Intangible Capital, Asset Prices, and Business Cycles

Transcription

1 Intangible Capital, Asset Prices, and Business Cycles Ryo Jinnai y Department of Economics, Princeton University Job Market Paper January 25, 2009 Abstract Recent empirical research motivated by the IT revolution has argued that intangible capital is an important missing piece in the economic analysis of asset prices and business cycles. In this paper, I introduce intangible capital in an otherwise standard real business cycle model with recursive utility, and show that the model s performance in matching data substantially improves. Speci cally, the model is consistent with (i) business cycle facts on the volatility of output, consumption, investment, labor hours, and intangible spending as well as their comovement with output at both high- and medium- frequencies, (ii) asset market facts on a high equity premium and a low and stable risk-free rate, and (iii) the joint dynamics of output and asset prices, namely, aggregate rm value leading the business cycle. Introduction Dynamic stochastic general equilibrium models have become the workhorse of modern macroeconomics partly because they perform well in accounting for aggregate quantity uctuations. Their asset pricing implications, however, are notoriously unsuccessful. Two remarkable challenges are the equity premium puzzle and the risk-free rate puzzle: namely, it is di cult to generate a six I would like to thank Nobu Kiyotaki for his advice and encouragement. Ricardo Reis and Chris Sims also deserve special recognition. I would also like to thank Roland Benabou, Saroj Bhattarai, Francesco Bianchi, Toni Braun, Kenichi Fukushima, Lars Hansen, Fumio Hayashi, Jakub Jurek, Jean-Francois Kagy, Per Krusell, Alisdair McKay, Marc Melitz, Tamas Papp, Woong Yong Park, Phillipe-Emmanuel Petalas, Esteban Rossi-Hansberg, Felipe Schwartzman, Satoru Shimizu, Takuo Sugaya, Jade Vichyanond, and Yuichiro Waki for their comments. y Tel: ; rjinnai@princeton.edu; URL:

2 percentage point equity premium and a low and stable risk-free rate as we see in the data. Since counterfactual asset pricing implications cast doubt upon the model s validity, xing the asset pricing implications while keeping the favorable quantity implications is a rst order priority for research. This paper contributes to accomplishing this goal. Speci cally, I propose a general equilibrium model that replicates a high equity premium, a low and stable risk-free rate, and standard business cycle statistics including the volatility of output, consumption, investment, labor hours, and their co-movement with output at both high- and medium- frequencies. In addition, my model accounts for an important empirical link between quantities and prices: stock prices lead the business cycle. Most standard macroeconomic models do not predict these joint dynamics, but rather have output and asset prices moving together. Intangible capital is at the center of my model. It is a production factor that includes knowledge created by research and development, brand names established by advertisement, organizational capital, and so on. Empirical research inspired by the IT revolution has suggested that the stock of intangible capital is large, and has argued that intangible capital is important to understanding asset prices and the aggregate economy. Motivated by the empirical work, I introduce intangible capital in an otherwise standard real business cycle model that includes recursive utility of Epstein and Zin (989). Production factors are labor, physical capital, and intangible capital. Two production technologies are available: one for nal goods production and the other for next period s intangible capital production. Productivities of the two technologies follow di erent stochastic processes. Productivity shifts in the goods producing technology cause high-frequency economic uctuations in my model. The model s high-frequency business cycle implications are consistent with the data because the model shares its core mechanism with a standard real business cycle model. Namely, improvements in the goods producing technology increase consumption, investment, and labor hours together because they increase output, the marginal product of physical capital, and the marginal product of labor at the same time. Examples are Hall (200a), Hall (200b), Bond and Cummins (2000), McGrattan and Prescott (200), Basu, Fernald, Oulton and Srinivasan (2003), Corrado, Hulten and Sichel (2005), Corrado, Hulten and Sichel (2006), and Fukao, Miyagawa, Mukai, Shinoda and Tonogi (2008). According to Corrado et al. (2005), average intangible spending in the U.S. economy is around 0% of GDP. 2

3 Productivity shifts in the intangible capital producing technology cause medium-frequency economic uctuations by a ecting the speed of intangible capital accumulation, which is the engine of economic growth in my economy. The model s medium-frequency business cycle is not only in line with the data, but is crucial for this paper s asset pricing implications. Improvements in the intangible capital producing technology immediately increase the intangible capital s price because today s intangible capital is an essential input to produce tomorrow s intangible capital. But the nal goods production increases sluggishly because the full impact of the technological improvements is realized only after the intangible capital is well accumulated. Because of this di erence in timing of the responses, the aggregate capital value leads the business cycle. Returns on intangible capital also lead the business cycle because the intangible capital s price leads the business cycle. At the same time, the representative consumer with recursive utility cares about his future prospect when he evaluates asset returns. Therefore, the representative consumer perceives intangible capital as a particularly risky asset since it pays a poor return when the economy falls in a prolonged low growth period. As a result, intangible capital has to pay a high risk premium in equilibrium, which leads to a high equity premium. Finally, the risk-free rate is low because of a self-insurance saving motive. The representative consumer wants to hold the risk-free bond despite its low return because he needs savings to selfinsure against negative productivity shocks, especially the one in the intangible capital producing technology. Related Papers Tallarini (2000) is an early study that explores asset pricing implications in a general equilibrium model. He introduces recursive utility in a prototypical real business cycle model, and raises risk aversion while keeping the intertemporal elasticity of substitution constant. High risk aversion improves asset pricing implications while keeping business cycle moments almost unchanged, but the improvement on the equity premium is not big enough to match the observed equity premium. His study shows that generating a high equity premium in a general equilibrium model is even more elusive than in an endowment economy because optimizing agents use many margins to self-insure 3

