Frank Cowell Microeconomics November 2006 Exercise 7 8

Frank Cowell: Microeconomics November 2006 Exercise 7. 8 MICROECONOMICS Principles and Analysis Frank Cowell

Ex 7. 8: Question Frank Cowell: Microeconomics n n purpose: to show to find equilibrium allocation in a GE model method: standard construction and solution of excess demand functions.

Ex 7. 8: approach Frank Cowell: Microeconomics n Step 1: model behaviour of each type as a price taker u u n Step 2: get excess demand function for one of the goods u u n Write down budget constraint for the unknown price p Set up Lagrangean for each type Find the FOCs Get demand functions from the FOCs 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

Ex 7. 8: type-a problem Frank Cowell: Microeconomics n 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 p. R 1 The utility function is: n So the Lagrangean of type a is: n n n

Ex 7. 8: type-a demand Frank Cowell: Microeconomics n Given the Lagrangean for a: n FOCs for interior maximum: n Rearrange and use the budget constraint: n Demand by a for good 2:

Ex 7. 8: type-b problem Frank Cowell: Microeconomics n 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: n So the Lagrangean of type b is: n n n

Ex 7. 8: type-b demand Frank Cowell: Microeconomics n Given the Lagrangean for b: n FOCs for interior maximum: n Rearrange and use the budget constraint: n Demand by b for good 2:

Ex 7. 8: excess demand Frank Cowell: Microeconomics n Demand by the two types for good 2: n Excess demand for good 2 is defined as x 21 + x 22 R 2 n So the excess demand function for good 2 is: n Letting q: = 2 R 1/R 2 excess demand is zero where

Ex 7. 8: how many equilibria? Frank Cowell: Microeconomics §Graph of p 2/3 pq 1 p 2/3 §Graph of pq 1 §Equilibrium § Excess demand is zero where p 2/3 = pq 1 p* p § There is clearly one equilibrium p*

Ex 7. 8: the equilibrium Frank Cowell: Microeconomics n To find the equilibrium we need the resource values u u u n n Equilibrium price must satisfy p 2/3 = (5/8) p 1 Use trial and error to find solution u u n n R 1 = 5 R 2 = 16 So q : = 2 R 1/R 2 = 5/8 check whethere 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 u u u LHS: p 2/3 = 4 RHS: (5/8) p 1 = 4 so this is the equilibrium

Ex 7. 8: Points to note Frank Cowell: Microeconomics n Step by step approach gets you very close to the solution u u u n n 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 u trial-and-error method quickly gives you the answer