Review of the H-O model. Problem 1. Assume that the production functions in the standard H-O model are the following:

Transcription

1 Revie of the H-O model Poblem 1 Assume that the poduction functions in the standad H-O model ae the folloing: f 1 L 1 1 ) L 1/ 1 1/ 1 f L ) L 1/3 /3 In addition e assume that the consume pefeences ae given by the folloing utility function: UC 1 C ) C 1 C 1) Deive the unit-cost functions fo each industy c i ) and labo and capital used to poduce one unit of each poduct: a il a i Which industy is labo and hich one is capital intensive? By definition e have c i ) min L i i {L i + i f i i L i ) 1} In ou case the poblem looks as follos: { } c 1 ) min L L 1/ 1 1/ 1 1 L 1 1 { } c ) min L + L 1/3 /3 1 L Let us solve the fist minimization poblem The Lagange function fo the poblem is given by L λl 1/ 1 1/ 1 1) The fist ode conditions ith espect to L 1 1 λ imply 1 1/ 1 λl 1/ λl1/ L 1/ 1 1/ 1 1 By solving this system of equations e deive that 1 1/ 1 / 1 /L 1 L

2 hich means that L 1 / 1 / That is a 1L / a 1 / c 1 ) L / 1/ At least to things should be noticed Fist the capital to labo atio is popotional to / If / goes up labo becomes elatively moe expensive) the industy uses elatively moe capital 1 /L 1 goes up The same is tue fo the othe industy see belo) Second the Cobb-Douglas poduction functions imply the Cobb-Douglas unit-cost functions ith the same eights Solving the minimization poblem fo the second industy e deive the algeba is exactly the same so I omit some intemediate deivations) / 05 /L ) /3 a L a ) 1/3 c ) L + ) /3 + ) 1/3 1/3 /3 /3 + 1/3 ) 3 /3 1/3 /3 As can be seen the labo to capital atio in industy 1 is a 1L /a 1 / in industy a L /a 05/ This means that fo any pai ) a 1L /a 1 > a L /a That is industy 1 is labo intensive hile industy is capital intensive ) Conside the equilibium in the closed economy ithout tade) Wite don the conditions that detemine the equilibium Find the elative age / and the elative pice in the equilibium The equilibium in the closed economy is chaacteized by thee objects: the elative pice p p 1 /p the facto pices ) and the outputs of each industy y 1 y ) The conditions that allo us solving the model can be itten as follos Fist in the closed economy both poducts ae poduced meaning that p 1 c 1 ) p c ) Second e have the maket cleaing conditions: a 1 y 1 + a y L a 1L y 1 + a L y

3 Finally e need to take into account that in the closed economy demand is equal to supply The utility function implies that demand fo poduct i is given by C i L + p i hee L + stands fo the total income in the economy homothetic pefeences imply that thee is a epesentative consume) Thus e deive one moe equilibium equation: C 1 C y 1 y p p 1 The above equations ae suffi cient to find the equilibium in the model The algeba behind the equilibium conditions is elatively staightfoad but cumbesome To find the elative age one can do the folloing: Substituting fo y 1 /y e deive a 1 y 1 + a y L a 1L y 1 + a L y L a 1y 1 + a y a 1L y 1 + a L y L L a p 1 p 1 + a p a 1L p 1 + a L a 1 y 1 /y + a a 1L y 1 /y + a L Then e need to substitute fo p /p 1 and a i a il In this case e deive that L / c ) c 1 ) + ) 1/3 / c ) c 1 ) + ) /3 / 3 /3 1/3 /3 + 1/ 1/ / 3 /3 1/3 /3 + 1/ 1/ 3 /3 ) 1/3 ) + 1/3 3 /3 ) /3 + ) /3 const hee const is some "cazy" constant e do not cae about As can be seen the outcome is quite simple: the elative age is popotional to the elative endoment of capital The highe is /L meaning that thee ae elatively less labo in the economy) the highe is the elative age As soon as e kno the elative age e can find the elative pice: p c ) p 1 c 1 ) 3 /3 1/3 /3 1/ 1/ 3 /3 ) ) 1/6 L 1/6 const1 hee const1 is again some constant e do not cae about As can be seen a ise in the elative endoment of labo leads to the highe elative pice of the capital intensive poduct The intuition is staightfoad A ise in the elative endoment of labo inceases the elative 3 ) 1/3 ) /3

