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Exercise 7.8

Exercise 7.8. MICROECONOMICS Principles and Analysis Frank Cowell. November 2006. Ex 7.8: Question. purpose : to show how to find equilibrium allocation in a GE model method : standard construction and solution of excess demand functions. Ex 7.8: approach.

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Exercise 7.8

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  1. Exercise 7.8 MICROECONOMICS Principles and Analysis Frank Cowell November 2006

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

  3. 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

  4. Ex 7.8: type-a problem • The endowment for type a is R1 • Let price of good 1 in terms of good 2 be p • The income of type a is then pR1 • The utility function is: • So the Lagrangean of type a is:

  5. 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:

  6. Ex 7.8: type-b problem • The endowment for type b is R2 • Recall that values are measured in terms of good 2 • So the income of type b is just R2 • The utility function is: • So the Lagrangean of type b is:

  7. 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:

  8. Ex 7.8: excess demand • Demand by the two types for good 2: • Excess demand for good 2 is defined as x21 + x22 R2 • So the excess demand function for good 2 is: • Letting q:= 2R1/R2 excess demand is zero where

  9. Ex 7.8: how many equilibria? • Graph of p2/3 pq  1 • Graph of pq 1 p2/3 • Equilibrium • Excess demand is zero wherep2/3= pq  1 p • There is clearly only one equilibrium p* p*

  10. Ex 7.8: the equilibrium • To find the equilibrium we need the resource values • R1 = 5 • R2 = 16 • So q := 2R1/R2 = 5/8 • Equilibrium price must satisfy p2/3= (5/8) p 1 • 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: p2/3= 4 • RHS: (5/8) p 1= 4 • so this is the equilibrium

  11. 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

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