4 against productivity shocks. 2 Bansal and Yaron (2004) demonstrate that persistent growth rate shocks in consumption and dividend processes in conjunction with recursive utility have strong asset pricing implications. This long-run risk is so powerful that even a reasonably small volatility in growth rate processes can generate a high equity premium. Bansal and Yaron (2004), however, study a partial equilibrium model in which consumption and dividends are exogenously given. A contribution of my paper is to o er a general equilibrium framework in which the optimizing agent faces endogenously arising long-run risk, which in my model is the medium-term economic uctuation. Comin and Gertler (2006) are the intellectual antecedents of medium-term business cycles. The authors develop a general equilibrium model that explains both high- and medium- frequency variation in the data. However, they concentrate only on the quantity implications. Finally, McGrattan and Prescott (2007) study a general equilibrium model with intangible capital. Assuming perfect foresight, the authors show that their model can explain seemingly puzzling observations in the U.S. economy in the 990s including low measured productivity, high aggregate stock value, and high labor hours. McGrattan and Prescott (2007), however, focuses on the 990s and do not address the role of intangible capital in accounting for quantity and asset market data over a longer time span. 2 Model The economy consists of a representative consumer, a unit mass of atomistic producers, and a government. All the production factors are owned by the consumer. 3 The nal good is numeraire. 2 Other recent studies that introduces recursive utility in a general equilibrium framework are Kaltenbrunner and Lochstoer (2008), Croce (2008), and Pananikolaou (2008). Kaltenbrunner and Lochstoer (2008) generate a high equity premium in an economy with low intertemporal elasticity of substitution, strong investment adjustment cost, and constant labor hours, which is e ectively very similar to the model of Jermann (998), who has low intertemporal elasticity of substitution with habit formation preference; Kaltenbrunner and Lochstoer (2008) also has a counterfactually volatile risk-free rate as Jermann (998) does. Pananikolaou (2008) studies a two-sector model similar to Boldrin, Christiano and Fisher (200), but introduces investment speci c technological change like Greenwood, Hercowitz and Krusell (2000). An important di erence between his model and my model is the timing of the asset price movement: in Pananikolaou (2008), a stock market crash caused by a positive investment speci c productivity shock predicts future economic boom, while in my model, a stock market boom caused by intangible capital speci c productivity shock predicts future economic boom, which is more in line with the data as I show in this paper. 3 Assuming that the producers own intangible capital does not change the results. 4

5 2. Producer A producer of index! 2 [0; ] maximizes pro t max fy t (!) + q N;t n t+ (!) [W t (l Y;t (!) + l N;t (!)) + r K;t k t (!) + (p N;t t ) n t (!)]g l Y;t (!);l N;t (!);k t(!);n t(!) subject to y t (!) = A t n t (!) N k t (!) K l Y;t (!) N K () n t+ (!) = S t N t n t (!) l N;t (!) (2) The producer purchases labor service (l Y;t (!) + l N;t (!)) at wage W t, rents physical capital k t (!) at rental price r K;t, and purchases intangible capital n t (!) at subsidized price (p N;t t ), where p N;t is market price and t is a subsidy from the government. Using these factors, he produces nal goods y t (!) and intangible capital n t+ (!) ; where the latter is usable only in the next period. Equation () is the goods producing technology, where A t is the productivity level in the goods producing technology. Equation (2) is the intangible capital producing technology that combines maintenance and creation of intangible capital, where S t is the productivity level in the intangible capital producing technology and N t is aggregate intangible capital stock. In equilibrium, N t = Z 0 n t (!) d!: Because each atomistic producer takes N t as exogenous, the intangible capital producing technology is constant returns to scale from the individual producer s point of view. The producer sells y t (!) at price in the goods market and sells n t+ (!) at price q N;t in the asset market. The production functions () and (2) capture intangible capital s three distinctive features. First, the aggregate intangible capital stock has positive externality in intangible capital production as Romer (990) emphasizes in the context of knowledge creation. For example, an innovative design of a product inspires product designers in other rms. Second, an individual producer needs his private intangible capital input when he produces new intangible capital. For example, Apple computer s private knowledge about the ipod is essential 5

6 when the company develops a new product, the iphone. 4 Because the exponent of N t and the exponent of n t (!) sum to one in equation (2), this economy has endogenous growth. Because the private contribution is crucial for this paper s asset pricing implications, I demonstrate its importance in section 5.2 by showing results in an alternative model in which the economy grows only with pure externality as in Romer (990). Third, private intangible capital input can be used in the two sectors at the same time as in McGrattan and Prescott (2007). For example, a patent of a medicine is useful for both producing the medicine and conducting new research related to the medicine. 2.2 Consumer The representative consumer maximizes the utility function 5 U t = 2 4( ) [C t ( L t ) ' ] + Et hu t+ i 3 5 subject to C t + I t + q N;t N t+ = W t L t + r K;t K t + p N;t N t T t (3) K t+ = ( It ) K t + K t : (4) K t E t is an operator taking expectation conditional on the period t information set. Notice that if the relative risk aversion equals the inverse of the intertemporal elasticity of substitution =, the recursive utility function reduces to the standard CRRA utility function. The separation between risk aversion and intertemporal substitution is a virtue of recursive utility. Equation (3) is the ow budget constraint. The consumer earns income by selling labor service L t at the wage W t, renting out physical capital K t at the rental price r K;t, and selling intangible capital N t at the market price p N;t. T t is a lump-sum tax. The consumer spends the income on consumption C t, physical capital investment I t, and purchases of the next period s intangible capital N t+ at the price q N;t. The physical capital stock K t evolves according to a standard 4 Recent empirical studies using product-level micro data also support the importance of private input in product creation. Broda and Weinstein (2007), for example, nd that 92% of product creation occurs within an existing rm. Similarly, Bernard, Redding and Schott (2006) nd that 94% of product addition occurs within an existing facility. 5 I follow Backus, Routledge and Zin (2007) for the functional form. 6

