Econ 5201 Lecture 3

Econ 5201 Lecture 3 - CONSUMERS PROBLEM III Consider...

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CONSUMER’S PROBLEM III Consider another utility function Ux x =+ 12 Recall that the constrained maximization problem is max U(x 1 , x 2 ) s.t. p 1 x 1 +p 2 x 2 =m The Lagrangian is Lxx m p x p x =++ () ( 1 1 2 λ ) 2 The FOCs are: λ L x x p 1 1 1 1 2 0 =− = (A) λ L x x p 2 2 2 1 2 0 = (B) ∂λ L mp x p x = 11 2 2 0 (C) Thus, 1 2 1 1 x p ( A 1 ) 1 2 2 2 x p ( B 1 ) p 1 x 1 +p 2 x 2 =m (C1) Hence, 1
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x x p p x x p p 2 1 1 2 2 1 1 2 2 2 =⇒= Thus, px p 22 1 2 1 2 = From (C1) m 11 2 2 += p m 1 2 1 2 ⇒+ = + = p p pp p p xm 1 2 2 2 1 1 () Thus, x mp 1 2 2 = + (D) Similarly x p p x 2 1 2 2 2 1 =⇒ x mp 2 1 21 2 = + (E) Notice that equations (D) and (E) express the optional quantities of x 1 and x 2 as functions of the variables (p 1 , p 2 , m). These two formulas can be utilized to determine the quantities of the two goods the consumer would buy in order to maximize utility given the price (p 1 , p 2 ) and income (m). These are the Marshallian demand functions. Note that if m and p 2 are held fixed at m 0 and p 2 0 respectively, we can express x 1 as a function of p 1 only. xp p m mp (, , ) 12 0 0 0 2 0 2 0 = + 2
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For example, let p 2 0 =2, m 0 =10, then xpp m pp (, ) 12 1 2 1 21 0 20 2 == = + x p m m mp p 1 1 22 0 0 0 2 0 1 2 0 2 0 20 ) () = + +< Similarly, x p m m m p p 1 2 11 0 0 0 1 0 1 0 1 0 2 0 ) ( ) [ ] = +- + = + > pp p 0 1 0 1 0 1 0 2 2 0 x m p 1 0 0 2 0 1 0 1 0 2 0 0 ) = + > Note that x 1
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Econ 5201 Lecture 3 - CONSUMERS PROBLEM III Consider...

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