O Senay, A Sutherland /Joumal of International Economic 117(2019)196-208 Combining the analysis of optimal policy and endogenous portfolio Table 1 choice presents some new technical challenges. These challenges arise Benchmark parameter values. because there is an interaction between policy choices and portfolio Discount factor B=0.99,n=0.005 choice. Monetary policy affects the stochastic behaviour of income and Elasticity of substitution: individual goods the hedging properties of assets and therefore affects optimal portfolio Elasticity of labour supply choice. In turn, the equilibrium portfolio affects consumption and labour Risk aversion supply choices and thus affects macroeconomic outcomes and welfare. Share of home goods in consumption basket Elasticity of substitution: home and foreign goods hus, in addition to the standard routes via which policy affects the Share of labour in production ====三 count of the welfare effects of policy that occur via the effects of policy TP shan setting macro economy, the optimal choice of monetary policy must take ad Calvo 095,A=0006 on portfolio allocation As will be demonstrated below, this mechanism turns out to play a key role.g Our solution approach follows the recent portfolio literature based on Devereux and Sutherland (201la)in computing equilibrium portfo. taste shock processes are based on Corsetti et al. (2010, 2018)and lios using a second order approximation to the portfolio selection equa Smets and Wouters(2003, 2005, 2007). tions for the home and foreign country in conjunction with a first order In this section we focus on optimal policy based on the simple policy approximation to the home and foreign budget constraints and the ve rule given in(8). This allows us to illustrate in detail the economic mechanism behind the effects we wish to emphasise. Given the As already explained, we model monetary policy as a simple targeting simplithed policy rule, the only policy parameter that needs to be tor of excess returns. ule(8). We optimise the choice of coefficient in the targeting rule by ferent values of 8p on welfare, portfolio allocation and the variances of value for op in the targeting rule and for each grid point there is an equi- key variables. librium portfolio allocation and a corresponding general macroeconomic Table 2 presents some key results for a range of values of the inter- equilibrium and level of welfare. We use the devereux and Sutherland national trade elasticity, 0. For comparison, this table shows the results 011b) portfolio solution approach to evaluate the equilibrium portfolio for the two-bond case together with the financial autarky and single at each grid point. This equilibrium portfolio is then used to compute real-bond version of the model For each value of e and for each financial macroeconomic equilibrium and evaluate welfare at each grid point. In conducting this analysis it necessary to be mindful of orders of al difference between optimal policy and strict inflation targeting, the derstood in the literature, this requires that the overall model must also icy and inflation targeting and for the two-bond case)equilibrium principles outlined in Samuelson(1970), an order n approximation of utility(in our case welfare)depends only on the order n-2 behaviour sumption Standard deviations are reported in percentage terms And of portfolios. Thus, in computing a second order approximation of wel- portfolio holdings are measured relative to steady state GDP. Given that, in this simple model, there are just two assets that can be traded fare, we only require the zero-order(or steady state)equilibrium porto- internationally, it is possible to represent portfolio positions in terms lio. Hence the technique outlined in Devereux and Sutherland(2011a) of a single number. In this case we focus on the home country's portfolio for computing the zero-order portfolio is sufficient for our purposes. position in the foreign nominal bond. As in Devereux and Sutherland 5. Optimal monetary policy in the basic model (2011a), we compute the zero-order(i.e. steady state) portfolio hold ing In the steady state it is assumed that net foreign assets are zero, so a positive holding of foreign bonds must be matched by an equivalent The benchmark parameter values used in the following analysis are negative(external )holding of home bonds. I isted in Table 1. Many of these parameter values are taken directly from First consider the autarky and single-bond cases. These two cases Corsetti et al (2010, 2018). The values ofA(the elasticity of substitution correspond to the financial market structures considered by Corsetti between individual final goods)and u( the Cobb-Douglas coefficient on et al. 2010, 2018). For both these cases, and for all the values of e labour in the production function of intermediate goods)are chosen to shown, the optimal value of p( derived numerically using the search yield a steady state monopoly mark-up of 11% and share of capital in procedure outlined above)differs from zero. This indicates a deviation output of.33. The implied steady state share of dividends in GDP is ap- from strict inflation targeting(which corresponds to 6p=O). But notice proximately 0. 