Adjusting Q: Vintage Capital And Irreversibility

Dmitry Livdan

Alexander Nezlobin

Haas School of Business,

Haas School of Business

University of California, Berkeley

University of California, Berkeley

[email protected]

[email protected] July 6, 2017

Abstract This paper extends the Q-theory of investment to capital goods with arbitrary efficiency profiles. Under the assumption of geometric economic depreciation employed by the traditional Q-theory, the firm’s replacement cost of assets-in-place is independent of their vintage composition and can be measured by their current productive capacity, i.e., the firm’s current capital stock. When the economic depreciation is nongeometric, the replacement cost of assets depends not only on their current productive capacity but also on the stream of capacity those assets will generate in future periods. We define a new measure of Q, which we call Adjusted Q, as the difference between the firm’s current value and the replacement cost of its assets-in-place divided by its next period’s capital stock. In both reversible and irreversible investment settings, we provide a closed-form expression for the Adjusted Q and demonstrate that it is a better predictor for the firm’s net investment rate than the traditional Tobin’s Q. JEL Codes: E22, G31. Keywords: Investment, Tobin’s Q, Vintage Capital.

1

Introduction

Traditional Q-theoretic models of investment rely on the assumption that the economic depreciation of capital goods is either geometric (in discrete-time) or exponential (in continuous time).1 This assumption is analytically convenient because it leads to a homogeneous capital stock whose future depreciation is independent of its current vintage composition. While geometric depreciation may be descriptive for some assets, its general applicability has long been challenged on both empirical and theoretical grounds.2 Moreover, one of the most commonly used empirical procedures of estimating Tobin’s Q, suggested by Lewellen and Badrinath (1997), explicitly relies on the alternative specification of straight-line economic depreciation over a finite useful life. Similarly, the assumption of straight-line depreciation is overwhelmingly used by firms in calculating the book value of their fixed assets. The book value of assets is, in turn, often used by econometricians as a proxy for their replacement cost in estimating Tobin’s Q. In this paper, we extend the Q-theory of investment to capital goods with arbitrary efficiency (or economic depreciation) profiles. We propose a new measure of Tobin’s Q, labeled Adjusted Q, which, for a known efficiency profile, can be readily calculated from the history of the firm’s investments and its current value. In the special case of geometric depreciation, our Adjusted Q reduces to Tobin’s Q minus one. We next consider capital goods conforming to the so-called one-hoss shay efficiency pattern, i.e., having equal efficiency over a finite useful life. In this case, Adjusted Q is equal to the firm value net of the replacement cost of its assets-in-place divided by the inflation-adjusted gross investment surviving to the current date. We show analytically that for any efficiency pattern, Adjusted Q has better predictive power than the traditional Tobin’s Q for future revenue growth and net investment. In addition, our model demonstrates that, in accordance with earlier investment literature, the firm’s normalized operating cash flow and lagged investment can be positively associated with future investment.3 Similar to Abel and Eberly (2011), this result arises naturally in our setting even in the absence of adjustment costs or financing constraints. Once one departs from the geometric economic depreciation assumption, the firm’s current capital stock, i.e., its current productive capacity, ceases to be a sufficient statistic for its past investment path. Consider, for example, a firm investing in capital goods with one-hoss 1

See, for example, the seminal paper by Hayashi (1982). See, for instance, Feldstein and Rothschild (1974). Ramey and Shapiro (2001) strongly reject the geometric depreciation model in their analysis of equipment-level data from closed aerospace plants. 3 See, for instance, Fazzari et al. (1988), Gilchrist and Himmelberg (1995), and Eberly et al. (2012). 2

1

shay efficiency. Future investment decisions of such a firm depend not only on its current productive capacity but also on the vintage composition of its capital stock. While the same effective capacity today can be generated by either old or new assets, the firm will need to replace its assets sooner if they are old. Generally, to predict the firm’s investment in a vintage capital setting, it is convenient to separate it into its net and replacement components. The firm’s replacement investment is the amount the firm needs to invest today to maintain its current capacity for another period. Replacement investment does not depend on the output market conditions and is determined solely by the vintage composition of the current capital stock. In contrast, the firm’s net investment component only reflects growth in output and is unaffected by the current vintage structure. Consistent with much of the neoclassical investment literature, the firm’s value in our model is equal to the sum of the replacement cost of its assets-in-place and the present value of expected future economic profits.4 However, in our vintage capital setting, an important difference arises between the firm’s replacement cost of assets-in-place and its current capital stock. We formally define the firm’s current capital stock to be a measure of its current productive capacity regardless of the age composition of its assets. In contrast, the replacement cost of the assets-in-place reflects not only their current-period aggregate capacity but also the capacity levels these assets are going to generate in all remaining periods of their useful lives. While the firm’s capital stock is the appropriate measure of the current scale of operation, the replacement cost of assets-in-place reflects the value of the current and future capacity that the assets will provide. Accordingly, we define Adjusted Q as the ratio of the difference between the firm’s value and the replacement cost of its assets-in-place to the current capital stock. Given the decomposition of the firm’s value described above, it follows that Adjusted Q is equal to the present value of future expected economic profits divided by the current capital stock. We provide a closed-form expression for Adjusted Q and show that it is positively related to future revenue growth and net investment. In particular, just as the firm’s expected future net investment rate, Adjusted Q is independent of the current vintage composition of the firm’s asset base. In contrast to Adjusted Q, traditional Tobin’s Q is shown to be path-dependent even when demand for the firm’s output is not, and therefore it is a noisier measure of expected future revenue growth and net investment than Adjusted Q. According to our model, the firm’s future replacement investment can be estimated di4

See, for instance, Lindenberg and Ross (1981), Salinger (1984), and Abel and Eberly (2011). In a model without uncertainty, Nezlobin (2012) extends this result to the vintage capital setting of Rogerson (2008).

2

rectly from its investment history without a reference to the firm’s current equity value. For example, consider again the case of capital goods with one-hoss shay efficiency over a useful life of T years. In this case, the firm’s replacement investment in the next period is simply equal to the (inflation-adjusted) investment made T − 1 periods ago since this is precisely the vintage that will be going offline after the next period, and the efficiency of all other vintages will remain unchanged. We provide a general expression for the firm’s future replacement investment for capital goods with arbitrary productivity profiles. We further show that the ratio of the firm’s cash flow to the replacement cost of assets predicts its future aggregate replacement investment over a sufficiently long horizon. This finding is similar to the one in Abel and Eberly (2011) who show that the normalized cash flow can be positively associated with the replacement component of the firm’s future investment even in the absence of liquidity constraints.5 The main difference between the two findings is that, whereas Abel and Eberly (2011) assume that the economic depreciation of assets is stochastic but equal across vintages, we show that the same effect can arise in a setting with deterministic yet vintage-specific economic depreciation. The modeling of vintage capital in our paper is based on Rogerson (2008) who provided a simple closed-form expression for the user cost of capital for assets with arbitrary efficiency profiles.6 In a setting without uncertainty, Rogerson’s (2008) framework has been adopted by McNichols et al. (2014) and Nezlobin et al. (2016) to study the behavior of Tobin’s Q in the presence of vintage capital effects. None of these papers, however, have explicitly considered the problem of predicting the firm’s investment in a dynamic stochastic framework. To introduce time-series variation in the firm’s investment opportunities and Q, we employ the approach of Abel and Eberly (2011), who consider a firm facing regime-switching uncertainty about demand for its output. As in Abel and Eberly (2011), the role of Q in our model is to reveal the current growth regime in the firm’s output market. The user cost of capital expressions in Rogerson (2008) and Abel and Eberly (2011) do not incorporate the effect of irreversibility on the firm’s investment decisions.7 It is 5

Starting with Fazzari et al. (1988), many empirical studies have included normalized cash flow together with Q in different variants of the investment regression. A significant and positive coefficient on normalized cash flow is often associated with the presence of financing constraints. Several studies have demonstrated that the cash flow effect can arise even in the absence of such constraints; see, e.g., Gomes (2001) and Cooper and Ejarque (2003). The empirical analysis of Erickson and Whited (2000) attributes the cash flow effect to measurement error. 6 For a more general, but arguably less analytically tractable, model of vintage capital, see, e.g., Benhabib and Rustichini (1991). 7 Abel and Eberly (2011) assume that investment is fully reversible, whereas Rogerson (2008) studies a

3

well-known that models with irreversible investment in capital goods with non-geometric economic depreciation are generally not analytically tractable.8 In this paper, we propose a model of irreversibility that admits closed-form solutions for the firm’s equity value, its user cost of capital, and Adjusted Q. Albeit stylized, our model exhibits features familiar in the continuous-time irreversible investment literature. In particular, consistent with the Bernanke (1983) “Bad News” principle, the user cost of capital is shown to be increasing in the probability of bad events. Furthermore, the firm’s optimal investment policy is “lumpy” in the sense that it is characterized by intermittent periods of zero and positive investment. To accommodate irreversibility, we complement the regime-switching structure of Abel and Eberly (2011) with an additional “competitive” regime: at each point in time, the firm can lose its monopoly (pricing) power in the output market and be forced to sell its output at a price below its long-run marginal cost of production. The competitive regime is generally not permanent, and the firm’s monopoly power can be restored at a later date. An alternative interpretation of our model is that the firm participates in two product markets, - premium (in which the firm prices its product) and generic (in which the firm is a price-taker), - with the access to the premium market being sporadic. For analytical tractability, we assume that the premium market is weakly expanding over time even when the firm is shut out of it. We show that in the setting described above, the firm’s optimal investment policy is characterized by two conditions: i) the firm does not invest in the competitive regime, and ii) its investment in the monopoly regime is governed by a user cost of capital that is higher than that in the case of reversible investment. As a consequence, the firm’s lagged investment takes on an additional predictive role in the setting with irreversibility: it indicates whether the firm was in the competitive or monopoly regime in the latest period and thus helps to predict the regime (and investment) next period. We proceed by providing closed-form expressions for the firm’s Adjusted Q in both regimes and demonstrating that Q is positively associated with the expected future net investment. The rest of the paper is organized as follows. Section 2 presents the main model of the paper. Our measure of Adjusted Q is introduced in Section 3. Section 3 also characterizes the firm’s value and its optimal investment policy in the setting with reversible investment. We extend our result to the case of irreversible investment in Section 4. Section 5 concludes. model in which the demand curves weakly shift out over time, thus rendering the irreversibility constraint non-binding at all dates. 8 See, for instance, Chapter 11 in Dixit and Pindyck (1994). Additional complications arise in models with regime-switching uncertainty about demand and in discrete-time.

