Dynamic Inefficiency in Decentralized Capital Markets Andr´e Kurmann∗

Stanislav Rabinovich†

Drexel University

University of North Carolina Chapel Hill October 19, 2017

Abstract We study the efficiency implications of bargaining in frictional capital markets in which firms match bilaterally with dealers in order to buy or sell capital. We show how two of the distinguishing characteristics of capital – ownership and the intertemporal nature of investment – give rise to a dynamic inefficiency. Firms that anticipate buying capital in the future overinvest because this increases their outside option of no trade in negotiations with dealers in the future, thereby lowering the bargained purchase price. Vice versa, firms that anticipate selling capital in the future strategically underinvest because this increases the bargained sale price. If the only motive for trade is capital depreciation, there is overinvestment. With stochastic productivity, high-productivity firms underinvest and low-productivity firms overinvest. In equilibrium, the inefficiency interacts with the externality from dealer entry and implies that no bargaining power achieves the constrained-efficient allocation. We propose a regressive tax on capital that can restore efficiency. Finally, we calibrate the model to data on physical capital markets and show that depending on bargaining power, the welfare loss from the inefficiency can be large. Keywords: Search, capital markets, over-the-counter markets, bargaining. JEL codes: C78, D83, E22, G1.

∗

School of Economics, LeBow College of Business, Drexel University, 3220 Market Street, Philadelphia, PA 19103. Email: [email protected]. † Corresponding author. Department of Economics, Gardner Hall CB 3305, University of North Carolina - Chapel Hill, Chapel Hill, NC 27599. Email: [email protected]

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1

Introduction

Many forms of capital trade in frictional decentralized markets. This is true both for physical capital, such as structures and equipment, and for financial assets, which frequently trade in so-called overthe-counter (OTC) markets.1 A burgeoning literature, reviewed below, has emerged that uses search and bargaining models to characterize the frictions involved in these markets and to rationalize important phenomena such as trade volume, bid-ask spreads, or trading delays. This departure from the Walrasian paradigm raises a classic question: Do these markets achieve efficiency, and if not, is there a tax policy that can restore efficiency? In this paper, we study this question in a model of frictional capital markets with bargaining. Trade in capital takes place through dealers who pay an entry cost and then match bilaterally with production firms in order to buy or sell capital. Our framework nests both the standard model of investment in physical capital, augmented with trading frictions, and the model of financial asset trade in OTC markets considered by Lagos and Rocheteau (2009). We show how two of the distinguishing characteristics of capital – ownership and the intertemporal nature of investment – generate a dynamic inefficiency. The distortion arises from the fact that firms’ current investment decision affects future bargaining outcomes. In particular, firms that anticipate buying capital in the future strategically overinvest because this increases their outside option of no trade in negotiations with dealers in the future, thereby lowering the bargained purchase price. Vice versa, firms that anticipate selling capital in the future strategically underinvest because this increases the bargained sale price. As a result of this strategic investment motive, the presence of capital depreciation leads, all else equal, to overinvestment. Stochastic productivity, on the other hand, can lead to either overinvestment or underinvestment. For a mean-reverting productivity process, high-productivity firms underinvest since they expect to be sellers of capital in the future, and low-productivity firms overinvest since they expect to be buyers of capital in the future. As a consequence, the dispersion of capital across firms and trade volume are too low relative to the social optimum, and investment is insufficiently responsive to shocks. 1 See Dell’Ariccia and Garibaldi (2005), Duffie et al. (2005, 2007); Green et al. (2010); or Afonso and Lagos (2012) for examples of decentralized financial asset markets that are subject to trading frictions and bargaining. See Rauch (1999) and Nunn (2007) for a list of real asset markets without organized exchange nor reference prices in trade publications. Also see Pulvino (1998) and Gavazza (2011) for the quantitative relevance of trading frictions and bargaining in commercial aircraft markets – presumably one of the most homogenous and frictionless real asset markets

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In equilibrium, the distortion from the firm’s strategic overinvestment motive interacts with the externality from dealer entry and implies that no bargaining power achieves the constrained-efficient allocation. This parallels the result of Acemoglu and Shimer (1999) who show that in a frictional labor market with bargaining, the standard Hosios (1990) condition no longer achieves efficiency when there are distortions on more than one margin. We then examine the policy implications of this inefficiency and show that a tax on capital can restore the efficient allocation. The tax is regressive, with a positive marginal tax rate for low levels of capital and a negative marginal tax rate for high levels of capital. For the case of zero depreciation – which is the case relevant for the literature on financial asset trade in OTC markets exemplified by Lagos and Rocheteau (2009) – the optimal corrective tax is equivalent to a wealth tax. Finally, we calibrate the model to data on physical capital markets to quantify the welfare loss implied by the inefficiency. Under our preferred calibration of the dealer’s bargaining power, the welfare loss turns out to be small. For either small or large bargaining powers, however, welfare losses can be substantial. Intuitively, as dealer bargaining power tends to zero, the strategic investment motive disappears but dealer entry vanishes and the market breaks down. Vice versa, as dealer bargaining power tends to one, both the strategic investment motive and the externality from dealer entry combine to result in a severely distorted capital market. This highlights the importance of modeling the interaction between the investment and entry margins. Related Literature. The paper contributes to the now extensive literature on search theoretical models of financial asset trade in OTC markets, pioneered by Duffie et al. (2005) and extended by Lagos and Rocheteau (2009) to allow for unrestricted asset holdings.2,3 The focus of this literature has so far been primarily on the positive implications of trading frictions. By contrast, our focus is on analyzing the normative implications and potential policy interventions. In particular, we establish that a regressive tax on capital – or equivalently a wealth tax for the case of zero depreciation – can restore the efficient allocation.4 2 Other important contributions to this literature include Duffie et al. (2007), Weill (2007), Vayanos and Weill (2008), Afonso and Lagos (2015), Lagos et al. (2011), Lester et al. (2015), and Hugonnier et al. (2015) among many others. Also see Williamson and Wright (2010) for a review of search-theoretic models of trade in OTC markets. 3 The search-theoretic literature on trading frictions in OTC markets typically considers asset trade between riskaverse investors with stochastic valuation for the asset and risk-neutral dealers. In terms of mechanics, this is equivalent to our setup with firms as long as the production technology is concave in capital. Also see Nosal and Rocheteau (2011) for such a reinterpretation of the Lagos-Rocheteau model. 4 It is important to contrast the inefficiency highlighted here to the misallocation of resources that occurs in firstgeneration models of OTC markets, such as Duffie et al. (2005), in which asset holdings are indivisible. In those models, search frictions reduce welfare by preventing immediate reallocation of capital from low-valuation to high-valuation

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The inefficiency result of our paper shares the common insight with the literature on overhiring in frictional labor markets initiated by Stole and Zwiebel (1996) and Smith (1999) that firms have a strategic incentive to distort input factor choices so as to improve their bargaining position.5 To our knowledge, we are the first to point out the relationship between the overhiring result in the Stole-Zwiebel-Smith environment and the inefficiency present in Lagos-Rocheteau type models of OTC markets. The source of the inefficiency is quite different, however. In the Stole-Zwiebel-Smith environment, firms decide on employment and then bargain pairwise with each worker over the wage. This leads to a within-period inefficiency: firms have an incentive to overhire so as to increase the outside option in negotiations with workers within the period, thereby lowering the wage. In our environment, by contrast, the firm bargains with only one dealer per period in a way that is bilaterally efficient. If the firm invested only once or if the environment was static, there would be no distortion, whereas this is not the case in the Stole-Zwiebel-Smith environment.6 Instead, the distortion in our environment arises because firms purchase capital, making the capital stock a state variable for the firm’s outside option in negotiations with dealers in the future. The inefficiency in our environment therefore stems from the intertemporal nature of investment, which is why we call it a dynamic inefficiency. The results of our paper – including the non-monotonic dependence of inefficiency on depreciation, the overinvestment of low-productivity firms vs. the underinvestment of high-productivity firms, and the rationale for a regressive tax on capital – hinge on the dynamic nature of this distortion. Lastly, by setting the analysis in an otherwise standard intertemporal investment model with capital depreciation, our insights can be applied to a modern dynamic macroeconomic context. As such, the paper relates to the emerging literature on trading frictions in physical capital markets, exemplified by Kurmann and Petrosky-Nadeau (2007), Gavazza (2011), Kurmann (2014), Shi and Cao (2014), or Ottonello (2017). This literature is considerably smaller than the above-mentioned body of research on OTC financial markets, and has focused, with the exception of Kurmann (2014), investors. However, the equilibrium in such a setting is usually constrained efficient: there is no intensive margin to be distorted, and conditional on matching, any potential mutually beneficial trades take place. In contrast, with an operative intensive margin as in Lagos and Rocheteau (2007), Lagos and Rocheteau (2009), or Lagos et al. (2011), the equilibrium is constrained inefficient, despite the fact that each trade between a firm and a dealer is bilaterally efficient. As a result, a government intervention can improve welfare by taxing or subsidizing capital, even though the government cannot eliminate the search friction. 5 Also see Cahuc and Wasmer (2001) and Cahuc et al. (2008), or Elsby and Michaels (2013). 6 Indeed, the overhiring result in the Stole-Zwiebel-Smith environment would not arise if workers bargained as a single coalition.

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on positive implications. Our paper contributes to this literature by showing that in frictional capital markets with bargaining, the assumption of firms purchasing and owning capital has substantially different normative implications than the assumption of firms renting capital on a period-by-period basis, which is the case analyzed by Kurmann (2014).7 Note, in particular, that firms in our model buy capital, rather than rent it as they do in Kurmann (2014). It is this feature that makes capital holdings appear as a state variable in the firm’s problem and thereby affect its future bargaining position, leading to a strategic investment motive. As explained above, our results depend crucially on the ownership of capital and the dynamic nature of the problem, and hence do not appear in Kurmann (2014) just as they do not appear in the Stole-Zwiebel-Smith intrafirm bargaining environment. Moreover, our paper is among the first to attempt a meaningful calibration of a model with bargaining in frictional capital markets, and to quantify the welfare implications of the inefficiency arising from this friction. The paper proceeds as follows. In Section 2, we show the dynamic inefficiency result in a deterministic version of the model, in which depreciation is the only motive for trade. Section 3 extends the model to incorporate stochastic productivity and studies how the magnitude and direction of the inefficiency depends on both depreciation and the productivity process. Section 4 considers the optimal corrective tax that restores the efficient allocation. Section 5 provides a quantitative exercise in which we calibrate the model to data on physical capital markets and evaluate the magnitude of the inefficiency. Section 6 concludes. All proofs are in Appendix A.

2

Deterministic model

This section analyzes the dynamic inefficiency result in a deterministic model of investment with a trading friction. The environment is deliberately simple, and the trading friction represents a minimal departure from the standard intertemporal competitive investment problem. Robustness to alternative modeling assumptions is discussed at the end of the section. 7

The inefficiency in Kurmann (2014) is, from a mechanical point of view, the same as in Stole and Zwiebel (1996) and Smith (1999). Firms bargain pairwise with all suppliers. The resulting overinvestment incentive stems from the inability to contract efficiently within the period (e.g. suppliers cannot form a coalition and bargain over both rental rate and capital simultaneously).

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2.1

Environment

Time is discrete and discounted at rate r = 1/β − 1. The economy is populated by a continuum of two types of agents: firms and dealers. The measure of firms is normalized to 1, while the measure of dealers active in the market is θ, which is determined endogenously by free entry. In the context of a physical capital market, dealers should be interpreted as suppliers of capital, but we refer to them as dealers throughout the paper, because our analysis also encompasses OTC financial markets. There is a single consumption good, and both firms and dealers derive linear utility from consumption. Dealers have a linear technology for converting consumption goods into capital and vice versa.8 Without loss of generality, we assume that the marginal cost of capital in units of consumption is one.9 Firms produce the consumption good with technology f (k), where f is continuous, twice differentiable, strictly increasing and strictly concave in capital k with f (0) = 0, limk→0 f 0 (k) = ∞ and limk→∞ f 0 (k) = 0. Capital is perfectly divisible and depreciates at rate δ ∈ [0, 1]. Every period, dealers active in the market need to pay operating cost c > 0. Matching of firms to dealers is random.10 Meetings are governed by a constant returns to scale matching function, with implied per period probability λ (θ) ∈ [0, 1] of a firm matching with a dealer and per period probability λ (θ) /θ ∈ [0, 1] of a dealer matching with a firm. We assume that λ (θ) is twice continuously differentiable with λ0 (θ) > 0, λ00 (θ) < 0, limθ→0 λ0 (θ) = 1, limθ→∞ λ0 (θ) = 0. Denote by nt (s) the beginning-of-period t measure of firms that last met a dealer s + 1 periods ago. The law of motion for nt (s) is described by nt+1 (0) = λ (θt ) , nt+1 (s + 1) = (1 − λ (θt )) nt (s) for s ≥ 0

(1) (2)

In case of a match, the firm and the dealer Nash bargain over the price of capital ρt for the quantity of new capital xt that the firm demands.11 If negotiation is successful, the dealer delivers 8

We show in Appendix B that the inefficiency result carries over naturally to the case where dealers have a convex cost of producing capital. 9 In the neoclassical growth model, the marginal cost of capital goods is the marginal utility of consumption, which equals 1 in our case. Nothing would change in our results if we introduced concave consumer preferences. Alternatively, in Lagos and Rocheteau (2009), the tradable asset is available in fixed supply and there is a frictionless competitive inter-dealer market that determines the marginal cost. In both cases, dealers take the marginal cost as exogenous and so, normalizing it to one does not matter for our results. 10 In Appendix C, we discuss how the environment and results would change when search is directed. See also Lester et al. (2015) for a detailed analysis. 11 An alternative but equivalent formulation is one in which the firm and the dealer Nash bargain simultaneously

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xt units of capital in exchange for ρt xt units of consumption, and the firm continues into the next period with capital stock

kt+1 = (1 − δ) kt + xt .

