2x2x2 Heckscher-Ohlin-Samuelson (H-O-S) model with factor substitution

Transcription

1 2x2x2 Heckscher-Ohlin-amuelson (H-O- model with factor substitution The HAT ALGEBRA of the Heckscher-Ohlin model with factor substitution o far we were dealing with the easiest ossible version of the H-O- model with no factor substitution (i.e. with fixed factor requirements er unit of outut, indeendent of relative factor rices. Now we will allow for factor substitution in resonse to changes in relative factor rewards. The aealing feature of this simle general equilibrium model is its ability to show how easily some famous theorems can be derived from a simle model. We will start with a small oen economy that takes the relative rice ratio as given, and later discuss a large oen economy case. If the technology is given and factor endowments and commodity rices are treated as arameters, the model serves to determine 8 unknowns: The level of commodity oututs (, The factor allocation to each industry (L,L,, actor rices (w,r We need 8 equations to be able to solve the model analytically. These equations can be given by: the roduction functions (2, the requirement that each factor receives the value of its marginal roduct (4, the requirements that each factor is fully emloyed (2.

2 The requirement that both factors are fully emloyed is given by equations: L L L a L a above relationshis emhasize the dual relationshis between factor endowments and goods oututs. Unit costs of roduction in each industry are given by the erfect cometition conditions: a L a c c a a r a r a L L w w The above relationshis emhasize the dual relationshis between factor rices and goods rices. Is it enough to solve the model? No, not in the general case when unit factor requirements change in resonse to relative factor rice changes! Therefore, we must sulements the above equations by four additional relationshis determining the inut coefficients. These are rovided by the requirement that in a cometitive equilibrium each a ij deends solely on the ratio of factor rices (w/r. Let s use the cost minimization condition of a tyical entrereneur (we saw in the factor secific model. In the manufacturing sector the unit roduction costs are given by: C = a r + a L w 2

3 The entrereneur treats factor rices as fixed and varies a s so as to set the derivative of costs equal to zero: dc = 0 = da r + da L w Now, exress it in terms of the rates of changes (dividing by sides by c : C dc C % change rateofgrowth ar da alw dal C a C a L L L L L 0 imilarly, in the food roducing sector we have: C L L 0 Alterations in factor roortions must balance out such that the Θ weighted average of the changes in inut coefficients in each industry is zero. This imlies that the relationshi between changes in factor rices and changes in goods rices is identical in the variable and fixed coefficient cases (Wong-Viner theorem. To see it differentiate totally the erfect cometition conditions and then exress them in the rates of changes. 3

4 In the manufacturing sector we have: d d dc da a imilarly, in the food sector we have: da a r a r a r r dr da dr r L da a L L L w a L a L L L dw w a L w dw w r L L L The relationshis between changes in goods rices and changes in factor rices can be written in the matrix form: L L r ( ( L L L L Interretation: factor rices deend only on commodity rices this is our factor rice equalization theorem! Our equations rove the factor rice equalization theorem (between countries, even though it to show only one country. However, we can easily reinterret each change as a rate of change between countries rather than as a change over time. Two countries are the same in some key resects. They have the same rice ratio because they trade freely without transort costs. 4

5 Unfortunately, a similar kind of argument does not aly to the case of the factor market clearing conditions (that do not simlify so easily. Let us exress our factor market clearing conditions using the rates of change. irst, take a total differential of the full emloyment condition for labor to obtain: da L a L d da L a L d dl Then divide both sides by L to get: dal al al d dal al al d dl al L L al L L L L L L L L L L L L Therefore, we have: L L L [ L L L L ] In the same way we obtain: [ ] 5

6 The relationshis between changes in factor sulies and changes in outut levels can be written in the matrix form: L L L ( ( L L L L The term ( L L L L shows the ercentage change in the total quantity of labor required by the economy as a result of changing factor roortions in each industry at unchanged oututs (constant λs. Crucial feature: If factor rices change, factor roortions alter in the same direction in both industries. The extent of this change deends on the elasticities of substitution between factors in each industry (assume constant elasticity of substitution between factors. The elasticity of substitution between labor and caital in the manufacturing sector is defined as: d( / L /( / L d( w/ r /( w/ r d( a / al /( a / a d( w/ r /( w/ r L r L This elasticity tells us how the caital-labor ratio will change if relative wage (wage-rental ratio changes by %. 6

7 imilarly, for the food sector we can write the elasticity of substitution between labor and caital as: d( / L /( / L d( w/ r /( w/ r d( a / al /( a / a d( w/ r /( w/ r L r L Now we need to find changes in unit factor requirements as functions of changes in factor rices. To do so let us use the above definitions of the elasticity of substitution and combine them with cost minimization conditions: C L L 0 L L C L L 0 L L r L ( r L ( L r L ( r L ( L Hence, we get: 7