7 accumulation rule (4), where is the physical capital depreciation rate and () is an adjustment cost function. Following Jermann (998), I consider the following adjustment cost; It = It + K t K t : Note that () satis es standard restrictions () = and 0 () = : I de ne q t as q t ; 0 I t K t which is Tobin s q, the ratio of the market value of new additional investment good to its replacement cost. Because q t = = 0 I t K t It K t ; is the elasticity of the investment-capital ratio with respect to Tobin s q. 2.3 Equilibrium I assume that the government keeps a balanced budget every period; Z t n t (!) d! = T t : 0 In equilibrium, the following market clearing conditions hold every period: Y t Z 0 y t (!) d! = C t + I t ; L t = L N;t + L Y;t ; where L Y;t Z 0 l Y;t (!) d!, L N;t Z 0 l N;t (!) d!; K t = N t = Z 0 Z 0 k t (!) d!; n t (!) d!; and N t+ = Z 0 n t+ (!) d!: 7

8 2.4 E cient Subsidy If the government does not use any subsidy, the equilibrium allocation is ine cient because of the externality. Namely, additional intangible capital supply increases social welfare because an increase of the intangible capital stock bene ts all producers intangible capital production. The ine ciency is corrected by subsidizing intangible capital usage. The optimal subsidy level is t = q N;t N t+ N t. To serve as a benchmark and also for computational tractability, I assume that the government adopts the optimal subsidy. Knowing that the equilibrium is e cient, I can nd the allocation by numerically solving a planner s problem. The formal planner s problem and the solution method are discussed in the appendix. To investigate the extent to which the results depend on the subsidy, I study the economy without subsidy in section Calibration Parameter values are summarized in Table. The time unit is a quarter of a year. Five parameters (; ; ; ; ) are chosen directly either at a standard value or based on empirical evidence. The discount rate is set at = :995. The physical capital depreciation rate is set at = :05. Following Bansal and Yaron (2004), I set the intertemporal elasticity of substitution at = :5 and the coe cient of risk aversion at = 0. = :5 is consistent with a number of empirical studies including Hansen and Singleton (982) and Vissing-Jorgensen (2002). = 0 is the upper bound of what Mehra and Prescott (985) consider as a reasonable parameter range. The elasticity of the investment-capital ratio with respect to Tobin s q is set at = 8; which is consistent with a recent empirical study of Hall (2004). For the productivity processes, I make the following parametric assumptions: log A t follows a stochastic process log A t+ = log A t + t+ ; t+ i:i:d:n 0; 2 and log S t follows a nite-state Markov process that approximates an AR() process 6 log S t = ( ) E [log S t ] + log S t + " t ; " t i:i:d:n 0; 2 " 6 The approximation method is the one proposed by Tauchen (986). I approximate with a 9-node Markov chain. The nodes are equally spaced between 2:6 "=( 2 ) =2. 8

9 Table : Calibration Calibrated directly Parameter Description Value time discounting :995 physical capital depreciation :05 intertemporal elasticity of substitution (IES) :5 relative risk aversion 0 elasticity of I=K w.r.t. Tobin s q 8 persistence of log S t :98 Calibrated using simulated moments Parameter Description Value N intangible capital share in goods production :27 K physical capital share in goods production :23 ' curvature on leisure 3:3 curvature on l N :0023 E [log S t ] unconditional mean of log S t :022 " standard deviation of " t :0035 standard deviation of t :0069 Targeted moments Description Value (i) average labor hours :20 (ii) average intangible spending over GDP :0 (iii) average labor compensation over GDP :60 (iv) average implied markup :20 (v) average GDP growth :005 (vi) standard deviation of GDP growth :0097 (vii) standard deviation of risk-free rate :0097 9