(3)

In case there is no match or if negotiations are unsuccessful, the firm continues into the next period with capital stock (1 − δ) kt .

2.2

Efficient investment and matching

Consider first the social planning problem. For the purpose of explaining the dynamic inefficiency result in Section 2.3, it is useful to start by considering the planner’s optimal investment level for an individual firm assuming a fixed match probability λ (θ) = λ. Under this assumption, the planner’s value for a firm that enters the period with capital stock k can be described recursively as Υ (k) = f (k) + λ max [− (k 0 − (1 − δ) k) + βΥ (k 0 )] + (1 − λ) βΥ ((1 − δ) k) . 0 k

(4)

A firm with current capital level k produces f (k). It then matches to a dealer with probability λ. If a match occurs, a new k 0 is chosen at cost of k 0 − (1 − δ) k. With probability 1 − λ, no match occurs, and the firm continues with (1 − δ) k units of capital. We guess – and verify below – that the planner’s optimal choice of k 0 is independent of current k. Then, by iterating forward, (4) can be expressed as

Υ (k) = Υ (0) +

∞ X


 (β (1 − λ))t f (1 − δ)t k + λ (1 − δ)t+1 k ,

(5)

t=0

where Υ (0) = λ max (−k 0 + βΥ (k 0 )) + (1 − λ) βΥ (0) 0 k

(6)

is the planner’s value of a firm with zero capital. Given (5) and (6), the planner’s problem can be characterized as follows. over both price and quantity. In particular, the bargaining outcome between the firm and the dealer in the model is bilaterally Pareto efficient, and thus our model does not give rise to the standard holdup problem. We prefer the present formulation because it affords a direct comparison to the standard intertemporal investment problem.

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Lemma 1. Υ(k) is continuous, twice differentiable, strictly increasing and strictly concave. The solution k 0 to the maximization problem in (4) is given by

1=β

∞ X


 (β (1 − λ) (1 − δ))t f 0 (1 − δ)t k P + λ (1 − δ)

(7)

t=0

Lemma 1 shows that k 0 is independent of current k, which verifies the above guess.12 For λ = 1, the condition reduces to the standard intertemporal investment problem whose solution is described by the well-known Euler equation 1 = β [f 0 (k 0 ) + (1 − δ)]. The presence of additional terms in (9) reflect the planner’s concern that the firm does not match with a dealer in the future, which occurs with probability (1 − λ). It can be shown that this matching friction leads, all else equal, to higher investment today. Lemma 2. Let k 0 be implicitly defined as a function of λ through equation (7). Then, ∂k 0 /∂λ ≤ 0, with strict inequality as long as δ ∈ (0, 1). Lemma 2 will be useful to understand the dynamic inefficiency result below. Intuitively, the trading friction gives rise to a “precautionary investment motive.” When λ is small, there is a high probability that the firm will be unable to purchase capital in the future, which means that there will be periods when the firm’s marginal product of capital is higher than the marginal cost. Due to the concavity of the production function, the planner has the firm insure against this possibility by accumulating a higher capital stock today, thereby reducing the firm’s trading needs in the future.13 The higher λ, the smaller this precautionary investment motive and therefore the smaller the efficient level of capital. Given this partial equilibrium result, let us now analyze the full planner’s problem, which consists of choosing sequences for kt+1 , θt , nt+1 (s) to maximize ∞ X t=0

( βt

−cθt +

∞ X

) 
 nt (s) f ((1 − δ)s kt−s ) − λ (θt ) kt+1 − (1 − δ)s+1 kt−s

(8)

s=0

subject to the law of motion (1)-(2) on the size distribution of firms.14 We will focus our analysis The fact that k 0 is independent of k is a consequence of the linearity of the dealer’s cost of production. In Appendix B, we show how the analysis can be generalized to convex production costs on the part of the dealer, or, equivalently, convex investment adjustment cost on the part of the firm. 13 This precautionary investment motive is called “liquidity hedging” by Lagos and Rocheteau (2009). 14 Since aggregate production is linear in the output of individual firms, the efficient investment level of an individual 12

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throughout on steady state solutions. Given the independence of optimal capital from past capital  ∞ stocks, the support of the stationary distribution of capital is (1 − δ)s k P s=0 where k P is the socially optimal capital decision in steady state. The solution to the planner’s problem can therefore be described as follows. Lemma 3. The necessary conditions for k P , θP to maximize (8) are

1=β

∞ X

β t 1 − λ θP



t



 (1 − δ)t f 0 (1 − δ)t k P + (1 − δ) λ θP

(9)

t=0

and

∞

X
 c s+1 P
 s+1 P

P P P − k − (1 − δ) k − Υ (1 − δ) k = n (s) β Υ k λ0 (θP ) s=0 where nP (s) = λ θP




1 − λ θP



s

(10)


and Υ (k) is given by (5) evaluated at λ = λ θP .

Equation (9) describes the efficient capital stock in the event of trade as described above. Equation (10) describes efficient matching. Adding a dealer has marginal cost c and increases the match
probability of firms by λ0 θP . The expected marginal benefit of an additional match is the weighted sum of additional surplus across firms with current capital equal to (1 − δ)s k P . If matched with a

dealer, such firms generate surplus of βΥ k P − k P − (1 − δ)s+1 k P . If unmatched, their surplus
is instead βΥ (1 − δ)s+1 k P .

2.3

Investment and matching in the decentralized economy

Consider now the decentralized equilibrium under Nash bargaining. Every period, dealers decide whether to pay cost c to operate, which determines the measure of dealers θ and in turn the match probability λ (θ). The firm takes λ (θ) as given when deciding on its level of investment. In steady state when θ is constant, the value of a firm entering the period with capital k can be described as v (k) = f (k) + λ(θ) max [−ρ (k, k 0 ) (k 0 − (1 − δ) k) + βv (k 0 )] + (1 − λ(θ)) βv ((1 − δ) k) , 0 k

(11)

where the price of capital ρ (k, k 0 ) solves the generalized Nash bargaining problem ρ (k, k 0 ) = arg max Sdφ Sf1−φ , ρ

firm does not depend on the capital stock distribution of other firms.

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(12)

with Sd and Sf representing the dealer’s and the firm’s surplus, respectively, from trading x = k 0 − (1 − δ) k units of new capital at price ρ. The dealer’s surplus from trade at price ρ is Sd = (ρ − 1) (k 0 − (1 − δ) k) .

(13)

Sf = −ρ (k 0 − (1 − δ) k) + βv (k 0 ) − βv ((1 − δ) k) .

(14)

The firm’s surplus from trade is

Substituting (13) and (14) into (12) and solving the bargaining problem yields  βv (k 0 ) − βv ((1 − δ) k) ρ (k, k ) = 1 + φ −1 . k 0 − (1 − δ) k 0



(15)

The bargained price of capital equals the dealer’s marginal cost of producing capital, which is unity, plus the dealer’s share φ of the match surplus per unit of investment, which is the difference between the firm’s average gain from continuing with k 0 as opposed to (1 − δ)k and the supplier’s marginal cost. The expression confirms that, as stated in (11), the price of capital is a function of the firm’s capital stock k with which it enters the period and the capital stock k 0 it would like to have in the next period. Equation (15) can be used to substitute out ρ (k, k 0 ) in (11) to obtain v (k) = f (k) + λ(θ) (1 − φ) max [− (k 0 − (1 − δ) k) + βv (k 0 )] 0 k

+ (1 − λ(θ) (1 − φ)) βv ((1 − δ) k)

(16)

Comparison of (16) with (4) shows that the firm’s problem is of the same form as the planner’s problem, with the only difference that the match probability λ (θ) is replaced by λ (θ) (1 − φ). Hence, the firm’s optimal k 0 is independent of current k, and the functional equation (16) can be expressed in iterated form as ∞   t h i X
ˆ ˆ (1 − δ)t+1 k , v (k) = v (0) + β 1 − λ(θ) f (1 − δ)t k + λ(θ) t=0

10

(17)

where ) ( ∞   i  h X ˆ t
λ(θ) ˆ (1 − δ)t+1 k 0 ˆ v (0) = max −k 0 + β β 1 − λ(θ) f (1 − δ)t k 0 + λ(θ) 0 k 1−β t=0

(18)

ˆ and λ(θ) = λ (θ) (1 − φ). By Lemma 1, the firm’s value v (k) has the same properties as the planner’s value Υ (k), and there exists a unique decentralized level of capital k D , determined by the first-order condition

∞    t h i X
ˆ ˆ (1 − δ) . 1=β β 1 − λ(θ) (1 − δ) f 0 (1 − δ)t k 0 + λ(θ)

(19)

t=0

Note that (19) is identical to (9) with λ(θ) replaced by λ (θ) (1 − φ) < λ (θ). Lemma 2 then immediately implies that for a given λ (θ), firms choose a higher capital level than the planner would. To understand this (partial equilibrium) overinvestment result, note that the firm’s problem is payoff equivalent to the one in an alternative environment in which the firm had all the bargaining power but the probability of meeting with a dealer was λ (θ) (1 − φ) < λ (θ).15 Both the social planner and the firm in the decentralized economy care about the possibility of not trading in a particular period, which gives rise to the precautionary investment motive discussed above. The firm, however, also cares about the value of its capital in the off-equilibrium case when negotiations with the dealer break down, because this outside option affects the terms of trade. This “strategic investment motive” effectively reduces the rate at which the firm discounts future payoffs in the event of no trade. To determine matching, consider the dealer’s decision to enter. In equilibrium, all the firms that contact a dealer leave the meeting with k D units of capital, where k D is the solution to (19). We can then write the rent that a dealer obtains from trading with a firm whose current capital is k as follows 

 $ (k) = φ βv k D − βv ((1 − δ) k) − k D − (1 − δ) k ,

(20)

where v is the value function of a firm with capital level k. The expected profit from operating depends on the distribution of k in the population of firms. The stationary distribution of firms across capital  ∞ levels has support (1 − δ)s k D s=0 . In steady state, the equilibrium measure of dealers, which we 15

This equivalence is the same as the one noted by Lagos and Rocheteau (2009) and Lagos et al. (2011).

