1

1

Introduction

What determines the growth rate of a firm? The average growth rate of capital stockof firms that continuously appear in the COMPUSTAT data set between 1970 and1999 was about 16% during the 1970s, 14% during the 1980s and 15% during the
1990s1 . Major companies in the U.S. continued to grow for the 30 years covered byour data set.
In order to explain the continuous expansion of a firm, traditional investment
theories typically assume the existence of a convex adjustment cost function [e.g.,
Lucas and Prescott (1971)]. This explicitly links the growth rate of capital to Tobin’s
Q. The derived growth rate is independent of the firm’s size, which arguably serves
as a first approximation for the growth of a large firm [e.g., Sutton (1997)].

The

dynamics of a firm are described as a process of adjustment toward the desired capitalstock.
However, this adjustment cost function is questioned by evidence that investment
is lumpy at the plant level [e.g., Doms and Dunne (1998)].

Lumpy investment ismore consistent with alternative theories that emphasize the importance of a nonconvex adjustment cost function [ e.g., Caballero (1999)].

Diﬀerent from the Q

theory of investment, adjustment is immediate once a firm decides to invest. Hence,
continuous growth must be explained by expanding exogenous external investment
opportunities2 .
In contrast to previous literature, this paper completely dismisses conventional
adjustment cost functions from our model and examines an alternative friction: a
1

The average growth rate of capital is the net value of property, plant and equipment (Compustat# 8) over the deflator for nonresidential investment.

The deflator is taken from the Bureau of

Economic Analysis.
2
Abel and Eberly (1994) incorporate two frictions into one model and discuss its relation to
Q.

Cooper and Haltiwanger (2000) argue that a model that mixes both convex and nonconvex

adjustment costs with irreversibility fits plant-level data well.

2

firm cannot optimize more than one input3 .

Hence, the immediate adjustment of

more than one input is prohibited. It is shown that each investment is lumpy as insunk cost models, but adjustment is gradual as in convex adjustment cost models.
The particular friction that this paper aims to capture is the limited capacityof human attention, the importance of which was originally emphasized by Simon(1947). It is a common observation that organizations demand millions of immediate decisions, but individual CEOs are not able to handle all of them at the sametime.

As it is diﬃcult to delegate strategic decisions, such as finding investment

projects, to other managers, CEOs must spend much of their time making and implementing an investment plan. Because their attention capacity is limited, CEOs
must postpone other strategic decisions such as developing a trained management
group. We investigate how the allocation of attention influences the growth rate ofa firm.
In order to determine the growth rate, we need one more assumption: there isinteraction among inputs, where the interaction means that inputs are strict complements or strict substitutes. When there is no interaction among inputs, adjustment
is immediate: the firm immediately chooses the optimal level of inputs and maintains
it. However, when interaction exists, the current optimal decision depends on theamount of other inputs. It is shown that adjustment then becomes gradual.
In this paper, we examine two-input cases, in which the inputs are called capitaland labor.

Two decisions are called investment and employment.

To give an

intuitive reason for gradual adjustment, consider the case where a firm’s attention isalternately allocated to investment and employment. Suppose that capital and labor
are complements and that the amounts of capital and labor are less than optimal. In
this case, profits would be larger if the firm could increase both capital and labor at
the same time. Hence, when the firm makes its investment decision, it invests more
3This assumption is the same as the one that a firm must incur infinite fixed costs when morethan one decision is made. Hence, the model can be interpreted as a variant of a sunk cost model,
though the firm never incurs sunk costs.

3

than needed for current production in order to increase the marginal product of laborand to enhance employment in the next period. Similarly, when the firm makes its
employment decision, it employs more than needed for current production in order
to increase the marginal product of capital and to enhance its investment in the next
period. This capital and labor hoarding increases cost for current production. This
static loss plays the role of adjustment cost.
Similar to the Q theory of investment, the adjustment cost derived has the property that the accumulated capital stock lowers the marginal cost of capital hoarding.
Hence, it causes a positive feedback mechanism that enhances the continuous adjustment of a firm: an increase in capital stock lowers the marginal cost of capital
hoarding, which gives more incentive to accumulate capital.
The following example is illustrative. Consider a firm that needs both an oﬃce
and skilled managers as productive inputs.

When the firm constructs a building,it will keep spare space. The spare space is not needed for current production, but
it allows the firm to employ and train more managers in future.

Similarly, excess

trained managers are needed to set up a new oﬃce in a diﬀerent region in the future.
Although the spare spaces prepare for future expansion, it is costly to build a skyscraper when a firm currently employs only three managers. In this way, adjustment
cost is endogenized. Note that once the firm employs 30 managers, it may be worth
constructing a building for them. That is, the existence of many skilled managers
as a result of the previous capital accumulation lowers the static loss due to capital
hoarding and enhances further capital accumulation.
The above adjustment process may not occur when a firm can optimally allocate
its attention. It may be optimal to stick to an employment decision. In this case,because a firm does not need to hoard labor, there is no growth. This might happen
when the owner of a restaurant cares only about employing or training skilled chefs
but not about investing in a new restaurant.
A natural question is: What might be the conditions under which the firm starts

4its adjustment? This paper derives necessary and suﬃcient conditions for the firm
to adjust to the desired level under a constant-returns-to-scale production function.
Intuitively, the firm must compare two types of rents: rent from changing its attention
and that from sticking to one decision. The firm starts its adjustment if and only if
the former rent is larger than the latter. We argue that when many workers quit a
job and a firm cannot accumulate skills, the firm sticks to employment and stops its
adjustment to the desired capital stock4 .
Assuming that the condition for the adjustment is satisfied, this paper derives thegrowth rate of a firm when the production function has constant returns to scale, there
is no adjustment cost function and markets are competitive.

The derived growthrate is shown to be a positive function of the modified user cost and the modified
wage rate, both of which are correlated with Tobin’s Q. It is shown that even if there
is no adjustment cost function and the production function has constant returns toscale, Tobin’s Q can deviate from one and contains information about investment
opportunities.
The dynamics of investment have some empirical support.

On the one hand,because there is no convex adjustment cost function in our model, each investment
is lumpy, which is similar to a model with sunk cost.

On the other hand, once

decisions are aggregated over time, the investment—capital ratio is independent of
the firm’s size and correlated with Tobin’s Q, which is similar to the Q theory ofinvestment. In contrast to the Q theory of investment, the correlation between thederived investment—capital ratio and Tobin’s Q is imperfect. Our calibration results
show that the investment—capital ratio has a weak positive correlation with Q, but
once we control for the cash flow—capital ratio, the correlation becomes negative.
That is, the investment—capital ratio has a stronger positive relationship with the4

The results may interest readers for an alternative reason, because if a firm sticks to one decision,

it cannot reach the standard optimal solutions. It implies that the standard profit maximization
problem may not be a good approximation of reality if limited attention is important. The derived
condition clarifies when the standard model is valid.

5

cash flow—capital ratio. The weak correlation between Q and investment is found in
the literature [e.g., Chirinko (1993)], and the robust impact of cash flow on investment
is also well known [e.g., Fazzari, Hubbard and Petersen (1988)].
Although we examine a model of limited attention, agents are perfectly rational,
which is diﬀerent from Simon (1947).
plement a plan.

In our model, attention is required to im-

A similar strategy was previously used by Giﬀord (1992).

She

investigated the role of limited attention on the growth of a firm in a dynamic programming framework such as ours. However, Giﬀord (1992) does not investigate theinteraction among inputs and, therefore, adjustment is immediate5 .
Investment decisions with complementary capital are investigated by Dixit (1997),
Eberly and Mieghem (1997) and Jovanovic and Stolyarov (2000).
Jovanovic and Stolyarov (2000) also produce spare spaces.

In particular,

However, their model

does not deal with the limited capacity of attention and, therefore, the source ofgrowth is exogenous movement of frontier technology.
This paper can be also interpreted as constructing a micro foundation for the
adjustment cost of investment.

