Elastic Attention, Risk Sharing, and International Comovements

Transcription

1 Elastic Attention, Risk Sharing, and International Comovements Wei Li University of Hong Kong Yulei Luo University of Hong Kong Jun Nie Federal Reserve Bank of Kansas City April 18, 2015 Abstract In this paper we examine the effects of elastic information-processing capacity (rational inattention or RI) proposed in Sims (2010) on international consumption correlations in a tractable small open economy model with exogenous income processes. We find that in the presence of capital mobility in financial markets, elastic attention (or elastic capacity) due to fixed information-processing cost lowers the international consumption correlations by generating heterogeneous consumption adjustments to income shocks across countries facing different macroeconomic uncertainty. In addition, we show that RI can also improve the model s predictions for the other key moments of the joint dynamics of consumption and income. Keywords: Rational Inattention, Elastic Capacity, Risk Sharing, International Consumption Correlations. JEL Classification Numbers: D83, E21, F41, G15. We are grateful to Hongyi Chen, Martin Ellison, Chris Otrok, Chris Sims, Tao Zha, Eric R. Young, and seminar participants at University of Hong Kong and Hong Kong Institute of Monetary Research for helpful suggestions and discussions. Luo thanks the Hong Kong General Research Fund (#HKU791913) for financial support. Part of this work was conducted while Li was visiting Hong Kong Institute for Monetary Research, whose financial support and hospitality is greatly appreciated. All errors are the responsibility of the authors. The views expressed here are the opinions of the authors only and do not necessarily represent those of the Federal Reserve Bank of Kansas City or the Federal Reserve System. Faculty of Business and Economics, The University of Hong Kong, Hong Kong. verawli@hku.hk. Faculty of Business and Economics, The University of Hong Kong, Hong Kong. yulei.luo@gmail.com. Economic Research Department, Federal Reserve Bank of Kansas City. jun.nie@kc.frb.org.

2 1 Introduction A common assumption in canonical international business cycles models is that world financial markets are complete in the sense that individuals in different countries are able to fully insure country-specific income risks using international financial markets. Under this assumption, the models predict that consumption (or consumption growth) is highly correlated across countries, and in some cases the international consumption correlation is equal to 1 regardless of income or output correlations. 1 The key reason behind this result is that since consumers are risk averse they will choose to smooth consumption over time by trading in international financial markets. However, in the data cross-country consumption correlations are very low and are generally lower than corresponding income or output correlations in many countries. 2 For example, Backus, Kehoe, and Kydland (1992) solve a two-country real business cycles (RBC) model and argue that the puzzle that empirical consumption correlations are lower than output correlations is the most striking discrepancy between theory and data. 3 In the literature, the empirical low international consumption correlations have been interpreted as indicating international financial markets imperfections for examples, see Kollman (1996), Baxter and Crucini (1995), Lewis (1996), and Kehoe and Perri (2002). Some extensions have been proposed to make the models better fit the data. For example, Devereux, Gregory, and Smith (1992) show that in the perfect risk-sharing model nonseparability between consumption and leisure has the potential to reduce the cross-country consumption correlation. Stockman and Tesar (1995) show that the presence of nontraded goods in the completemarket model can also improve the models prediction on this dimension. Kollman (1996) shows in a two-country RBC model that financial market incompleteness can generate significantly lower cross-country consumption correlations. Wen (2007) shows that adding country-specific demand shocks can also help explain the cross-country business cycle comovements within a completemarket framework. In addition, Fuhrer and Klein (2006) find that habit formation play an important role in affecting international consumption correlations. In particular, they show that with shocks to labor income and the interest rate habit formation by itself can generate positive consumption correlations across countries even in the absence of international risk sharing and common income shocks. They then argue that if habit is a good characterization of individual consumers behavior, the absence of international risk sharing is even more striking than standard 1 See Chapter 6 in Obstfeld and Rogoff (1996) for a textbook treatment on this topic. 2 Table 1 reports the cross-country consumption and income correlations using the G-7 data. 3 Pakko (1996) shows that, in the presence of complete asset markets, consumption should be more highly correlated with total world income than with domestic income, while the data shows the opposite. provides an alternative standard for evaluating models of international business cycles. This result 1

