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APPENDIX 1

Derive cost equation as a function of R₁, R₂, Q

Consider the system:

C = R₁ X₁ + R₂X₂ (1)
X₁/ X₂ = k R₂/ R₁ (2)

where (1) is the cost function and (2) is the condition for cost minimisation (ratio of variables proportionate to the inverse ratio of their own costs). As the factor of proportionality “k” may take values lower, equal or greater than one it can be represented by the ratio a/b. Its value is lower, equal or greater than one when “a” is lower, equal or greater than “b”. Using (2), write (1) as a function of X₁ or X₂ alone:

C = (a + b)R₁X₁/a (3)
C = (a + b)R₂X₂/b (4)

Raising (3) and (4) at the power “a” and “b” respectively, we get:

C^a = (a + b)^a(R₁ X₁)^a/a^a (5)
C^b = (a + b)^b(R₂X₂)^b/b^b (6)

Multiplying (5) and (6):

C^a+b = (a+b)^a+b(R₁/a)^a(R₂/b)^b(X₁)^a(X₂)^b

that, for a + b = 1 (elasticity), becomes:

C = (R₁/a)^a(R₂/b)^b(X₁)^a(X₂)^b

Bahia/casestudy 26/2/2002