562_lecture3_production_cost

# 562_lecture3_production_cost - SI 562 Lecture 3 Production...

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1 Page 1 SI 562 Lecture 3 1 Production and Cost Professor Yan Chen Fall 2009 AGENDA MODULE 1: Production Optimization Production Functions Marginal Returns 2 MODULE 2: Short-Run Cost Concepts Long-Run Cost Concepts MODULE 3: Network externalities Module 1: Production Economics 3

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2 Page 2 Production Analysis Example: Food Gatherers Founded 1988; 1 st MI food rescue program USDA: Almost 1 / 2of all food is thrown out; yet 1 / 4 children, 1/6 senior have inadequate nutrition Transport 2 tons per day, providing 2000 meals Problem: FG wants to expand stat wide Gets grant for 4 Problem: FG wants to expand state-wide. Gets grant for info processing system to provide logistics Budget: \$100,000 Inputs: Networked workstations @ \$5000/yr part-time operator/administrator \$10,000/yr How to configure? 1) Objective? Max meals served daily 2) How do inputs affect objective? () A W, M M 0 > W M 0 > A M 0 2 > A W M 0 > W M A (1) - - - 000 , 100 . . ) , ( max , = + A P W P t s A W M A W A W (a) (b) (c) So: Method 1: From (1): W = (100,000-P A A)/P W Plug into objective function: 5 ) , / ) 000 , 100 (( max A P A P M W A A 0 = + A M P P W M W A W A P P W M A M = W A P P W M A M = FOC: , W MP W M marginal product of W A W A W P P MP MP = ~ like consumer choice Method 2: Lagrangian: ) 000 , 100 ( ) , ( + = A P W P A W M L A W λ Lagrangian Multipier 0 = λ = W P M L (1) FOC: 6 W W 0 = λ = A P A M A L 0 000 , 100 = + = λ A P W P L A W (2) (3) From (1) and (2): A W A W P P P P A M W M = λ λ = Same as Method 1.
3 Page 3 Production Optimization I: How to choose the optimal combination of inputs? Necessary Condition: Given a budget, want to maximize output. (Not sufficient: need to also choose optimal budget) Consider two inputs: M = M(W, A) Different combinations of (W, A) give different outputs Various combinations give the same output: 7 isoquant (compare to indifference curves) W A M 1 M 2 =1,500 M 3 =4,000 M 4 =5,000 Budget Constraint as before Goal: Go northeast Solution: find isoquant tangent to budget W W P B , B W P A P W A = + cost line isoquant M * 8 A A B P W A P P W M A M dA dW = W A W A P P MP MP W M A M = = Slope of budget (cost line): Slope of isoquant: Tangent: marginal rate of technical substitution Production optimization II: Given an output level, how to choose inputs to minimize cost? Duality in production and cost theory A isocost curves A P W P C + i W M * 9 A W A W = , min s.t. M(W,A)=M * ) ) , ( ( * M A W M A P W P L A W λ + = Lagrangian: 0 = λ = W M P W L W 0 = λ = A M P A L A FOC: A M W M P P A W =

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4 Page 4 Production Functions : A relationship or mapping between inputs and outputs Could be represented as table, graph or analytical functions Inputs : Fixed: Held constant even if output changes Variable: Amount used is adjusted as output changes Example: Fixed Variable (info econ) 10 building electricity certain staff contract rent R&D Copy to CDs
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562_lecture3_production_cost - SI 562 Lecture 3 Production...

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