Risk, Returns, and Multinational Production Chapter 2

3.4 Earnings-to-Price Ratios and Returns The solution of the model delivers quasi-closed form solutions (up to multiplicative parameters) for the value functions VS (a, Q, Q∗ ) (S ∈ {D, X, I}), and allows us to compute easily the earnings-to-price ratios and returns generated by the model. Our earnings yields measure in the model is given by the ratio πt /Vt , where πt represents per-period profits and Vt is the market value of the firm. In a static model, or in a dynamic but deterministic model, or in a dynamic model with non-persistent shocks, πt /Vt is constant and independent on the firm’s status, since per-period profits and value of the firm are proportional. Dynamics and uncertainty introduce a wedge between these two magnitudes, which reflects the option value. Let epS (a, Q, Q∗ ) denote the earnings yields of a firm with productivity 1/a in status S when the realization of aggregate demand is (Q, Q∗ ). Earnings yields in the model are given by: epD (a, Q, Q∗ ) = epX (a, Q, Q∗ ) = epI (a, Q, Q∗ ) = πD (a, Q) VD (a, Q, Q∗ ) πD (a, Q) + πX (a, Q∗ ) VX (a, Q, Q∗ ) πD (a, Q) + πI (a, Q∗ ) . VI (a, Q, Q∗ ) (29) (30) (31) The empirical evidence presented in Section 2 suggests the following ordering in aggregate earnings yields across groups: Z ωD (Q∗ ) epD (a, Q, Q∗ )dG(a) < Z ωX (Q∗ ) epX (a, Q, Q∗ )dG(a) < Z ωI (Q∗ ) epI (a, Q, Q∗ )dG(a). (32) While is not possible to prove analytically that the model generates this ordering, the results of our numerical simulations confirm that the calibrated model is consistent with it, and that the ordering is robust to a number of variations in the calibration. Returns in the model are given by the earnings yields plus the expected change in the valuation of the firm: retS (a, Q, Q∗ ) = epS (a, Q, Q∗ ) + E[dVS (a, Q, Q∗ )] , f or S ∈ {D, X, I}. VS (a, Q, Q∗ ) (33) Also in this case, the model does not have clear-cut analytical predictions for the ordering 22 of E[dVS (a,Q,Q∗ )] VS (a,Q,Q∗ ) . The value of this object depends on the curvature of the value functions and it also critically depends on the calibration, since different firms exhibit different value functions and respond differently to the same realizations of the shocks. For the model to reproduce the ordering found in the data: Z ωD (Q∗ ) retD (a, Q, Q∗ )dG(a) < Z ωX we need the differences in (Q∗ ) E[dVS (a,Q,Q∗ )] VS (a,Q,Q∗ ) retX (a, Q, Q∗ )dG(a) < Z ωI (Q∗ ) retI (a, Q, Q∗ )dG(a) (34) not to overturn the ordering of the earnings yields, which is what our baseline calibration delivers. The robustness exercises in Section 6 highlight the sensitivity of this component of the returns to selected parameters of the model. The results of the calibrated model are presented in Section 5. Before moving to the quantitative results, in the next section we show a series of qualitative properties of the model that illustrate its amenability to reproduce features of the trade dynamics data. 4 Qualitative Properties of the Solution In this section we illustrate the workings of the model with a series of analytical properties of the solution. These properties highlight the potential of the model to account for export and FDI dynamics of heterogeneous firms, and the departure from the deterministic model which allows to match the ranking in the financial variables. 4.1 Ordering of the Quantity Thresholds The relationship between the sunk costs of exporting and FDI, FI > FX , implies a precise ordering of the quantity thresholds that are solution of (20)-(26). Theorem 1. If FI > FX , fI = fX , and τ w = w∗ , the quantity thresholds Q∗RS (a), for R, S ∈ {D, X, I} and for a given productivity level 1/a, satisfy the following ordering: Q∗IX (a) < Q∗ID (a) < Q∗XD (a) < Q∗DX (a) < Q∗DI (a) < Q∗XI (a). (35) Proof: See Appendix B. The ordering in (35) is easy to prove in the extreme case in which operating costs are equal across modes of serving the foreign market. However, the ordering holds in 23 our numerical results for a wide range of parameterizations. As long as the differences in operating costs of exports versus FDI are “not too large” the thresholds will be ordered according to (35). The intuition behind this property of the model is straightforward. Like in Dixit (1989), the pure presence of sunk