To our knowledge, the literature that derives theadjustment cost of investment relies on gradual learning [e.g., Prescott and Visscher (1980)]. This paper does not include an information problem but still derives
adjustment cost because hoarding behavior causes static losses.
Recently, Abel and Eberly (2003) proposed an investment model without an adjustment cost function. Similarly to ours, their model also predicts that Tobin’s Q
can be greater than one without adjustment cost, and the investment—capital ratio
is correlated with Tobin’s Q but not with marginal Q. In their model, investment
5

More recently, some researchers investigated limited attention in a rational agent model to ex-

plain the stickiness of prices [Reis (2004)]. They describe limited attention by assuming information
processing friction . In contrast, this paper limits the number of decisions an agent can take immediately.

This strategy allows us to investigate interaction among inputs, which is the key toderiving a positive feedback mechanism for capital accumulation. We hope that our strategy nicelycomplements theirs.

6opportunities are provided by exogenous rent due to monopoly power or decreasingreturns-to-scale technology.

By contrast, our model does not need any exogenous

shocks and is based on constant-returns-to-scale technology and a perfectly competitive market. Because the firm cannot optimize two inputs at the same time, there
is a fixed input at each moment that creates quasi-rent, which provides an incentive
for the firm to grow.
The paper is organized as follows.

The next section examines the firm that

allocates its attention alternately to investment and employment decisions.

This

exercise clarifies an intuitive mechanism to induce the dynamics of a firm. Section 3
analyzes the optimization of the allocation of attention. Section 4 investigates theconditions under which the firm adjusts to the desired level.

Assuming that thefirm satisfies the conditions for adjustment, Section 5 derives the growth rate of afirm. Section 6 calibrates the model and discusses quantitative relationships between
investment, Q and cash flow in this model. Section 7 concludes and discusses possible
extensions.

2

The Alternate Allocation of Attention

In this section, we examine a standard firm’s profit maximization problem, exceptthat a firm can optimize only one input at a time. In order to illustrate our idea,
we assume that the firm alternates its attention. This assumption is relaxed in thenext section.
As a benchmark, we first consider a stylized static problem. A firm maximizes its
profits by choosing capital and labor given the wage rate, w, the user cost of capital,u, and the production function F (K, L), where K is capital stock and L is labormeasured in eﬃciency units. Although we call K and L capital and labor for the
purpose of explanation, our mechanism works for any inputs.
Assuming that the production function is twice continuously diﬀerentiable, bounded,7and strictly concave, and that the solutions are interior, it is known that the firm’s
decisions are characterized by two first-order conditions:u = FK (K ∗ , L∗ ) , w = FL (K ∗ , L∗ ) ,where K ∗ and L∗ are the solutions to this static optimization problem.

Since themarginal products of capital and labor are equal to user cost and wage rates, the firm
has no incentive to change its decision.
One of the important assumptions behind this result is that the firm can maximize
labor and capital at the same time. However, this may not be true in reality. Whena firm invests, it must consider thousands of matters: which machine to buy, where
to set it, who should be assigned as an operator and so on.

Our economic model

is abstracted from these details. However, it is natural to think that planning andimplementation are time consuming. More specifically, we assume that the firm can
optimize only one input at a time.
When the firm optimizes an input, another input is assumed to be constant. This
assumption aims to capture a role of routine operation. When the firm makes optimal
decisions in investment, they presume that they maintain the same amount of laboras before6 . After investment is made, the previous level of labor may not be optimal.
However, as the previous level of labor is the result of past optimization, it would
be one of the best alternatives given the restriction of limited attention.

In thisway, maintaining the same level can be interpreted as a very primitive descriptionof “remembering by doing” which Nelson and Winter (1982, p.99) emphasize as animportant function of a routine.
Given these assumptions, the standard problem must be rewritten as the following
dynamic programming problem:
6Note that we measure labor in eﬃciency units. Hence, labor is considered to be the set of skills

accumulated in a firm. The development of more skilled workers would demand time and attention.
However, it would be an innocuous assumption that the firm can maintain the same level of humancapital without attention.

8Vl (K) =Vk (L) =max {F (K, L0 ) − wL0 − uK + βVk (L0 )} ,L0 ∈R+

max {F (K 0 , L) − wL − uK 0 + βVl (K 0 )} ,K 0 ∈R+

where β ∈ (0, 1) is a discount factor, and Vl (K) and Vk (L) are the value of the firm
when it currently makes an employment decision and when it currently makes aninvestment decision, respectively. Because routinely chosen inputs are assumed tobe constant, when the firm makes an investment decision, labor is considered a stock
variable; when the firm makes an employment decision, capital is a stock variable.
Two first-order conditions are
FL (K, λ (K)) = w − βVk0 (λ (K)) ,(1)

FK (κ (L) , L) = u − βVl0 (κ (L)) ,(2)

where λ (K) and κ (L) are optimal policy functions. Compared with the first-orderconditions for simple static optimization, marginal cost deviates from input prices
because of the additional terms, βVk0 (λ (K)) and βVl0 (κ (L)). Because the current
investment decision (employment decision) can influence future profits, this possibility
must be taken into account. Hence, the marginal products of capital and labor must
be equal to input prices minus the eﬀects on future profits. We call w − βVk0 (λ (K))

the modified wage rate and u − βVl0 (κ (L)) the modified user cost.

To illustrate the main idea, we assume that the value function is twice continuously
diﬀerentiable. Then it is shown that

FKL (K, λ (K)),
FLL (K, λ (K)) + βVk00 (λ (K))
FKL (κ (L) , L)
κ0 (L) = −.
FKK (κ (L) , L) + βVl00 (κ (L))

λ0 (K) = −

(3)
(4)

Because we assume a strict concave production function, the value functions are also
strictly concave. Therefore, the denominators of the two equations are negative. This9means that both policy functions are strictly increasing functions when two inputs
are strict complements ( FKL (K, L) > 0 for all K and L ) and that they are strictly
decreasing functions when two inputs are strict substitutes (FKL (K, L) < 0 for all K
and L ). We assume that the production function is either strictly complementaryor strictly substitutionary.

These assumptions are suﬃcient but not necessary forour results, but it makes our explanation clearer.

Figure 1: The growth and decline of a firm
We are interested in the following dynamics:
Kt = κ (λ (Kt−1 )) .
10

Equations (3) and (4) imply that current capital stock is strictly increasing in theprevious capital stock if two inputs are either strict complements or strict substitutes.
Figure 1 depicts the dynamics of the firm, which are shown to be representative,
below. If the initial capital stock is lower than K ∗ , it grows and converges to K ∗ .
On the other hand, if the initial capital stock exceeds K ∗ , the firm declines andconverges to K ∗ . It is shown below that K ∗ coincides with an optimal solution to
the static problem.

Hence, the process of growth can be seen as the adjustment

process to the desired capital stock.
Note that if FKL (K, L) = 0 for all K and L, the derivative of the function κ (λ (·))
is 0. Hence, κ (λ (Kt−1 )) in figure 1 is horizontal. This means that adjustment isimmediate.

It shows that interaction between two inputs for some K and L isnecessary for gradual adjustment.
In order to understand the mechanism of the dynamics, two Euler equations arederived:MCLH (Lt , Lt−1 ) = MRLH (Lt ) ,MCKH (Kt , Kt−1 ) = MRKH (Kt ) , where
MCLH (Lt , Lt−1 ) ≡ w − FL (κ (Lt−1 ) , Lt ) , MRLH (Lt ) ≡ β [FL (κ (Lt ) , Lt ) − w] ,MCKH (Kt , Kt−1 ) ≡ u − FK (Kt , λ (Kt−1 )) , MRKH (Kt ) ≡ β [FK (Kt , λ (Kt )) − u] .
MCLH (Lt , Lt−1 ) ( MCKH (Kt , Kt−1 ) ) represents the marginal costs of labor hoarding (capital hoarding); MRLH (Lt ) ( MRKH (Kt ) ) represents the marginal rent
from labor hoarding (capital hoarding). Euler equations show that it is optimal toequate the marginal cost and marginal rent from hoarding. Since the intuitive logic
is the same, we confine our discussion mainly to capital. However, the reader canapply the same logic to labor.
MCKH (Kt , Kt−1 ) is defined as the deviation of user cost from the marginal product of capital. When user costs are larger than the marginal productivity of capital,
11

the firm employs more capital than is needed for current production.

