Frank Cowell: Microeconomics Exercise 7.8 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

Frank Cowell: Microeconomics Ex 7.8: Question purpose: to show how to find equilibrium allocation in a GE model purpose: to show how to find equilibrium allocation in a GE model method: standard construction and solution of excess demand functions. method: standard construction and solution of excess demand functions.

Frank Cowell: Microeconomics Ex 7.8: approach Step 1: model behaviour of each type as a price taker   Write down budget constraint for the unknown price 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

Frank Cowell: Microeconomics Ex 7.8: type-a problem The endowment for type a is R 1 Let price of good 1 in terms of good 2 be p The income of type a is then pR 1 The utility function is: So the Lagrangean of type a is:

Frank Cowell: Microeconomics Ex 7.8: type-a demand Given the Lagrangean for a: FOCs for interior maximum: Rearrange and use the budget constraint: Demand by a for good 2:

Frank Cowell: Microeconomics Ex 7.8: type-b problem The endowment for type b is R 2 Recall that values are measured in terms of good 2 So the income of type b is just R 2 The utility function is: So the Lagrangean of type b is:

Frank Cowell: Microeconomics Ex 7.8: type-b demand Given the Lagrangean for b: FOCs for interior maximum: Rearrange and use the budget constraint: Demand by b for good 2:

Frank Cowell: Microeconomics Ex 7.8: excess demand Demand by the two types for good 2: Excess demand for good 2 is defined as x x 2 2  R 2 So the excess demand function for good 2 is: Letting  := 2R 1 /R 2 excess demand is zero where

Frank Cowell: Microeconomics Ex 7.8: how many equilibria?   Graph of p 2/3   Graph of p   1   Equilibrium p p 2/3 p  p*p*   Excess demand is zero where p 2/3 = p    There is clearly only one equilibrium p *

Frank Cowell: Microeconomics Ex 7.8: the equilibrium To find the equilibrium we need the resource values   R 1 = 5   R 2 = 16   So  := 2R 1 /R 2 = 5/8 Equilibrium price must satisfy p 2/3 =  5/8  p  Use trial and error to find solution   check whether there is excess demand/supply at certain prices   try easy numbers that have integer cube roots: p = 1? 8? 27? … Clearly p = 1 is too low and p = 27 is too high Try p = 8   LHS: p 2/3 = 4   RHS: (5/8) p  1 = 4   so this is the equilibrium

Frank Cowell: Microeconomics Ex 7.8: Points to note Step by step approach gets you very close to the solution   work out individual demands   set excess demand to zero   get a condition to determine equilibrium price Graphical intuition helps you get the form of the solution Don’t get fazed by awkward numbers   trial-and-error method quickly gives you the answer