INSTRUCTIONS : For credit you may support your answer with mathematical approach

along with verbal justification.

CASE: Within a small nation, there is a fixed population choosing between two locationsGwagwalada and Yanyan. The population consumes a numeraire good and housing. Within each area, there is free entry of firms producing the numeraire good using both land and labour. The productivity level in Gwagwalada is a function of Gwagwalada's population. The productivity level in Yanyan is fixed. There is a fixed amount of land in each area which can be used to produce either housing or the numeraire good. There is also free entry of firms producing housing, also using land and labour, and the productivity of firms in the housing sector is the same in each of the two areas. The numeraire good is traded freely on international markets and thus has a normalized market price of one. Housing, however, can only be purchased locally and its price is determined in equilibrium. Individual welfare is a function of consumption of the numeraire good, housing, and a public amenity. The amenity is produced by the public sector, where the level of the amenity is a function only of labour devoted to the public amenity divided by total population. This spending is financed by a lump sum tax on residents and potentially also a housing tax.

(1) Assume that government spending is exogenous and funded with area-specific lump sum taxes. Characterize the competitive equilibrium. Note that the above setup provides many, but not all, of the assumptions necessary to solve the model. Be sure to specify any other assumptions you need.

(2) Derive the conditions for a socially optimal spatial distribution of population. Describe a minimal set of policy interventions that will produce such an optimum.

(3) Assume that localities can set tax levels using lump sum taxes. Derive the tax rates if they are trying to maximize area population and compare with the setting where they are maximizing average welfare of their citizens.

(4) Assume localities are providing the local public good using local lump sum taxes to maximize population as in (3). Is it ever Pareto improving to tax housing? Expl