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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

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Exercise 7.7

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  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

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