Tobin's q

Chapter 10

Investment and Tobins q

All the models considered so far (the OLG models as well as the represen-

tative agent models) have ignored capital adjustment costs. In the closed-

economy version of the models aggregate investment is merely a reflection

of aggregate saving and appears in a “passive” way as just the residual of

national income after households have chosen their consumption. We can

describe what is going on by telling a story in which firms just rent capital

goods owned by the households and households save by purchasing addi-

tional capital goods. In these models only households solve intertemporal

decision problems. Firms just maximize current profits. This may be a legit-

imate abstraction in some contexts, but not if capital investment is of central

importance for the problem studied.

We shall now consider the so-called q-theory of investment, which assumes

that (a) firms make the investment decisions and install the purchased cap-

ital goods in their own business, and (b) there are certain adjustment costs

associated with investment in capital goods; by this is meant that in addition

to the direct cost of buying new capital goods there are also costs of installa-

tion, costs of reorganizing the plant, costs of retraining workers to operate the

new machines etc.; finally, (c) the adjustment costs are strictly convex, that

is, the marginal adjustment cost is increasing in the level of investment. This

strict convexity of adjustment costs is the crucial constituent of the q-theory

of investment; it is that element which assigns investment decisions an active

role in the model. There will be both a well-defined saving decision and a

385

386 CHAPTER 10. INVESTMENT AND TOBINS Q

well-defined investment decision, separate from each other. As a result, in a

closed economy the whole spectrum of current and future interest rates has

to adjust so that aggregate saving can be equal to aggregate investment at all

points in time; or, what amounts to the same, the term structure of interest

rates1 has to adjust so that the aggregate demand for goods (consumption

plus investment) is equalized to the aggregate supply of goods. This implies

that even when ignoring uncertainty (assuming perfect foresight), the rela-

tionship between the (real) rate of interest and the (net) marginal product

of capital becomes less firm, at least in the short run.

When faced with adjustment costs, the optimizing firm has to take the fu-

ture into account. Therefore, firms’ expectations become important. Further,

the firm will adjust its capital stock only gradually when new information

arises. In particular, for a small open economy with perfect mobility of goods

and financial capital we avoid the counterfactual result that the capital stock

is instantaneously adjusted when the interest rate in the world financial mar-

ket changes. Indeed, sluggishness in investment is what the data show. Some

empirical studies conclude that only a third of the difference between the cur-

rent and the “desired” capital stock tends to be covered within a year (Clark

1979).

Tobin’s q-theory of investment (after the American Nobel laureate James

Tobin, 1918-2002) is one theoretical approach to the explanation of this slug-

gishness in investment (Tobin 1969). Under certain conditions, to be de-

scribed below, the theory gives a remarkably simple operational macroeco-

nomic investment function, in which the key variable explaining aggregate

investment is the valuation of the firms by the stock market relative to the re-

placement cost of the firms’ physical capital (the book value of firms). Other

explanations of sluggish or lumpy capital adjustment focus on uncertainty,

indivisibility and irreversibility (Zeira 1987, Dixit and Pindyck 1994, Adda

and Cooper 2003) or financial problems due to bankruptcy costs (Nickell

1978). Here we shall concentrate on the q-theory of investment.

1The term structure of interest rates is the relationship between the internal rate ofreturn on a bond and its term (= time to maturity). Another name for the same thing isthe yield curve.

10.1. Convex capital adjustment costs 387

10.1 Convex capital adjustment costs

The technology of the representative firm is

Y = F (K,TL),

where F is a neoclassical production function with CRS, and Y,K, and L

are output, capital input and labor input per time unit, respectively, while

T is the technology level growing at the constant rate γ ≥ 0 (Harrod-neutraltechnical progress). Time is continuous. The increase per time unit in the

firm’s capital stock is given by

K = I − δK, δ ≥ 0, (10.1)

where I is gross investment per time unit and δ is the rate of wearing down

of capital (physical capital depreciation).

Let J denote the capital adjustment costs (measured in units of output)

per time unit. Assuming the price of investment goods is one (the same as

that of output goods), then total investment costs are I + J, i.e., the direct

purchase costs, 1 · I, plus the indirect cost associated with installation etc.The q-theory of investment assumes that the capital adjustment cost, J, is a

strictly convex function of investment I and generally also depends negatively

on the current capital stock. Thus,

J = G(I,K),

where the adjustment cost function G satisfies

G(0,K) = 0, GI(0, K) = 0, GII(I,K) > 0, and GK(I,K) ≤ 0. (10.2)

For fixed K = K the graph is as shown in Fig. 10.1. Also negative in-

vestment, i.e., sell off of capital equipment, involves costs (to dismantling,

reorganization etc.). ThereforeGI < 0 for I < 0. The important assumption,

however, is that GII > 0 (strict convexity in I), implying that the marginal

adjustment cost is increasing in the level of investment. If the firm wants

to accomplish a given installation project in only half the time, then the

installation costs are more than doubled (the risk of mistakes is larger, the


J

I

( , )G I K

0

Figure 10.1:

problems with reorganizing work routines are larger etc.). Think of building

a new plant in a month instead of a year.

The strictly convex graph in Fig. 10.1 illustrates the essence of the matter.

Assume the current capital stock in the firm is K and that the firm wants

to increase it by a given amount ∆K. If the firm chooses the investment

level I > 0 per time unit, then, in view of (10.1), it takes approximately

∆t = ∆K/(I − δK) units of time to accomplish the desired increase ∆K.

If, however, the firm slows down the adjustment and invests only half of I

per time unit, then it takes approximately 2∆t units of time to accomplish

∆K. The total cost of the two approaches are approximately G(I , K)∆t

and G(12I, K)2∆t, respectively (ignoring discounting). By drawing a few

straight line segments in Fig. 10.1 the reader will be convinced that the last-

mentioned cost is smaller than the first-mentioned (see Problem 8.1). On

the other hand, the benefits of installed capital in the future are discounted

and changes in the firm’s environment take place continually, so that it is

not advantageous to postpone the investment too much.

