Uncertainty-Driven Growth Koki Oikawa Nihon University Population Research Institute March 5th, 2008

Abstract In this paper, I present a model in which uncertainty raises growth. The mechanism for this is learning by doing in the research sector: …rms undertake research to reduce uncertainty, which results in social knowledge accumulation that improves the productivity of future research. The model explains the positive correlation between TFP growth and dispersion in manufacturing industries. The model also …ts aggregate TFP behavior in U.S. manufacturing during the last two decades.

1

Introduction

This paper analyzes the e¤ect of …rm-level uncertainty on aggregate productivity growth. In the literature, many researchers have examined the e¤ect of macro-level uncertainty, such as disturbances in output levels or growth rates on economic growth. And there seems to be consensus that there exists a negative correlation between growth and ‡uctuation (Ramey and Ramey (1995), Martin and Rogers (2000), Imbs (2004)). On the other hand, only a few papers investigate the e¤ect of micro-level uncertainty. If micro-level uncertainty does not bring any impact at the aggregate level, or if it is analogous to macro-level uncertainty, it would be su¢ cient to study macro-level uncertainty. However, the story is not that simple. First, Comin and Mulani (2006) document that micro- and macro-level volatility have opposite trends in the United States. So these two types of volatility are not related in a simple way. Second, even under no aggregate uncertainty, …rm-level uncertainty may in‡uence aggregate growth. A typical story is that …rms faced with high uncertainty tend to hesitate in I am grateful to Diego Comin, Boyan Jovanovic, Katsuya Takii, and Gian-Luca Violante for helpful suggestions and discussions. All errors are mine. Email: [email protected]

1

investing in irreversible investment under uncertainty, which lowers aggregate capital accumulation, and thus output growth (cf. Dixit and Pindyck (1994)). Bertola (1994) introduces an idiosyncratic shock in labor unit cost and derives a negative e¤ect of …rm-level uncertainty on output growth in an environment in which labor mobility is costly. However, is micro-level uncertainty always harmful to growth, as seen in the above theories? In this paper, I show that this is not true, in the light of the fact that inherent uncertainty can be reduced by research e¤orts of …rms. Uncertainty has many aspects; some impacts of uncertainty are negative, but others are positive— uncertainty may raise growth. I focus on this positive aspect of …rm-level uncertainty. In Section 2, I will show the basic one-sector model where uncertainty raises productivity growth. The intuitive story is that, in an uncertain environment, …rms carry out research to reduce that uncertainty. Such e¤orts induce new discoveries and enlarge the social knowledge base, which in turn improves the productivity of research thereafter. As a result, an increase in the level of uncertainty decreases the expected productivity level in the short run, but increases the expected productivity growth rate and, moreover, increases the expected productivity level in the long run.1 Section 3 is the analysis of social planner’s problem of the model. In Section 4, I apply this mechanism to the TFP growth in the U.S. manufacturing sector during the last two decades to examine how uncertainty has a¤ected the behavior of TFP. The estimation result implies that the uncertainty-driven growth mechanism explains about 20% of the growth in TFP during the sample period. There is also a signi…cant upward trend in …rm-level uncertainty, which is consistent with the recently observed positive trend in idiosyncratic risks (Campbell, Lettau, Malkiel, and Xu (2001), Comin and Philippon (2005), etc.). Section 5 presents an extension of the basic model to analyze multiple industries. By this extension, I explain the positive relationship between productivity growth and dispersion. Dwyer (1998) observes this phenomenon among the 4-digit U.S. textile industries, and I show that the same relation is true among the 2-digit U.S. manufacturing industries, using the Compustat data set. Section 6 is concluding comments. 1

Comin (2000) presents a related story in which a rise in uncertainty induces replacement of old capital with ‡exible capital (such as information techonology capital), which …nally realizes a higher productivity growth after the slowdown of productivity growth. The mechanism of resurgence in productivity growth in Comin (2000) comes from the assumption that productivity growth is higher with ‡exible capital than with old capital. In that sense, its model is di¤erent from the model in this paper, where uncertainty itself raises growth.

2

2

The Model

In this section, I …rst present the basic structure of the model and then describe the equilibrium within an instant. Later, I link the instants.

2.1

Preliminary Explanation of Building Blocks

There exists a unit mass of risk-neutral agents. Each can choose to be a producer or a researcher in each instant without any cost. Production.— One producer owns one plant. A plant produces output of P without any input. However, the productivity of the plant depends on the number of researchers it hires. E (P ) = Af (n; s); where A is the technology level, n is the amount of research, s is a parameter, and f (n; s) is the expected technical e¢ ciency (which is increasing, concave, and converging to 1 as n goes to in…nity for any s). Research.— A new research output is created in a linear form. If a producer hires R researchers, then the amount of research n that he obtains is kR. The research productivity, k, is increasing in the level of knowledge capital in the whole economy because new knowledge is created through combinations of existing knowledge. For simplicity, I assume that the research productivity is identical to the social knowledge capital. If an agent chooses to be a researcher, he is hired by a producer. If a producer obtains n research outputs, it demands n=k researchers. I assume that the research performed for one …rm is not helpful for another. Researcher market equilibrium.— Both producers and researchers take the researcher wage w as given. The producer’s optimization problem is as follows. nw : n 0 k By solving this maximization problem, the optimal n is obtained and the expected pro…t is computed. The researcher wage w is pinned down by the occupational choice. Let e be the producer’s expected pro…t. Because there is no cost for changing career in this model, the following should hold in equilibrium: max Af (n; s)

e

= w:

To derive the total amount of research in the economy, I use the following researcher market clearing condition, where is the fraction of researchers, = (1

3

n ) : k

The research created in one instant is accumulated in knowledge capital, which is publicly available in the next instant. Denoting as the obsolescence rate of knowledge, k_ = k

2.2

k:

Technical E¢ ciency Function

In this subsection, I describe the speci…cs of the function f and the rest of the details of the model, which was brie‡y sketched in the previous subsection. I formulate the production function similarly to Jovanovic and Nyarko (1996). P =A 1

z)2

s(

The shock parameter is drawn from a standard normal distribution, which is common knowledge, and is idiosyncratic among …rms. z is the control variable. The parameter s is the level of uncertainty. If a manager by chance sets z as the same as , production reaches the maximum level for any s.2 With the quadratic form and risk neutrality, the optimal z equals the expected value of . Research is made or bought to revise …rm’s belief about through Bayesian updating. Suppose a …rm obtains n units of research. The integration of research outputs turns out to be a signal about , such that3 +p ; n

N (0;

2 r ):

According to the posterior variance, say h(n; s), the technical e¢ ciency in expectation, f (n; s), is s : f (n; s) = 1 h(n; s); where h(n; s) = 1 + n= 2r Below, I will refer this h as the expected technical ine¢ ciency, and use it as opposed to f since it is more convenient in later arguments. Finally, I rewrite the expected pro…t function of …rms: e

= A(1

h(n; s))

nw ; k

(1)

where the superscript e indicates expectation. 2

Because the ine¢ ciency term is quadratic, the distribution of ine¢ ciency is asymmetric and riskneutral …rms will thus try to reduce the variance. This formulation may be interpreted as a model of Stochastic Frontier Production Function (see, for example, Kumbhakar and Lovell (2000)) in which the technical ine¢ ciency is distributed as a 2 distribution. 3 This setting is for avoiding a non-integer number of signals. If I allow a continuous number of signals, n is the number of signals and each signal has the value of + .

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2.3

Instantaneous Equilibrium

Pro…t maximization.— A producer maximizes (1) about the amount of research, n 0, taking other variables and parameters as given. Then the optimal research outputs and the expected pro…ts can be described as follows: 8 for w > skA < 0 2 ; r q n= (2) s : 2r 1 otherwise; 2 w=kA r

e

=

(

A(1 s); p 2 swA=k + A 2 r

2 r w=k;

for w > skA 2 ; r otherwise.

(3)

Researcher market equilibrium.— The researcher market equilibrium may have both producers and researchers who obtain the same pro…ts in expectation, or only producers who do not demand any research. In the latter case, the researcher market collapses. Because an active researcher market never occurs at k = 0, there exists some range of k in which the second type of equilibrium occurs. Proposition 1 shows there is unique equilibrium with the active researcher market if k is su¢ ciently large. Proposition 1 If 2 r (1

s)

; s then there exists a unique equilibrium with an active researcher market. k>

Proof. In equilibrium,

e

= w implies 0 Ak 1 q w= 2@ s r 1+ k

s

2 r 2 r

+1

(4)

12

A ;

2 s). When s 1, this inequality always which is well-de…ned if and only if k r (1 holds. When s < 1, (4) guarantees it. Even if w is uniquely well-de…ned, it may be too high for any producer to choose n > 0 (the inactive researcher market). Such a case occurs if, from (2), 0 12 Ak @ 1 skA A q 2 : 2 2 k s r 1+ r r +1 s

However, this inequality implies k exists unique equilibrium under (4).

2 r

2 r (1

5

s)=s, exclusive to (4). Hence, there

So, if k is su¢ ciently large, the researcher market equilibrium is uniquely determined at the price that makes the expected pro…ts of producers and researchers equivalent. 2 s)=s, the only equilibrium is that everybody is a producer, and When k r (1 demands no research. Therefore, (4) is su¢ cient and necessary for the existence of an active researcher market equilibrium. Note that this condition always holds if s 1. No research implies negative expected pro…t in such a case. Below, I assume k is su¢ ciently large. Equilibrium pro…le.— Under (4), the variables in instantaneous equilibrium are derived as follows. For later convenience, I …rstly solve the equilibrium technical inef…ciency, h(k; s), as 1 q 2 : (5) h(k; s) = 1 + ks 2r + 1 r

From

e

= w in equilibrium, the researcher wage is written in terms of h(k; s) as w(k; s) =

Akh(k; s)2 : s 2r

(6)

The other variables can be also expressed as functions of technical ine¢ ciency, h. The demand for research from each producer re‡ects the ratio between the uncertainty level and the posterior variance, n(k; s) =

2 r

s h(k; s)

1 :

(7)

According to the equilibrium researcher wage, the measure of each group of specialists is set so that the researcher market is cleared. Denote the measure of researchers as . From the researcher market clearing condition: (1 )n = k, (k; s) = 1

k h(k; s)2 : s 2r 1 h(k; s)

(8)

When s is extremely large, the proportion of researchers is almost 1. So, a producer obtains an in…nite number of units of research, and its expected pro…t is positive in equilibrium, even under in…nite s. Comparative statics, given k and A.— To observe the e¤ect of the …rm-level uncertainty on the above instantaneous equilibrium pro…le, I present the comparative statics, keeping k and A constant. Proposition 2 Suppose (4) holds and k and A are given. An upward shift of s increases n, h, and , and decreases w.

6

Proof. The e¤ects on n and h are immediately observed by looking at equations (5) and (7). About w, @w Ak h3 = < 0; 2 s2 (1 @s h) r since h 2 (0; 1) under (4). Since n increases and k is constant, clear the researcher market.

has to increase to

An increase in the level of uncertainty stimulates research demand (higher n) and supply (higher ). However, the increase in research is not enough to make up for the negative impact of an increase in s. The expected productivity of each producer is thus lower (higher h and lower y e ). The e¤ect on the researcher wage could be counter-intuitive. When the level of uncertainty rises, producers demand more research. This is an upward pressure on the researcher wage, which results in many producers switching to being researchers, which in turn increases the supply of research. This downward pressure overcomes the …rst impact. If the researcher wage increases as a result, the producers’pro…ts must be less than those before the change, because of higher uncertainty and a higher researcher wage. Thus w > e . Therefore, switching must occur until the expected pro…ts of producers and researchers are equal, and the researcher wage …nally settles at a lower level.

