Surplus and Information in a Firm’s Make-or-Buy Decision∗
Ingela Alger1
Ching-to Albert Ma2
Régis Renault3
February 15, 2008
Abstract
What determines firm boundaries? We propose a hitherto unexplored trade-off between making and buying
for a firm. In our model a firm needs a manager to oversee production: the firm chooses between hiring
a manager as an employee, and contracting with an external production unit run by this manager. The
manager gathers information about production costs and acquires on-the-job experience that will yield
additional surplus in her future career. Under external production, the manager discloses production costs
information to the firm only if she is given proper incentives, whereas the firm may freely access that
information if production is in-house. When selecting external production, the firm trades off the access to
cost information against a better ability to extract the manager’s job experience rent: a manager running
her own business may bear higher losses than if she is employed by the firm, thanks to collateralizable
assets. External production is more profitable when the magnitude of the job experience rent is large or
when production costs are likely to be low. There may be excessive or insufficient external production as
compared to what would be socially optimal.
Keywords and phrases: Theory of the firm, job experience rent, informational rents.
JEL classifications: D23, L22
∗
Preliminary and incomplete.
1
Economics Department, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada;
Ingela Alger@carleton.ca
2
3
Economics Department, Boston University, 270 Bay State Road, Boston, MA 02215, USA; ma@bu.edu
THEMA, Université de Cergy-Pontoise, 33 Bd du Port, 95011 Cergy-Pontoise Cedex, and Institut Universitaire de France, France; regis.renault@eco.u-cergy.fr
1
Introduction
Why do firms exist, and why do they vary in size?4 In the classic microeconomics model the
firm is defined as an exogenously given production function, a fact that Coase (1937) rightly criticized. Early attempts to depart from this black-box approach by identifying variables affecting
firm boundaries in systematic ways, relied on factors such as technology, market power, and externalities.5 Starting in the 1970s the focus shifted towards information-based explanations. For
many years the most prominent theories were based on the hold-up problem, whereby contracting
parties face various kinds of losses due to contract incompleteness. The transaction cost theory,
initiated by Williamson (1975), first emphasized losses in the form of ex post inefficiencies arising
in the bargaining process. Later, however, it also put forward losses in the form of inefficient ex
ante investments (Williamson, 1985); inefficient investments are also central in the property rights
literature, first developed by Grossman and Hart (1986).6
Hold-up theories generally assume that inefficiencies increase with the degree of relationshipspecificity of the assets. Moreover, it is argued that the integration of assets in one single firm
reduces the inefficiencies. Hence, a higher degree of asset specificity should lead to a higher degree
of integration. A large number of empirical studies have found a positive correlation between
relationship specificity and vertical integration (Klein, 2005). However, these studies do not prove
that the correlation is driven by a desire to avoid inefficiencies related to contractual incompleteness.
Recently several authors, including Holmström and Roberts (1998), and Gibbons (2005), have
advocated research efforts beyond theories based on the hold-up problem.
In this paper we present a hitherto unexplored trade-off between making and buying. We introduce an aspect that has been neglected before in this literature, namely, that collateralizable
4
There are many reasons for why we should care about this question. The degree of competition in an
industry may depend on the number of firms, and therefore also the size of firms, with consequences for
prices, output, and welfare. Furthermore, the division of the surplus generated by a firm depends on the
bargaining power of employees, which in turn may hinge on the size of their employer. The costs of collecting
the taxes from the production of a final good, as well as the timing of these taxes, also depends on the number
of firms that were involved in its production. Industry structure may further affect urban structure.
5
See Holmstrom and Tirole (1989), and Joskow (2005) for surveys.
6
For surveys, see Holmström and Tirole (1989), Holmström and Roberts (1998), and Gibbons (2005).
Recent contributions in this literature include Ramey and Watson (2001), and Legros and Newman (2008).
1
assets enable their owners to reduce credit constraints.7 Our theory also builds on earlier ideas by
including a measure of relationship specificity, as well as uncertainty—both of which Williamson
(1985) saw as key features of contractual relationships between and/or within firms—but it disregards inefficiencies linked to the hold-up problem.8 The objective is twofold. First, we ask whether
a firm would rather make an intermediate good itself, or buy it from an outside firm. Second, we
analyze whether this choice is socially optimal.
