Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Exercise 7.8

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank CowellFrank Cowell

November 2006 November 2006

Frank C

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Frank C

owell: M

icroeconomics

Microeconom

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Ex 7.8: Question

purposepurpose: to show how to find equilibrium allocation in a GE model : to show how to find equilibrium allocation in a GE model methodmethod: standard construction and solution of excess demand functions. : standard construction and solution of excess demand functions.

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

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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 C

owell:

Frank C

owell: M

icroeconomics

Microeconom

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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:

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

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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 C

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Frank C

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icroeconomics

Microeconom

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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:

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

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 C

owell:

Frank C

owell: M

icroeconomics

Microeconom

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Ex 7.8: excess demand Demand by the two types

for good 2:

Excess demand for good 2 is defined as x2

1 + x22 R2

So the excess demand function for good 2 is:

Letting := 2R1/R2 excess demand is zero where

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

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Ex 7.8: how many equilibria?Graph of p2/3

Graph of p 1

Equilibrium

p

p2/3

p

p*

Excess demand is zero

where p2/3 = p

There is clearly only one

equilibrium p*

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

Ex 7.8: the equilibrium To find the equilibrium we need the resource values

R1 = 5 R2 = 16 So := 2R1/R2 = 5/8

Equilibrium price must satisfy p2/3 = 5/8p 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

Frank C

owell:

Frank C

owell: M

icroeconomics

Microeconom

ics

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