2.4

Dynamics and Steady State

I now describe the dynamics of the model. To avoid complication, I assume the unknown parameter for each …rm is i.i.d. over time. Therefore, the past research outputs are not directly helpful when predicting the current , and so each …rm’s initial belief about is unchanged over time. However, those knowledge can be made use of in subsequent research activities as ingredients or hints of a new idea. I assume that there is no intellectual property rights and all research outputs are built into publicly available social knowledge just after being created. And this accumulation of social knowledge capital improves the research productivity. In other words, the productivity of research improves through learning by doing (similar to Arrow (1962), Matsuyama (2002), etc.). The accumulation process of knowledge capital k is gk

k_ = (k; s) k

;

(9)

where 2 (0; 1) is the obsolescence rate of existing knowledge. Here, I use comparative dynamics to show the main result of this paper: steady state knowledge capital and average productivity is higher under more uncertainty.

7

First, I discuss the behavior of the proportion of researchers, (k; s), which is essential for determination of the growth path. Then, I will derive and analyze the steady state. The curve of .— Since the accumulation of knowledge capital follows (9), the curve of (k; s) determines the growth path. The e¤ect of an increase in k on is negative for su¢ ciently large k because, if plenty of knowledge exists in the economy, additional demands for research by each producer in response to a decrease in the researcher wage (which results from an increase in k) is relatively small, and research productivity increases proportionally to knowledge capital. Moreover, if knowledge capital diverges, nearly all …rms are producers and a researcher whose measure is zero supplies in…nite units of research using in…nite knowledge capital. In Figure 1, I draw curves of , or (8), when s < 1. If s < 1, there exists a range of k in which is increasing. When k is su¢ ciently small, the marginal e¤ect of knowledge accumulation on each producer’s demand for research is so large that the proportion of researchers must increase. When s 1, there is no such range; is monotonically decreasing for any k > 0. Since I focus on the decreasing part of to consider the steady states, I do not dwell on the change of the shapes of curves according to the size of s. Steady state knowledge capital and long-run growth.— The relevant steady state of knowledge capital is reached at k , at which knowledge creation and obsolescence are equal: (k ; s) = : (10) As depicted in Figure 1, if s < 1 and is too high, the knowledge capital is always shrinking. Consider the case where is su¢ ciently low. Then the non-monotonicity of in k gives rise to two distinct k’s that satisfy (10). The lower one is an unstable steady state. If the initial k is su¢ ciently large, the knowledge accumulation leads the economy to the stable steady state k . Otherwise, knowledge shrinks and approaches the other stable steady state with k = 0. Rather than sticking to the latter collapsing case, I assume the initial k is su¢ ciently high and is su¢ ciently small to activate the knowledge accumulation. Comparative dynamics.— As illustrated in Figure 1, the steady state level of knowledge capital k is increasing in the uncertainty level, s, because greater uncertainty increases the proportion of researchers for any k (Proposition 2). Greater uncertainty stimulates research activities and thus knowledge capital is accumulated more. I now show the main result of this section. Higher uncertainty enlarges not only the knowledge capital, but also the average productivity in the steady state. As stated in Proposition 2, an increase in s leads to an increase in h in static equilibrium, implying that uncertainty is harmful to average productivity in the short run. However, an increase in s causes a decrease in h h(k ; s). In other words, higher uncertainty leads to higher average productivity in the long run. The next proposition shows this property.

8

Figure 1: Steady states of k

9

Proposition 3 For any given initial k su¢ ciently large, higher s implies higher productivity and total outputs in the stable steady state. Proof. Suppose s1 > s2 . If k (s1 )=s1 > k (s2 )=s2 , h (s1 ) < h (s2 ). I will show ~ k (s1 )=s1 > k (s2 )=s2 . Let k~ = k=s 2r . Rede…ne as a function of k. ( 0 if k~ 1s2s ~ ~ 2 (k) = k) 1 k~ 1h(h( otherwise, ~ k) where ~ = h(k) 1+

1

q

k~

: 1 s

+1

~ s1 ) < h(k; ~ s2 ) for any k. ~ So an increase in s shifts (k) ~ curve When s1 > s2 , h(k; ~ = , this upward shift causes upward. Since the steady state of k~ is also de…ned as (k) ~ ~ ~ an increase in the steady state k, i.e., k (s1 ) > k (s2 ). Since the productivity sequence is yt (si ) = At (1 h (si )) on the long-run growth path (i = 1; 2), yt (s1 ) > yt (s2 ). Moreover, total output, (1 )y is higher under greater uncertainty since (s1 ) = (s2 ) = in the steady states. An upward shift in the level of uncertainty negatively a¤ects productivity in the short run as shown in Proposition 2. However, in the long run, the impact of the same change is reversed after knowledge capital is su¢ ciently accumulated. This property comes from the nonexclusiveness of knowledge use in research and the setting that managers can change their occupations between researchers and producers. Under greater uncertainty, there are more researchers in the economy and, using existing knowledge capital as input, they create new knowledge, which becomes nonexclusive input of knowledge creation in the subsequent instants. Therefore, even if the initial knowledge capital is the same, knowledge accumulation is accelerated under greater uncertainty. Moreover, when there are smaller measure of producers, the marginal change in total demands for researchers in response to a change in k is larger, which limits the size of decline in the payment per research output (w=k) in response to the increase in knowledge capital for a given k.4 This e¤ect keeps research more pro…table, and the measure of researchers is thus relatively high in the transition path under greater uncertainty. In total, the long-run e¤ect of uncertainty overcomes the shortrun negative impact of uncertainty. To see what happens clearly, I simulated the model and plotted the results in Figure 2. These graphs illustrate the e¤ect of changes in the level of uncertainty, s, on the 4

After some calculation, the elasticity of w=k to k equals 1

10

.

path of knowledge accumulation (the top), average productivity (the middle), and total outputs (the bottom). When s = 0:5, knowledge capital hardly accumulates and then both average productivity and total outputs held up. On the other hand, when s = 1:1, knowledge capital accumulates rapidly, and the other two variables exceed those under s = 0:5 somewhere while they are lower in the beginning .

3

Empirical Application: TFP Dynamics and Uncertainty Level in the U.S. Manufacturing Sector

In this section, I apply the one-sector model presented in Section 2 to the TFP behavior of the U.S. manufacturing sector during two recent decades.

