Hi Reuben,

I willl try to explain you, why I think its important two have both UIP conditions instead of only 1 when we go for higher order approximations.

The resource constraint is given by the following equation

( S_{t}*B_{f,t}^{H} is the nominal value of Bonds issued by foreign households and held by domestic households converted to domestic currency, ,CA is the current account):

S_{t}*(B_{f,t}^{H}/ R_{f,t} - B_{f,t-1}^{H}) = B_{h,t}^{F} /R_{t} - B_{ht-1}^{F} + CA

–> The difference between the nominal Flows of Bonds is the current account.

The optimal allocation of foreign bonds held by domestic households is given by (UC is the lagrangian)

UC_{t} = R_{f,t}*exp(-riskpremium(B_{f,t}^{H}) * E_{t} [UC_{t+1} * S_{t+1}/S_{t}]

Here we added the risk premium to pin down the level of foreign households held by domestic agents in steady state (0).

The optimal allocation of home bonds held by foreign households is given by:

UC_{f,t} = R_{t}*exp(-riskpremium(B_{f,t}^{H}) * E_{t} [UC_{f,t+1} * S_{t}/S_{t+1}]

Again, we added the risk premium to pin down the level of bonds held by foreign households.

We could combine the two equations with the FOC for domestic bonds to get the two UIPs, but I think its easier to make the point here.

What you suggest is to set one of the level of bond holdings to 0 (its determinstic steady state) or simply not deriving the FOC for the optimal level of bonds held by foreigners, such that the resource constraint becomes.

S_{t}*(B_{f,t}^{H}/R_{f,t} - B_{f,t-1}^{H}) = 0 - 0 + CA

Given the steady of B_{f,t}^{H} of 0, this unique identifies the level of bond holdings. Up to a first order approximation, this is fine. But for higher order approximation you will get a covariance term in the FOCs (thanks to you I got that).

UC_{f,t} = R_{t}*exp(-riskpremium(B_{f,t}^{H}) * E_{t} [UC_{f,t+1} * S_{t}/S_{t+1}]

–>

UC_{f,t} = R_{t}*exp(-riskpremium(B_{f,t}^{H}) * E_{t} [UC_{f,t+1}] * E_{t}[S_{t}/S_{t+1}]] + Cov(UC_{f,t+1}, S_{t}/S_{t+1}])

As you can see, if Cov(UC_{f,t+1}, S_{t}/S_{t+1}]) is unequal to 0, the stochastic steady state of bond holdings is no longer 0. In fact in my models, Bond holdings in the stochastic steady state are positiv, as the expected depreciation of the exchange rate is positvely correlated with a fall in consumption (an increase utility).

By effectively setting the level of foreign bond holdings to 0 (which is equivalent to not deriving the equation), you neglect the effect of uncertainty on bond holdings for this country. In a symetric model, you will have asymetric stochastic steady states for instance.