costs implies that entry thresholds are higher than exit thresholds: Q∗DX > Q∗XD , Q∗DI > Q∗ID , and Q∗XI > Q∗IX . A higher quantity demanded Q∗ is needed to induce a firm to pay the sunk entry cost of FDI with respect to the quantity necessary to induce the firm to export: Q∗DI > Q∗DX . An even larger positive shock is needed to induce and exporter to become a multinational, since it is already serving the foreign market with exports and already paid a sunk cost: Q∗XI > Q∗DI . Similarly when considering exit, a larger negative shock is needed to induce a multinational to exit the foreign market with respect to the shock needed to induce an exporter to exit: Q∗ID < Q∗XD . Finally, an even larger negative shock is needed to induce a multinational to divest but still serve the foreign market as an exporter: Q∗IX < Q∗ID . The ordering of expression (1) applies for a given productivity level 1/a. The following subsection illustrates the behavior of the value functions and quantity thresholds for different values of firm-level productivity. 4.2 Comparative Statics: Value and Productivity System (20)-(26) makes clear that both the quantity thresholds and the parameters of the value functions depend on the productivity level 1/a. Figure 2 shows the value function of a domestic firm as a function of the aggregate quantity demanded in the foreign market Q∗ and of productivity 1/a. VD is increasing in Q∗ , as the option value of entering the foreign market is increasing in the quantity demanded. VD is also increasing in firm’s productivity, as more productive firms can get higher profits from entering the foreign market. Figure 3 shows the value functions of an exporter and of a multinational firm as functions of Q∗ and 1/a. VX and VI are U-shaped functions of Q∗ , indicating the high option value of exiting for low realizations of Q∗ and the high option value of not changing status for high realizations of Q∗ .29 The behavior of the value functions for Q∗ → 0 does not vary much across the productivity dimension: when Q∗ is low, the value is high as firms of 29 Notice that for Q∗ → ∞, the value function of an exporter is steeper than the one of a multinational, because the exporter gets high value both from staying in the market as an exporter and from the option value of becoming a multinational. 24 V D 600 400 200 0 0 10 2 8 4 6 6 4 8 2 10 * Q 12 0 1/a Figure 2: Value function of a domestic firm. all productivity levels associate a high value to the option of exiting. Conversely, the behavior of the value functions when Q∗ is “large” varies with individual productivity: the value function is steeper for higher productivity firms, indicating that more productive firms obtain higher returns from staying in the foreign market when the realized aggregate demand is high. From Figure 3, the qualitative behavior of VX and VI appears very similar. Figure 4 plots the difference between the value functions of firms serving the foreign market and firms selling only domestically, VX − VD and VI − VD . For each productivity level 1/a, each plot has two stationary points, a local maximum and a local minimum. The value matching and smooth pasting conditions imply that the local maxima correspond to the “entry” thresholds (Q∗DX and Q∗DI in the left and right plot respectively), while the local minima correspond to the “exit” thresholds (Q∗XD and Q∗ID ). The picture shows that both entry and exit thresholds are decreasing in 1/a, indicating that more productive firms enter the foreign market for lower realizations of aggregate demand Q∗ with respect to less productive firms. Similarly, more productive firms need larger negative shocks to demand to be induced to exit the foreign market with respect to less productive firms. Notice that for Q∗ → 0, VX − VD and VI − VD tend to infinity, because the option value of exiting the 25 VX VI 500 500 0 0 0 0 10 5 5 5 10 5 10 1/a 0 Q* 10 1/a 0 Q* Figure 3: Value functions of an exporter and of a multinational firm. V −V X V −V D I 20 20 10 10 0 0 10 −10 −20 0 D 5 5 10 0 10 −10 −20 0 1/a * 5 5 10 0 1/a * Q Q Figure 4: Difference between the value functions of exporters and multinationals and the value function of domestic firms. 