Employing

one more unit of capital increases expenses by user cost and output by the marginal
product of capital. Hence, the deviation of user cost from the marginal productivity
can be considered the marginal cost of capital hoarding. Although we do not haveany adjustment cost function, the cost of capital hoarding serves as adjustment costin this paper.
On the other hand, hoarding capital generates a profit opportunity for the firmin the next period, because when the firm invests in capital, it knows that capital isfixed in the next period: it creates rent from capital. In order to increase this rent,
the firm has an incentive to invest more than needed. Employing one more unit ofcapital increases output by the marginal product of capital and fixed cost by user
cost.

Hence, the marginal rent from capital hoarding is the present value of thisdiﬀerence, which is how MRKH (Kt ) is defined.
When the firm reaches the steady state, K ∗ and L∗ , the firm keeps the same levelof capital stock and labor. Hence, there is no reason to maintain capital hoarding.
Substituting K ∗ = Kt = Kt−1 and L∗ = Lt = Lt−1 into two Euler equations, it
is shown that capital and labor in the steady state are the same as under static
optimization:
FK (K ∗ , λ (K ∗ )) = u, FL (κ (L∗ ) , L∗ ) = w.
By taking the derivative with respect to K around the steady state, the slope ofκ (λ (K)) at the steady state in Figure 1 is shown to be less than one:dκ (λ (K))[FKL (K ∗ , λ (K ∗ ))]2
|K=K ∗ =∈ (0, 1) .
dK
FKK (K ∗ , λ (K ∗ )) FLL (K ∗ , λ (K ∗ ))Since κ (λ (K)) is a strictly increasing function, figure 1 shows that the dynamics must
be globally stable. Hence, the dynamics depicted by figure 1 are representative.
In order to understand the mechanism of growth in detail, let us examine MRKH (Kt )and MCKH (Kt , Kt−1 ). Because of the strict concavity of the production function,
12

MCKH (Kt , Kt−1 ) is strictly increasing in Kt . Similarly, because MRKH (Kt ) =βVl0 (Kt ) for all Kt and the value function is strictly concave, MRKH (Kt ) is strictly
decreasing in Kt . Hence, the intersection is unique, which is depicted in Figure 2.
The dynamics occur because MCKH (Kt , Kt−1 ) depends on Kt−1 . We can derive
the following condition from the definition of MCKH (Kt , Kt−1 ):∂MCKH (Kt , Kt−1 )= −FKL (Kt , λ (Kt−1 )) λ0 (Kt−1 ) < 0.
∂Kt−1
Note that when FKL (K, L) > 0, λ0 (K) > 0, while when FKL (K, L) < 0, λ0 (K) < 0.
Hence, marginal cost declines as capital accumulates. This provides a nice positive
feedback mechanism: an increase in capital stock reduces the marginal cost of capital
hoarding, which generates incentive for the firm to make further investment. Figure 2
describes the feedback mechanism. If MCKH (Kt , Kt−1 ) and MRKH (Kt ) intersect
at a positive value, capital accumulates. Larger capital lowers the marginal cost ofcapital hoarding and provides further incentive to accumulate capital. On the other
hand, if MCKH (Kt , Kt−1 ) and MRKH (Kt ) intersect at a negative value, the firm
disinvests. This increases the marginal cost of capital hoarding and forces the firm
to give up more capital. Since MCKH (Kt , Kt−1 ) and MRKH (Kt ) are equal to 0
when Kt = K ∗ , the dynamics stop.
In the next section, we assume that the production function has constant returns
to scale in K and L and analyze the optimal allocation of attention. In the case of
constant returns to scale, the production function is not strictly concave, and therefore
there is no steady state in general. Without any proof, the dynamics and mechanismof the growth of the firm with a constant-returns-to-scale production function are
depicted in Figure 3 and 4. Figure 4 shows that MRKH (Kt ) is constant when the
production function has constant returns to scale.

Hence, as Figure 3 shows, thefirm can grow as far as market conditions allow or can decline until it vanishes.

13

Figure 2: The mechanism of growth and decline of a firm

14Figure 3: The growth and decline of a firm when the production function
has constant returns to scale

15

Figure 4: The mechanism of the growth and decline of a firm when the
production function has constant returns to scale16

3

The Optimal Allocation of Attention

In the previous section, the order of decisions is given. However, the alternate allocation of attention may not be optimal. Some firms may concentrate on training andemploying workers and not care much about new physical investment. In this section, we relax this restriction and assume that the firm optimally chooses investment
or employment.

This makes our analysis more complicated.

In order to provide

a tractable model, we assume that F (K, L) exhibits constant returns to scale in K
and L. This assumption has other advantages, too. First, it allows us to link ourtheory to Tobin’s Q. Second, since constant returns to scale and a competitive economy eliminate any economic rent, the only source of rent in this model becomes theexistence of quasi-fixed input due to routine operation. In this section we show thatthe rent from the quasi- fixed factor causes Tobin’s Q to deviate from 1 7 .
Let us first set up our model:

V (K, L) = max {Vl (K) , Vk (L; K)} ,

(5)

0
0
0
max
h
i {F (K, L ) − wL + V ((1 − δ k ) K, (1 − δ l ) L )} ,Vl (K) = β(6)αKL0 ∈ 0, 1−δlVk (L; K) =

0
0
max
,
h
i {β [F (K , L) − wL + V ((1 − δ k ) K , (1 − δ l ) L)] − pk I}(7)αL
K 0 ∈ 0, 1−δk0s.t. I = K − K,1and i is an interest rate.
where α < β = 1+i

The parameters δ k and δ l are thedepreciation rate of capital and eﬀective labor, respectively.

The parameter pk is

the price of investment goods, and the price of output is normalized to 1. Equation(5) describes the optimal choice of projects, where the projects are investment or
employment, and the value function V (K, L) represents the value of the firm when
7

There is also a cost to assuming constant-returns-to-scale technology. As the production func-tion is constant returns to scale in K and L, we must restrict our attention to the case of complementarity, below.

17the firm currently selects projects. The function Vl (K) represents the value of thefirm when the firm currently makes an employment decision, which is defined by
Equation (6) and the function Vk (L; K) represents the value of the firm when thefirm currently makes an investment decision, which is defined by Equation (7).
There are three technical diﬀerences from the previous section. First, we explicitly
model investment decisions in Equation (7) and exclude rental expenses of capitalfrom the model. The explicit expression of an investment decision clarifies the link
between investment and Tobin’s Q.

The timing of discounting and depreciation

rate is arranged so we can later derive Jorgenson’s user cost. Second, we allow thedepreciations of capital and labor.

The depreciation of labor is unusual.

Since

we consider L as human capital rather than the number of workers, it has to beinterpreted as the depreciation rate of human capital in a firm. Finally, there are
explicit upper bounds on choice variables, which are influenced by α < β.

Thisguarantees that the solution does not explode and that there exists a unique valuefunction V (K, L). The proof is an application of Stokey and Lucas (1989, p87) and
we omit it8 .
Since the production function has constant returns to scale, the value function isL
expected to be linear in K, V (K, L) = Q (l) K where l = K. Define the deviationof Tobin’s Q from 1 by D (l) ≡

h

i

Q(l)
−1
pk

pk . The standard guess and verify method

proves that D (l) must recursively satisfy the following equation:.
D (l) = β max {π k l, π l }
(

Ã

!

