lecture16kg

# lecture16kg - Introduction to Microeconomics Lecture 16...

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Introduction to Microeconomics Lecture 16 Unconstrained Optimization Kathryn Graddy

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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

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2 2 To solve the optimisation problem above, we can look for a value of that satisfies two conditions: The first-order condition: 0 The second-order condition: 0 y d dy d dy Π = Π <
Review: Profit Maximization y ( ) ( ) ( ) max ( ) ( ) ( ) 0 y R y C y y d R y C y dy Π = - Π Π = - =

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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 0 2 Is this the maximum? 8 8 0 so yes d y dy y d dy Π = - = = Π = - - <

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Perfect Competition In perfect competition ( ) (A firm is a price taker) ( ) ( ) max ( ) A perfectly competitive firm still maximizes profits! d FOC: '( ) 0 dy P = C'(y) y P y P y Py C y y P C y = Π = - Π Π = - =
Perfect Competition - : - 0 ( ) - ( ) : ( ) '( ) R C FOC MR MC P Q q C q FOC P Q C q π π = = = = Monopoly - : - 0 ( ) - ( ) ( ) : ( ) * '( ) ( ) ( ) ( ) * * '( ) ( ) ( ) ( ) - '( ) 1 (Inverse elasticity rule) R C FOC MR MC P q q C q P q FOC P q q C q q P q P q P q q C q q P q P q P q C q P MC p π π ε ε = = = + = + = + = - =

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Cournot Model (simultaneous quantity setting) 1 2 ( )
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