【正文】 Hence, RPT is approximately I have never succeeded in deriving an analytical expression for all these cases. I have, however, used puter simulations (for example with Excel) to derive approximations to these production possibility frontiers. These tend to show that increasing returns to scale is patible with concavity providing factor intensities are suitably different (case [e]), but convexity arises when factor intensities are similar (case [d]). a. Draw the production possibility frontier and the Edgeworth box diagram. Find where P line is tangent to PPF。 then go back to the box diagram to find input ratio. See Corn Law Debate example in the text. b. P given, land/labor ratio is constant. Equilibrium moves from E to E 39。. Cloth (OC E39。 OC E) Wheat (OW39。E39。 OWE) a. b. If wage = 1, each person’s ine is 10. Smith spends 3 on x, 7 on y. Jones spends 5 on x, 5 on y. Since and demands are we have So Smith demands 6x, 21y. Jones demands 10x, 15y. c. Production is x = 16, y = 36. 20 hours of labor are allocated: 8 to x production, 12 to y production. a. Functions are obviously homogeneous of degree zero since doubling of p1, p2 and p3 does not change ED2 or ED3. b. Walras’s Law states Hence, if ED2 = ED3 = 0, p1ED1 = 0 or ED1 = 0. Can calculate ED1 as p1ED1 = –p2ED2 – p3ED3 Notice that ED1 is homogeneous of degree zero also. c. ED2 = 0 and ED3 = 0 can be solved simultaneously for p2 /p1 and p3 /p1 . Simple algebra yields p2 /p1 = 3 p3 /p1 = 5. If set p1 = 1 have p2 = 3, p3 = 5 at these absolute prices ED1 = ED2 = ED3 = 0. PPF = f2 + c2 = 200 a. For efficiency, set MRS = RPT PPF: 2c2 = 200, c = 10 = f = U, RPT = 1. b. Demand: PF /PC = 2/1 = MRS = c/f so c = 2f. Budget: 2f + 1c = 30 the value of production. Substituting from the demand equation: 4f = 30 f = 30/4, c = 15. 。 an improvement from (a) (the demand effect). c. Set RPT = 2/1 f = 2c. PPF: 5c2 = 200, Budget now is: Spend A further improvement (the production specialization effect) d. f = Food c = Cloth a. Labor constraint f + c = 100 (see graph below) b. Land constraint 2f + c = 150 (see graph below) c. Heavy line in graph below satisfies both constraints. d. Concave because it must satisfy both constraints. Since the RPT = 1 for the labor constraint and 2 for the land constraint, the production possibility frontier of part (c) exhibits an increasing RPT。 hence it is concave. e. Constraints intersect at f = 50, c = 50. f 50 f 50 f. If for consumers g. If pf /pc = or pf /pc = , will still choose f = 50, c = 50 since both price lines “tangent” to production possibility frontier at its kink. h. .8f + .9c = 100 Capital constraint: c = 0 f = 125 f = 0 c = Same PPF since capital constraint is nowhere binding. a. Same for region b. RPT’s should be equal. c. Therefore, , hence But so substituting for yields xA = 2xB also yA = 2yB If Note: Can also show that more of both goods can be produced if labor could move between regions. a. Contract curve is straight line with slope of 2. The only price ratio in equilibrium is 3 to 4 (pc to ph). b. 40h, 80c is on . Jones will have 60h and 120c. c. 60h, 80c is not on . Equilibrium will be between 40h, 80c (for Smith) and 48h, 96c (for Smith), as Jones will not accept any trades that make him worse off. UJ = 4(40) + 3(120) = 520. This intersects the contract curve at 520 = 4(h) + 3(2h), h = 52, c = 104. d. Smith grabs everything。 trading ends up at OJ on . (for diagram, see Problem ) a. Core is OS, OJ between points A and B. b. Offer curve for Smith is portion of OS OJ above point A (since requires fixed proportions). For Jones, offer curve is to consume only C for pc/ph 3/4 and consume only h for pc/ph 3/4. For pc/ph = 3/4, offer curve is the indifference curve UJ. c. Only equilibrium is at point B. pc/ph = 3/4 and Smith gets all the gains from tradethe benefits of being inflexible. Chapter 6/Demand Relationships Among Goods v 36CHAPTER 5 INCOME AND SUBSTITUTION EFFECTS Problems in this chapter focus on parative statics analyses of ine and ownprice changes. Many of the problems are fairly easy so that students can approach the ideas involved in shifting budget constraints in simplified settings. Theoretical material is confined mainly to the Extensions where Shephard39。s Lemma and Roy’s Identity are illustrated for the CobbDouglas case. Comments on Problems An example of perfect substitutes. A fixedproportions example. Illustrates how the goods used in fixed proportions (peanut butter and jelly) can be treated as a single good in the parative statics of utility maximization. An exploration of the notion of homothetic functions. This problem shows that Giffen39。s Paradox cannot occur with homothetic functions. This problem asks students to pursue the analysis of Example to obtain pensated demand functions. The analysis essentially duplicates Examples and . Another utility maximization example. In this case, utility is not separable and crossprice effects are important. This is a problem focusing on “share elasticities”. It shows that more customary elasticities can often be calculated from share elasticities—this is important in empirical work where share elasticities are often used. This is a problem with no substitution effects. It shows how price elasticities are determined only by ine effects which in turn depend on ine shares. This problem illustrates a few simple cases where elasticities are directly related to parameters of the utility function. This problem shows how the aggregation relationships described in Chapter 5 for the case of two goods can be generalized to many goods. A revealed