πk
πl
0

0

0

α
α
, αl = 1−δ
and f (·) = F (·, 1).
where k0 = KL , l0 = LK , αk = 1−δkl

is Jorgenson’s (1963) user cost, u ≡ (i + δ k ) pk .
8)

1 − δl
≡ 0max f (k ) − uk − w + D(1 − δ k ) k0 ,k ∈[0,αk ](1 − δ k ) k0(∙ µ ¶Ã!
)¸
1(1 − δ l ) l00
≡ 0maxf 0 −w l −u+D
(1 − δ k ) ,l ∈[0,αl ]l
1 − δk0(8)(9)

The variable u

Variable, π k ( π l ) is the present

Technically, it is also assumed that there is BF ∈ (0, ∞), which satisfies |F (K, L)| ≤BF (|K| + |L|) for any K ∈ R+ and L ∈ R+ .

18

value of the stream of the discounted rent per unit labor (per unit capital) when
the firm currently makes an investment decision (employment decision). The rent
exists because there are quasi-fixed factors in the model. We call π k and π l theaverage rent when the firm makes an investment decision and an employment decision,respectively.
Substituting the definition of D (l) into the definition of π k and π l , we can express
π k and πl as the solutions to the following two equations. The proof of the following
proposition is given in the Appendix.
Proposition 1 The deviation of Tobin’s Q from 1, D (l) =of two measures of average rent, π k and π l :

h

i

Q(l)
−1
pk

pk , is a function

D (l) = β max {πk l, πl } ,where π k and πl are the solutions to the following two equations:
π k = max {Πk (u − β k π l ) , Πk (u) + β l π k }(10)πl = max {Πl (w − β l π k ) , Πl (w) + β k πl } .

(11)max [f (k0 ) − xk0 ] − w(12)1Πl (x) = 0max f 0 − x l0 − ul ∈[0,αl ]l(13)

Πk (x) =k0 ∈[0,αk ]

∙ µ ¶

¸

where β k = (1 − δ k ) β and β l = (1 − δ l ) β.
The function Πk (x) ( Πl (x) ) is the average instantaneous profit function when
the firm invests ( employs ) and maps from the prices for the variable inputs to
instantaneous profits per unit labor (per unit capital).
The proposition shows that Tobin’s Q can deviate from 1 when max {πk l, πl } is notequal to 0 and that π k and π l are the maximums of diﬀerent types of the average rent.
Let us explain the meaning of equation (10). We can apply the same arguments toEquation (11). When the firm makes an investment decision and the next decision is
employment, the firm invests more than is needed for current production to increase
19the marginal product of labor at the next period. Because of this additional benefit,
the relevant user cost for this investment is the modified user cost, u − β k π l , whichis explained in Section 2.

Hence, the average rent from changing its attention toemployment is Πk (u − β k π l ).

However, if the firm continues to invest during thenext period, there is no benefit from capital hoarding. Hence, the relevant user cost
is equal to Jorgenson’s.

After maximizing profits, given an input price of u, thefirm expects to receive π k again at the next period. Hence, the average rent fromholding to an investment decision is Πk (u) + β l πk . Equation (10) shows that π k is
the maximum of the two types of the average rent.
The next section solves Equations (10) and (11), and derives the conditions underwhich the firm alternates its attention and adjusts to the desired level.

4

When Does a Firm Adjust to the Desired Level?

This section investigates the condition under which it is optimal to alternate its
attention.

Firstly, exploiting the benefits of stationarity, the sequential problem

expressed in Equations (10) and (11) is modified into a one shot problem. Let us
define Gl (π k ), Gk (π l ), Rl , and Rk as
Gl (πk ) = Πl (w − β l π k ) , Gk (πl ) = Πk (u − β k πl ) ,Πl (w)Πk (u), Rk =.
Rl =1 − βk1 − βlNew functions Gl (π k ) and Gk (π l ) are mapping from π k to π l and π l to π k , respectively. New variables Rl ( Rk ) are the present discounted value of the stream of rent
per unit capital (per unit labor) when the firm sticks to employment ( investment). We call Rl and Rk the reservation value of average rents when the firm sticks toemployment and investment, respectively.
We define the following new problem:
π k = max {Gk (π l ) , Rk } ,
20

(14)

π l = max {Gl (π k ) , Rl } .

(15)

The new problem says that the firm adjusts to the desired level when the two averagerents from changing attention are larger than their reservation values.

Since theenvironment is stationary, this one shot problem is expected to be equivalent to the#original problem. Let π#k and π l denote the solutions to Equations (10) and (11),and let π ∗k and π ∗l denote the solutions to Equations (14) and (15). The following
lemma shows that the new problem is equivalent to the original one. The proof isestablished in the Appendix.
Lemma 2 The solutions to Equations (10) and (11) are equivalent to the solutions
to Equations (14) and (15).
#∗π ∗l = π #l , πk = πk .

The lemma permits us to work with Equations (14) and (15). In order to characterize the conditions under which adjustment takes place, it is convenient to define
mappings Hl (π l ) and Hk (π k ) :Hk (πk ) = π k − Gk (Gl (π k )) ,(16)

Hl (π l ) = π l − Gl (Gk (π l )) .

(17)

∗∗
∗∗
∗∗
∗∗
∗∗
Let π ∗∗
k and π l denote the solutions to π k = Gk (π l ) and π l = Gl (π k ). Then∗∗
Hl (π ∗∗l ) = 0, Hk (π k ) = 0.

(18)

That is, the solutions to Equations (18) characterize the average rents when the firm
adjusts to the desired level. We first state the property of H - functions. The proof
is given in the Appendix.
Lemma 3 The functions Hl (·) and Hk (·) are strictly increasing and concave:
Hl0 (π l ) > 0, Hk0 (π k ) > 0(19)

Hl00 (π l ) ≤ 0, Hk00 (πk ) ≤ 0.

(20)

21

Since Lemma 3 shows that Hl (·) and Hk (·) are continuous and strictly increasing∗∗
functions, the existence and uniqueness of π ∗∗l ( π k ) can be proved by showing thatfor some small πl ( π k ), Hl (π l ) < 0 ( Hk (π k ) < 0 ) and for other large π l ( π k ),Hl (πl ) > 0 ( Hk (πk ) > 0 ). The formal proof of the following theorem is given in
the Appendix.
∗∗
Theorem 4 There exists a unique π ∗∗l and π k .

Now, we are ready to provide conditions under which the firm alternates its attention.

The following theorem achieves this.

The proof is established in the

Appendix.
Theorem 5 The firm alternates its attention and adjusts to the desired level, if andonly if Hl (Rl ) < 0 and Hk (Rk ) < 0:∗∗∗
Hl (Rl ) < 0, Hk (Rk ) < 0 if f π∗l = π ∗∗l > Rl , π k = π k > Rk .

(21)

Note that Hl (Rl ) and Hk (Rk ) are not influenced by endogenous variables. Thismeans that Theorem 5 characterizes technological and market conditions when thefirm adjusts to the desired level. The intuition behind this condition is explained asfollows. Note that
Hl (Rl ) < 0, if f Rl < Gl (Gk (Rl )) ,(22)

Hk (Rk ) < 0, if f Rk < Gk (Gl (Rk )) .

(23)

The left-hand side of Equation (22), Rl , is the average rent when the firm holds to employment decision from now on. The right-hand side of Equation (22), Gl (Gk (Rl )),is the average rent when the firm holds to employment after the firm changes its
attention once. Since the environment is stationary, if the firm changes its attention
once, it will continue to do so. A similar interpretation is applied to Equation (23).
Hence, Theorem 5 says that when the benefit from a one-time change in its attention
22

Figure 5: Hl (Rl ) < 0, Hk (Rk ) < 0: In this case, the firm alternates its attention
between investment and employment and adjusts to the desired level. The solid line
represents max {Gk (π l ) , Rk } and max {Gl (π k ) , Rl } and the intersection of the twosolid lines is the solution.