In addition to the strict convexity (10.2) imposes the conditionGK(I,K) ≤0. Indeed, it seems often realistic to assume that GK(I,K) < 0 for I 6= 0.

A given amount of investment may require more reorganization in a small

firm than in a large firm (size here being measured by K). When installing

a new machine, a small firm has to stop production altogether, whereas a

large firm can to some extent continue its production by shifting some work-

ers to another production line. A further argument is that the more a firm


has invested historically, the more experienced it is now. So, for a given I

today, the associated adjustment costs are lower. From now, we shall speak

of “adjustment costs” and “installation costs” interchangeably.

10.1.1 The decision problem of the firm

With the output good as unit of account, let cash flow (before interest pay-

ments, if any) at time t be denoted Rt. Then

Rt ≡ F (Kt, TtLt)− wtLt − It −G(It,Kt), (10.3)

where wt is the wage at time t. The adjustment cost G(It,Kt) implies that a

part of output Yt is used up in transforming investment goods into installed

capital, and therefore less output is available for sale. Assume the firm is a

price taker and that there is no uncertainty. The interest rate is rt, which

we assume to be positive, at least in the long run. The decision problem, as

seen from time 0, is to choose a plan (Lt, It)∞t=0 to maximize the value of the

firm,2

V0 =

Z ∞

0

Rte− t

0 rsdsdt s.t. (10.4)

Lt ≥ 0, It free (i.e., no restriction on It), (10.5)

Kt = It − δKt, K0 given, (10.6)

Kt ≥ 0 for all t. (10.7)

There is no specific terminal condition but we have posited the natural feasi-

bility condition (10.7) that the firm can never have a negative capital stock.

To solve the problem we use the Maximum Principle. The problem has

two control variables, L and I, and one state variable, K. We set up the

current-value Hamiltonian (for ease of exposition we suppress the time sub-

script when not needed for clarity):

H(K,L, I, q, t) ≡ F (K,TL)− wL− I −G(I,K) + q(I − δK), (10.8)

2The relationship between this formulation of the firm’s problem and the static max-imization of current profits considered in the previous chapters is explained in AppendixA.


where q (to be interpreted below) is the adjoint variable associated with the

dynamic constraint (10.6). For each t ≥ 0 we maximize H wrt. the control

variables. Thus, ∂H/∂L = F2(K,TL)T − w = 0, i.e.,

F2(K,TL)T = w; (10.9)

and ∂H/∂I = −1−GI(I,K) + q = 0, i.e.,

1 +GI(I,K) = q. (10.10)

Next, we partially differentiate H wrt. the state variable and set the result

equal to rq − q, where r is the discount rate in (10.4):

∂H

∂K= FK(K,TL)−GK(I,K)− δq = rq − q. (10.11)

Now, by the Maximum Principle, for an interior optimal path (Kt, Lt, It)3

there exists an adjoint variable qt such that for all t ≥ 0 the conditions (10.9),(10.10), and (10.11) hold. Further, we conjecture that the path also satisfies

the transversality condition

limt→∞

qte− t

0 rsdsKt = 0. (10.12)

The optimality condition (10.9) is the usual employment condition equal-

izing the marginal product of labor to the real wage. The left-hand side of

(10.10) gives the cost of acquiring one extra unit of installed capital at time

t (the sum of the cost of buying the marginal investment good and the cost

of its installation). That is, the left-hand side is the marginal cost, MC, of

increasing the capital stock. Since (10.10) is a necessary condition for op-

timality, the right-hand side of (10.10) must be the marginal benefit, MB,

of increasing the capital stock. Hence, qt represents the value to the firm of

having one more unit of (installed) capital at time t. To put it differently:

the adjoint variable qt can be interpreted as the shadow price (measured in

current output units) of capital along the optimal path.4 This interpretation

is confirmed when we solve the differential equation (10.11), see below.

3The path (Lt, It,Kt) is called interior, if, for all t > 0, Lt > 0 and Kt > 0.4We remember that a shadow price, measured in some unit of account, of a good is the

maximum number of units of account that the optimizing agent is willing to offer for oneextra unit of the good.


The conjectured transversality condition (10.12) says that, if along the

optimal path Kt remains positive and asymptotically finite, then the dis-

counted shadow price in the far future must be asymptotically zero, i.e.,

limt→∞

qte− t

0 rsds = 0; (10.13)

and if along the optimal path, Kt grows forever, then the discounted shadow

price must approach zero faster thanKt grows. Intuitively, otherwise the firm

would gain by decreasing its ultimate capital stock “left over” for eternity.

Multiplying by e−t0 (rs+δ)ds on both sides of (10.11), we get by integration

and application of (10.13)5

qt =

Z ∞

t

[FK(Kτ , TτLτ)−GK(Iτ ,Kτ )] e− τ

t (rs+δ)dsdτ. (10.14)

The right-hand side of (10.14) is the present value, as seen from time t, of

expected future increases of the firm’s cash-flow that would result if one extra

unit of capital were installed at time t. Indeed, FK(Kτ , TτLτ) is the direct

contribution to output of one extra unit of capital, while −GK(Iτ , Kτ) ≥ 0is the adjustment cost reduction of future investment projects (capital ad-

justment costs are decreased by |GK |, when Kτ is one unit larger). However,

future increases of cash-flow should be discounted at a rate equal to the rate

of interest plus the capital depreciation rate (because, from one extra unit of

capital at time t there are only e−δ(τ−t) units left at time τ).

To concretize our interpretation of qt as representing the value to the

firm at time t of having one extra unit of capital, let us perform a thought

experiment. Assume that a extra units of (installed) capital at time t drops

down from the sky. At time τ > t there are a · e−δ(τ−t) units of these still inoperation. Now, replace t by τ in (10.3) and consider the firm’s cash-flow Rτ

as a function of Kτ , Lτ , Iτ , t, τ , and a. Taking the partial derivative wrt. a,

we get∂Rτ

∂a |a=0= [FK(Kτ , TτLτ)−GK(Iτ , Kτ)] e

−δ(τ−t).

