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Duration and Renovation of Contracts in the Presence of a Holdup Problem –
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A Dynamic Approach
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Abstract
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The owner of an asset or natural resource often transfers the right to use or exploit it to a firm in
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exchange of a rent. The limited time of the license and the failure of the owner to commit to
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compensate the agent for any improvement in the asset or resource, is likely to lead to the holdup
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problem. So far, for overcoming the holdup problem the economic literature paid little attention to
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time itself. However, time is a crucial element while the contractual relationship unfolds over time,
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and in practice it often forms a building element of the contract. Within a dynamic principal agent
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framework this paper analyzes under which conditions and at which point in time the holdup
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problem may arise. It determines the contract duration that maximizes the net benefits of the
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principal and its relationship to the occurrence of the holdup problem. For overcoming the holdup
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problem the paper suggests to construct a sequence of overlapping short-term contracts. For this
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purpose it determines the minimal length of these short-term contracts and the length of the
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overlapping periods. Overlapping periods itself are achieved by advance notice of the renewal of the
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contract. Finally, the studys finds that the owner can adjust the rent over time to extract the entire
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cooperative benefits.
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Key words: Resource management, holdup problem, contract design, dynamic optimization.
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1. Introduction
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Firms have to decide which transactions take place within the firm and for which transactions they
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want to rely on markets. In other words firms have to define their boundaries. Nobel prize winner
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Oliver Williamson (1979) proposed transaction costs as a critical element for deciding where to draw
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the boundary line between the firm and the market. One problem related to this question is that market
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partners often have to make relationship specific investment which can only be recovered within the
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relationship. If one market partner decides unilaterally to end the relationship market transactions may
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become very costly since investments are often sunk costs that cannot be recovered. As a remedy one
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may suggest to write a complete contingent contract that safeguards the interest of each partner. Yet,
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as put forward by Grossman and Hart (1986) the economic agents’ rationality is bounded and does not
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allow foreseeing all possible future contingencies. Moreover, certain future states or acts cannot form
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part of a written contract since they cannot be verified by a third party, e.g. intangible goods like
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originality or trendiness of the product or human capital. Likewise, the non-investing partner has little
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incentive to renegotiate the contract if a third party (court) cannot verify the cooperative investment.
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Finally, the investment already benefited the non-investing partner while the contract was in place and
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therefore this partner has little incentive to accept any demands ex post. The threat that the non-
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investing partner may take advantage of this situation may induce the investing partner to undersupply
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cooperative investments which is commonly known as the holdup problem.
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The previous literature on the holdup problem has considered time explicitly but only in a stylized
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manner. Normally, agents can invest in the first period and benefits can only be obtained in the
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second period. The standard model assumes that each period is of equal length but it does specify it.
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Moreover investments are frequently not modeled as a stock but as a flow variable. Looking at real
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world examples, however, shows that periods of investments and benefits are often intertwined and
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the dynamics of the investment behavior and the realization of the benefits is more complex that
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depicted by the standard model. Given this observation our point of departure is that the investment
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behavior of the agents depends on the length of the period or contract duration, in particular if
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investment is not a one-time event but a continuous process while the contract is in vigor. Moreover,
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the benefits of investments are often not immediate and can only be recovered over time, i.e., there is
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a continuous time lag which is often not uniformly distributed. Hence, we focus in this study on the
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description of the dynamics of the optimal investment behavior in order to determine the optimal
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contract duration in the presence of a holdup problem. Additionally we analyze the question whether
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an efficient long-run equilibrium can be achieved by a contractual relationship.
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2. Literature Review
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To overcome the holdup problem the literature proposes the assignment of property rights and
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changes in the governance structure so that the incentives of the partners are aligned (Grossman and
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Hart 1986) and (Aghion and Tirole 1997). Alternatively Aghion et al. (1994) suggested the design of
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contracts that allow guidance of the ex-post renegotiation process, Felli and Roberts (2011) and
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MacLeod and Malcomson (1993) the improvement of market contracts and Baker et al. (2002) the
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consideration of the value of future relationships (relational contracts). The hold-up problem is
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frequently analyzed in form a principal agent model if one of the agents is in the position to make a
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“take it or leave it offer”, or in form of a bargaining game where no ex ante contract exists and the
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two partners are on equal terms. Both approaches, however, have in common that the agents can
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realize only a one-time investment at a pre-specified date, and exchange or bargaining does not occur
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before all investments have been realized. In practice, however, the timing of the investment, the
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duration of the relationship and the sharing rule for the cooperative surplus are often negotiated before
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investments are completed.
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Within the context of a bargaining game Che and Sákovics (2004) maintained the previous employed
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modeling approach but allow for an infinite sequence of investments. The authors find an
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asymptotically efficient equilibrium where investment takes place all at once. If the time horizon were
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no infinite this equilibrium would not be supported. Likewise Gul (2001) shows that there exists an
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asymptotic efficient equilibrium in a holdup model without ex ante contracts. Yet, the analysis does
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not consider repeated investment decisions since the model only allow for investment in the first
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period. For the case of a contract which relies on renegotiation with symmetric information and
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production and trade activities that are verifiable by court Evans (2008) established the existence of
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an efficient equilibrium. The work that comes closest to our work is by Guriev and D.Kvasov (2005).
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Investment, benefit sharing and renegotiation take place in continuous time. Like in our work time is
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not only a dimension along which the relationship unfolds, but also a continuous verifiable variable
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that can be employed for the design of the contract. To overcome the holdup problem it relies on
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renegotiation, advance notice and third-party enforcement, i.e., the authors assume that both the
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provision of a service and its timing can be verified by a third party.
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A different strand of the literature is based on multiperiod principal agent model. Rey and Salanie
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(1996) show for the case of asymmetric information that renegotiable short-term contracts can be as
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efficient as long-term contracts when agents have no incentives to renegotiate, and physical and
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monetary variables can be transferred between the time periods. Instead of renegotiation Levin(2003)
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Kvaløy (2006), Board (2011), and Halac (2012) study the design of self-enforcing relational contracts
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to mitigate the holdup problem. Thus, the contract must be designed such that the agents have
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economic incentives to honor the contract in all contingencies.
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In light of the previous literature this study explores, within the principal agent framework, the
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optimal trajectory of the agent’s investment behavior during the validity of the signed contract.
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Following the spirit of Che and Sákovics (2004) we initially analyze whether and from when onwards
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a holdup problem arises. For this purpose, following the literature (Lichtenberg 2007; Jacoby and
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Mansuri 2008), we determine the socially and private optimal solutions and analyze possible solutions
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to overcome the gap between the private and social solutions, i.e., to overcome the holdup problem.
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The present study aims to extend the previous results by considering the dynamics of the agent in
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continuous time where investment and sharing of benefits are time-wise intertwined, the contract
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duration is determined endogenously and investments affect the corporative benefits in an
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accumulative way. In contrast to the work by Guriev and D.Kvasov (2005) our study does not built on
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the fact that the provision of a service and its timing can be verified by a third party. Instead we use
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the rationality constraint of the agent to determine the provision and timing of the investment. We
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consider this point to be important since precisely the absence of the verifiability makes it difficult to
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include the provision of a service and its timing in the contract.
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2. Statement of Problem
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We assume that the principal is the owner of an asset and makes a “take it or leave it offer” to an
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agent. Within this context we find for instance landlords that do not cultivate their land themselves but
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lease it out to agricultural producers in form of a fixed rent or sharecropping contract. An equally
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frequently found example is given by franchise agreements. The franchiser is the owner of the asset,
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often in form of a brand and the contract authorizes the franchisee to exploit the brand. Likewise one
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can think of a public asset or permission that is rented out to an agent in order to exploit it. For
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instance the exploitation of transport infrastructure of the right to issue a public certificate (medical,
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technical). Within this setup the principal receives a rent paid by the agent. In this respect the agent is
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the residual claimant of the part of the corporative benefits that are not taken away with the payment
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of the rent.
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We start our analysis by looking at the problem of the owner of the resource.
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2.1. The principal´s problem:
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Let us denote calendar time by t and assume that the length of the contract T is finite and endogenous.
The principal’s objective is to maximize the income, I, from the payment of the rents. For this purpose
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it is necessary to find the value T  0 and the rent R  t   0,   , that maximizes