3.1

A Source of Uncertainty

Before getting into the empirics, I slightly change the setting to make it closer to the real economy. First of all, the Hicks neutral technological parameter A is also growing at an exogenous rate, > 0. At = A 0 e t : Moreover, I assume s is increasing in A. If the Hicks-neutral technology level A is large, producers produce more complicated goods and there exists a higher probability of failure in a production process, as in Kremer (1993). Speci…cally, 2 (0; 1):

s(A) = A ;

Unlike the case in Section 2, uncertainty is increasing as the technology improves, which keeps shifting the curve of in Figure 1 upward over time, and knowledge capital k is diverging. However, the informational value of knowledge capital, k~ k=s 2r , converges to a …nite number. Rewriting the technical ine¢ ciency, h, and the proportion ~ of researchers, , as functions of k, ~ s) = h(k; 1+

1

q

k~

; 1 s

+1

~ k; ~ s)2 kh( : ~ s) 1 h(k;

~ s) = 1 (k; The dynamics of k~ are determined by ~ s) gk = (k;

11

:

(11)

2 0

1 5

1 0

5

0 0

5

1 0

1 5

2 0

2 5

3 0

5

1 0

1 5

2 0

2 5

3 0

5

1 0

1 5

2 0

2 5

3 0

0 . 8

0 . 7

0 . 6

0 . 5 0

0 . 7

0 . 6

0 . 5

0 . 4

0

Figure 2: Simulation under distinct levels of uncertainty ( k1 = 1). The horizontal axes are the number of periods.

12

2 r

= 0:5; A = 1;

= 0:05;

~ s(A) = A Figure 3: Steady state of k:

~ s) is increasing in s for any k~ when the researcher market is The function (k; active. In the limit of time, it converges to the following function: ~ s) = 1 lim (k;

s!1

p

k~ k~ + 1 1 +

p

k~ + 1

;

which is depicted as the downward sloping curve in Figure 3. As shown in the …gure, the steady state k~ is determined by = + . In the steady state, the expected technical ine¢ ciency converges to h =

1 p : 1 + k~ + 1

(12)

One concern in this subsection is whether the technological growth rate, , positively a¤ects the average productivity growth rate. This is not obvious because an increase in also implies an increase in the growth rate of uncertainty. Another concern is the comparative dynamics regarding and . Proposition 4 Suppose that + is su¢ ciently small. The growth rate of average productivity is increasing in parameter for any k. gy = in the steady state. Also in the steady state, the average technical ine¢ ciency is higher if or is higher.

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Proof. gy is written as gy =

1

h(1 2h) 2 (1 h)2

+

1 h (1 21 h

)gk ;

which is increasing in for any k since h(1 2h)=2(1 h)2 < 1 and < 1. Next, from (12), converges to h =1 h in the limit. Substitute this and gk = in the long run (since k=s must be constant) into the above equation, gy =

1 h 21 h

1

h 1

h

(1

)gk :

Therefore, lim gy = :

t!1

Next, from Figure 3, a higher or implies lower k~ . Then, from (12), the average technical ine¢ ciency is higher in the steady state. A higher implies more rapid technological growth, so it accelerates productivity growth. However, at the same time, it generates more uncertainty in production, which negatively a¤ects the level of productivity. The …rst statement of Proposition 6 states that the former e¤ect dominates the latter in total. The last statement of Proposition 4 says that the higher growth rate of uncertainty is harmful also in the long run. This is because the negative short-run e¤ect in Proposition 2 occurs in each instant, which dominates the positive long-run e¤ect in Proposition 3.

3.2

TFP Dynamics and Uncertainty Level: the U.S. Manufacturing

Here I apply the one-sector model used in the last section to the TFP behavior of the U.S. manufacturing sector during two recent decades. To do so, I use the data of IT quality index described in Cummins and Violante (2002) as the proxy of knowledge capital k. Knowledge capital in this paper corresponds to productivity of research, which is, here, information collection and processing. Put di¤erently, knowledge capital in this model can be interpreted as a general-purpose technology that a¤ects almost all industries. In that sense, information technology may be considered as a proxy of knowledge capital.5 5

In response to the productivity resurgence in the United States after the mid-1990s, a number of papers on growth accounting conclude that IT capital deepening substantially contributes to the growth of labor productivity and TFP. Stiroh (2001) investigates the contribution of IT assets to

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Table 1: IT quality and manufacturing TFP indexes. Year 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000

IT quality (k= 1.000 1.078 1.131 1.264 1.384 1.508 1.674 1.845 1.979 2.144 2.349 2.541 2.819 3.032 3.252 3.541 3.996 4.449 5.005 5.539 5.921

2 r)

TFP index (1996=100) 81.2 81.7 83.0 85.1 87.7 89.2 90.7 93.6 95.3 93.5 93.3 92.4 94.0 95.1 97.3 99.2 100.0 103.1 105.7 108.7 111.3

To create the IT quality index, I follow the methodology described in Cummins and Violante (2002). The IT quality index is de…ned as the gap between qualityadjusted consumption price index and quality-adjusted investment price index among Information Processing Equipment and Software (IPES) assets, which are de…ned by the Bureau of Economic Analysis (BEA).6 Intuitively, the IT quality index indicates the productivity of e¢ ciency units of IPES investment goods. Table 1 lists the IT quality index during 1980-2000 (normalized as k1980 = 2r = 1) and the data on TFP are taken from the Bureau of Labor Statistics (BLS) multi-factor productivity in all manufacturing industries. labor productivity both in IT-producing industries and in IT-using industries, and concludes that the contribution is signi…cant in both groups of industries. This empirical observation supports the view that IT is a new general-purpose technology (Basu, Fernald, and Shapiro (2001)). IT improvement has made information collection and processing less costly, and therefore reduces costs in the production process. 6 A detailed de…nition and explanation of these indices can be found in Cummins and Violante (2002).