Let the value of the firm, i.e., the value of the integral (10.4), as seen from a

5It is presupposed that the firm considered is active permanently in the sense thatlimt→∞Kt > 0. For details of the derivation, see Appendix B.


fixed point, t, in time, be called Vt. Then, at any point where Vt is differen-

tiable, we have

∂Vt∂a |a=0

=

Z ∞

t

(∂Rτ

∂a |a=0)e−

τt rsdsdτ (10.15)

=

Z ∞

t

[FK(Kτ , TτLτ )−GK(Iτ ,Kτ)]e− τ

t (rs+δ)dsdτ = qt,

when the firm moves along the optimal path; the last equality sign comes

from (10.14).6 This confirms that qt is the shadow price of capital at time t,

and that qt represents “marginal benefit” by increasing the capital stock.

On this background it becomes understandable that the control variables

at any point in time should be chosen so that the Hamiltonian function

is maximized. This amounts to maximizing the sum of the current direct

contribution to the criterion function and the indirect contribution, which

is the benefit of having a higher capital stock in the future (as measured

approximately by qt∆Kt). In the same way we now understand the last

equality sign in (10.11); a condition for optimality must be that the firm

acquires capital up to the point where the “net marginal product of capital”,

FK − GK − δqt, equals “net capital costs”, rtqt − qt. Here we look at qtas the “overall” price at which the firm can buy and sell the marginal unit

of installed capital. Continuing along this line of thought, by reordering in

(10.11) we get the “no-arbitrage” condition

FK −GK − δq + q

q= r, (10.16)

saying that the rate of return on the marginal unit of capital must equal the

interest rate.

As we know, the Maximum Principle gives only necessary conditions for

an optimal path, not sufficient conditions. We use the principle as a tool

for finding candidates for a solution. Having found in this way a candidate,

one way to proceed is to check whether, e.g., Mangasarian’s sufficient condi-

tions are satisfied. Given the transversality condition (10.12) and the non-

negativity of the state variable, K, the only additional condition to check6Not only is the value of the adjoint variable, q, at time t equal to the marginal

contribution to the value of the firm at time t of a hypothetical “injection” of capital attime t, but the discounted value of the adjoint variable, qte−

t0rτdτ , measures, in optimum,

the marginal contribution to V0 of such an injection at time t (see Lèonard 1987).


is whether the Hamiltonian function is jointly concave in the endogenous

variables (here I, K, and L). If it is jointly concave in these variables, then

the candidate is an optimal solution.7 In our context inspection of (10.8)

reveals that the Hamiltonian function is jointly concave in (I,K,L) if −G(I,K) is jointly concave in (I,K). This condition is equivalent to G(I,K) being

jointly convex in (I,K), an assumption allowed within the confines of (10.2).

For example, G(I,K) = (12)βI2/K (where β > 0) will do.

10.1.2 The implied investment function

From condition (10.10) we can derive an investment function. Rewriting

(10.10) we have that an optimal path satisfies

GI(It, Kt) = qt − 1. (10.17)

Combining this with the assumption (10.2) on the adjustment cost function,

we see that

It T 0 for qt T 1, respectively.

Indeed, in view of GII > 0, (10.17) implicitly defines optimal investment, It,

as a function of the shadow price, qt, and Kt :

It =M(qt,Kt), where Mq = 1/GII > 0. (10.18)

It follows that optimal investment is an increasing function of the shadow

price of installed capital. And since GI(0,K) = 0, we haveM(1, K) = 0. Not

surprisingly, the investment rule is: invest now, if and only if the marginal

value to the firm of installed capital is larger than the price of the capital good

(which is 1, excluding adjustment costs). At the same time, the rule says

that, because of the convex adjustment costs, invest only up to the point

where the marginal adjustment cost, GI(It, Kt), is equal to the difference

between qt and 1, cf. (10.17).

Condition (10.18) shows the remarkable information content that the

shadow price qt has. As soon as qt is known (along with the current cap-

ital stock Kt), the firm can decide the optimal level of investment through

7Strict concavity ensures uniqueness.


knowledge of the adjustment cost function G alone (since, when G is known,

so is its inverse, the investment function M). All the information about the

production function, input prices, and interest rates now and in the future

that is relevant to the investment decision is summarized in one number, qt.

The form of the investment function, M, depends only on the adjustment

cost function G. These are very useful properties in theoretical and empirical

analysis.

10.1.3 A convenient special case

We now introduce the convenient case where the installation function G is

homogeneous of degree one so that we can, for K > 0, write

J = G(I,K) = G(I

K, 1)K ≡ g(

I

K)K, or (10.19)

J

K= g(

I

K),

where g(0) ≡ G(0, 1) = G(0, K) = 0.

LEMMA 1 We have: (i) g0(I/K) = GI(I,K); (ii) g00(I/K) = KGII(I,K) >

0 for K > 0; and (iii) GK(I,K) = g(I/K)− g0(I/K)I/K < 0 for I 6= 0.

Proof. (i) GI = Kg0/K = g0; (ii) GII = g00/K; (iii) GK = ∂(g(I/K)K)/∂K

= g(I/K)− g0(I/K)I/K < 0 for I 6= 0 since g00 > 0. ¤

The graph of g(I/K) is qualitatively the same as that in Fig. 10.1 (imag-

ine we have K = 1 in that graph). The linear homogeneity assumption

implies that the adjustment cost relative to the existing capital stock is a

strictly convex function of the investment-capital ratio, I/K; an example:

J = G(I,K) = 12βI2/K is convex in (I,K) and gives J/K = 1

2β(I/K)2.

A further important property is that (10.19) makes the revenue function in

(10.3) homogeneous of degree one wrt. K, L, and I. This has two key impli-

cations. First, Hayashi’s theorem applies (see below). Second, the q-theory

can easily be incorporated in a model with economic growth.8

Does the hypothesis of linear homogeneity in K, L, and I make economic

sense? According to the replication argument it does. Suppose a given firm8The relationship between our function g and other ways of formulating the theory is

commented on in Appendix C.


has K units of installed capital and produces Y units of output with L units

of labor. When at the same time the firm invests I units in new capital, it

obtains the revenue R after deducting the installation costs, G(I,K). Then

it makes sense to assume that the firm could do the same thing at another

place. (Of course, here we ignore that also land is a necessary input. But

in manufacturing and service industries this source of departure from linear

homogeneity seems minor and is perhaps offset by synergy effects from larger

size.)