3 tests suggest; that is, existing studies may overstate the extent to which common consumption movements across countries reflect international risk sharing because some of them are due to habit. 4 In addition, Luo, Nie, and Young (2014) examine the effects of model misspecification (robust control) on international consumption correlations in an small open economy real business cycles model. In this paper, we follow Sims (2003, 2010) and assume that consumers face fixed informationprocessing costs and thus only have limited information-processing capacity when making optimal decisions. Consequently, they cannot observe the state of the economy perfectly and learn the true state using noisy observations. Specifically, we find that elastic attention due to the fixed information-processing cost proposed in Sims (2010) has the potential to generate heterogenous adjustments in consumption across countries. In contrast, in the RI model with fixed capacity (e.g., Sims 2003 and Luo 2008), there are two competing forces on consumption dynamics. First, this model generates gradual responses of consumption growth to income shocks due to imperfect state observation. Just like the habit formation hypothesis, this channel increases cross-country consumption correlations. Second, the noise due to imperfect observation reduces consumption correlations across countries because it increases consumption volatility but has no effect on the covariance of consumption across countries. 5 It is worth noting that in the presence of fixed information-processing cost, both the adjustment speed (measured by the Kalman gain) and the variance of the RI-induced noise depend on the amount of fundamental uncertainty. Since different countries have different amounts of fundamental uncertainty, the two channels lead to greater heterogeneity and thus lower cross-country consumption correlations. The remainder of the paper is organized as follows. Section 2 presents the standard Fullinformation Rational Expectations (FI-RE) small open economy (SOE) model and discusses the model s puzzling implications for international consumption correlations. 3 introduces RI into this SOE model and examines the theoretical implications of elastic attention. Section 4 presents the main findings about how elastic attention improves the model s performance on the joint dynamics of consumption and income. Section 5 generalizes the benchmark model by considering an economy with a continuum of inattentive consumers. Section 6 concludes. 4 Baxter and Jermann (1997) also argue that the international diversification puzzle is worse than you think due to the correlation between nontraded labor income and the returns to domestic risky assets. 5 Note that following the RI literature, we assume that the noise shocks are independent over time and across countries. 2

4 2 A Full-information Rational Expectations Small Open Economy Model 2.1 Model Setup In this section we present a full-information rational expectations (FI-RE) version of a small open economy (SOE) model and will discuss how to incorporate rational inattention (RI) into this stylized model in the next section. Following the incomplete financial market literature, we consider the representative agent case and assume that the only asset that is traded internationally is a risk-free bond. (In Section 5, we consider the model in which there are a continuum of consumers.) Following Glick and Rogoff (1995), we formulate the full-information RE-SOE model as [ ] subject to the flow budget constraint max {c t} E 0 t=0 β t u(c t ) (1) b t+1 = Rb t c t + y t, (2) where u(c t ) = (c c t ) 2 /2 is the utility function, c t is consumption, c is the bliss point, R 1 is the exogenous and constant gross world interest rate, b t is the amount of the risk-free world bond held at the beginning of period t, and y t is real income in period t. 6 Here we assume that the household sector takes y t as given to keep our model tractable. Incorporating the firm sector and modelling investment decisions explicitly does not change the main results in this paper. The model assumes perfect capital mobility in that domestic consumers have access to the bond offered by the rest of the world and that the real return on this bond is the same across countries. In other words, the world risk-free bond provides a mechanism for domestic households to smooth consumption using the international capital market. Finally we assume that the no-ponzi-scheme condition is satisfied. A similar problem can be formulated for the rest of the world (ROW). We use an asterisk ( ) to represent the rest of world variables. For example, we assume that yt is the aggregate income of the rest of the world (G-7, OECD, or EU). Furthermore, we assume that domestic endowment and the ROW endowment are correlated. We will specify the structure of the income processes later after we discuss how to determine y t endogenously. Let βr = 1; optimal consumption is then determined by permanent income: 6 Here we ignore the investment and government spending components. c t = (R 1) s t, (3) 3

5 where s t = b t + 1 R R j E t [y t+j ] j=0 is the expected present value of lifetime resources, consisting of financial wealth (the risk free foreign bond) plus human wealth. In order to facilitate the introduction of the rational inattention hypothesis, we follow Luo (2008) and Luo, Nie, and Young (2015), and reduce the multivariate model with a general income process to a univariate model with iid innovations to permanent income s t that can be solved analytically. 7 Letting s t be defined as a new state variable, we can reformulate the SOE model as v(s 0 ) = max {c t} t=0 { [ ]} E 0 β t u(c t ) t=0 (4) subject to s t+1 = Rs t c t + ζ t+1, (5) where the time (t + 1) innovation to permanent income can be written as ζ t+1 = 1 R j=t+1 ( ) 1 j (t+1) (E t+1 E t ) [y j ] ; (6) R v(s 0 ) is the consumer s value function under FI-RE. Under the FI-RE hypothesis, this model with quadratic utility leads to the well-known random walk result of Hall (1978): c t = R 1 ( ) R (E t E t 1 ) 1 j (y t+j) (7) R = (R 1) ζ t, which relates the innovations to consumption to income shocks. 8 j=0 In this case, the change in consumption depends neither on the past history of labor income nor on anticipated changes in labor income. In addition, certainty equivalence holds, and thus uncertainty has no impact on optimal consumption. 7 See Luo (2008) for a formal proof of this reduction. Multivariate versions of the RI model are numerically, but not analytically, tractable, as the variance-covariance matrix of the states cannot generally be obtained in closed form. 8 Under FI-RE the expression for the change in individual consumption is the same as that for the change in aggregate consumption. 4

6 2.2 Estimating Income Processes We follow consumption literature (Quah, 1990; Pischke, 1995; Luo, Nie and Young, 2014) and assume that income y t consists of two components, a random walk and a white noise component. y t+1 = y p t+1 + yi t+1, (8) y p t+1 = yp t + ε t+1, (9) y i t+1 = y + ɛ t+1. (10) where y p t+1 is the permanent income component, and yi t+1 is the transitory income component, ε t+1 and ɛ t+1 are orthogonal iid shocks with with mean 0 and variance ω 2 and ωɛ 2. For the rest of the world (ROW), we use asterisk to denote all the variables. y is the aggregate income of ROW. And income process is specified with two shocks y t+1 = y p t+1 + yi t+1, (11) y p t+1 = yp t + ε t+1, (12) y i t+1 = y + ɛ t+1, (13) where ε t+1 and ɛ t+1 are orthogonal iid shocks with with mean 0 and variance ω 2 and ω 2 ɛ. We allow contemporaneous correlations between the SOE and the ROW, corr(ε, ε ) = η > 0, corr(ɛ, ɛ ) = ρ > 0. Since y t = ε t + ɛ t ɛ t 1 and yt = ε t + ɛ t ɛ t 1, the innovation to permanent income can be written as ζ t = 1 R 1 ε t + 1 R ɛ t N(0, ωζ 2), where ω2 ζ = ω2 /(R 1) 2 + ωɛ 2, and the international correlation of income growth can be written as: corr( y t, yt E[ε t ε t ] + 2E[ɛ t ɛ t ] ) =. (14) ω 2 + 2ωɛ 2 ω 2 + 2ωɛ 2 Quah (1990) propose this specification proposed and argues that it has the potential to solve the excess smoothness puzzle in consumption. He finds that the volatility of consumption is mainly due to the permanent component, with transitory component accounts for 1% to 2% of total variance of consumption. Using annual per capita GDP data in from Penn World Table (version 7.1), we find that income volatility is mainly due to permanent shocks for G-7 countries, for example, ω 2 = and ω 2 ɛ = for the U.S., ω ɛ /ω = 0.002, which is consistent with literature. Quah (1990) estimates that 1% 2% of consumption variation is contributed by the variation of the transitory component in the income process. Luo, Nie and Young (2015) finds that the ratio is using quarterly US data over the period of For the other G-7 countries, the ratio is , , , , , and for Canada, France, UK, 5

7 Italy, Japan and Germany, respectively. The result is robust for different versions of PWT GDP data. 9 Therefore, we will focus on the permanent income component in the rest of our discussion, ωζ 2 = ω2 /(R 1) 2 + ωɛ 2 ω 2 /(R 1) 2. In this case, the cross country income correlation can be approximated as follows: corr( y t, y t ) E[ε tε t ] ωω = corr(ε t+1, ε t+1). 2.3 Implications for Consumption Correlations In the FI-RE model proposed in Section 2.1, consumption growth can be written as c t = (R 1) ζ t, which means that consumption growth is white noise and the impulse response of consumption to the income shock is flat with an immediate upward jump in the initial period that persists indefinitely. It is worth noting that this consumption behavior does not fit the data well. The reason is that as has been well documented in the consumption literature, the impulse response of aggregate consumption to aggregate income takes a hump-shaped form, which means that aggregate consumption growth reacts to income shocks gradually. In the next two sections, we show that introducing informational frictions can help generate more realistic impulse responses of consumption to income. Given that consumption dynamics in the rest of the world is c t = (R 1) ζt, the international consumption correlation can thus be written as corr( c t, c t ) corr(ε t, ε t ) corr( y t, yt ). (15) Note that this prediction would not be consistent with the empirical evidence, as international consumption correlations are lower than output correlations for most pairs of countries, see Table 1. 9 The ratio ranges from to

8 3 Theoretical Implications of RI for Consumption-Income Comovements 3.1 Introducing RI We assume that consumers in the model economy cannot observe the true state s t perfectly and only observes the noisy signal s t = s t + ξ t, (16) when making decisions, where ξ t is the iid Gaussian noise due to imperfect observations. The specification in (16) is standard in the signal extraction literature and captures the situation where agents happen or choose to have imperfect knowledge of the underlying shocks. 10 Since imperfect observations on the state lead to welfare losses, agents use the processed information to estimate the true state. 11 Specifically, we assume that households use the Kalman filter to update the perceived state ŝ t = E t [s t ] after observing new signals in the steady state in which the conditional variance of s t, Σ t = var t (s t ), has converged to a constant Σ: ŝ t+1 = (1 θ) (Rŝ t c t ) + θ (s t+1 + ξ t+1 ), (17) where θ = 1 exp ( 2κ) is the Kalman gain (i.e., the observation weight). 12 Combining (5) with (17), we obtain the following equation governing the perceived state ŝ t : ŝ t+1 = Rŝ t c t + η t+1, (18) where η t+1 = θr (s t ŝ t ) + θ (ζ t+1 + ξ t+1 ) (19) is the innovation to the mean of the distribution of perceived permanent income, s t ŝ t = (1 θ) ζ t 1 (1 θ)r L θξ t 1 (1 θ)r L (20) is the estimation error where L is the lag operator, and E t [η t+1 ] = 0. Note that η t+1 can be rewritten as [( η t+1 = θ ζ t+1 1 (1 θ)r L ) ( + ξ t+1 )] θrξ t, (21) 1 (1 θ)r L 10 Note that this noisy signal specification is consistent with that adopted in the traditional signal extraction models with exogenous noises. Angeletos and La O (2009) and Luo, Nie, and Young (2014) for its recent applications. 11 See Luo (2008) and Luo, Nie, and Young (2015) for details about the welfare losses due to information imperfections within the partial equilibrium permanent income hypothesis framework. 12 θ measures how much uncertainty about the state can be removed upon receiving the new signals about the state. 7

9 where ωξ 2 = var (ξ t+1) = 1 1 θ ω 2 1/(1 θ) R 2 ζ. Expression (21) clearly shows that the estimation error reacts to the fundamental shock positively, while it reacts to the noise shock negatively. In addition, the importance of the estimation error is decreasing with θ. More specifically, as θ increases, the first term in (21) becomes less important because (1 θ) ζ t in the numerator decreases, and the second term also becomes less important because the importance of ξ t decreases as θ increases. 13 Following Sims (2010) and Luo and Young (2014), we assume that consumers minimize the distance between true state and perceived state under finite information-processing capacity. Assuming a constant information cost of λ, { [ ( )]} R 2 Σ t 1 + ωζ 2 Σ t + λ ln min {Σ t} t=0 Σ t is variance of s t after collecting time t information, while R 2 Σ t 1 + ωζ 2 is the variance before information collection. This minimization problem demonstrates consumer s trade-off between uncertainty of perceived state and the cost attached to reduction in uncertainty. In an extreme case when information is costless, i.e., λ 0, there is no informational friction as in the FI-RE model, Σ = 0; on the contrary, when λ, Σ, i.e., no information will be collected. The optimal steady state conditional variance can be solved as Σ = (1 R(R 1) λ) + (1 R(R 1) λ) 2 + 4R 2 λ 2R 2 ωζ 2, (22) where λ λ/ω 2 ζ. ŝ t is governed by the Kalman filtering equation Σ t ŝ t+1 = (1 θ)(rŝ t c t ) + θ(s t+1 + ξ t+1 ). (23) The Kalman gain θ measures how much uncertainty can be removed. It is positively related to the capacity chosen to process information. Following Luo and Young (2014), θ can be expressed endogenously as θ = 1 1 R [ ] 1 R(R 1) λ 2 [ R(R 1) λ] + 4R 2 λ 1. (24) Figure (1) clearly shows that the value of θ increases with the level of macroeconomic uncertainty measured by ωζ 2 (i.e., θ/ ω2 ζ > 0). That is, the higher the income uncertainty, the more capacity is devoted to monitoring evolution of the state. With a fixed information-processing cost λ, the 13 Note that when θ = 1, var (ξ t+1) = 0. 8

10 agent is allowed to adjust the optimal level of capacity and attention in such a way that the marginal cost of information-processing for the problem at hand remains constant. This result is consistent with the concept of elastic capacity proposed in Kahneman (1973). Under RI, optimal consumption is and the change in consumption can be expressed as [ c RI t = θ(r 1) c t = (R 1)ŝ t+1, (25) ζ t 1 (1 θ)r L + Similarly, we can get the ROW consumption evolution [ c RI t = θ (R 1) ζ t 1 (1 θ )R L + ( ξ t ( ξ t )] θrξ t 1. (26) 1 (1 θ)r L θ Rξt 1 )] 1 (1 θ. (27) )R L Consumption adjusts gradually to income shocks instead of adjusting fully immediately. When θ < 1, the true state is no longer observable due to the existence of consumer s endogenous information noises ξ. Through gradual learning and adjustment, inattention opens up for past income shocks and information noises to affect current consumption decision. As the consumer pays less attention (smaller θ), these shocks become more important. When θ = 1, the true state can be observed and past shocks are not informative, hence the above expression reduces to c t = (R 1)ξ t, which is the same as in the full information model. For different countries, their domestic fundamental uncertainty can affect optimal consumption decisions through θ and its interaction with ξ. Different volatility of income shocks leads to different level of θ. The higher θ is, the more new information a country processes. Different level of θ then leads to different adjustments of consumption. This helps explain why consumption correlation is in general lower than income correlation. 3.2 Implications for Cross-Country Consumption Correlations Given (26) and (27), we have the following proposition about the cross-country consumption correlation: Proposition 1 The consumption correlation between the two economies under RI can be written as with corr( c RI t, c RI t ) Π corr(ε t, ε t ) Π corr( y t, yt ), (28) Π = θθ (1 (1 θ)r 2 )(1 (1 θ )R 2 ) 1 (1 θ)(1 θ )R 2 1. (29) 9

11 Proof. See Appendix 7.1. When θ = θ = 1, Π = 1, which means the consumption correlation should be as high as the income correlation and contradicts the empirical findings. Introducing elastic attention (λ > 0 and θ < 1) has three distinct effects on the consumption correlation: 1. The slow propagation channel: Without the endogenous noises, RI reduces the variance of consumption and thus increases the consumption correlation. 2. The noise channel: The presence of endogenous noises (ξ), which are uncorrelated across countries, increases the variance of consumption without changing the cross-country covariance, and thus reduces the consumption correlation. 3. The elastic attention channel: The consumption correlation is further reduced by the θ θ difference. As θ and θ deviate further from each other, the consumption correlation becomes smaller relative to the income correlation. The last channel is identified uniquely in our elastic attention model, in which attention level, θ and θ, are optimally chosen by consumers based on their own domestic countries income uncertainty. Income uncertainty in two economies are different by nature, which leads to different levels of attention. Figure 3 shows how Π varies with the value of θ for given θ. 4 Main Findings 4.1 Data The annual data between 1950 and 2010 from Penn World Table (PWT), both version 7.1 and version 8.0, are used to study the consumption-income correlation between each of the four smaller economies of G-7 (Canada, Italy, UK and France) and the rest of the world economy. All variables are at 2005 constant prices. The ROW economy with respect to each country is constructed using the weighted average of the G-7 countries excluding the domestic country. The correlation between Canada and the U.S. is also studied as an special case. The U.S. is treated as ROW to Canada as over 70% of Canada s international trade is contributed by the U.S. (Miyamoto and Nguyen, 2014). We will discuss more about this pair of countries in Section 4.3. Table 1 demonstrates the international consumption correlation puzzle. The puzzle persists in both per capita data or aggregate data. In this paper, we choose to discuss in per capita form to eliminate differences in size among the entities and thus make the variables more comparable. The variance of per capita output can better measure the fundamental uncertainty of a country 10

12 after eliminating the size effect. Table 2 summaries the key empirical findings. The numbers in the parentheses are GMM-corrected standard errors. 4.2 Parameter Values We first assume a fixed Kalman gain to the rest of the world θ = 0.8, that is, 80% of the state uncertainty is removed upon receiving the new signals. It is worth noting that a less-than-one value of θ can be rationalized by examining the welfare effects of limited capacity. 14 In the RI literature, to explain the observed aggregate fluctuations and the effects of monetary policy on the macroeconomy, the calibrated values of θ are lower and deviate more from the FI-RE case. For example, Adam (2007) found θ = 0.4 based on the response of aggregate output to monetary policy shocks. Luo (2008) found that if θ = 0.5, the otherwise standard permanent income model generates realistic relative volatility of consumption to labor income. The value of λ can be recovered by solving Equation (24). Given that λ λ /ωζ 2 and that both domestic country and ROW consumers are facing a stable information cost, i.e., λ = λ, we have λ = λ ω 2 ζ = λ ωζ 2 ωζ 2. Plugging the expression for λ back into Equation (24), we can get θ. θθ (1 (1 θ)r 2 )(1 (1 θ )R 2 ) Now we can calculate Π = and then use Equation (28) to determine corr( c RI 1 (1 θ)(1 θ )R 2 t, c RI t ). The interest rate is set to follow the existing literature (e.g., Glick and Rogoff, 1995): R = The depreciation rate is δ = Main Results Table 3 reports simulated results alongside data findings. The numbers corr ( y, y ) and corr ( c, c ) are the simple correlation coefficients between the annual change a country s real per capita output (or consumption) and the annual change of the rest of the world s real per capita consumption (or output), with the world defined as the output-weighted average of the rest G7 countries in the Penn World Table (version 7.1). The Canada-US correlations are the consumption-output correlations between Canada and the U.S. First two columns are empirical findings from data. In the third column, the benchmark theory predicts consumption correlations almost as high as income correlations. The performance of the model is improved when elastic attention is introduced. As shown in Table 3, the RI model 14 See Luo (2008) and Luo, Nie, and Young (2015) for details about the welfare losses due to imperfect observations within the linear-quadratic-gaussian permanent income framework; they are uniformly small. 11

13 generates much lower consumption correlations, which fits the data better. For example, the consumption correlation between Canada and the ROW is 0.78 predicted by the FI-RE case, and is reduced to 0.53 in the RI model. In all cases, RI helps reduce the consumption correlation and makes the model match the data better. We also check whether the findings are robust by different levels of ROW inattention, θ. Table 4 displays the results. Consumption correlations are in general lower at all θ values. Even small deviation from the full information rational expectation model (θ = 0.9) drives down the consumption correlation. For the Canada-ROW case, the correlation is reduced from 0.78 in benchmark to In Tables 5-7, we use the three versions of real output from PWT version 8.0 to repeat the exercise and find that the results are quite robust. 4.4 Canada-US Case The Canada-US case is of interest because Canada and the U.S. have one of the world s closest bilateral relationship. Total trade of goods (imports and exports) in 2014 amount to billion dollars. In addition, Canada is a typical small open economy studied in the literature. We now apply our model to study their correlations, treating the U.S. as the rest of the world to Canada. The U.S. is a reasonable approximation of the rest of the world to Canada since their relationship is highly asymmetric. First, Canada relies on the U.S. as its principal trading partner. 71% of the total trade of goods of Canada was conducted with the U.S in Over the period , on average, export 75% of exports and 68% of imports of Canada was traded with the U.S (See Minamoto and Nguyen, 2014; data from Statistics Canada). Second, US is overwhelmingly larger than Canada, with an economy more than 10 times the size of Canadian economy. The cross-country consumption correlation puzzle also exists in this group, the correlation coefficient between the changes in Canada s and the U.S. annual real per capita output is 0.84, while the corresponding consumption correlation is only The FI-RE model predicts that the consumption correlation should be approximately the same as the output correlation, A small deviation from the FI-RE case, assuming the representative consumer in the U.S. has limited capacity and chooses to remove 80% of the uncertainty upon receiving the new signal, the elastic attention model predicts a 0.52 consumption correlation, which is close to the data (0.54). The number is lowered through two channels. One is due to the information noise that is endogenously induced by inattention, which introduces heterogeneity to the two representative agents. For the Canadian and the US citizen, the information noises have zero covariance but increases the variance of their own consumption innovation, therefore decrease the correlation. 12