Our approach is simple. We take two basic laws, one governing asset ownership, and the other
governing employment and non-employment contractual relations, and study their implications for
the costs and benefits of producing a good within a firm, versus buying it from a subcontractor.
Throughout we define a firm as the collection of assets which are owned by the same set of individuals, together with these owners. This is consistent with the legal definition, “a corporation
is a legal person that may own property, but a division or branch of the corporation may not”
(Iacobucci and Triantis, 2007, p.518), and with the property rights literature (Grossman and Hart,
1986).
First, we note that a “property right is the exclusive authority to determine how a resource is
used” (Alchian, 2002). In particular, the owner of an asset decides whether it should be used as
collateral9 to raise credit. Moreover, theory suggests, and empirical studies confirm (refs needed)
that the use of collateral helps individuals and firms gain greater access to credit, presumably due
to information asymmetries between lenders and borrowers. Thus, the first key assumption in our
theory is that, ceteris paribus an individual who owns an asset faces less severe liquidity constraints
than an individual who does not.
7
There is substantial empirical evidence that a firm’s fraction of tangible assets to total assets is positively
correlated with its leverage; see, e.g., Rajan and Zingales (1995).
8
Some other authors have proposed theories sharing the latter feature. Crémer’s (1995) model suggests
that a firm may prefer to buy a good from a subcontractor rather than making it, because it enhances its
ability to commit to carry out punishments in case of a bad outcome. Crémer, Garicano and Prat (2007)
argue that an organization must develop an internal language to enable precise communication about the
various production processes it handles, and that developing a precise language is costly. According to their
theory the synergy gains from including a larger number of production processes within a single firm must
be traded off against costs in the form of a lower language precision.
9
At http://legal-dictionary.thefreedictionary.com, collateral is defined as “property pledged to secure a
loan or debt, usually funds or personal property.”
2
Turning now to laws governing employment and non-employment contractual relations,10 “every
employee accepts an implied duty to ‘yield obedience to all reasonable rules, orders, and instructions
of the employer’ (53 American Jurisprudence (2nd) §97; Restatement of Agency (2nd), §§2, 220.
385)” (Masten, 1988, p.185; see also Harvey, 2003). Furthermore,
[c]omparison of commercial and employment law also provides support for the informational advantage commonly attributed to internal organization [...]. In commercial
transactions, laws regarding the transfer of information are fairly liberal. As a rule,
‘one party to a business transaction is not liable to the other for harm caused by his
failure to disclose to the other facts of which he knows the other is ignorant and which
he further knows the other, if he knew them, would regard as material in determining
the course of his action in the transaction in question’ (Restatement (2nd) of Torts, §51;
also see Restatement of Contracts (2nd), §303). [...] (Masten, 1988, p.186)
By contrast, an employee
is obliged ‘to communicate to [his employer] all facts which he ought to know’ (56 CJS
67 [...]). Again, the law distinguishes between employment and commercial transactions
in a way that apparently supports [...] superior access to information [...]. (Masten,
1988, p.186)
This leads us to our second key assumption: the owner of an asset is also the owner of any
information pertaining to it.
We incorporate these assumptions in a very simple model, with one retailer and one production
manager. The manager oversees the production of a good (it may either be the final good that the
retailer sells, or an intermediate good that the retailer uses in the assembly of the final good), which
requires some productive asset. There is one source of uncertainty: the true cost of production
is not known ex ante. The production of the good generates benefits for the agents as follows:
the retailer obtains sales revenues, and the manager acquires experience, which may have market
10
To illustrate the model we rely on features in the U.S. legal system, but we will argue below that these
features are present in many other countries, too.
3
value. This market value is modeled by way of a parameter β: a higher β indicates that the
experience has a higher market value. We argue that this value is determined both by the degree of
relationship specificity, and by the thickness of the market in which the production manager may
sell his experience.
First, the market value of the manager’s production experience depends on the extent to which
the production that the manager oversees is relationship specific: for instance, if the production
involves an intermediate good, the production experience is all the more marketable if the good may
be used by other firms than the retailer. As an illustration, in his study of component production
in the aerospace industry, Masten (1984) identifies “electrical piece parts” as being standard (i.e.,
potentially used by many firms), and “circuits designed to individual specifications” as highly
specialized (i.e., used by one firm only). But the potential use by many other firms only defines
the potential market for the manager to market his experience: clearly, a second factor that would
affect the value of the manager’s production experience is the actual thickness of this market.11
We consider two different organizational forms, in-house production and procurement. With
in-house production, the retailer owns the productive asset, and hires the production manager as
an employee. In this setting both the retailer and the production manager observe the realized
production cost (it is reasonable to assume that the manager always observes this cost, by virtue of
his involvement in the production process). With procurement, the production manager owns the
productive asset, and the retailer buys the good from the production manager. In this setting only
the manager observes the realized production cost. Furthermore, with procurement the manager
may use the asset as collateral, and therefore faces a more relaxed credit constraint than when the
retailer owns the production asset: we model this by assuming that with in-house production there
is a lower limit on the transfer from the retailer to the manager.