23

is greater than sticking to the same decision for both investment and employment, it
is optimal for the firm to allocate its attention and make its adjustment.
A graphical explanation of this result is depicted in Figure 5. Note that Theorem(5) is restated as
−1
∗
∗
G−1
l (Rl ) < Gk (Rl ) , Gk (Rk ) < Gl (Rk ) , if f π l > Rl , π k > Rk .
−1
The conditions, G−1l (Rl ) < Gk (Rl ) and Gk (Rk ) < Gl (Rk ), are satisfied in Figure

5. The two solid lines represent max {Gk (π l ) , Rk } and max {Gl (π k ) , Rl }. Hence,
the intersection of the two solid lines is the solution to the original problem. As you
can see, π ∗k > Rk and π ∗l > Rl .
How do economic parameters influence the conditions for adjustment? Unfortunately, changes in most of the parameters do not bring clear results. We compare theaverage rent from sticking to one decision and that from changing attention. Most of
the parameters, such as w and u, influence both rents in the same direction. Hence,the eﬀect of w and u on the conditions for adjustment depends on which rents are
more influenced by w and u. However, a relatively simple condition is obtained forthe eﬀect of the depreciation of labor.
Proposition 6 Suppose that the employment decision is solved by an interior solution. Then³

dHl Rl ; δl
dδ l

´

> 0.

The condition implies that the larger the depreciation, the more likely the condition for adjustment is violated. As argued below, when the depreciation rate oflabor is large, the firm is likely to hold to the employment decision.

A companythat cannot maintain human capital does not expect high profits from investment.
Hence, it must spend much time recruiting and training workers.
Note that there is an asymmetry between the depreciation of labor and capital
because the depreciation of capital also influences user cost, u. This additional eﬀect
24

obscures the result. If we can keep u constant, the eﬀect of δ k is similar to that ofδl .
When the firm holds to one decision, there are three possible cases: the firmholds to investment, the firm holds to employment, or the firm’s attention depends
on the initial capital—labor ratio. The following theorem establishes the necessary
and suﬃcient conditions under which each case occurs. The proof is established in
the Appendix.
Theorem 7 Suppose Hl (Rl ) ≥ 0 or Hk (Rk ) ≥ 0.
Gl (Rk ) > Rl , Gk (Rl ) ≤ Rk , if f π ∗l = Gl (Rk ) > Rl , π ∗k = Rk ,(24)

Gl (Rk ) ≤ Rl , Gk (Rl ) > Rk , if f π ∗l = Rl , π ∗k = Gk (Rl ) > Rk ,(25)

Gl (Rk ) ≤ Rl , Gk (Rl ) ≤ Rk , if f π ∗l = Rl , π∗k = Rk .

(26)

Equation (24) shows the condition under which the firm holds to investment,
Equation (25) shows the condition under which the firm holds to employment andEquation (26) shows the condition under which the allocation of attention depends
on the initial capital—labor ratio.
Figure 6, 7 and 8 provide the examples for each case. In figure 6, Hk (Rk ) ≥ 0( G−1
k (Rk ) ≥ Gl (Rk ) ), Gl (Rk ) > Rl , and Gk (Rl ) ≤ Rk are satisfied.

It shows

that π ∗l = Gl (Rk ) > Rl and π ∗k = Rk . Since Gl (Rk ) > Rl , when a firm makes an
employment decision, the next decision is investment. However, because Gk (Rl ) ≤Rk , once the firm makes an investment decision, it does not change its attention.
Hence, the firm holds to investment decisions.
−1
In Figure 7, Hl (Rl ) ≥ 0 ( G−1
k (Rk ) ≥ Gl (Rk ) ), Hk (Rk ) ≥ 0 ( Gl (Rl ) ≥ Gk (Rl )), Gl (Rk ) ≤ Rl and Gk (Rl ) > Rk are satisfied.
Gk (Rl ) > Rk .

It shows that π∗l = Rl and π∗k =The condition Gk (Rl ) > Rk means that when the firm makes aninvestment decision, the next decision is employment; Gl (Rk ) ≤ Rl means that once
the firm makes an employment decision, it does not change its attention. Hence, thefirm holds to employment.
25

Figure 6: Hk (Rk ) ≥ 0, Gl (Rk ) > Rl , Gk (Rl ) ≤ Rk : In this case, the firm holds toinvestment. The solid line represents max {Gk (πl ) , Rk } and max {Gl (π k ) , Rl } andthe intersection of the solid lines is the solution.

26

Figure 7: Hl (Rl ) ≥ 0, Hk (Rk ) ≥ 0, Gl (Rk ) ≤ Rl , Gk (Rl ) > Rk : In this case, thefirm sticks to employment.

The solid line represents max {Gk (π l ) , Rk } andmax {Gl (π k ) , Rl } and the intersection of the solid lines is the solution.

27

Figure 8: Hl (Rl ) ≥ 0, Hk (Rk ) ≥ 0, Gl (Rk ) ≤ Rl , Gk (Rl ) ≤ Rk : In this case, an initial capital—labor ratio determines the allocation of the firm’s attention. The solidline represents max {Gk (π l ) , Rk } and max {Gl (π k ) , Rl } and the intersection of thesolid lines is the solution.

28

Note that when the employment decision is solved by an interior solution, an
increase in the depreciation rate of labor lowers Gl (Rk ) and Rk , while it keeps Gk (Rl )and Rl the same. Hence, it is likely that the large depreciation rate forces the firm
to hold to employment. This confirms our previous argument about the eﬀect of achange in the depreciation rate of labor.
−1
In figure 8, Hl (Rl ) ≥ 0 ( G−1
k (Rk ) ≥ Gl (Rk ) ), Hk (Rk ) ≥ 0 ( Gl (Rl ) ≥Gk (Rl ) ), Gl (Rk ) ≤ Rl and Gk (Rl ) ≤ Rk are satisfied. Because Gl (Rk ) ≤ Rl and
Gk (Rl ) ≤ Rk are satisfied, once the firm allocates its attention either to investment or
to employment, it never changes its attention. The allocation of the firm’s attention
depends on its initial capital—labor ratio.

If its initial capital—labor ratio is large

enough, the marginal productivity of labor is larger than the marginal productivityof capital and the firm holds to employment. If an initial capital—labor ratio is smallenough, the opposite is true: the firm holds to investment.

5

Investment and the Growth of a Firm

Suppose that Hl (Rl ) < 0 and Hk (Rk ) < 0. The firm alternates its attention. Thissection derives the growth rate of the capital stock and discusses its property. Notethat
I
(1 − δ l ) L0
K0
=
− 1,
K
(1 − δ l ) L0K= κ (u − β k π ∗l ) (1 − δ l ) λ (w − β l π ∗k ) − 1,
where κ (·) and λ (·) are the optimal policy functions corresponding to Equations(12) and (13), respectively. Using Hotelling’s lemma, the optimal policy functions
are related to the profit functions:−κ (u − β k π ∗l ) = Π0k (u − β k π ∗l ) ,−λ (w − β l π∗k ) = Π0l (w − β l π ∗k ) .
The following theorem summarizes the above arguments.
29

Theorem 8 Suppose that Hl (Rl ) < 0 and Hk (Rk ) < 0. Then the average investment is a decreasing function of the modified user cost and the modified wage rate:I= (1 − δ l ) Π0k (u − β k π ∗l ) Π0l (w − β l π ∗k ) − 1.
K

(27)

The theorem shows that the firm’s investment is fully determined by the modified
user cost and modified wage rate when δ l = 0. Recall that these two measures of
average rents, π ∗l and π ∗k , have a clear relationship with Tobin’s Q:
Q (l)
β
=
max {π ∗k l, π ∗l } + 1.
pk
pk

(28)

Hence, Tobin’s Q is correlated with investment. However, the correlation between
growth rate and Tobin’s Q is indirect.

This might explain the weak correlation

between investment and Tobin’s Q found in the literature. This point is examined
quantitatively in the next section.
Similarly to Abel and Eberly (2003), marginal Q is not correlated with investment.
Since the marginal cost of investment is pk in this model, if the solution is interior,the marginal benefit of investment has to be pk .

Hence, the marginal Q does nothave any connection to the investment decision.

