A dynamic approach to the tendency of industries to cluster

From (7) it follows that,

This function says that the cost of the manufacturing sector is a function of the
location of the producer service sector. In particular, the relative cost of manufacturing
decreasing in the relative location of producer services, l^u. Hence, the larger the producer
services sector, the lower are the costs of the manufacturing sector.

Next, since consumer expenditure is the only source of demand for the manufacturing
industry’s output, we have that η ^d = md/md , which is exogenous. The expenditure on
the output of the producer service sector comes from the manufacturing industry alone.
This implies that the producer service sector’s share of the manufacturing sector’s total
expenditure (total cost), γ, determines the demand for the output of the producer service
sector.

d d ( d d )

(₁₁)     m^u = V nd c^d (z^d _{+ z}^d ₊ f )   ^ n^u = n² p² > ^{22 +} z ²¹< ≡ l^d

^{(11)     m}1   / ⁿ1 ^c1 zZ 11 ^{+ z} 12 ⁺ И   ^ l∣          ( d ) ≡ ^l

ⁿ1 ^p 1 ∖^z 11 ^{+ z} 12/

Using Equations (8), (10), (11) as well as keeping in mind that the relative cost and
the relative prices for the producer service sector is ω , l^u and ld can be expressed as in
Equation (12) and (13).

(12a)      l^u = f^u (p^u, η^u, t ) = f^u (ω, l^d, t )

(12b)      l^d = f^d (p^d, η^d, t )= f^d (g ( ω, l^u, t )), η^d, t )

These equations can be solved for the endogenous variables, l^u and l^d. Equation (12a)
shows the dependence of the producer service sector’s location on the demand from the
manufacturing industry and gives l^u as an increasing function of l^d. Equation (12b)
captures the dependence of the manufacturing sector’s location on supply from the
producer service sector and gives l^d as an increasing function of l^u. Hence, the model
shows that the locations of the sectors are simultaneously determined.

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