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denote by θD , must satisfy the free entry condition
∞ λ θD X D s D
c= n (s) $ (1 − δ) k θD s=0 where nD (s) = λ θD




1 − λ θD



s

(21)

denotes the equilibrium probability of meeting a firm with

capital stock (1 − δ)s k D . Using the expressions for $ and v, we then immediately obtain the following characterization.
Lemma 4. The steady-state equilibrium allocation with free entry, k D , θD , satisfies

1=β

∞ X



t 

 β t 1 − λ θD (1 − φ) (1 − δ)t f 0 (1 − δ)t k D + λ θD (1 − φ) (1 − δ)

(22)

t=0

and

∞

X 
cθD s+1 D

s+1 D
 D D D = n (s) β v k − v (1 − δ) k − k − (1 − δ) k φλ (θD ) s=0

(23)


ˆ=λ ˆ θD . where v (k) for any k is given by (17) evaluated at λ Comparison of (9) and (10) with (22) and (23) then immediately yields the following inefficiency result. Proposition 5. k D 6= k P or θD 6= θP or both for any φ as long as δ ∈ (0, 1). In other words, no bargaining power can simultaneously achieve efficiency on both investment and entry margins. For example, suppose that φ is such that θD = θP . This φ must be greater than 0, implying by Lemma 2 that k D > k P . The inefficiency result has two components. The first is an externality common in the search literature: when deciding on entry, dealers do not take into account that the resulting change in market tightness affects the rest of the economy. The second is the strategic investment motive described earlier, which is the focus of this paper: firms make inefficient investment decisions to improve their bargaining position in negotiations with dealers in future matches. We can disentangle the two components by considering the following limiting cases. First, suppose that the matching rate is constant; i.e. λ (θ) = λ for all θ. This shuts down the entry externality. Corollary 6. If λ (θ) = λ ∀θ, k D ≥ k P , with strict inequality as long as φ > 0 and δ ∈ (0, 1). 12

Corollary (6) illustrates the strategic investment motive described above and evidenced by (19). Facing the same match probability with dealers as in the planner’s case, firms overinvest because a higher capital level increases the outside option of no trade, thereby lowering the negotiated price.16 This overinvestment motive falls away only if firms have all the bargaining power; i.e. φ = 0; or if either δ = 0 (no depreciation) or δ = 1 (full depreciation). For δ = 0, the firm is not concerned about bargaining with dealers in the future because the firm purchases capital only once. For δ = 1, the dynamic incentive to overinvest falls away because the firm’s entire capital stock is exhausted in each period and therefore does not affect the firm’s outside option in future trades. The two cases illustrate that the degree of overinvestment is non-monotonic in δ: a higher δ increases the motive for trade in the future, but reduces the strategic value of current investment for future negotiations. Second, return to the general case with λ (θ) endogenous and assume that δ = 1.
Corollary 7. For δ = 1, the decentralized equilibrium satisfies 1 = βf 0 k D and k D = k P indepenλ0 (θP )θP dent of φ. Furthermore, θD = θP if and only if φ = λ(θP ) . When δ = 1, the strategic investment motive is not present, and the standard Hosios (1990) condition obtains: if φ equals the elasticity of the matching function under the planner’s solution, then the dealer’s bargaining power exactly reflects the externality from adding another dealer.17

2.4

An example with a standard production function

In general, the strategic investment motive of the firm does not necessarily imply k D > k P since θD is typically different from its efficient counterpart θP . In particular, if θD > θP , the firm’s precautionary investment motive is smaller than the planner’s, which may overturn k D > k P . More generally, there is no guarantee that the equilibrium is unique. Intuitively, suppose for a moment that λ (θ) is high, which reduces the precautionary motive for firms to invest since they expect to meet a dealer again in the near future. By concavity of the firm’s value function, this increases the expected rent of dealers, thereby spurring entry. If this complementarity is sufficiently strong, multiple equilibria can arise. βv (kD )−βv((1−δ)k) To see this, consider price equation in (15) in steady state. By the strict concavity of v(k), > kD −(1−δ)k 0 D D βv (k ) = 1. Hence ρ(k , k) > 1 for all k as long as δ ∈ (0, 1) and φ ∈ (0, 1]. Further, again by the strict concavity of v(k), ρ(k D , k) is decreasing in k. 17 Of course, for δ = 0, the strategic investment motive would also fall away. However, in this case, there would be no motive for trade except in the first period, implying zero dealer entry in steady state. 16

13

For the following standard production function, it is possible to establish uniqueness of the equilibrium and sign the inefficiency on both margins. Example 8. Assume f (k) = k α with α ∈ (0, 1). Then the decentralized equilibrium is unique. Furthermore, suppose that
λ0 θP θP φ= λ (θP ) Then θD > θP and k D > k P . This special case is illustrated graphically in Figure 1. To construct this example, the dealer’s λ0 ( θ P ) θ P bargaining power is set to φ = λ(θP ) . The downward sloping lines show the firm’s optimal capital Figure 1: Equilibrium for a special production function 1000 900

Efficient Decentralized

800 700

k

600 500 400 300 200 100 0 0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

3

choices implied by the first-order condition for capital for the planner’s solution (equation (9), shown by the blue solid line) and the decentralized economy (equation (22), shown by the red dashed line) as a function of θ. The higher θ, the lower the optimal k because a higher match rate λ(θ) reduces the precautionary investment motive. The decentralized economy’s capital decision lies above the planner’s capital decision because of the additional strategic investment motive. Next, the upward sloping lines show optimal entry conditions for θ, for the planner’s solution (equation (10), shown by the blue solid line) and the decentralized economy (equation (23), shown 14

by the red dashed line) as a function of capital to be provided by the dealer. For the special case of f (k) = k α , this relationship between k and θ is always positive, implying uniqueness of the λ0 (θP )θP equilibrium. For φ = λ(θP ) , entry in the decentralized economy for a given k is larger than in the planner’s solution (the upward sloping red dashed line is to the right of the upward sloping blue solid line). Intuitively, when φ satisfies the Hosios condition, dealers internalize the entry externality, but still face a different surplus from the social planner, e.g. the perceived surplus

from a match with a particular firm is βv (k) − βv (1 − δ)s+1 k − k − (1 − δ)s+1 rather than

βΥ (k) − βΥ (1 − δ)s+1 k − k − (1 − δ)s+1 . The social planner cares about matching a firm to a dealer only to the extent that the marginal product of the additional capital exceeds the marginal cost; the firm in the decentralized economy, however, also benefits from the additional capital because it improves its future bargaining position. Thus, the dynamic overinvestment incentive can lead not only to excessive investment conditional on meeting a dealer, but also to excessive entry of dealers, as shown in the figure. In equilibrium, the distortions on the investment and the entry margin interact with each other. A higher θ, all else equal, mitigates the overinvestment incentive, by Lemma 2. In other words, if θD > θP , overinvestment (the discrepancy between k D and k P ) is lower than it would have been had we fixed θ at θP . The result in Example 8 that k D > k P occurs because, although θD > θP , it is still

the case that λ θD (1 − φ) < λ θP . Conversely, the strategic investment incentive distorts entry. In fact, as this particular numerical example illustrates, despite φ being at the Hosios condition, the discrepancy between equilibrium k D and k P is small, but entry is severely distorted (θD > θP ). This illustrates that the firm’s strategic investment motive can contaminate other margins and lead to potentially important inefficiencies, a point to which we return in the numerical exercise in Section 5.

3

Stochastic model

In the deterministic model, depreciation is the only motive for trade and the firm is always the buyer of capital. We now extend the model to allow for stochastic shocks to productivity, which generates a second motive for trade. The resulting framework nests both the model in Section 2 and a discretetime analogue of the OTC asset market considered in Lagos and Rocheteau (2009) and Nosal and Rocheteau (2011), which has stochastic asset valuation but no depreciation. 15

3.1

Environment

The environment is the same as in Section 2 except for the firm’s production function, which now takes the form zf (k) where z denotes the firm’s idiosyncratic productivity level, with z ∈ {z1 , z2 , ..., zI }. With probability γ, a firm retains its current productivity the next period. With probability 1 − γ, it draws a new productivity from an invariant distribution with P rob (zj ) = πj . Denote by Πij the conditional probability that z = zj in period t + 1 conditional on z = zi in period t, so that P Πij = (1 − γ) πj for i 6= j, and Πii = γ + (1 − γ) πi . Also, let z = j πj zj be the unconditional mean of z, and let ξit = γ t zi + (1 − γ t ) z be the expected value of productivity t periods from now conditional on current productivity being zi . Denote by nt (i, j, s) the beginning-of-period-t measure of firms who met a dealer last in period t − 1 − s, whose productivity at the time of meeting a dealer was zi , and whose current productivity is zj . The law of motion of nt (i, j, s) is described by nt+1 (i, j, 0) = λ (θt )

X

nt (m, i, s) Πij

m,s

nt+1 (i, j, s + 1) = (1 − λ (θt ))

X

nt (i, m, s) Πmj .

(24)

m

3.2

Efficient investment and matching

As before, it is useful to first consider the planner’s investment problem with λ exogenously fixed. Let Υi (k) denote the social value of a firm with current productivity zi and current capital k. This value can be written recursively as "

#

Υi (k) = zi f (k) + λ max − (k 0 − (1 − δ) k) + β 0

X

k

+ (1 − λ) β

X

Πij Υj (k 0 )

j

Πij Υj ((1 − δ) k) .

(25)

j

Guessing – and verifying later – as for the deterministic case that the optimal choice of k 0 is independent of current k, (25) can be iterated forward and expressed as

Υi (k) = Υi (0) +

∞ X


 β t (1 − λ)t ξit f (1 − δ)t k + λ (1 − δ)t+1 k ,

t=0

16

(26)

where

"

#

Υi (0) = λ max −k 0 + β 0 k

X

Πij Υj (k 0 ) + (1 − λ) β

X

j

Πij Υj (0) .

(27)

j

Also analogous to the deterministic case, the solution k 0 to the maximization problem for each i satisfies 1=β

∞ X


  β t (1 − λ)t (1 − δ)t ξi,t+1 f 0 (1 − δ)t k 0 + (1 − δ) λ ,

(28)

t=0

which establishes that k 0 is independent of k. Note that Lemma 2 does not carry over to the stochastic case in the sense that a lower λ does not necessarily raise the optimal k 0 for every productivity level zi . Instead, the precautionary investment motive is positive only if the firm’s current productivity is sufficiently low so that it is expected to be a buyer of capital in the future. If instead, the firm’s current productivity is high so that it is expected to be a seller of capital in the future, the precautionary investment motive is negative. We return to this intuition below when characterizing the dynamic inefficiency of the decentralized economy. Consider now the full planner’s problem in which dealer entry is endogenous. Denote by kj,t+1 the capital stock chosen by the planner in period t for a firm whose current productivity is zj . Then the planner’s problem consists of choosing sequences of kj,t+1 , θt , nt+1 (i, j, s) to maximize ∞ X

( βt

)

−cθt +

t=0

X


 nt (i, j, s) zj f ((1 − δ)s ki,t−s ) − λ (θt ) kj,t+1 (j) − (1 − δ)s+1 ki,t−s (i)

(29)

i,j,s

subject to (24). As before, we will focus on the steady-state solution. Lemma 9. The necessary conditions for kjP , θP to solve the planner’s problem are

1=β

∞ X

β 1 − λ θP






t 

 (1 − δ) ξj,t+1 f 0 (1 − δ)t kjP + λ θP (1 − δ) ∀j

(30)

t=0

and # " X X



c (31) = nP (i, j, s) β Πjm Υm kjP − Υm (1 − δ)s+1 kiP − kjP − (1 − δ)s+1 kiP λ0 (θP ) m i,j,s

17

where

s


1 − γ s+1 πi πj , i 6= j




s nP (i, i, s) = λ θP 1 − λ θP 1 − γ s+1 πi + γ s+1 πi

nP (i, j, s) = λ θP

1 − λ θP




(32)


and Υi (k) is given by (26) evaluated at λ = λ θP .

3.3

Investment and matching in the decentralized economy

The derivation for the decentralized equilibrium is also similar to the deterministic case. Let vi (k) be the value function of a firm with current productivity zi and current capital level k. The generalization of (16) to the stochastic environment is "

#

vi (k) = zi f (k) + λ (θ) (1 − φ) max − (k 0 − (1 − δ) k) + β 0 k

+ (1 − λ (θ) (1 − φ)) β

X

Πij vj (k 0 )

j

X

Πij vj ((1 − δ) k)

(33)

j

Analogous to the deterministic case, iterating on (33) establishes that the solution to the functional equation (33) has the form ∞   t h i X t
t+1 ˆ ˆ β 1−λ ξit f (1 − δ) k + λ (1 − δ) k , vi (k) = vi (0) +

(34)

t=0

ˆ = λ (θ) (1 − φ). with λ In steady state, all firms with productivity zi at the time of matching with a dealer leave the meeting with kiD units of capital, where kiD is the solution to the maximization problem in (33). The flow profit for a dealer from trading with a firm whose current productivity is zi and whose current capital is k is therefore " $i (k) = φ β

# X


D

Πij vj ki

− βvj ((1 − δ) k) − kiD − (1 − δ) k


j

18




,

(35)

and the free entry condition becomes
∞ λ θD X X D s D
c= n (i, j, s) $ (1 − δ) ki , j θD s=0 i,j

(36)

where nD (i, j, s) denotes the equilibrium probability of meeting a firm with capital level (1 − δ)s ki and current productivity zj , which is given by (32) evaluated at θ = θD . Equilibrium must now satisfy the optimality of kiD for each i, taking θD as given, and the free entry condition (36). This gives the following result.
Lemma 10. The steady-state equilibrium allocation with free entry, k D , θD , satisfies

1=β

∞ X




t 

 β 1 − λ θD (1 − φ) (1 − δ) ξj,t+1 f 0 (1 − δ)s kjD + λ θD (1 − φ) (1 − δ) ∀j

(37)

t=0

and " # X X



cθD Πjm vm kjP − vm (1 − δ)s+1 kiP − kjP − (1 − δ)s+1 kiP = nD (i, j, s) β (38) D φλ (θ ) m i,j,s
ˆ = λ θD (1 − φ). where v (k) for any k is given by (34) evaluated at λ Similar to Proposition 5, comparison of (22)-(23) with (30)-(31) makes clear that the decentralized equilibrium is inefficient for any φ. The direction of the inefficiency cannot in general be characterized analytically. We will explore the direction and magnitude of the distortion numerically in Section 5. Here, we focus on the case when λ (θ) = λ so that the standard entry externality resulting from violating the Hosios condition disappears and the only distortion comes from the strategic investment motive. This allows us to establish the following result. Lemma 11. For δ < 1, a sufficient condition for kiD > kiP is ξit ξi,t+1