4

2

Model Setup

Production Technology Consider a firm that uses a single type of capital goods to produce a single non-storable output good. Capital goods have a useful life of T periods and their efficiency declines with age. Specifically, a unit of capital good purchased in period t comes online in period t + 1 and allows the firm to produce xτ units of the output good in period t + τ , where 1 = x1 ≥ ... ≥ xT . The vector x = (x1 , . . . , xT ) will be referred to as the efficiency pattern of the firm’s assets.9 The firm’s effective capital stock in period t, i.e., its aggregate production capacity, can then be written as: T X xτ · It−τ , (1) Kt = τ =1

where It−τ is the firm’s gross investment in period t − τ .10 Let Θt ≡ (It−1 , ..., It−T ) denote the firm’s relevant investment history in period t. We normalize the purchase price of new capital goods to unity, so that the direct cost of investment in period t is measured by It .11 There are two efficiency patterns commonly considered in the earlier literature: geometric economic depreciation and one-hoss shay efficiency. In the geometric depreciation scenario, assets are infinitely lived, T = ∞, and the amount of investment surviving to date τ of its life is declining exponentially in τ : xτ = (1 − δ)τ −1

(2)

for some 0 ≤ δ ≤ 1. An important property of this pattern is that the rate by which the productive capacity of a unit of capital good decreases over a given period is independent of the age of that unit. Under this assumption the firm’s capital stock becomes homogeneous, i.e., the vintage composition of the firm’s current stock is irrelevant for future investment choices. 9

For notational convenience, let xT +1 ≡ 0. Our model of vintage capital builds on Rogerson (2008). Similar to that paper, we assume that the firm purchases only new capital goods. For models with investment in used capital goods, see, e.g., Eisfeldt and Rampini (2007) and Jovanovic and Yatsenko (2012). 11 Our results can be extended to a setting where the price of new capital goods changes over time. 10

5

In the one-hoss shay case, often invoked in the regulation literature (see, e.g., Fisher and McGowan 1983, Laffont and Tirole 2000 and Rogerson 2011), the productivity of assets is constant over their useful life, 1 = x1 = ... = xT , for some finite T . An assumption similar to one-hoss shay efficiency underlies a popular empirical procedure for estimating Tobin’s Q suggested by Lewellen and Badrinath (1997).12 Let Kt,t+j denote the capacity provided in period t + j by assets already in place at the beginning of period t . Formally, it can be expressed as: Kt,t+j =

T −j X

xj+τ · It−τ

τ =1

for j ≥ 0.13 If depreciation is geometric, then we have: Kt,t+j = (1 − δ)j Kt , i.e., the current capacity of the firm’s assets in place determines how much capacity those assets will provide in each future period. Consequently, in this case, we obtain the usual law of motion for the firm’s capital stock: Kt+1 = (1 − δ)Kt + It . However, in general, Kt does not uniquely determine Kt,t+j , which depends on the full vintage composition of the capital stock at date t, Θt .14 Then, the firm’s capital stock in period t + 1 can be written as: Kt+1 = Kt,t+1 + It . (3) We assume that capital is the only input required for production.15 The inverse demand 12

We discuss the relation between the assumptions in Lewellen and Badrinath (1997) and one-hoss shay efficiency in greater detail below. More recently, a pure one-hoss shay assumption has been utilized by McNichols et al. (2014) in developing an alternative methodology for Tobin’s Q estimation. 13 Note that Kt = Kt,t . 14 Consider, for example, the following two investment histories in the scenario with one-hoss shay efficiency (1) (2) (1) (2) and assets with a two-period useful life: Θt = (1, 0) and Θt = (0, 1). Then, we have: Kt = Kt = 1, yet (1) (2) Kt,t+1 = 1 > 0 = Kt,t+1 . 15

It is straightforward to extend our results to a setting with a constant returns-to-scale Cobb-Douglas

6

function for the firm’s output good takes the following form:  P (Zt , Kt ) =

Kt Zt

α−1 ,

where 0 < α < 1, and Zt is a stochastic demand shift parameter. In this specification, α − 1 is the inverse of the price elasticity of demand, with α → 1 corresponding to the case of perfect competition. The firm’s revenue in period t, R (Zt , Kt ), can then be written as: R (Zt , Kt ) = Zt1−α · Ktα , and the firm’s net cash flow is equal to R (Zt , Kt ) − It . As in Abel and Eberly (2011), we model the demand shift parameter as following a regime-switching process. Specifically, in each period Zt+1 = (1 + µt+1 ) · Zt ,

(4)

where with probability λ, µt+1 is drawn from some time-invariant distribution with a finite support in [µmin , µmax ], and with probability 1 − λ, the growth rate remains the same as in the previous period, µt+1 = µt . To ensure that the firm’s value is always finite, we impose that r > µmax . Recall that investments “come online” with a lag of one period, i.e., when the firm decides on the investment level It it effectively chooses its capital stock for period t + 1, Kt+1 . To simplify the exposition, we assume that the firm and the equity market observe µt+1 and, therefore, Zt+1 just before the choice of It is made.16 The firm is all-equity financed and all cash flows are disbursed to (or supplied by) the 1 be firm’s shareholders immediately. Let r > 0 be the firm’s discount rate and γ ≡ 1+r the corresponding discount factor. The firm makes its investments so as to maximize the expected present value of its future cash flows. We initially consider a setting where the firm’s investments are fully reversible, i.e., It can be less than zero. In periods where It is negative, the firm sells a capacity stream which is equivalent to |It | units of new capital goods for the price of |It | units of new capital goods. Equivalently, the firm can be assumed production function with two inputs, capital and labor. 16 In Appendix B, we extend the main result of the next section to a setting where µt+1 is observed only after investment It is made. However, the comparisons with earlier literature, e.g., Abel and Eberly (2011), are more straightforward under the current assumption that µt+1 is observed at date t. Furthermore, the assumption of a perfect one-period foresight is crucial for analytical tractability in the setting with irreversible investments.

7

to be able to sell its used capital goods at their perfectly competitive price, which is formally defined below. We consider a setting with irreversible investments in Section 4 of this paper.

User Cost of Capital and Replacement Cost of Assets To characterize the optimal investment policy, it is helpful in our case to use the notion of the user cost of capital, i.e., a hypothetical perfectly competitive rental rate per unit of capital stock (see, e.g., Jorgenson 1963 and Arrow 1964). The user cost of capital will be denoted by c. It is well-known that in the geometric depreciation scenario, c is equal to r + δ. The generalization of the concept of the user cost of capital to the setting with arbitrary efficiency patterns is due to Rogerson (2008). Following the approach of that paper, consider a hypothetical perfectly competitive rental market for capital goods.17 In this market, a provider of rental services can buy one unit of the capital good (at a cost of one dollar) and then rent out its capacity in future periods. In period τ of the asset’s useful life, the asset will generate xτ · c in rental income. Then, the net present value of the rental firm’s investment project will be: −1 + γ · x1 · c + ... + γ T · xT · c. Since the rental market is assumed to be perfectly competitive, the quantity above must equal zero, i.e., T X γ τ xτ c. (5) 1= τ =1

Then, it follows that the user cost of capital is given by: c = PT

1

τ =1

γ τ xτ

.

(6)

It is straightforward to verify that when the economic depreciation of assets is geometric and given by (2), the right-hand side of (6) reduces to r + δ. On the other hand, for asset with one-hoss shay efficiency, the user cost of capital is given by: c = PT

1

τ =1

γτ

17

=

r . 1 − γT

(7)

The formal results in our paper do not rely on the existence of such a market; however, the notion of the rental market for capital goods will be useful for expositional purposes.

8

We will refer to c · Kt as the current cost of the capital stock in period t. Note that the current cost of capital can be expressed as a function of past investment cash outflows: c · Kt = c

T X

xτ · It−τ ,

(8)

τ =1

Furthermore, let πt be the firm’s economic profit in period t defined as the difference between its operating cash flow and the current cost of the capital stock in that period: πt ≡ R (Zt , Kt ) − c · Kt . In our model, an important difference arises between the firm’s effective capital stock in a given period and the replacement cost of assets-in-place at the beginning of that period.18 To formally define the latter quantity, consider a unit of asset purchased in period t − τ from the perspective of date t. This asset will provide the following stream of capacity in the remaining periods of its useful life: {xτ +1 , ..., xT }. If we again consider a hypothetical rental market for capital goods, the value of such a stream in that market would be:19
vτ ≡ c γxτ +1 + ... + γ T −τ xT .

(9)

Note that v0 = 1 (the replacement cost of an asset just acquired is equal to its price) and vT = 0. Then, the total replacement cost of assets in place at date t (just after the new investment It is made) is equal to: RCt ≡

T −1 X

vτ · It−τ .

(10)

τ =0

It can be verified that in the scenario with geometric depreciation, vτ = (1 − δ)τ . It follows that the replacement cost of assets under the assumption of geometric depreciation is ∞ X RCt = (1 − δ)τ · It−τ = Kt+1 , τ =0 18 19

As shown below, the two quantities are equal in the geometric depreciation scenario. So defined vτ can be interpreted as the perfectly competitive price for a unit of capital good of age τ .

9

i.e., the replacement cost of assets in place at the beginning of period t + 1 is simply equal to that period’s capital stock. However, for arbitrary efficiency profiles, the replacement cost of assets in place and the effective capital stock will be two different linear aggregates of the relevant investment history. Specifically, according to (8), the weights on past investments in the calculation of Kt+1 are proportional to the current efficiency of those investments. On the other hand, equations (9-10) show that the weight on It−τ in the expression for RCt is proportional to the present value of all capacity levels that a unit of capital good of that vintage is yet to generate in the future periods. The difference between the effective capital stock and the replacement cost of assets-inplace is best illustrated in the scenario with assets conforming to the one-hoss shay efficiency profile. For such assets, the capital stock in period t + 1 is given by Kt+1 =

T −1 X

It−τ .

(11)

τ =0

On the other hand, applying the expression for the user cost of capital for one-hoss shay assets in (7), vτ from equation (9) can be calculated as:
1 − γ T −τ . vτ ≡ c γ + ... + γ T −τ = 1 − γT

(12)

Therefore, the replacement cost of assets in place at date t is: RCt =

T −1 X 1 − γ T −τ τ =0

1 − γT

· It−τ .

(13)

While RCt and Kt+1 are determined by the same investments, the two quantities are no longer equal to each other. Figure 1 illustrates how the replacement cost of one unit of the capital good declines with age for assets with one-hoss shay efficiency. Note that while the current capacity of such assets, xτ , is constant over their useful life and vanishes instantaneously at T , their replacement cost declines to zero gradually. Figure 1 further suggests that the rate of decline in replacement cost is higher for older assets. This is indeed the case: for assets with one-hoss efficiency, the ratio vτ /vτ −1 is decreasing in τ . Another observation that can be drawn from Figure 1 is that for relatively low values

10

1 r = 0.05 r = 0.12 r = 0.20

0.9

0.8

0.7

v

=

0.6

0.5

0.4

0.3

0.2

0.1

0

0

5

10

15

=

Figure 1: One-Hoss Shay Efficiency - Replacement Cost per Unit Investment Note: The useful life of assets is 15 years. The discount rates are 5% (solid line), 12% (dash-dotted line), and 20% (dashed line).

of the discount rate, r, the replacement cost of one-hoss shay assets declines to zero almost linearly, i.e., it becomes consistent with the straight-line depreciation rule.20 The assumption of straight-line economic depreciation is used as a starting point in constructing an empirical measure of Tobin’s Q in Lewellen and Badrinath (1997). While the two assumptions, onehoss shay efficiency and straight-line economic depreciation, are consistent for low values of r, Figure 1 suggests that they diverge from each other as r increases. In fact, it can be verified that the assumption of straight-line economic depreciation translates into the following linear efficiency pattern:21 xτ = 1 −

r (τ − 1) . 1 + rT

Therefore, to justify the assumption of straight-line depreciation, one needs to assume that the efficiency of assets declines each period by an amount that depends on the firm’s discount 20 21

Applying L’Hˆ opital’s rule to (12) yields vτ = 1 − See, e.g., Rajan and Reichelstein (2009).