Although the marginal Q is not

informative, Tobin’s Q is still informative since it contains information about future
rent. This point is emphasized by Abel and Eberly (2003).
Note that investment is periodic. When the firm makes its employment decision,
there is no investment; when the firm makes its investment decision, the investment
is lumpy. Lumpy investment is consistent with evidence in Doms and Dunne (1998).
Equation (27) appears when we aggregate them over time.
Since Equation (27) shows gross investment, the net growth rate must take into
account the depreciation rate of capital stock. Note that
(1 − δ k ) K 0−1
K= (1 − δ k ) (1 − δ l ) κ (u − β k π∗l ) λ (u − β k π ∗l ) − 1.
(1 − δ l ) L0
− 1 ≡ gl .
=
L

gk ≡

30

Hence, the following corollary is immediate.
Corollary 9 Suppose that Hl (Rl ) < 0 and Hk (Rk ) < 0. The net growth rate of a
firm isg ≡ gl = gk = (1 − δ k ) (1 − δ l ) Π0k (u − β k π ∗l ) Π0l (w − β l π ∗k ) − 1.
Corollary 9 shows that the growth rate is independent of firm size. We would liketo examine what influences the growth rate. Applying the implicit function theorem
to Equations (18), the following propositions are immediate.
Proposition 10 Suppose that Hl (Rl ) < 0 and Hk (Rk ) < 0.

Then there exist

functions π l (w, u), π k (w, u) and g (w, u) such thatπ ∗l = π l (w, u) , π l1 (w, u) < 0, π l2 (w, u) < 0,π ∗k = π k (w, u) , π k1 (w, u) < 0, π k2 (w, u) < 0,g = g (w, u) , g1 (w, u) ≤ 0, g2 (w, u) ≤ 0.
Moreover, assume that κ (u − β k π ∗l ) and λ (w − β l π ∗k ) are interior solutions. Then
there exist functions, π lδ (δ l , δ k ), π kδ (δ l , δ k ) and g δ (δ l , δ k ) such thatlδπ∗l = π lδ (δ l , δ k ) , π lδ
1 (δ l , δ k ) < 0, π 2 (δ l , δ k ) < 0,kδπ ∗k = π kδ (δ l , δ k ) , π kδ
1 (δ l , δ k ) < 0, π 2 (δ l , δ k ) < 0,g = g δ (δl , δ k ) , g1δ (δ l , δ k ) < 0, g2δ (δ l , δ k ) < 0.
The results in the proposition are intuitive. Since an increase in wage rate, user
cost, the depreciation rate of capital stock and labor input all lower the average rents,
they lower the growth rate of the firm.

6

Calibration

In this section, we specify the production function and calibrate the model.

This

exercise is aimed at examining the quantitative relationship between investment, Tobin’s Q and cash flow.

1

We assume that F (K, L) = z [θK ρ + (1 − θ) Lρ ] ρ , where31ρ < 1.

Before showing our calibration results, it may be instructive to look at

analytical solutions given this specification.

For this purpose, we assume for the

moment that the solutions, κ (u − β k π ∗l ), λ (w − β l π∗k ), κ (u) and λ (w) are interior.
It is shown from Equations (14) and (15) that
π ∗l
π ∗k

⎧
⎨

1−ρ

θ

w [λ (w)]θ(1−θ)[w − β l π∗k ] [λ (w − β l π ∗k )]1−ρ − u,
= max ⎩
(1 − θ)1 − βk(1−ρ

1−θ
u [κ (u)]1−θ
[u − β k π∗l ] [κ (u − β k π ∗l )]1−ρ − w, θ= max
θ
1 − βlwhereλ (p) =

"

#1

ρ

θ

−w

)

⎭

#1

ρ

1−θ

for any p. The variablesp( z(1−θ))[ zθp ]π ∗l and π ∗k are the solutions of these two complex equations. If the conditions forρ
1−ρ −(1−θ)

and κ (p) =

"

⎫

− u⎬

ρ
1−ρ −θ

adjustment are satisfied, the following investment function is derived:
⎡

⎤1 ⎡

⎤1

ρ

ρ

1−θ
θ
I
⎥ ⎢
⎥
= (1 − δ l ) ⎢⎣h⎦ ⎣³⎦ − 1.
i ρ
´ ρ
1−ρ
1−ρ
u−β k πl
w−β l π kK−θ− (1 − θ)zθz(1−θ)

Note that after controlling u − β k π l and w − β l π k , z still has a positive impact on thegrowth rate. This suggests that cash flow may positively influence investment after
controlling Q, which is examined quantitatively later.

Calibration: We assume that a firm makes decisions twice a year.

Hence, thederived investment is considered to be an annual value.
We assume that the price of investment goods, pk , is equal to 1 as a benchmark,the depreciation rate of capital, δ k , is 0.05 and the interest rate, i, is 0.08.

Thismeans that user cost, u = 0.13 and β = 0.93. The interest rate, i = 0.08, implies
that the annual interest rate is equal to 0.1664, which is higher than the risklessrate of return. This number corresponds roughly to an annual interest rate of 0.16,
which was assumed in Abel and Eberly (2002).

They argued that the firm uses

risk-adjusted hurdle rates of return, which correspond to their number. We follow
their argument. Since we define L as the eﬃciency unit of labor, it is not clear what32

Table 1: Benchmark Parameters
w pk i
δk
θ
δl
α
1

1

0.08 0.05 0.3 0.1 1.07

might be a reasonable w. As a benchmark, we assume that w = 1. We also assume
that θ = 0.3, which means the capital share of 0.3 when the production function isapproximated by Cobb—Douglas (ρ = 0). For the depreciation of human capital, δl , isassumed to be 0.1, which corresponds to an annual value of 19 percent9 . We assume
that α = 1.07, which means α = β1 − 0.01.

Table 1 summarizes our benchmark

parameters.
We have generated the investment—capital ratio, Tobin’s Q and cash flows forseveral values of z and ρ, which largely influence investment behavior.

For this

purpose, we first discuss what is the reasonable range of z and ρ. Since w is assumed
h ³

to be 1, we can estimate z from z = θ

K
Y

´ρ

+ (1 − θ)³wLY´ρ i− 1ρ. Hence, we need toknow the reasonable range for ρ to find a reasonable range for z.
Berndt (1991, p455) reviews the literature and claims that empirical findings
from two-digit cross-sectional studies support a Cobb—Douglas production function(ρ = 0), though time series estimates of the elasticity show a range of 0.3 to 0.5, whichroughly corresponds to the range of ρ = −1 to ρ = −2.3. However, these estimates
are not based on the eﬃciency unit. Krusell, Ohanian Ríos-Rull and Violante (2000)
estimate the elasticity of substitution between equipment and skilled labor as 0.67,
which corresponds to ρ = −0.5. From this evidence, we expect that a plausible rangeof ρ is roughly 0 to −2.5.
9

It is not clear what might be a reasonable number. One possible source of information about

the depreciation rate can be found from the separation rate between a firm and a worker. Yashiv(2000) assumes that the separation rate is 1.7 percent a month, which corresponds to an annual
value of 19 percent. Christensen, Lentz, Mortensen, Neumann and Werwatz (2005) report that the
separation rate of managers in the largest firms is 22 percent, where managers are the group who
accumulate the most firm-specific human capital.

These numbers provide a justification for ourassumption. In any case, our results do not change much by using other numbers.

33

Using data from the National Income and Product Accounts, the capital—output
ratio in the U.S., K/Y , is roughly 2.11 in 2004.

Assuming that the labor share,

wL/Y , is 0.7, the estimated value of z is 1.03 when ρ = 0 and 1.25 when ρ = −2.5.
Hence, we consider that a reasonable range of z is between 1 and 1.25.
For given ρ, we have generated 100 observations that correspond to diﬀerent z
that increase by 0.0025 starting from 1. Hence, the maximum value of z is 1.2475.
The parameter ρ ranges from −7 to 0 in 0.5 intervals. Hence, we have 15 diﬀerent

values of ρ. We have 1500 observations10 . As we have argued, the plausible valueof ρ is expected to be greater than −2.5. However, we find that it is instructive toreport the results when ρ is less than −2.5.
Figures 9, 10 and 11 show our calibration results. Figure 9 shows the relationshipbetween the investment—capital ratio ( I/K ) and the productivity parameter ( z )for the production functions with diﬀerent elasticity of substitution between capital1
and labor, ( 1−ρ). When the production function is Cobb—Douglas ( ρ = 0 ), I/K isconstant at 0 or 0.205. The zero I/K means that the firm holds to an employment
decision in that range of z, while a positive constant value of I/K implies that the firmgrows and both investment decisions and employment decisions are constrained bytheir upper bounds. In other words, when the production function is Cobb—Douglas
and the firm grows, there is no interior solution.
When ρ = −2, negative investment appears.