10 where t and " t are independent. I set = :98, but results are robust to a reasonably high value of, say larger than :95. A-productivity has a direct impact on the goods producing e ciency. Therefore, it is aimed at capturing changes in regulations, the climate, the public health, and so on. log A t is modeled as a random walk mainly for computational convenience. S-productivity does not have a direct impact on the goods producing e ciency, but only does so with a time lag through intangible capital accumulation. Therefore, it is aimed at capturing fundamental innovations such as a breakthrough in the basic scienti c research or the advent of an embryonic creative idea about organizational structure or a business model. I assume that log S t is persistent because these fundamental innovations seem to have a long-lasting impact, but also assume that log S t is stationary because otherwise the growth rate of the economy becomes non-stationary. With these parametric assumptions, we have seven parameters ( N ; K ; '; ; E [log S t ] ; ; " ) left to be chosen. I choose them so that simulated moments from the model match the following seven moments in the data: (i) average hours worked, (ii) average intangible spending over GDP, (iii) average labor compensation over GDP, (iv) average implied markup, i.e., measured economic pro t over labor and physical capital compensations, (v) average per capita GDP growth rate, (vi) standard deviation of per capita GDP growth rate, and (vii) standard deviation of annualized riskfree rate. The measured economic pro t in the model is Y t r K;t K t W t L t. The intangible spending data is taken from Corrado et al. (2005). The standard deviation of the annualized risk-free rate is taken from Bansal and Yaron (2004) and Bansal and Yaron (2000), who derive the risk-free rate by subtracting a trailing 2-month average of in ation from the one month Treasury bill. 3 Main Results 3. A History from the Model I start with a plot of a sample history from the model to see the basic performance of the model. In Figure, I plot realizations of log output and S-productivity. Output grows in the long-run because the economy has endogenous growth. At the same time, output swings both at medium- and highfrequencies. A casual observation shows that the medium-frequency business cycle is generated by 0

11 Figure : A sample history from the model. In the top panel, a realization of log output and a linear time trend regressed on log Y t are plotted. In the bottom panel, a realization of the S-productivity in the same simulation is plotted.

12 the S-productivity. For example, S t is high for the rst 70 quarters, and output growth is above trend in this period. S t suddenly decreases at around 80 quarters and remains low for the next 0 quarters; output growth is below trend in this period, and so on. Because the medium-term cycle is generated by persistent swings in S-productivity, the high-frequency movements in output must be due to A-productivity. 3.2 Asset Prices I report asset market facts in Table 2 under the heading Data. As is well known, the risk-free rate is low and stable, and the equity premium, de ned as the value-weighted return on the New York Stock Exchange minus the risk-free rate, is high and volatile. I am interested in the model s ability to replicate these facts. I de ne the implied risk-free rate R f;t as R f;t E t [M t+ ] where M t+ is the stochastic discount factor de ned as M t+ = Ct+ ( Lt+) ' Ct ( L t ) ' h E t U t+ U t+ i because re-arranging rst order conditions shows that net return of any asset R t+ satis es = E t [M t+ ( + R t+ )] : I de ne return on physical capital R K;t+ as R K;t+ r K;t+ + q t+ + It+ K t+ 0 I t+ It+ K t+ K t+ : q t The denominator q t is the marginal cost to acquire additional physical capital; the numerator is the marginal gain from the additional physical capital in the next period, which consists of the rental price r K;t+ and the capital gain that includes the contribution through the adjustment cost. 2

13 Table 2: Asset Pricing Implications a Data b Variable Mean Standard Deviation Risk-free rate :86 (:42) :97 (:28) Equity premium 6:33 (2:5) 9:4 (3:07) Benchmark model c Variable Mean Standard Deviation Risk-free rate :94 (:88) :97 (:32) Unleveraged equity premium 2:8 (:70) Leveraged equity premium ( = :33) 3:27 (:05) Leveraged equity premium ( = :67) 6:54 (2:0) Return on physical capital :95 (:84) Return on intangible capital 4:74 (:72) 2:78 (:36) 4:7 (:54) 8:34 (:08) :05 (:27) 3:88 (:53) Constant-S model c Variable Mean Standard Deviation Risk-free rate 3:33 (:04) :0 (:02) Unleveraged equity premium :23 (:6) Leveraged equity premium ( = :33) :35 (:24) Leveraged equity premium ( = :67) :69 (:48) Return on physical capital 3:43 (:0) Return on intangible capital 3:59 :5 (:22) (:3) a Returns are annualized percentage return. Standard deviations are presented in percentage. b Sample period: Numbers in parentheses are standard errors with Newey and West (987) correction using 0 lags. Source: Bansal and Yaron (2004). c Means across,000 simulations of sample size 280 are reported; numbers in parentheses are standard deviations across the simulations. Returns are time aggregated into 70 year-long annual data before calculating moments. :37 (:2) 2:06 (:8) 4: (:36) :53 (:05) 3

14 I de ne return on intangible capital R N;t+ as R N;t+ p N;t+ q N;t : The denominator q N;t is the purchasing price of intangible capital N t+ in period t; the numerator p N;t+ is the resale price of the intangible capital in period t +. I de ne two equity returns. Unleveraged equity return R e;t+ is the value-weighted return on the two capitals R e;t+ V K;t V N;t R K;t+ + R N;t+ V K;t + V N;t V K;t + V N;t where V K;t q t K t+ is aggregate physical capital value and V N;t q N;t N t+ is aggregate intangible capital value. Because the empirical measure of the equity return is the value-weighted return on the New York Stock Exchange, unleveraged equity return is the comparable return if all the rms listed on the New York Stock Exchange were 00% equity nanced. But this is far from the reality: rms issue a variety of assets with a continuum of risk characteristics, and stockholders bear disproportionate risk in the sense that they are usually residual claimants. Hence, I introduce leverage to study the impact of corporate capital structure on equity returns. I assume that the representative consumer owns the aggregate stocks of physical and intangible capital through a competitive mutual fund. The mutual fund issues two forms of assets: a bond that pays the risk-free rate R f;t to the bond holder and an equity that pays the residual to the equity holders. The fund s value is the summation of the aggregate physical capital value V K;t and the aggregate intangible capital value V N;t irrelevant to the corporate capital structure (the Modigliani- Miller theorem). The corporate capital structure, however, does a ect the equity return; if the debt-equity ratio is, the leveraged equity return is ~R e;t+ () = R e;t+ R f;t: Using the debt-equity ratio reported in the Flow-of-Funds account, = :5, as a rough guide, I report the leveraged equity premiums at = :33 and = :67 to show sensitivity. 7 Asset return implications of the model are reported in Table 2 under the heading Benchmark 7 = :5 is the debt-equity ratio for the U.S. non nancial corporate sector. 4