<

f 0 (1 − δ)t k




f 0 (1 − δ)t−1 k

19


∀t ≥ 1, ∀k

Similarly, a sufficient condition for kiD < kiP is ξit ξi,t+1

>

f 0 (1 − δ)t k




f 0 (1 − δ)t−1 k


∀t ≥ 1, ∀k

Lemma 11 shows that whether a firm overinvests or underinvests depends on a simple tradeoff between expected future productivity and the depreciation rate of capital. The intuition for this inefficiency is similar to the deterministic case. In contrast to the planner, the firm cares not only about the possibility of not trading in the future, but also about the effect its capital stock has on the outside option when bargaining with future dealers. This is the firm’s strategic investment motive, which effectively reduces the rate at which the firm discounts future payoffs in the event of no trade. If the firm anticipates buying additional capital in the future, which occurs if its marginal product of capital in the event of no trade is expected to rise, the strategic investment motive is positive and thus kiD > kiP . Vice versa, if the firm anticipates selling capital in the future, which occurs if its marginal product of capital in the event of no trade is expected to decrease, the strategic investment motive is negative and thus kiD < kiP . Going one step further, for a mean-reverting productivity process such as the one we have assumed here, firms with low productivity expect to buy capital in the future, and firms with high productivity expect to sell it. In what follows, we use this logic to explicitly characterize the productivity regions for which either overinvestment or underinvestment occurs. We do so for two specific cases. For δ = 0 – which, notably, encompasses the literature on OTC markets following Lagos and Rocheteau (2009) – Proposition 13 fully characterizes the inefficiency. For general δ, Proposition 15 characterizes the inefficiency under the assumption that the production function takes the power form zf (k) = zk α . Consider first the case δ = 0. The following provides explicit formulas for the optimal capital levels. Lemma 12. Suppose δ = 0. kiP solves

f0



β γ − γβ (1 − λ) 1 = γ˜ P zi + 1 − γ˜ P z , where γ˜ P = . (k) 1−β 1 − γβ (1 − λ)

(39)

Similarly, kiD solves

1 β γ − γβ (1 − λ (1 − φ)) D D = γ ˜ z + 1 − γ ˜ z , where γ˜ D = . i 0 f (k) 1−β 1 − γβ (1 − λ (1 − φ)) 20

(40)

From comparing (39) and (40), the following inefficiency result obtains: Proposition 13. Suppose δ = 0. kiD > kiP for zi < z; kiD = kiP for zi = z; and kiD < kiP for zi > z. For δ = 0, productivity shocks are the only reason to buy or sell capital. Since, as explained above, the firm cares about the effect its capital stock has on the outside option when bargaining with future dealers, it effectively discounts future payoffs in the event of no trade at a lower rate than the planner. As a result, the firm puts too low of a weight on current productivity relative to future expected productivity, which is reflected by γ˜ D < γ˜ P . So, whenever z < z, the firm overinvests and whenever z > z, the firm underinvests. Intuitively, firms with below-average z expect their productivity to grow, and hence anticipate buying capital in the future. By increasing their current capital level, they improve their outside option in future capital purchases. Conversely, firms with above-average z expect their productivity to fall, and hence anticipate selling capital in the future. By reducing their capital holdings today, they improve their terms of trade in future capital sales. Lemma 14. Suppose δ ≥ 0, and the production function has the form zf (k) = zk α , α ∈ (0, 1). kiP solves k 1−α =



αβ ω P γ˜ P zi + 1 − γ˜ P z , 1 − β (1 − δ)

where ωP =

1 − β (1 − λ) (1 − δ) , 1 − β (1 − λ) (1 − δ)α

γ˜ P =

γ − γβ (1 − λ) (1 − δ)α 1 − γβ (1 − λ) (1 − δ)α

(41)

(42)

Similarly, kiD solves k 1−α =



αβ ω D γ˜ D zi + 1 − γ˜ D z , 1 − β (1 − δ)

where 1 − β (1 − λ (1 − φ)) (1 − δ) ω = , 1 − β (1 − λ (1 − φ)) (1 − δ)α D

γ − γβ (1 − λ (1 − φ)) (1 − δ)α γ˜ = 1 − γβ (1 − λ (1 − φ)) (1 − δ)α D

(43)

(44)

From this, we derive the following threshold result for overinvestment. Proposition 15. Suppose δ ≥ 0, and the production function has the form zf (k) = zk α , α ∈ (0, 1). If γ (1 − δ)α−1 > 1, then kiD > kiP for all z. If γ (1 − δ)α−1 ≤ 1, then kiD > kiP for zi < zˆ, kiD = kiP for zi = zˆ, and kiD < kiP for zi > zˆ, where the threshold zˆ is defined by

ω D 1 − γ˜ D − ω P 1 − γ˜ P zˆ = z ω P γ˜ P − ω D γ˜ D 21

(45)

and is strictly greater than z. For δ > 0, kiP and kiD differ for two reasons. First, ω D > ω P , which captures the fact that the presence of depreciation creates an incentive to overinvest. Second and as discussed above, γ˜ D < γ˜ P , which captures the fact that the firm discounts future payoffs in the event of no trade at a lower rate than the planner and therefore puts too low of a weight on current versus future expected productivity. This leads to overinvestment for low z and underinvestment for high z. If γ (1 − δ)α−1 > 1, the depreciation channel is so strong that the firm always overinvests.18 Otherwise, the firm overinvests for z below some threshold zˆ, and underinvests for z above the threshold. This threshold is strictly larger than z. The presence of depreciation makes it more likely that, in the future, the firm will be a buyer of capital, thus increasing the set of productivity realizations for which the firm overinvests. Propositions 13 and 15 state that in the event of trade, the capital stock of high-productivity firms is inefficiently small and the capital stock of low-productivity firms is inefficiently high. This implies that the distribution of capital across firms in the decentralized equilibrium is inefficiently concentrated. Intuitively, if either δ = 0 or the production function is of the power form, optimal k depends on a weighted average of z and z. The weight on z is smaller in the decentralized equilibrium than in the planner’s solution (i.e. γ˜ D < γ˜ P ) because firms discount future payoffs in the event of no trade at a lower rate than the planner. Hence, the firms put too low of a weight on current versus future expected productivity and the distribution of capital is less disperse than the one under the planner’s solution. Similarly, because firms take less extreme trading positions than the planner, it can be shown that trade volume in the decentralized equilibrium is inefficiently small. These results are consistent with Lagos and Rocheteau (2009) who show that an increase in dealer bargaining power results in lower dispersion of asset holdings and lower trade volume. Moreover, by the same reasoning, a firm’s investment decision is less sensitive to productivity shocks in equilibrium than it would be in the social optimum. We close this section by characterizing the optimal capital choice for extreme values of the productivity persistence parameter, γ. The parameter determines how likely productivity is to change over time, making it a key determinant of the gains from trade. Recall that in Section 2 we argued 18

Intuitively, for underinvestment to ever occur, there must be a sufficiently high probability that a high-productivity firm will eventually become low-productivity; i.e. γ must be sufficiently small. The higher δ, the lower γ must be for this to occur.

22

that the degree of inefficiency was non-monotonic in δ (which, in the deterministic model, is the only determinant of the gains from trade). The following result establishes, analogously, that the degree and direction of inefficiency is also non-monotonic in γ. Corollary 16. If either γ = 0 or γ = 1, then kiD ≥ kiP for all i, with equality if and only if δ = 0 or δ = 1. When γ = 0, current productivity is unimportant in determining the optimal capital choice for future periods. When γ = 1, a firm expects its productivity to remain at its current level forever, and hence does not anticipate selling or buying capital as a result of changing productivity. In either case, depreciation becomes the only motive for trade. In particular, all firms overinvest unless δ = 0 or δ = 1, as established in Section 2.19 Thus, underinvestment can only occur for intermediate values of persistence.

4

Policy implications

Since the decentralized equilibrium is constrained inefficient, a natural question is how taxes can improve welfare. In what follows, we establish that a regressive tax on capital restores efficiency. Intuitively, this result obtains because the firm does not put enough weight on current productivity in its optimal decision. We also show that the proposed tax is generally distinct from an investment tax or a wealth tax. The general argument for finding the optimal corrective tax-subsidy scheme is as follows. First, taking as given the efficient level of λ, we calculate the optimal tax-subsidy scheme on capital. Second, given this tax-subsidy scheme, it is easy to compute the lump-sum per-transaction tax or subsidy to dealers such that the efficient level of λ is consistent with free entry.

4.1

A regressive tax on capital

The capital tax we propose takes the particular form that it is only imposed in the event of trade; i.e. when a firm matches and increases its capital to k 0 , it is subject to a tax τ (k 0 ). The firm’s problem, 19

This result holds, in fact, for more general stochastic processes, since either i.i.d. or permanent.

23

ξit ξi,t+1

= 1 whenever productivity shocks are

taking λ as given, is therefore ! vi (k) = zi f (k) + λ max β 0

X

k

+ (1 − λ) β

X

Πij vj (k 0 ) − ρi (k 0 , k) (k 0 − (1 − δ) k) − τ (k 0 )

(46)

j

Πij vj ((1 − δ) k) ,

j

where the Nash-bargained price of capital is now " 0

ρi (k, k ) = 1 + φ

X j

# βvj (k 0 ) − βvj ((1 − δ) k) − τ (k 0 ) Πij −1 . k 0 − (1 − δ) k

(47)

As a result, the firm’s value function becomes vi (k) = zi f (k) + λ (1 − φ) (1 − δ) k ! + λ (1 − φ) max β 0

X

k

+ (1 − λ (1 − φ)) β

Πij vj (k 0 ) − k 0 − τ (k 0 )

(48)

j

X

Πij vj ((1 − δ) k)

j

Comparison of (48) with (33) shows that the firm’s problem with capital taxation differs from the problem without taxation only by the cost of capital for the firm-dealer pair, which is now k 0 + τ (k 0 ) instead of k 0 . In particular, v still has the form

vi (k) = v (0) +

∞   t h i X
ˆ ˆ (1 − δ)t+1 k β 1−λ ξi,t f (1 − δ)t k + λ

(49)

t=0

ˆ = λ (1 − φ). The firm’s optimal capital choice is therefore determined by the first-order with λ condition ∞    t h i X
ˆ (1 − δ) , ˆ (1 − δ) 1 + τ (k ) = β β 1−λ ξj,t+1 f 0 (1 − δ)t k 0 + λ 0

0

(50)

t=0

We are interested in finding a marginal tax rate function τ 0 (k 0 ) such that, for each zi , the k 0 solving (50) coincides with kiP given by (30). Such a function exists. Define k P (z), for any z, to be the

24

solution to 1=β

∞ X

(β (1 − λ) (1 − δ))t






 γ t+1 z + 1 − γ t+1 z f 0 (1 − δ)t k P (z) + λ (1 − δ)

(51)

t=0

Since (51) gives a one-to-one relationship between z and k P (z), and since ξj,t+1 is simply γ t+1 zj + (1 − γ t+1 ) z, the solution to (50) with k 0 replaced by k P (z) gives a marginal tax schedule as a function of z. In other words, for any z, define τˆ0 (z) to be the solution to ∞   i  t h X


0 t P t+1 t+1 ˆ ˆ z f (1 − δ) k (z) + λ (1 − δ) (52) 1 + τˆ (z) = β β 1 − λ (1 − δ) γ z+ 1−γ 0

t=0

To express this marginal tax schedule as a function of k 0 instead, define z P (k 0 ) = k P


−1

(k 0 ) as the

inverse of k P (z) = k 0 . Then, for each k 0 , the required marginal tax schedule is
τ 0 (k 0 ) = τˆ0 z P (k 0 )

(53)

Following a similar logic to Propositions 13 and 15, we can explicitly characterize the required marginal tax rate for any production function when δ = 0, and for the power production function when δ > 0. For both cases, we show that the optimal tax is regressive, in other words, τ 0 (k) is decreasing in k. Proposition 17. Suppose that δ = 0. The marginal tax implementing the efficient capital allocation is given by !
γ˜ D z P (k 0 ) + 1 − γ˜ D z −1 , γ˜ P z P (k 0 ) + (1 − γ˜ P ) z

1−β τ 0 (k 0 ) = 1 − β (1 − λ (1 − φ))

(54)

where γ˜ P and γ˜ D are given by (39) and (40), and z P (k 0 ) is the solution to k P (z) = k 0 . The tax is regressive: τ 00 (k 0 ) < 0. Furthermore, τ 0 (k 0 ) > 0 for k 0 < k P (z) and τ 0 (k 0 ) < 0 for k 0 > k P (z). The optimal tax is regressive, since firms’ capital choices are insufficiently responsive to productivity without the tax. Moreover, since firms with above-average productivity underinvest and firms with below-average productivity overinvest, it is optimal to subsidize investment by above-average productivity firms, and tax it for below-average productivity firms. Proposition 18.

Suppose that δ ≥ 0 and the production function has the form zf (k) = zk α . The 25

marginal tax implementing the efficient capital allocation is given by 1−β τ 0 (k 0 ) = 1 − β (1 − λ (1 − φ)) (1 − δ)

!