In view of (i) of Lemma 1, the linear homogeneity assumption allows us

to write (10.17) as

g0(I/K) = q − 1. (10.20)

This equation defines the investment-capital ratio, I/K , as an implicit func-

tion, m, of q − 1, or, more conveniently, a function, m, of q :

ItKt= m(q−1) ≡ m(qt), wherem(1) = m(0) = 0 and m0 = m0 = 1/g00 > 0.

(10.21)

In this case, q encompasses all information that is of relevance to the decision

about the investment ratio I/K. Fig. 10.2 illustrates the implied investment

function m (the specific values marked out on the axes are explained below).

In general the function is non-linear. In the specific example above, however,

where g(I/K) = 12β(I/K)2, (10.20) gives I/K = (q − 1)/β ≡ m(q), a linear

function. We see that the parameter β can be interpreted as the degree of

sluggishness in the capital adjustment. The degree of sluggishness reflects

the degree of convexity of adjustment costs.9

To see how the shadow price q changes over time we rewrite (10.11) as

qt = (rt + δ)qt − FK(Kt, TtLt) +GK(It, Kt). (10.22)

In the special case (10.19) we have

GK(I,K) =∂J

∂K=

∂£g( I

K)K¤

∂K= g(

I

K) +Kg0(

I

K)−IK2

= g(I

K)− I

Kg0(

I

K) = g(m(q))−m(q)(q − 1)

9Indeed, for given q, the degree of sluggishness is proportional to the degree of convexityof adjustment costs: the degree of convexity of g(·) is g00/g0 = (I/K)−1 = β(q−1)−1, whichgives β = (q − 1)g00/g0.


q 1

/I K

*q

( )m q

nδ γ+ +

Figure 10.2:

from (10.21) and (10.20). Inserting this into (10.22) gives

qt = (rt + δ)qt − FK(Kt, TtLt) + g(m(qt))−m(qt)(qt − 1). (10.23)

This differential equation is very useful in macroeconomic analysis, as we see

below.

In a macroeconomic context, for steady state to obtain, gross investment

must be large enough to compensate not only for capital depreciation, but

also for growth in the effective labor input (TL). That is, the investment-

capital ratio, I/K,must be equal to the sum of the depreciation rate, the rate

of technical progress, and the growth rate of the labor force, i.e., δ + γ + n.

That level of q which is required to motivate such an investment-capital ratio

is called q∗ in Fig. 10.2.

10.2 Marginal q and average q

Our q above, determining investment, should be distinguished from what is

usually called Tobin’s q or average q. In a more general context, let pIt denote

the purchase price (in terms of output units) per unit of the investment good.

Then Tobin’s q or average q is defined as qat ≡ Vt/(pItKt), that is, Tobin’s

10.2. Marginal q and average q 397

q is the ratio of the market value of the firm to the book value of the firm,

that is, the replacement cost of its capital stock (the top index “a” stands

for “average”). In our simplified context we have pIt ≡ 1 (the price of theinvestment good is the same as that of the output good). Therefore Tobin’s

q can be written

qat ≡VtKt

, (10.24)

Conceptually this is different from the shadow price on capital, our q

above. In the language of the q-theory of investment our q is called marginal

q, representing the value to the firm of one extra unit of capital relative to the

replacement cost. Indeed, the term marginal q is natural in view of condition

(10.15) saying that along the optimal path we have qt = (∂Vt/∂Kt)/pIt ≡ qmt(“m” for “marginal”). Since we have pIt ≡ 1, we can now write

qmt = qt = ∂Vt/∂Kt. (10.25)

The two concepts, average q and marginal q, have not always been clearly

distinguished in the literature. What is relevant to the investment decision is

marginal q. Indeed, the analysis above showed that optimal investment is an

increasing function of qm. Further, the analysis showed that a “critical” value

of qm is 1 and that only if qm > 1, is positive gross investment is warranted.

The importance of average q, qa, is that it can be measured empirically

(as the ratio of the sum of the share market value of the firm and its debt

to the replacement cost of its total capital). Since qm is much harder to

measure than qa, it is important to know the relationship between qm and

qa. Fortunately, we have a simple theorem giving conditions under which

qm = qa.

THEOREM (Hayashi, 1982) Assume the firm is a price taker, that the

production function F is concave in (K,L) and that the adjustment cost

function G is convex in (I,K) (that is, in addition to (10.2), we assume

GKK ≥ 0 and GIIGKK −G2IK ≥ 0). Then, along the optimal path we have:

(i) qmt = qat for all t ≥ 0, if F and G are homogeneous of degree 1.

(ii) qmt < qat for all t, if F is strictly concave in (K, L) and/or G is strictly

convex in (I, K).

Proof. See Appendix D.


The assumption that the firm is a price taker may, of course, seem critical.

The Hayashi theorem has been generalized, however. Also a monopolistic

firm, facing a downward-sloping demand curve and setting its own price, may

have a cash flow which is homogeneous of degree one in the three variables

K,L, and I. If so, then the condition qmt = qat for all t ≥ 0 still holds (Abel1990).

In any case, when qm is approximately equal to (or proportional to)

qa, the theory gives a remarkable simple operational investment function,

I = m(qa)K, cf. (10.21). At the macro level we interpret qa as the market

valuation of the firms relative to the replacement cost of their total capital

stock. This market valuation is an indicator of the future earnings potential

of the firms. Under the conditions in (i) of the Hayashi theorem the market

valuation also indicates the marginal earnings potential of the firms, hence,

it becomes a determinant of their investment. This establishment of a re-

lationship between the stock market and firms’ aggregate investment is the

basic point in Tobin (1969).

10.3 Applications

Capital adjustment costs in a Ramsey model for a closed economy

It is straightforward to allow for convex capital adjustment costs in the Ram-

sey model for a closed economy (see Abel and Blanchard, 1983, Lim andWeil,

2003, and Groth and Madsen, 2008). Among the insights are that investment

decisions attain an active role in the economy. And expectations become

important for firms’ investment. For example, expected future changes in

corporate taxes and depreciation allowance tend to change firms’ investment

today.

