Unequal cities: Self-selection, matching, and the distribution of income

We develop a model of a city populated by heterogeneous agents. Agents self-select into entrepreneurship, and entrepreneurs set up firms which hire workers. We characterize the equilibrium matching between firms and workers, as well as the within-city assignment of agents to locations. We then explore the implications of city size and the characteristics of the underlying skill distribution for selection into entrepreneurship, rent gradients, and city-wide inequality in disposable incomes. Agents self-select into occupations based on their earnings, we assume that highly productive agents have a comparative advantage in entrepreneurship, so that there exists some endogenously determined unique cutoff that separates workers from entrepreneurs. Workers are hired by entrepreneurs and are paid a match-specific wage, whereas entrepreneurs are the residual claimants to their firms' profits. All agents live in a linear city that stretches out on the finite interval. Both workers and entrepreneurs commute to the central business district (CBD) for work. Commuting entails costs, which we model parimoniously using an `iceberg' specification. In our model, agents differ by income. A commuting costs are paid as a fraction of income, it follows that richer agents will want to locate closer to the CBD to minimize their costs. We thus have to find the spatial distribution of agents in the city- the mapping of agent's talent to city locations, such that every agent picks his preferred location. We look at the case with a fixed lot size assumption: maximizing utility is equivalent to maximizing disposable income - income net of land rent and commuting costs. First, we will derive the comparative statics of the equilibrium variables with respect to exogenous parameters of the model like total population, commuting costs, and various moments of the talent distribution of the population. We expect that reducing commuting costs will lead to an increase in the density in a city, and that it will increase the share of entrepreneurs in the city. A larger share of entrepreneurs leads to tougher selection, which should magnify income inequality in the city. We have no a'priori intuition for the direction of change in "real" income inequality. Furthermore, a distribution of talent that is more skewed towards highly talented workers should increase the steepness of the rent gradient towards the city center. The reason is that more productive agents live closer to the center and compete for land there, and that a larger mass of highly talented agents will increase competition for land towards the center. Depending on how fast land prices go up for the rich compared to land prices for the poor, "real" income inequality may a'priori rise or fall as the distribution of talent gets more unequal within the city.