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110 100 TFP Index 90 80

1980

1985

1990 Year Observation

1995

2000

Prediction

Figure 4: TFP index: Data v.s. Prediction

The equation to be estimated by maximum likelihood is 0 1 1 A; q y t = At @ 1 kt = 2r 1 1+ +1 st

where

At = A0 (1 + )t ; s t = At : yt is the TFP index in the manufacturing sector and is the technological change rate. The error term has a normal distribution with a mean of 0. The estimation results are summarized in Table 2, and Figure 4 plots the observations and predictions about the TFP index from this estimation. This result implies the following. First, since is signi…cantly positive, the level of uncertainty has a signi…cant upward trend as the technology parameter A increases. This is consistent with the upward trend in idiosyncratic risk observed in several papers (Campbell, Lettau, Malkiel, and Xu (2001), Comin and Philippon (2005), Comin and Mulani (2006), etc.). Second, a sizable part of TFP growth is explained by the uncertainty story. To test this, I did the benchmark estimation of setting k= 2r = 1 over the sample period and = 0 (when the change in A is the unique source of TFP growth). The benchmark

16

Table 2: MLE results Coef. S.E. .551 .252 .011 .003

is estimated as 0.014. Compared with this benchmark growth rate, about 20% of TFP growth is driven by …rm-level uncertainty.7

4

TFP dispersion and growth: Multiple Industry Model

So far, the model has only one sector. To obtain a cross-sectional view among industries, I extend the model to include many industries and one research sector. The aim of this section is to analyze the relation between …rm-level TFP dispersion and growth.8 Firm- or plant-level productivity dispersion is widely observed in many industries (Bartelsman and Doms (2000), Dwyer (1998), Syverson (2003), etc.). Dwyer (1998) reports a positive relationship between the …rm-level TFP dispersion and the …rmlevel TFP growth within U.S. 4-digit textile industries, too.9 This relation is not a special feature of the textile industries. In Figure 5, TFPgrowth and Dispersion during 1969-2003 in SIC 2-digit manufacturing industries (except the tobacco industry) are 7

Based on this estimation, the obsolescence rate of knowledge capital, , is estimated using the accumulation function (9). The estimated parameter is more than 0.8, which seems too high. The linear function form of knowledge accumulation (k_ = N k) may be too speci…c, but I keep it in the model for simplicity because all propositions in this paper hold if the gross gowth rate of social knowledge is increasing in the total amount of research N . 8 Oikawa (2007) is another application of the multiple industry model which is essentially the same as the one described here. 9 Imbs (2004) shows a similar relation in terms of volatility. It points out that volatility and growth could be positively related at the sector level, even though the correlation between them is negative at the aggregate level. If the sector-level growth rates are negatively correlated, even though volatile sectors grow fast, aggregate volatility will be low in fast-growing economies. The aggregate relation is obscured by covariance among sector-speci…c shocks. Based on this idea, Imbs uses panel data from the United Nations Industrial Development Organization (47 countries and 28 sectors) to show that variances in growth rates of value-added at the sector levels are positively correlated with mean growth rates. He also reports the same result among OECD countries.

17

4

35

3

30

23

TFP growth 2

36 37 25 22 33

34

39 38 28

1

24 32

0

29

26

1.5

2

27 20

31

2.5 Dispersion

3

3.5

Figure 5: TFP growth and dispersion: 1969-2003 (the 2-digit numbers are SIC codes)

scattered10 , where TFPgrowth is the annual growth rates of the weighted average …rmlevel TFP in each industry and Dispersion is the logged average (during the sample period) of the weighted yearly standard deviations of …rm-level TFPs in each industry. I used the Compustat data set for the …rm-level data. All data plotted in Figure 5 are listed in Table 3 in the Appendix with detailed data construction. Figure 5 suggests a positive correlation between productivity growth and dispersion. The OLS regression indicates TFPgrowth =

1:749 + 1:247 Dispersion: (1:394)

(:537)

where the numbers in parentheses are standard errors. Thus, there exists a signi…cant positive correlation among those variables. Below, I present an explanation of this relationship with using the multiple-industry version of the model.11 In the one-sector model, high uncertainty leads to greater knowledge accumulation, which in turn results in higher average productivity by further reducing 10

I omit the tobacco industry because it has extremely low growth and dispersion. If I were to include it, the positive relation is much stronger, but the relation among other industries is obscured. I omit data before 1969 because the Producer Price Index for transportation equipment isn’t available until then. 11 Dwyer (1998) and Jovanovic and Tse (2006) explain the positive correlation by vintage models: new entrants equip a new technology that is the most productive, so an industry with more R&D expenditure has larger dispersion and faster growth.

18

uncertainty. This implies low dispersion of productivity in the one-sector model. However, in the multiple-industry model where one research sector provides information to all industries, a highly uncertain industry has high dispersion because knowledge accumulation is not independent among industries. Industries.— Suppose there exists a continuum of industries i 2 [0; 1], producing distinct …nal goods. Each industry has its uncertainty level si , where s0 = 0 and si > sj if i > j.12 I assume that all industries have the same technology parameter A. I denote the price of good i as qi and set q0 = 1. Equilibrium prices.— In this multiple-industry setting, a …rm may choose in which industry it operates and also may choose its research specialization. Because there is no switching cost in the model, every producer in every industry and every researcher must have the same expected pro…ts. In equilibrium, the researcher wage and …nal goods prices are determined such that their expected pro…ts are equal. To settle the expected pro…ts, consider industry 0 …rst. Because there is no uncertainty in industry 0, producers earn an assured pro…t of A. Therefore, the expected pro…t of any …rm has to be A in equilibrium. Then the equilibrium researcher wage is w = A: Let qi be the …nal goods price in industry i. The expected pro…t of a producer in industry i is wni : qi A(1 h(ni ; si )) k Before derivation of the equilibrium …nal goods prices, qi s, let me denote the minimum s at which a positive amount of research is demanded as s(k) =

2 r

k+

2 r

:

The industries with s 2 [0; s(k)] do not require any researcher. Therefore, among those industries, the expected productivity is A(1 si ) and qi = 1=(1 si ). In industries with s > s(k), producers employ some researchers and the expected pro…ts then turn out to be ! r 2 2q s r i e 2 + r : i = A qi k k This expected pro…t must be equal to A. Therefore, qi =

2 r si kh2i

if si > s(k);

12

The assumption that there exists no uncertainty in industry 0 is just for simplicity of calculation. All propositions hold even under s0 > 0.