For each organizational form there is a contract that specifies a transfer from the retailer to
the manager, as well as a quantity to be produced. We solve for the contract that maximizes the
retailer’s expected profit, and that ensures the manager’s voluntary participation. Each organi11
We follow McLaren (2003): “Define a rise in market thickness as any increase in the effective number
of firms in a given market, in the sense that there is an increase in the probability that any given agent will
be able to find in a given length of time an agent with whom it will be possible to realize gains from trade.”
(p.328)
4
zational form then imposes different additional constraints. With in-house production there is a
lower limit on the transfer from the retailer, due to the manager’s restricted access to credit. This
minimum payment constraint implies that the retailer may not fully extract the manager’s experience rent. With procurement this minimum payment constraint is absent; instead, the retailer
faces a classic information extraction problem, and must therefore leave an informational rent to
the manager.
The trade-off between making and buying that appears in this model is then as follows: while
in-house production allows the retailer to economize on informational rents, it limits his ability to
extract the manager’s experience rent. We use the analysis to determine whether the retailer would
prefer to make or to buy the good, and we obtain three sets of results.
First, our theory predicts that ceteris paribus the retailer prefers to procure the good if and
only if β, the parameter measuring the value of the manager’s experience, is large enough. This is
consistent with the large empirical literature on specificity, starting in the 1980’s with Monteverde
and Teece (1982). For instance, in his study on the production of aerospace components referred
to above, Masten (1984) finds that standard components (i.e., components that may easily be used
by many other firms) are more likely to be bought, while highly specialized components are more
likely to be produced in-house. Furthermore, in their analysis of data from the U.S. auto industry
Masten et al. (1989) report that a higher degree of human capital specificity is correlated with a
higher degree of in-house production. See also numerous other references cited in Klein (2005).
This prediction is also consistent with a smaller literature on the effects of market thickness on
vertical integration. For instance, using a large dataset on U.S. manufacturing firms, Holmes (1999)
establishes that a plant situated in an area with a high own-industry employment has a significantly
higher proportion of purchased inputs than a similar plant located in an area where the employment
in the same industry is low; see also Pirrong (1993) and Hubbard (2001) for analyses of shipping
industries.
Second, our model predicts that, ceteris paribus, a higher degree of cost uncertainty makes
in-house production more attractive relative to procurement, due to the larger informational rent
involved with procurement. This is consistent with a change in the way Toyota obtained electronic
5
car components. According to Ahmadjian and Lincoln (2001) Toyota used to purchase 70% of
its electronic car components from one independent supplier (called Denso), but this figure had
declined to 50% by the end of the 1990’s. The following excerpts from Ahmadjian and Lincoln
(2001) suggest that this is consistent with our model: for most auto parts “an auto assembler [has]
access to a supplier’s cost structure and understand intimately its manufacturing process. (. . . )
[K]nowledge asymmetries between customer and supplier posed few problems when the technology
behind the parts never strayed far from the assembler’s core knowledge base. (. . . ) As electronics
technology grew more complex and integral to automotive design and manufacturing, information
asymmetries increased between Toyota and Denso. (. . . ) Toyota was candid in interviews with us
and with the Japanese press in saying that one factor in motivating its decision to manufacture
electronics components was an interest in boosting bargaining leverage over Denso with a firm grasp
of Denso’s real costs.” (p.688)
Third, we obtain predictions related to the tangibility of assets. In our model, ceteris paribus,
a production process involving collaterizable assets is more likely to be outsourced to a separate
firm than one involving no or few such assets.
In the second part of the paper we ask whether the retailer’s decision is socially efficient. We
address this question by doing the following thought experiment: given the contracts that the
retailer would offer to the manager conditional on organizational form, would social surplus be
higher if the retailer hired the manager as an employee, or as an external contractor? We find
that sometimes the retailer chooses to make when it would be socially efficient to buy, and that
sometimes it is the reverse.