26 foreign market is extremely high for very low realizations of Q∗ (and irrespective of firm’s productivity). Conversely, for Q∗ → ∞, VX − VD and VI − VD tend to negative infinity, because the domestic firms’ option value of entering the foreign market is extremely high, compared to the flow profits of staying for firms that are already serving that market. The difference between the value functions of a multinational firm and of an exporter displays similar properties. Figure 4 suggests a systematic relationship between the quantity thresholds Q∗RS and the firm productivity level 1/a. Theorem 2 establishes this result. Theorem 2. ∂Q∗RS (a) > 0, f or R, S ∈ {D, X, I}, ∀a. ∂a (36) Proof: See Appendix B. Theorem 2 establishes that the six thresholds Q∗RS (a) are decreasing in productivity 1/a, indicating that more productive firms need smaller positive shocks to demand to enter the foreign market, and larger negative shocks to exit. The one-to-one correspondence between productivities and quantity thresholds established by Theorem 2 implies that the functions Q∗RS (a) are invertible, hence for each realization of aggregate foreign demand Q∗ we can compute six productivity thresholds aRS (Q∗ ), for R, S ∈ {D, X, I}, that determine the selection of heterogeneous firms into the three statuses and their likelihood of switching across statuses. This redefinition of the thresholds in terms of productivity is extremely helpful to compute the model numerically. The shares of firms belonging to each status can be written as functions of the productivity thresholds aRS , so the law of motion of the status distribution is given by: ωDt+1 = ωDt · [1 − G(aDX )] + ωXt · [1 − G(aXD )] + ωIt · [G(aIX ) − G(aID )] (37) ωXt+1 = ωDt · [G(aDX ) − G(aDI )] + ωXt · [G(aXD ) − G(aXI )] + ωIt · [1 − G(aIX )] (38) ωIt+1 = ωDt · G(aDI ) + ωXt · G(aXI ) + ωIt · G(aID ) (39) where we omitted the dependence of ωSt and aRS on Q∗ to ease the notation. For a given initial distribution Ω0 , equations (37) - (39) describe its evolution depending on the productivity thresholds ruling firms’ allocation into statuses. Notice that the sets ωSt vary with the realization of Q∗ , as firms may switch status, but only depend on the firms’ status in the previous period, due to the Markov property of Brownian motions. 27 10 QDX QXD 9 QXI QIX 8 QDI QID 7 1/a 6 5 4 3 2 1 0 2 4 6 8 10 Q 12 14 16 18 20 Figure 5: Quantity thresholds as functions of firm’s productivity. Figure 5 illustrates Theorem 2. For an arbitrary parametrization, we plot the six quantity thresholds as functions of firm-level productivity. The picture also shows an additional property of the thresholds: hysteresis, defined as the horizontal distance between entry and exit thresholds, is also decreasing in productivity, indicating that – for the same choice of status – more productive firms suffer less from being locked into a market by the presence of sunk entry costs, and exhibit less hysteresis than less productive firms. On the other hand, more productive firms self-select into the status (I) which is associated with more hysteresis.30 This ambiguous result generates an imperfect sorting of productivities into status, which is a well-documented feature of the data. 5 Empirical Results The objective of this section is an evaluation of the model performance in matching qualitatively and quantitatively features of the data on trade and FDI dynamics, and the pattern of earnings yields and returns across firms. In Section 5.1 we calibrate the parameters of the model to quantitatively match the switching pattern and the relative presence of the 30 Dixit and Pindyck (1994) show that hysteresis is increasing in the sunk cost of investment: entry (exit) thresholds are increasing (decreasing) in the sunk cost, hence the difference between entry and exit thresholds is also increasing in the sunk cost. 28 three types of firms in the data. In Section 5.2 we use the calibrated version of the model to compute earnings yields and returns, which are not targeted moments in the calibration. We show that the quantitative theory is able to replicate the observed ranking in financial variables that we documented in Section 2. 