The lower bound of I/K is −1,

which means that either the investment or employment decision is constrained by thelower bound of 011 . In this case, the firm immediately disappears. There is a rangeof z that supports interior solutions with negative investment.
solutions with positive investment still do not appear.
10

However, interior

When ρ = −4, there is a

Of course, strictly speaking, ρ and z must satisfy a certain relationship. For the purpose of thisexercise, the strict regulation is not particularly important. We allow the variation of ρ and z withina certain range, which results in the heterogeneity of firms.
11
Technically, we cannot allow 0 inputs. Hence, the lower bound is assumed to be 1 × 10−17 ,which is approximately 0.

34

Figure 9: The relationship between investment—capital ratio (I/K) and productivity (z) for production functions with diﬀerent elasticities of substi1
tutions between capital and labor ( 1−ρ).

35Figure 10: The relationship between Tobin’s Q and productivity (z) for
production functions with diﬀerent elasticities of substitutions betweenπ∗1lcapital and labor ( 1−ρ): Tobin’s Q is calibrated by 1+i
+ 1.

36

Figure 11: The relationship between cash flow—capital ratio and productivity (z) for production functions with diﬀerent elasticities of substitutions
1
between capital and labor ( 1−ρ): cash flow—capital ratio is calibrated by the
ρ

1
ρ ρ

] −wL
summation of z[θK +(1−θ)(L)for two consecutive periods.
K37

range of z that does support interior solutions that bring positive investment. When
ρ = −6, the range of z that supports interior solutions with positive investment
becomes larger.

To summarize, we can generate interior solutions with positive

investment, but capital and labor must be more complementary than the estimates
within our plausible range. Possible extensions to mitigate this problem are discussed
in our conclusion.
Figure 10 demonstrates the relationship between Tobin’s Q and z.

We reportlTobin’s Q when a firm makes an employment decision, which is denoted by Qpk(l) andπ∗lcalibrated from 1+i+ 1. For our range of parameters, there is no case for which thelfirm will hold to an investment decision. Hence, every observation generates Qpk(l) .
For all ρ, Tobin’s Q is close to 1 until z reaches a threshold. Above the thresholdlevel of z, Tobin’s Q expands greatly for a slight increase in z and deviates far from1.

Note that the threshold level of z coincides with that of z above which I/K

hits its upper bound in Figure 9.

Once I/K hits its upper bound, an increase inz cannot increase capital stock further, although it increases market value. That is
why Tobin’s Q becomes sensitive after I/K hits its upper bound.
Figure 11 shows the relationship between the cash flow—capital ratio and z forthe production functions with diﬀerent ρ. The cash flow—capital ratio is calibrated
ρ

1
ρ ρ

] −wL
by the summation of z[θK +(1−θ)(L)for two consecutive periods. For all ρ, theKcash flow—capital ratio increases as z increases except for the region in which either
an employment decision or an investment decision has interior solutions that bringpositive investment. When the firm expands and has an interior solution, adjustment
cost is generated by static losses as explained before. These static losses make the
cash flow—capital ratio smaller.

Once both decisions hit upper bounds, the firmcannot increase investment any further.

Hence an increase in z simply increases

output and therefore increases the cash flow—capital ratio.
In order to examine the eﬀect of Tobin’s Q and cash flow on investment, we
conduct a simple regression using data generated from our calibration.

38

Table 2

Table 2: Regression on Investment—Capital Ratio:ρ ∈ [−7, 0] ρ ∈ [−7, 0] ρ ∈ [−2.5, 0] ρ ∈ [−2.5, 0]
Q

0.079

-0.031

0.056-0.020(0.003)(0.001)(0.003)(0.003)

CASH/K

Constant

Adjusted − R2

4.041

3.579

(0.040)(0.088)-0.429-1.126-0.270-1.059(0.013)(0.008)(0.017)(0.021)

0.333

0.915

0.316

0.820

# of observation 1500
1500
600
600
Every coeﬃcient is significant at a 0.5 % level.
reports our results.

The first column shows that the investment—capital ratio is

positively correlated with Tobin’s Q. However, the coeﬃcient and R2 is small. Thisis consistent with evidence [e.g., Chirinko (1993)]. Moreover, once we include cash
flow, the coeﬃcient on Tobin’s Q is negative and cash flow has a strong positiveeﬀect on investment[ the second column].

The robust impacts of cash flow, after

controlling for Tobin’s Q, is found in the investment literature [e.g., Fazzari, Hubbard
and Petersen (1988)]. The results do not change even when we restrict our attention
to the observations with ρ ≥ −2.5. The calibration results show that our model cangenerate stylized facts found in the literature.

7

Conclusion and Extensions

This paper applies a simple idea to investment theory: a person cannot make manydecisions at a time, but an organization needs millions of interrelated decisions. The
growth rate of the firm in our model is derived when the production function displays
constant returns to scale, there is no adjustment cost function and markets are competitive. We show that each investment is lumpy, but adjustment is not immediate.

39

Furthermore, the growth rate of a firm is independent of firm size and imperfectly
correlated with Tobin’s Q.
One drawback found in our calibration exercises is that in order to support aninterior solution that has positive investment, the model demands more complementarity than the estimates, which are expected to be plausible. This indicates that weneed some modifications for the purpose of empirical research. The current model isdesigned to distinguish a novel mechanism for the growth of a firm from other models
in the literature. For this purpose, we make two extreme assumptions. First, we
completely dismiss any conventional adjustment costs. However, it is natural to thinkthat not only physical capital accumulation but also human capital accumulation is
likely to involve a certain adjustment cost.

Incorporating these additional adjust-ment costs makes it easier to support interior solutions.

Second, we assume that

there is no price adjustment and that the production function has constant returns
to scale. When the production function displays constant returns to scale and there
is no price adjustment, an optimal decision should be on the boundary in a standard
static optimization. Hence, it is understandable to find many boundary solutions in
our model. This reasoning suggests that extension to a general equilibrium model
will make the model support interior solutions with more reasonable parameters. Alternatively, the assumption of a decreasing-returns-to-scale production function will
also mitigate problems. These are interesting future extensions.
One may also object to our assumption on routine operation. We assume thata firm maintains its previous level of input when it does not give any attention to
optimizing the input. However, it is clear that an alternative decision rule might bepossible. We also assume that a firm can costlessly change its routine if it wishes to do
so. Common observation tells us that this is not the case. Our assumptions should
be considered as simplifications to clarify the messages of our paper. However, we
believe that we point out important aspects of a routine: limited attention demands
routine operation, and routine operation causes rent, which can be both the source40

of growth and a barrier to change.
Developing more reasonable models of a routine is beyond the scope of this paper,but it is also an interesting research agenda. We hope that our model can be extended
to incorporate more realistic features of a routine.

