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How to derive the equal marginal principle with a general equation and general prices? Good x1,...

How to derive the equal marginal principle with a general equation and general prices? Good x1, x2. Price p1, p2, Income m.

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Answer #1

Let's consider a Cobb-Douglas utility function of following form:

u = x11/2x21/2

Generalized Budget line: M = p1.x1 + p2.x2

Utility is maximized when MU1/MU2 = p1/p2

MU1 = u/x1 = (1/2).(x2/x1)1/2

MU2 = u/x2 = (1/2).(x1/x2)1/2

MU1/MU2 = [(1/2).(x2/x1)1/2] / [(1/2).(x1/x2)1/2] = x2 / x1 = p1 / p2

p1.x1 = p2.x2

Plugging into generalized budget line,

M = p1.x1 + p1.x1 = 2p1.x1

x1 = M / (2p1)

Again,

M = p2.x2 + p2.x2 = 2p2.x2

x2 = M / (2p2)

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