15 model. The simulation results are by and large consistent with the data. The risk-free rate is low and stable. The equity premium is high and volatile. Because intangible capital pays a high return while physical capital pays a low return, the high equity premium is due to the high return on intangible capital. In Table 2 under the heading Constant-S model, I report asset price implications in a model in which S t is always constant at its mean. Because the rest of the parameters are the same in the two models, the di erence between the constant-s model and the benchmark model is the e ect of the stochastic S-productivity. The constant-s model does not replicate the asset market facts. The risk free rate is too high and too stable. The equity premium is too small even with strong leverage. Unlike the benchmark model, expected returns on the risk-free bond, physical capital, and intangible capital are roughly the same. I also experiment with a calibrated constant-s model, the model in which S t is constant but parameters are calibrated using simulated moments from the constant-s model. The calibrated constant-s model does not match the asset market facts either. The details are in the appendix. The fact that the model generates a high equity premium with a stable risk-free rate is a strong feature of my model because a counterfactually volatile risk-free rate is a common problem of asset pricing models in the production economy. This problem is a downside of a utility function that strongly favors a smooth consumption path (e.g., Jermann (998), Boldrin et al. (200), and Kaltenbrunner and Lochstoer (2008)). Such a utility function helps to increase the equity premium since consumers demand a high risk compensation for an uncertain equity return, but the utility function also leads to a counterfactually volatile risk-free rate since the equilibrium interest rate is very sensitive to the slope of a consumption path. Therefore Cochrane (2005) stresses, checking interest rate volatility is an important question to ask of any general equilibrium model in nance. My model makes progress along this line of research; the reason my model does not su er from the same problem is that the mechanism to have a high equity premium is very di erent from the one in the literature. 8 8 Campbell and Cochrane (999) study external habit preference which has the implication that the risk-free rate is constant. But reconciling the external habit with a standard real business cycle model in which labor hours are endogenously chosen is di cult (Lettau and Uhlig (2000) and Uhlig (2004)). 5

16 3.3 Understanding the Asset Pricing Implications This section discusses the mechanism behind the asset pricing implications reported in Table 2. Remember net return of any asset R t+ satis es = E t [M t+ ( + R t+ )] (5) where M t+ is the stochastic discount factor M t+ = Ct+ ( Lt+) ' Ct ( L t ) ' h E t U t+ U t+ i : If the relative risk aversion equals the inverse of the intertemporal elasticity of substitution, = ; the second term of the stochastic discount factor disappears; hence, the rst term is the stochastic discount factor of the CRRA utility function. The inability of a standard real business cycle model with the CRRA utility function to explain the equity premium-risk free rate puzzle is well documented. The same is true for my model with CRRA utility because its consumption and labor hour implications are similar to those in the standard real business cycle model. My model s improvement on the asset pricing implications, therefore, comes from the second term. With the second term, the representative consumer cares about his future prospect when he evaluates the asset return. Because = 0 > = 2 3 ; the exponent of the second term is negative with a large absolute value, implying that the return in a state in which the representative consumer enjoys high continuation utility is heavily discounted. Therefore, an asset whose return is positively correlated with the representative consumer s continuation utility has to pay a high risk premium to satisfy equation (5). 9 9 Because the risk free rate also satis es equation (5), we have 0 = E t [M t+ (R t+ R f;t )] : Using the de nition of covariance, we have E t [R t+ R f;t ] = ( + R f;t ) cov t (M t+; R t+) : This equation tells that expected excess return is higher for assets that have a large negative covariance with the stochastic discount factor M t+. In terms of the continuation utility, expected excess return is higher for assets that have a large positive covariance with the continuation utility U t+: 6

17 Figure 2: Responses to A shock. An increase in A-productivity occurs in period 0: Percentage change from the economy without the increase in A are plotted. In Figure 2, I plot the impact of an increase in A-productivity in the benchmark model. 0 Both intangible capital and physical capital enjoy a high return because the increase in A-productivity increases their marginal products. The increase in A-productivity increases the representative consumer s continuation utility as well. In Figure 3, I plot the impact of an increase in S-productivity in the benchmark model. The representative consumer s continuation utility dramatically increases because the positive innovation in S-productivity signals that the economy enters into a prolonged high growth period. Intangible capital s return also dramatically increases because of the large capital gain. The price of intangible capital jumps because the price re ects intangible capital s contribution to create new 0 Responses depend on the state of the economy. I plot responses in a particular state, but similar responses are seen in di erent states as well. 7

18 Figure 3: Responses to a positive S shock. An increase in the S-productivity occurs in period 0. Percentage change from the economy without the increase are plotted. 8