14 The other is due to the different level of uncertainty in the two countries. Variance of annual change of output in the U.S. is 1.4 times of that of Canada, see Figure 4. As a result, the attention level would be chosen to be different. If we assume the attention level for the U.S. is 0.8, given the same information cost, attention level for Canada would be 0.75, which further lowers the consumption correlation. 4.5 Hong Kong-China Case Similarly for China and Hong Kong, over 47% of exports and 50% of imports of Hong Kong was traded with China in PPP converted Real GDP of Hong Kong is 2.7% of that of China in In this pair of relationship, the small open economy, Hong Kong, is more volatile in terms of variation of annual output change. Our model also improves the prediction for the consumption correlation between the two. If we set θ = 0.7, the model predicts corr( c, c ) = 0.24, which matches the the empirical evidence perfectly. 5 Generalization to an Economy with a Continuum of Consumers Suppose the economy has a continuum of identical consumers instead of a single consumer, we now need to discuss the aggregation problem. Sims(2003) argues that a considerable part of the idiosyncratic responses is common across individuals despite of the heterogeneity of information noises induced by each individual s own inattention. Aggregating across all individual consumers facing the same aggregate income process using (26) yields the expression of the change in aggregate consumption: [ ( )] θζ t c t = (R 1) 1 (1 θ)r L + θ θrξ ξ t t 1, (30) 1 (1 θ)r L where i denotes a particular individual, E i [ ] is the population average, and ξ t = E i [ξ t ] is the common noise. Assume that ξ t consists of two independent noises: ξ t = ξ t + ξ i t, where ξ t = E i [ξ t ] and ξ i t are the common and idiosyncratic components of the error generated by ζ t, respectively. Define a single parameter, µ var ( ξ t ) var (ξ t ) [0, 1], to measure the common source of coded information on the aggregate component (or the relative importance of ξ t vs. ξ t ). 15 Idiosyncratic noises are cancelled out after aggregation while the 15 In a recent paper, Angeletos and La O (2009) show how dispersed information about the aggregate productivity 13

15 common noise remains. 16 The following proposition summarizes how the aggregation factor affects cross-country consumption correlation: Proposition 2 Given µ, the cross-country consumption correlation can be written as where corr( c, c ) Ξ corr( y, y ), (31) 1 Ξ = [1 (1 θ)(1 θ )R 2 ]σ(θ, µ)σ(θ, µ), { 1 σ(θ, µ) = 1 [(1 θ)r] µ2 (1 (1 θ) R 2 )θ 1 1 [(1 θ)r] 2 { σ(θ 1, µ) = 1 [(1 θ )R] 2 + µ2 Proof. See Appendix 7.2. }, 1 (1 (1 θ ) R 2 )θ 1 1 [(1 θ )R] 2 Expression (31) shows that the higher the value of µ (i.e., the common noise is more important), the higher the variance of consumption growth, and the lower the international consumption correlation. Figure (5) and (6) shows that the consumption correlation is increasing with µ for any given values of θ. }. 5.1 A Special Case: No common Noise (µ = 0) Suppose all noises are idiosyncratic (µ = 0), we have the following expression for the change in aggregate consumption c RI Similarly, we can get the ROW consumption evolution c RI ζ t t = θ(r 1) 1 (1 θ)r L, (32) ζ t t = θ (R 1) 1 (1 θ )R L. (33) The consumption correlation between the two countries becomes corr( c RI t, c RI t ) Π corr( y t, yt ), (34) shock contributes to significant noise in the business cycle and helps explain cyclical variations in Solow residuals and labor wedges. 16 Black (2010) also argues that an idiosyncratic shock along a given dimension (for different types of agents) might not be independent from agent to agent and it can have a substantial aggregate effect. See Part III of Black (2010) for a detailed discussion on a topic on the law of large numbers. 14