T ( j 1)
j 0
Tj
I  
e rt R(t )dt  e rTjV (T ),
(1)
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where the index j indicates the number of repetition of the contract, i.e. j  0 is the current contract
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and j  1 is the first repetition of the contract. The function V (T ) describes the principal’s costs
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related with the signature of a new contract. Naturally, such expenses and losses depend on the length
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of contracts. Short term contracts imply the frequent repetition of search and transaction costs and
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long term contracts impede the adjustment of the contract to changes in the environment. Thus, we
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assume that the function V T   0 decreases initially with T
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larger. As a result the function V (T ) is u-shaped with a unique minimum.
but increases again as T is getting
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2.2. The agent´s problem:
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For a fixed number
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R  t  , t  Tj , T  j  1  , the agent determines the input m(t ) and the investment n  t  ,
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t Tj, Tj  T  , that maximizes
I j  max
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
T ( j 1)
m ( t ), n ( t ) Tj
j  0,1, 2,
a given contract duration
T , and a given rent
e  rt [ B(m(t ), s (t ))  cm m(t )  cn n(t )  R (t )]dt
(2)
where the “asset quality” indicator s(t) satisfies the equation:
s(t )  h(m(t ), s(t ))  g (n(t ),
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s(0)=s0 , t[Tj, T(j +1)],
(3)
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under the inequality-constraints
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0  n  n , m  t   0, s  t   0, B(m(t ), n(t ), s(t ))  cm m(t )  cn n(t )  R(t )  u0  0 ,
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where u0 denotes the agent’s reservation utility, and s the resource/asset quality with its initial
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condition s (0)  s0 . The parameters cm , cn , s0 and the functions u0 (t ) , B, h, g, are given. The
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agent’s benefits, B(m,s), increase with the input m and the resource quality s. The changes of the
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resource quality over time are described by equation (3). The more intensive is the use of the resource,
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i.e., the higher are m and s, the stronger is the deterioration of the resource captured by the function
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h(m,s). Hence, we obtain that h  m, s  , hm , hs > 0. 1 The agent has the possibility to improve the
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quality of the resource by investing n which is described by the function g ( n ) . The parameters cm
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and cn denote the costs of the input m and the investment n respectively.
(4)
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Here and thereafter, the subscript of a function with respect to a variable denotes the partial derivative of the
function with respect to this variable.
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Since the holdup problem may arise, the optimal dynamics of the agent’s choices may include a
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decrease of m* (t ) and n* (t ) and/or s*(t) over the interval t  T  j  1  , T  j  1  , towards the
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end of the planning horizon Tj , T  j  1  . 2 Consequently, the optimal rent R *  t  may not be
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constant over time, so that time cannot be excluded from the principal’s problem in section 2.1. The
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qualitative behavior of the optimal of m* , n* , s* and R* over Tj , T  j  1  will be clarified in
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section 4 - the analysis of the principal’s problem.
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The problem (1) - (4) is a dynamic optimization problem. For the sake of clarity, we will focus on its
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sustainable dynamics and assume, as it can be seen from our formulation of the objective function (1),
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that the optimal value of T is identical for the entire sequence of contracts. Consequently, we look for
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a sustainable solution that follows the same dynamics for all j.
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3. Analysis of the agent’s problem
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In this section, we study the dynamics of the optimization problem (1) - (4) with j  0 for a given
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function R . The general formulation of the optimization problem allows obtaining optimality
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conditions in its generic form, however, not the description of the qualitative behavior of the optimal
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trajectories. For this purpose, let us choose a specific applied problem. Consider the following
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specifications of the functions.
B(m, s)  Y (m, s)  Am s, with 0    1 ,
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h(m, s)  Y (m, s),
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(5)
g ( n)  g 0 n ,
(6)
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where Y is the output. Since the price of the output is normalized to one Y is also equal to earnings
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produced by the input m (at a given level of the resource s),
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(concavity) of the production process and  the deterioration of the resource as a result of its use
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with 0    1 . Over the entire range of s the production function is nonlinear and should be
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specified as a concave function, for example by s  However, often the exploitation of the resource is
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only economically viable within a limited range of s. Given that this range is relatively small we
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assume that  is equal to 1 over this range. Another justification for this assumption corresponds to
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the case where the principal makes the important investments while the agent realizes in comparison
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only smaller ones. For instance the principal invest in the physical infrastructure (transport, water) or
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in the brand name (franchisee ) while the agent maintains the resource (infrastructure, service
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quality). Let g0 denote the improvement of the quality of the asset by investing n .
2

