lecture16kg - Introduction to Microeconomics Lecture 16...

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Unformatted text preview: Introduction to Microeconomics Lecture 16 Unconstrained Optimization Kathryn Graddy Outline Optimization Perfect Competition vs. Monopoly Cournot Von Stackelberg Maxima and Minima with Two Variables Differentiated Products Optimization In lecture 8 when we maximized profits, we were solving an optimization problem. More formally, the agent's optimisation problem is: max ( ) ( )is called the agent's y y y objective f Π Π- the thing he wants to optimise. unction 2 2 To solve the optimisation problem above, we can look for a value of that satisfies two conditions: The first-order condition: The second-order condition: y d dy d dy Π = Π < Review: Profit Maximization y ( ) ( ) ( ) max ( ) ( ) ( ) y R y C y y d R y C y dy Π =- Π Π ′ ′ =- = Example ( 29 2 2 2 Demand: ( ) 4 2 ( ) 2 12 ( ) ( ) ( ) (4 2 ) 2 12 4 4 12 P y y C y y y yP y C y y y y y y =- =- Π =- =--- =- + Example (cont.) 2 2 FOC: 4 8 2 Is this the maximum? 8 8 0 so yes d y dy y d dy Π =- = = Π = -- < Perfect Competition In perfect competition ( ) (A firm is a price taker) ( ) ( ) max ( ) A perfectly competitive firm still maximizes profits! d FOC: '( ) dy P = C'(y) y P y P y Py C y y P C y = Π =- Π Π =- = Perfect Competition- : - ( ) - ( ) : ( ) '( ) R C FOC MR MC P Q q C q FOC P Q C q π π = = = = Monopoly- : - ( ) - ( ) ( ) : ( ) * '( ) ( ) ( ) ( ) * * '( ) ( ) ( ) ( )- '( ) 1 (Inverse elasticity rule)...
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This note was uploaded on 04/12/2010 for the course ECON DEAM taught by Professor Vines during the Spring '10 term at Oxford University.

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lecture16kg - Introduction to Microeconomics Lecture 16...

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