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hw8-06-2 - expenditure function B ∗ = B p x p y U...

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ECONOMICS 331 Mathematical Economics Kevin Wainwright Homework Assignment 8 Skip has the following utility function: U ( x, y ) = x ( y + 1) , where x and y are quantities of two consumption goods whose prices are p x and p y respectively. Skip has a budget of B. Therefore the Skip’s maximization problem is x ( y + 1) + λ ( B p x x p y y ) a) From the fi rst order conditions fi nd expressions for the demand functions x = x ( p x , p y , B ) y = y ( p x , p y , B ) Carefully graph x and y . Graph Skip’s indi ff erence curves. What kind of good is y? b) Verify that skip is at a maximum by checking the second order conditions. c) By substituting x and y into the utility function fi nd an expressions for the indirect utility function, U = U ( p x , p y , B ) d) By rearranging the indirect utility function, derive an expression for the
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Unformatted text preview: expenditure function, B ∗ = B ( p x , p y , U ) Interpret this expression. Find ∂ B / ∂ p x and ∂ B / ∂ p y . Skip’s maximization problem could be recast as the following minimization problem: p x x + p y y s.t. U = x ( y + 1) e) Write down the lagrangian for this problem. f) Find the values of x and y that solve this minimization problem and show that the values of x and y are equal to the partial derivatives of the expendi-ture function, ∂ B / ∂ p x and ∂ B / ∂ p y respectively. (Hint: use the indirect utility function) 1...
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