note2 - Microeconomic Theory Econ 101A Fall 2008 GSI: Eva...

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Microeconomic Theory Econ 101A Fall 2008 GSI: Eva Vivalt Section Notes 2: Consumer Theory 1 Utility Maximization Problem Problem set-up: Two goods: x 1 and x 2 with prices p 1 and p 2 . Consumers have a fixed income, I , and thus are constrained to spend less or equal to the amount I on goods x 1 and x 2 . Thus the budget constraint can be expressed as: p 1 x 1 + p 2 x 2 I . It is impossible to consume negative amounts of goods, so x 1 0 ,x 2 0. Consumers maximize utility U ( x 1 ,x 2 ) which is increasing in both x 1 and x 2 and quasi-concave in ( x 1 ,x 2 ). Consumers are price takers, which means that they take prices as fixed (i.e. no impact of their individual demands on prices. This can be justified by postulating a large number of consumers). Since utility is increasing in both x 1 and x 2 , we can safely assume that the budget constraint holds with equality. Also, we will assume that the non-negativity constraints are slack, so that x 1 > 0 and x 2 > 0. Setting the inequalities to equality, combining the two constraints, and rearranging a bit we get the standard utility maximization problem (or UMP): max x 1 ,x 2 U ( x 1 ,x 2 ) s.t. p 1 x 1 + p 2 x 2 = I We can solve this constrained maximization problem by writing the Lagrangian: L ( x 1 ,x 2 ) = U ( x 1 ,x 2 ) - λ [ p 1 x 1 + p 2 x 2 - I ] and finding the FOCs: ∂L ∂x 1 = U 1 ( x * 1 ,x * 2 ) - λ * x 1 = 0 ∂L ∂x 2 = U 2 ( x * 1 ,x * 2 ) - λ * x 2 = 0 ∂L ∂λ = - [ p 1 x * 1 + p 2 x * 2 - I ] = 0 The solution yields the Lagrange multiplier λ * = λ * ( p 1 ,p 2 ,I ) and the ordinary demand functions x * 1 = x * 1 ( p 1 ,p 2 ,I ) and x * 2 = x * 2 ( p 1 ,p 2 ,I ). 1 1 These are actually uncompensated demand functions. The functions are “uncompensated” because price changes will cause utility changes – a situation which does not occur with compensated demand curves. 1
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1.1 Example: Linear Expenditure System The linear expenditure system (LES) is a generalization of the Cobb-Douglas utility function that incor- porates the idea that individuals require some minimum amount of each good ( a,b ). In this case, the consumer’s UMP is: max x 1 ,x 2 ( x 1 - 1) α ( x 2 - b ) β s.t. p 1 x 1 + p
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note2 - Microeconomic Theory Econ 101A Fall 2008 GSI: Eva...

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