5.1 Calibration The calibration exercise that we present in this section is designed to match a series of facts on exports and FDI sales dynamics. We do not make any use of financial variables in this exercise. In the next section we show that the model calibrated by targeting trade facts only performs well also in matching non-targeted moments like earnings yields and returns. We present a bilateral calibration exercise, that describes export and FDI activity between the U.S. and an aggregate set of trading partners. Due to data availability, we impose a series of symmetry assumptions. In particular, we assume that preferences and productivity distributions are identical in the U.S. and in the other countries, and that the cost structure (the values of τ , fX , FX , fI , FI ) is also the same across countries.31 To calibrate the model, we need to choose a functional form for the cost distribution G(a), and assign values to its parameters. We need to parameterize the Brownian motions, and choose values for preference parameters and parameters describing trade and FDI costs. We refer to the literature to assign parameters to the preferences and to the firms’ productivity distributions. The parameters ruling the Brownian motions are chosen to match data on standard deviations and correlation of GDP growth between the U.S. and its major trading partners. We choose the remaining parameters so that the model matches a series of moments computed from the data. We start describing the calibration with the parameters we adopt from the literature. Several studies document that the tail of the empirical firm size distribution is well approximated by a Pareto distribution (see for example Luttmer (2007)). Since firm size (sales) is linked to the productivity distribution in the model, we assume that firms’ productivities 1/a are distributed according to a Pareto law with location parameter b and shape parameter k.32 We normalize b = 1, and choose k to match the coefficient of the empirical 31 Compustat records data of firms with activities in the U.S., among which there are both U.S. based firms and foreign firms. However, only data of foreign firms with activities in the U.S. are reported (in other words, we have no data about foreign firms with activities only in their domestic market), which implies that we cannot construct shares of foreign firms in each status or their dynamic behavior. 32 The Pareto distribution is also a convenient choice for computational reasons, since it allows to solve 29 sales distribution: if productivity is Pareto-distributed with shape parameter k, sales in the model are also Pareto-distributed with shape parameter k/(η − 1). By regressing firm rank on firm size, Luttmer (2007) finds that k/(η − 1) = 1.06. We then choose k accordingly, given a value for η. There is little agreement in the literature on the value to attribute to the elasticity of substitution across differentiated varieties, η. Many papers that focus on long-run macroeconomic predictions use a standard value of 2. Other papers that focus on matching data at business cycle frequencies choose much higher values. Alessandria and Choi (2007), for example, set η ≈ 10 (to match markups of about 11%). We set η = 2.65, equal to the median value in Broda and Weinstein (2006) SITC 5-digit estimates.33 This choice implies k = 1.749. Finally, we set the risk aversion parameter to γ = 20, as an intermediate value among those proposed by the literature.34 We abstract from labor cost and market size differences, and set both wages to w = w∗ = 1 and the mass of firms in each country to n = n∗ = 1. We impose that the drifts of the Brownian motions ruling the evolution of Q, Q∗ have value µ = µ∗ = 0. The need to impose zero expected demand growth arises from the fact that we abstract from firms’ productivity growth.35 We compute σ and σ ∗ as the standard deviations of consumption growth for the U.S. and for a set of OECD countries: σ = st.dev.(gtus ) = 0.031 σ ∗ = st.dev.(gtoecd ) = 0.023 explicitly for the aggregate prices P , P ∗ as functions of the productivity thresholds aRS and of the other parameters of the model. 33 Broda and Weinstein (2006) estimate sectoral elasticities of substitution from price and volume data on U.S. consumption of imported goods. By using data at the 10-digit Harmonized System, they estimate how much demand shifts between 10-digit varieties when relative prices vary, within each 3-digit SITC sector. 