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Exercise 7.7. MICROECONOMICS Principles and Analysis Frank Cowell. November 2006. Ex 7.7(1): Question. purpose : to build up four examples of solving CE using the offer-curve approach - PowerPoint PPT Presentation
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Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Exercise 7.7
MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis
Frank CowellFrank Cowell
November 2006 November 2006
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(1): Question
purposepurpose: to build up four examples of solving CE using the : to build up four examples of solving CE using the offer-curve approach offer-curve approach
methodmethod: use examination of preference map as a shortcut to : use examination of preference map as a shortcut to getting offer curves. Then use offer curves in Edgeworth box getting offer curves. Then use offer curves in Edgeworth box
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(1): Case A log log xx11 + [1 + [1 ] log ] log xx22 Cobb-Douglas preferencesCobb-Douglas preferences
x1
x2
x1
x2
> ½ = ½
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(1): Case B xx11 + + xx22 Linear indifference curvesLinear indifference curves
x1
x2
x1
x2
> 1 = 1
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(1): Case C xx11
22 + + xx2222
If If =1 indifference curves are quarter circles =1 indifference curves are quarter circles
x1
x2
x1
x2
> 1 = 1
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
x1
x2
x1
x2
Ex 7.7(1): Case D min {min {xx11, , xx22}} Leontief preferencesLeontief preferences
> 1 = 1
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(2): Question
MethodMethod: : Use standard Lagrangean approach Use standard Lagrangean approach Then plot locus of optimal points as price is varied.Then plot locus of optimal points as price is varied.
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(2): Demand, case A Set up the Lagrangean:
Differentiate w.r.t. x1, x2, to get the FOC:
Solve these three equations to get = 1 / 10
So demand is: This will give us the offer
curve…
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(2): Offer curve, case A
x1
x2
•
Preferences
EndowmentIncrease the price
The offer curve
••
•
10
Offer curve is the vertical line x1
1 = 10
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(3): Question
Method Can get the solution to type A by adapting part 2 Types B-D follow by using the diagrams in Part 1
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(3): Offer curve, case A
x1
x2
•
Preferences
Endowment
Increase the price
The offer curve
• • •
20
Offer curve is the horizontal line x2
2 = 20[1−]
Use the demand function from part 2. Income is 20 now (instead of 10)
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
x′
Ex 7.7(3): Offer curve, case B
x1
x2
•
Preferences
EndowmentIncrease the price
The offer curve
•
• 20
Offer curve is the line segment with a kink at x″.
Key point is whether budget constraint lies on line joining x′ :=(0,20) and x″:=(20/, 0)
•x″
We can infer demands and offer curve directly from diagram.
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
x′
Ex 7.7(3): Offer curve, case C
x1
x2
•
Preferences
EndowmentIncrease the price The offer curve
•
20
Offer curve is blob at x′ and line segment from x″.
Again, key point is whether budget constraint lies on line joining x′ :=(0,20) and x″:=(20/, 0)
•x″
Again infer demands and offer curve directly from diagram.
•
•
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(3): Offer curve, case D
x1
x2
•
Preferences
EndowmentIncrease the price
The offer curve
20
Offer curve is line through the all the corners
Solution must lie on corner of the indifference curve where x2 = x1. Use this fact and the budget constraint x2 + x1=20
Again use the diagram directly.
•
•
•
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(4): Question
Method Again re-use previous results, this time from parts 2 and 3 Substitute in the parameter values Check where the offer curves intersect
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(4): Equilibrium, case A Group 1 has type A preferences:Group 1 has type A preferences:
given income 10given income 10 offer curve is vertical line offer curve is vertical line xx111 1 = 10 = 10
substitute in substitute in = = ½ and we find ½ and we find xx111 1 = 5 = 5
from materials-balance conditionfrom materials-balance condition xx112 2 = 10 = 10 5 = 5 5 = 5
Group 2 also has type A preferences:Group 2 also has type A preferences: given income 20 given income 20 offer curve is the horizontal line offer curve is the horizontal line xx22
2 2 = 20[1 = 20[1−−]] substitute in substitute in = = ¾ and we find ¾ and we find xx22
2 2 = 5 = 5 from materials-balance conditionfrom materials-balance condition xx22
11 = 20 = 20 5 = 15 5 = 15 So equilibrium allocation isSo equilibrium allocation is xx1 1 = (5, 15), = (5, 15), xx2 2 = (5, 5) = (5, 5) Also use the demand functions to solve for equilibrium Also use the demand functions to solve for equilibrium
for examplefor example xx2211 = = 10 10[1 [1 ] = 5] = 5 (recall that (recall that = = ½½))
given that, in equilibrium,given that, in equilibrium, xx2211 = 15… = 15…
… … we must have, in equilibrium,we must have, in equilibrium, = 3 = 3
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(4): Equilibrium, case A
O1
O2
20
10Draw the Edgeworth boxOffer curve for type 1Offer curve for type 2A
=3
x1 = (5,15) x2 = (5,5)
Equilibrium allocation• Equilibrium price
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(4): Equilibrium, case B
O1
O2
=3
x1 = (5,15) x2 = (5,5)
Equilibrium allocationEquilibrium price
Offer curve for type 2B
Offer curve for type 1
•
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(4): Equilibrium, case D
O1
O2
=3
x1 = (5,15) x2 = (5,5)
Equilibrium allocationEquilibrium price
Offer curve for type 2D
Offer curve for type 1
•
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(5): Question
Method Reexamine intersection of the offer curves Consider point about numbers in groups
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7(4): Equilibrium? Case C
O1
O2
Look at the box again
Offer curve for type 2C
Offer curves do not intersect Will there be a solution?
Offer curve for type 1
Mimic effect of large numbers
If the groups are large then on average result looks like case B
Solution will be as in case B
Frank Cow
ell: Frank C
owell: M
icroeconomics
Microeconom
ics
Ex 7.7: Points to remember Use graphics to find the “shape” of the solution
for example types B, C, D in part 2 follow directly from thinking about the indifference curves
Reuse the solutions from one part in another for example we got the solution to type A in part 3 by
adapting part 2 Be careful of discontinuous response functions
wording of part 5 allows you to consider a “mixture” solution
Don’t do more than is necessary part 5 just asked you to discuss the issue you don’t have to produce a numerical solution