The paper is structured as follows. In the next section we describe the model and the first best.
We then derive the optimal contract conditional on the retailer buying the good from an outside
manager, followed by the optimal contract conditional on the retailer producing the good. In the
following section we use these results to determine whether the retailer prefers to make or to buy
the good. Finally, before concluding we analyze whether the retailer’s preferred organization mode
is socially efficient.
6
2
The Model
A principal would like to produce some outputs and this must use the service of an agent. We
consider two cases. In the first case, the agent is the principal’s employee, and we call him the
inside manager; in the second, the agent is an outsider working for the principal on a contract, and
we call him the outside manager. We can imagine under these two cases that, respectively, the
principal owns some asset which an employee is required to work with for production, while the
outside manager owns the required production asset. We use the variable q to denote the output,
and its price is normalized at 1. The output q is verifiable and contractible, and the principal
always retains output ownership.
There is uncertainty about the cost of producing output q. This uncertainty is described by
the random variable α distributed on a support [α, ᾱ], with α > 0, and distribution and density
functions F and f , respectively. The cost of producing q units of output is αc(q), where c is a
twice continuously differentiable, strictly increasing and strictly convex function, with c(0) = 0.
We assume that f (α) > 0 for all α ∈ [α, ᾱ]. We define the function h by h(α) ≡
F (α)
f (α) ,
and assume
that it is strictly increasing.
A manager gets to observe the cost parameter α. For the principal, the first difference between
using an inside manager and an outside manager is that an inside manager must truthfully reveal
this information. The employment relationship entitles the principal to have access to whatever
information the manager gets to observe; one can imagine that an internal accounting or audit
system is in place for discovery. On the other hand, if the principal uses an outside manager, the
cost parameter remains the private information of that manager; the principal simply is not privy
to that information.
The principal is risk neutral, and her utility is the output less any payment she has to make to
cover production cost or a manager’s compensation. We let a manager assume the production cost
initially. The principal later covers this cost, either through a compensation to the inside manager
or a contractual payment to the outside manager.12
12
We could let the principal assume the production cost if she hires the inside manager, but since the
principal knows the cost parameter, this alternative is equivalent.
7
The preferences of the inside and outside managers are identical. Both are risk neutral, responsible for the production cost, and benefit from any payment from the principal. In addition, each
manager values output according to a known parameter β ≥ 0, and this is common knowledge. If t
denotes a transfer net of production cost from the principal to the manager, a manager’s utility is
βq − t. The parameter β measures the manager’s private benefits from each unit of output. This
includes enjoyment, any reputation or investment future return from producing output q. The
linear specification conveniently captures the idea that this return is increasing in the manager’s
performance in the current project with the principal. Each manager has a reservation utility
U ≥ 0.
For the principal, the second difference between using an inside manager and an outside manager
is a minimum income constraint. The inside manager’s (net) monetary compensation must be at
least M , where 0 ≤ M ≤ U ; the outside manager does not have a minimum income constraint.
Each of the managers has the reservation utility U that may include the future returns from the
experience with the principal. The financial constraint for the inside manager may be interpreted
as a minimal financial obligation, but the inside manager has inadequate access to the credit market
so that he must use the income from the principal to fulfill his financial obligation. The outside
manager presumably owns more assets, and does not face any such constraint.
Next we define the contracts that the principal may offer a manager. A contract is denoted
by C ≡ {[q(α), t(α)], α ∈ [α, α]}: the principal specifies that q(α) is the output and the manager
receives t(α) when the cost parameter is α. If an inside manager is used, the principal and the
manager have perfect information about the cost parameter α so the above is well defined. If
an outside manager is used, the manager possesses private information about α. Without loss
of generality, we let a contract be a direct revelation mechanism where the production level and
compensation are functions of the manager’s report on α, and where the manager optimally reports
the value of α truthfully.
We study two settings. The first is Outside Contractig where the principal uses an outside
manager. The second is Internal Contracting where the principal uses an inside manager. Under
Outside Contracting, the extensive form is:
8
Stage 1: The outside manager observes α, but the principal does not.
Stage 2:
The principal offers a contract C ≡ {[q(α), t(α)], α ∈ [α, α]} (a revelation mechanism)
to the manager, and the manager chooses either to accept or reject it.
