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slides_lect5 - Lecture 5 1 Summary of Lectures 1 2 and 3...

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Lecture 5 1 Summary of Lectures 1, 2, and 3: Technology, Pro°t maximization, The cost function and duality 2 Summary of Lecture 4: Consumption theory 2.1 Preference orders 2.2 The utility function 2.3 The utility maximization problem 2.3.1 Solving the UMP 2.3.2 Walrasian demand 2.3.3 Indirect utility function 2.4 The expenditure minimization problem 2.4.1 Compensated or Hicksian demand Instead of maximizing utility given a budget constraint we can consider the dual problem of minimizing the expenditure necessary to obtain a given utility level. Speci°cally, if we would like to reach the utility level that results in the °rst problem it turns out that the bundle that minimizes the cost of doing so coincides with the solution to the °rst problem. The FOC for expenditure minimization imply the same relation between the prices and the marginal utilities as the FOC for utility maximization. The solution to this problem is the optimal consumpion bundles as functions of p and u . Income is adjusted so the consumer can a/ord the cheapest possible bundle that yields u . These demand functions (one for each good) are called compensated or Hicksian demand functions and are denoted h ( p; u ) . 2.4.2 Expenditure function The minimal expenditure necessary to reach u is the expenditure function : X i p i ° h i ( p; u ) = e ( p; u ) Remark 1 Local non-satiation This assumption implies that v ( p; m ) is strictly increasing in m . Thus we can derive the minimal expenditure necessary to reach u , e ( p; u ) , simply 1
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by inverting v ( p; m ) . It follows that e ( p; u ) is strictly increasing in u . Properties of the expenditure function 1. e ( p; u ) is nondecreasing in p . 2. e ( p; u ) is homogeneous of degree 1 in p. 3. e ( p; u ) is concave in p .
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