exercise 7.8
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
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
ics
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
ics
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
ics
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
owell:
Frank C
owell: M
icroeconomics
Microeconom
ics
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
ics
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
ics
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