The sign * indicates the evaluation of the variable along the optimal trajectory.
describes the nonlinearity
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The maximization problem for the agent leads for an interior solution of m to the following equation:
 Am(t ) 1 
cm
.
1   (t )  s(t )
(7)
However n(t) is bounded so that the first order condition takes the form :
 0, g0 (t )  cn  0

n(t )  [0, n ], g 0 (t )  cn  0 .
 n , g  (t )  c  0
0
n

(8)
Moreover, the solution has to comply with the state and costate equation given by
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s (t )    Am (t ) s(t )  g 0 n(t ) ,
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 (t )   Am(t )  (t )  r (t )  Am(t ) ,  (T )  0 .
s(0)  s0 ,
(9)
(10)
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The costate variable  has a natural interpretation as the future value (shadow price) of a marginal
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increase in the quality of the resouce. Since this future value is equal to zero once the contract expires
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it holds that  (T )  0 .
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The first order condition (8) has a clear economic interpretation as it suggests that the agent does not
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invest at all if the costs of the investment cn are smaller than the resulting benefits g0 . These
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benefits are given by the improvement of the resource times its shadow value. For cn  g 0  we obtain
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that it is optimal to realize the maximum possible investment n
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invested within the range of  0, n  is not defined.
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The solutions s,  of the linear ordinary differential equations of (9) and (10) for a given m yield
t
s (t )  g 0 
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t
 Am ( )
e u


d
and for cn  g 0  the amount
t
 Am ( )
n(u )du  s0e 0


d
,
(11)
0

  r   Am   d

 (t )   e 
Am  u  du .
T
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u
t
(12)
t
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Albeit the specifications presented in (5) and (6), the optimization problem (2) - (4) still remains
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nonlinear. However, since we we are left only with one nonlinear function - Y ( m, s ) - we can
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immediately obtain meaningful ranges for the parameters of the model when m follows an interior
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trajectory.
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Observation 1 
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The future value (t) of the unit increase in the soil quality is limited and always smaller than 1  ,
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 1  
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
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Proof: The evaluation of equation (12) for r > 0 yields directly the result that
T
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
 (t )   e 
u
t
 Am ( ) d

1
1  e t


T
Am (u )du 
t
 Am ( ) d
 1
 .


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To analyze possible ranges of parameters and develop an investigation technique for the agent´s
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problem (2) - (4), we start with the first-best sustainable solution i.e. a solution where the optimal
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behavior of the agent is also optimal from the perspective of the principal. It describes the situation
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where the agent´s contract is of infinite duration. In this case the the interest of the principal and the
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agent are lined up, the principal has no need to commit, and consequently the holdup problem does
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not emerge.
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Analysis of the steady state solution (first-best solution)
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Mathematically the first-best solution of the optimization problem (2) - (4) corresponds to the case
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where T   in other words it corresponds to the the steady-state solution.
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The infinite-horizon version of the optimization problem (2) - (4), given the specifications (5) and (6)
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can be stated as
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
I   max  e rt [ Am s  cm m  cn n  R]dt
0
(13)
m  0, s  0 , 0  n  n , lim ˆe rt  0.
s    Am s  g 0 n ,
t 
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This formulation does not include time explicitly. Its long-term steady-state regime ( m̂ , n̂ ŝ , ̂ ) is
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constant over time and is defined by the equations:
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A mˆ  1 
cm
1  ˆ  sˆ
,
 0, g0 (t )  cn  0

nˆ (t )  [0, n ], g 0 (t )  cn  0
 n , g  (t )  c  0
0
n

sˆ 
g 0 nˆ
,
 Amˆ 
(14)
(15)
(16)
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Amˆ  
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1
r ˆ
.
1  ˆ
(17)
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From theorem 1 we know that  
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1
r

   ˆ 

ˆ
 Am
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Hence the term
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sustainable value ̂ . It is easy to see that a positive solution of the nonlinear system (14) - (17) exists
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only for certain values of the parameters. In order to obtain an approximate analytic solution and
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understand in more depth the dynamics of the model, we consider some special but realistic cases
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below. Before we analyze these cases we show that a unique positive solution of the steady state
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solution exists.
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Theorem 1 (on the steady-state first-best regime).
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The nonlinear system (14) - (17) has a unique positive solution ( m̂ , ŝ , ̂ ) . The solution is positive:
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ˆ  nˆ  sˆ  0 if ˆ  cn g0 .
mˆ  0, sˆ  0, and nˆ  n if ˆ  cn g0 and m
1
r

Am


so that eq. (17) shows that
1

r
determines the difference between the maximum value of  and its
Am
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Proof: See appendix A 
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Analysis of the dynamic solution.
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Let us return to the analysis of the full dynamic version (2) - (4) of the agent problem on the finite
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horizon [0,T]. The formulas (9) and (10) lead to
t
s(t )  e
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T

 (t )   e 
u
t
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t
0
  A m ( ) d
r   Am ( ) d
t  A m ( ) d