34 Assigning a value to γ is a difficult choice to make in this setting. In their seminal contribution, Mehra and Prescott (1985) report evidence from several micro and macro studies suggesting a value of γ between 1 and 4. They also show that a model with CRRA preferences and such a low value of γ can match the risk-free rate, but generates returns that are too low compared with the data (the equity premium puzzle). Hansen and Singleton (1982) estimate the value of γ that matches returns, obtaining a value of γ ≈ 60. This high value is implausible based on the empirical evidence, and generates too high a risk-free rate. Our model features CRRA preferences for the differentiated good, and is hence subject to the same problem: we are not able to match quantitatively both the risk-free rate and the returns. For this reason, we choose an “intermediate” value in our baseline calibration. The choice of γ = 20 is consistent with empirical estimates of the Sharpe ratio: S ≡ E(ret−r) . It is possible to prove that S ≤ (1 + r)(γσ∆C ). Estimates of the Sharpe σret ratio for the U.S. found S ≈ 0.4. Our chosen value of γ is in line with these estimates, generating a Sharpe ratio of 0.47. 35 If µ > 0, E(dQ/Q) would be increasing over time and for t → ∞ all firms would become multinationals. By setting µ = 0, we are implicitly assuming that Q, Q∗ and b grow at a rate such that the distribution of firms in the three groups does not degenerate over time. 30 where gtoecd (gtus ) is the average growth rate of consumption across OECD countries (in the U.S.) in year t (t=1979-2006).36 We set the risk-free rate in both economies (r and r ∗ ) to match a long-run average of 3-month T-bills rate of 2%. For the calibration of σM and ∗ , notice that CRRA preferences with risk-aversion coefficient γ imply that σ σM M = γσ ∗ = γσ ∗ ).37 Hence σ ∗ (σM M = 0.46 and σM = 0.62. The correlation coefficient between the two Brownian motions, ρ, is a key parameter in this exercise. To select a value for ρ, we computed correlations in GDP growth rates between the U.S. and a set of countries that include the U.S. major trading partners (Canada, Mexico, France, Germany, the U.K., Spain, Japan, Hong Kong, Korea, Malaysia and Taiwan) and the largest developing economies (China, Brazil and India). For the sample period (1979-2006) correlation vary from a minimum of -0.02 (between U.S. and Brazil) to a maximum of 0.786 (between U.S. and Canada). Brazil is the only country in the sample displaying a negative correlation. The mean correlation is 0.19, and the median is 0.12. Based on this numbers, for our baseline calibration we choose a value of ρ = 0.12. It remains to calibrate the variable trade cost τ , fixed operating costs fX , fI , sunk costs FX , FI (that we assume to be equal in the U.S. and in its aggregate trading partners), and the initial values for the aggregate demand levels, Q(0) and Q∗ (0). We follow the methodology of the calibration in Alessandria and Choi (2007), and select values for these parameters to match a set of moments related to trade and FDI dynamics. We target data on firms’ persistence in the same status and on the shares of the three types of firms in the data. We compute all the moments from Compustat data. In terms of status persistence, on average, 93.11% of domestic firms remain domestic the following year, while 3.73% of them become exporters, and the remaining become multinationals. 90.27% of exporters continue exporting the following year, while 5.96% of them become multinationals, and the remaining exit the foreign market to sell domestically only. Multinational firms exhibit even higher persistence, with 98.31% of them continuing being multinationals the following year, and only 0.82% of them becoming exporters the following year. Domestic firms’ and exporters’ persistence moments are close to the ones reported in Alessandria and Choi (2007), but we are unaware of other papers computing moments related to persistence in multinational activity. 36 37 Data source: OECD Statistics. See footnote 16. 