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This paper introduces persistent productivity shocks in a continuous-time mononopolistic competition model of trade with hetererogenous firms similar to Melitz (2003). In the model, the presence of sunk costs and uncertainty have three main consequences: first, firms export decisions become history-dependent. Second, the model generates firm dynamics and allows for substantial heterogeneity in export growth conditional on survival. Policy experiments modify the equilibrium along both the cross-sectional and time dimensions. Third, both the generated equilibrium firm size distribution and sales distribution of exporters into a foreign market are Pareto in the upper tail.
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