0
s

g
n(u )du  ,
0 0 e
 0


u

1
1  e t


T
Am (u )du 
r   Am ( ) d
T
u
T

  r   Am ( )  d 
1    r   Am ( ) d
  e t
 r  e t
du 
 
t

1
(18)
 r T  t r   Am ( ) d
du
   e

t
u
(19)
10
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Equation (18) shows that the optimal s(t) does not strive to a constant limit that coincides with the
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sustainable solution sˆ 
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small compared to  Am (t ) , then  (t ) decreases monotonically from  (0)  0 to  (T )  0 . In
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particular for m(t )  constant one finds that
 (t ) 
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g 0 nˆ
as defined in (16). At least if (a) m(t ) is non-increasing, or (b) r is
 Amˆ 
Am
1  e r   Am ( )T t   ,

r   Am (t ) 
(20)
And the optimal n(t ) is bang bang given by
n , 0  t  t *
n(t )  
 0, t*  t  T
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(21)
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where t * is determined from the equation  (t )  cn g0 . The instant t * depends on  which in turn
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depends on m by (19).
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For an interior trajectory of m equation (7) can be presented as:
m1 (t ) 
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A
s (t ) 1   (t ) .
cm
(22)
The substitution of (18) and (19) into (22) gives
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1
m
t
u
T
u
T
t  A m ( ) d

   r   Am ( ) d
   t  r   Am ( )d
A   A0 m ( ) d 

0
t
(t ) 
e
s0  g0  e
n(u )du  e
 r e
du  .

0
cm


t

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(23)
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Equation (23) is a nonlinear integral equation with respect to m(t ), t  [0, T ] . Because of the special
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structure of the nonlinear integral operator, the existence of a unique solution in m under a given n
266
can be shown using the contracting mapping principle , at least for small r<<1. Without further
267
specification of the value of T the qualitative solution of the equations (18) - (21) cannot be obtained.
268
Thus, we analyze the qualitative solution of the optimization problem (2) - (4) for different lengths of
269
the contract length T.
270
271
Short-term contracts (T small)
11
272
Intuitively there exist a small value of T that does not provided sufficient incentives for the agent to
273
invest. Obviously, this value depends on the values of the parameters and is determined in this
274
subsection. We assume that r  1 and make use of the mean value theorem for integral so that the
275
shadow value of the resource, equation (19), is approximately equal to
276
 (t )  
277
where t   (t )  T and  (T )  0 . For the case of no investment, i.e.  (t )  cn g 0 , one obtains that
278
279
T
u
T
 1
   r   Am ( )  d
 r   Am  ( t ) (T t )  
1   t  r   Am ( )d
t
e

r
e
du   1  e
,






t


1 e

 r   Am ( ( t )) T t 
(24)
1
c
 r   Am ( (t )) T t   1   cn  .
 n e


g0
g0 

(25)
The equation (25) can be solved for T which yields
cn
c
)
 ln(1   n )
g0
g0
T

.

r   Am ( (t ))
r   Amˆ 
 ln(1  
280
281
(26)
Hence, if equation (26) holds the exact solution for the control, state and co-state variables is given by

cn
, for 0  t  T ,  (T )  0
g0
n0
282
s (t )  s0e
 A
t
0 m

( ) d 
(27)
1
 A
1
m
1    s  .
 cm

283
Equation (27) shows that in the absence of investment short-term contracts lead to a degradation of
284
the stock and a decrease in the productive input. This pattern is repeated if the contract is renovated
285
with the same short contract duration. In other words a sequence of short term contract leads to a
286
continuous degradation of the stock. For the design of contracts equation (26) is of interest as it
287
provides guidance for the choice of the optimal contract duration in order to avoid that agents do not
288
invest at all. If the deterioration rate  is high, and/or the productivity of the resource is small, Amˆ  ,
289
equation (26) holds, and even a longer contract length T does not avoid that the agent does not invest
290
while the contract is in vigor. Similarly, if the values of the parameters  cn g0 are such that this term
291
is close to 1 the absolute value of the logarithm is large. Hence, only very longtime contracts can
12
292
avoid the absence of investment. However, if the values of the parameters  cn g0 are such that the
293
term is close to 0 the absolute value of the logarithm is close to one. Thus, even contracts that have a
294
short duration are able to avoid the lack of investment. Equation (26) can be written as
295
 r   Amˆ  T  ln(1   gc

n
) and allows drawing a separation line between no investment and
0
296
investment as a function of the parameters and the choice of the contract duration - see Figure 1. For
297
instance if the absolute value of the logarithm corresponds to the level of point A then any contract
298
duration less than 8 years leads to no investment. Likewise for point B any contract duration less than
299
2 years leads to no investment. In other words the owner of the resource has to choose T sufficiently
300
large so that the straight line intersects with the given point of the downward sloping curve in order to
301
avoid that the agent does not invest at all.
302
303
Fig. 1 Investment failure as a function of the contraction duration and the parameter values for t *  0
304
305
306
Intermediate-term contracts (T is neither short nor long)
307
Let us assume that the optimal solution of the agent’s problem for s (t ),  (t ), m(t ) and
308
n(t ), t  [0, T ] exists and T is neither very short nor very long, or in other words that equation (26)
309
does not hold. Moreover, it seems reasonable to assume that r  1 . Hence, by equation (19) it holds
13
310
  r   Am ( )  d 
1
that  (t )  1  e t
 is monotonically decreasing for any m( )  0. Therefore, the