16 with (1 (1 θ) Π = 2 R 2 )(1 (1 θ ) 2 R 2 ) 1 (1 θ)(1 θ )R 2 1. (35) Since µ = 0, the endogenous noise part disappears. The heterogeneity across countries introduced by elastic capacity depends on the difference, θ θ. When θ = θ, Π = 1, which has the same predictions as the standard FI-RE model. becomes smaller relative to income correlation. As θ and θ drift apart, consumption correlation 5.2 Consumption Correlation under RI Comparing the implications on consumption correlation obtained in the representative agent case, the model with a continuum of consumers (µ (0, 1)), and the no-common-noise case (µ = 0), we have Ξ [Π, Π], where θθ Π = (1 (1 θ)r 2 )(1 (1 θ )R 2 ) 1 (1 θ)(1 θ )R 2, (1 (1 θ) Π = 2 R 2 )(1 (1 θ ) 2 R 2 ) 1 (1 θ)(1 θ )R 2. We proceed to vary the two parameters µ and θ to show the implications from different models quantitatively. We have two interesting findings. First, given θ θ, consumption correlation is decreasing with µ. A high µ means more common noises, and the noise channel plays a major role in reducing the correlation. Second, given µ, corr( c, c ) is increasing in θ, which is the same as in the representative agent model. As we can see from Table 8, our model can better fit the data in many combinations of the two parameter values. For example, for France, when θ = 0.8 and µ = 0.8 (i.e., 80% of uncertainty is removed upon new signals and 80% of the noise information is remained), the RI model predicts that corr( c, c ) = 0.42, which is very close the empirical counterpart, Implications for Other Stochastic Properties of Consumption Given the exogenous income process and the consumption rule, we can readily obtain other key stochastic properties of consumption under elastic capacity model. The following proposition summarizes the implications of elastic capacity for the relative volatility, persistence, and correlation with output of consumption in the home country: Proposition 3 Under RI, the relative volatility of consumption change to income change (i.e., the excess smoothness ratio) can be written as: rv = sd( c) sd( y) = θ 2 1 [(1 θ)r] 2 + µ2 15 { θ 1 (1 θ) R 2 θ 2 } 1 [(1 θ)r] 2, (36)

17 the first-order autocorrelation of consumption change is ρ c (1) = (1 µ 2 )(1 θ)r { (1 (1 θ) 2 R 2 1 )) + µ 2 1 [(1 θ)r] 2 1 (1 (1 θ)r 2 )θ 1 1 [(1 θ)r] 2 }, (37) and the contemporaneous correlation between consumption change and income change is corr( c, y) = Proof. See Appendix 7.3. { 1 + µ [(1 θ)r] 2 (1 (1 θ)r 2 )θ [(1 θ)r] 2 }. (38) As we can see from Table 9, for most of the combinations, our model fits data better. Compared to the benchmark model, we can have a positive first-order consumption autocorrelation and relative volatility of consumption to income less than 1. When µ is sufficiently high, the consumption correlation is decreasing in θ. That is when the noise channel (the presence of ξ t ) dominates the slow propagation channel (the 1 (1 θ)r L term). Volatility of consumption is decreasing in θ due to less induced noises. When µ is not sufficiently high, rv is increasing in θ since slow propagation channel takes control and increase volatility as θ goes up. Note that in the representative agent model, the excess smoothness ratio is θ 1. Imperfectly observing the state reduce the ability to smooth consumption thus 1 (1 θ)r 2 result in excess volatility of consumption. On the other hand, rv = θ 2 1 when µ = 0. 1 (1 θ) 2 R [ 2 Therefore, given θ, rv θ 2, θ ]. For example, if θ = 40% and µ = 0.1, the 1 [(1 θ)r] 2 1 (1 θ)r 2 model predicts that rv = 0.52, which is close to its empirical counterpart in the U.S. data (around 0.54). Given a fixed µ, the autocorrelation of consumption growth is decreasing with θ since the response of consumption to the noise has a negative relationship with consumption growth over time. The consumption-income correlation is increasing in θ given a fixed µ. A reduction in µ leads to higher autocorrelation and higher consumption-income correlation. The intuition is that more idiosyncratic noises are cancelled out and the noise channel reduces the variance of consumption growth by a smaller amount while the covariance between consumption growth and income growth remains the same. 6 Conclusion We have examined how introducing rational inattention into an otherwise standard small open economy model changes the international consumption-income correlations and the the joint dynamics of consumption and income. We have shown that a rational inattention model with agents 16

18 whose attention is elastic to exogenous income processes (Sims, 2010) has the potential to better account for the international diversification and consumption correlations puzzles (Backus, Kehoe, and Kydland, 1992). Specifically, we have found that elastic attention due to fixed informationprocessing cost can produce consumption correlations lower than income correlations, which makes the model fit the data better. 7 Appendix 7.1 Deriving International Consumption Correlations under RI TBA 7.2 Deriving International Consumption Correlations under RI in a Model TBA with a Continuum of Consumers 7.3 Deriving Other Stochastic Properties of Consumption under RI TBA References [1] Adam, Klaus (2007), Optimal Monetary Policy with Imperfect Common Knowledge, Journal of Monetary Economics 54(2), [2] Angeletos, George-Marios and Jennifer La O (2010), Noisy Business Cycles, NBER Macroeconomics Annual 2009 Vol. 24, [3] Backus, David K., Patrick J. Kehoe, and Finn E. Kydland (1992), International Real Business Cycles, Journal of Political Economy 84(1), [4] Baxter, Marianne and Mario J. Crucini (1993), Explaining Saving-Investment Correlations, American Economic Review 83(3), [5] Baxter, Marianne and Urban J. Jermann (1997), The International Diversification Puzzle Is Worse Than You Think, American Economic Review 87(1), [6] Black, Fischer (2010), Exploring General Equilibrium. The MIT Press. 17