19 intangible capital in the future, and the positive innovation in S-productivity expands its potential. The assumption that an individual producer needs his private intangible capital input to produce tomorrow s intangible capital is crucial here because the owner of today s intangible capital can claim reward from its contribution only if it is private. The fact that the intertemporal elasticity of substitution is greater than one in my model is also crucial because otherwise a large increase in the real interest rate o set the positive e ect on the future pro t stream (Bansal and Yaron (2004)). Physical capital s return slightly decreases due to the wealth e ect. Becoming more optimistic, the representative consumer increases consumption and decreases physical investment. The decrease in physical investment then decreases the physical capital s price. This capital loss leads to the low physical capital return. We can now interpret the asset pricing implications. Intangible capital has a high risk premium in the benchmark model because it pays high returns when the representative consumer enjoys high continuation utility. Physical capital has a low risk premium because its return is weakly correlated with the agent s continuation utility. In fact, the representative consumer uses physical capital as a cushion against negative productivity shocks. This self-insurance saving pushes down physical capital s return. The physical capital return in the constant-s model is higher than that in the benchmark model because the self-insurance saving motive is weaker in the constant-s model. The intangible capital return in the constant-s model is lower than that in the benchmark model because intangible capital is a less risky asset in the constant-s model. 3.4 Business Cycle Statistics Panel A of Table 3 reports the volatility of output, consumption, physical investment, and labor hours, and their co-movement with output. The data and simulation are ltered using the bandpass lter of Christiano and Fitzgerald (2003) before calculating the statistics. Under the heading High-frequency component, I report results of cyclical components with frequency less than 32 quarters; this frequency is often associated with the business cycle frequency. The benchmark model successfully replicates the data. The model understates the volatility of 9

20 Table 3: Business Cycle Statistics High-frequency component Medium-frequency component 2-32 quarter quarter Statistic a Data b Benchmark c Constant-S c Data Benchmark c Constant-S c Panel A Standard variables Y :58 (:8) : (:3) :99 (:0) 2:47 (:48) 3:43 (:05) :60 (:42) C :77 (:09) I 4:63 (:57) L :74 (:20) (Y; C) :73 (:4) (Y; I) :77 (:6) (Y; L) :87 (:7) W LN :37 (:8) (Y; W L N ) :57 :72 (:09) 2:49 (:3) :35 (:04) :98 (:02) :99 (:0) :79 (:06) :66 (:07) 2:27 (:22) :24 (:02) :00 (:00) :00 (:00) :99 (:00) :70 (:34) 5:07 (:99) 2:67 (:52) :90 (:9) :64 (:5) :90 (:7) Panel B Intangible spending :67 (:30) :39 :72 (:) :00 8:2 (:97) :57 2:63 (:85) 6:63 (2:04) :9 (:27) :98 (:0) :96 (:03) :67 (:0) 3:80 (:8) :70 :22 (:33) 3:32 (:84) :34 (:09) :98 (:00) :96 (:0) :87 (:04) :40 (:42) :00 (:00) (:22) (:9) (:00) (:22) (:6) a x denotes the standard deviation of variable x, and (x; y) denotes the correlation between variable x and variable y: Standard deviations are presented in percentage. b Panel A; Sample period: 950:Q to 2008:2Q. Numbers in parentheses are standard errors with Newey and West (987) correction using 0 lags. Panel B; Sample period: 959 to Numbers in parentheses are standard errors with Newey and West (987) correction using 3 lags. c Means across,000 simulations are reported; numbers in parentheses are standard deviations across the simulations. Sample size of each simulation is the same as the sample size of the corresponding data. 20

21 labor hours, but this is a common feature of most RBC models (King and Rebelo (2000)). The benchmark model and the constant-s model generate similar results at this frequency. Hence, the high-frequency economic uctuations are mostly attributed to A-productivity. Under the heading Medium-frequency component, I report results of cyclical components with frequency between 33 to 28 quarters. The benchmark model replicates the data at this frequency as well. The constant-s model generates much smaller volatility than the benchmark model. Hence S-productivity is important for medium-term economic uctuations. Business cycle implications for endogenously chosen labor hours are a strength of my model compared to many other models in the literature. First of all, the representative consumer in my model freely chooses labor hours, while many papers in the literature assume that labor hours are constant (Jermann (998), Guvenen (2005), Kaltenbrunner and Lochstoer (2008), and Croce (2008)). A reason they assume exogenous labor supply is that they adopt strong investment adjustment cost to generate a large action in Tobin s q, but labor hours tend to move countercyclically in a model with strong investment adjustment cost as Boldrin et al. (200) point out. 2 In my model, the endogenously chosen labor hours are pro-cyclical because investment adjustment cost is small. My model can generate high equity premium despite the small investment adjustment cost because the driving force that generates the high equity premium is not Tobin s q but the intangible capital s price q N. In Panel B, I report the volatility of intangible spending and its co-movement with output. Because intangible spending uctuations are at the heart of my model, this exercise is an important check on the model s validity. I use annual R&D spending data as a proxy of intangible spending because they are the best quality data available. The benchmark model performs well in replicating the data, except for volatility in the medium-frequency component. Because the constant-s model is o the mark, stochastic S-productivity is important for matching the volatility of intangible spending and its co-movement with output. A notable exception is Boldrin et al. (200), who instead adopt considerable adjustment frictions. 2 The mechanism goes as follows. Suppose a positive TFP shock hits the economy. Other things being equal, output increases. Because investment adjustment cost is steep, the representative consumer increases investment only a little, and consumes most of the increased output. Labor hours decrease because consumption and leisure are complement, but not enough to o set the productivity increase. The result is counter-cyclical labor hours. 2