311
switching time t * is unique, and the structure of the optimal n(t ) is bang-bang:
312
The unique instant t * , 0  t *  T , and formulas (18), (19) and (22) define the unique solution of the
313
agent’s problem (2) - (4) given the specifications of the functions in (5) and (6). However, it cannot
314
be determined in explicit form for the general case and in order to advance in its analysis we resort to
315
approximate solutions.
T
316
317
Fig. 2 Evolution of the shadow price of the resource and of the investment over time
318
319
320
Let assume  A  1 , where A can be large or small. Then s (t ),  (t ) and m(t ) are slowly changing
321
functions such that s(t ),  (t ), m(t )  1. In this case we can assume that m( (t ))  m(t ) in
322
equations (18), (19) and (22). It allows to obtain an approximate solution of the evolution of the
323
shadow price of the resource given by
14
u
T
324
 (t )   e


  r   Am (t)u t d
t
Am (u )du 
t
1
1  e r   Am (t )T t  

  r Am (t ) 
(28)
325
Equation (28) comply with the transversality condition  (T )  0 as t approaches T . From equation
326
(8) we know that switching from investment to no investment occurs at
327
328
cn
1
1  e  r   Am (t* ) T t*   .
  (t * ) 

g0
  r Am (t * ) 
Consequently, we can determine the optimal switching time. It is given by
 c

1
t *  T  ln 1  n    r Am (t * )  
 *
 g0
 r   Am (t )
329
330
and the optimal investment pattern by
331

 cn

1
 *
, no investment
   ln 1     r Am (t )  
 *
 g0
 r   Am (t )


 c

1

T  t *    ln 1  n    r Am (t * )  
, switching time
 *
g
r


Am
(
t
)
0




1
   ln 1  cn    r Am (t * )  
, investment.
 *

 g0
 r   Am (t )

332
The closer the expression 1 
333
the period of no investment and consequently the less important is the holdup problem.
(29)
(30)
cn
  r Am (t * )  is to one the closer is t * to T , i.e. the shorter is

g0
334
335
Observation 2 (on the moderation of the holdup problem)
336
Low cost for investment, cn , a large effect of the investment on the resource quality, g 0 , a high
337
productivity of the input Am or a low value of the deterioration rate  moderate the holdup
338
problem.
339
340
Proof: by inspection of equation (30). 
341
342
Observation 3 (on the evasion of the holdup problem)
343
The holdup problem does not arise if the principal offers the agent a new contract of length T at time
344
t * . The new contract starts at T  t * and has a duration of T.
15
345
Proof: Agents will not change their investment behavior at time t * if the contract is renewed at this
346
point in time since the time period T  t * of the current contract is completely covered by the time
347
period 0  t * of the new contract. Thus, for overcoming the holdup problem the principal has to give
348
advance notice of the renewal of the contract of T  t * . Equation (29) shows that the period of
349
advance notice is given by
 c

1
T  t *   ln 1  n    r Am (t * )  
.
 *
g
r


Am
(
t
)
0


350
(31)
351
The principal can calculate the advance notice period for a given T or for a preferred renewal interval
352
t * of the contract by adjusting the contract length T . The result is a sequence of overlapping contracts
353
with renewal frequency t * and advance notice time T  t * . 
354
355
Equation (29) is highly policy relevant since it provides a concrete help for overcoming the holdup
356
problem as decribed in observations 2 and 3. The origin of the holdup problem is the fact that the
357
principal cannot commit to compensate the agent for his investment. The principal cannot honour any
358
commitment because the investments of the agent are not verifiable by a third party. Changes in the
359
resource quality may be the result of the investment behavior of the agent but they also be the result of
360
external influences (for example the weather or market influences). Hence, the principal cannot
361
deduce the investment behavior of the agent from the evolution of the resource quality. For this
362
reason, although the quality of the resource is observable, it is not contractible. Advance notice
363
however is verifiable by both parties and consequently the principal can also commit to give advance
364
notice. The time dimension of the relationship can therefore form an integral part of a contract.
365
366
Observation 4 (on the sustitution of the long-run optimum by a sequence of short-term
367
contracts)
368
(i) If t *  0 the long-run optimum cannot be achieved by a sequence of contracts that are renovated at
369
time t * . (ii) If t *  0 it is possible to achieve the long-run optimum by a sequence of contracts that
370
are renovated at time t * .
371
Proof: If the length of the contract T is relativelyy short the optimal value of t * is equal to zero.
372
Hence, it is optimal for the agent not to invest at all and consequently there does not exist a sequence
373
of short-term contracts that approximate the optimal long-run contract. As soon as the value of t * is
374
strictly positive agent starts investing and a sequence of short-term contracts of length T renovated at
375
time t * can be employed to establish the long-run optimum.
376
16
377
Provided that certain conditions are fulfilled Rey and Salanie obtain for the case of symmetric and
378
asymmetric information (moral hazard (1990) and adverse selection (1996) similar conclusions as in
379
part (i) of observation 4. Although the model presented here does not consider moral hazard or
380
adverse selection, principal and agent do not have identical information. Both have information about
381
the quality of the resource but not about the agent’s investment behavior. The findings of this paper
382
emphasize the importance of the length of the short-term contracts so that the long-run optima can be
383
achieved. In other words the sequence of short-term contract has to qualify before it can serve as a
384
substitute for the long-run optimum. Che and Sákovics (2004) postulated that dynamic investment
385
alone and not the signature of contracts can solve the incentive problem of the holdup problem – at
386
least asymptotically. Their model is based on “contribution games” and as such difficult to compare
387
with ours. Yet, our findings suggests that dynamic alone does not provide a solution to the holdup
388
problem and contracts are necessary for overcoming it.
389
390
Observation 5 (on the length of the no investment period)
391
If the productivity of the resource is high the length of the no investment period hardly changes with
392
the length of the contract. In the opposite case the length of the no investment period varies with the
393
length of the contract.
394
Proof: Equation (30) shows that a high productivity Am has nearly no effect on the difference of
395
T  t * . Consequently changes in the length of the contract lead to changes in the optimal switching
396
time but not a change in the length of the no-investment period. 
397
 r   Am

 c

(t * ) T  t *    ln 1  n    r Am (t * )  
 g0

398
Writing
399
illustrating the determination of t * as a function of the parameter values – see Figure 3. Suppose that
400
the absolute value of ln 1 
401
of T  t , investment will not take if r   Am (t * )(T  t * ) is below or equal to the level of B. For
402
the case of Figure 3 any value of T  t *  2 leads to no-investment. The no investment period is
403
T  t * years long and the period of investment lasts t * years.
404
In case that the contract is not renewed the holdup problem still exists.
405
Fig. 3 Investment failure as a function of the contraction duration and the parameter values for t *  0
equation
(31)


as
allows

cn
  r Am (t * )   is equal to the level of point B. Than for any value

g0

*
17
406
407
With respect to the evolution of the resource we depart from equation (18) and obtain for the
408
investment and no-investment period
409
s(t )  s0e  Am
410

(t )t

s(t )  s0e  Am

(t )t

g0 n 
1  e  Am (t )t  , for 0  t  t *


 Am (t ) 



g0 n 
1  e  Am (t )t *  e  Am (t )(t t *) , for t*  t  T .