The essence of the matter is that current and future interest rates have

to adjust so that aggregate saving can be equal to aggregate investment at

all points in time or, what amounts to the same, so that the output market

clears at all points in time. Given full employment (Lt = Lt), the output

market clears when

F (Kt, TtLt)−G(It,Kt) = value added ≡ GDPt = Ct + It,

10.3. Applications 399

where Ct is determined by the intertemporal utility maximization of the

forward-looking households, and It is determined by the intertemporal value

maximization of the forward-looking firms. This is the first time in this book

that clearing in the output market is assigned a role. In the earlier models

investment was just a passive reflection of household saving. Investment

was automatically equal to the residual of national income left over after

consumption had taken place. Nothing had to adjust to clear the output

market, neither the interest rate nor output. In contrast, in the present model

continuous clearing in the output market is decisive for the determination of

the macroeconomic dynamics. On the other hand, a separate market for the

stock of existing capital goods is no longer considered. Indeed, the interest

rate is no longer tied down by a requirement that such a market clears.

We said that current and future (short-term) interest rates have to adjust

so that the output market clears. We could also say that an adjustment of

the whole structure of interest rates (the yield curve) takes place and consti-

tutes the equilibrating mechanism in the output market. By having output

market equilibrium playing this role in the model, a first step is taken to-

wards medium-run and short-run macroeconomic theory. We take further

steps in later chapters, by allowing different forms of imperfect competition

and other market imperfections to enter the picture. Then the demand side

gets a dominating role both in the determination of q (and thereby invest-

ment) and in the determination of aggregate output and employment. This

is what the Keynesian and New Keynesian theory is about. But for now we

still assume perfect competition at all markets including the labor market;

indeed, by instantaneous adjustment of the real wage, labor demand con-

tinuously matches labor supply. In this sense the present model remains a

member of the neo-classical family (supply-dominated models).

A small open economy with capital adjustment costs

A simpler set-up is (as usual) that of a small open economy (henceforth SOE).

Introducing convex capital adjustment costs, we avoid the counterfactual

result that the capital stock adjusts instantaneously when the interest rate

at the world financial market changes.


We assume:

1. Perfect mobility across borders of goods and financial capital.

2. Domestic and foreign financial claims are perfect substitutes.

3. No mobility across borders of labor.

4. Labor supply is inelastic and constant and there is no technical progress

(i.e., n = γ = 0).

5. The capital adjustment cost function G(I,K) is homogeneous of degree

1.

In this setting the SOE faces an exogenous interest rate, r, which is given

from the world financial market and which we assume constant. With L > 0

denoting the constant labor supply in our SOE, continuous clearing on the

labor market under perfect competition gives Lt = L for all t ≥ 0 and

wt = F2(Kt, L) ≡ w(Kt). (10.26)

At any time t, Kt is predetermined in the sense that due to the convex

adjustment costs, changes in K take time. Thus (10.26) determines the

market real wage wt.

Since the capital adjustment cost function G(I,K) is assumed homoge-

neous of degree 1, the analysis of Section 10.1 applies and we can write (10.23)

as

qt = (r + δ)qt − FK(Kt, L) + g(m(qt))−m(qt)(qt − 1). (10.27)

Here r and L are exogenous so that the capital stock,K, and its shadow price,

q, are the only endogenous variables in this differential equation. Another

differential equation with these two variables can be obtained by inserting

(10.21) into (10.6) to get

Kt = (m(qt)− δ)Kt. (10.28)

Fig. 10.3 shows the phase diagram for these two coupled differential

equations. Let q∗ be defined by the requirement m(q∗) = δ. Suppressing for

convenience the explicit time subscripts, we then have

K = 0 for m(q) = δ, i.e., for q = q∗.


K

0K =

q

0q =

*q

0K *K

E

B

Figure 10.3:

Note that when δ > 0, we have q∗ > 1. This is so because also mere rein-

vestment to offset capital depreciation requires an incentive, namely that the

marginal value to the firm of replacing worn-out capital is larger than the

purchase price of the investment good (since the installation cost must also

be compensated). From (10.28) is seen that

K ≷ 0 for m(q) ≷ δ, respectively, i.e., for q ≷ q∗, respectively,

cf. the horizontal arrows in Fig. 10.3.

From (10.27) we have

q = 0 for 0 = (r + δ)q − FK(K, L) + g(m(q))−m(q)(q − 1). (10.29)

If, in addition K = 0 (hence, q = q∗ and m(q) = m(q∗) = δ), this gives

0 = (r + δ)q∗ − FK(K, L) + g(δ)− δ(q∗ − 1), (10.30)

where the right-hand-side is increasing in K, in view of FKK < 0. Hence,

there exists at most one value of K such that the steady state condition

(10.30) is satisfied;10 this value is called K∗, corresponding to the steady

10And assuming that F satisfies the Inada conditions, (10.30) shows that such a valuedoes exist.


state point E in Fig. 10.3. The question now is: what is the slope of the

q = 0 locus? In Appendix E it is shown that at least in a neighborhood

of the steady state point E this slope is negative in view of the assumption

r > 0. From (10.27) we see that

q ≶ 0 for points to the left and to the right, respectively, of the q = 0 locus,

since FKK(Kt, L) < 0. The vertical arrows in Fig. 10.3 show these directions

of movement.

Altogether the phase diagram shows that the steady state E is a saddle

point, and since there is one predetermined variable, K, and one jump vari-

able, q, the steady state is saddle-point stable. At time 0 the economy will

be at the point B in Fig. 10.3 where the vertical line K = K0 crosses the

saddle path. Then the economy will move along the saddle path towards the

steady state. This solution satisfies the transversality condition (10.12) and

is the unique solution to the model (for details, see Appendix F).

