Homework #6: Answers. a. If both goods are produced, what must be their prices?


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1 Text questions, hpter 7, problems 12. Homework #6: Answers 1. Suppose there is only one technique tht cn be used in clothing production. To produce one unit of clothing requires four lborhours nd one unit of cpitl; in food production ech unit requires single lborhour nd one unit of cpitl. At n initil equilibrium suppose the wge rte nd the cpitl rentl re ech vlued t $2.. If both goods re produced, wht must be their prices? Note tht these re Leontief (i.e. fixed coefficient) production functions. The zero profit conditions for these industries re: P w + r ( 1 2) + ( 1 2) 4 K P w + r ( 4 2) + ( 1 2) 10 L K (1) The second equlity follows from substitution. The, if both goods re produced, P 4 nd P 10. b. Now keep the price of food constnt nd rise the price of clothing to $15. Trce through the effects on the distribution of income. Rnk the reltive chnges in the wge rte, the price of clothing, the price of food (unchnged by ssumption), nd the rent on cpitl. Relte your results to the StolperSmuelson theorem. This problem ssume 50% increse in P, i.e. ssumed tht clothing is the Lintensive good, i.e. StolperSmuelson theorem sserts tht: P " P ' P ' L K 4 1 > K.. 5. It is Thus, the w$ > p$ 50% > p$ 0 > r$. (2) Thus, lbor gins unmbiguously nd cpitl loses unmbiguously from this price chnge. With Leontief production functions, equtions (1) re liner equtions in w nd r, so they cn be solved simultneously for ny vlues of P nd P. Tht is: P 4 w + r ( 1 w) + ( 1 r) K P 15 w + r ( 4 w) + ( 1 r) L K (3) One wy to do this is is to trnsform equtions (3) into n explicit reltionship between w
2 nd r, plot both lines, nd find the wge nd rentl tht stisfy these two equtions. Alterntively, one cn solve the equtions directly by substitution or using rmer s rule. To do the ltter, note tht (3) cn be rewritten s: P P L K K w r w. 4 1 r The determinnt of the mtrix A [ ij ], A 3, A w 11, A r 1. Thus: w A w / A 11/3; r A x / A 1/3. These cn be compred to w 2 nd r 2 to give the rnking, given in (2), derived from the StolperSmuelson theorem: w$. 83 > p$. 5 > p$ 0 > r$. 83. (5) 2. Retin the ssumptions bout technology in problem 1: L 4, K 1, 1, K 1.. Drw digrm with cpitl on the verticl xis nd lbor on the horizontl xis. Drw ry through the origin with slope of unity, nd show how outputs of food nd clothing cn be mesured long this ry. Drw fltter ry, with slope of 1/4, nd show how outputs of clothing cn be mesured long this ry. The rys with slope 1 nd 1/4 re the fixed expnsion pths for these Leontief technologies. Becuse these re constnt returns to scle technologies, once we decide on unit, ll outputs re mesured, in terms of those units, long the expnsion pths. b. Suppose the economy possesses 1000 units of lbor. ind the fullemployment output of ech good if the cpitl stock is 500 units. As in the textbook, we use the fullemployment conditions to nswer this question. L y + y L K y + y K K (6) As in question 1, the fct tht the technologies re Leontief, mens the fullemployment conditions re liner reltions. We cn write these in mtrix form s:
3 L K K L K y y y y. (7) Now we just use rmer s rule gin, this time on A T. A T 3, A T L 1000, A T K y A T L/A T 1000/3, y A T /A T 500/3. c. ind the lowest nd highest cpitl stocks tht still llow full employment of both fctors. The lowest nd highest cpitl stocks consistent with fullemployment re determined by speciliztion in the Kintensive good nd the Lintensive good. Thus, the highest cpitl stock consistent with fullemployment involves speciliztion in the Kintensive good (food), which uses K nd L in 1/1 rtio. Since there re 1000 units of L, speciliztion in food would use 1000 units of K. The lowest cpitl stock consistent with fullemployment involves speciliztion in clothing, which uses K nd L in 1/4 rtio. This mens tht speciliztion in would require 250 units of K. d. Drw the trnsformtion schedule for ech of the cses in 2 nd 2b.
