The total consumption of the homogeneous good, G, is residual and given
by:
G = θN - c1D - (αw + ad + ah) X tiDi - n (F + K).
i=1...n
In the symmetric case (considered in most of this paper), ti = t, i = 1...n
and the average individual consumption of homogeneous good g (n) wkere there
are n subcenters, is given by:
g (n) = θ - c1 - (aw + ad + αh¢ t - n (F + K)/N . (2)
Note that at least one center (center 1) is sustainable provided that:
g(1) = θ - c1 - (aw + ad + ah¢ t - (F + K)/N > 0 .
2.2 Market structures and taxes
The homogenous good is produced competitively in the center. The wage in
this industry is normalized to one. As the market is competitive and marginal
cost is equal to one, the price of the homogenous good is also one (in this case,
the transport cost is not incurred by the producers of the homogeneous good).
As a consequence, the value of time is one1 and the transport cost equals the
travel time, ti.
The price of the differentiated good i is denoted by pi and the wage offered
by firm i producing the differentiated good i is denoted by wi, i =1...n.The
government finances the public infrastructure input by imposing a head-tax T
and a fixed levy on the firms S : nK = NT + nS .
2.3 Household preferences
The household consumes the homogeneous good (at the city center) and one unit
of the differentiated good in one of the n subcenters. Each household supplies θ
units of labour in the city center for the production of the homogeneous good and
one unit of labour in one subcenter for the production of the differentiated good.
As labour supply is fixed, and as the quantity of the differentiated good is also
fixed, the consumption of the homogeneous good is the residual. We consider
that each household chooses a single place of employment (besides the city
center) and a single shopping destination (besides the city center). Therefore
the only choice of interest for the household is the choice of the employment
location (where the differentiated good is produced) and the choice of the type
of differentiated good to consume (where to shop).
The direct utility function of a household who supplies one unit of labour
to the differentiated industry i and buys one unit of the differentiated good of
type k is:
1 One unit of time allows the production of one unit of the homogeneous good, which has
a price equal to one, so the opportunity cost of one unit of time spent on the road is one.