22 3.5 Cross Correlations of Output and Aggregate Capital Value In Figure 4, I plot correlations of output and the aggregate capital value Corr [log Y t ; log (V K;t+k + V N;t+k )] for k = 5; 4; ; 5: The top panel shows the correlations in the data. As a measure of the aggregate capital value, I use a broadly de ned total rm value calculated by Hall (200a). 3 only is the aggregate capital value pro-cyclical, but the correlation is stronger in negative k. Not We expect to see this pattern if an increase in the current aggregate capital value signals an increase in future output. For this reason, the asymmetry in this direction is often used as a de nition of a leading indicator (e.g., Backus et al. (2007)). In the middle panel of Figure 4, I plot the cross correlations in the benchmark model. The aggregate capital value is pro-cyclical, and the cross correlations are asymmetric with stronger correlations in negative k. These features are consistent with the data. In the bottom panel of Figure 4, I plot the cross correlations in the constant-s model. The cross correlations observed in the constant-s model are very di erent from the actual cross correlations; in particular, we see a counterfactual peak at the contemporaneous correlation. Hence, stochastic S-productivity is important for the cross correlation of output and the aggregate capital value. In Figure 5, I plot the impact of S-productivity and A-productivity on output and the aggregate capital value in the benchmark model. A positive S-shock has asymmetric impacts on output and the aggregate capital value. An increase in S-productivity initially drives down output and then gradually pushes it up. The initial drop is caused by the wealth e ect; output then starts to increase sluggishly because intangible capital accumulation gradually improves the good producing sector s e ciency. The positive S shock, however, increases the aggregate capital value immediately because it increases the aggregate intangible capital value immediately. A positive A-shock has symmetric impacts on output and the aggregate capital value. An increase in A-productivity increases both output and the aggregate capital value immediately. Hence, 3 Hall (200a) measures the total rm value as the market value of outstanding equities plus the imputed market value of bonds plus the reported value of other nancial liabilities less nancial assets for all nonfarm, non nancial corporations. 22

23 Figure 4: Cross correlation of log output, log Y t ; and the log aggregate capital value, log (V K;t+k + V N;t+k ) ; for k = 5; 4; ; 5: The top panel is the actual observation; sample period is 950:Q-999:3Q. Dotted band presents 95% con dence intervals with Newey and West (987) correction using 0 lags. The middle panel and the bottom panel are the cross correlations observed in the benchmark model and the constant-s model, respectively. Solid lines present means of,000 simulations of sample size 99, and dotted bands present 2.5% and 97.5% quantiles. The data and simulation are ltered using the band-pass lter of Christiano and Fitzgerald (2003) with less than 28 quarters before calculating the statistics. 23

24 Figure 5: Responses of log output, log Y t ; and the log aggregate capital value, log (V N;t + V K;t ) ; to an increase in S-productivity (left column) and to an increase in A-productivity (right column). Percentage change from the economy without the shock are plotted. 24

25 the di erence in the timing of responses against S-productivity is the source of the asymmetric cross correlations in the benchmark model. 4 VAR Evidence Cross correlation is a convenient summary statistic, but a vector autoregression (VAR) gives a more detailed picture of the dynamic relation between the variables. In this section, I investigate joint dynamics of output and the aggregate capital value using a VAR. 4. Reduced Form VAR I rst document dynamics of output and the aggregate capital value in the data. The econometric framework is a bi-variate VAR model log Y t log (V N;t + V K;t ) = c + 5X j= B j log Y t log (V N;t j + V K;t j ) j u t ; u t i:i:d:n (0; ) : (6) I plot impulse response functions to orthonormal shocks e t = (e ;t ; e 2;t ) 0 that satisfy u t = P e t ; P P 0 = where P is a 2 2 lower triangular matrix; namely, the second shock e 2;t does not have contemporaneous impact on output. This is a reduce form VAR analysis and no structural interpretation of e is intended; showing a clean MA () representation of the bi-variate system (6) is the sole purpose of this exercise. Figure 6 shows the results. The rst shock e increases output immediately and persistently while it has negligible e ect on the aggregate capital value. The second shock e 2 increases the aggregate capital value immediately and persistently while it increases output gradually. These results show that an increase in the aggregate capital value predicts an increase in future output. 4 To see if my model is consistent with this VAR evidence, I generate arti cial data from my 4 The formal Granger causality test rejects the hypothesis that the lagged aggregate capital values do not enter into the rst equation at % signi cance level. 25

26 Figure 6: Orthogonal impulse response functions (actual data). e 2 does not have contemporaneous impact on output by construction. The rst column plots percentage responses to a one standard deviation positive e shock. The second column plots percentage responses to a one standard deviation positive e 2 shock. The output response is in the rst row, and the aggregate capital value response is in the second row. Solid lines present means of 0,000 Monte Carlo draws; dotted bands are 68% error bands. Y Responses of V E Responses to E2 26