ω D γ˜ D z P (k 0 ) + 1 − γ˜ D z −1 , ω P (˜ γ P z P (k 0 ) + (1 − γ˜ P ) z)

(55)

where ω P , γ˜ P and ω D , γ˜ D are given by (42) and (44), and z P (k 0 ) is the solution to k P (z) = k 0 . The tax is regressive: τ 00 (k 0 ) < 0. If γ (1 − δ)α−1 > 1, then τ 0 (k 0 ) > 0 for all k 0 . If γ (1 − δ)α−1 ≤ 1, then τ 0 (k 0 ) > 0 for k 0 < k P (ˆ z ) and τ 0 (k 0 ) < 0 for k 0 > k P (ˆ z ), where zˆ is given by (45). The message of Proposition 18 is two-fold. First, the optimal tax is again regressive since firms’ capital choices are insufficiently responsive to productivity without the tax. Second, when the depreciation channel always dominates, the tax should always be positive. Otherwise, the tax should be positive at low capital levels and negative (i.e. a capital subsidy) at high capital levels, since low-productivity firms are overinvesting and high-productivity firms are underinvesting. A natural question to ask is how the proposed capital tax compares to alternative taxes such as an investment tax or a wealth tax. Consider first a tax on investment x = k 0 − (1 − δ)k. Such a tax cannot restore the planner’s allocation because what matters for the inefficiency is not the additional capital purchased but the total capital that the firm wants to have as an outside option in future bargaining. Consider next a wealth tax; i.e. a tax on the firm’s capital stock independent of whether trade occurs or not. In this case, the first-order condition for the firm’s optimal capital choice in the event of trade becomes 1=β

∞  X

ˆ β(1 − λ)(1 − δ)

t 

 ˆ ξj,t+1 f (1 − δ) k − τ (1 − δ) k + λ (1 − δ) . t

0

0




0

t

0




(56)

t=0

For δ > 0, wealth depends on whether trade occurs in the future. This makes the problem intractable because it is no longer possible to show that there is a unique τ 0 = τˆ0 (z) that can be mapped into
τ 0 (k 0 ) = τˆ0 z P (k 0 ) . Hence, it is not clear whether a unique marginal tax function exists and, if it exists, what its properties are.20 For δ = 0, by contrast, wealth is constant and the present value of marginal taxes in the event 20

For a tax on income, zf (k), the problem becomes intractable for very much the same reason as for the wealth tax when δ > 0. The firm’s first-order condition contains an infinite sequence of marginal tax rates, one for each future marginal product of capital in the event of no trade. As a
result, it is no longer possible to argue that there is a unique τ 0 = τ 0 (z) that can be mapped into a τ 0 (k 0 ) = τ 0 z P (k 0 ) .

26

of no trade can be expressed in closed-form. Hence, the same techniques as above can be applied to establish that a regressive tax establishes efficiency. This is important, because the no-depreciation case encompasses the entire literature on financial asset trade in OTC markets with unrestricted asset holdings, starting with Lagos and Rocheteau (2009). Our results thus imply that a regressive wealth tax restores the efficient allocation in this environment. The above derivation of the optimal tax on capital still leaves open the question, for endogenous λ, of what tax or subsidy to dealers delivers the efficient level of entry. Fortunately, this is straight
forward. Let τ (k 0 ) be a tax schedule such that τ 0 (k 0 ) satisfies (52) and (53) at λ = λ θP , and let
v be given by (49) with λ = λ θP . Then, θP is consistent with free entry if there is a per-meeting subsidy S to dealers (which may of course be positive or negative) that is given by " X X


cθP Πjm vm kjP − vm (1 − δ)s+1 kiP = S + φ n (i, j, s) β P λ (θ ) m i,j,s #

− kjP + τ kjP − (1 − δ)s+1 kiP

(57)

with n (i, j, s) given by (32).

4.2

A tax on dealer profits

Since the misallocation of capital arises due to dealers having non-negative bargaining power, this suggests that an alternative way to restore efficiency is by taxing dealer profits. Specifically, suppose that there is a 100% tax on dealer profits: in other words, a dealer with profits $ is subject to a tax T ($) = $. This effectively reduces the dealer bargaining power to zero: the Nash bargaining problem faced by the firm-dealer pair is now max β 0 ρ,k

X m

Πjm vj (k 0 ) − β

X

Πjm vj ((1 − δ) k) − ρ (k 0 − (1 − δ) k)

(58)

m

subject to (ρ − 1) (k 0 − (1 − δ) k) − T ((ρ − 1) (k 0 − (1 − δ) k)) = 0

(59)

Clearly, the pair would set ρ = 1; if ρ > 1, a decrease in ρ would make the firm strictly better off without changing the dealer’s after-tax profits. This yields an allocation of capital equivalent to the

27

one that would occur if φ = 0. Since dealer profits are now always zero, the efficient level of θ is then implemented by providing a lump-sum per-meeting subsidy S to dealers, where S=

5

cθP λ (θP )

(60)

A numerical example

To provide a sense of the importance of the distortions implied by the dynamic inefficiency, we calibrate the stochastic model of Section 3 to match data on physical capital markets. While the model could also be calibrated to fit characteristics of OTC markets for financial assets, physical capital markets offer a natural starting point since we have relatively good calibration targets for many of the parameters and aggregate consumption utility offers a well-defined welfare objective. There is also recent evidence that even relatively standardized forms of capital such as commercial aircraft are subject to substantial trading frictions (e.g. Gavazza, 2011). Similarly, as documented by Kurmann and Petrosky-Nadeau (2007) and Ottonello (2017), commercial non-residential real estate markets are characterized by important vacancy rates that vary inversely with the business cycle similar to the unemployment rate, suggesting that search and matching frictions apply not only for labor markets but also for the (re-)allocation of capital. We evaluate the model based on a quarterly frequency and assume the following functional forms: zf (k) = zk α for the firm’s production function; and λ(θ) = θ/(1 + θχ )1/χ for the matching function as proposed by Den Haan et al. (2000) for discrete-time environments.21 Following Hosios (1990) and many other studies, the discount rate is assumed to be approximately zero. This simplifies the welfare calculations because it allows us to directly compare steady state consumption utilities instead of computing the discounted value of the change in consumption utility along the path from the decentralized allocation to the planner’s solution. The elasticity of firm profits to capital is set to α = 0.5 consistent with the estimates reported in Cooper and Haltiwanger (2006) for the U.S. manufacturing sector, and the idiosyncratic productivity process z is specified to fit the large and persistent differences in productivity observed in firm-level data (e.g. Syverson, 2011).22 Specifically, 21

This matching function has the advantage that the match probabilities λ(θ) and λ(θ)/θ are bounded between 0 and 1 for any θ, which is important for the ensuing welfare cost analysis. 22 Following Cooper and Haltiwanger (2006), the calibration of the firm’s production function takes into account the firm’s endogenous labor decision and a non-zero markup of price over marginal cost.

28

we assume that z is distributed lognormally, with the arrival rate of new productivity shocks 1 − γ and the standard deviation σln(z) being selected to match the estimates reported by Foster et al. (2007) for the U.S. manufacturing sector.23 The remaining parameters of the model are calibrated for the decentralized solution to match the following targets: (i) an annual gross investment rate of 0.18 and a ratio of used capital investment to gross investment of 0.24 consistent with the estimates by Ramey and Shapiro (1998) and Eisfeldt and Rampini (2006) based on Compustat data; (ii) a quarterly match rate for dealers of 0.5, which lies in between the average allocation rate of new machinery and equipment in the U.S. calculated by Kurmann and Petrosky-Nadeau (2007) from the BEA’s Survey of Current Business and the average match rate of office and retail space for rent or sale implied by the estimates in Ottonello (2017) based on Spanish data; and (iii) and an average absolute price dispersion of 0.17 as reported for the used commercial aircraft market by Gavazza (2011). As opposed to the deterministic case, the power production function in the stochastic case does not necessarily imply a unique equilibrium. For the chosen calibration, however, the equilibrium turns out to be unique.24 The appendix provides details about computing the equilibrium and the calibration procedure to match the different targets. Table 1 summarizes the resulting calibration values as well as the implied match probability for firms and total operating costs of dealers as a fraction of aggregate output. Table 1: Calibration α

γ

σln(z)

δ

χ

c

φ

λ(θD )

cθD /Y D

0.50

0.90

0.22

0.03

0.95

0.10

0.52

0.46

0.005

The implied calibration values look reasonable overall. The quarterly depreciation rate of δ = 0.034 is a bit higher than the average value of 0.025 typically used based on the BEA depreciation 23

The lognormal distribution is approximated by a truncated lognormal over 100 evenly spaced grid points on the   interval [ ln(z h )/100 ln(z h ) with parameters parameters µ and σ for z < z h and probability mass at z h such that cumulate density for z h equals one. The mean of the lognormal distribution µ is normalized to zero and z h is set to 5. Results are robust to alternative values. 24 This is different from the numerical example reported in the working paper version of Lagos and Rocheteau (2009) and in Nosal and Rocheteau (2011). An important reason for uniqueness in our numerical example is that we approximate the idiosyncratic productivity process over a fine grid of states instead of two states only. The lognormality assumption is not crucial for the result. We would also obtain a unique equilibrium for a uniform distribution.

29

tables. This difference is due to our targeting of gross investment and used capital reallocation rates in the Compustat data. Depreciation in our case therefore implicitly includes not only physical wear and tear but also economic obsolescence.25 To quantify the distortions implied by the inefficiency, we compute steady state aggregate consumption of the decentralized solution C D = −cθD +

X



 
α nD (i, j, s) zj (1 − δ)s kiD − λ θD kjD − (1 − δ)s+1 kiD

i,j,s

for different values of φ, keeping the other parameters fixed, and compare it to the planner’s counterpart

C P = −cθP +

X


α

 nP (i, j, s) zj (1 − δ)s kiP − λ θP kjP − (1 − δ)s+1 kiP .

i,j,s

Figure 2 reports the results of this exercise. For the calibrated value of φ = 0.52, the distortion turns out to be small. However, for other calibrations that are less in line with our targets, the welfare losses would be larger. Next, if φ is varied towards either 0 or 1, the distortion becomes more important. Intuitively, for low dealer bargaining power, the strategic investment incentive of the firm is reduced but at the same time, the entry externality from dealers becomes more important, to the point that as φ → 0, the market breaks down. For large dealer bargaining power, the dynamic overinvestment incentive increases and, in combination with the dealer entry externality, results in large inefficiencies. For example, for φ = 0.95, consumption in the decentralized equilibrium would be 12 percent lower than what the planner could achieve.

25

The model not only fits the average absolute price dispersion of used commercial aircraft – i.e. one of the targets – but also does ar reasonable job matching the standard deviation of the absolute price dispersion of 0.19 reported by Gavazza (2011).

30

Figure 2: Distortions implied by the inefficiency 1

0.98

CD / C P

0.96

0.94

0.92

0.9

0.88 0

0.1

0.2

0.3

0.4

0.5

?

31

0.6

0.7

0.8

0.9

1

6

Conclusion

This paper studies how bargaining in capital markets with trading frictions distorts market participants’ incentives and leads to inefficient equilibrium outcomes. The key result coming out of our analysis is that if firms anticipate buying capital in the future, they have a strategic incentive to accumulate more than the efficient level of capital today. This increases the outside option in negotiations with future dealers, thereby lowering the price at which firms can purchase capital in the future. Conversely, if firms anticipate selling capital in the future, they have a strategic incentive to accumulate an inefficiently low level of capital today. This strategic investment incentive is present in many recent models of frictional capital markets. It provides a novel rationale for capital taxation and, more generally, suggests that search frictions may have important implications for regulation of OTC asset markets. Indeed, the quantitative exercise based on a calibration of the model with data from physical capital markets implies that in combination with endogenous dealer entry, the welfare loss arising from the firm’s strategic investment motive can be substantial. The analysis also highlights the importance of capital depreciation in determining the direction of inefficiency. In the absence of depreciation, high-productivity firms underinvest while lowproductivity firms overinvest. The presence of depreciation increases the set of productivity realizations for which firms overinvest, because it raises the chance that any firm will become a buyer of capital. A similar argument can be made for other channels that create an incentive to buy capital in the future, such as trend growth in productivity. This suggests that the dynamic investment distortion is greater in faster-growing economies, an extension that we leave for future research. Finally, the dynamic inefficiency that we describe for the case of frictional capital markets may also be present in other input markets. For instance, a similar dynamic inefficiency could arise in a frictional labor market with bargaining over long-term wage contracts.26 Frictional markets for intermediates provide another application. Analyzing these cases seem worthwhile avenues to pursue. 26

Analysis of this dynamic inefficiency might be more complicated because such a frictional labor market enviroment could at the same time give rise to the within-period inefficiency that leads to overhiring, as analyzed by Stole and Zwiebel (1996) and Smith (1999).