τ T

11

when r → 0.

rate, r/ (1 + rT ). While our general model can be applied in both setting, for our future examples we will maintain the assumption of one-hoss shay efficiency .

3

Adjusted Q and Reversible Investment

Optimal Investment Policy and Firm Valuation In this section, we jointly characterize the firm’s optimal investment policy and its equity value on the optimal investment path. Let V (Zt , µt+1 , Θt ) denote the firm’s cum-dividend value of equity at date t.22 Note that the value function depends on the current value of the demand shift parameter, Zt , as well as on the current growth regime, µt+1 . The function V (Zt , µt+1 , Θt ) must satisfy the following Bellman equation: V (Zt , µt+1 , Θt ) = Zt1−α · Ktα + max {γ · Et [V (Zt+1 , µt+2 , Θt+1 )] − It } . It

(14)

Note further that Et [V (Zt+1 , µt+2 , Θt+1 )] = (1 − λ) V ((1 + µt+1 ) Zt , µt+1 , Θt+1 ) + λEµ˜ [V ((1 + µt+1 ) Zt , µ ˜, Θt+1 )] , where Eµ˜ [·] denotes the expectation operator conditional on a new growth regime (˜ µ) arriving next period. In the proof of Proposition 1, we show that the optimal investment policy is to choose It so as to maximize the firm’s economic profit in the following period:  1−α α It∗ = arg max Zt+1 Kt+1 − c · Kt+1 . I

(15)

The first-order condition for the optimal level of capital stock in period t + 1 then is: 1−α α−1 αZt+1 Kt+1 = c,

from which it follows that the optimal capital stock is the given by 1

∗ Kt+1 = M α · Zt+1 , 22

Recall that at date t, when It is chosen, the firm already observes µt+1 .

12

(16)

α

where the constant M is defined as M ≡ (α/c) 1−α . Furthermore, it is straightforward to check that the maximized economic profit in period t + 1 is: 1−α ∗ · c · Kt+1 α = (1 − α) · M · Zt+1 .

∗ = πt+1

(17)

Let V ex (Zt+1 , µt+1 , Θt+1 ) denote the firm’s ex-dividend value at date t:23 V ex (Zt+1 , µt+1 , Θt+1 ) ≡ V (Zt , µt+1 , Θt ) +

It −Zt1−α · Ktα . |{z} part of Θt+1

Our first result, summarized in Proposition 1, characterizes the firm’s equity value on the optimal investment path.

Proposition 1 On the optimal investment path, 1

∗ Kt+1 = M α · Zt+1

(18)

∗ It∗ = Kt+1 − Kt,t+1 .

(19)

and

The firm’s ex-dividend equity value at date t is given by V ex (Zt+1 , µt+1 , Θt+1 ) = RCt + Zt+1 · ν (µt+1 ) , where ν (µt+1 ) ≡

(1 − α) · M · ω r + λ − µt+1 + λµt+1

  ω ≡ Eµ˜

r−µ ˜ r+λ−µ ˜ + λ˜ µ

(20)

and ω is a constant given by −1 .

Proposition 1 shows that the firm’s value is equal to the sum of two components: the 23

Since V ex (Zt+1 , µt+1 , Θt+1 ) depends on It , we write the last argument of this function as Θt+1 , not Θt . Moreover, the firm’s ex-dividend value at date t does not depend on Zt (since cash flow Zt1−α · Ktα has already been incurred), therefore the first argument of V ex (·, ·, ·) is now Zt+1 .

13

replacement cost of its assets in place and the present value of future expected economic profits. Specifically, we can rewrite equation (20) as: V ex (Zt+1 , µt+1 , Θt+1 ) = RCt +

ω r + λ − µt+1 + λµt+1

∗ πt+1 .

(21)

If the firm could rent its capacity on an as-needed basis at cost c per unit per period of time, then its value would be simply given by the second term of the equation above. The replacement cost of assets in place, RCt , can be viewed as the present value of future rental costs that the firm has effectively “prepaid” by investing in long-lived assets in the past. The two shaded areas in Figure 2 illustrate the two components of the firm value: the darker shaded (orange) area represents the future economic profits whereas the lighter shaded (blue) area corresponds to the replacement cost of assets. Recall that when the economic depreciation of assets is geometric, the replacement cost of assets in place at date t is equal to the capital stock in period t + 1, which in turn is proportional to state variable Zt+1 on the optimal investment path. Then, the ex-dividend value of the firm in (20) can be expressed as a function of state variables alone regardless of its investment history up to date t: (1 − α) · M · ω · Zt+1 r + λ − µt+1 + λµt+1 1 (1 − α) · M · ω · Zt+1 = M α · Zt+1 + . r + λ − µt+1 + λµt+1

∗ V ex (Zt+1 , µt+1 , Θt+1 ) = Kt+1 +

(22)

The valuation function above is the discrete-time analogue of equations (24-25) in Abel and Eberly (2011).24 ∗ For arbitrary efficiency profiles, RCt and Kt+1 will generally be two different linear aggregates of the firm’s investment history up to date t. Accordingly, the firm value will depend on state variables Zt+1 and µt+1 , as well as on RCt . Furthermore, the firm’s optimal investment policy is path-dependent: it follows from equations (18-19) that It∗ depends on Zt+1 and on 24

In Abel and Eberly (2011), the depreciation rate of assets in place is stochastic and, therefore, M is stochastic and is indexed by t. Importantly, at each instant of time, the same depreciation rate is applied to capital goods of all vintages, and therefore the replacement cost of assets can still be measured by their current capacity. We note also that the denominator of the second term in Abel and Eberly (2011) is r + λ − µt+1 , whereas in our model it is r + λ − µt+1 + λµt+1 ; a similar difference also exists in the definition 2 of ω. The extra terms disappear in the continuous-time limit of our model as they are of the order of (dt) .

14

∗ π𝑡𝑡+1 ∗ 𝑅𝑅 𝑍𝑍𝑡𝑡+1 , 𝐾𝐾𝑡𝑡+1

∗ π𝑡𝑡+4

∗ π𝑡𝑡+3

∗ π𝑡𝑡+2

∗ π𝑡𝑡+5

∗ 𝑅𝑅 𝑍𝑍𝑡𝑡+𝑠𝑠 , 𝐾𝐾𝑡𝑡+𝑠𝑠 ∗ 𝑐𝑐𝐾𝐾𝑡𝑡+𝑠𝑠

∗ ∗ − 𝐾𝐾𝑡𝑡+1 Net Investment at date 𝑡𝑡 + 1: 𝐾𝐾𝑡𝑡+2

∗ 𝐼𝐼𝑡𝑡+1

Replacement Investment at date 𝑡𝑡 + 1: ∗ 𝑐𝑐𝑥𝑥4 𝐼𝐼𝑡𝑡−3 𝐾𝐾𝑡𝑡+1 − 𝐾𝐾𝑡𝑡+1,𝑡𝑡+2

𝑐𝑐𝑥𝑥3 𝐼𝐼𝑡𝑡−2 𝑐𝑐𝑥𝑥4 𝐼𝐼𝑡𝑡−2

∗ 𝑐𝑐𝐾𝐾𝑡𝑡+1

𝑐𝑐𝑥𝑥2 𝐼𝐼𝑡𝑡−1 𝑐𝑐𝑥𝑥3 𝐼𝐼𝑡𝑡−1 𝑐𝑐𝑥𝑥4 𝐼𝐼𝑡𝑡−1

𝑡𝑡

𝑐𝑐𝑥𝑥1 𝐼𝐼𝑡𝑡

𝑐𝑐𝑥𝑥2 𝐼𝐼𝑡𝑡

𝑡𝑡 + 1

𝑐𝑐𝑥𝑥3 𝐼𝐼𝑡𝑡

𝑡𝑡 + 2

𝑐𝑐𝑥𝑥4 𝐼𝐼𝑡𝑡

𝑡𝑡 + 3

𝑡𝑡 + 4

𝑡𝑡 + 5

Figure 2: Optimal Investment and Valuation with Vintage Capital ∗ Note: Assets have a useful life of four periods. In period t + 1, the firm’s capital stock, Kt+1 , consists of four

vintages corresponding to investments It−3 through It , depicted along the vertical axis. The firm generates ∗ ∗ revenues of R(Zt+1 , Kt+1 ) and has a current cost of capital of cKt+1 , leaving it with optimal economic profits ∗ of πt+1 . At the end of period t+1, the oldest vintage, It−3 , is fully retired, and the firm experiences a positive demand shock. The firm’s total investment at date t + 1 is decomposed into its replacement component, ∗ ∗ ∗ Kt+1 − Kt+1,t+2 , and net investment, Kt+2 − Kt+1 .

past investments It−τ entering Kt,t+1 . In the geometric depreciation setting, one would have: 1

Kt,t+1 = (1 − δ) Kt∗ = (1 − δ) M α · Zt , and therefore Zt would be a sufficient statistic for the history of investments. This is not the case for other depreciation patterns. Accordingly, the problem of predicting the next period’s investment is now more complex because at each point in time one needs to take into account the full vintage composition of assets in place.

15

Adjusted Q and Net Investment Let i∗t+1 be the firm’s investment rate in period t + 1: i∗t+1 ≡

∗ It+1 . ∗ Kt+1

We now turn to the main question of the paper: how can one estimate i∗t+1 based on the observable information at time t? To address this question, we will decompose the gross investment rate into two separate components: the net investment rate, int+1 , and the replacement investment rate, irt+1 , defined, respectively, as: K∗ − K∗ int+1 ≡ t+2 ∗ t+1 Kt+1 and irt+1 ≡

∗ Kt+1 − Kt+1,t+2 . ∗ Kt+1

Under these definitions, i∗t+1 = int+1 + irt+1 . The replacement investment rate depends only on the history of investments and the economic depreciation schedule. As we will show below, it can therefore be directly calculated from the observable history of the firm’s investments. The net investment rate will now be the main focus of our discussion. Figure 2 depicts the ∗ decomposition of It+1 into its replacement and net components. It immediately follows from Proposition 1 that the net investment rate in period t + 1 is equal to the growth rate in operating cash flows in period t + 2: 1

int+1

K∗ − K∗ M α · Zt+2 − 1 = µt+2 . ≡ t+2 ∗ t+1 = 1 Kt+1 M α · Zt+1

(23)

Therefore, the conditional expectation at date t of the net investment rate in period t + 1 can be calculated as:   Et int+1 = Et [µt+2 ] = (1 − λ) µt+1 + λEµ˜ [˜ µ] .

