ions is a measure of the conflict be- cks. So, if the shadow portfolio al- cy is a figucases there is, by definition, no portfolio decision for agents to make. In the autarky case there are no traded assets so there is no portfolio. In the single-bond case there is only one traded asset so portfolio composi￾tion is fixed. But in the two-bond case there are two traded assets so there is a portfolio allocation decision. The existence of the portfolio allocation decision gives the policymaker a qualitatively different channel by which to influence the equilibrium degree of risk sharing and this new channel has a qualitatively and quantitatively important effect on the op￾timal policy rule. In the two-bond case the policymaker can implicitly use monetary policy to alter the risk sharing properties of the two bonds, and can therefore influence the equilibrium portfolio, in such a way that the equilibrium portfolio provides a significantly higher degree of risk shar￾ing compared to an inflation targeting equilibrium. This can be a powerful incentive for the policymaker to deviate from strict inflation targeting. This portfolio allocation incentive, by definition, does not exist in the au￾tarky or single-bond cases and this explains the significant differences be￾tween the two-bond case and the autarky and single-bond cases. This process is illustrated in more detail in Fig. 1. This figure is based on the benchmark parameter set given in Table 1 with the international trade elasticity, θ, set equal to 1.5. It therefore corresponds to one of the cases reported in Table 2. So it can be noted from Table 2 that the optimal value of δD is approximately −1.686. Panel (a) in Fig. 1 plots welfare against δD and shows that welfare at the optimal value of δD is approxi￾mately 0.0036% above welfare at strict inflation targeting (δD ¼ 0 ) (which again corresponds to the number reported in Table 2).15 Panel (b) shows the standard deviation of PPI inflation at the optimal value of δD is approximately 0.01% while Panel (c) shows that the standard de￾viation of Dis reduced by optimal policy by a factor of 4 when compared to strict inflation targeting (i.e. from 1.39% to 0.33%). Panel (c) also shows that optimal policy implies a very significant reduction in the vol￾atility of the real exchange rate gap compared to strict inflation targeting (i.e. from 1.13% for strict inflation targeting to 0.27% for optimal policy). The main mechanism behind these effects is illustrated in relation to portfolio allocations and asset returns in Panels (d) and (e) of Fig. 1. Panel (d) shows how equilibrium portfolio allocation depends on the choice of the parameter δD: Panel (d) shows that this portfolio position rises from approximately 23 times steady state GDP at δD ¼ 0 (strict in- flation targeting) to approximately 31 times steady state GDP at δD ¼ − 1:686 (optimal policy), so monetary policy has a significant impact on portfolio holdings. This is clearly an important part of the mechanism which allows optimal policy to reduce deviations from perfect risk sharing. But the role played by portfolio allocation can be understood more clearly if one considers the nature of imperfect risk sharing in this sim￾ple model. There are two independent sources of risk faced by house￾holds - TFP shocks and taste shocks. But there is only one degree of freedom in portfolio choices to allow hedging against these two sources of risk. The equilibrium portfolio is therefore a compromise between hedging the two different sources of risk. The degree of compromise can be illustrated by comparing two alternative ‘shadow worlds’ - one where there are only TFP shocks and one where there are only taste shocks. In each of these worlds there is a ‘shadow portfolio allocation’ which maximises welfare.16 The divergence between these two alternative shadow portfolio allocations is a measure of the conflict be￾tween hedging the two sources of shocks. So, if the shadow portfolio al￾locations are very different, one can say that there is a significant ‘hedging conflict’ in the choice of a portfolio designed to hedge both shocks simultaneously, i.e. the compromise portfolio is likely to offer limited hedging against one or other (or both) shocks. While, if the shadow portfolios are very similar, one can say that the ‘hedging con- flict’ is less severe - i.e. a compromise portfolio is likely to be a good hedge against both shocks. Panel (d) illustrates how these two shadow portfolio allocations de￾pend on the policy parameter δD:In particular, these plots show that the degree of divergence between the two shadow portfolio allocations, and thus the degree of conflict between hedging the two sources of shocks, depends on δD: Panel (d) shows how the policy maker can ease the con- flict between hedging TFP shocks and hedging taste shocks by making an appropriate choice of the policy parameter δD: It is apparent from Panel (d) that