 Am (t )
(32)
(33)
411
For the trajectory of the input m we obtain
412
m1 (t ) 
413
The duration of the contract is long (T large)
414
The final case is where the contract duration is large, i.e. let T  1 and for the values of t  [0, T ]
415
we assume that t  0 and T  t  0. Hence, by equation (32) we obtain
416



 r   Am ( t ) T t   
g0 n 
A
A
s0e   Am (t )t 
1  e  Am (t )t   . (34)
1    s(t )  1  e 



 
cm
cm 
 Am (t ) 
s(t ) 
g0 n
 Am (t )
(35)
18
417
and by equation (34) and (35)
418
m1 (t ) 
419
Hence, we observe that
420
 g0 n
 g0 n
, or m 
(1   )

cm  m (t )
cm 
m(t )  mˆ , s(t )  sˆ and  (t )  ˆ
(36)
(37)
421
while t  t * and deviate thereafter.The evolution of the stock of the resource and its associated
422
shadow price is depicted in Figure 4 and 5 respectively.
423
Fig. 4 Evolution of the resource over time
424
425
426
427
Fig. 5 Evolution of the shadow price of the resource over time
19
428
429
430
Using equations (32) and (34) we consider two solutions, denoted by s1 (t ) and s 2 (t ) , with
431
corresponding initial values for the resource, s01  sˆ and s02  sˆ .Moreover we can write equation
432
(32) as
433
s(t ) 


g0 n
g0 n 
 e  Am (t )t  s0 

 Am (t )
 Am (t ) 

434
435
436
Fig. 6 Evolution of the resource and the corresponding shadow price over time
(38)
20
437
438
439
440
441
The trajectories of s(t ) and  (t ) are depicted in the Figure 6 . For the case where s01  sˆ we find that
g0 n
gn
 0  . Hence the sum of the two terms in equation (32) decreases and approaches

 Am(0)
 Amˆ
gn
g0 n
gn
sˆ  0  . For the case where s02  sˆ we see that
 0  so that the sum of the two

 Amˆ
 Am(0)
 Amˆ
g0 n
from below while 0  t  t * . At

 Amˆ
442
terms in equation (32) increases and approaches sˆ 
443
t *  t  T , s (t ) decreases exponentially in both cases. In contract the first best solution is
21
444
445
n(t )  n , t  [0, T] and the corresponding trajectory of s (t ) approaches asymptotically ŝ in both
cases.
446
447
“Periodic” solution of the agent problem.
448
In the original problem (2) - (4), the final value s(T) of the optimal s(t) is the initial value s(T) for the
449
next interval [T, 2T]. Since we are interested in a sustainable solution of the original principal-agent
450
problem (2) - (4), we need to find the “perfect” initial condition that satisfies s(0)= s(T). By (18), the
451
ideal initial value s0 that produces a sustainable solution in the agent problem (2) - (4) is determined
452
from
T
453
A m
s0e 0

( ) d 
u
t
A m
 s0  g0  e 0
*

( ) d 
0
n du
(39)
454
One may think that the principal may overcome the holdup problem if he/she restores the asset at the
455
beginning of the contractual relationship so that the agent’s initial condition for the stock is given by
456
s (0)  sˆ , i.e. it is given by the steady state value. Yet, the first-best solution ( mˆ , sˆ) given in (14) and
457
(16) corresponds to the case where t *  T which will never be optimal for the agent if there is no
458
advance notice of the renewal of the contract. If s0 
459
is characterized by

sˆ0

s(t )     Am (t ) (t  t* )

 sˆ0e
460
461
464
0  t  t*
t*  t  T
.
(40)
The switching instant is found from
  Amˆ  T t *  
cn
1
   t *   1  e

g0
 
462
463
g0 n
 sˆ0  sˆ then the evolution of the stock
 Amˆ 
(41)
which shows that
t*  T 
  cn 
1
ln
1 

 Amˆ  
g0 
with g 0   cn
(42)
465
The smaller are the costs cn or the smaller are deterioration rate of the resource  , or the bigger is
466
the improvement effect of the investment g 0 the closer the ln is to zero, i.e. the optimal switching
467
time t * is close to T . The later effect is less strong for  since it also appears in the denominator of
22
1
. Yet for the parameters related to production, A and  , one can conclude that the
 Am
468
factor
469
optimal switching time t * comes closer to T if the productivity increases.
470
Equation (40) shows that the stock is maintained over the interval of time  0, t * but thereafter it
471
decreases so that s  0  s T  . Hence, the choice of the initial value s (0)  sˆ does not lead to a
472
sustainable use of the resource.
473
474
Observation 6 (on the maintance of the resource)
475
The principal can maintain the resource by the choice of the contract length T and the initial state of
476
the resource s0 .
477
478
Proof: Provided t * is sufficiently large so that ŝ is reached a periodic solution is obtained if we
479
choose the value s (0) such that it corresponds to the accumulated loss of the resource once the agent
480
stops investing. In other words it should matches the difference between the steady state value
481
sˆ 
g0 n
and the terminal value s (T ) . The value of s (0) is given by
 Amˆ 