15. The Calvo parameter for price setting, K, is Chosen to that the difference between the welfare level yielded by optimal policy imply an average period between price changes of 4 quarters. The and the welfare level yielded by strict inflation targetingis very small for values of (inverse labour elasticity) and p (risk aversion )are consi all values of e, except for 0=1/2. The variance of the real exchang tent with the estimates of Smets and Wouters(2003, 2005, 2007). The gap and the variance of PPl inflation are also only marginally different parameters of the endogenous discount factor, B and n, are chosen to between the optimal policy and strict inflation targeting equilibria for yield a steady state rate of return of approximately 4%. The TFP and all values of 0, except for 0=1/2 These results broadly match the results emphasised by Corsetti et al policy choices are made in advance of(2010, 2018)who find that the differences between the optimal rule mane ta y sei ry rule fn ano tme pes eh. senay and su herland s oea. w a sh emphasise and inflation targeting are likely to be very small except for low values how monetary policy can interact with portfolio choice. But in that earlier paper we ana lyse non-cooperative policy in a world where financial markets are complete. The inter tion that occurs there is an explicitly distortionary effect that is quite different to the n Note that, for all values of e, gross portfolio positions are very large ote, t der realisation of asset retum differentials. These terms. however, drop the expectations operator is applied and therefore do not enter the exp nd taxation ancial regulation issues which go wusts.informa- nd-order approximation of expected utility. See Devereux and Sutherland(2010b)for a scope of the analysis in this paper. Note that throughout our analysis (again for the pur- more detailed discussion of orders of approximation in the analysis of portfoli poses of simplification) we also abstract from short selling constraints.Combining the analysis of optimal policy and endogenous portfolio choice presents some new technical challenges. These challenges arise because there is an interaction between policy choices and portfolio choice. Monetary policy affects the stochastic behaviour of income and the hedging properties of assets and therefore affects optimal portfolio choice. In turn, the equilibrium portfolio affects consumption and labour supply choices and thus affects macroeconomic outcomes and welfare. Thus, in addition to the standard routes via which policy affects the macro economy, the optimal choice of monetary policy must take ac￾count of the welfare effects of policy that occur via the effects of policy on portfolio allocation. As will be demonstrated below, this mechanism turns out to play a key role.9 Our solution approach follows the recent portfolio literature based on Devereux and Sutherland (2011a) in computing equilibrium portfo￾lios using a second order approximation to the portfolio selection equa￾tions for the home and foreign country in conjunction with a first order approximation to the home and foreign budget constraints and the vec￾tor of excess returns. As already explained, we model monetary policy as a simple targeting rule (8). We optimise the choice of coefficient in the targeting rule by means of a grid search algorithm. Each grid point represents a different value for δD in the targeting rule and for each grid point there is an equi￾librium portfolio allocation and a corresponding general macroeconomic equilibrium and level of welfare. We use the Devereux and Sutherland (2011b) portfolio solution approach to evaluate the equilibrium portfolio at each grid point. This equilibrium portfolio is then used to compute macroeconomic equilibrium and evaluate welfare at each grid point. In conducting this analysis it necessary to be mindful of orders of ap￾proximation. We approximate welfare up to second order. As is well-un￾derstood in the literature, this requires that the overall model must also be solved up to second-order accuracy. But note that, according to the principles outlined in Samuelson (1970), an order n approximation of utility (in our case welfare) depends only on the order n − 2 behaviour of portfolios. Thus, in computing a second order approximation of wel￾fare, we only require the zero-order (or steady state) equilibrium portfo￾lio. Hence the technique outlined in Devereux and Sutherland (2011a) for computing the zero-order portfolio is sufficient for our purposes.10 5. Optimal monetary policy in the basic model The benchmark parameter values used in the following analysis are listed in Table 1. Many of these parameter values are taken directly from Corsetti et al. (2010, 2018). The values of λ (the elasticity of substitution between individual final goods) and μ (the Cobb-Douglas coefficient on labour in the production function of intermediate goods) are chosen to yield a steady state monopoly mark-up of 11% and share of capital in output of 0.33. The implied steady state share of dividends in GDP is ap￾proximately 0.15. The Calvo parameter for price setting, κ, is chosen to imply an average period between price changes of 4 quarters. The values of ϕ (inverse labour elasticity) and ρ (risk aversion) are consis￾tent with the estimates of Smets and Wouters (2003, 2005, 2007). The parameters of the endogenous discount factor, β and η, are chosen to yield a steady state rate of return of approximately 4%. The TFP and taste shock processes are based on Corsetti et al. (2010, 2018) and Smets