the optimal choice of δD (δD ¼ −1:686) significantly re￾duces the divergence between the shadow portfolio allocations and thus allows the equilibrium portfolio allocation to become simulta￾neously a good hedge against both sources of risk. It is this that allows optimal policy to generate a significant reduction in deviations for per￾fect risk sharing.17 Note, of course, that any deviation of policy from strict inflation targeting generates inflation volatility, which (given sticky nominal prices) has a negative effect on welfare, so optimal policy is a compromise between inflation volatility and deviations from risk sharing. Panel (d) illustrates the impact of policy on portfolio and shadow portfolio allocations. But further insight into the effects of policy on portfolio allocations can be found in Panel (e). This panel shows the im￾pact effect of each shock on the return differential between the two bonds. The size and sign of these effects are key determinants of the equilibrium and shadow portfolio allocations. Panel (e) shows that the policy parameter has a significant influence on the impact effects of shocks on the return differential so it is evident that this link is an im￾portant part of the mechanism that allows monetary policy to affect portfolio allocations.18 Fig. 1 shows clearly how portfolio allocation plays an important role in the policy transmission mechanism. In particular, these results show that policy has a potentially important role in easing the hedging con- flict that arises when there are more sources of risk than degrees of free￾dom in portfolio allocation. However, it should be clear from this explanation, that this mechanism can only be important if a hedging conflict exists in the first place. The example illustrated in our simple benchmark model is one where TFP shocks and taste shocks create a sig￾nificant hedging conflict. We will show below that these two sources of shocks continue to create a hedging conflict in a more general model. But we also show that other combinations of shocks may not give rise to such a significant hedging conflict. In these cases, the optimal policy rule is much closer to strict inflation targeting - so it will become clear 15 Where welfare is measured as a percentage of steady state equivalent consumption. In Figure 1 Panel (a) this is normalised to be zero at inflation targeting. 16 Note that, for any single shock or combination of shocks, the welfare maximising port￾folio is not necessarily equal to the market equilibrium portfolio (as yielded by the Dever￾eux and Sutherland (2011a) approach). This is because the endogenous discount factor defined in (2) implies an externality in portfolio selection. Individual households treat ag￾gregate consumption as exogenous when selecting their portfolio position while a welfare maximising policymaker would internalise this effect. The shadow portfolios shown in the figure are based on maximising welfare and therefore differ from the portfolios that would be selected by households in the presence of each shock. The welfare maximising shadow portfolios are relevant here because, in this exercise, we are illustrating the incentives faced by the policymaker in choosing the policy rule. The shadow portfolios are computed using a search algorithm that maximises welfare (as measured by (7)) in the presence of each individual source of shock and for each value of the policy parameter, δD. 17 Note that the externality contained within the endogenous discount factors (which generates a difference between the equilibrium portfolio allocation and the welfare maximising portfolio allocation) implies that it is not possible for policy to achieve perfect risk sharing by choice of δD: Note also that this same externality also implies that the mar￾ket equilibrium portfolio will not yield full risk sharing even in a case where there are suf- ficient assets to span the space of shocks. In principle in the two-bond model this implies that there would be a policy trade-off between inflation stabilisation and risk sharing even if there is only one source of shocks (and thus no hedging conflict). Numerical experi￾ments with the basic model show that this effect is quantitively extremely small and therefore has no significant quantitive implications for the main focus of analysis in this paper. 18 The hedging properties of a portfolio position depend on how the return differential (i.e. the difference between the return on the home bond and the foreign bond) responds on impact to the two shocks. Note, however, that there are a number of interacting factors which determine the equilibrium and shadow portfolios, so it is not possible to draw a simple link between the effects of the policy parameter on return differentials in Panel (e) and the shadow portfolio allocations shown in Panel (d). 202 O. Senay, A. Sutherland / Journal of International Economics 117 (2019) 196–208