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This paper develops a two-country DSGE model for a monetary union in which each country is populated by two types of households - savers and borrowers - and two types of production sectors - a consumption goods sector and a housing sector. Households trade nominal private debt in equilibrium, with the borrowers being subject to a collateral constraint, which is tied to the value of the stock of housing. The analysis focuses on the implications of housing and credit market heterogeneities for the design of optimal monetary policy of a common central bank. The results indicate that as long as the only heterogeneity between the two countries is the degree of nominal rigidity, the Benigno (2004) result according to which the common central bank puts a higher weight on stabilizing inflation in the country with a higher degree of nominal rigidity continues to hold. However, due to the introduction of collateralized household debt and borrowing constraints the effects of cross-country disparities in the degree of price rigidities on the volatility of inflation are amplified. While the volatility of inflation in the economy with a higher degree of price rigidity is almost as low as in the model without borrowers, inflation in the economy with more flexible prices becomes much more volatile. As a result, under optimal policy the central bank allows union-wide inflation to fluctuate more in response to productivity shocks than in an economy without credit constrained borrowers. In addition we find that housing and credit market heterogeneities have an impact on the central bank's goal to stabilize inflation. Finally, the paper shows that even in the presence of a common productivity shock already small deviations of some of the credit market parameters from the symmetry assumption are sufficient to create sizeable reductions in the cross-country correlation of inflation rates.