31 Table 4: Summary of Calibrated Parameters. Parameter Definition Value Source drift of Q, Q∗ variance of Q variance of Q∗ correlation of Q, Q∗ risk-free rate variance of s.d.f. variance of s.d.f. 0 0.023 0.031 0.12 0.02 0.46 0.62 no productivity growth st.dev. of U.S. cons. growth st.dev. of OECD cons. growth GDP growth correlations 3-month T-bills rate model restriction (σM = γσ) ∗ = γσ ∗ ) model restriction (σM lower bound shape parameter 1 1.749 normalization Luttmer (2007) el. of substitution risk aversion 2.65 20 Broda and Weinstein (2006) Sharpe ratio iceberg export cost fixed export cost fixed FDI cost sunk export cost sunk FDI cost 1.32 100 210 600 1500             initial demand 0.5 Brownian motions µ, µ∗ σ σ∗ ρ r, r ∗ σM ∗ σM Pareto distribution b k Preferences η γ Trade and FDI costs τ fX fI FX FI Aggregate demand Q(0), Q∗ (0) to match data            Next, we look at the average share of firms in each status. In Compustat, the average share of firms selling only domestically is 34.81% of the sample, while exporters are 29.2% of the sample, and multinational firms account for the remaining 35.99% of the sample. As previously noted, these numbers reflect selection of the largest firms in the data set.38 Despite the divergence of our moments with the ones reported in other papers, we decided to match Compustat data in this exercise to be internally consistent. While not representative of the entire population of U.S. firms, Compustat offers a detailed portrait of the largest firms in the economy, which are the major players in determining volumes of trade and 38 Bernard et al. (2007), among others, report that the average share of manufacturing firms that export is about 18%, while Bernard and Jensen (2007) report that multinational firms represent only 1% of manufacturing firms. 32 FDI.39 Table 4 summarizes the calibrated parameters. The calibrated iceberg cost is 32%, consistent with a medium-range estimate in Eaton and Kortum (2002). Sunk costs of export and FDI equal to 600 and 1500, respectively, indicate that a domestic firm must spend on average 3.68 times its per-period revenue to enter the foreign market as an exporter, and about 9.2 times its per-period revenue to start FDI operations there. Fixed costs of export and FDI equal to 100 and 210, respectively, indicate that an exporter must spend on average 39.9% of its per-period revenue in operating costs, and that a multinational firms must spend on average 6.54% of its per-period revenue in operating costs. Aggregate demand parameters are set at Q = Q∗ (0) = 0.5. Table 5: Moments. Comparison of the moments, model versus data. (Source: Compustat). Data Model D → D (%) 93.11 95.06 D → X (%) 3.73 2.68 X → X (%) 90.27 90.91 X → I (%) 5.96 5.31 I → I (%) 98.31 97.74 I → X (%) 0.82 2.26 Data Model X (%) 29.2 26.73 I (%) 35.99 29.74 Table 5 displays jointly the moments computed from the data and the moments generated by the calibrated model. Overall, the model performs well in matching the moments from the data: the calibration error, computed as the sum of squared differences between the model-generated moments and the moments computed from the data, is equal to 0.0053. In particular the persistence parameters are precisely matched, while the shares of exporters and multinational are slightly smaller than in the data. 39 The choice of a Pareto distribution for firms’ productivity is robust to the selection problem associated with dealing with Compustat data. The Pareto distribution is invariant to lower truncations, hence if we assume that the entire productivity distribution is Pareto with parameters (b, k), the distribution of firms in Compustat will also be Pareto, with parameters (b0 , k), b0 > b. As the lower bound of the distribution does not enter the computation of the moments, we normalize it to one. 33 5.2 Quantitative Results: Earnings Yields and Returns With the calibrated version of the model, we compute earnings yields and returns across the three groups of firms. In our calculations, we follow the construction of the portfolios we used in the data analysis presented in Section 2. We generate an artificial dataset of 100 firms with productivities drawn from a Pareto distribution with parameters (b, k) = (1, 1.749), and we simulate a 20-period economy 100 times.40 In each simulation, we initialize the firms’ distribution into status by assuming that all firms start domestic, and we generate a sample process for (Q, Q∗ ). Given the process of the shocks, we simulate the economy, recording the distributions of firms into status in each period. For each firm and period, we compute