(24)

Thus the problem of predicting int+1 amounts to estimating µt+1 from the information observable at date t.25 25

One artifact of the assumption of the perfect one-period foresight in the setting with fully reversible ∗ investment is that the firm’s choice of Kt+1 is always ex-post optimal given the realization of Zt+1 . Taken literally, the model then implies that the econometrician can estimate µt+1 by simply calculating the last

16

In Abel and Eberly (2011), the firm’s investment rate and Tobin’s Q comove positively over time since both variables change in the same direction as the expected growth in operating cash flows. To restore this result in our vintage capital setting, we introduce the following adjusted measure of Tobin’s Q: Adjusted Qt ≡

V ex (Zt+1 , µt+1 , Θt+1 ) − RCt , ∗ Kt+1

(25)

i.e., Adjusted Q is defined as the difference between the firm’s value at date t and the replacement cost of its assets divided by its effective capital stock in period t + 1. An inspection of equations (1) and (10) confirms that so defined Adjusted Q can be measured based on the history of investments observed by date t. We postpone the discussion of some of the measurement issues till the end of this section. In the special case when the economic ∗ and depreciation is geometric, RCt = Kt+1 Adjusted Qt ≡

V ex (Zt+1 , µt+1 , Θt+1 ) − 1 = T obin0 s Qt − 1. ∗ Kt+1

Applying our result in Proposition 1 for arbitrary efficiency patterns, we obtain: Adjusted Qt =

Zt+1 · ν (µt+1 ) 1

M α · Zt+1 1

(1 − α) · M 1− α · ω = . r + λ − (1 − λ) µt+1

(26)

It follows that the Adjusted Q is increasing in µt+1 and thus positively associated with the   expected net investment rate, Et int+1 , as calculated in equation (24). An important property of the Adjusted Q is that it is history independent: it varies only in the current realization of µt+1 but not in the past growth rates in operating cash flows or period’s net investment rate, int . In Appendix B, we show in a variant of our model without one-period   foresight that the firm’s latest net investment rate provides a less precise measure of µt+1 and Et int+1 than the Adjusted Q measure defined below. Furthermore, this problem does not arise in the setting with irreversible investment: int does not perfectly reveal µt+1 when the latest investment, It∗ , is zero due to a binding irreversibility constraint.

17

investments. In contrast, note that Tobin’s Q can be expressed as: V ex (Zt+1 , µt+1 , Θt+1 ) RCt 1 ∗ (1 − α) · M 1− α · ω Kt+1 =1+ · r + λ − µt+1 + λµt+1 RCt K∗ = 1 + Adjusted Qt · t+1 . RCt

T obin0 s Qt ≡

(27)

∗ to the replacement cost While Adjusted Q is a function of µt+1 alone, the ratio of Kt+1 of assets, RCt , depends on the firm’s investment history and the economic depreciation schedule: PT −1 ∗ Kt+1 xτ +1 It−τ . = Pτ =0 T −1 RCt τ =0 vτ It−τ

Nezlobin et al. (2016) study the behavior of Tobin’s Q in a model with constant growth in demand and arbitrary efficiency patterns. They show that Tobin’s Q decreases in past investment growth rates. In our setting, past investment growth rates depend on past realizations of µt−τ , and, therefore, past values of µt−τ will have an effect on Tobin’s Q making it a noisier measure of µt+1 . To illustrate this point, consider two firms, A and B, that have been identical up to and including investment It−3 and the stochastic demand shift parameter Zt−2 . In periods t − 1 and t, the stochastic demand shift parameter for Firm A grows at rates µA t−1 = µH and A µt = µL , with µL < µH . In contrast, for Firm B, the growth rates in those periods are: B µB t−1 = µL and µt = µH . Note that in period t, the stochastic demand shift parameter is again the same for both firms: ZtA = Zt−2 (1 + µH ) (1 + µL ) = ZtB . To summarize, the two firms experienced the same demand shocks in periods t − 1 and t, albeit in a different order. Lastly, assume that the growth rate in period t + 1 is the same B for both firms, µA t+1 = µt+1 . B Since µA t+1 = µt+1 , the two firms will have an equal expected net investment rate in period t + 1 and equal Adjusted Q at date t. Moreover, since ZtA = ZtB , the optimal capacity stock in period t + 1 is also the same for the two firms: 1

1

∗ B B ∗ α Kt+1 (A) = M α ZtA (1 + µA t+1 ) = M Zt (1 + µt+1 ) = Kt+1 (B) .

18

However, Firm A and Firm B will generally have a different replacement cost of assets in place at date t. It immediately follows from (27) that their Tobin’s Q will also be different. Specifically, let ∆RCt denote the difference between the replacement cost of assets in place for Firm A and Firm B. In Appendix, we show that if the useful life of assets is at least three periods,  1 (28) ∆RCt = M α Zt−2 (µH − µL ) v0 (x22 − x3 ) − v1 x2 + v2 . Recall that when the economic depreciation is geometric and the useful life is infinite, xτ = (1 − δ)τ −1 and vτ = (1 − δ)τ . Then, ∆RCt = 0, and the two firms have the same Tobin’s Q. However, for assets with one-hoss shay efficiency, 1

∆RCt = M α Zt−2 (µH − µL ) {−v1 + v2 } < 0, i.e., the replacement cost of assets in place is lower and Tobin’s Q is higher for Firm A. Recall that Firm A is the one that experienced the higher shock first and the lower shock second. Accordingly, its net investment rate was higher in period t − 2 than in period t − 1. Firm B, on the other hand, made a relatively larger investment in period t − 1. Hence, while in period t + 1 the two firms have equal effective capital stocks, the capital stock of Firm B is generated by newer capital goods. This leads to a higher replacement cost of assets in place for Firm B. This example demonstrates that Tobin’s Q can assume different values for two firms even when their expected net investment rates coincide.26

Effects of Inflation and Measurement Issues So far we have presented the model under the assumption of constant price of capital goods, thus ignoring the effects that inflation or technological progress may have on the firm’s profits over time. Our model can be readily extended to account for these effects. Let nt denote the price of the capital good in period t and CF It ≡ nt It be the total investment cash outflow in that period. Following Rogerson (2008), consider a setting where the price of new capital goods changes geometrically over time: nt = n0 t , 26

In fact, in our example, the two firms will also have equal replacement investment rates in period t + 1 provided that the useful life of assets is longer than three periods.

19

where 0 <  < 1 + r is a parameter that reflects the net effect of technological progress and inflation on input prices.27 Rogerson (2008) shows that the user cost of capital in this setting is proportional to the price of new capital goods in each period: nt

ct = PT

τ =1

(γ)τ xτ

.

Furthermore, the replacement cost of assets in place can be calculated from the observable history of investment cash outflows according to the following expression:28 RCt = v0 · CF It + ... + vT −1 · CF It−T +1 , where vτ is now defined as:

(29)

PT −τ τ

vτ ≡ 

j j=1 (γ) xτ +j . PT j (γ) x j j=1

(30)

In the one-hoss shay scenario, the expression for vτ above simplifies to: h i τ 1 − (γ)T −τ vτ =

1 − (γ)T

.

(31)

It can be verified that in the setting with geometrically changing asset prices, the following variant of the Adjusted Q measure will have predictive power for the firm’s net investment rate: Adjusted Qt ≡

V ex (Zt , µt+1 , Θt ) − RCt , ∗ nt Kt+1

(32)

∗ where RCt is given by (29-30), and nt Kt+1 is the amount that the firm would have to pay in period t to replicate its effective capital stock in period t + 1 with new assets. The denominator of the ratio above can be calculated from the history of investment cash outflows: ∗ nt Kt+1 = x1 · CF It + ... + xT · T −1 · CF It−T +1 .

(33)

∗ Lastly, note that the replacement investment in period t+1 is equal to nt+1 Kt+1 − Kt+1,t+2 27 28

Our model can further be extended to accommodate a stochastic price of new capital goods. See, for instance, Nezlobin (2012).

20




and can also be written as a linear combination of past investment cash outflows: ∗ nt+1 Kt+1




− Kt+1,t+2 =

T −1 X

(xτ +1 − xτ +2 ) CF It−τ .

(34)

τ =0

We now turn to discussing how the Adjusted Q approach can be applied in practice to predict firm-level investment. We assume that the econometrician observes the firm’s value as well as the history of investments of the firm. The next step is to determine (or make an assumption about) the efficiency pattern of the firm’s assets. If assets can be assumed to adhere to geometric depreciation with a very long useful life, then there is no difference between the Tobin’s Q and Adjusted Q approaches. If one of these two assumptions is violated, i.e., the useful life of assets is short or the one-hoss shay efficiency is more descriptive, then the econometrician needs to estimate the following annual/quarterly ∗ quantities: the replacement cost of assets, RCt ; the effective capital stock, nt Kt+1 ; and
∗ the replacement investment, nt+1 Kt+1 − Kt+1,t+2 . As we have shown above, all of these quantities are linear combinations of past investment cash outflows with weights determined by the efficiency pattern of the firm’s assets. Consider, for example, the case of one-hoss shay efficiency. Lewellen and Badrinath (1997) describe a procedure that uses Gross PP&E disclosures to estimate the useful life of a firm’s assets, T .29 Then, the replacement cost of assets-in-place can be estimated using ∗ is equal to the equations (29) and (31). Furthermore, in the one-hoss shay case, nt Kt+1 30 firm’s inflation-adjusted gross investment over the past T periods: ∗ nt Kt+1 = CF It +  · CF It−1 + ... + T −1 · CF It−T +1 .

Adjusted Q in equation (32) reduces then to the difference between the firm’s value and its replacement cost of assets divided by the inflation-adjusted gross investment. While Adjusted Q predicts the future growth in sales and the net investment rate, the firm’s replacement investment in period t + 1 can be calculated directly from the firm’s observable investment history. For assets with one-hoss shay efficiency, the replacement 29

This procedure relies only on the assumption of finite useful life, equal in duration for capital goods of all vintages, and not on the assumption of straight-line economic depreciation. For an alternative procedure of estimating T , see, e.g., McNichols et al. (2014). 30 For PP&E, firms report gross investment directly in their financial statements; however, the reported gross PP&E values are not adjusted for inflation. For other expenditures that have long-term economic benefits but are not recognized as investments under GAAP (i.e., R&D spending), gross investment can usually be inferred from past financial statements if an estimate of their useful life is available.

21

investment in (34) is given by:
∗ − Kt+1,t+2 = T · CF It−T +1 . nt+1 Kt+1 Accurately predicting the amount above may be difficult in practice because it requires a precise estimate of the assets’ useful life.31 However, it is the firm’s net investment rate that plays a more important economic role in our model since it is directly related to the future growth in firm’s output. The problem of predicting replacement investment simplifies considerably if the goal is to predict investment over a sufficiently long horizon. For example, consider again the case of assets with one-hoss shay productivity and zero inflation. We can show that the following two ratios are positively related in our model: the ratio of the firm’s operating cash flow in period t + 1, R (Zt+1 , Kt+1 ), to the replacement cost of assets, RCt , and the ratio of the present value of replacement investments over the next T periods to the current capital stock, ∗ Kt+1 . Let P V RIt denote the present value of replacement investments from period t + 1 to t + T . Under the one-hoss shay assumption, the firm’s replacement investment in t + τ is simply equal to It+τ −T . Therefore P V RIt takes the following form:

P V RIt ≡

T X

T
X ∗ γ τ Kt+τ − Kt+τ,t+τ +1 = γ τ It+τ −T .

τ =1

τ =1

Using (12), the replacement cost of assets-in-place can be expressed as: RCt =

T −1 X 1 − γ T −τ τ =0

1 − γT

It−τ

∗ Kt+1 − P V RIt = . 1 − γT

1

∗ Now observe that on the optimal investment path R (Zt+1 , Kt+1 ) = M 1− α Kt+1 , leading to the following expression for normalized cash flow:



1 1 − γ T M 1− α 1 − γT 1 R (Zt+1 , Kt+1 ) 1− α ∗ = ∗ M Kt+1 = . ∗ RCt Kt+1 − P V RIt 1 − P V RIt /Kt+1 31

On the other hand, it is straightforward to estimate the replacement investment ex-post under the onehoss shay assumption: it is equal to the change in inflation-adjusted gross investment over a given period minus the new investment made in that period.