19

where hi is the expected ine¢ ciency in industry i,13 hi = 1+

q

1 k si

2 r 2 r

: +1

The inter-industry relationship between productivity dispersion and growth depends on the pro…les of hi and qi . Below, I will examine their relationship. Productivity dispersion and growth. — Since the e¢ ciency loss z follows a 1 2 z) =hi follows normal distribution with mean 0 and variance hi =si = 1+ni = 2 , si ( r 2 14 -distribution with d.f. 1. Therefore, V ar A(1

z)2 ) = 2A2 h2i :

(

(13)

On the other hand, the growth rate of the average productivity in industry i is gy

y_i 1 h3i k_ = : yi 1 hi 2(1 hi ) si 2r

(14)

k and gk are common across industries. The …rst term in the right hand side is strictly increasing in h. The larger h is, the more room there is for improvement. The second fraction describes to what extent the researcher wage declines in response to a one-unit increase in k=s 2r , the informational value of knowledge. The third fraction is the amount of increase in the informational value of knowledge in an instant. The …rst two fractions are increasing in s (h is increasing in s), and the last term is decreasing in s. However, when k is su¢ ciently large, the …rst two e¤ects dominate the third. Proposition 5 shows this property. Proposition 5 The average productivity growth is higher in an industry with a larger variance of productivity if k is su¢ ciently large. Proof. Suppose that the variance in TFP is greater in industry i than in industry j, which implies si > sj from (13). Since gy is increasing in s if k>

2 r (1

+ 3s);

gy > gy if k is su¢ ciently large. 13

Since the relative wage w=qi corresponds to the researcher wage in the one-sector model for each si , the technical ine¢ ciency has the same form between the one-sector and the multiple-industry models. 14 If I introduce the growth of A, this model is also consistent with the persistency of productivity dispersion, even though the technical ine¢ ciency h is decreasing over time.

20

Productivity growth and dispersion are indirectly related by way of the uncertainty level. There is no direct causality between these two variables. A higher s makes h higher because di¢ culty in production cannot be perfectly canceled out by research. On the other hand, the productivity growth rate is higher under a higher s because additional information about shocks improves technical e¢ ciency more. This proposition might seem to be inconsistent with the result from the one sector model; high uncertainty implies low technical ine¢ ciency in the long run, which corresponds to low variance in productivity. If each industry has a distinct research sector and its own knowledge accumulation, the result should be the same as the one-sector model. In this extension, there is only one research sector and the accumulation of knowledge is the same in all industries.

5

Social Optimality

In this section, I will compare the market equilibrium path in the previous section with the social optimal path to clarify the characteristics of the current model. Since there exists intertemporal positive externality in research, research in market equilibrium is underinvested. Then, I will provide a research subsidy policy which achieves the social optimal steady state in the market.

5.1

The Social Optimal Steady State

Social planner’s problem.— Consider the social planner’s problem that maximizes the sum of the present discounted total output, Yt , over the in…nite horizon with a constant interest rate of r. Z 1 max e rt Yt dt (15) 1 fnt gt=0

0

subject to

k_ t = (

t

(16)

)kt

nt : kt + n t The value function of the maximization problem de…ned in the above is Yt = (1

rv(k) = max n

t )Af (nt ; s);

t

k Af (n) + v 0 (k) k+n

21

=

n k+n

k

(17)

(18)

Figure 6: The competitive steady state and the social optimum steady state

Then, usual steps lead the following two di¤erential equations determine the social optimal dynamics: rf 0 (n) f (n; s) n_ = n n(k + n)f 00 (n) fn (n; s) k_ n = : k k+n

r+ k r

n ; (19)

Hence, the social optimal steady state is pinned down by the intersection of the following curves: n_ r f (n; s) n ; (20) =0 , k= n r+ fn (n; s) k_ 1 =0 , k= n: (21) k Figure 6 illustrates these curves and the social optimal steady state is indicated by SO, or (n ; k ). (Note that f (n; s)=fn (n; s) is strictly convex in n.)

22

The external e¤ect in the market steady state.— Now compare the social optimum to the market equilibrium in the previous section. Similar to Lucas (1988) and Romer (1990), this model has an external e¤ect about knowledge accumulation. In their papers, human capital or technology has an external e¤ect which causes ine¢ ciency in market equilibrium. In this model, any research outputs are accumulated as social knowledge and they make future research activity more creative. However, each individual …rm considers only bene…ts to the current productivity from research. On the social optimal path, more research is done and knowledge accumulation is accelerated. To see this external e¤ect more clearly, I show that the total amounts of research is always smaller in market equilibrium for any given knowledge level. Proposition 6 For a given k su¢ ciently large, the total amount of research in the social planner’s problem is greater than that in the market equilibrium. Proof. It su¢ ces to show that n > n for any given k since the total amount of research is k, where is increasing in n. From Equation (18), the …rst order condition about n is as follows: Af 0 (n; s)

@ 0 @ [ Af (n; s)] + [v (k) ( @n @n

) k] = 0:

(22)

The …rst term is the marginal private bene…t and the last term is positive intertemporal externality. The second term is marginal cost and it is equivalent to w=k in market equilibrium since @ [ Af (n; s)] = A (1 @n

21

)

h k

+

h2 s

2 r

w Ah2 = : = 2 s r k

Therefore, the …rst two terms in (22) corresponds with the …rst order condition of an individual …rm in market equilibrium. The social planner perceives more marginal bene…ts from research activity by the positive intertemporal externality of the last term in (22), which is positive. So, n > n . Steady state comparison.— Figure 6 also draws the market steady state, M . It is de…ned in the current diagram by arranging the equation = as k = (1

)

f (n; s) fn (n; s)

(23)

and (21). Figure 6 illustrates the steady state knowledge capital is greater in the social optimal steady state. The next proposition shows this is always the case under some condition and also shows that the average productivity and total output are greater in the social optimal steady state.