8

Appendix

The Proof of Proposition 1: We derive only Equation (10). Since the derivationof Equation (11) is similar, we omit the proof. Substitute D (l) = β max {π k l, π l }into Equation (8),π k = 0max {f (k0 ) − uk0 − w + max {β l π k , β k π l k0 }} .
k ∈[0,αk ](29)Define π1k and π 2k ,π 1k =
π 2k =

max [f (k0 ) − (u − β k π l ) k0 − w] ,k0 ∈[0,αk ]max [f (k0 ) − uk0 − w] + β l π k .

k0 ∈[0,αk ],Note that if β l π k ≤ β k π l k, π k = π 1k and if β l π k ≥ β k π l k, π k = π 2k , where k is asolution of Equation (29).
Suppose that arg maxk0 ∈[0,αk ] [f (k0 ) − [u − β k π l ] k0 − w] = k∗ and that β l π k ≥β k π l k∗ . Thenπ 1k ≤ (k∗ ) − [u − β k π l ] k∗ − w ≤ f (k∗ ) − uk∗ − w + β l π k ≤ π 2k .
Similarly, suppose that arg maxk0 ∈[0,αk ] [f (k0 ) − uk 0 − w] + β l π k = k∗ and that β l π k ≤β k π l k∗ . Thenπ2k ≤ f (k∗ ) − uk ∗ − w + β l π k ≤ f (k∗ ) − [u − β k π l ] k∗ − w ≤ π1k .
Hence,noπk = max π 1k , π 2k .
41

Q.E.D.
#∗
The Proof of Lemma 2: We only prove π ∗l = π #l . Since π k = π k can be provedby the same method, we omit its proof. Define Rl∗ (π k ) such that
Gl (π k ) = Πl (w) + β k Rl∗ (π k ) .

(30)

Note that π l ≥ Rl∗ (π k ), if and only if Πl (w) + β k π l ≥ Gl (π k ). Rearranging Equation(30),Rl∗ (π k ) =Gl (π k ) − (1 − β k ) Rl.
βk

Necessity: Suppose that π l = max {Gl (π k ) , Πl (w) + β k π l }.
Rl∗ (π k ). Then π l = Rl . Hence, it is shown that
π l − Rl∗ (π k ) =Suppose that π l ≥Rl − Gl (π k ).
βkThis means that π l ≥ Rl∗ (π k ) and π l = Rl imply Rl ≥ Gl (π k ) and π l = Rl . Similarly,
suppose that π l < Rl∗ (π k ). Then π l = Gl (π k ). Hence, it is shown that
π l − Rl∗ (π k ) =(1 − β k ) {Rl − Gl (π k )}.
βkThis means that π l < Rl∗ (π k ) and πl = Gl (π k ) imply Rl < Gl (π k ) and πl = Gl (π k ).
Therefore, π l = max {Gl (π k ) , Rl }.
Suﬃciency: Suppose that π l = max {Gl (π k ) , Rl }. Suppose that Gl (π k ) ≤ Rl . Thenπ l = Rl . Hence, it is shown thatRl − Gl (πk ) = β k [π l − Rl∗ (π k )] .
This means that Gl (π k ) ≤ Rl and π l = Rl imply π l ≥ Rl∗ (π k ) and π l = Rl . Similarly,
suppose that Gl (π k ) > Rl . Then π l = Gl (πk ). Hence, it is shown thatRl − Gl (π k ) =βk[π l − Rl∗ (π k )] .
1 − βkThis means that Gl (π k ) > Rl and π l = Gl (π k ) imply π l ≥ Rl∗ (π k ) and π l = Gl (π k ).
Therefore, π l = max {Gl (π k ) , Πl (w) + β k π l }. Q.E.D.
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The Proof of Lemma 3: We prove Hl0 (π l ) > 0 and Hl00 (π l ) ≤ 0. Since Hk0 (πk ) > 0

and Hk00 (π k ) ≤ 0 is proved by the same method, we omit it.

Hl0 (π l ) = 1 − β k β l λ [w − β l Gk (π l )] κ [r − β k πl ] ≥ 1 − β 2 α2 > 0,
where λ [·] is an optimal policy function of Equation (13) and κ [·] is an optimal policy
function of Equation (12).
hi2Hl00 (πl ) = − G00l (Gk (π l )) [G0l (Gk (π l ))] + G0l (GK (π l )) G00k (π l ) ≤ 0.
The second inequality comes from the fact that Gl (·) and Gk (·) are strictly increasing
and convex functions.

The Proof of Theorem 4: We prove the existence and uniqueness of π ∗∗l . Since∗∗
the proof of the existence and uniqueness of π ∗∗k is the same as that of π l , we omit

it. Note that Hl (π l ) can be written as follows:
Hl (πl ) = (1 − β l β k κ (u − β k πl ) λ (w − β l GK (πl ))) π l + D (π l )⎧⎪⎨D (πl ) ≡ u − ⎪

pf

³

1
λ(w−β l GK (π l ))

´

⎫
⎪
⎬

⎩ − [(1 + β ) w − β (pf (κ (u − β π l )) − uκ (u − β π l ))] ⎪
⎭
l

l

k

k

λ (w − β l GK (π l )) .

Note that since λ (w − β l GK (π l )) and κ (u − β k π l ) are bounded, D (π l ) is also bounded
for any π l . Note also that 1 > β l β k κ (u − β k π l ) λ (w − β l GK (π l )) for any πl becauseof the constraints on choice variables.

Hence, there exists large π l ∈ R such that

Hl (πl ) > 0, and there also exists small π l ∈ R such that Hl (π l ) < 0. Since Hl (π l )is continuous and strictly increasing in π l , there exists an unique π ∗∗l . Q.E.D.

The Proof of Theorem 5:
Necessity: Suppose that Hl (Rl ) < 0 and Hk (Rk ) < 0. Suppose that Gl (π ∗k ) ≤ Rl .

Then π∗l = Rl . Hence, π∗k = max {Gk (Rl ) , Rk } and Gl (max {Gk (Rl ) , Rk }) ≤ Rl .
0 > Rl − Gl (Gk (Rl )) ≥ Gl (max {Gk (Rl ) , Rk }) − Gl (Gk (Rl )).
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Hence, Gk (Rl ) >max {Gk (Rl ) , Rk }. Contradiction. Similarly, suppose that Gk (π∗l ) ≤ Rk . Thenπ ∗k = Rk .

Hence, π ∗l = max {Gl (Rk ) , Rl } and Gk (max {Gl (Rk ) , Rl }) ≤ Rk .

0 > Rk − Gk (Gl (Rk )) ≥ Gk (max {Gl (Rk ) , Rl }) − Gk (Gl (Rk )). Hence, Gl (Rk ) >max {Gl (Rk ) , Rl }. Contradiction.

∗∗∗
Suﬃciency: Suppose that π ∗l = π ∗∗l > Rl , π k = π k > Rk . The result is immediatefrom the following lemma.
Lemma 11Hl (Rl ) ≥ 0 if f π∗∗l ≤ Rl ,Hk (Rk ) ≥ 0 if f π∗∗k ≤ Rk .
Proof. Note that since Hl (·) and Hk (·) are strictly increasing functions, theresult is obvious.
Q.E.D.

The Proof of Theorem 7:
Necessity: Note that from theorem 5, either π ∗k = Rk or π ∗l = Rl .
Suppose Gl (Rk ) > Rl and Gk (Rl ) ≤ Rk . Suppose that π ∗k > Rk . Then π ∗l =Gl (π ∗k ) > Gl (Rk ) > Rl . Contradiction. Hence, π ∗k = Rk . Then π ∗l = Gl (Rk ) > Rl .
Suppose that Gl (Rk ) ≤ Rl and Gk (Rl ) > Rk .

π ∗k = Gk (π ∗l ) > Gk (Rl ) > Rk .

Contradiction.

Suppose that π ∗l > Rl .
Hence, π∗l = Rl .

Then

Then π ∗k =Gk (Rl ) > Rk .
Suppose that Gl (Rk ) ≤ Rl and Gk (Rl ) ≤ Rk .

Suppose that π ∗l = Rl .

Thenπ ∗k = max {Gk (Rl ) , Rk } = Rk . Suppose π ∗k = Rk . π∗l = max {Gl (Rk ) , Rl } = Rl .
Suﬃciency:
Suppose π∗l = Gl (Rk ) > Rl , π ∗k = Rk . Then Gk (Rl ) < Gk (π ∗l ) ≤ Rk .
Suppose π∗l = Rl , π∗k = Gk (Rl ) > Rk . Then Gl (Rk ) < Gl (π ∗k ) ≤ Rl .

Suppose π∗l = Rl , π∗k = Rk . Then the result is obvious. Q.E.D.
44

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