The effect of a shift in the interest rate Assume that until time 0 the

economy has been in the steady state E in Fig. 10.3. Then, an unexpected

shift in the interest rate occurs so that the new interest rate is a constant r0

> r (and this interest rate is rightly expected to remain at this level forever

in the future). From (10.28) we see that q∗ is not affected by this shift, hence,

the K = 0 locus is not affected. However, the q = 0 locus and K∗ shift to

the left, in view of FKK(K, L) < 0.

Fig. 10.4 illustrates the situation for t > 0. At time t = 0 the shadow price

q jumps down to a level corresponding to the point B in Fig. 10.4. There is

now a more heavy discounting of the future benefits that the marginal unit

of capital can provide. As a result the incentive to invest is diminished and

gross investment will not even compensate for the depreciation of capital.

Hence, the capital stock decreases gradually. This is where we see a crucial

role of convex capital adjustment costs in an open economy. For now, the

adjustment costs are the costs associated with disinvestment (disassembling

and selling out of machines). If these convex costs were not present, we

would get the same the counterfactual prediction as from the previous open-

economy models in this book, namely that the new steady state is attained


*q

K

q

* 'K *K

0K =

0q =

E'E

0new q =B

Figure 10.4:

immediately after the shift in the interest rate.

As the capital stock is diminished, the marginal productivity of capital

rises and so does q. The economy moves along the new saddle path and

approaches the new steady state E’ as time goes by.

Suppose that for some reason such a decrease in the capital stock is

not desirable from a social point of view; this could be because of positive

external effects of capital and investment, e.g., a kind of “learning by doing”.

Then the government could decide to implement an investment subsidy σ,

0 < σ < 1, so that to attain an investment level I, purchasing the investment

goods involves a cost of (1− σ)I. Assuming the subsidy is financed by some

tax not affecting firm’s behavior (for example a constant tax on households’

consumption), investment is increased again, and the economy may in the

long run end up in the old steady state, E, again.

A growing small open economy with capital adjustment costs

The basic assumptions are the same as in the previous section except that

now labor supply, L, grows at the rate n ≥ 0, while there is technical progressat the rate γ ≥ 0 (both rates exogenous and constant).With full employment,as before, L = L = L0e

nt. Now we need the production function on intensive


form defined through

Y = F (K,TL) = F (K

TL, 1)TL ≡ f(k)TL,

where k ≡ K/(TL) is the effective capital intensity, and f satisfies f 0 >

0, f 00 < 0. The market-clearing real wage at time t is determined as

wt = F2(Kt, TtLt)Tt =hf(kt)− ktf

0(kt)iTt ≡ w(kt)Tt,

where both kt and Tt are predetermined. By log-differentiation of k ≡K/(TL) we get

·kt

kt=

Kt

Kt− (γ + n) = m(qt)− (δ + γ + n),

from (10.28). Thus

·kt = [m(qt)− (δ + γ + n)] kt. (10.31)

The change in the shadow price of capital is now described by

qt = (r + δ)qt − f 0(kt) + g(m(qt))−m(qt)(qt − 1), (10.32)

from (10.23).

The differential equations (10.31) and (10.32) constitute our new dynamic

system. Fig. 10.5 shows the phase diagram, which is qualitatively similar to

that in Fig. 10.3. We have

·k = 0 for m(q) = δ + γ + n, i.e., for q = q∗,

where q∗ now is defined by the requirement m(q∗) = δ + γ + n. Notice, that

since γ + n > 0, we get a larger steady state value q∗ than in the previous

section. This is so because now a higher investment-capital ratio is required

for a steady state to be possible.

From (10.32) we see that q = 0 now requires

0 = (r + δ)q − f 0(k) + g(m(q))−m(q)(q − 1).


*q

q

*k *'k

ˆ 0k =

0q =

E 'E

0new q =

B

k

Figure 10.5:

If, in addition·k = 0 (hence, q = q∗ and m(q) = m(q∗) = δ + γ + n), this

gives

0 = (r + δ)q∗ − f 0(k) + g(δ + γ + n)− (δ + γ + n)(q∗ − 1).

Here, the right-hand-side is increasing in k (in view of f 00(k) < 0). Hence,

the steady state value k∗ of the capital intensity is unique, cf. the steady

state point E in Fig. 10.5.

Assuming r > γ + n we have, at least in a neighborhood of E in Fig.

10.5, that the q = 0 locus is negatively sloped (see Appendix E).11 Again

the steady state is a saddle point, and the economy moves along the saddle

path towards the steady state. In Fig. 10.5 it is assumed that until time 0

the economy has been in the steady state E. Then, an unexpected shift in

the interest rate to a lower constant level r0 takes place. The q = 0 locus

is shifted to the right, in view of f 00 < 0. The shadow price q immediately

jumps up to a level corresponding to the point B in Fig. 10.5. The economy

moves along the new saddle path and approaches the new steady state E’

11In our perfect foresight model we in fact have to assume r > γ + n for the firm’smaximization problem to be well-defined. If instead r ≤ γ + n, the market value of therepresentative firm would be infinite, and maximization would loose its meaning.


with a higher capital intensity as time goes by. In Problem 10.2 the reader is

asked to examine the analogue situation where an unanticipated downward

shift in the rate of technical progress takes place.

10.4 Concluding remarks

There has been made many econometric tests of the q theory of investment,

often with quite critical implications. Movements in qa, even taking account

of changes in taxation, seem capable of explaining only a minor fraction of the

movements in investment. And the estimated equations relating investment

to qa typically give strong auto-correlation in the residuals. Other variables

such as changes in aggregate output, the degree of capacity utilization, and

corporate profits seem to have explanatory power independently of qa (see

Abel 1990, Chirinko 1993). So there is reason to be somewhat sceptical to-

wards the notion that all information of relevance for the investment decision

is reflected by the market valuation of firms. This throws doubt on the basic

assumption in Hayashi’s theorem or its generalization, the assumption that

firms’ cash flow tends to be homogeneous of degree one wrt. K, L, and I.

Going outside the model, there are further circumstances relaxing the

link between qa and investment. In the real world with many production

sectors, physical capital is heterogeneous. If for example a sharp increase

in the price of energy takes place, a firm with energy-intensive technology

will fall in market value. At the same time it has an incentive to invest in

energy-saving capital equipment. Hence, we might observe a fall in qa at the

same time as investment increases. Imperfections on financial markets may

loosen the relationship between qa and investment further and help explain

the observed positive correlation between investment and corporate profits.