27 Figure 7: Orthogonal impulse response functions (simulated data). e 2 does not have contemporaneous impact on output by construction. The rst column plots percentage responses to a one standard deviation positive e shock. The second column plots percentage responses to a one standard deviation positive e 2 shock. The output response is in the rst row, and the aggregate capital value response is in the second row. Maximum likelihood estimate of orthogonal impulse response functions are calculated with each of,000 arti cial data sets; solid lines present means of,000 simulations; dotted bands present 2.5% and 97.5% quantiles across simulations. model and apply the same econometric technique to the simulated data. Results are plotted in Figure 7. We observe two limitations. One is the aggregate capital s impact response against the rst shock e ; the impact is negligible in the actual data while the impact is positive and statistically signi cant in the simulated data. The other is the aggregate capital value s volatility; that is, my model cannot fully account for high volatility of the aggregate capital value observed in the data. The impulse response functions are, however, qualitatively consistent with those observed in the data. The rst shock e increases output immediately and persistently with the same magnitude as we see in the data while it has statistically insigni cant e ect on the aggregate capital value except for rst seven quarters. The second shock e 2 increases the aggregate capital value immediately and persistently while it increases output gradually with the same magnitude as we see in the data. 27

28 4.2 Structural VAR This section shows a structural VAR analysis. Namely, the two structural shocks, innovation t in the A-productivity and innovation " t in the S-productivity, are identi ed from actual data using a restriction that is consistent with my model economy, and empirical impulse response functions to the identi ed shocks are drawn. The econometric framework is the bi-variate VAR system (6). The fundamental innovations ( t ; " t ) and the VAR shock u t are related with an impact matrix ; u t = " t= " t = : An identi cation problem arises because we have four unknowns in while the moment restriction 0 = imposes only three restrictions. We need an additional restriction to identify the fundamental innovations from the data. Because the model economy does not imply a binding zero restriction either on impact or in the long-run, popular zero restrictions are not available. The model economy, however, does have a fairly robust implication: a positive innovation in the A-productivity increases output and the aggregate capital value immediately. Relying on this implication, I identify the two fundamental innovations by the pure-sign restriction of Uhlig (2005); the restriction is that a positive A-productivity innovation increases output and the aggregate capital value on impact. Because the restriction is too weak to uniquely pin down the impact matrix, I treat all the possible that satis es the sign restriction equally likely. Hence, the resulting error bands of impulse responses contain not only the uncertainty in the VAR parameters (c; B ; ; B 5 ; ) but also the uncertainty in identifying the impact matrix. A technical discussion appears in the appendix. The impulse responses to the identi ed fundamental innovations are plotted in Figure 8. right column shows the responses to a positive innovation in t, the A-productivity factor. The The positive A shock increases output and the aggregate capital value immediately and persistently. 28

29 Figure 8: Impulse response functions (SVAR model). The rst column plots percentage responses to a one standard deviation positive S shock. The second column plots percentage responses to a one standard deviation positive A shock. The output response is in the rst row, and the aggregate capital value response is in the second row. Solid lines present means of 0,000 Monte Carlo draws from posterior distribution; dotted bands are 68% error bands Y Responses of V S shock Responses to A shock The responses are consistent with the model s predictions in Figure 5. The left column shows the responses to a positive innovation in " t, the S-productivity factor. The positive S shock increases the aggregate capital value immediately and persistently, and causes an initial dip and subsequent recovery in output. Though the recovery part is not strong in the structural VAR, the responses are qualitatively consistent with the model s predictions in Figure Link to the News Shock Literature The burgeoning news shock literature explores propagation mechanisms with which the economy enjoys a boom when a positive news shock, a signal that predicts future exogenous productivity improvement, hits the economy (Beaudry and Portier (2004), Jaimovich and Rebelo (2008), and Christiano, Ilut, Motto and Rostagno (2007)). This is not an easy task because in a standard real business cycle model, good news about future productivity causes a recession contemporaneously; optimism leads to a rise in current consumption due to the wealth e ect, which then reduces labor supply and output. An empirical support of the news shock approach is given by Beaudry and 29

30 Portier (2006), who nd that a stock price innovation that is contemporaneously orthogonal to TFP predicts gradual increase in future TFP. My model o ers a structural interpretation of a news-driven business cycle story. An innovation in the S-productivity in my model looks like a news shock because it does not have contemporaneous e ect on traditionally measured TFP the Solow residual after controlling the contributions of labor and physical capital but it gradually increases the measured TFP as intangible capital is accumulated. But the innovation in the S-productivity in my model is a real shock that changes the current intangible capital producing technology, and the future gain in the measured TFP is the outcome of hard work in the intangible capital producing sector, while in the news shock story, news shock is a pure signal that does not have any real e ect on the contemporaneous technology, and future productivity gain is manna from heaven. Finally, stock price dynamics in my model are consistent with the data, including the empirical evidence of Beaudry and Portier (2006). 5 Inspecting the Mechanisms Two model comparisons are conducted. First, I compare economies with and without the e cient subsidy. Then, I compare economies with two di erent intangible capital producing technologies. Ine ciency arises in both exercises. Finding an ine cient allocation in an economy with recursive utility is di cult because neither a social planner s problem nor a local approximation method is readily available. Therefore I impose an additional parameter restriction that the coe cient of relative risk aversion equals the inverse of intertemporal elasticity of substitution, = = 2 3, throughout this section. This restriction enables me to use a local approximation method because recursive utility reduces to familiar CRRA utility. I keep the intertemporal elasticity of substitution unchanged because this parameter is crucial for determining the quantity implications (Tallarini (2000)). The downside of this restriction, however, is that the model is no longer able to generate a high equity premium since the coe cient of relative risk aversion is substantially lowered. 30