22

Similar to the main finding in Abel and Eberly (2011), the equation above shows that normalized cash flow is positively associated with future replacement investment, which is consistent with the cash flow effect in investment regressions. Whereas in Abel and Eberly (2011) the result is due to the stochastic depreciation rate of assets in place (which is the same for capital goods across all vintages), in our model it is driven by deterministic economic depreciation that it is vintage-specific. In both models, the cash flow effect arises even in the absence of liquidity constraints or adjustment costs.

4

Irreversible Investment

Production Technology It is well-known that irreversible investment models with non-geometric economic depreciation do not generally allow for analytical solutions for the firm’s optimal investment policy and valuation.32 Moreover, even with geometric depreciation, irreversible investment models are less tractable in discrete time. The model we propose here, while stylized, allows for rich economic intuition. Specifically, consistent with earlier literature relying on the geometric depreciation specification, we recover the following two results regarding the firm’s optimal investment policy. First, the user cost of capital is shown to increase in the probability of bad events, which is consistent with the Bernanke (1983) “Bad News” principle. Second, the firm’s investment on the optimal path is lumpy, in the sense that the firm finds it optimal to refrain from investment for multiple consecutive periods. Assume that the firm cannot scale back its investment, It ≥ 0 for all t. We maintain the assumption that the demand shift parameter Zt follows the regime-switching process in (4). However, we now require that the output market be weakly expanding over time, i.e., the minimum possible realization of µt is non-negative, µmin ≥ 0.33 It is well-known that if 32

For a more comprehensive discussion, see, for instance, Dixit and Pindyck (1994), p. 374. For general characterizations of the optimal investment policy in the absence of uncertainty, see, e.g., Arrow (1964) and Benhabib and Rustichini (1991). In Rogerson (2008), investments are irreversible but the product market is assumed to be weakly expanding thus precluding the irreversibility constraint from binding. As a consequence, in that paper, irreversibility does not affect the user cost of capital. 33 In fact, a weaker condition is sufficient: the output market should not contract at a rate faster than the minimum depreciation rate of the firm’s capital goods: µmin ≥ − min

τ =2,...,T

xτ −1 − xt . xτ −1

In the geometric depreciation case, the above condition simplifies to µmin ≥ −δ.

23

the output market is weakly expanding over time, then the irreversibility constraint is never binding and the optimal investment policy is governed by the same user cost of capital as in the case of the reversible investment. Therefore, to make the investment problem more interesting, one needs to allow for the possibility of unfavorable demand shocks. We model such “bad news” with an additional regime-switching process. Specifically, assume that each period the firm can lose its pricing (monopoly) power in the product market with probability λC . If the firm does lose its pricing power, it can still sell its output in a perfectly competitive market at an exogenous price p per unit, not exceeding c as given by equation (6).34 Once in the competitive regime, the firm may return back to the monopolistic regime with probability λM . Effectively, the firm has access to two output markets, a “generic market”, which is perfectly competitive, and a “premium market”, in which the firm earns monopoly rents. The firm’s access to the premium market is intermittent and determined by the regime-switching process described above, while the firm’s access to the generic market is continuous. The premium market is always weakly expanding, even when the firm itself is in the competitive regime. Lastly, we assume that the firm has a perfect one-period foresight of its access to the markets: specifically, we say that the firm is in the competitive regime (C) in period t if it can sell its output only in the generic market in period t + 1; conversely, the firm is said to be in the monopoly regime (M) in period t if it has access to both markets in period t + 1.35 To summarize, the firm’s operating cash flow in period t can now be written as:

R (Zt+1 , Kt+1 ) =

 Z 1−α K α

in state M at t;

pK

in state C at t.

t+1

t+1

t+1

We now turn to characterizing the user cost of capital in the scenario with irreversible investment.

User Cost of Capital As we show in the proof of Proposition 2, since p ≤ c, the firm will never invest in the competitive regime. Let c˜ denote the user cost of capital in the monopoly regime, and let φτ (j, k) be the probability that the firm is in regime k ∈ {C, M} τ periods from now if it 34

Our model can be generalized to allow for a stochastic perfectly competitive price of output provided that its supremum does not exceed c. 35 By construction, the price p is low enough to ensure that the firm never finds it optimal to participate in the generic market if the premium market is also accessible.

24

starts in regime j ∈ {C, M} today. For example, in this notation, we have: φ1 (M, M) = 1 − λC , φ1 (M, C) = λC , φ2 (M, M) = (1 − λC )2 + λC λM . It is useful to note that φτ (j, M) + φτ (j, C) = 1. To calculate c˜, suppose the firm is in the competitive regime in period t and consider the marginal unit of investment on the optimal path. The benefit generated by this unit in period t + 2 is equal to x1 · c˜. The marginal benefit of this investment in period t + τ depends on whether the firm is in the competitive or monopoly regime in period t + τ − 1. The expected value of that benefit is given by (φτ −1 (M, M) c˜ + φτ −1 (M, C) p) · xτ . Now we can equate the marginal cost of one unit of investment in period t to its expected marginal benefit:

1 = γ · c˜ · x1 + γ 2 · (φ1 (M, M) c˜ + φ1 (M, C) p) · x2 + ...

(35)

+ γ T · (φT −1 (M, M) c˜ + φT −1 (M, C) p) · xT . The equation above is linear in c˜ and thus uniquely defines the user cost of capital. Comparing equations (5) and (35), a few observations are in order. First, note that φτ (M, M) = 1 − φτ (M, C) and p ≤ c imply that c˜ ≥ c. As expected, the user cost of capital in the setting with irreversible investment exceeds that in the setting with reversible investment. Moreover, it is straightforward to see that if p = c, then c˜ = c, i.e., irreversibility affects the firm’s user cost of capital only if the bad news are sufficiently bad. Generally, the user cost of capital is monotonically decreasing in p. Consider further the following special case: λM = 0, p = 0 and λC > 0, i.e., the competitive state is permanent and the value of the firm’s assets in that state is zero. Then, we have: φτ (M, M) = (1 − λC )τ and c˜ = PT

1

τ −1 τ xτ τ =1 γ (1 − λC )

25

.

In this special case, the probability of bad events effectively serves as a hazard rate by reducing the periodic discount factor for periods t + 2 and onwards from γ to γ (1 − λC ). For further discussion, it will be convenient to rewrite equation (35) in matrix form. Let I be the 2 × 2 identity matrix and A be the following state transition matrix: A≡

1 − λC λC λM 1 − λM

! .

(36)

  Then, it is straightforward to verify that the vector of probabilities φτ (M, M) φτ (M, C) is given by   φτ (M, M) φτ (M, C) = e1 Ai ,   where e1 is the following 1 × 2 basis vector e1 ≡ 1 0 . Now we can rewrite the definition of c˜ in (35) as: 1 = γe1 I

c˜ p

!

c˜ p

x1 + γ 2 e 1 A

! x2 + ... + γ T e1 AT −1

c˜ p

! xT .

(37)

To characterize c˜ in the general setting, we introduce the following auxiliary quantity:36 cˆ ≡ PT

τ =1

1 γ τ (1 − λC − λM )τ −1 xτ

.

(38)

We note that for all efficiency patterns, cˆ ≥ c. Straightforward algebra shows that for assets with one-hoss shay efficiency, cˆ =

r + λC + λM 1 − {γ (1 − λC − λM )}T

,

whereas for assets conforming to geometric depreciation, cˆ reduces to cˆ = r + δ + (1 − δ) (λC + λM ) . In the general case, c˜ is given by Lemma 1. 36 Intuitively, the definition of cˆ in (38) differs from the definition of c in (6) in that the periodic discount factor for periods 2, ..., T of the asset’s useful life is lowered from γ to γ (1 − λC − λM ).

26

Lemma 1 The user cost of capital that solves (37) is given by: (c − p) (ˆ c − c) λC cλC + cˆλM .

c˜ = c +

To conclude our discussion of the user cost of capital in this setting, let us note that it takes a particularly compact form for assets with geometric economic depreciation. Specifically, one readily obtains: c˜ = r + δ +

(r + δ − p) (1 − δ)λC . r + δ + (1 − δ) λM

As discussed above, the user cost of capital increases in the probability of “bad” states (λC ) as long as the marginal return on capital in those states, p, is less than c = r + δ. In contrast, when the firm is in the monopoly regime, the growth rate in demand for the firm’s output, µt , does not affect the firm’s user cost of capital, provided that µt ≥ 0. This is a manifestation of the “bad news” principle in our model.

Optimal Investment Policy and Valuation Proposition 2 shows that the firm’s optimal investment policy is to invest in the only  1−α Zt+1 monopoly regime so as to equate the next period marginal revenue, α Kt+1 , to the user cost of capital c˜:  1−α ∗ Kt+1 = arg max Zt+1 Kt+1 − c˜ · Kt+1 . Kt+1

The optimal capital stock in period t + 1 is then given by: 1

∗ Kt+1 = M (˜ c) α Zt+1 , α

where M (˜ c) ≡ (α/˜ c) 1−α , while the corresponding optimal investment level, It∗ , is determined ∗ from It∗ = Kt+1 −Kt,t+1 . Note that since the premium market is always expanding and assets are weakly depreciating, It∗ specified above is non-negative. To fully characterize firm’s value, we need to extend the notion of the replacement cost of assets in place to the setting with regime-switching demand and irreversibility. In the previous section, the replacement cost of a unit of capital good of age τ , vτ , was calculated as a function of marginal benefits that such unit was to generate in the remaining periods of 27

its useful life.37 In the present setting, the future benefits of each unit of capital good depend on the future realizations of the market state variable. In turn, the probability distribution of future regimes depends on the current regime. Consider, for example, the expected marginal benefit that a unit of investment of age τ today provides two periods from now, assuming that the firm is currently in the monopoly regime. This amount can be calculated as follows: xτ +2 (φ1 (M, C) · p + φ1 (M, M) · c˜) . In contrast, if the firm is currently in the competitive regime, the same expected benefit will be given by: xτ +2 (φ1 (C, C) · p + φ1 (C, M) · c˜) . Therefore, the replacement cost of assets now depends not only on the history of investments but also on the current market regime. Let vτ,M and vτ,C denote the replacement cost per unit of investment of age τ in the monopoly and competitive regimes, respectively, and let vτ =

vτ,M vτ,C

! .

We define vτ as follows: vτ ≡ Iγxτ +1 + Aγ 2 xτ +2 + AT −τ −1 γ T −τ xT · 



c˜ p

! (39)

Intuitively, vτ,j for j ∈ {C, M} is set equal to the expected value of output that the asset will produce in future periods (given that the current market state is j), where each output unit is valued at c˜ if the firm has access to the premium market in the corresponding period and at p if it does not. Then, the total replacement cost of assets in state j ∈ {C, M} is equal to: T −1 X RCt,j = vτ,j · It−τ . τ =0

Note that the definition of c˜ in (37) implies that v0,M = 1. Therefore, the replacement cost of newly acquired assets is equal to their price if the assets have been purchased in 37

See equation (9) above.

