We determine the properties of the core-periphery model with three regions and compare our results with those of the standard 2-region model. The conditions for the stability of dispersion and concentration are established. As in the 2-region model, dispersion and concentration can be simultaneously stable. We show that the 3-region (2-region) model favours the concentration (dispersion) of economic activity. Furthermore, we provide some results for the n-region model. We show that the stability of concentration of the 2-region model implies that of any model with an even number of regions. JEL classification: R12, R23

We study the long-run spatial distribution of industry using a multi-region coreperiphery model with quasi-linear log utility (Pflüger, 2004). We show that a distribution in which industry is evenly dispersed among some of the regions while the other regions have no industry cannot be stable. A spatial distribution where industry is evenly distributed among all regions except one can be stable, but only if that region is significantly more industrialized than the other regions. When trade costs decrease, the type of transition from dispersion to agglomeration depends on the fraction of workers that are mobile. If this fraction is low, the transition from dispersion to agglomeration is catastrophic once dispersion becomes unstable. If it is high, there is a discontinuous jump to partial agglomeration in one region and then a smooth transition until full agglomeration. Finally, we find that mobile workers benefit from more agglomerated spatial distributions whereas immobile workers prefer more dispersed distributions. The economy as a whole shows a tendency towards over-agglomeration for intermediate levels of trade costs.

We consider a version of the intertemporal general equilibrium model of Cox et al. (Econometrica 53:363-384, 1985) with a single production process and two correlated state variables. It is assumed that only one of them, Y 2 , has shocks correlated with those of the economy's output rate and, simultaneously, that the representative agent is ambiguous about its stochastic process. This implies that changes in Y 2 should be hedged and its uncertainty priced, with this price containing risk and ambiguity components. Ambiguity impacts asset pricing through two channels: the price of uncertainty associated with the ambiguous state variable, Y 2 , and the interest rate. With ambiguity, the equilibrium price of uncertainty associated with Y 2 and the equilibrium interest rate can increase or decrease, depending on: (i) the correlations between the shocks in Y 2 and those in the output rate and in the other state variable; (ii) the diffusion functions of the stochastic processes for Y 2 and for the output rate; and (iii) the gradient of the value function with respect to Y 2 . As applications of our generic setting, we deduct the model of Longstaff and Schwartz (J Financ 47:1259(J Financ 47: -1282(J Financ 47: , 1992) for interest-rate-sensitive contingent claim pricing and the variance-risk price specification in the option pricing model of Heston (Rev Financ Stud 6:327-343, 1993). Additionally, it is obtained a variance-uncertainty price specification that can be used to obtain a closed-form solution for option pricing with ambiguity about stochastic variance.

This paper discusses the literature on horizontal mergers between multisided platforms and argues that the Cournot model can provide useful insights into the welfare effects of such mergers. To illustrate those insights, we develop a simple model in which two-sided platforms offer a homogeneous service and compete à la Cournot, and derive the effects of "average-marginal-costpreserving" mergers on consumers on both sides of the market. We conclude with a discussion of several research avenues that could be explored to understand better the impact of horizontal mergers between multisided platforms.

Literature on dynamic portfolio choice has been finding that volatility risk has low impact on portfolio choice. For example, using long-run US data, Chacko and Viceira [2005. "Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets." The Review of Financial Studies 18 (4): 1369-1402] found that intertemporal hedging demand (required by investors for protection against adverse changes in volatility) is empirically small even for highly risk-averse investors. We want to assess if this continues to be true in the presence of ambiguity. Adopting robust control and perturbation theory techniques, we study the problem of a long-horizon investor with recursive preferences that faces ambiguity about the stochastic processes that generate the investment opportunity set. We find that ambiguity impacts portfolio choice, with the relevant channel being the return process. Ambiguity about the volatility process is only relevant if, through a specific correlation structure, it also induces ambiguity about the return process. Using the same long-run US data, we find that ambiguity about the return process may be empirically relevant, much more than ambiguity about the volatility process. Anyway, intertemporal hedging demand is still very low: investors are essentially focused on the short-term risk-return characteristics of the risky asset.