23

Proposition 7 The market steady state levels of knowledge and total outputs are lower than those in the social optimal steady state when + r is su¢ ciently small. Proof. The curve (23) places above the corresponding curve (21) if + r is su¢ ciently small since (1

)

f (n; s) fn (n; s)

r +r

f (n; s) fn (n; s)

n

=

1 +r

(1

r)

f (n; s) + rn fn (n; s)

>0

for su¢ ciently small + r. Thus, k > k for su¢ ciently small + r. Next. the expected technical e¢ ciency in each steady state is solved for as follow: f (n ; s) = 1

f (n ; s) = 1

2 1 q (2 + ) 1 + 1 2 1 q 1+ 1+ 1

4(1+ ) (2+ )2 s

4 (1+ )2 s

;

;

(24)

(25)

)(r + ) . r Since = = , the total output in SO is greater than that in M if f (n ; s) > f (n ; s). This inequality always holds since where

(1 +

)2

>

1+ (2 + )2

(1

for any

2 (0; 1) and r > 0.

(26)

The e¤ect of uncertainty on the social optimal steady state.— As illustrated in Figure 7, a rise in the level of uncertainty s (s1 > s2 ) shifts the curve (20) downward since f (n; s)=fn (n; s) n is decreasing in s for a given n. Hence, an increase in s delivers higher research and knowledge capital in the steady state under the social optimum. And, moreover, the steady state average productivity (and total output) is higher with a higher s from (24). This result is the same as in the market steady state. However, the mechanism is totally di¤erent. In the market steady state, a rise in the level of uncertainty increases the scale of intertemporal externality through expanding the measure of researchers. Now, there is no externality by de…nition. Instead, there is a trade-o¤ between research and production for the social planner. Actually, the social planner of the economy with lower s (say s2 ) can choose the path to SO(s1 ) in Figure 7, but does not choose it and leads the economy to SO(s2 ) simply because to get SO(s1 ) imposes too much cost of research investment in terms of the present discounted value.

24

Figure 7: The e¤ect of the level of uncertainty on the social optimal path. s1 > s2 .

5.2

Research Subsidy Policy

Here, I will show a simple redeem resolving ine¢ ciency in the market. The social optimal steady state can be achieved under competition by an appropriate subsidy policy for research. Suppose that a researcher receives a subsidy of bw per each research (b > 0), i.e., researcher’s income is (1 + b)w while producers pay only w to each researcher. The expected technical ine¢ ciency in instantaneous equilibrium with such a subsidy is 1 q : (27) h= 2 r 1 + (1+b)k + 1 s 2 r

As seen in the above equation, a subsidy reduces the expected technical ine¢ ciency since there are more researchers in equilibrium and a cheaper researcher wage. For a given k, producers demand more researchers than before the subsidy policy is executed. As a result, the curve of n_ = 0 determining the market steady state shifts downward

25

in the n-k diagram. In other words, (23) changes to k = (1

)

f (n; s) fn (n; s)

b 1+b

2 r

(28)

;

which lies below the curve (23). Because one can drag down the curve by increasing b, the appropriate adjustment of b achieves the social optimal levels of knowledge capital, the measure of researchers, average productivity, and the total output under competition in the steady state. The e¤ect of a change in the level of uncertainty on the appropriate subsidy.— Then, how the appropriate b changes in response to an increase in the level of uncertainty? As mentioned before, the both n_ = 0 curves in SO and M shift downwardly in response to an increase in s. So, it is ambiguous whether the subsidy should be larger for a higher level of uncertainty. To see this, I derive an equation that b should satisfy to attain the social optimum. 1 (1 + )(1 + )a0 1 + (2 + ) 2 (2 + ) 4 (1 + ) = ; a1 = : 1+ (2 + )2

b = 1+b where a0

r 1

a1 s

s +1 ; a1

Since the left hand side is increasing in b and the right hand side is increasing in s, a higher level of uncertainty requires higher level of research subsidy to achieve the social optimum under competition.

6

Concluding Remarks

In this paper, I investigated the positive impact of …rm-level uncertainty on productivity growth. Based on …rms’optimization, uncertainty stimulates research activity, and hence accumulation of knowledge capital. High uncertainty implies a low level of average productivity in the short run, but faster accumulation of knowledge. In the long run, an economy with more uncertainty has a higher productivity level than one with less uncertainty. Empirically, the model …ts the U.S. manufacturing TFP behavior during the last two decades. The estimation result implies that the model explains about 20% of TFP growth during the sample period, and that there exists a signi…cant upward trend in …rm-level uncertainty, which is observed in recent papers. In the multiple-industry model, I derive the cross-sectional productivity dispersion and the growth rate of average productivity within industries to explain the observed positive correlation among those variables in the U.S. manufacturing sector. In the

26

model, they are positively correlated because the inherent …rm-level uncertainty raises both dispersion and growth rates of productivity under certain conditions. One interesting aspect of the characteristic of knowledge capital in this paper is that it is de…ned as publicly available. This is, of course, an extreme case, but, even if this assumption is relaxed, the same argument holds at least to some degree when there exists information spillover. Related to this point, the model in this paper provides a new intuition about intellectual property rights. The publicly available knowledge implies no intellectual property rights. This model thus shows that the positive e¤ect of uncertainty works at its maximum when there are no intellectual property rights. As discussed in Boldrin and Levine (2002), intellectual property rights are not crucial for knowledge creation, and it is better to remove them if creators still have incentives to innovate. In this context, the model in this paper suggests a new cost of intellectual property rights.

