Regional Intergration and Migration: An Economic Geography Model with Hetergenous Labour Force

The second step of the consumer’s problem is to choose the optimal allocation of income
between A and M so that the utility is maximised

Maximising U = M^μA^{1 μ} subject to. PM + PaA=Y , gives us the uncompensated demand
for A and for each variety, m_j :

A = (1 - μ )Y / Pa

From the consumer’s utility maximisation problem we can also express the indirect utility
function, substituting (7) and (8) in (1) yields:

U = μ^μ (1 - μ )^1-μY
PμP (¹^^μ)

the term P^μPA°^^μ) can be interpreted as the regional cost-of-living index in the economy.
What is the welfare effect of an increase in the number of variety? Assuming that all varieties
are available at the same price p_i = p_j, ∀j ∈ [1,..., n ], we can rewrite the manufacturing goods
price index as:

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