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Economics Problem Solving: Equilibrium Analysis, Lagrangean Functions, and Demand Curves

This presentation explains equilibrium analysis in economics by using Lagrangean functions and demand curves to find optimal prices. Learn how to model behavior, set up Lagrangean equations, and calculate excess demand functions. Discover how to find equilibrium prices through mathematical calculations.

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Economics Problem Solving: Equilibrium Analysis, Lagrangean Functions, and Demand Curves

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  1. EC202: Worked Example #3.15 Frank Cowell April 2005 • This presentation covers exactly the material set out in the file WorkedExamples.pdf, but with the addition of a few graphics and comments • To start the presentation select Slideshow\View Show or click on icon below left. • Mouse click or [Enter] to advance through slide show

  2. WX3.15: problem

  3. WX3.15: approach • Step 1: model behaviour of each type as a price taker • Write down budget constraint for the unknown p • Set up Lagrangean for each type • Find the FOCs • Get demand functions from the FOCs • Step 2: get excess demand function for one of the goods • Use the demand functions for each type from step 1 • Other EDF follows by Walras’ law • Step 3: find equilibrium price(s) as root(s) of EDF

  4. WX3.15: type-a person • Lagrangean for a: • FOCs for interior maximum: • Rearrange and use the budget constraint: • Demand by a for good 2:

  5. WX3.15: type-b person • Lagrangean for b: • FOCs for interior maximum: • Rearrange and use the budget constraint: • Demand by b for good 2:

  6. WX3.15: excess demand • Demand by the two types for good 2: • So the excess demand function for good 2 is • Letting q:= 2R1/R2 excess demand is zero where

  7. WX3.15: how many equilibria? • Excess demand is zero where p2/3= pq  1 pq  1 p2/3 p p* • There is clearly only one equilibrium p*.

  8. WX3.15: the equilibrium • Given R1 = 5 R2 = 16 • So q := 2R1/R2 = 5/8 • Equilibrium price must satisfy p2/3= (5/8) p 1 • Clearly p = 1 is too low • Try p = 8 (which has an integer cube root) • LHS = 4; so does RHS

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