3 Solve the following two dimensional maximization problems subject to the

3 solve the following two dimensional maximization

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3. Solve the following two dimensional maximization problems subject to the linear constraint w = p 1 x 1 + p 2 x 2 , p 1 , p 2 > 0. Sketch contour lines of the objective function and the constraint set. Then compute the change in x 1 with respect to p 2 , and a change in x 1 with respect to w . Assume α 1 > α 2 > 0. Foriandii, when do the SOSCs hold? = p 1 p 2 x 1 x 1 = w
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0 p 1 p 2 p 1 α 1 ( α 1 1) x α 1 2 1 x α 2 2 α 1 α 2 x α 1 1 1 x α 2 1 2 (1) So a set of sufficient conditions to ensure this are that 0 < α 1 < 1 and 0 < α 2 < 1, so that the left-hand side is positive but the right-hand side is negative. ii. Stone-Geary f ( x ) = ( x 1 γ 1 ) α 1 ( x 2 γ 2 ) α 2 The Lagrangian is L ( x, λ ) = ( x 1 γ 1 ) α 1 ( x 2 γ 2 ) α 2 λ ( p 1 x 1 + p 2 x 2 w ) The FONCs are α 1 ( x 1 γ 1 ) α 1 1 ( x 2 γ 2 ) α 2 λp 1 = 0 α 2 ( x 1 γ 1 ) α 1 ( x 2 γ 2 ) α 2 1 λp 2 = 0 ( p 1 x 1 + p 2 x 2 w ) = 0 The first two equations imply α 1 ( x 2 γ 2 ) α 2 ( x 1 γ 1 ) = p 1 p 2 4
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Solving for x 2 yields x 2 = p 1 α 1 p 2 α 2 ( x 1 γ 1 ) + γ 2 Substituting this into the constraint yields p 1 x 1 + p 1 α 1 α 2 ( x 1 γ 1 ) + p 2 γ 2 = w Solving for x 1 yields x 1 = α 1 w p 2 α 1 γ 2 + p 1 α 2 γ 1 p 1 ( α 1 + α 2 ) x 2 = α 2 w p 1 α 2 γ 1 + p 2 α 1 γ 2 p 2 ( α 2 + α 1 ) and the comparative statics are ∂x 1 ∂w = α 1 p 1 ( α 1 + α 2 ) > 0 ∂x 1 ∂p 2 = α 1 γ 2 p 1 ( α 1 + α 2 ) < 0 The bordered Hessian is 0 p 1
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