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The winners and losers of economic competition are not evenly distributed across geographic space. This holds, almost as a rule, regardless of the scope in which the geographic space is defined. Starting with the scope of the national economy, at any point in time, competition between urban areas bestows economic growth on the winners while leaving economic stagnation or even decay to the losers. Viewing a single urban area, the same can be said of the municipalities that make up the urban area: the most competitive receive the bulk of the good fortune, while the less competitive are left behind. This multitude of outcomes within a scope presents interesting incentives and welfare calculations: a municipalities fortune is tied not only to its competitive position among the constituent groups of the urban area, but also to the competitive position of the urban area within the national market.
Any intervention considered within an urban area must then account for incentives on both levels of competition. Consider an often debated policy instrument, municipal mergers. Advocates of mergers claim that the merging parties will gain access to greater economies of scale, but the effect of changing the competitive position of the urban area is often left unexamined. This could be seen as a cause for the lack of normative implications from that literature.
Incorporating the effect on the urban area into policy prescriptions requires estimates of the future competitive environment and how the municipalities incorporate that information into their decisions. In this paper, we employ a two stage model. The first stage captures firms locational decision in terms of urban areas. Due to agglomeration forces, a firms optimal urban area choice is determined through the joint process of all firms making locational choices. As locational decisions represent a large fixed investment, implicit in each firms optimization problem is a prediction regarding future states of urban areas. Further, these beliefs are self-reinforcing as expectations regarding optimal locations elicit behavior that confirms this belief. Examining these locational choices allows us to infer what firms expect the future competitive environment of the urban area market to be. In the second stage, we switch from firms to municipalities as the agents in our model. We allow municipalities to choose an expenditure mix to compete for the firms and residents residing in the urban area, but with the restriction that the outcome of this competition must result in the quantities that elicited the behavior found in the first stage. In our modeling, we assume that a firms urban areas choice is made based upon quantities that arrise from the municipal level competition. This feature is allows us to incorporate the incentives of the urban area into the municipal decision process. Further it is attractive computationally, as a informational restriction on firms, and motivates the equilibrium approaches we use in the model.
To solve for firms expectations in the first stage, we employ the Oblivious Equilibrium (OE) approach from Weintraub, Benkard, & Van Roy (2008) and take firm heterogeneity as their locational choice. This approach ties closely with the economic problem at hand and handles the large state space inherent with modeling all firms. Both advantages are most easily seen in comparison to an alternative solution concept, Markov Perfect Equilibrium (MPE). Within the context of this problem, an MPE would maintain that firms strategies are functions of all other firms strategies in the national market. In comparison, the OE approach requires that firms only have knowledge of the the number of firms in their own urban area and the stationary equilibrium number of firms in the other urban areas. This is directly compatible with our current context, as the stationary equilibrium of the urban area market can be seen as a both a motivation for and a consequence of firms locational decisions. The computational advantages stem from a identical argument.
Further, the more recent work of Weintraub, Benkard, & Van Roy (2010) and Adlakha, Johari & Weintraub (2013) point to the strong connection between these two equilibrium concepts. In general, the OE solution will converge to the MPE solution if the number of firms grows with the market size and if the industry state has “light tails”, meaning that no dominant firms emerge in equilibrium. In our model, the number of firms is exactly the market size and all firms are assumed symmetric such that no firm can ever become dominant.
In the second stage, we note that these conditions are much less likely to be met. This is because the agglomeration effects within an urban area tend to create dominant municipalities, which directly contradicts the conditions for an OE. To handle this issue, we turn to a refinement of OE, partial oblivious equilibrium Benkard, Jeziorski & Weintraub (2015). Partial oblivous equilbirium (POE) allows for a strategically important set of dominant municipalities. This choice of equilibrium concept has two main advantages. First, it is compatible with the standard Urban Economic theory of mono/polycentrism where a few dominant employment subcenters dictate the equilbirum outcomes for the entire urban area. Second, it provides an ideal avenue in which to preform our counterfactual municipal merger. We can frame the question of municipal mergers as the effect of merging municipalities such they combine to yield a municipality that enters into the “dominant set” of municipalities.
In this section, we outline the multi-stage competition structure. As discussed above, the upper level competition between urban areas is entirely conceptual. The urban areas do not directly make decisions, but play strategies that are endogenously determined by municipal competition.
The modeling follows Weintraub, Benkard, & Van Roy (2008) who in turn follow Ericson and Pakes (1995). We assume discrete time periods indexed by \(t \in 0,1,\dots\). Each urban area is assigned a unique integer index \(i\) and is described by its only state variable, integer valued quality \(x_{it}\). The industry state, \(s_t\), is a vector containing a count of all urban areas at a given quality level. For a total number of firms in the United States \(m\), in each period, urban area \(i\) earn profits given by \(\pi_{m}(x_{it},s_{-it})\), where \(s_{-it}\) denotes the number of urban areas that share its quality level.
Let \(\iota_{it}\in \mathcal I \subset \mathbb R_+\) be the investment level of urban area \(i\) in period \(t\). Note here that the investment level is only conceptually choosen at the urban area level, and is best described as the endogenous result of the aggregate investment decisions of the municipalities. Given an investment level, the urban areas quality at time \(t+1\) evolves according to
\[ x_{it+1} = x_{it} + w(x_{it},\iota_{it}, \zeta_{i,t+1}) \] Where \(w\) captures the impact of investment on urban area quality and \(\zeta_{it+1}\) is an idiosyncratic shock reflecting the uncertainty in investment. Investment costs are governed by a function \(d:\mathcal I \to \mathbb R_+\).
Each urban area is modeled to maximize expected net present value with a constant discount factor \(\beta\). In each period, the following events occur in order: investment decisions, competition for firms and residents, then quality transition.
Equilibrium in this model
The modeling of the subcenter competition follows closely to that of the urban area competition. The subcenter model takes the same discrete time as the urban area model. Each subcenter in urban area \(i\) is index by \(j\). Each subcenter has a single state variable \(x_{ijt}\) that captures its quality in period \(t\). The industry state is then the count of all subcenters at this In each period, each subcenter earns \(\pi(x_{ijt},s_{i-jt})\)