Exercise 7.7

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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

Transcript of Exercise 7.7

  • Exercise 7.7MICROECONOMICSPrinciples and Analysis Frank Cowell November 2006

  • Ex 7.7(1): Questionpurpose: 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

  • Ex 7.7(1): Case Aa log x1 + [1 a] log x2 Cobb-Douglas preferencesa > a =

  • Ex 7.7(1): Case Bb x1 + x2 Linear indifference curvesb > 1b = 1

  • Ex 7.7(1): Case Cg x12 + x22 If g =1 indifference curves are quarter circlesg > 1g = 1

  • Ex 7.7(1): Case Dmin {dx1, x2}Leontief preferencesd > 1d = 1

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

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

  • Ex 7.7(2): Offer curve, case Ax1x2PreferencesEndowmentIncrease the price rThe offer curve10Offer curve is the vertical line x11 = 10a

  • Ex 7.7(3): QuestionMethodCan get the solution to type A by adapting part 2Types B-D follow by using the diagrams in Part 1

  • Ex 7.7(3): Offer curve, case Ax1x2PreferencesEndowmentIncrease the price rThe offer curve20Offer curve is the horizontal line x22 = 20[1a] Use the demand function from part 2. Income is 20 now (instead of 10r)

  • Ex 7.7(3): Offer curve, case Bxx1x2PreferencesEndowmentIncrease the price rThe offer curve20Offer 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/b, 0)xWe can infer demands and offer curve directly from diagram.

  • Ex 7.7(3): Offer curve, case Cxx1x2PreferencesEndowmentIncrease the price rThe offer curve20Offer 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/g, 0)xAgain infer demands and offer curve directly from diagram.

  • Ex 7.7(3): Offer curve, case Dx1x2PreferencesEndowmentIncrease the price rThe offer curve20Offer curve is line through the all the cornersSolution must lie on corner of the indifference curve where x2 = dx1. Use this fact and the budget constraint x2 + rx1=20Again use the diagram directly.

  • Ex 7.7(4): QuestionMethodAgain re-use previous results, this time from parts 2 and 3Substitute in the parameter valuesCheck where the offer curves intersect

  • Ex 7.7(4): Equilibrium, case AGroup 1 has type A preferences:given income 10r offer curve is vertical line x11 = 10asubstitute in a = and we find x11 = 5from materials-balance condition x12 = 10 5 = 5Group 2 also has type A preferences:given income 20 offer curve is the horizontal line x22 = 20[1a] substitute in a = and we find x22 = 5from materials-balance condition x21 = 20 5 = 15So 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 = ) given that, in equilibrium, x21 = 15 we must have, in equilibrium, r = 3

  • Ex 7.7(4): Equilibrium, case AO1O22010Draw the Edgeworth boxOffer curve for type 1Offer curve for type 2Ar =3x1 = (5,15) x2 = (5,5) Equilibrium allocationEquilibrium pricer

  • Ex 7.7(4): Equilibrium, case BO1O2r =3x1 = (5,15) x2 = (5,5) Equilibrium allocationEquilibrium priceOffer curve for type 2BOffer curve for type 1r

  • Ex 7.7(4): Equilibrium, case DO1O2r =3x1 = (5,15) x2 = (5,5) Equilibrium allocationEquilibrium priceOffer curve for type 2DOffer curve for type 1r

  • Ex 7.7(5): QuestionMethodReexamine intersection of the offer curvesConsider point about numbers in groups

  • Ex 7.7(4): Equilibrium? Case CO1O2Look at the box againOffer curve for type 2COffer curves do not intersectWill there be a solution?Offer curve for type 1Mimic effect of large numbersIf the groups are large then on average result looks like case BSolution will be as in case B

  • Ex 7.7: Points to rememberUse graphics to find the shape of the solutionfor example types B, C, D in part 2 follow directly from thinking about the indifference curvesReuse the solutions from one part in anotherfor example we got the solution to type A in part 3 by adapting part 2Be careful of discontinuous response functionswording of part 5 allows you to consider a mixture solutionDont do more than is necessarypart 5 just asked you to discuss the issueyou dont have to produce a numerical solution