Lec05-Expenditure_Function

# Lec05-Expenditure_Function - Lecture 5 The Expenditure...

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Unformatted text preview: Lecture 5 - The Expenditure Function: An Application to the Economics of Food Stamps David Autor 14.03 Microeconomic Theory and Public Policy, Fall 2005 1 1 The Expenditure Function We are next going to look at a potentially richer (and better) application of consumer theory: the value of Food Stamps. Before that, we need some more machinery. So far, we’ve analyzed problems where income was held constant and prices changes. This gave us the Indirect Utility Function. Now, we want to analyze problems where utility is held constant and expenditures change. This gives us the Expenditure Function. These two problems are closely related — in fact, they are ‘ duals .’ Most economic problems have a dual problem , which means an inverse problem. For example, the dual of choosing output in order to maximize pro f ts is minimizing costs at a given output level; cost minimization is the dual of pro f t maximization. Similarly, the dual of maximizing utility subject to a budget constraint is minimizing expenditures subject to a utility constraint. Minimizing costs is the dual of maximizing utility. 1.1 Setup of expenditure function Consumer’s primal problem : maximize utility subject to a budget constraint. Consumer’s dual problem : minimizing expenditure subject to a utility constraint (i.e. a level of utility you must achieve) This dual problem yields the “expenditure function”: the minimum ex- penditure required to attain a given utility level. Setup of the dual 1. Start with: max U ( x, y ) s.t. p x x + p y y ≤ I 2 2. Solve for x ∗ , y ∗ ⇒ v ∗ = U ( x ∗ , y ∗ ) given p x , p y , I . V ∗ = V ( p x , p y , I ) V is the indirect utility function. 3. Now solve the following problem: min p x x + p y y s.t.U ( x, y ) ≥ v ∗ gives E ∗ = p x x ∗ + p y y ∗ for U ( x ∗ , y ∗ ) = v ∗ . E ∗ = E ( p x , p y , V ∗ ) 1.2 Graphical representation of dual problem U = v* x y 3 The dual problem consists in choosing the lowest budget set tangent to a given indi f erence curve. Example: min E = p x x + p y y s.t. x . 5 y . 5 ≥ U p where U p comes from the primal problem. L = p x x + p y y + λ ¡ U p − x . 5 y . 5 ¢ ∂L ∂x = p x − λ. 5 x − . 5 y . 5 = 0 ∂L ∂y = p y − λ. 5 x . 5 y − . 5 = 0 ∂L ∂λ = U p − x . 5 y . 5 = 0 The f rst two of these equations simplify to: x = p y y p x We substitute into the constraint U p = x . 5 y . 5 to get U p = μ p y y p x ¶ . 5 y . 5 x ∗ = μ p y p x ¶ . 5 U p , y ∗ = μ p x p y ¶ . 5 U p E ∗ = p x μ p y p x ¶ . 5 U p + p y μ p x p y ¶ . 5 U p = 2 p . 5 x p . 5 y U p 4 1.3 Expenditure function: What is it good for? The expenditure function is an essential tool for making consumer theory operational for public policy analysis....
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Lec05-Expenditure_Function - Lecture 5 The Expenditure...

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