32

Acknowledgments We thank Ricardo Lagos, an Associate Editor, and two anonymous referees, as well as Jun Ishii, Miguel Leon-Ledesma, Benjamin Lester, Theodore Papageorgiou, Guillaume Rocheteau, and participants at numerous conferences and seminars for helpful comments and discussions.

33

References [1] Acemoglu, D. and R. Shimer (1999). ”Holdups and Efficiency with Search Frictions.” International Economic Review 40(4), 827-849. [2] Afonso, G. and R. Lagos (2012). ”An Empirical Investigation of Trade Dynamics in the Market for Federal Funds.” FRB of New York Staff Report. [3] Afonso, G. and R. Lagos (2015). ”Trade Dynamics in the Market for Federal Funds.” Econometrica, 83(1), 263-313. [4] Cahuc, P. and E. Wasmer (2001). ”Does Intrafirm Bargaining Matter in the Large Firm’s Matching Model?” Macroeconomic Dynamics 5, 742-747. [5] Cahuc, P., F. Marque and E. Wasmer (2008). ”A Theory of Wages and Labour Demand with Intra-Firm Bargaining and Matching Frictions.” International Economic Review 49, 943-972. [6] Cao, M. and S. Shi (2014). ”Endogenous Procyclical Liquidity, Capital Reallocation and Q.” Working paper. [7] Cooper, Russell W., and John C. Haltiwanger (2006). “On the nature of capital adjustment costs.” The Review of Economic Studies, 73(3), 611-633. [8] Den Haan, Wouter J., Garey Ramey, and Joel Watson (2000). “Job Destruction and Propagation of Shocks.” American Economic Review, 90(3), 482-498. [9] Dell’Ariccia, G. and P. Garibaldi (2005). ”Gross Credit Flows.” Review of Economic Studies 72, 665-685. [10] Duffie, D., N. Garleanu and L. H. Pedersen (2005). ”Over-the-counter Markets.” Econometrica 73, 1815-1847. [11] Duffie, D., N. Garleanu and L. H. Pedersen (2007). ”Valuation in Over-the-counter Markets.” Review of Financial Studies 20(5), 1866-1900. [12] Eisfeldt, A.L. and A. A. Rampini (2006). ”Capital Reallocation and Liquidity.” Journal of Monetary Economics 53(3), 369-99. [13] Foster, L., J. Haltiwanger, and C. Syverson (2008). “Reallocation, Firm Turnover, and Efficiency: Selection on Productivity or Profitability?” American Economic Review, 98(1), 394-425. [14] Gavazza, A. (2011). ”The Role of Trading Frictions in Real Asset Markets.” American Economic Review 101, 1106-1143. [15] Green, R. C., D. Li, and N. Schuerhoff (2010). ”Price Discovery in Illiquid Markets: Do Financial Asset Prices Rise Faster Than They Fall?” Journal of Finance 65(5), 1669-1702 [16] Kurmann, A. (2014). ”Holdups and Overinvestment in Capital Markets.” Journal of Economic Theory 151, 88-113.

34

[17] Lagos, R. and G. Rocheteau (2007). “Search in Asset Markets: Market Structure, Liquidity, and Welfare.” American Economic Review P&P 97(2), 198-202. [18] Lagos, R. and G. Rocheteau (2009). ”Liquidity in Asset Markets with Search Frictions.” Econometrica 77(2), 403-26. [19] Lagos, R., G. Rocheteau and P.-O. Weill (2011). “Crises and Liquidity in Over-the-Counter Markets.” Journal of Economic Theory 146(6), 2169-2205. [20] Lester, B., G. Rocheteau and P.-O. Weill (2015). “Competing for order flow in OTC markets.”Journal of Money, Credit and Banking 47 (S2), 77-126. [21] Hugonnier, J., B. Lester and P.-O. Weill (2015). ”Heterogeneity in Decentralized Asset Markets.” Federal Reserve Bank of Philadelphia, Working Paper 15-22. [22] Nosal, E. and G. Rocheteau (2011). Money, Payments, and Liquidity. Cambridge: MIT Press. [23] Nunn, N. (2007). ”Relationship-Specificity, Incomplete Contracts, and the Pattern of Trade.” Quarterly Journal of Economics, 569-600. [24] Ottonello, P. (2017). ”Capital Unemployment.” Working paper. [25] Pulvino, T. C. (1998). ”Do Asset Fire Sales Exist? An Empirical Investigation of Commercial Aircraft Transactions.” Journal of Finance 53(3), 939-92. [26] Ramey, Valerie, and Matthew D. Shapiro (1998). “Capital Churning.” Working paper. [27] Rauch, J. E. (1999). ”Networks versus Markets in International Trade.” Journal of International Economics 48, 7-35. [28] Smith, E. (1999). ”Search, Concave Production, and Optimal Firm Size.” Review of Economic Dynamics 2, 456-471. [29] Stole, L. A. and J. Zwiebel (1996). ”Intra-Firm Bargaining under Non-Binding Contracts.” Review of Economic Studies 63(3), 375-410. [30] Syverson, Chad (2011). “What Determines Productivity?” Journal of Economic Literature, 49(2), 326-365. [31] Vayanos, D. and P.-O. Weill (2008). “A Search-Based Theory of the On-the-Run Phenomenon.” Journal of Finance 63(3), 1361–98. [32] Weill, P.-O. (2007). ”Leaning against the wind.” Review of Economic Studies 74(4),1329–1354. [33] Williamson, S. and R. Wright (2010). ”New Monetarist Economics: Models.” Handbook of Monetary Economics, second edition, B. Friedman and M. Woodford, eds., Elsevier.

35

Appendix: Not for Publication A

Proofs

Proof of Lemma 1. First, observe that if Υ takes the form in (5), then it inherits the following properties of f : it is continuous, twice-differentiable, strictly increasing, and strictly concave. As a result, the optimal k 0 is indeed  given by the first-order condition (7) and, in particular, does not depend on k. Restrict k to lie in 0, k , where the upper bound k is large enough not to bind at the optimum; then 

f (1 − δ)t k is bounded for any t, and hence lim (β (1 − λ))t f (1 − δ)t k + λ (1 − δ)t+1 k = 0; t→∞

the sum in (5) is therefore well-defined. Second, the mapping defined in (4) maps functions of the form (5) into other functions of this form, and the set of such functions is closed. Since the mapping is a contraction, the unique fixed point must have the form (5), by Corollary 1 to the Contraction Mapping Theorem in Stokey et al. (1989). To find Υ (0), simply substitute k = 0 into the Bellman equation for Υ (k). and subtracting λ(1 − δ) from each side, we see Proof of Lemma 2. Multiplying (7) by 1−β(1−λ)(1−δ) β
1 P P that k is determined by Γ λ, k = β − (1 − δ), where " Γ λ, k


P

≡ (1 − β (1 − λ) (1 − δ))

∞ X

# (β (1 − λ) (1 − δ))t f 0 (1 − δ)t k


P

(61)

t=0

By the implicit function theorem,

with

∂Γ/∂λ dk P =− , dλ ∂Γ/∂k

(62)

"∞ # X
∂Γ t P
2 t 00 = (1 − β(1 − λ)(1 − δ)) β(1 − λ)(1 − δ) f (1 − δ) k ∂k t=0

(63)

and

∞


∂Γ X t (β (1 − δ))t (1 − λ)t−1 f 0 (1 − δ)t−1 k P − f 0 (1 − δ)t k P . = (64) ∂λ t=1


By the strict concavity of f , f 00 (1 − δ)t k < 0 and f 0 (1 − δ)t−1 k < f 0 (1 − δ)t k for any k as long as δ ∈ (0, 1). Hence, ∂Γ/∂k < 0 and ∂Γ/∂λ ≤ 0, which in turn implies dk/dλ ≤ 0, strict as long as δ ∈ (0, 1).


s Proof of Lemma 3. First, in steady state, the law of motion (1)-(2) becomes nP (s) = λ θP 1 − λ θP . Next, the first-order condition for kt+1 is

λ (θt ) = β

∞ X

β s nt+1+s (s) (1 − δ)s [f 0 ((1 − δ)s kt+1 ) + (1 − δ) λ (θt+1+s )]

(65)

s=0




s Imposing steady state and using nP (s) = λ θP 1 − λ θP immediately yields (9). Next, let t+1 t+1 β µt+1 (0) and β µt+1 (s + 1) be the Lagrange multipliers on (1) and (2). The first-order condition

36

for θt can then be written as ∞

X 
 c nt (s) − kt+1 − (1 − δ)s+1 kt−s + β (µt+1 (0) − µt+1 (s + 1)) = 0 λ (θt ) s=0

(66)

The first-order condition for nt (s) is
µt (s) = f ((1 − δ)s kt−s ) − λ (θt ) kt+1 − (1 − δ)s+1 kt−s + β (1 − λ (θt )) µt+1 (s + 1)

(67)

Iterating on (67) and imposing steady state yields µ (s) =

∞ X

i   h  s+i P s+i P P β (1 − λ (θ)) f (1 − δ) k − λ (θ) k − (1 − δ) k i

i

(68)

i=0

which immediately implies

µ (0) − µ (s + 1) = Υ k P − Υ (1 − δ)s+1 k P

(69)

Substituting this into (66) and imposing steady state yields (10). Proof of Corollary 6. For φ ∈ (0, 1) and δ ∈ (0, 1), the result k D > k P is a direct consequence of Lemma 2. For δ = 0, both the planner’s solution in (7) and the decentralized solution in (19) reduce to 1 = β (f 0 (k 0 ) + 1), which is independent of the match probability λ. For δ = 1, both the planner’s solution in (7) and the decentralized solution in (19) reduce to 1 = βf 0 (k 0 ), which is likewise independent of λ. Thus, in both cases, the equilibrium coincides with the planner’s solution, independently of φ. Proof of Corollary 7. Setting δ to 1 in the value function (16) and in equilibrium expressions (22) and (23) yields
1 = βf 0 k D (70) and


cθD D = βf k − kD φλ (θD )

(71)

Moreover, setting δ to 1 in the planner’s value function (5) and the planner’s solution (9) and (10) yields
1 = βf 0 k P (72) and


c P = βf k − kP λ0 (θP )

(73)

The first-order conditions for capital are equivalent, so k D = k P . The first-order conditions for θ imply that θD = θP if and only if
θP λ0 θP φ= (74) λ (θP ) Proof of Example 8. When f (k) = k α , the equations (9) and (10) for the planner’s solution simplify 37

to

βα 1 − β (1 − δ) k α−1 = α 1 − β (1 − δ) (1 − λ (θ)) 1 − β (1 − δ) (1 − λ (θ))

(75)

and c 1 − (1 − δ)α β = · kα α α 0 λ (θ) 1 − (1 − δ) (1 − λ (θ)) 1 − β (1 − δ) (1 − λ (θ)) δ 1 − β (1 − δ) − · k 1 − (1 − δ) (1 − λ (θ)) 1 − β (1 − δ) (1 − λ (θ))

(76)

Equations (75) and (76) can be further simplified. Notice that (75) can be written as βα 1 − β (1 − δ) kα = k α 1 − β (1 − δ) (1 − λ (θ)) 1 − β (1 − δ) (1 − λ (θ))

(77)

Solving for k gives  k=

βα 1 − β (1 − δ) (1 − λ (θ)) · 1 − β (1 − δ) 1 − β (1 − δ)α (1 − λ (θ))

1  1−α

(78)

and substituting (77) into (76) gives us 1 − (1 − δ)α αδ k= − α 1 − (1 − δ) (1 − λ (θ)) 1 − (1 − δ) (1 − λ (θ)) 

−1

cα 1 − β (1 − δ) (1 − λ (θ)) · 0 1 − β (1 − δ) λ (θ)

(79)

Thus, the system (75), (76) is equivalent to (78), (79). We will verify that this system has a unique solution. (78) clearly gives k as a decreasing function of θ. So, it is enough to check that (79) gives k as an increasing function of θ. The second and third terms are clearly increasing in θ. We need to check that the term in brackets is decreasing in θ:   αδ 1 − (1 − δ)α d − dθ 1 − (1 − δ)α (1 − λ (θ)) 1 − (1 − δ) (1 − λ (θ))   (1 − δ)α (1 − (1 − δ)α ) αδ (1 − δ) 0 = λ (θ) − (1 − (1 − δ) (1 − λ (θ)))2 (1 − (1 − δ)α (1 − λ (θ)))2   (1 − δ)α α 0 < λ (θ) (80) 2 (αδ − (1 − (1 − δ) )) (1 − (1 − δ)α (1 − λ (θ))) < 0, where the first inequality follows from the fact that 1 − δ < (1 − δ)α and the second from the fact α −aα that αbα−1 < b b−a whenever b > a. Next, turn to the decentralized equilibrium conditions (22) and (23). For the functional form f (k) = k α , they reduce to  k=

βα 1 − β (1 − δ) (1 − λ (θ) (1 − φ)) · 1 − β (1 − δ) 1 − β (1 − δ)α (1 − λ (θ) (1 − φ))