4 Workbook problems, 1, Isoqunts nd ctor Intensity:. In digrm with cpitl on the verticl xis nd lbor on the horizontl xis, drw smoothly bowedin isoqunts which stisfy the following two conditions: i. Unit isoqunts pss through the points given by: K 4 K 2 L 2 4 nd the wgerentl rtio is the sme in the two industries t these points. ii. No fctorintensity reversls. b. Which Industry uses lbor reltively intenstively? rom the dt given in the question we hve: K 4 L > 2 K 2 4, so food is lbor intensive in the initil equilibrium. However, by the no fctorintensity reversl ssumption, we know tht this must be true t every w/r, so food is the lbor intensive industry. c. In your digrm, show tht for ny chnge in the wgerentl rtio, the industry you chose in prt (b) is lwys the lborintensive industry. Show lso tht when the wgerentl rtio rises, both industries choose more cpitlintensive techniques. As long s the unit isoqunts intersect once, nd only once, given the ssumption tht they re constnt returns to scle nd permit smooth substitution between K nd L, for ny common w/r, k > k. In prticulr, s illustrted, n increse in the common w/r results in both industries becoming more Kintensive. This mkes sense, since K nd L cn be substituted, when the reltive price of L rises rtionl entrepreneurs will seek to substitute K for the now more expensive L. 4. hnging fctor use: onsider two countries with the technology shown in the two figures nd
5 ssume tht the (initil) equilibrium world price is p P /P.. Suppose the world price of clothing rises by 10% while the price of food remins constnt. Drw in the new production point on the production possibilities frontier. An increse in P with P held constnt will rise the reltive price of (i.e. p rises, sy pn > p). At the initil equilibrium, pn > MRT so the reltive price of exceeds its opportunity cost in production, leding to incresed output. The new equilibrium will occur t tngency between the PP nd ntionl income line with slope pn i.e. where MRT pn. b. Applying the StolperSmuelson theorem, wht should hppen to the wgerentl rtio? Drw in the tngencies long the unit isoqunts t the new wge rentl rtio. Wht hppens to the cpitllbor rtio in ech industry? The isoqunt digrm identifies s the Kintensive good. Thus, from the Stolper Smuelson theorem, we hve: r$ > p$. 1 > p$ 0 > w$. (8) Let ω : w nd denote the originl wgerentl rtio T nd the new one TN, the r reltionships in (8) imply tht T > TN. Reclling tht both sectors fced T nd now both fce TN, both sectors will optiml input choices will respond in the sme direction. At the initil input bundle TN < MRTS, tht is, t new fctor prices, the reltive cost of lbor is less thn the mrginl physicl product of lbor reltive to the mrginl physicl product of cpitl. Applying the rbitrge rgument, entrepreneurs in both sectors will substitute lbor for cpitl. Thus, the cpitl lbor rtio will fll in both sectors. c. Drw in the new lloction of lbor nd cpitl between the two industries in the production box. (Mke sure tht the slope long the unit isoqunts is the sme in both digrms.) Drw in the cpitllbor rtios t the new equilibrium point nd the contrct curve. In ddition to getting the slopes right between digrms, the essentil thing is to find qulittively correct new point on the efficient locus ( contrct curve ). Tht is, the new expnsion pths, reflecting more Lintensive production in both sectors, must intersect t point of tngency between higher vlued isoqunt nd lower vlued isoqunt. It is the slope of this tngency tht needs to be the sme s tht of the slopes in the unit isoqunt digrm from prt (b).
