1 / 22

Exercise 7.7

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

jmaribel
Télécharger la présentation

Exercise 7.7

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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. x2 x2 x1 x1 Ex 7.7(1): Case A • a log x1 + [1 a] log x2 • Cobb-Douglas preferences a > ½ a = ½

  4. x2 x2 x1 x1 Ex 7.7(1): Case B • bx1 + x2 • Linear indifference curves b > 1 b = 1

  5. x2 x2 x1 x1 Ex 7.7(1): Case C • gx12 + x22 • If g =1 indifference curves are quarter circles g > 1 g = 1

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

  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 10roffer curve is vertical line x11 = 10a • substitute in a = ½ and we find x11 = 5 • from materials-balance conditionx12 = 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 conditionx21 = 20  5 = 15 • So equilibrium allocation isx1 = (5, 15), x2 = (5, 5) • Also use the demand functions to solve for equilibrium r • for examplex21 = 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 A 10 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 B O2 • 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 D O2 • 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 C O2 • 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

More Related