A dynamic approach to the tendency of industries to cluster

Appendix A. Derivation of the cost function for manufacturing

The production function for the manufacturing firm is given by:
(1)

Z is a sub-production function over the producer service sector’s composite output of
and is of the CES-type:

n₁                    n 2

The Lagrangean associated with the problem of minimizing ∑ p_{i 1} χ_i11 ÷ ∑ p_{i 2}tχ_{i 21}

i =1                     i =1

subject to the constraint that the production level is constant at Z is:

The 1^st order conditions are:

σ

a/       ( ʌ 1-ɪ   n 2  1-т A^σ-1

(6) ɪ = Z -Σχ_m^σ ÷Σχ_m^σ   = 0

^cλ         к i=1          i=1       )

Dividing (4) with ðf/ðχ_{j 11} and (5) respectively, leads (after some manipulation) to

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