1. MICROECONOMICS Principles and Analysis Frank Cowell
Exercise 7.7
MICROECONOMICS
Principles and Analysis
Frank Cowell
November 2006

2. Ex 7.7(1): Question
purpose: to build up four examples of solving CE using the offer-curve approach
method: use examination of preference map as a shortcut to getting offer curves. Then use offer curves in Edgeworth box

3. Ex 7.7(1): Case A a log x1 + [1 a] log x2 Cobb-Douglas preferences x1

4. Ex 7.7(1): Case B b x1 + x2 Linear indifference curves x1 x2 x1 x2

5. Ex 7.7(1): Case C
g x12 + x22
If g =1 indifference curves are quarter circles x1 x2 x1 x2
g > 1
g = 1

6. Ex 7.7(1): Case D min {dx1, x2} Leontief preferences x1 x2 x1 x2

7. Ex 7.7(2): Question Method: Use standard Lagrangean approach
Then plot locus of optimal points as price is varied.

8. Ex 7.7(2): Demand, case A Set up the Lagrangean:
Differentiate w.r.t. x1, x2, l to get the FOC:
Solve these three equations to get l = 1 / 10r
So demand is:
This will give us the offer curve…

9. Ex 7.7(2): Offer curve, case A
Preferences x2
Endowment
Increase the price r
The offer curve
Offer curve is the vertical line x11 = 10a • • • • x1 10

10. 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

11. Ex 7.7(3): Offer curve, case A
Preferences x2
Endowment • 20
Increase the price r
The offer curve
Use the demand function from part 2. Income is 20 now (instead of 10r) • • • x1
Offer curve is the horizontal line x22 = 20[1−a]

12. Ex 7.7(3): Offer curve, case B
Preferences x2
Endowment x′ • • 20
Increase the price r
The offer curve
We can infer demands and offer curve directly from diagram.
Key point is whether budget constraint lies on line joining x′ :=(0,20) and x″:=(20/b, 0) •
Offer curve is the line segment with a kink at x″. x1 • x″

13. Ex 7.7(3): Offer curve, case C
Preferences x2
Endowment x′ • • 20
Increase the price r
The offer curve
Again infer demands and offer curve directly from diagram.
Again, key point is whether budget constraint lies on line joining x′ :=(0,20) and x″:=(20/g, 0) • • x1 • x″
Offer curve is blob at x′ and line segment from x″.

14. Ex 7.7(3): Offer curve, case D
Preferences x2
Endowment • 20
Increase the price r
The offer curve •
Again use the diagram directly. •
Solution must lie on corner of the indifference curve where x2 = dx1. Use this fact and the budget constraint x2 + rx1=20 • x1
Offer curve is line through the all the corners

15. 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

16. Ex 7.7(4): Equilibrium, case A
Group 1 has type A preferences:
given income 10r offer curve is vertical line x11 = 10a
substitute in a = ½ and we find x11 = 5
from materials-balance condition x12 = 10  5 = 5
Group 2 also has type A preferences:
given income 20 offer curve is the horizontal line x22 = 20[1−a]
substitute in a = ¾ and we find x22 = 5
from materials-balance condition x21 = 20  5 = 15
So equilibrium allocation is x1 = (5, 15), x2 = (5, 5)
Also use the demand functions to solve for equilibrium r
for example x21 = 10r[1  a ] = 5r (recall that a = ½)
given that, in equilibrium, x21 = 15…
… we must have, in equilibrium, r = 3

17. Ex 7.7(4): Equilibrium, case A10 O2
Draw the Edgeworth box
Offer curve for type 1
Offer curve for type 2A
Equilibrium allocation •
Equilibrium price
x1 = (5,15)
x2 = (5,5) 20
r =3 r O1

18. Ex 7.7(4): Equilibrium, case BO2
Offer curve for type 1
Offer curve for type 2B
Equilibrium allocation •
Equilibrium price
x1 = (5,15)
x2 = (5,5)
r =3 r O1

19. Ex 7.7(4): Equilibrium, case DO2
Offer curve for type 1
Offer curve for type 2D
Equilibrium allocation •
Equilibrium price
x1 = (5,15)
x2 = (5,5)
r =3 r O1

20. Ex 7.7(5): Question Method Reexamine intersection of the offer curves
Consider point about numbers in groups

21. Ex 7.7(4): Equilibrium? Case CO2
Look at the box again
Offer curve for type 1
Offer curve for type 2C
Mimic effect of large numbers
Offer curves do not intersect
Will there be a solution?
If the groups are large then on average result looks like case B
Solution will be as in case B O1

22. 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