earnings, equilibrium value (our model-based measure of market capitalization), and variation in the equilibrium value of the firm. For each year we create three portfolios of domestic firms, exporters, and multinationals, and we compute portfolio earnings, prices, value changes, earnings yields (earnings-to-price ratios), and returns (earnings-to-price ratios plus average percentage changes in value). For each simulation, we compute the mean and standard deviation of earning yields and returns over time.41 We repeat this process for the 100 Monte Carlo simulations, and we average the results across simulations. Table 6: Earnings Yields. Summary statistics of earnings yields computed from simulated data, and comparison with real data. All values are in percentage terms. Mean (model) Mean (data) Std. Dev. (data) DOM 0.909 2.12 3.54 EXP 0.978 3.7 2.85 MN 1.217 4.85 2.76 Table 6 reports the results for earnings yields. The model generates average earnings yields of 1.22% for multinational firms, 0.98% for exporters, and 0.91% for firms selling only domestically, which are consistent with the ordering we found in the data, albeit lower in levels. Table 7 reports the results for the returns. The model generates average returns of 1.3% 40 The small number of Monte Carlo simulations is justified by the fact that for µ = 0 there are small differences across simulations. The entire computation of the model for a given parametrization takes about 3 hours on a cluster of 8 CPUs. 41 When averaging over time, we discard the first three periods of each simulation to reduce the importance of the fact that we initialize the firms’ distribution into status at Ω0 = Ω∗0 = (0, 0). 34 Table 7: Returns. Summary statistics of returns computed from simulated data, and comparison with real data. All values are in percentage terms. Mean (model) Mean (data) Std. Dev. (data) DOM 0.916 7.37 15.03 EXP 1.057 10.69 15.34 MN 1.3 11.62 12.86 for multinational firms, 1.057% for exporters, and 0.916% for firms selling only domestically, which are also consistent with the ordering we found in the data. While the model succeeds in matching the order of earnings yields and returns across group, it fails in matching their magnitudes. As we mentioned previously, this comes at no surprise in light of the well known inability of models based on CRRA preferences to match quantitatively stock market excess returns for reasonable values of the parameter ruling risk aversion. Our robustness exercises will shed light on this aspect of the computation by showing counterfactual calculations for different values of γ. 6 Robustness The objective of this section is to illustrate how the numerical results of the model change when letting vary the value of some relevant parameters. In particular, we focus our attention on alternative values of ρ (the correlation between the Brownian motions ruling the evolution of (Q, Q∗ )) and γ (the risk aversion). 6.1 Negative correlation between aggregate shocks across countries A model with stochastic demand in both countries implies that the income of domestic agents depends on both the realization of foreign and domestic demand. Fluctuations in domestic demand Q enter directly into consumers’ utility. Fluctuations in foreign demand Q∗ imply fluctuations in the profit shares that consumers receive from the stocks they hold; hence, they affect their income and their consumption of the homogeneous good H, which also enters the utility function. Fluctuations in Q and Q∗ have similar qualitative effects on domestic consumers’ utility (and symmetrically on foreign consumers). The baseline value of ρ = 0.12 was chosen as the median correlation between the GDP 35 growth of the U.S. and a set of major trading partners and other large countries. The choice of the value of ρ is important because it determines the relative strengths of two channels that operate in the model. On the one hand, selling in foreign markets increases the riskiness of a firm’s profit flows via the combination of aggregate persistent shocks and sunk costs, as we emphasize throughout the paper. On the other hand, one could think of exporters and multinational firms entering foreign markets to hedge against the fluctuations of demand in their home market. If this second channel prevailed, exporters