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22 θ R=1.02 R=1.03 R= ω x 10 4 ζ Figure 1: Effect of Fundamental Uncertainty on the Kalman Gain 21

23 θ =0.5 θ =0.8 θ = Figure 2: Impulse Responses of Consumption to Income Shock θ =0.9 θ =0.6 θ = Π θ Figure 3: Representative Agent Model Implications 22

24 $ Canada y United States y Year Figure 4: Fundamental Uncertainty of Canada and US Π µ=1 µ=0.5 µ= θ Figure 5: Effects of the Common Noise (θ = 0.9) 23

25 Π θ µ=1 µ=0.5 µ=0 Figure 6: Effects of the Common Noise (θ = 0.5) 24

26 Table 1: The Puzzle corr ( y, y ) corr ( c, c ) Canada France UK Italy Japan Germany Canada-US G7 average * The numbers corr ( y, y ) and corr ( c, c ) are the simple correlation coefficients between the annual change a country s real per capita output (or consumption) and the annual change of the rest of the world s real per capita consumption (or output), with the world defined as the output-weighted average of the rest G7 countries in the Penn World Table (version 7.1). Canada-US correlations are between Canada and US. Average correlations are averages of individual country correlations.

27 Table 2: Summary of statistics corr ( y, y ) corr ( c, c ) sd( c)/sd( y) corr( c, y)) autocorr( c) Canada (0.06) (0.09) (0.05) (0.06) (0.09) Italy (0.17) (0.16) (0.09) (0.03) (0.10) UK (0.11) (0.10) (0.05) (0.02) (0.13) France (0.15) (0.12) (0.06) (0.05) (0.10) US (0.04) (0.03) (0.10) Hong Kong (0.04) (0.05) (0.14) Canada-US (0.04) (0.09) HK-China (0.20) (0.17) Table 3: Summary of Data and model predictions Data (corr ( y, y )) Data(corr ( c, c )) RE(corr ( c, c )) RI(corr ( c, c )) Canada Italy UK France Canada-US HK-China

28 Table 4: Comparing consumption correlations from data and Rational Inattention model PWT7.1 corr ( y, y ) D corr ( c, c ) D RI(θ = 0.9) θ = 0.8 θ = 0.7 θ = 0.6 θ = 0.5 Canada Italy UK France Canada-US HK-China Table 5: Comparing consumption correlations from data and Rational Inattention model PWT8.0 RGDPna corr ( y, y ) D corr ( c, c ) D RI(θ = 0.9) θ = 0.8 θ = 0.7 θ = 0.6 θ = 0.5 Canada Italy UK France Canada-US Table 6: Comparing consumption correlations from data and Rational Inattention model PWT8.0 RGDPe corr ( y, y ) D corr ( c, c ) D RI(θ = 0.9) θ = 0.8 θ = 0.7 θ = 0.6 θ = 0.5 Canada Italy UK France Canada-US

29 Table 7: Comparing consumption correlations from data and Rational Inattention model PWT8.0 RGDPo corr ( y, y ) D corr ( c, c ) D RI(θ = 0.9) θ = 0.8 θ = 0.7 θ = 0.6 θ = 0.5 Canada Italy UK France Canada-US

30 Table 8: Theoretical corr ( c, c ) from different models Data RE RI RI (θ = 0.8) (θ = 0.5) Canada (µ = 1) (µ = 0.9) (µ = 0.8) (µ = 0.5) (µ = 0.1) Italy (µ = 1) (µ = 0.9) (µ = 0.8) (µ = 0.5) (µ = 0.1) UK (µ = 1) (µ = 0.9) (µ = 0.8) (µ = 0.5) (µ = 0.1) France (µ = 1) (µ = 0.9) (µ = 0.8) (µ = 0.5) (µ = 0.1) Canada-US (µ = 1) (µ = 0.9) (µ = 0.8) (µ = 0.5) (µ = 0.1) Hong Kong-China (µ = 1) (µ = 0.9) (µ = 0.8) (µ = 0.5) (µ = 0.1)

31 Table 9: Comparing consumption moments from different models Data RE RI(µ = 1) RI(µ = 0.5) RI(µ = 0.1) θ = 0.8 θ = 0.5 θ = 0.8 θ = 0.5 θ = 0.8 θ = 0.5 Canada sd( c)/sd( y) autocorr( c) corr( c, y) Italy sd( c)/sd( y) autocorr( c) corr( c, y) UK sd( c)/sd( y) autocorr( c) corr( c, y) France sd( c)/sd( y) autocorr( c) corr( c, y) Canada (as in Canada-US) sd( c)/sd( y) autocorr( c) corr( c, y) HK sd( c)/sd( y) autocorr( c) corr( c, y)