6 5. omprtive Advntge in the HeckscherOhlin Model: The mount of cpitl nd lbor required to produce one unit of food nd one unit of clothing re given by: K 1 K 2 L 3 2 Assume tht the techniques of production re invrint to the wgerentl rtio.. Suppose the home country is endowed with 160 units of lbor nd 100 units of cpitl. i. Drw the lbor nd cpitl constrints in output spce. (Put y on the horizontl xis nd y on the verticl xis.) Identify the slopes of the two constrints. Strt with the fullemployment conditions: L 160 y + y 2y + 3y L K 100 y + y 2y + 1y K K (9) Trnsform the equtions so tht they re in slopeintercept form: y y L L y y y 2 2 K K y y y 2 2 K K (10) These re esily plotted in y y spce. Note tht the the lbor constrint hs higher intercept (80 > 50) nd steeper slope (3/2 > ½). Recll from the text tht the reltive slopes of these two constrints re determined by the fctorintensity ssumption, i.e.: K L K K L < < (11) K ii. At full employment, how much clothing nd food does the home country produce? As in the previous questions, we cn use rmer s rule to solve for outputs. Thus, write the system from (9) in mtrix form to get:
7 L K K L K y y y y. (12) A T 4, A T L 140, A T K y A T L/A T 140/4 35, y A T /A T 120/4 30. b. Suppose the oreign country is endowed with 120 units of lbor nd 80 units of cpitl. i. Drw the lbor nd cpitl constrints in output spce. (Put y on the horizontl xis nd y on the verticl xis.) Identify the slopes of the two constrints. Agin, we strt with the fullemployment conditions: L 120 y + y 2y + 3y L K 80 y + y 2y + 1y K K (13) Trnsform the equtions so tht they re in slopeintercept form: y y L L y y y 4 2 K K 80 1 y y y 2 2 K K (14) These re esily plotted in y y spce. Note tht the the cpitl constrint hs higher intercept (40 > 30) nd steeper slope (2 > ½). Note, becuse the technicl coefficients re the sme in Home nd oreign, the slopes of the K nd L constrints must be the sme. However, becuse the endowments differ, the intercepts differ. ii. At full employment, how much clothing nd food does the home country produce? As in the previous questions, we cn use rmer s rule to solve for outputs. Thus, write the system from (13) in mtrix form to get:
8 L K K L K y y y y. (15) A T 4, A T L 120, A T K 80. y A T L /A T 120/4 30, y A T /A T 80/4 20. c. ompre the outputs of the two countries. Wht theorem would hve predicted this pttern of production? Under the ssumptions of this question, the Rybczynski theorem predicts tht the country tht is more bundntly endowed with L will produce reltively more of the good tht whose production is Lintensive. We hve lredy seen, in (11), tht clothing is L intensive. To find the reltively Lbundnt country we need to check: L K > L K 160 > 120 ( i.e. 16. > 15. ) Thus, the Rybczynski theorem predicts tht the Home country will produce reltively more clothing nd less food thn the oreign country. Since the prediction is borne out by the nlysis. 20 >, 30 d. Assume tht the two countries hve the sme tstes. i. Which country will export food? lothing? With identicl, homothetic preferences, t common prices both countries will consume food nd clothing in the sme proportions. Home is producing reltive more clothing thn oreign t common prices, the only wy for this to be true is if Home exports clothing nd oreign exports food. ii. ould you hve predicted this result on the bsis of fctor endowments nd the fctorintensities of the two industries? Wht theorem re you using to mke this prediction?
9 The HO theorem predicts tht country will hve comprtive dvntge, nd thus (by the lw of comprtive dvntge) export, the good whose production uses the country s bundnt fctor intensively. Becuse the Home country is reltively Lbundnt, nd clothing is reltively Lintensive, the HO theorem predicts tht Home will export clothing. 6. Reltive ctorabundnce nd Trde Ptterns: The production possibilities frontiers below re drwn under the ssumption tht the Home country is endowed with higher proportion of cpitl to lbor thn the oreign country. pitl is used reltively intensively in the production of food. Preferences in the two countries re identicl nd homothetic.. Suppose these countries begin to trde. Drw in the new production nd consumption points for ech country. Note tht, in these drwings, A (A) is the utrky equilibrium point, while P (P) nd () re the post trde production nd consumption points for the Home (oreign) country. b. Wht hppened to the pttern of production? Hve the countries become more or less specilized s result of trde? If both countries re lrge, so posttrde world reltive price settles strictly between the two utrky prices, both countries become more specilized in production of the commodity in which they hve comprtive dvntge. Thus, their ptterns of production become more dissimilr. Even if one country is lrge nd the other smll, becuse the smll country will become more specilized, the countries become more
10 dissimilr. c. Wht hppened to the pttern of consumption? Hve the bundles in the Home nd foreign countries become more or less similr s result of trde? With identicl, homothetic preferences, if trde equlizes commodity prices, the reltive mounts of food nd clothing in consumption must be identicl. In the digrm, this should be reflected in common slopes for the incomeconsumption lines in the two digrms. If countries hppened to hve the sme posttrde incomes, the bundles consumed would be identicl. d. Wht do you think hs hppened to the wgerentl rtio in ech country? Since trde equlizes commodity prices, the ctorprice equliztion theorem tells us tht the wgerentl rtios in the two countries hve become more similr. If both countries remin unspecilized in the new equilibrium, s they do in the bove digrm, the wgerentl rtios must be identicl (i.e. equl).
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