28

the monopoly regime. Furthermore, in the proof of Proposition 2, we demonstrate that vτ,C ≤ vτ,M for all τ . It follows that for a given realization of investment history, the replacement cost of assets in place is greater in the monopoly state than in the competitive state. As a special case of this inequality, note that v0,C ≤ vτ,M = 1, i.e., if the firm were to invest in the competitive regime, the replacement cost of assets just acquired would be less than their acquisition cost, which indicates that investing in the competitive regime is indeed value-destroying for the firm. Proposition 2 summarizes these results.

Proposition 2 If the firm is in the monopoly regime in period t, it invests so that 1

∗ Kt+1 = M (˜ c) α · Zt+1 ∗ and It∗ = Kt+1 − Kt,t+1 . The firm does not invest in the competitive regime. The firm’s ex-dividend equity value at date t in regime j ∈ {M, C} is given by

Vjex (Zt+1 , µt+1 , Θt+1 ) = RCt,j + Zt+1 · νj (µt+1 ) ,

(40)

where νj (·), j ∈ {M, C}, is monotonically increasing in its argument.

Similar to our result in Proposition 1, equation (40) decomposes the firm’s value into the replacement cost of its assets and the present value of the future expected economic profits. However, in the current setting the two components of the firm value are regime-dependent. To illustrate how the intermittent access to the premium output market affects the present value of future economic profits, consider the special case of λ = 1, i.e., when a new growth regime in Zt arrives every period, and let µ ¯ = Eµ˜ [˜ µ]. Then, it can be verified that functions νM (µt+1 ) and νC (µt+1 ) are constants taking the following form:38 νM = (1 − α) M (˜ c)

(r − µ ¯ + (1 + µ ¯) λM ) , (r − µ ¯) (r − µ ¯ + (1 + µ ¯) (λC + λM ))

38

(41)

In this case, functions νM (·) and νC (·) do not depend on µt+1 because a new regime µt+2 arrives with probability one.

29

and νC = (1 − α) M (˜ c)

λM (1 + µ ¯) . (r − µ ¯) (r − µ ¯ + (1 + µ ¯) (λC + λM ))

(42)

When λC = 0, equation (41) reduces to the usual Gordon growth formula. As λC increases, the firm is anticipated to be shut out of the premium market in some periods and the present value of future expected economic profits is reduced accordingly.

Adjusted Q and Investment ∗ Predicting investment It+1 in the setting with irreversibility entails two tasks: first, one needs to determine the likelihood of the firm being in the monopoly regime in period t + 1; second, one needs to estimate the investment amount conditional on the firm being in the monopoly regime. To start with the former task, note that the current regime in period t is a sufficient statistic for the next period’s regime. There are two observable variables correlated with the firm’s current regime. First, the firm’s latest investment, It∗ , is different from zero only if the firm is currently in the monopoly regime. Therefore, consistent with much of the empirical literature, lagged investment will have predictive power for future investment in our model.39 Second, the econometrician can calculate the firm’s economic profit in the latest period relative to the user cost of capital, c, that ignores the effect of irreversibility: R (Zt , Kt∗ )−cKt∗ . This economic profit will be non-negative only if the firm is in the monopoly regime in period t−1, i.e., the firm has access to the premium market in period t. Therefore, a high (exceeding the current cost of capital) operating cash flow in the latest period will be associated with a higher probability of the firm being in the monopoly regime in the next period. This is a potential channel through which the cash flow variable might load significantly in investment regressions. In estimating the firm’s investment amount, it is important to consider whether the firm is in the monopoly or competitive state in the latest period. First, assume that the firm is in the monopoly regime in period t as indicated by It∗ > 0. Then, the following variant of Adjusted Q will be informative about the next period’s net investment rate:40

Recall also that when It∗ is greater than zero, the net investment rates are serially correlated as long as λ > 0. 40 As before, the firm’s replacement investment rate can be estimated directly from the history of investment cash outflows. 39

30

Adjusted Qt,M ≡

ex VM (Zt+1 , µt+1 , Θt+1 ) − RCt,M . ∗ Kt+1

Indeed, Proposition 2 implies that the ratio above simplifies to: 1

Adjusted Qt,M = M (˜ c)− α · νM (µt+1 ) , which is an increasing function of µt+1 . The firm’s net investment rate in period t + 1 will be non-zero only if the firm is still in the monopoly regime that period, in which case the net investment rate will be given by: int+1 =

∗ ∗ − Kt+1 Kt+2 = µt+2 . ∗ Kt+1

As before, Et [µt+2 ] is monotonically increasing in µt+1 and, as a result, it is correlated with Adjusted Q. Now let us consider an arguably more interesting case when the firm is in the competitive regime in period t (It∗ = 0). Now the firm’s capital stock in period t + 1 is no longer informative about the state of demand in the premium market, Zt+1 . However, equation (40) reveals that the firm’s equity value still carries information about Zt+1 since investors value the firm taking into account the probability of it returning to the premium market. It is now sensible to calculate the net investment rate and Adjusted Q relative to latest period in which the firm was expanding its productive capacity. Specifically, let η (t) be the last period in which the firm’s investment was positive:  η (t) ≡ max η ≤ t|Iη∗ > 0 . Consider the following definition of Adjusted Q in the competitive regime: Adjusted Qt,C

VCex (Zt+1 , µt+1 , Θt+1 ) − RCt,C . ≡ ∗ Kη(t)+1

It follows from Proposition 2 that Adjusted Q defined above reduces to: 1 −α

Adjusted Qt,C = M (˜ c)

· νM (µt+1 ) ·

t Y j=η(t)+1

31

(1 + µj+1 ) .

(43)

Now assume that the firm enters the monopoly regime in period t + 1. Then, its net investment rate measured relative to period η (t) + 1 will be given by: ∗ ∗ − Kη(t)+1 Kt+2 ∗ Kη(t)+1

=

t+1 Y

(1 + µj+1 ) − 1.

j=η(t)+1

The date-t expectation of the above rate can then be calculated as: {(1 − λ) (1 + µt+1 ) + λEµ˜ [˜ µ]}

t Y

(1 + µj+1 ) − 1.

(44)

j=η(t)+1

Comparing expression (44) to Adjusted Q in (43), we observe that both Adjusted Q and the expected period t+1 net investment rate (measured relative to period η (t)+1) are increasing functions of growth rates µη(t)+2 , ..., µt+1 . It is again straightforward to verify that the firm’s replacement investment (measured relative to period η (t) + 1) can be inferred directly from the history of investments. To conclude this section, we note that while the estimation process for the model with irreversible investment is generally more complex than for the model outlined in the previous section, in certain special cases, the models lead to similar predictions. Consider, for example, a scenario where 0 < λC , λM < 1 yet the competitive output price p is sufficiently close to c. It follows from Lemma 1 that under these conditions, c˜ will be close to c as well. Furthermore, as we show in the proof of Proposition 2 (equation 74), the replacement cost of assets-in-place in both monopoly and competitive regimes will approach the replacement cost of assets as calculated in the setting with reversible investment. However, since λC and λM are greater than zero, the firm’s investment policy will still be of the intermittent nature.

Conclusion In this paper, we have extended the Q-theory of investment to a setting with vintage capital. Outside of the special case of geometric economic depreciation, the firm’s replacement cost of assets-in-place and its current capital stock are two different linear aggregates of its investment history. While the firm’s capital stock captures the current scale of operations, the replacement cost of assets-in-place depends on their current and future productive capacity. We have defined the Adjusted Q as the difference between the firm’s current value and the replacement cost of its assets divided by the capital stock and shown that Adjusted 32

Q predicts future net investment better than the traditional Q. The distinction that we draw between the replacement cost of assets-in-place and the firm’s current capital stock may prove important for econometricians in future studies. For example, the common practice of deflating firm-level economic variables by the book value of assets may artificially introduce undesirable path-dependency in those variables. The reason for this is that the straight-line depreciation rule, which is overwhelmingly applied in practice, leads to book values with vintage-composition dependency. Our analysis suggests that the firm’s capital stock may be a more appropriate deflator for firm-level economic variables. Related to this, it may be interesting to re-examine some of the prior studies that relied on deflation by book values. Consider, for example, a stylized observation in the finance literature that the market-to-book ratio has explanatory power for the cross-section of stock returns. Should a higher value of the market-to-book ratio be attributed to the presence of growth options or the vintage-composition effect introduced in the denominator of this ratio? The model of vintage capital that we have presented in this paper is analytically tractable in discrete-time in both reversible and irreversible investment settings. As such, it may prove useful in future theoretical studies of investment behavior. Our current analysis in this paper has relied on several simplifying assumptions. For example, we have assumed that the firm does not face capital adjustment costs or financing constraints. Studying the effects of such constraints on firm value and investment in the presence of vintage capital is a promising avenue for future research.

Appendix A Proof of Proposition 1. Recall that the cum-dividend value function must satisfy the following Bellman equation: V (Zt , µt+1 , Θt ) = Zt1−α · Ktα + max {γ · Et [V (Zt+1 , µt+2 , Θt+1 )] − It } . It

(45)

Consider the following candidate solution for V (Zt , µt+1 , Θt ): V (Zt , µt+1 , Θt ) =

Zt1−α

·

Ktα

+

T −1 X τ =1

where ν (·) is some function yet to be determined. 33

vτ It−τ + Zt+1 · ν (µt+1 ) ,

(46)

Substituting the anzats in (46) into the Bellman equation (45), we obtain: T −1 X

vτ It−τ + Zt+1 · ν (µt+1 ) =

τ =1

(

"

α 1−α + · Kt+1 max γ · Et Zt+1 It

T −1 X

# vτ It−τ +1 + Zt+2 · ν (µt+2 ) − It

) . (47)

τ =1

It is straightforward to verify that for vτ given by (9), the following condition holds: γvτ − vτ −1 = −cγxτ .

(48)

Applying (48) and recalling that µt+1 and Zt+1 are realized at date t, equation (47) can be rewritten as ( ! ) T X 1−α α Zt+1 · ν (µt+1 ) = max γ Zt+1 · Kt+1 −c xτ It−τ +1 + γEt [Zt+2 · ν (µt+2 )] , It

τ =1

which is further equivalent to:
1−α α Zt+1 · ν (µt+1 ) = max γ Zt+1 · Kt+1 − cKt+1 + γEt [Zt+2 · ν (µt+2 )] . It

(49)

∗ It follows that Kt+1 is given by (16). Substituting the optimized value of economic profit in period t + 1 into (49), we obtain:

Zt+1 · ν (µt+1 ) = γ (1 − α) · M · Zt+1 + γEt [Zt+2 · ν (µt+2 )] . Let us now expand the last term in the right-hand side of the equation above as: γEt [Zt+2 · ν (µt+2 )] = γ (1 − λ) Zt+1 (1 + µt+1 ) · ν (µt+1 ) + γλZt+1 Eµ˜ [(1 + µ ˜) · ν (˜ µ)] . The first term in the right-hand side above reflects the value of γZt+2 · ν (µt+2 ) if the current regime stays in place for another period, i.e., µt+2 = µt+1 , multiplied by the probability of that event, 1 − λ. The second term is is the expected value of γZt+2 · ν (µt+2 ) conditional on a new regime arriving. Let B ≡ Eµ˜ [(1 + µ ˜) · ν (˜ µ)] . Note that B is a constant that depends on the distribution

34

F (˜ µ) and the function ν (·) . Then, we have: Zt+1 · ν (µt+1 ) = γ (1 − α) · M · Zt+1 + γ (1 − λ) Zt+1 (1 + µt+1 ) · ν (µt+1 ) + γλZt+1 B. It follows that ν (µt+1 ) =

(1 − α) · M + λB . r + λ − µt+1 + λµt+1

(50)

It is straightforward to see that if B is positive, ν (µt+1 ) increases in µt+1 . We can now determine the constant B. Using (50), we get: 

 (1 − α) · M + λB B = E (1 + µ ˜) · . r+λ−µ ˜ + λ˜ µ Therefore, 

(1 + µ ˜) B = (1 − α) · M · E r+λ−µ ˜ + λ˜ µ

  E

r−µ ˜ r+λ−µ ˜ + λ˜ µ

−1 .

