# Determining Foreign Assets and Exchange Rate in a New Keynesian Model

Dear all,

I wish to write down an open economy model where the domestic agent can use foreign assets to smooth consumption and risk aversion (using third order prunned solution for the non linear system), in a New Keynesian economy. However, i am specially interested in also determining exchange rate in this environment.

The class of models that uses foreign assets for this purpose (like Schmitt-Grohe and Martin Uribe work) do not determine exchange rate. Actually, reading some papers, I am observing that it is quite hard to determine both foreign assets and exchange rates in these models.

Usually, it seems that foreign assets are pinned down by some external account “extra” equation, so exchange rate can be determined. Conversely, other papers forget about exchange rates and deal only with the “same” consumption unit in both countries (i.e. real exchange rate is the unity).

I am wondering if there is a way to determine both foreign assets and exchange rate without including the “external account” equation, so that agent can eventually use output to finance foreign assets.

In practical terms, I want a more general market-clear condition to hold:

C_t + Q_t NX_t + Q_t F_t = Y_t + (1+r)Q_t F_{t-1}

Instead of the more restrictive two equation market clear:

C_t = Y_t
NX_t + F_t = (1+r)F_{t-1}

Where F_t are foreign assets, Q_t is the real exchange rate, NX_t are net exports and r is a given world-interest rate.

Of course, it is necessary to include one more equation so we can close the model. I was thinking maybe something like an Euler Equation of an external agent (exogenous SDF) deciding to buy shares of domestic firms, since in NK economies profits flows \{\Pi_{t}\}_t usually have a positive mean.

I would appreciate any help,