ˆ   Am
s(0)  s0  se
482
483
484
(T t *)
.
(43)
For the initial condition of (43) the dynamics of the stock yield
s (t ) 
g0 n

gn
  s0  0 

 Amˆ
 Amˆ

   Amˆ  (T t *)
.
e

485
Albeit the possibility of the principal to maintain the resource over time by the appropriate choice of T
486
and s0 at the beginning of the contractual relationship, the periodic solution is not optimal since it
487
does not replicate the first-best solution.
488
After having analyzed the agent’s problem we now turn to the decision problem of the principal.
489
490
4. Analysis of the principal’s problem
491
The problem of the principal is given by the equations (1) - (4) and consists of finding the optimal
492
value T  0 and the function R  t   0,   , that maximizes net income I . Since the principal can
493
choose R it is possible to extract the entire cooperative benefits and the optimal R(t), t[0, ∞), will be
23
494
a corner solution along the inequality-constraint B(m(t ), s(t ))  cm m(t )  cn n(t )  R(t )  u0  0 of
495
(4),
496
considered in (5) and (6) by
i.e., the optimal is given by R(t )  B(m(t ), s(t ))  cm m(t )  cn n(t )  u0 ,
or for the case
R(t )  Am (t ) s (t )  cm m(t )  cn n(t )  u0 , t[0, ∞).
497
(44)
498
Thus, if the solution of the agent’s problem is known it is straightforward finding a sustainable
499
solution of the agent’s problem. We discuss three cases. The first case considers the first-best, the
500
second case the finite horizon and the third case the periodic solution.
501
502
First best solution (steady state solution)
503
Given a steady state solution the agent’s problem (2) - (4) has a unique constant sustainable
504
ˆ , sˆ  , the optimal sustainable rent R̂ over [Tj, T(j+1)]
trajectory ( m̂ , ŝ ). Along the this trajectory  m
505
is also constant and is the same for all periods j:
506
ˆ  sˆ  cmm
ˆ  c´n n  uo
Rˆ  Am
(45)

1
1
  1 11
  sˆ  cm mˆ  cn n  u0 .
 cm 
507
Rˆ  A
508
ˆ , sˆ) is
In this case, the total discounted principal’s profit (1) along the sustainable trajectory ( m
509


T ( j 1)

Rˆ V (T )
V (T )
Iˆ(T )  Rˆ  
e  rt dt   e  rTjV (T )  Rˆ  e  rt dt 


Tj
0
1  e  rT
r 1  e  rT
j 0
j 0
(46)
(47)
510
and the maximum of Iˆ(T ) is reached at the same value T* as the minimum of the function V(T).
511
Thus, the solution of the original principal-agent problem (1) - (4) is given by
T *  argmax V T  and R*  Rˆ .
512
513
514
Finite horizon and periodic solution.
515
In the case of a finite horizon and periodic solution the problem (1) - (4) has the unique sustainable
516
~ (t ) , t[0,T], given by (41). Correspondingly,
trajectory determined by (40) and slowly changing m
517
the path of the optimal sustainable rent (44) over [Tj, T(j+1)] is the same for all periods j and is also a
518
slowly changing function R(t ) . It is described by:
519
R(t )  Rˆ   (t ) .
(48)
24
520
The term  (t ) denotes the reduction in net benefits as a result of the divergence of the agent’s
521
solution from the first best solution. The reduction in net benefits is given by
522
ˆ  sˆ  Am(t ) s   cm  m
ˆ  m(t )   c n  nˆ  n(t )  .
 (t )  ˆ   (t )   Am
523
Observation 7 (on th share of the cooperative benefits)
524
525
The principal is the residual claimant of the variations of the net benetfits over time. The agent is
always left just with the reservation utility.
526
Proof: According to Figure 6, if s0  sˆ , the resource and consequently also the net benefits increases
527
over the time interval  0,t *  and decrease over the interval t * , T  . If s0  sˆ the resource decreases
528
529
over time. Consequently the principal adjust the rent so that if follows the same pattern in order to
extract the benefits that exceeds the reservation utility. 
530
531
This result points to a difference between the findings related to principal agent problem as discussed
532
in the literature (Laffont and Martimort 2002) and observation 7. A standard result in this literature is
533
that the agent is fully incentived as a residual claimant if the rent is a fixed amount and does not
534
depend on the output or net benefits. Yet, observation 7 states the opposite result. This difference can
535
be explained by the fact that literature does not consider a stock variable and relationship between the
536
actions of the agent and the net benefits is stochastic. Hence, one finds variation of the net benefits at
537
each moment of time that can be claimed by the agent since he/she just has to pay a fix amount for the
538
use of the resource. However, since our model does not consider moral hazard but the holdup problem
539
the variation over time is not stochastic and the principal can adjust the rent over time in order to
540
extract any additional benefit.
541
One may think that the agent may act strategically. Knowing that he/she is always left just with the
542
reservation utility he/she may decide not to invest. Yet, the principal has anticipated this behavior by
543
adjusting the rent so that the agent only obtains the reservation utility if he/she invests. Otherwise
544
he/she will obtain lower net benefits. Therefore the adjustment of the rent over time is the principal’s
545
instrument that forces the agent to invest until time t * . Beyond this point of time the instrument is
546
not effective anymore and the principal need to adjust the rent with the objective to meet participation
547
constraint.
548
549
Let us denote the principal’s discounted total rent over the contract of duration T as
550
J (T )   e rt R(t )dt
T
0
(49)
25
551
Along the periodic solution the dynamics of the rent R  t  over Tj , T  j  1  will be the same for
552
all j. Therefore, we will obtain from (1) that
553
I (T )   e rTj J (T )  e rTjV (T )   J (T )  V (T )  / 1  e rT 