Stage 3:
If the manager has accepted the contract C, he reports a value of α and the terms of
the contracts are executed.
Under Internal Contracting, the extensive form is
Stage 1: The inside manager and the principal observe α.
Stage 2:
The principal offers a contract C ≡ {[q(α), t(α)], α ∈ [α, α]} to the manager, and the
manager chooses either to accept or reject it.
Stage 3: If the manager has accepted the contract C, the terms of the contracts are executed.
The two extensive forms exhibit two differences. First, the outside manager obtains private
information about the cost parameter, whereas the inside manager and the principal share the cost
information. Second, the outside manager and the inside manager decide on accepting the contract
using different criteria. Stage 2 of the extensive forms are the participation constraints. Here, the
outside manager accepts the contract if and only if his utility is at least U . The inside manager,
however, accepts the contract if and only if his utility is at least U and he obtains a net monetary
compensation at least M .
We have let the outside manager observe the cost parameter before contracting. Because the
manager is risk neutral, the principal may “sell the firm” to the manager at the first-best (expected)
price if a contract can be offered before the manager acquires any cost information. We wish to
consider the effect of asymmetric information, hence rejecting contracts that are designed before
information acquisition.
We now derive the first-best benchmark; this would be the outcome if the principal were able
to hire an inside manager who did not have a minimum income constraint. Suppose that the
cost parameter is α, and that the principal implements production level q with a transfer t. The
principal’s profit is q −t. The manager’s utility is βq +t−αc(q), which must be at least U . The first
9
best is a pair of q and t that maximize the principal’s profit subject to the manager’s reservation
utility constraint. Clearly, the optimal value of t satisfies t = U − βq + αc(q). Substituting this
value of t to the principal’s objective function, we characterize the first best by the value of q that
maximizes the social surplus:
(1 + β)q − αc(q) − U,
(1)
The social surplus takes into account both the principal’s and the manager’s benefit from the
output, (1 + β)q, as if the cost αc(q) generated a total output (1 + β)q. Given α, let the firstbest quantity be q ∗ (α) that maximizes total surplus (1) (by equating social marginal benefit 1 + β
and marginal cost αc0 (q)), and the first-best transfer be t∗ (α) = U − βq ∗ (α). Observe that the
manager’s benefit βq ∗ (α) would be extracted by the transfer. Because the cost parameter varies
over the interval [α, α], we can compute the first-best expected profit, and express it as a function
of β and U :
(2)
π ∗ (β, U ) =
Z
α
[(1 + β)q ∗ (α) − αc(q ∗ (α))]f (α)dα − U.
α
It will be useful to define a function q̃ as follows:
(3)
q̃(x) = arg max q − xc(q),
or
q
The first best q ∗ (α) can be written as q̃
α
1+β
c0 (q̃(x)) =
1
.
x
. The manager’s benefit parameter β alternatively
can be thought of as reducing the cost parameter α.
Because the second-order cross partial derivative of q − xc(q) is −c0 (q) < 0 from the strict
convexity of c, q̃ is strictly decreasing in x.
3
Optimal Contract for Outside Manager
Here we characterize the optimal contract for the outside manager. An optimal contract maximizes
the principal’s expected profit subject to the constraint that the outside manager obtains at least
the reservation utility U , and reports α truthfully:
Z
(4)
ᾱ
[q(α) − t(α)]f (α)d(α)
max
q(·),t(·)
α
10
subject to the manager’s participation constraints
βq(α) + t(α) − αc(q(α)) ≥ U
(5)
∀α ∈ [α, ᾱ]
and incentive constraints
(6)
βq(α) + t(α) − αc(q(α)) ≥ βq(α̂) + t(α̂) − αc(q(α̂))
∀(α, α̂) ∈ [α, ᾱ] × [α, ᾱ].
The method for solving for the optimal contract is by now well-known (see for example Laffont
and Martimort, 2003). The difference between the optimal contract under asymmetric information
and the first best is Myerson’s “virtual cost” adjustment. The manager’s information rent leads
to an adjustment of the manager’s cost function from α to α + h(α). We state the following
proposition, but omit its proof.
Proposition 1 The optimal contract for the outside manager is the quantity-transfer pair
(q o (α), to (α)) where
• the quantity is q o (α) = q̃
α+h(α)
1+β
• the transfer is to (α) = U +
R ᾱ
α
,
c(q o (x))dx.