4 supply of the labo intensive poduct the Rybczynski theoem) hich in tun leads to a loe elative pice of this poduct p 1 /p in the equilibium 3) Assume that no thee ae to counties that tade Home is endoed ith L units of labo and units of capital Foeign is endoed ith L and espectively We assume that L L hile > We also assume that the facto endoments in the counties ae such that FPE holds Find the elative age / in the fee tade equilibium The key point hee is to emembe that the fee tade equilibium is exactly the same as the equilibium in the integated old see the lectue) As a esult e need to find the elative age in the old integated equilibium hee factos ae pefectly mobile beteen counties) This equilibium is in tun the equilibium in the closed economy hee the endoments of labo and capital ae L + L and + As a esult e have see the deivations above) + L + L const 4) Compae the home autaky elative pice of the poducts ith that in the fee tade equilibium The idea is the same as befoe The elative pice in the closed economy is given by ) p L 1/6 const1 p 1 In the fee tade equilibium the elative pice is ) p L + L 1/6 const1 p 1 + ) old ) autaky So e need to compae L L+L ith + Since L L and > p p 1 < p p 1 This is exactly hat the H-O theoem states Home is a labo intensive county As a esult afte opening up to tade the elative pice of the labo intensive poduct poduct 1 goes up) o altenatively the elative pice of the capital intensive poduct goes don 5) Who gains fom tade in the foeign county? Foeign is a capital intensive county As a esult opening up to tade benefits the capital holdes hile okes lose This is the Stolpe-Samuelson theoem 4

5 Poblem The poduction function of computes in Neveland is f C L C C ) L 1/3 C /3 C The poduction function of steel is f S L S S ) L 1/ S 1/ S Assume that Neveland is a small open economy and the elative pice of steel / 1 The labo endoment in the county is 1000 same fo capital: L ) Detemine the amounts of computes and steel poduced in Neveland To find the poduction of both poducts e use the facto maket cleaing conditions: a C y C + a S y S L a CL y C + a SL y S Thus one needs to compute a i and a il hee i {C S} Fom the pevious poblem one can see that these coeffi cients ae detemined by the elative age / Thus e need to find the elative age assuming that Neveland is a small open economy that is the poduct pices ae given) To do so e can do the folloing see also the solution of the pevious poblem) c S ) c C ) ) 1/6 3 /3 Thus ) 6 3 /3 071 No e can find the labo and capital coeffi cients in both industies again see the solution of the pevious poblem): a SL ) 6 ) 3 / 3 /3 3 / a S ) /3 6 ) 3 3 /3 / 084 Same fo the compute industy: a CL a C ) ) ) /3 6 / /3 079 ) ) 6 1/3 3 /3 1/3 113 As can be seen the compute industy is capital intensive hile the steel industy is labo intensive 5

6 To find the output in each industy e need to solve: y C + 084y S It is easy to check that y C 515 and y S 500 L y C + 119y S ) Compute the eal age in tems of steel and in tems of computes The eal age in tems of steel and computes is given by Anothe ay : c S ) c C ) 1/ 1/ 05 ) ) 6 1/ 05 3 /3 3 /3 1/3 /3 1 ) /3 1 3 /3 3 / / 04 ) 6 3 /3 /3 04 3) No suppose the elative pice of steel ises to 105 Compute the eal age in tems of steel and in tems of computes Compae the esults in ) and 3) and elate them to the Stolpe-Samuelson theoem Let us edo the execise in 1) and ) ) c S ) c C ) ) 6 6 1/6 3 /3 3 / ) /3 095 In this case c S ) ) 1/ 1/ 1/ ) 1/ As can be seen the eal age goes up This is exactly in line ith the S-S theoem: a ise in the elative pice of the labo intensive poduct steel) leads to highe eal etuns to labo highe eal age) 6