8 L L L L ( r ( r ( r ( r ubstituting the exressions for the changes in unit factor requirements in resonse to changes in factor rices into the set of equilibrium conditions we obtain: L L L L( r ( r where L L L is the aggregate ercentage saving in labor inuts at unchanged oututs associated with a % increase in the relative wage (the saving resulting from the adjustment to less laborintensive techniques in both industries as relative wages rise, and similarly. JONE (965 AGNIICATION EECT If commodity rices are unchanged factor rices are constant and the system of equations tells us that changes in commodity oututs are related to changes in factor endowments. If both endowments exand at the same rate both commodity oututs exand at identical rates. L L L 8

9 This can be demonstrated as follows: det L L ( L L L ( L 0 (negative when is caital intensive L L L L L L L L ( L ( L L L Now we can easily notice that when L then. However, if both factor endowments grow at different rates, the good intensive in the use of the fastest growing factor exands at a greater rate than either factor, and the other commodity grows (if at all at a slower rate than either factor. or examle, suose that labor exands more raidly than caital. With caital intensive comared to we have then: L 9

10 This is called the AGNIICATION EECT of factor endowments on commodity oututs at unchanged commodity rices. or simlicity consider a secial case when the endowment of only one factor increases, say labor L 0. det L L ( L L L 0 ( L 0 (negative when is caital intensive L 0 L L Lsince - L L L ( L L L L 0 L (you can notice that the numerator is bigger than the denominator since by assumtion >λ L 0

11 Hence, we observe the following magnification effect: L This is our Rybczyński theorem which can be restated as follows: At the unchanged commodity rices an exansion in one factor results in an absolute decline in the commodity intensive in the use of the other factor. imilarly, the magnification effect is also the feature of the link between commodity rices and factor rices. In the absence of technological change or taxes/subsidies, if the rice of caital intensive good grows more raidly than the rice of the labor intensive good, then the reward to factor used intensively in the roduction of manufactures (caital grows more than the rice of manufactures and we have: r Intuition: The source of the magnification effect is easy to detect. ince the relative change in the rice of either commodity is a ositive weighted average of factor rice changes it must be bounded by these changes. or simlicity consider a secial case when the rice of only one good increases, say 0. In this case the increase in the rice of raises the return to the factor used intensively in (caital by an even greater amount (and lower the return to the other factor.

12 Now we have: L L L L r L L L Hence, we observe the following magnification effect: r 0 0 r w This is our toler-amuelson theorem which can be restated as follows: 0 An increase in the rice of a caital intensive good raises the return to the factor used intensively in (caital by an even greater amount and lowers the rice of the other factor. 0 0 inally, we are ready to study the large economy case. 2

13 3 Endogenous demand To close the model we assume that consumer trade atterns are homothetic and ignore any differences between the workers and caitalists. Thus, the ratio of quantities of goods consumed deends only on the relative commodity rice ratio: f Let us exress this relationshi in terms of the rates of change using the elasticity of substitution between two commodities on the demand side σ D : D D d d / /( / / / / since ( Previously we considered the effect of a change in factor endowments at unchanged commodity rices. With the model closed by the demand relationshi commodity rices will have to adjust so as to clear the commodity markets. Recall that in the general case when commodity rices change also factor rices change:

14 L L L L( r ( r Hence, ( ( L ( L ( r L L We can notice that on the suly side the change in the ratio of oututs roduced deends on the change in factor endowments and the change in factor rices. Let us concentrate for the moment on the change in the relative factor rices which can be obtained from: L L r Hence, r L L Now we can substitute the relationshi between the changes in the ratio of factor rices and the changes in the ration of goods rices into our relationshi between the change in the ratio of outut roduced and the change in the ratio of factor rices: 4

15 ( L where L L L L L ( L L L ( L (the elasticity of substitution between the goods on the suly side along the roduct transformation curve Equilibrium In equilibrium, the change in the ratio of outut roduced has to be equal to the change in the ratio of outut consumed. This allows us to determine the change in the commodity rice ratio as we can notice that this change is given by the mutual interaction of demand and suly. L L ( ( Hence L L L ( L ( D D 5

16 6 Having determined the change in the commodity rice ratio we can determine the change in the outut ratio as a function of the change in the factor endowment ratio: ( ( ( ( ( ( ( L L L L D L D D L L L When commodity rices adjust to the initial changes in outut brought about by the change in factor endowments, the comosition of oututs may in the end not change by as much as the factor endowments. This deends whether the elasticity exression σ D /(σ +σ D is smaller than the factor-intensity exression (λ L - λ. Large values of damen the sread of outut, small values of work in the similar way.

17 These effects can be summarized in the table below: D D D D D D ( ( ( L L L Less than : change : change (comlete damening No magnification effect ore than : change agnification effect exists although is damened D When D we observe the full magnification effect. Conclusion: The only art of the Rybczyński theorem which is challenged by the introduction of the demand side is the one that concerns the magnification effect. 7