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Lagrangean Equation and Maximizing Subject

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Consider the problems of maximizing u(x) subject to px = y and maximizing v(u(x)) subject to px = y, where v(u) is strictly increasing over the range of u. Prove that x* solves the first problem if and only if it also solves the second problem.

If this proof is in the Mas-Collel text or Varian text, let me know and I can look it up.

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Solution Summary

The solution applies the Lagrangean equation for maximizing problems. The Mas-Collel and Varian text is used as a proof.

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Consider the problems of maximizing u(x) subject to px = y and maximizing v(u(x)) subject to px = y, where v(u) is strictly increasing over the range of u. Prove that x* solves the first problem if and only if it also solves the second problem. If this proof is in the Mas-Collel text or Varian text, let me know and I can look it up.

To solve the first problem: maximizing u(x) subject to px = y
Assuming x is a one variable commodity bundle,
Lagrangean equation: L = u(x) - b(px-y)
First order ...

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