(50)
j 1
554
The optimal T is determined by
555
dI (T )
  J (T )  V (T )    J (T )  V (T )  re rT / 1  e  rT   0
dT
556
which implies that
557
J (T )  V (T )  I (T )re rT
558
The exact dynamics of J T  will depend on the optimal behavior of the agent. The optimal duration
559
of the contract is obtained when the net benefits from an extension of the contract by one year is equal
560
to the interest paid on the present value of the contract. In other words the higher is the discount rate
561
the higher it has to pay to extend the contract by one year. Therefore periods of high discount rates
562
favor shorter contract durations and vice versa for period of lower discount rates.
563
Finally, we calculate the rent for a contract of length T which is not characterized by a periodic
564
565
solution. Let us denote the rent of the first-best solution by R* . For the time period [Tj, T(j+1)] it is
given by
566
R*  t   Am* s*  cmm*  c´n n*  uo   *  uo
567
J j (T )  
568
569
T  j 1
Tj
e  rt R*  t  dt   j (T ) ,
(51)
(52)
(53)
(54)
where  j (T ) relates to the j-th contract and is given by
 j (T )  
T  j 1
Tj
e  rt  *   (t )  dt
570
and thus for the sum of the principal´s rent we obtain
571
I (T )   e rTj J j (T )  e rTjV (T ) .
(55)

(56)
j 0
572
573
Hence, the maximum of I (T ) is reached at


e  rTj J j (T )  V (T )  rj  J j (T )  V (T )  .
(57)
26
574
In other words the optimal value of T depends on the renovation of the contract and is variable.
575
576
Conclusions
577
This study analyzes, within an dynamic principal agent framework, the optimal trajectory of the
578
agent’s investment behavior over time It considers the fact that investment and sharing of benefits are
579
time-wise intertwined and that investments affect the corporative benefits in an accumulative way.
580
Initially it determines the conditions under which the socially and private optimal solutions differ.
581
This difference gives rise to the holdup problem. The study finds that the determination of the optimal
582
contract length allows to moderate the holdup problem but it does not allow to avoid it. Likewise a
583
periodic solution is not able to bypass the holdup problem. However, a sequence of overlapping short-
584
term contract is able to replicate the behavior of the agent that has a long-term contract. In this way it
585
is possible to overcome the holdup problem. The study determines the minimal length of the short-
586
term contract and the length of the overlapping periods. The overlapping itself results from the
587
renewal of the contract before it has expired. Thus, the principal has to give advance notice of the
588
renewal if he/she wants to approach the first-best solution. The adjustment of the rent over time
589
allows the principal to leave the agent only with his/her reservation utility at every moment of time.
590
591
Appendix A
592
593
Proof of theorem 1 (on the steady-state first-best regime).
594
Using (16) in (14) we obtain that
595
mˆ 
596
597
g 0 n
1  ˆ ,
cm 


(58)
Employing this results in (17) we obtain a single equation in ̂ given by

A  g 0 n 

 1  ˆ
r  cm  
598


1
 ˆ
(59)
599
By Theorem 1, we know that the unknown ̂ satisfies ˆ  1 /  . Then, 0  x  1  ˆ  1 and the
600
equation Cx 1  1  x has a unique solution 0  x  1 given by the intersection of the functions

601
1
Cx
A  g 0 n 
and 1  x where C 

 . Obviously from an economic point of view only the case
 r  c1 
27
602
ˆ  0, sˆ  0, is interesting and not the case where m
ˆ  nˆ  sˆ  0 . Thus, it makes sense to
where m
603
concentrate on the condition ˆ  cn g0
604
605
References
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607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
Aghion, P., M. Dewatripont, et al. (1994). "Renegotiation Design with Unverifiable Information."
Econometrica 62(2): 257-282.
Aghion, P. and J. Tirole (1997). "Formal and Real Authority in Organizations." Journal of Political
Economy 105(1): 1-29.
Baker, G., R. Gibbons, et al. (2002). "Relational Contracts and the Theory of the Firm." The Quarterly
Journal of Economics 117(1): 39-84.
Board, S. (2011). "Relational Contracts and the Value of Loyalty." American Economic Review 101(7):
3349-3367.
Che, Y.-K. and J. Sákovics (2004). "A Dynamic Theory of Holdup." Econometrica 72(4): 1063-1103.
Evans, R. (2008). "Simple Efficient Contracts in Complex Environments." Econometrica 76(3): 459.
Felli, L. and K. Roberts (2011). Does Competition Solve the Hold-up Problem?, London School of
Economics STICERD.
Grossman, S. J. and O. D. Hart (1986). "The Costs and Benefits of Ownership: A Theory of Vertical and
Lateral Integration." Journal of Political Economy 94(4): 691-719.
Gul, F. (2001). "Unobservable Investment and the Hold-Up Problem." Econometrica 69(2): 343-376.
Guriev, S. and D. Kvasov (2005). "Contracting on Time." American Economic Review 95(5): 13691385.
Halac, M. (2012). "Relational Contracts and the Value of Relationships." American Economic Review
102(2): 750-779.
Jacoby, H. G. and G. Mansuri (2008). "Land Tenancy and Non-Contractible Investment in Rural
Pakistan." The Review of Economic Studies 75(3): 763-788.
Kvaløy, O. (2006). "Self-enforcing contracts in agriculture." European Review of Agricultural
Economics 33(1): 73-92.
Laffont, J.-J. and D. Martimort (2002). The Theory of Incentives. Princeton Princeton University Press.
Levin, J. (2003). "Relational Incentive Contracts." The American Economic Review 93(3): 835-857.
Lichtenberg, E. (2007). "Tenants, Landlords, and Soil Conservation." American Journal of Agricultural
Economics 89(2): 294-307.
MacLeod, W. B. and J. M. Malcomson (1993). "Investments, Holdup, and the Form of Market
Contracts." The American Economic Review 83(4): 811-837.
Rey, P. and B. Salanie (1990). "Long-term, Short-term and Renegotiation: On the Value of
Commitment in Contracting." Econometrica 58(3): 597-619.
Rey, P. and B. Salanie (1996). "On the Value of Commitment with Asymmetric Information."
Econometrica 64(6): 1395-1414.
Williamson, O. E. (1979). "Transaction-Cost Economics: The Governance of Contractual Relations."
Journal of Law and Economics 22(2): 233-261.
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