ADONGO'S-RYBCZYNSKY'S THEOREM

ADONGO’S-RYBCZYNSKI’S THEOREM

The relative change of the industries x₀, x₁,x₂,……,x_j-1,…,x_r-1 depend the effect on outputs of change in the lands H₁,H₂,…., H₀,H_j-1,…,H_r and labors, L₀,L₁,L₂,….,L_j-1,….,L_{r -1} endowments.

This theorem states that if the endowments of some resources increase, the industries use the resources most intensively will relatively increase their outputs while the others industries will relatively decrease their outputs. The relative factor intensity of each industry is measured by the ratio of factors use in each industry in comparison to others industries 0,1,2,...., and 1,2,3,..., are higher are relatively initial-orders labors-lands-intensively and final-orders labors-lands-intensively respectively.

Algebraically, the Adongo’s-Rybczynski’s theorem is derived from the simple solution of multi-combinational system equations. In the example above, the solution of the model are composed by the lands and labors constraints in their relative order of sequences.

(a_0*a_1*….a_j-1*…a_r-1)_1*(x_0*x_1*…_*X_j-1*…._*x_r-1)+(a_1*a_2*…a_j*…a_r)_1*(x_1*x2_*….x_j*…._*x_r)=H ...1

(a_0*a_1*….a_j-1*…a_r-1)_2*(x_0*x_1*…_*X_j-1*….x_r-1)+ (a_1*a_2*…_*a_j*…a_r)₂(x_1*x_2*….x_j*….x_r)=H.....2           

Applying Adongo’s rule of determinants, we have

X₀=x_r[detA₁/detA₂]

Where;

H=H_1*H_2*…_*H_j*…._*H_r+H_0*H_1*…._*H_j-1*….H_r-1

L=L_1*L_2*…_*L_j*…._*L_r+L_0*L_1*…_*L_j-1*…._*L_r-1

DetA₁=H(a_1*a_2*..a_j*…a_r)₂-L(a_1*a_2*…_*a_j*…_*a_r)₁

DetA₂=L(a_0*a_1*…._*a_j-1*…._*a_r-1)₁-H(a_0*a_1*…_*a_j-1*…a_r-1)₂

NOTE: Further modification of this theorem will be considered later.



REFERENCE
Rybczynski, T.M(1955). "Factor Endownent and Relative Commodity Prices"