References Arrow, K. (1962): “The Economic Implication of Learning by Doing,” Review of Economic Studies, 29, 155–173. Bartelsman, E., and M. Doms (2000): “Understanding Productivity: Lessons from Longitudinal Microdata,”Journal of Economic Literature, 38, 569–594. Basu, S., J. G. Fernald, and M. D. Shapiro (2001): “Productivity Growth in 1990s: Technology, Utilization, or Adjustment,” Carnegie-Rochester Conference Series on Public Policy, 55(1), 117–165. Bertola, G. (1994): “Flexibility, Investment, and Growth,” Journal of Monetary Economics, 34(2), 215–238. Boldrin, M., and D. K. Levine (2002): “Perfectly Competitive Innovation,”Federal Reserve Bank of Minneapolis, Research Department Sta¤ Report 303. Campbell, J. Y., M. Lettau, B. G. Malkiel, and Y. Xu (2001): “Have Individual Stocks Become More Volatile? An Empirical Exploration of Idiosyncratic Risk,” Journal of Finance, 56(1), 1–43. Comin, D. (2000): “An Uncertainty-Driven Theory of the Productivity Slowdown: Manufacturing,” C.V. Starr Center for Applied Economics, New York University, Working Paper No. 00-16. Comin, D., and S. Mulani (2006): “Diverging Trends in Aggregate and Firm Volatility,”Review of Economics and Statistics, 88(2), 374–383.

27

Comin, D., and T. Philippon (2005): “The Rise in Firm-Level Volatility: Causes and Consequences,”NBER Working Paper 11388. Cummins, J. G., and G. L. Violante (2002): “Investment-Speci…c Technological Change in the US (1947-2000): Measurement and Macroeconomic Consequences,” Board of Governors of the Federal Reserve System, Finance and Economics Discussion Series 2002-10. Dixit, A. K., and R. S. Pindyck (1994): Investment under Uncertainty. Princeton University Press. Dwyer, D. W. (1998): “Technology Locks, Creative Destruction, and Nonconvergence in Productivity Levels,”Review of Economic Dynamics, 1, 430–473. Imbs, J. (2004): “Growth and Volatility,”CEPR Discussion Paper 3561. Jovanovic, B., and Y. Nyarko (1996): “Learning by Doing and the Choice of Technology,”Econometrica, 64(6), 1299–1310. Jovanovic, B., and C.-Y. Tse (2006): “Creative Destruction in Industries,”NBER Working Papers 12520. Kremer, M. (1993): “The O-Ring Theory of Economic Development,” Quarterly Journal of Economics, 108(3), 551–575. Kumbhakar, S. C., and C. A. K. Lovell (2000): Stochatic Frontier Analysis. Cambridge Univ. Press. Lucas, R. E. (1988): “On the Mechanics of Economic Development,” Journal of Monetary Economics, 22, 3–42. Martin, P., and C. A. Rogers (2000): “Long-Term Growth and Short-Term Economic Instability,”European Economic Review, 44, 359–381. Matsuyama, K. (2002): “The Rise of Mass Consumption Societies,” Journal of Political Economy, 110(5), 1035–1070. Oikawa, K. (2007): “Final Goods Substitutability and Economic Growth,”Economics Bulletin, 15(16), 1–7. Ramey, G., and V. A. Ramey (1995): “Cross-Country Evidence on the Link between Volatility and Growth,”American Economic Review, 85, 1138–1151. Romer, P. M. (1990): “Endogenous Technological Change,” Journal of Political Economy, 98(5).

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Stiroh, K. (2001): “Information Technology and the U.S. Productivity Revival: What Do the Industry Data Say?,”American Economic Review, 92, 1559–1576. Syverson, C. (2003): “Product Substitutability and Productivity Dispersion,” NBER working paper 10049.

Appendix: Estimation of Production Functions and TFP Computation The data come from Compustat North America. To calculate …rm-level TFP, I de…ne outputs as sales plus changes in …nal goods inventories (for a …rm failing to report its change in …nal goods inventories, I use the average change in …nal goods inventories in the industry to which that …rm belongs). The outputs are transformed to real terms by PPI (by major commodities), which is available from BLS. Labor is de…ned as the number of employees.15 Capital is de…ned as gross property, plant, and equipment, transformed to real terms by the price index of nonresidential investment goods, which is available from BEA. The form of the production function is assumed to be CobbDouglas in each SIC 2-digit industry. To eliminate time and regional e¤ects, I add time dummies and state dummies. After the production function estimation, …rmlevel TFPs are computed as y=k k l l , abusing notations. To avoid the in‡uence of possible human errors in the data, I omit outliers. An observation is de…ned as an outlier if TFP is …ve standard deviations away from the mean of TFPs within the same industry and in the same year. I omit 538 observations out of 89,554.16 Next, the weighted means and standard deviations of TFPs are calculated by years and industries (weighted by the sales shares). Finally, the degree of dispersion in one industry is de…ned as the logged mean of weighted standard deviations over the sample period in that industry. The annual growth rates of the mean TFPs are estimated by projecting the logs of the weighted mean TFP on the time variable. Table 3 lists the results.

15

The Compustat dataset does not require that …rms report working hours. Even if I include those observations, the relation mentioned in this paper is still positive, but relatively weak. 16

29

Table 3: Mean-TFP growth and Dispersion (1969-2003). SIC code 20 (Food and kindred products) 22 (Textile mill products) 23 (Apparel and other textile products) 24 (Lumber and wood products) 25 (Furniture and …xtures) 26 (Paper and allied products) 27 (Printing and publishing) 28 (Chemicals and allied products) 29 (Petroleum and coal products) 30 (Rubber and miscellaneous plastics products) 31 (Leather and leather products) 32 (Stone, clay, glass, and concrete products) 33 (Primary metal industries) 34 (Fabricated metal products) 35 (Industrial machinery and equipment) 36 (Electrical and electronic equipment) 37 (Transportation equipment) 38 (Instruments and related products) 39 (Miscellaneous manufacturing industries)

30

Dispersion 2.86 2.79 2.69 2.43 1.69 1.92 2.90 2.74 2.64 3.03 2.24 1.84 2.41 2.11 3.45 2.77 2.75 2.68 2.73

TFP growth (%) 0.07 1.68 2.65 0.99 2.05 -0.11 0.43 1.21 0.38 3.22 -0.01 0.61 1.47 1.38 3.92 2.40 2.39 1.31 1.39