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APPENDIX 2

Derive production function from cost function

The system is given:

C = R₁ X₁ + R₂X₂ (1)
f₁/f₂ = R₁/R₂ (2)
pf₁ = R₁. (3)

where (1) is the cost function, (2) is the condition for cost minimisation (output maximisation) and (3) is the condition for profit maximisation.
The (1) can be written as follows in the plane X₁X₂:

X₁ = C/R₁ - R₂X₂/R₁
X₁ = C/R₁ - f₂X₂/f₁

f₁X₁ + f₂X₂ = f₁C/R₁ (4)

Using the Euler’s Theorem (f₁X₁ + f₂X₂ = Q) and (3), the (4) becomes:

C = pQ (5)

where “p” is the price of service. The relationship (5) is the long-run condition in a competitive market and states that revenue equal cost, so that profit equals zero. Recalling the expression for cost function from Appendix 1 (case: a + b = 1):

Q = (1/p)(R₁/a)^a(R₂/b)^b(X₁)^a(X₂)^b

If we assume:

A = (1/p)(R₁/a)^a(R₂/b)^b

We obtain:

Q = A(X₁)^a(X₂)^b

Bahia/casestudy 26/2/2002