and Wouters (2003, 2005, 2007). In this section we focus on optimal policy based on the simple policy rule given in (8). This allows us to illustrate in detail the economic mechanism behind the effects we wish to emphasise. Given the simplified policy rule, the only policy parameter that needs to be determined is δD: It is therefore simple to investigate the effects of dif￾ferent values of δD on welfare, portfolio allocation and the variances of key variables. Table 2 presents some key results for a range of values of the inter￾national trade elasticity, θ. For comparison, this table shows the results for the two-bond case together with the financial autarky and single￾real-bond version of the model. For each value of θ and for each financial market structure the table shows the optimal value of δD; the welfare difference between optimal policy and strict inflation targeting, the standard deviations of a number of variables in the case of optimal pol￾icy and inflation targeting and (for the two-bond case) equilibrium portfolios for the case of optimal policy and inflation targeting. Welfare is measured in terms of the equivalent percentage of steady state con￾sumption. Standard deviations are reported in percentage terms. And portfolio holdings are measured relative to steady state GDP. Given that, in this simple model, there are just two assets that can be traded internationally, it is possible to represent portfolio positions in terms of a single number. In this case we focus on the home country's portfolio position in the foreign nominal bond. As in Devereux and Sutherland (2011a), we compute the zero-order (i.e. steady state) portfolio hold￾ing. In the steady state it is assumed that net foreign assets are zero, so a positive holding of foreign bonds must be matched by an equivalent negative (external) holding of home bonds.11 First consider the autarky and single-bond cases. These two cases correspond to the financial market structures considered by Corsetti et al. (2010, 2018). For both these cases, and for all the values of θ shown, the optimal value of δD (derived numerically using the search procedure outlined above) differs from zero. This indicates a deviation from strict inflation targeting (which corresponds to δD ¼ 0). But notice that the difference between the welfare level yielded by optimal policy and the welfare level yielded by strict inflation targeting is very small for all values of θ, except for θ = 1/2. The variance of the real exchange rate gap and the variance of PPI inflation are also only marginally different between the optimal policy and strict inflation targeting equilibria for all values of θ, except for θ = 1/2. These results broadly match the results emphasised by Corsetti et al. (2010, 2018) who find that the differences between the optimal rule and inflation targeting are likely to be very small except for low values of θ. The results in Table 2 go somewhat further than Corsetti et al. Table 1 Benchmark parameter values. Discount factor β ¼ 0:99; η = 0.005 Elasticity of substitution: individual goods λ = 6 Elasticity of labour supply 1/ϕ = 0.5 Risk aversion ρ = 2 Share of home goods in consumption basket γ = 0.875 Elasticity of substitution: home and foreign goods θ = 0.25 - 6.00 Share of labour in production μ = 0.67 Calvo price setting κ = 0.75 TFP shocks ηA = 0.95, σA = 0.006 Taste shocks ηΨ = 0.9, σΨ = 0.01 9 In this paper we are making an assumption that policy choices are made in advance of trade in asset markets. This implies that equilibrium portfolios depend on the choice of monetary policy rule. In another paper, Senay and Sutherland (2013), we also emphasise how monetary policy can interact with portfolio choice. But in that earlier paper we ana￾lyse non-cooperative policy in a world where financial markets are complete. The interac￾tion that occurs there is an explicitly distortionary effect that is quite different to the mechanism being analysed in this paper. 10 Note, the fact that welfare is based on expected utility is crucial in allowing us to focus on the zero-order portfolio. A second-order approximation of realised utility may include terms that depend on the first-order behaviour of portfolio holdings multiplied by the first-order realisation of asset return differentials. These terms, however, drop out when the expectations operator is applied and therefore do not enter the expression for the sec￾ond-order approximation of expected utility. See Devereux and Sutherland (2010b) for a more detailed discussion of orders of approximation in the analysis of portfolios. 11 Note that, for all values of θ, gross portfolio positions are very large relative to steady state GDP. Portfolio positions of this magnitude are obviously very unrealistic. It is only for very few countries (usually tax havens) where external portfolio positions exceed 4 or 5 times GDP. It is not the purpose of this analysis to match the data on international portfolio positions. Such an exercise is likely to require consideration of transaction costs, informa￾tional asymmetries, and taxation and financial regulation issues which go well beyond the scope of the analysis in this paper. Note that throughout our analysis (again for the pur￾poses of simplification) we also abstract from short selling constraints. 200 O. Senay, A. Sutherland / Journal of International Economics 117 (2019) 196–208