(51)

Now substituting the expression above into (50) and simplifying, we obtain

ν (µt+1 ) =

n h io−1 r−˜ µ (1 − α) · M · E r+λ−˜ µ+λ˜ µ r + λ − µt+1 + λµt+1

.

To conclude the proof note that the ex-dividend price of the firm is equal to: V ex (Zt+1 , µt+1 , Θt+1 ) = V (Zt , µt+1 , Θt ) − Zt1−α · Ktα + It = RCt + Zt+1 · ν (µt+1 ) .

Proof of Equation 28. From equations (18-19), investment It−2 for Firm j (where j ∈ {A, B}) is given by: 1

j It−2 = M α Zt−2 · (1 + µjt−1 ) − Kt−2,t−1 ,

(52)

where Kt−2,t−1 is the same for both firms since the firms have been identical up to investment It−3 .

35

j can be written as: Recall that Zt is the same for both firms. Therefore, It−1 1

j j It−1 = M α Zt − Kt−1,t 1

j = M α Zt − Kt−2,t − x2 It−2 1

1

= −x2 M α Zt−2 · (1 + µjt−1 ) + x2 Kt−2,t−1 + M α Zt − Kt−2,t ,

(53)

where the last equality follows by applying (52). Note that only the first term in the righthand side of (53) is different across firms. Lastly, we have:

1

j j − x3 It−2 Itj = M α Zt+1 − Kt−2,t+1 − x2 It−1 1

1

= M α Zt+1 − Kt−2,t+1 + x22 M α Zt−2 · (1 + µjt−1 ) − x22 Kt−2,t−1 1

1

−x2 M α Zt + x2 Kt−2,t − x3 M α Zt−2 · (1 + µjt−1 ) + x3 Kt−2,t−1 .

(54)

The only two terms in the right-hand side that are different between the two firms are 1 1 x22 M α Zt−2 · (1 + µjt−1 ) − x3 M α Zt−2 · (1 + µjt−1 ). Since the two firms were identical up to investment It−3 and T ≥ 3, we obtain:


A B A B ∆RCt = v0 ItA − ItB + v1 It−1 − It−1 + v2 It−2 − It−2 . Now applying (52-54), we get: 
1 ∆RCt = M α Zt−2 (µH − µL ) v0 x22 − x3 − v1 x2 + v2 .

Proof of Lemma 1. Observe that the matrix A can be diagonalized as follows: A = UDU−1 , where U≡

C 1 − λλM

1

36

1

(55)

! (56)

and 1 0 0 1 − λC − λM

D≡

! .

(57)

For future reference, note that U−1

λM ≡ λM + λC

1

λC λM

−1

1

! .

(58)

Let Π ≡ γIx1 + γ 2 Ax2 + ... + γ T AT −1 xT .

(59)

Then, equation (37) can be written as: 1 = e1 Π

c˜ p

! .

(60)

Applying the diagonalization in (55) to (59), we obtain: Π = U(γx1 I + γ 2 x2 D + ... + γ T xT DT −1 )U−1 .

(61)

The matrix D is diagonal, so we have: Dτ =

1 0 0 (1 − λC − λM )τ

! .

Then, equation (61) simplifies to: PT

Π=U =U

τ 0 τ =1 γ xτ PT τ −1 τ 0 xτ τ =1 γ (1 − λC − λM ) ! c−1 0 U−1 . 0 cˆ−1

Let us now plug in the expression above into (60): 1 = e1 U

c−1 0 0 cˆ−1

! U−1

c˜ p

! .

The claim of the Lemma now follows by straightforward algebra. 37

! U−1

Proof of Proposition 2. We will first solve the investment and valuation problems assuming that the firm does not invest in the competitive regime and subsequently prove that investing in the competitive regime indeed cannot be optimal. Let ! VM (Zt , µt+1 , Θt ) Vt = , VC (Zt , µt+1 , Θt ) where Vj (Zt , µt+1 , Θt ) is the firm’s value in regime j ∈ {M, C} at date t right after the operating cash flow of period t, R (Zt , Kt ), is paid out but before the investment It is made. Then, we have: Vjex (Zt+1 , µt+1 , Θt+1 ) = Vj (Zt , µt+1 , Θt ) + It . We need to show that Vj (Zt , µt+1 , Θt ) = v1,j It−1 + ... + vT −1,j It−T +1 + Zt+1 νj (µt+1 ) , or, in the matrix form, Vt =

T −1 X

vτ It−τ + Zt+1 ν (µt+1 ) ,

(62)

τ =1

where νM (µt+1 ) νC (µt+1 )

ν (µt+1 ) =

! .

The valuation vector Vt must satisfy the following Bellman equation: ( Vt = max γ It

1−α α Zt+1 Kt+1 pKt+1

!

) + γAEt [Vt+1 ] − It e01

,

(63)

where the prime denotes the transposition operator. The last term in the maximization problem above is therefore equal to: It 0

It e01 =

! ,

which reflects our current assumption that the firm invests only in the monopoly regime.

38

Note further that (1 − λC ) Et [VM (Zt+1 , µt+2 , Θt+1 )] + λC Et [VC (Zt+1 , µt+2 , Θt+1 )] λM Et [VM (Zt+1 , µt+2 , Θt+1 )] + (1 − λM ) Et [VC (Zt+1 , µt+2 , Θt+1 )]

AEt [Vt+1 ] =

! .

The current investment It enters Θt+1 in both the top and the bottom element of this vector. However, since It is different from zero only in the monopoly state, it will be understood that It = 0 in Θt+1 in the calculation of the bottom element.41 Plugging in the anzats in (62) into (63) and rearranging, one obtains:

( Zt+1 ν (µt+1 ) = max γ It

!

1−α α Zt+1 Kt+1 pKt+1

+ γA

T −1 X

!

(1 − λC ) v1,M + λC v1,C 0

+γ

It )

vτ It+1−τ + γAE [Zt+2 ν (µt+2 )] −

It e01

τ =2

−

T −1 X

vτ It−τ . (64)

τ =1

Now observe that vectors vτ given by (39) satisfy: γAvτ − vτ −1 = −γ

c˜ p

! xτ

(65)

for τ ≥ 2. Furthermore, the definition of c˜ in (37) can be rewritten as: ( e1

c˜ p

γ

!

) x1 + γAv1

= 1,

which implies 1 0 0 0

!( γ

c˜ p

!

) x1 + γAv1

= e0 1 .

It then follows that γ

(1 − λC ) v1,M + λC v1,C 0

! − e0 1 = γ

c˜ 0

! x1 .

(66)

Collecting the coefficients on investments It−T +1 , ..., It and applying (65) and (66), equa41

Similarly, in the first term of the right-hand side of (63), pKt+1 is calculated at It equal to zero. To summarize, the second row of (63) is calculated at It = 0.

39

tion (64) reduces to: (

1−α α Zt+1 Kt+1 − c˜Kt+1 pKt+1 − pKt+1

Zt+1 ν (µt+1 ) = max γ It

!) + γAE [Zt+2 ν (µt+2 )] .

(67)

1−α α It follows that It∗ is chosen in the monopoly state so as to maximize Zt+1 Kt+1 − c˜Kt+1 and ∗ , is equal to: the optimal capital stock, Kt+1 1

∗ Kt+1 = M (˜ c) α Zt+1 .

(68)

Since the premium market is always expanding (Zt+1 ≥ Zt ), investment levels It∗ correspond∗ ing to Kt+1 given above are always non-negative. ∗ into (67) and dividing both sides by Zt+1 , we obtain that ν (·) must Substituting Kt+1 satisfy the following condition: ν (µt+1 ) = γ (1 − α) M (˜ c) e0 1 + γAE [(1 + µt+2 ) ν (µt+2 )] .

(69)

For a given function ν (·), let B denote the following constant vector: B=

BM BC

! ≡

Eµ˜ [(1 + µ ˜) νM (˜ µ)] Eµ˜ [(1 + µ ˜) νC (˜ µ)]

! .

(70)

Then, equation (69) can be expanded as: ν (µt+1 ) = γ (1 − α) M (˜ c) e0 1 + γ (1 − λ) (1 + µt+1 ) Aν (µt+1 ) + γλAB.

(71)

Now solving for ν (µt+1 ), one obtains: ν (µt+1 ) = γ {I − γ (1 − λ) (1 + µt+1 ) A}−1 [(1 − α) M (˜ c) e0 1 + λAB] . To fully characterize the solution, it now remains to find the constant vector B. Recall that B = Eµ˜ [(1 + µ ˜) ν (˜ µ)]. Let G denote the following matrix:   G ≡ Eµ˜ (1 + µ ˜) {I − γ (1 − λ) (1 + µ ˜) A}−1 .

40

(72)

Then, applying (72), we have: B = Eµ˜ [(1 + µ ˜) ν (˜ µ)] = γG [(1 − α) M (˜ c) e0 1 + λAB] . It then follows that B = γ (1 − α) M (˜ c) {I − γλGA}−1 Ge0 1 , which completes the characterization of the valuation function. To conclude the proof, we need to show that i) ν (µt+1 ) decreases in µt+1 and ii) it cannot be optimal for the firm to invest in the competitive regime. To prove the former claim, let us rewrite (71) as: ν (µt+1 ) − γ (1 − α) M (˜ c) e0 1 − γ (1 − λ) (1 + µt+1 ) Aν (µt+1 ) = γλAB. Differentiating the equation above with respect to µt+1 , we obtain: dν (µt+1 ) dν (µt+1 ) − γ (1 − λ) (1 + µt+1 ) A − γ (1 − λ) Aν (µt+1 ) = 0. dµt+1 dµt+1 Now we can solve for the derivative of ν (µt+1 ) with respect to µt+1 : dν (µt+1 ) = γ (1 − λ) {I − γ (1 − λ) (1 + µt+1 ) A}−1 Aν (µt+1 ) . dµt+1 It can be verified that {I − γ (1 − λ) (1 + µt+1 ) A}−1 A = κ

1 − λC − ρ λC λM 1 − λM − ρ

! ,

where ρ ≡ γ (1 − λ) (1 + µt+1 ) (1 − λC − λM ) and κ ≡ {(1 − γ (1 − λ) (1 + µt+1 )) (1 − ρ)}−1 > 0. All elements of the matrix above are positive, and therefore both components of ν (µt+1 ) must be increasing in µt+1 . To prove that it is suboptimal for the firm to invest in the competitive stage, observe 41

that under the investment policy described above, the firm’s marginal revenue is equal to c˜ in all periods where the firm has access to the premium market, and it is equal to p in periods when only the generic market is accessible. Now consider the following extended problem (EP ). Assume that the firm can rent (or rent out) its capacity on an as-needed basis at unit price c˜ when it has monopoly power and at unit price p when the market is competitive. The definition of c˜ in (37) ensures that the firm is indifferent between buying a unit of capital good in the monopoly regime and renting an equivalent stream of capacity in future periods at prices p or c˜, depending on future output market conditions. Since under the current investment policy, the marginal revenue is always equal to the rental rates, this policy remains an optimal one for problem EP among policies with zero investment in the competitive regime. We will now show that a policy cannot be optimal for problem EP if it allows for positive investment in the competitive regime. Specifically, the firm will be better off by renting a stream of capacity equivalent to one unit of capital good than purchasing such unit in the competitive regime. The present value of rental payments needed to replicate the capacity of one unit of capital good purchased in the competitive regime is given by: γe2 I

c˜ p

!

c˜ p

x1 + γ 2 e2 A

! x2 + ... + γ T e2 AT −1

c˜ p

! xT ,

where e2 ≡ (0, 1). The expression above is equal to v0,C , and, therefore we need to show that v0,C ≤ 1 = v0,M . We will prove a more general claim that vτ,C ≤ vτ,M for all τ . Applying the diagonalization (55-57) to vτ in (39), we obtain:
vτ = U γxτ +1 I + γ 2 x2 D + ... + γ T −τ xT DT −τ −1 U−1 =U

Sτ,1 0 0 Sτ,2

! U−1

where Sτ,1 ≡

c˜ p

Sτ,2 ≡

T X

T X

,

γ τ xτ

γ τ (1 − λC − λM )τ −1 xτ .

