MICROECONOMICS Principles and Analysis Frank Cowell 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 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 A a log x1 + [1 a] log x2 Cobb-Douglas preferences x1

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

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

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

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

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…

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

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

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]

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″

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″.

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

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

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

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

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

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

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

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

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