38

1  1−α

(81)

and   cαθ 1 − (1 − δ)α αδ 1 − β (1 − δ) = − k (82) α φλ (θ) 1 − (1 − δ) (1 − λ (θ)) 1 − (1 − δ) (1 − λ (θ)) 1 − β (1 − δ) (1 − λ (θ) (1 − φ)) The same argument used for the planner’s solution establishes that the solution to this system of equations is unique. Finally, we compare the decentralized equilibrium allocation k D , θD solving (81), (82) to the planner’s allocation k P , θP solving (78), (79). For any x, define g (x) by   1 βα 1 − β (1 − δ) (1 − x) 1−α 1 − β (1 − δ) · g (x) = 1 − β (1 − δ) (1 − x) 1 − β (1 − δ) 1 − β (1 − δ)α (1 − x)   1 βα (1 − β (1 − δ) (1 − x))α 1−α = · (1 − β (1 − δ))α 1 − β (1 − δ)α (1 − x) Note that g 0 (x) < 0 since

d (1 − β (1 − δ) (1 − x))α <0 dx 1 − β (1 − δ)α (1 − x)

(83)

(84)

Now, assume that θP λ0 θP φ= λ (θP )




The planner’s solution (78), (79) can be rewritten, after eliminating k P , as  −1

αδ cαθP 1 − (1 − δ)α P − = g λ θ α φλ (θ) 1 − (1 − δ) (1 − λ (θP )) 1 − (1 − δ) (1 − λ (θP ))

(85)

where the left-hand side is increasing in θ and the right-hand side is decreasing in θ. The decentralized equilibrium allocation given by (81), (82) can similarly be rewritten, after eliminating k D , as  −1

αδ 1 − (1 − δ)α cαθD − = g λ θD (1 − φ) α D D D φλ (θ ) 1 − (1 − δ) (1 − λ (θ )) 1 − (1 − δ) (1 − λ (θ )) We will now show that θD > θP . Suppose θD ≤ θP ; then





g λ θP ≤ g λ θD < g λ θD (1 − φ)

(86)

(87)

D which would then imply θP < θD by
combining (85)P
and (86). This Dis
a contradiction;P
so θ > P D θ . Next,
we will
show that

λ θ (1 − φ) < λ θ . Suppose λ θ (1 − φ) ≥ λ θ
. Then g λ
θD (1 − φ) ≤ g λ θP , implying that θD ≤ θP . This is a contradiction. So, λ θD (1 − φ) < λ θP , which then implies k D > k P by (78) and (81).

Proof of Lemma 9. In steady state, (24) becomes (32). (The full derivation is available upon request.) Next, the first-order condition for kj,t+1 is λ (θt )

X i,s

nt (i, j, s) = β

∞ X s=0

β s (1 − δ)s

X

nt+1+s (j, m, s) [zm f 0 ((1 − δ)s k) + λ (θt+1+s ) (1 − δ)] (88)

m

39

In steady state,

P

n (i, j, s) = πj , and, by (32),

i,s

X n (j, m, s) πj

m

and

X n (j, m, s) πj

m

= λ θP




zm = λP (θ) 1 − λ θP

1 − λ θP



s 



s

,


 γ s+1 zj + 1 − γ s+1 z


Canceling λ θP on both sides then yields (30). Next, let β t+1 µt+1 (i, j, s) be the multipliers on (24). The first-order condition for θt can be written as X
c = − nt (i, j, s) kt+1,j − (1 − δ)s+1 kt−s,i (89) 0 λ (θ) i,j,s ! X X +β nt (i, j, s) Πjm µ (j, m, 0) (90) j,m

i,s

X

X

i,m,s

j

! −β = −

X

nt (i, j, s) Πjm µ (i, m, s + 1)

(91)

nt (i, j, s) kt+1,j − (1 − δ)s+1 kt−s,i

(92)




i,j,s

! +

X

nt (i, j, s) ·

β

X

Πjm µ (j, m, 0) − β

m

i,j,s

X

Πjm µ (i, m, s + 1)

(93)

m

And the first-order condition for nt (i, j, s) states that
µt (i, j, s) = zj f ((1 − δ)s ki,t−s ) − λ (θt ) kj,t+1 (j) − (1 − δ)s+1 ki,t−s (i) X X Πjm µt+1 (j, m, 0) Πjm µt+1 (i, m, s + 1) + βλ (θt ) +β (1 − λ (θt ))

(94)

m

m

from which it is immediate that, in steady state, µ (i, j, s) = Υj ((1 − δ)s ki ). Replacing µ with Υ in (93) gives (31). Proof of Lemma 11. The proof is similar to the proof of Lemma 2. Consider the equation 1=β

∞ X


 (β (1 − λ) (1 − δ))t ξjt f 0 (1 − δ)t kj + λ (1 − δ)

(95)

t=0

This implicitly defines kj as a function of λ. The planner’s solution kjP solves (95) for λ = λ, and the equilibrium capital level kjD solves (95) for λ = λ (1 − φ), so a sufficient condition for kjD > kjP is

40

that kj be decreasing in λ, and vice versa. We can write (95) as Γj (λ, kj ) = " Γj (λ, k) ≡ (1 − β (1 − λ) (1 − δ))

∞ X

1 β

− (1 − δ), where

(β (1 − λ) (1 − δ))t ξj,t+1 f 0 (1 − δ)t k

#


(96)

t=0

By the implicit function theorem,

with

dk ∂Γj /∂λ =− , dλ ∂Γj /∂k

(97)

# "∞ X

∂Γj t = (1 − β (1 − λ) (1 − δ)) β (1 − λ) (1 − δ)2 ξj,t+1 f 00 (1 − δ)t kj ∂k t=0

(98)

and

∞


∂Γj X = t (β (1 − δ))t (1 − λ)t−1 ξjt f 0 (1 − δ)t−1 kj − ξj,t+1 f 0 (1 − δ)t kj . (99) ∂λ t=1
By the strict concavity of f , f 00 (1 − δ)t k < 0, and so ∂Γj /∂k < 0. A sufficient condition for ∂Γj /∂λ < 0 is
f 0 (1 − δ)t k ξjt
∀t, k < ξj,t+1 f 0 (1 − δ)t−1 k

Similarly, a sufficient condition for ∂Γj /∂λ > 0 is ξjt ξjt+1

>

f 0 (1 − δ)t k




f 0 (1 − δ)t−1 k


∀t, k

This proves the result. Proof of Lemma 12. Assume δ = 0. The optimal choice of capital, kiP , solves the first-order condition 1=β

∞ X

(β (1 − λ))t





 γ t+1 zi + 1 − γ t+1 z f 0 (k) + λ

(100)

t=0

Subtracting

βλ 1−β(1−λ)

from both sides of (100), we obtain

    1−β γ 1 γ 0 = βf (k) zi + − z . 1 − β (1 − λ) 1 − γβ (1 − λ) 1 − β (1 − λ) 1 − γβ (1 − λ)

(101)

1−β(1−λ) D Multiplying both sides by (1−β)f 0 (k) gives (39). Next, note that ki solves the same equation as (100) but with λ replaced everywhere by λ (1 − φ). Replacing λ with λ (1 − φ) gives (40).

Proof of Proposition 13. This follows directly from comparing (39) with (40). Proof of Lemma 14. Assume f (k) = k α . The first-order condition characterizing the optimal kiP is

41

now 1 = βαk

α−1

∞ X



t (β (1 − λ) (1 − δ)α ) γ t+1 zi + 1 − γ t+1 z +

t=0

Subtracting

βλ(1−δ) 1−β(1−λ)(1−δ)

βλ (1 − δ) 1 − β (1 − λ) (1 − δ)

(102)

from both sides, we get


  P 1 − β (1 − δ) 1 = αk α−1 ˜ zi + 1 − γ˜ P z , α γ 1 − β (1 − λ) (1 − δ) 1 − β (1 − λ) (1 − δ) where γ˜ P =

γ − γβ (1 − λ) (1 − δ)α 1 − γβ (1 − λ) (1 − δ)α

(103)

(104)

Multiplying both sides by 1−β(1−λ)(1−δ) k 1−α gives (41). Similarly, kiD is the solution to (41) but with 1−β(1−δ) λ replaced by λ (1 − φ); replacing λ by λ (1 − φ) gives (43). Proof of Proposition 15. First notice, from (41) and (43), that kiD > kiP if and only if



ω D γ˜ D zi + 1 − γ˜ D z > ω P γ˜ P zi + 1 − γ˜ P z ,

(105)

which can be rearranged as



ω D γ˜ D − ω P γ˜ P zi < ω D 1 − γ˜ D − ω P 1 − γ˜ P z

(106)

From (42) and (44), we have ω D > ω P and γ˜ D < γ˜ P , which implies that the right-hand side of (106) is always positive. If γ (1 − δ)α−1 > 1, straightforward algebra implies that ω D γ˜ D > ω P γ˜ P for all z; in this case, the left-hand side of (106) is negative for all z, and therefore kiD > kiP for all i. Otherwise, it is clear from (106) that kiD > kiP if and only if zi < zˆ, with zˆ defined by (45). Proof of Corollary 16. This is immediate from Lemma 11 and the fact that, both for γ = 0 and γ = 1,
f 0 (1 − δ)t k ξit
∀t, k, =1≤ ξi,t+1 f 0 (1 − δ)t−1 k with equality if and only if δ = 0 or δ = 1. Proof of Proposition 17. The expression in (54) is obtained by applying the first-order condition (50) and using the same simplification as in the of Lemma 12.
When δ = 0, the marginal tax is
proofs P P D P 0 D positive if and only if γ˜ z (k )+ 1 − γ˜ z > γ˜ z (k 0 )+ 1 − γ˜ P z, which, as shown in Proposition 13, is true if and only if z P (k 0 ) < z. It remains to verify that the tax is regressive. Since γ˜ D < γ˜ P , the fraction
γ˜ D z + 1 − γ˜ D z (107) γ˜ P z + (1 − γ˜ P ) z is decreasing in z. Since z P (k 0 ) is increasing in k 0 , this implies that τ 0 (k 0 ) is decreasing in k 0 . Proof of Proposition 18. The proof is similar to the proof of Proposition 17. The expression in (55) is obtained by applying the first-order condition (50) and using the same simplification as in the proof of Proposition 15. When f (k) = k α , the marginal tax is positive if and only if 42





ω D γ˜ D z P (k 0 ) + 1 − γ˜ D z > ω P γ˜ P z P (k 0 ) + 1 − γ˜ P z . As shown in Proposition 15, if γ (1 − δ)α−1 , this is true for all z P (k 0 ); otherwise, this is true if and only if z P (k 0 ) < zˆ. It remains to verify that the tax is regressive. Since γ˜ D < γ˜ P , the fraction
γ˜ D z + 1 − γ˜ D z (108) γ˜ P z + (1 − γ˜ P ) z is decreasing in z. Since z P (k 0 ) is increasing in k 0 , this implies that τ 0 (k 0 ) is decreasing in k 0 .

B

Convex costs of production

In this section we extend the basic deterministic model in Section 2 by assuming that dealer costs of production are convex rather than linear. We show that the same dynamic overinvestment incentive arises as with a linear cost of production. For simplicity, we ignore the entry margin and assume λ = 1. This will simplify the exposition and make the intuition as transparent as possible; in fact, under this assumption, the planner’s problem for this economy is simply the neoclassical investment problem with convex costs of adjustment. Every period, a firm meets a dealer and decides on the level of investment, x. If the capital level prior to the meeting was k, the firm exits the meeting with capital k 0 = (1 − δ) x + k and the dealer pays a cost ψ (x), where we assume ψ 0 (x) > 0, ψ 00 (x) > 0. The planner’s problem is then Υ (k) = f (k) + max (−ψ (k 0 − (1 − δ) k) + βΥ (k 0 )) 0 k

(109)

A standard contraction mapping argument establishes that this functional equation has a unique fixed point Υ, which is strictly increasing and strictly convex. Furthermore, letting k 0 = g (k) be the optimal policy function, the solution is characterized by the first-order condition ψ 0 (g (k) − (1 − δ) k) = βΥ0 (g (k))

(110)

Υ0 (k) = f 0 (k) + (1 − δ) ψ 0 (g (k) − (1 − δ) k)

(111)

and the envelope condition

Combining (110) and (111) gives the Euler equation for investment ψ 0 (g (k) − (1 − δ) k) = β [f 0 (g (k)) + (1 − δ) ψ 0 (g (g (k)) − (1 − δ) g (k))]

(112)

Note that, unlike the case with linear production costs, the optimal k 0 = g (k) is now a function of k, and it is no longer possible to obtain a closed-form analytic expression for k 0 . However, we can still obtain a partial characterization of the optimal investment choice. By convexity of ψ, there exists a unique k∗P satisfying


 (113) ψ 0 δk∗P = β f 0 k∗P + (1 − δ) ψ 0 δk∗P
This value is the unique solution to g k∗P = k∗P . The following result is standard. Lemma 19. Consider any k0 < k∗P , and define the sequence kn+1 = g (kn ) for all n ≥ 0. Then the sequence {kn } is a monotonically increasing sequence converging to k∗P .

