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APPENDIX 3

Derive input variables as a function of R₁, R₂, Q

Given the system:

Q = A(X₁)^a(X₂)^b
C = R₁X₁ + R₂X₂
X₂/X₁ = bR₁/aR₂

1. Express variable X₂ as a function of a, b, R₁, R₂, A, Q:

X₁ = (Q/A)^1/a(1/X₂)^b/a
X₂ = bX₁R₁/(aR₂) = _(bR₁) (Q/A)^1/a(1/(X₂)^b/a_/(aR₂)
(X₂)^(a+b)/a = bR₁(Q/A)^1/a/(aR₂)

X₂ = _(bR₁)^a/(a+b)(Q/A)^1/(a+b)_/(aR₂)^a/(a+b) (1)

2. Express variable X₁ as a function of a, b, R₁, R₂, A, Q:

X₂ = ((Q/A)^1/b(1/X₁)^a/b
X₁ = aX₂R₂/(bR₁) = _(aR₂) (Q/A)^1/b(1/(X₁)^a/b_/(bR₁)
(X₁)^(a+b)/b = aR₂(Q/A)^1/b/(bR₁)

X₁ = _(aR₂)^b/(a+b)(Q/A)^1/(a+b)_/(bR₁)^b/(a+b) (2)

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As a check, the relevant cost function coincides with the function in Appendix 1:

C = R₁X₁ + R₂X₂ =
= R₁(aR₂)^b/(a+b)(Q/A)^1/(a+b))/(bR₁)^b/(a+b) + R₂(bR₁)^a/(a+b)(Q/A)^1/(a+b)/(aR₂)^a/(a+b) =
= _(R₁)^a/(a+b)(R₂)^b/(a+b)(a/b)^b/(a+b) + (R₁)^a/(a+b)(R₂)^b/(a+b)(b/a)^a/(a+b)_(Q/A)^1/(a+b)

C = (a + b)_(R₁/a)^a/(a+b)(R₂/b)^b/(a+b)_(Q/A)^1/(a+b)

Bahia/casestudy 26/2/2002