τ =1

42

!

!

τ =1

and

c˜ p

(73)

It can be readily verified that Sτ,1 ≥ Sτ,2 ≥ 0. Multiplying the matrices in the right-hand side of (73), we obtain: 1 vτ = λC + λM

λC (Sτ,1 p + Sτ,2 (˜ c − p)) + λM Sτ,1 c˜ λC Sτ,1 p + λM (Sτ,1 c˜ − Sτ,2 (˜ c − p))

! .

Therefore, vτ,M − vτ,C = Sτ,2 (˜ c − p) ≥ 0. It is worthwhile to note that in the limiting case of p → c, c˜ given by Lemma 1 approaches c as well, and ! ! λC Sτ,1 p + λM Sτ,1 c˜ 1 1 vτ → = cSτ,1 , (74) λC + λM λC Sτ,1 p + λM Sτ,1 c˜ 1 i.e., the replacement cost of assets-in-place approaches that in the setting with fully reversible investment.

Appendix B In this Appendix, we outline the solution to our model with reversible investment in the setting where the firm does not observe the realization of growth regime µt+1 before choosing investment It . Since capital goods come online with a one-period lag, the firm must plan its capital stock one period ahead while facing uncertainty about the output market conditions. ∗ , chosen based on the expected value of Zt+1 , may end As a result, the firm’s optimal Kt+1 up being ex-post suboptimal given the realization of Zt+1 . Our main goal is to demonstrate that the date-t expectation of the firm’s net investment   rate in period t + 1, Et int+1 , and Adjusted Q are functions of µt alone. As a consequence,   Adjusted Q is informative about Et int+1 . Furthermore, in contrast with the model presented in the main text, the firm’s latest realized net investment rate int is a function of not only   µt , but also µt−1 . Therefore, Adjusted Q provides a more precise signal about Et int+1 than the net investment rate int . Assume that the firm does not observe µt+1 prior to choosing investment It . Then, the Bellman equation for the firm’s cum-dividend value takes the following form: V (Zt , µt , Θt ) = Zt1−α · Ktα + max {γ · Et [V (Zt+1 , µt+1 , Θt+1 )] − It } . It

43

(75)

We conjecture the following solution for V (Zt , µt , Θt ): V (Zt , µt , Θt ) =

Zt1−α

·

Ktα

+

T −1 X

vτ It−τ + Zt · ν (µt ) ,

(76)

τ =1

where the function ν (·) will be determined below. Substituting the above anzats into the Bellman equation and following the same steps as in the derivation of equation (49), equations (75-76) lead to   1−α α · Kt+1 − cKt+1 + γEt [Zt+1 · ν (µt+1 )] . Zt · ν (µt ) = max γEt Zt+1 Kt+1

(77)

The first-order condition for the maximization problem in the right-hand side above yields:  1−α  α−1 αEt Zt+1 Kt+1 = c. Then, the optimal capital stock in period t + 1 can be written as:

1   1  1−α  1−α 1 1  ∗ Kt+1 = M α Et Zt+1 = M α Zt Et (1 + µt+1 )1−α 1−α 1 α

= M ψ (µt ) Zt , where the function ψ (µt ) is given by:    1 ψ (µt ) ≡ (1 − λ) (1 + µt )1−α + λEµ˜ (1 + µ ˜)1−α 1−α . It is useful to note for future reference that ψ (µt ) is increasing in µt . ∗ Substituting Kt+1 into (77) we obtain after some algebra the equation for ν (µt ): ν (µt ) = γ (1 − α) M ψ (µt ) + γEt [(1 + µt+1 ) ν (µt+1 )] . Expanding the expectation in the right-hand side yields: ν (µt ) = γ (1 − α) M ψ (µt ) + γ (1 − λ) (1 + µt ) ν (µt ) + γλEµ˜ [(1 + µ ˜) ν (˜ µ)] .

44

(78)

Let B ≡ Eµ˜ [(1 + µ ˜) · ν (˜ µ)] . Then, ν (µt ) can be expressed as: ν (µt ) =

(1 − α) M ψ (µt ) + λB . r + λ − µt + λµt

(79)

Since the numerator is increasing in µt while the denominator is decreasing in µt , it follows that ν (µt ) increases in µt . Following the same steps as in the proof of Proposition 1, one can verify that B= where

(1 − α) M ω1 , 1 − λω2



 ψ (˜ µ) (1 + µ ˜) , r+λ−µ ˜ + λ˜ µ



 1+µ ˜ . r+λ−µ ˜ + λ˜ µ

ω1 ≡ Eµ˜ and ω2 ≡ Eµ˜

This concludes the verification of the conjectured solution in (76). The cum-dividend function in (76) corresponds to the following ex-dividend value function at date t:42 V ex (Zt , µt , Θt+1 ) = RCt + Zt · ν (µt ) . Using (78), the Adjusted Q can then be expressed as: Adjusted Q =

1 V ex (Zt , µt , Θt+1 ) − RCt = ν (µt ) M − α ψ (µt )−1 . ∗ Kt+1

Therefore, Adjusted Q is a function of µt alone. We can further use equation (78) to calculate the firm’s expected net investment rate in period t + 1: 42

Note that V ex (Zt , µt , Θt+1 ) is now a function of µt , not µt+1 , since µt+1 is realized at date t + 1 and it is unknown at date t.

45

  n  (1 + µt+1 ) ψ (µt+1 ) Et it+1 = Et −1 ψ (µt ) Eµ˜ [(1 + µ ˜) ψ (˜ µ)] = (1 − λ) (1 + µt ) + λ −1 ψ (µt ) Eµ˜ [(1 + µ ˜) ψ (˜ µ)] = (1 − λ) µt − λ + λ . ψ (µt )   It follows that, like Adjusted Q, Et int+1 depends only µt . However, the firm’s latest realized net investment rate is given by int =

(1 + µt ) ψ (µt ) − 1, ψ (µt−1 )

and it depends on both µt and µt−1 . Adjusted Q thus provides a more precise signal about int+1 than int .

References Abel, A. and J. Eberly. “How Q and Cash Flow Affect Investment Without Frictions: An Analytic Explanation,” Review of Economic Studies 78 (2011): 1179-1200. Arrow, K. “Optimal Capital Policy, Cost of Capital and Myopic Decision Rules,” Annals of the Institute of Statistical Mathematics 1-2 (1964): 21-30. Benhabib, J., and A. Rustichini. “Vintage Capital, Investment, and Growth,” Journal of Economic Theory 55 (1991): 323-339. Bernanke, B. “Irreversibility, Uncertainty, and Cyclical Investment,” Quarterly Journal of Economics 98,1 (1983): 85-106. Dixit A. and R. Pindyck. Investment Under Uncertainty. Princeton University Press, Princeton NJ. (1994). Eisfeldt, A., and A. Rampini. “New or Used? Investment with Credit Constraints,” Journal

46

of Monetary Economics 54,8 (2007): 2656-2681. Erickson, T., and T. Whited. “Measurement Error and the Relationship between Investment and Q,” Journal of Political Economy 108 (2000): 1027-1057. Fazzari, S., G. Hubbard, and B. Petersen “Finance Constraints and Corporate Investment,” Brookings Papers on Economic Activity 1 (1988): 141-195. Feldstein, M., and M. Rothschild. “Towards an Economic Theory of Replacement Investment,” Econometrica 42,3 (1974): 393-424. Fisher, F., and J. McGowan. “On the Misuse of Accounting Rates of Return to Infer Monopoly Profits,” American Economic Review 96 (1983): 82-97. Gilchrist, S., and C. Himmelberg. “Evidence on the Role of Cash Flow for Investment,” Journal of Monetary Economics 36 (1995): 541-572. Gomes, J. F. “Financing Investment,” American Economic Review 91, 5 (2001): 1263-1285. Hayashi, F. “Tobin’s Marginal q and Average q: A Neoclassical Interpretation,” Econometrica 50,1 (1982): 213-224. Jorgenson, D. “Capital Theory and Investment Behavior,” American Economic Review Papers and Proceedings 53 (1963): 247-259. Jovanovic, B., and Y. Yatsenko. “Investment in Vintage Capital,” Journal of Economic Theory 147,2 (2012): 551-569. Laffont, J.J. and J. Tirole (2000). Competition in Telecommunications, MIT Press, Cambridge, MA. Lewellen, W. G., and S. G. Badrinath. “On the Measurement of Tobin’s Q,” Journal of Financial Economics 44 (1997): 77-122.

47

Lindenberg, E. B., and S. A. Ross. “Tobin’s Q Ratio and Industrial Organization,” Journal of Business 54 (1981): 1-32. McNichols, M., M. Rajan, and S. Reichelstein “Conservatism Correction for the Marketto-Book Ratio and Tobin’s Q,” Review of Accounting Studies 19,4 (2014): 1393-1435. Nezlobin, A. “Accrual Accounting, Informational Sufficiency, and Equity Valuation,” Journal of Accounting Research 50 (2012): 233-273. Nezlobin, A., Rajan, M. and S. Reichelstein “Structural Properties of the Price-to-Earnings and Price-to-Book Ratios.” Review of Accounting Studies 21,2 (2016): 438-472. Rajan, M., and S. Reichelstein. “Depreciation Rules and the Relation between Marginal and Historical Cost,” Journal of Accounting Research 47 (2009): 823-867. Ramey, V., and M. Shapiro. “Displaced Capital: A Study of Aerospace Plant Closings,” Journal of Political Economy 109,5 (2001): 958-992. Rogerson, W. “Intertemporal Cost Allocation and Investment Decisions,” Journal of Political Economy 105 (2008): 770-795. Rogerson, W. “On the Relationship Between Historic Cost, Forward Looking Cost, and Long Run Marginal Cost,” Review of Network Economics 10, 2 (2011): 1-31. Salinger, M. “Tobin’s q, Unionization, and the Concentration-Profit Relationship,” The RAND Journal of Economics 15 (1984): 159-170.

48