We might also question that capital adjustment costs really have the

hypothesized convex form. It is one thing that there are costs associated with

installation, reorganizing and retraining etc., when new capital equipment is

procured. But should we expect these costs to be strictly convex in the

volume of investment? To think about this, let us for a while ignore the

role of the existing capital stock. Hence, we write total adjustment costs

J = G(I) with G(0) = 0. It does not seem problematic to assume G0(I) > 0

10.5. Appendix 407

for I > 0. The question concerns the assumption G00(I) > 0. According

to this assumption the average adjustment cost G(I)/I must be increasing

in I.12 But against this speaks the fact that capital installation may involve

indivisibilities, fixed costs, acquisition of new information etc. All these

features tend to imply decreasing average costs. In any case, at least at the

microeconomic level one should expect unevenness in the capital adjustment

process rather than the above smooth adjustment.

Because of the mixed empirical success of the convex adjustment cost

hypothesis other theoretical approaches that can account for sluggish and

sometimes non-smooth and lumpy capital adjustment have been considered:

uncertainty, investment irreversibility, indivisibility, financial problems due

to bankruptcy costs, etc. (Nickell 1978, Zeira 1987, Dixit and Pindyck 1994,

Caballero 1999, Adda and Cooper 2003). These approaches notwithstanding,

it turns out that the q-theory of investment has recently been somewhat

rehabilitated from the empirical point of view. For large firms, unlikely to be

much affected by financial frictions, Eberly et al. (2008) find that the theory

does a good job in explaining investment behavior. In any case, the q-theory

of investment is widely used in macroeconomics (in different versions) because

of its simplicity and the appealing link it establishes between asset markets

and firms’ investment. And the q-theory has also had an important role in

studies of the housing market and the role of housing prices for household

wealth and consumption.

10.5 Appendix

A.When value maximization is - and is not - the same as continuousstatic profits maximization

Text not yet available.

12Indeed, for I 6= 0 we have d[G(I)/I]/dI = [IG0(I)−G(I)]/I2 > 0, when G is strictlyconvex (G00 > 0) and G(0) = 0.


B. Proof of (10.14)

Brief version: we integrate (10.11) partially from 0 to t1, let t1 → ∞, and

use (10.13) with t replaced by t1. Finally, we replace t and 0 by τ and t,

respectively. Detailed text not yet available.

C. Comparison with other expositions of the q-theory

The simple relationship we have found between I and q can easily be gener-

alized to the case where the purchase price on the investment good, pIt, is

allowed to differ from 1 (its value above) and the capital adjustment cost

is pItG(It, Kt). In this case it is convenient to replace q in the Hamil-

tonian function by, say, λ. Then the first order condition (10.10) becomes

pIt + pItGI(It, Kt) = λt, implying

GI(It, Kt) =λtpIt− 1,

and we can proceed, defining qt by qt ≡ λt/pIt.

Sometimes in the literature adjustment costs J appear in a slightly dif-

ferent form compared to the above exposition. For example, Romer (2001,

p. 371 ff.) assumes the capital adjustment costs J depend only on I so that

GK ≡ 0. Abel and Blanchard (1983), followed by Barro and Sala-i-Martin(2004, p. 152-160), introduce a function, φ, representing capital adjustment

costs per unit of investment as a function of the investment-capital ratio.

That is, total adjustment cost is J = φ(I/K)I, where φ(0) = 0, φ0 > 0. This

implies that J/K = φ(I/K)(I/K). The right-hand side of this equation may

be called g(I/K), and then we are back at the formulation in Section 10.1.

Indeed, defining x ≡ I/K, we have installation costs per unit of capital equal

to g(x) = φ(x)x, and assuming φ(0) = 0, φ0 > 0, it holds that

φ(x)x = 0 for x = 0, φ(x)x > 0 for x 6= 0,g0(x) = φ(x) + xφ0(x) R 0 for x R 0, respectively, andg00(x) = 2φ0(x) + xφ00(x).

Here g00(x) must be positive for the theory to work. But the assumptions

φ(0) = 0, φ0 > 0, and φ00 ≥ 0, imposed on p. 153 and again on p. 154 in

10.5. Appendix 409

the Barro and Sala-i-Martin book, are not sufficient for this (since x < 0 is

possible). This is why we prefer the g(·) formulation rather than the φ(·)formulation.

It is sometimes convenient to let the capital adjustment cost G(I, K)

appear, not as a reduction in output, but as a reduction in capital formation

so that

K = I − δK −G(I,K).

This approach is used by Hayashi (1982) and Heijdra and Ploeg (2002, p. 573

ff.). For example, Heijdra and Ploeg write the rate of capital accumulation

as K/K = ϕ(I/K)−δ, where the “capital installation function” ϕ(I/K) canbe interpreted as defined by ϕ(I/K) ≡ [I −G(I,K)] /K = I/K − g(I/K);

the latter equality comes from assuming G is homogeneous of degree 1. In

one-sector models, as we usually consider in this text, this changes nothing

of importance. In more general models this installation function approach

may have some analytical advantages. What gives the best fit empirically is

an open question.

D. Proof of Hayashi’s theorem

For convenience we repeat:

THEOREM (Hayashi) Assume the firm is a price taker, that the productionfunction F is concave in (K, L), and that the adjustment cost function G is

convex in (I, K). Then, along the optimal path we have:

(i) qmt = qat for all t ≥ 0, if F and G are homogeneous of degree 1.

(ii) qmt < qat for all t, if F is strictly concave in (K, L) and/or G is strictly

convex in (I, K).

Proof. We introduce the functions

A = A(K,L) ≡ F (K,TL)− F1(K,TL)K − F2(K,TL)TL, and(10.33)

B = B(I,K) ≡ G1(I,K)I +G2(I,K)K −G(I,K). (10.34)

Then the cash-flow of the firm, given by (10.3), at time τ can be written

Rτ = Fτ − F2TL−Gτ − I

= Aτ + F1Kτ +Bτ −G2Kτ −G1Iτ − Iτ ,


where we have used first (10.9) and then the definitions of A and B above.

