S.1. INTRODUCTION THIS SUPPLEMENT CONTAINS DETAILED DERIVATIONS and supplementary material for the paper. A separate Technical Data Appendix, located together with the replication data and code, collects together additional empirical results and robustness tests.Section S.2 of this supplement presents a more detailed analysis of the theoretical model. We report the technical derivations of the expressions reported in the paper. We also establish a number of results about the properties of the general equilibrium with exogenous location characteristics and endogenous agglomeration forces. Section S.3 calibrates the model for known parameter values and shows that there is a one-to-one mapping from these known parameters and the observed data to unobserved location characteristics. Therefore, these unobserved location characteristics correspond to structural residuals that are functions of the parameters and the observed data.Section S.4 turns to the structural estimation of the model, where both the parameters and unobserved location characteristics are unknown and to be estimated. We derive the moment conditions used in the estimation and review the Generalized Method of Moments (GMM) estimator as applied to our setting. We discuss the computational algorithms used to estimate the model and report the results of a grid search over the parameter space that we use to characterize the properties of the GMM objective function. Section S.5 uses the model to undertake counterfactuals for the effects of division and reunification. Section S.6 contains further information about the data sources and definitions. S.2. THEORETICAL MODELIn this section, we develop in further detail the theoretical model outlined in the paper. We present the complete technical derivations for all the expressions and results reported in the paper. In the interests of clarity and to ensure that this section of the supplement is self-contained, we reproduce some material from the paper, but also include the intermediate steps for the derivation of expressions.We consider a city embedded within a wider economy. The city consists of a set of discrete locations or blocks, which are indexed by i = 1 S. The city is

This paper synthesises the state of knowledge on the economic effects of density. We consider 15 outcome categories and 347 estimates of density elasticities from 180 studies. More than 100 of these estimates have not been previously published and have been provided by authors on request or inferred from published results in auxiliary analyses. We contribute own estimates of density elasticities of 16 distinct outcome variables that belong to categories where the evidence base is thin, inconsistent or non-existent. Along with a critical discussion of the quality and the quantity of the evidence base we present a set of recommended elasticities. Applying them to a scenario that roughly corresponds to an average high-income city, we find that in per-capita present value terms (at a 5% discount rate), a 1%-increase in density implies an increase in wage

We analyze the economic impact of the German high-speed rail (HSR) connecting Cologne and Frankfurt, which provides plausibly exogenous variation in access to surrounding economic mass. We find a causal effect of about 8.5% on average of the HSR on the GDP of three counties with intermediate stops. We make further use of the variation in bilateral transport costs between all counties in our study area induced by the HSR to identify the strength and spatial scope of agglomeration forces. Our most careful estimate points to an elasticity of output with respect to market potential of 12.5%. The strength of the spillover declines by 50% ever 30 minutes of travel time, diminishing to 1% after about 200 minutes. Our results further imply an elasticity of per-worker output with respect to economic density of 3.8%, although the effects seem driven by worker and firm selection.