43

Proof. First, we show that, for any k < k∗P , g (k) > k. Suppose that g (k) ≤ k, then, since k < k∗P , we have
ψ 0 δk∗P > ψ 0 (δk) ≥ ψ 0 (g (k) − (1 − δ) k) = βΥ0 (g (k))


> βΥ0 g k∗P = βΥ0 k∗P , (114) which is a contradiction. So g (k) > k. Next, equation (110) implies that g (k) is increasing in k; this follows from
thePconvexity of ψ and P P concavity of Υ. Therefore, if k < k∗ , it is also the case that g (k) < g k∗ = k∗ . This shows that k∗P is an upper bound for any sequence {kn } as defined above. It is clearly the least upper bound, because of the continuity of g. In other words, starting from any k0 , the firm will gradually accumulate capital, converging to the steady state value k∗P . Next, we derive an analogous characterization for the decentralized equilibrium. The firm’s value function is v (k) = f (k) + max (−ρ (k, k 0 ) (k 0 − (1 − δ) k) + βv (k 0 )) , 0 k

(115)

where ρ (k, k 0 ) is determined by Nash bargaining to be ρ (k, k 0 ) (k 0 − (1 − δ) k) = φψ (k 0 − (1 − δ) k) + (1 − φ) (βv (k 0 ) − βv ((1 − δ) k))

(116)

Substituting (116) into (115) gives v (k) = f (k) + φv ((1 − δ) k) + (1 − φ) max (−ψ (k 0 − (1 − δ) k) + βv (k 0 )) 0 k

(117)

It is straightforward to show that the mapping defined in (117) is a contraction, and maps concave functions into concave functions. As a result, the unique fixed point v is a concave function, and the optimal k 0 = g D (k) is determined by the first-order condition

ψ 0 g D (k) − (1 − δ) k = βv 0 g D (k) (118) Together with the envelope condition 
 v 0 (k) = f 0 (k) + (1 − δ) φβv 0 ((1 − δ) k) + (1 − φ) ψ 0 g D (k) − (1 − δ) k

(119)

this yields the Euler equation for investment

ψ 0 g D (k) − (1 − δ) k =β f 0 g D (k) (120) 


 + (1 − δ) φβv 0 (1 − δ) g D (k) + (1 − φ) ψ 0 g D g D (k) − (1 − δ) g D (k) As with the planner’s solution, we can find the unique stationary level of capital k∗D as the solution
to g D k∗D = k∗D , which is determined by



 (121) ψ 0 δk∗D = β f 0 k∗D + (1 − δ) φβv 0 (1 − δ) k∗D + (1 − φ) ψ 0 δk∗D 44

We then have Lemma 20. Consider any k0 < k∗D , and define the sequence kn+1 = g D (kn ) for all n ≥ 0. Then the sequence {kn } is a monotonically increasing sequence converging to k∗D . Proof. The proof is the same as the proof of Lemma 19. It remains to compare k∗D to k∗D . The following result extends the main overinvestment result in the paper to the setting with convex costs. Proposition 21. k∗D > k∗P as long as φ ∈ (0, 1) and δ ∈ (0, 1).


Proof. By concavity of v, we have v 0 (1 − δ) k∗D > v 0 k∗D . Note that, by (118), ψ 0 δk∗D =
βv 0 k∗D , and so



 ψ 0 δk∗D = β f 0 k∗D + (1 − δ) φβv 0 (1 − δ) k∗D + (1 − φ) ψ 0 δk∗D 


 > β f 0 k∗D + (1 − δ) φβv 0 k∗D + (1 − φ) ψ 0 δk∗D 

= β f 0 k∗D + (1 − δ) ψ 0 δk∗D (122) Comparing (122) to (113) immediately reveals that k∗D > k∗P . In the decentralized equilibrium, as in the planner’s solution, the firm gradually accumulates capital until it reaches the steady-state level; however, this steady state level is higher in equilibrium than under the planner’s solution. Thus, starting from any k0 , the firm overaccumulates capital.

C

Directed search

In this appendix, we discuss an alternative environment in which search is directed: dealers post terms of trade, and firms choose which terms of trade to direct their search to, with the market tightness determined endogenously. This setting is similar to the model considered in Lester et al. (2015), except that we also allow for depreciation of capital. Consider dealers as posting contracts in markets indexed by (x, $) where x is the amount of capital (positive or negative) that the dealer will provide to a firm, and $ is the transfer, in units of consumption, that the requires in exchange. Posting a contract incurs a cost c, as in the main text. Firms then choose in which markets to search for capital. The market tightness θ in each market is determined endogenously, and must satisfy λ (θ) ($ − x) ≥ c θ

(123)

Otherwise dealers posting that contract would make negative profits. As is standard, we can formulate a firm’s problem as choosing x, $, θ to maximize its profits: ( " # X vi (k) = zi f (k) + max λ (θ) −$ + β Πij vj (x + (1 − δ) k) x,$,θ

j

) + (1 − λ (θ)) β

X

Πij vj ((1 − δ) k)

j

45

(124)

subject to the constraint (123). Since the constraint (123) will clearly bind at the optimum, we can eliminate $ using the constraint, obtaining ( " # X vi (k) = zi f (k) + max −cθ + λ (θ) −x + β Πij vj (x + (1 − δ) k) x,θ

j

) + (1 − λ (θ)) β

X

Πij vj ((1 − δ) k)

(125)

j

This is clearly equivalent to the problem of a social planner who chooses, for each firm, how many dealers should operate, and what investment it should undertake if it meets a dealer, both as a function of the firm’s current productivity and capital level.27 Therefore, if the decentralized equilibrium is unique, it produces the same allocation as the social planner’s solution.28

D

Computation of the numerical example

This section provides details for computing the equilibrium in the numerical example of Section 5 and the calibration of the model. For the functional form zf (k) = zk α , the first-order conditions (30) and (31) for the planner’s problem can be written as kjP =

αAPj BP

1 ! 1−α

(126)

for each j, and    X 1 1 − (1 − δ)α αδ c P = − B πj kjP α λ0 (θ) α 1 − (1 − δ) (1 − λ (θ)) 1 − (1 − δ) (1 − λ (θ)) j X (1 − γ) λ (θ) (1 − δ)α πj APj + (1 − (1 − λ (θ)) (1 − δ)α ) (1 − γ (1 − λ (θ)) (1 − δ)α ) j where we defined

! kjP α −

X

πi kiP α

(127)

i


γ˜ P zj + 1 − γ˜ P z , =β 1 − β (1 − λ (θ)) (1 − δ)α

(128)

γ − γβ (1 − λ (θ)) (1 − δ)α γ˜ = , 1 − γβ (1 − λ (θ)) (1 − δ)α

(129)

APj P

27

Note that assuming directed rather than random search changes the environment, and hence the planner’s solution, rather than just the decentralized equilibrium. In particular, the social planner in a random search economy was choosing a single θ for all firms, whereas the social planner in a directed search economy chooses a distinct θ for each type of firm. 28 Concavity of the problem (125) cannot be guaranteed in general. Lester et al. (2015) show that the equilibrium is unique when asset holdings are restricted be either 0 or 1.

46

and BP =

1 − β (1 − δ) 1 − β (1 − λ (θ)) (1 − δ)

(130)

For a given matching function λ(θ) and a given stochastic process z defined by γ and πj , equation (126) allows us to compute kjP as a function of θ. Substituting kjP from (126) into (127) then gives a single equation in one variable, θ, which is solved numerically for θP . The decentralized equilibrium is computed similarly. Conditions (37) and (38) for the decentralcθ ized equilibrium are identical except that the left-hand side of (127) is replaced by φλ(θ) instead of c P P P D D D , and Aj , B ,kj are replaced by Aj , B , kj where λ0 (θ)
γ˜ D zj + 1 − γ˜ D z =β , 1 − β (1 − λ (θ) (1 − φ)) (1 − δ)α

(131)

γ − γβ (1 − λ (θ) (1 − φ)) (1 − δ)α γ˜ = , 1 − γβ (1 − λ (θ) (1 − φ)) (1 − δ)α

(132)

AD j D

and BD =

1 − β (1 − δ) 1 − β (1 − λ (θ) (1 − φ)) (1 − δ)

(133)

Aggregate quantities in steady state for both the planner’s solution and the decentralized equilibrium are computed as follows. Aggregate steady state capital equals P X λ(θ∗ ) i πi ki∗ ∗ ∗ s ∗ ; (134) K = n (i, j, s)(1 − δ) ki = ∗ ))(1 − δ) 1 − (1 − λ(θ i,j,s where the ∗ exponent denotes values for either the planner’s solution or the decentralized equilibrium from hereon. Aggregate steady state net investment in new capital equals P X  ∗  δλ(θ∗ ) i πi ki∗ ∗ ∗ ∗ s+1 ∗ . (135) I = λ(θ ) n (i, j, s) kj − (1 − δ) ki = 1 − (1 − λ(θ∗ ))(1 − δ) i,j,s Notice that this implies I ∗ = δK ∗ as in the frictionless model. Aggregate steady state output equals X Y∗ = n∗ (i, j, s)zj ((1 − δ)s ki∗ )α i,j,s

P P λ(θ∗ )(1 − γ ∗ ) i πi zki∗α λ(θ∗ )γ ∗ i πi zi ki∗α = + 1 − (1 − λ(θ∗ ))(1 − δ)α 1 − (1 − λ(θ∗ ))(1 − δ)α where γ∗ =

(136)

γ − γ (1 − λ (θ∗ )) (1 − δ)α . 1 − γ (1 − λ (θ∗ )) (1 − δ)α

Steady state consumption is then given by C ∗ = −cθ∗ + Y ∗ − I ∗ .

(137)

To calibrate the model, we adopt the matching function of Den Haan, Ramey and Watson (2000),

47

which implies λ(θ) = θ/(1 + θχ )1/χ

(138)

and proceed as follows. First, notice that the fraction of firms λ(θt ) that meet a dealer, those with a sufficiently high productivity level purchase capital – i.e. kj,t+1 > ki,t−s (1 − δ)s+1 – and the remaining firms sell capital. Hence, acquisitions of capital by firms (gross investment) equal X   (139) Mt = λ(θt ) nt (i, j, p) kj,t+1 − ki,t−s (1 − δ)s+1 1kj,t+1 >ki,t−s (1−δ)s+1 i,j,s

Likewise, sales of capital by firms (gross separations) equal X   ϕt (1 − δ)Kt = λ(θt ) nt (i, j, p) kj,t+1 − ki,t−p (1 − δ)p+1 1kj,t+1
(140)

i,j,p

where ϕt is defined as the separation rate of undepreciated aggregate stocks. The difference between the two gross flows equals net investment in new capital X   It = λ(θt ) nt (i, j, s) kj,t+1 − ki,t−s (1 − δ)s+1 (141) i,j,s

and the evolution of the aggregate capital stock of firms can be expressed as Kt+1 = (1 − δ)Kt + Mt − ϕt (1 − δ)Kt = (1 − ϕt )(1 − δ)Kt + Mt

(142)

In steady state, the rate of gross investment relative to the capital stock of firms equals M = 1 − (1 − ϕ)(1 − δ) K

(143)

and investment in used capital must satisfy M used = ϕ(1 − δ)K. Hence, for a given rate of gross investment to capital and a given ratio of used capital investment to gross investment, we can compute ϕ(1 − δ) =

M used M used M = , K M K

which, by using (143), allows us to compute δ=

M + ϕ(1 − δ). K

Given ϕ(1 − δ), the decentralized economy versions of (126) and (134) can be used to numerically calculate the value for λ(θ) that solves (140) in steady state.29 Of course, this solution is conditional 29

Note that (126) and (134) are functions of λ(θ), not θ.

48

on a value for φ, which we take as given for now but set later so that the implied distribution of prices fits a given target. Then, given a value for λ(θ)/θ, we obtain θ and by (138) we obtain χ. Finally, we set the value of cis calculated by solving the free entry condition for the decentralized economy    X cθ 1 1 − (1 − δ)α αδ D πj kjD = − B φλ (θ) α 1 − (1 − δ)α (1 − λ (θ)) 1 − (1 − δ) (1 − λ (θ)) j ! α X X (1 − γ) λ (θ) (1 − δ) + πj A D kjDα − πi kiDα j (1 − (1 − λ (θ)) (1 − δ)α ) (1 − γ (1 − λ (θ)) (1 − δ)α ) j i (144) As mentioned above, this algorithm is conditional on a value for φ. We choose this value such that the implied steady state distribution of prices, computed numerically as " # Dj (kjD ) − Dj ((1 − δ)s+1 kiD ) ρ(i, j, s) = 1 + φ (145) kjD − (1 − δ)s+1 kiD where Dj (k) = β

X

α D πjm vm (k) − k = AD j k − B k,

m

matches a given property (in our case, the average absolute dispersion of transaction prices for the used commercial aircraft market).

49