Two-country Model and Foreign Exchange Dynamics

We establish the nature of the dynamics of the exchange rate in a two country model with heterogenous firms a la Abadir and Talmain (2002). We are picking an “off-the-shelf” two country (‘home’ or ‘foreign’) monopolistic competition model from Cook and Devereux (2016), denoted as CD later, and adapt it to our setting. We generalize their model by allowing heterogenous productivities among firms, and heterogeneity in the shares of demand for each commodity. Breaking the homogeneity of firms is required for important reasons. First, it is the reality, and it leads to a very different and nonlinear solution of the model. Second, the “representative firm” assumption has been shown in our previous work to be far from innocent, as it linearizes the dynamics of GDP and the other aggregate variables to a process that contradicts the data (loses the long-memory and turning-point features). We also simplify aspects of the model, as CD were interested in a very different problem: the implication of the zero lower-bound of interest rates on monetary policy. We assume no disutility of work (i.e., inelastic labour supply), no capital mobility, and no price friction. The first two restrictions and the Calvo price adjustment (as assumed in CD) simplify the calculations but do not change the spirit of the model. The home country [resp. foreign] is inhabited by an infintely lived dynasty withNt [resp. N t ] agents at time t. The home dynasty aims to maximize, at time t, the intertemporal utility: