Consider the utility function U(x 1 , x 2 ) x 1 x 2 with budget constraint p 1 x 1 p 2 x 2 c (a) Show that the maximum of U(x 1 , x 2 ) subject to the budget constraint is equal to c 2 (4p 1 p 2 ) (b) Calculate the value of the Lagrange multiplier occurring in (a) (c) Prove the following interpretation is the rate of increase in utility per unit increase in total budget c

Question: Consider the utility function U(x 1 , x 2 ) = x 1 x 2 with budget constraint p 1 x 1 + p 2

Consider the utility function U(x₁, x₂) = x₁x₂ with budget constraint p₁x₁ + p₂x₂ = c.
(a) Show that the maximum of U(x₁, x₂) subject to the budget constraint is equal to c²/(4p₁ p₂).
(b) Calculate the value of the Lagrange multiplier λ occurring in (a).
(c) Prove the following interpretation: λ is the rate of increase in utility per unit increase in total budget c.

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