and multinationals should exhibit lower returns in equilibrium, as the international status actually reduces their riskiness. The correlation between the processes for (Q, Q∗ ) is tightly linked to the relative strength of these two channels: a positive (negative) ρ reinforces the first (second) channel. Consistent with the channel emphasized in the paper, the data show that most GDP correlations between the U.S. and its major trading partners are positive and high. However, to illustrate the potential of the hedging mechanism, we present the results of the model for ρ = −0.02, equal to the correlation between U.S. and Brazil (the lowest in the sample used in the calibration). The other parameters are left unchanged with respect to the baseline scenario. The model with negative ρ fits the data almost equally well as the baseline case: the calibration error is 0.0062. The results for earning yields and returns are displayed in Table 8. While the ordering of the earnings yields is preserved and the magnitudes are very similar to the baseline case, the same is not true for the returns. Domestic firms exhibit the highest returns, consistent with the idea that a negative ρ strengthens the hedging motive, and exporters and multinationals are “safer” than firms selling only in their domestic market. However, comparing the returns of exporters and multinationals one can still see the effect of the sunk costs at work: albeit lower than the returns of domestic firms, multinationals exhibit higher returns than exporters, indicating the higher riskiness driven by higher sunk costs. 6.2 Higher Risk Aversion The baseline value of γ = 20 was chosen to be consistent with empirical estimates of the Sharpe ratio within a CRRA framework. As our baseline results show, the calibrated model generates earnings yields and returns that are much lower than the ones observed in the 36 Table 8: Baseline versus Negative Correlation. Summary statistics of earnings yields and returns computed from simulated data, baseline scenario and counterfactual scenario (ρ = −0.02). All values are in percentage terms. Baseline Earnings Yields Returns ρ = −0.02 Earnings Yields Returns DOM 0.909 0.916 0.938 1.789 EXP 0.978 1.057 1.111 0.872 MN 1.217 1.3 1.255 1.295 data. In light of the extensive literature on the equity premium puzzle,42 this hardly comes at a surprise. In this section we present the results of the model for a slightly higher value of γ and highlight the effect of the increase in γ for the returns. We run the simulation for γ = 30.43 The match of the model generated moments with the data is hardly affected by the change in γ: the calibration error is 0.0077. The results for earning yields, value changes and returns are displayed in Table 9. The orders of both earnings yields and returns are not affected by the change in γ. Earnings yields are lower, because a higher γ correspond to a lower discount term. The changes in the value functions (dV/V) are more pronounced than in the baseline scenario: this second effect is driven by the impact of γ on the concavity of the value functions. On aggregate, the level effect on the earnings yields dominates, and returns are lower than in the baseline scenario. This happens because in the calibration we set µ = µ∗ = 0, so that γ has no effect on the risk-adjusted drift, but only on the variance. Simulating the model for a positive value of µ would reduce the level effect on the earnings yields and magnify the concavity effect on dV/V. However, the choice of a positive value of µ would not be appropriate for the calibration, as it would imply computing the model-generated moments as time-series averages of non-stationary data. For this reason we do not follow this direction. Addressing the challenge of replicating earnings yields and returns of magnitudes consistent with the data would require different assumptions about preferences and the growth process, and is beyond the scope of this paper. 42 See Mehra and Prescott (1985) and Hansen and Singleton (1982) among others. The choice of γ for this experiment is constrained by the fact that the discount factors r − (µ + σσM ) and r − (µ∗ + ρσ ∗ σM ) must be positive, and σM = γσ. For this reason we cannot run the exercise for γ as high as – say – in Hansen and Singleton (1982). 43 37