivatives for each function and then verify Young's theorem.
Function 1: Derivatives and Young's Theorem For the first function, z = 6 x 7 y 2 + 3 x 3 y 3 + y 16 , we compute the first partial derivatives:
z x = ∂ x ∂ z = 42 x 6 y 2 + 9 x 2 y 3
z y = ∂ y ∂ z = 12 x 7 y + 9 x 3 y 2 + 16 y 15
Now, we compute the second partial derivatives:
z xx = ∂ x 2 ∂ 2 z = 252 x 5 y 2 + 18 x y 3
z yy = ∂ y 2 ∂ 2 z = 12 x 7 + 18 x 3 y + 240 y 14
z x y = ∂ x ∂ y ∂ 2 z = 84 x 6 y + 27 x 2 y 2
z y x = ∂ y ∂ x ∂ 2 z = 84 x 6 y + 27 x 2 y 2
Since z x y = z y x , Young's theorem is verified for the first function.
Function 2: Derivatives and Young's Theorem For the second function, Z = 5 x 4 y 8 , we compute the first partial derivatives:
Z x = ∂ x ∂ Z = 20 x 3 y 8
Z y = ∂ y ∂ Z = 40 x 4 y 7
Now, we compute the second partial derivatives:
Z xx = ∂ x 2 ∂ 2 Z = 60 x 2 y 8
Z yy = ∂ y 2 ∂ 2 Z = 280 x 4 y 6
Z x y = ∂ x ∂ y ∂ 2 Z = 160 x 3 y 7
Z y x = ∂ y ∂ x ∂ 2 Z = 160 x 3 y 7
Since Z x y = Z y x , Young's theorem is verified for the second function.
Function 3: Derivatives and Young's Theorem For the third function, z = 8 x 2 y − 11 x y 3 , we compute the first partial derivatives:
z x = ∂ x ∂ z = 16 x y − 11 y 3
z y = ∂ y ∂ z = 8 x 2 − 33 x y 2
Now, we compute the second partial derivatives:
z xx = ∂ x 2 ∂ 2 z = 16 y
z yy = ∂ y 2 ∂ 2 z = − 66 x y
z x y = ∂ x ∂ y ∂ 2 z = 16 x − 33 y 2
z y x = ∂ y ∂ x ∂ 2 z = 16 x − 33 y 2
Since z x y = z y x , Young's theorem is verified for the third function.
Function 4: Derivatives and Young's Theorem For the fourth function, z = 3 x 2 + 12 x y + 5 y 2 , we compute the first partial derivatives:
z x = ∂ x ∂ z = 6 x + 12 y
z y = ∂ y ∂ z = 12 x + 10 y
Now, we compute the second partial derivatives:
z xx = ∂ x 2 ∂ 2 z = 6
z yy = ∂ y 2 ∂ 2 z = 10
z x y = ∂ x ∂ y ∂ 2 z = 12
z y x = ∂ y ∂ x ∂ 2 z = 12
Since z x y = z y x , Young's theorem is verified for the fourth function.
Conclusion In conclusion, we have verified the first and second order derivatives and proved Young's theorem for all four given functions.
Examples
Young's theorem is crucial in economics, especially in production theory. Imagine a firm's output depending on two inputs: labor and capital. Young's theorem ensures that the effect of a small change in labor on the marginal product of capital is the same as the effect of a small change in capital on the marginal product of labor. This symmetry simplifies economic models and ensures consistent decision-making, helping firms optimize resource allocation and predict market responses accurately. Understanding these relationships allows economists to build more reliable and efficient models.