The value of the firm as seen from time t is now

Vt =

Z ∞

t

(Aτ +Bτ )e− τ

t rsdsdτ (10.35)

+

Z ∞

t

[(F1 −G2)Kτ − (G1 + 1)Iτ ]e− τ

t rsdsdτ

=

Z ∞

t

(Aτ +Bτ )e− τ

t rsdsdτ + qtKt,

when moving along the optimal path, cf. Remark 1 below. It follows that

qmt ≡ qt =VtKt− 1

Kt

Z ∞

t

[A(Kτ , Lτ ) +B(Iτ , Kτ)]e− τ

t rsdsdτ. (10.36)

Since F is concave, we have for all K and L, A(K,L) ≥ 0 with equalitysign, if and only if F is homogeneous of degree one. Similarly, since G is

convex, we have for all I and K, B(I,K) ≥ 0 with equality sign, if and onlyif G is homogeneous of degree one. Now the conclusions (i) and (ii) follow

from (10.36) and the definition of qa in (10.24). ¤

A different − and perhaps more illuminating − way of understanding (i)in Hayashi’s theorem is the following. Let xt ≡ It/Kt. Then Kt/Kt = xt− δ,

implying Kτ = Kte− τ

t (xs−δ)ds. Hence, when F and G are homogeneous

of degree one, not only are A and B, as defined in (10.33) and (10.34),

respectively, equal to zero, but Kt can be put outside the integral in (10.35).

We get

Vt = Kt

Z ∞

t

[f 0(kτ)− g(xτ)− xτ ]e− τ

t (rs−xs+δ)dsdτ,

along the optimal path. In this expression, f is the intensive production

function, kτ is the effective capital intensity (≡ Kτ/(TτLτ)), determined by

the market real wage wτ , and, finally, g(x) ≡ G(x, 1). In view of (10.21),

with t replaced by τ , the optimal investment ratio xτ depends, for all τ , only

on qτ , not on Kτ , hence not on Kt. Therefore,

∂Vt/∂Kt =

Z ∞

t

[f 0(kτ)− g(xτ)− xτ ]e− τ

t (rs−xs+δ)dsdτ = Vt/Kt,

and the conclusion qmt = qat follows from (10.25) and (10.24).

10.5. Appendix 411

Remark 1.

Here we prove that the last integral in (10.35) is equal to qtKt, when invest-

ment follows the optimal path. Keeping t fixed and using z as our varying

time variable we have

(F1 −G2)Kz − (G1 + 1)Iz = [(rz + δ)qz − qz]Kz − (G1 + 1)Iz

= [(rz + δ)qz − qz]Kz − qz(Kz + qKz) = rzqzKz − (qzKz + qzKz) = rzuz − uz,

where we have used (10.11), (10.10), (10.6) and the definition uz = qzKz,

respectively. We look at this as a differential equation: uz−rzuz = ϕz, where

ϕz ≡ −[(F1 − G2)Kz − (G1 + 1)Iz]. The solution of this linear differential

equation is

uz = utezt rsds +

Z z

t

ϕτezτ rsdsdτ,

implying, by multiplying through by e−zt rsds, reordering and inserting the

definitions of u and ϕ,Z z

t

[(F1 −G2)Kτ − (G1 + 1)Iτ ]e− τ

t rsdsdτ

= qtKt − qzKze− z

t rsds → qtKt for z →∞,

from the transversality condition (10.12) with t replaced by z and 0 replaced

by t.

Remark 2.

We have assumed throughout that G is strictly convex in I. This does not

imply that G is jointly strictly convex in (I,K). For example, the function

G(I,K) = I2/K is strictly convex in I (since G11 = 2/K > 0). But at the

same time this function has B(I,K) = 0 and is therefore homogeneous of

degree one. Hence, it is not jointly strictly convex in (I,K).

E. The slope of the q = 0 locus

First, we shall determine the sign of the slope of the q = 0 locus in the case

g + n = 0, considered in Fig. 10.3. Totally differentiating in (10.29) wrt. K


and q gives

0 = −FKK(K, L)dK + {r + δ + g0(m(q))m0(q)− [m(q) + (q − 1)m0(q)]} dq == −FKK(K, L)dK + [r + δ −m(q)] dq, (since g0(m(q)) = q − 1, by (10.20)).

Thereforedq

dK |q=0=

FKK(K, L)

r + δ −m(q)for r + δ 6= m(q).

From this it is not possible to sign dq/dK at all points along the q = 0

locus. But in a neighborhood of the steady state we have m(q) ≈ δ, hence

r + δ − m(q) ≈ r > 0. And since FKK < 0, this implies that at least in a

neighborhood of E in Fig. 10.3 the q = 0 locus is negatively sloped.

Second, consider the case g + n > 0 illustrated in Fig. 10.5. Here we get

in a similar way

dq

dk |q=0=

f 00(k∗)

r + δ −m(q)for r + δ 6= m(q).

From this it is not possible to sign dq/dk at all points along the q = 0 locus.

But in a neighborhood of the steady state we have m(q) ≈ δ + γ + n, hence

r + δ −m(q) ≈ r − γ − n. Since f 00 < 0, then, at least in a neighborhood of

E in Fig. 10.5, the q = 0 locus is negatively sloped, when r > γ + n.

F. Testing sufficient conditions

Text not yet available.

10.6 Problems

Problem 10.1 (induced sluggish capital adjustment). Consider a firm with

capital adjustment costs J = G(I,K), satisfying

G(0,K) = 0, GI(0,K) = 0, GII(I,K) > 0, and GK(I,K) ≤ 0.

The notation is standard.

a) Can we from this conclude anything as to strict concavity or strict

convexity of the function G? If yes, with respect to what argument or

arguments?

10.6. Problems 413

b) For given K = K illustrate the adjustment cost curve in a diagram.

c) By drawing a few straight line segments in the diagram, illustrate that

G(12I, K)2 < G(I, K) for any given I > 0.

Problem 10.2


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