From Proposition 1, the principal’s expected profit π o and the manager’s expected rent Ro may
be written as:
πo =
Z
ᾱ
[(1 + β)q o (α) − (α + h(α)) c (q o (α))] f (α)dα − U
α
Ro =
Z
ᾱ
h(α)c (q o (α)) f (α)dα,
α
where we have emphasized that the expected profit depends on β and U , while the expendted rent
depends on β.
The next proposition reports some comparative statics results which we will use later.
Proposition 2 In the optimal contract for the outside manager,
• the principal’s expected profit is increasing and convex in β, and decreasing and linear in U ;
11
• the manager’s expected rent Ro is increasing in β.
Proof: From the Envelope Theorem
dπ o (β)
=
dβ
Z
ᾱ
q o (α)f (α)dα,
α
which is positive. Furthermore, because q o (α) = q̃
α+h(α)
1+β
, q o is increasing in β. Furthermore
dπ o (U )
= −1.
dU
The outside manager’s expected rent is increasing in β because the optimal quantity q o is
increasing in β, and the cost function c is increasing.
4
Q.E.D.
Optimal Contract for Internal Manager
In this section we characterize the optimal contract if the principal uses the internal manager.
Contracting will be under full information, but the internal manager must receive a minimum
income from the principal as well as the reservation utility.
4.1
The optimal contract
A contract [q(α), t(α)] must satisfy the minimum income and reservation utility constraints, respectively t(α) ≥ M , and βq(α) + t(α) ≥ U , α ∈ [α, ᾱ]. Given a contact [q(α), t(α)], the principal’s
R ᾱ
payoff is α [q(α) − αc(q(α)) − t(α)]f (α)d(α). An optimal contract is one that maximizes the
principal’s payoff subject to the minimum income and reservation utility constraints. Pointwise
optimization can be used to solve for the optimal contract: for each α ∈ [α, ᾱ], choose (q, t) to
maximize
(7)
q − αc(q) − t
subject to
(8)
βq + t ≥ U
12
t ≥ M.
(9)
In the first best, only the reservation utility constraint (8) is relevant. The first-best quantity
α
is q ∗ (α) = q̃ 1+β
, and the corresponding transfer is t∗ (α) = U − βq ∗ (α). From the properties
of q̃, the first-best quantity q ∗ (α) is decreasing in the cost parameter α; low values of α mean low
marginal costs, and the efficient quantities are higher. As the cost parameter α increases, the firstbest quantity decreases, and the necessary transfer must increase to satisfy the resevation utility
constraint.
For very low values of the cost parameter α, the first-best transfer is so small that the minimum
income constraint becomes violated. That is, when α is small, most of the manager’s compensation
for fulfilling the reservation utilility constraint in fact derives from the output. As a result, the
first best is infeasible for low values of α. The minimum income constraint must bind while the
reservation utility constraint becomes irrelevant.
At very high values of α, the first-best quantity is so small that most of the compensation
needed to satisfy the reservation utility constraint must come from the transfer. Therefore, the
minimum income constraint is satisfied and only the reservation utility constraint binds. The first
best is feasible for very high values of the cost parameter α.
What about intermediate values of α? Here, the reservation utility and minimum income
constraints bind simultaneously. The cost parameter α is still low enough so that the first-best
transfer cannot satisfy the minimum income constraint. Nevertheless, the value of α is not so high
that the principal would want to reduce the quantity (and simultaneously raise the transfer) to
relax the minimum income constraint. Obviously when both constraints bind, the set of feasible
contracts degenerates into a singleton (q, t) = ( U −M
β , M ), which is the solution of (8) and (9) as
equalities.
The next proposition describes the optimal contract for the internal manager. It refers to two
threshold values α̂ and (1 + β)α̂, where α̂ is defined by:
U −M
0 U −M
(10)
= 1, or equivalently
= arg max q − α̂c(q) = q̃(α̂).
α̂c
q
β
β
(Recall that when both constraints are binding, (q, t) = ( U −M
β , M ).) The lower threshold α̂ is when
the reservation utility constraint begins to bind, while the upper threshold (1 + β)α̂ is when the
13
minimum income constraint begins to be slack. For values of α between these thresholds, both
constraints bind. Figure 1 shows the graph of the optimal quantity as a function of α for M = 0,
U = 1, β = 1, and c(q) = q 2 /2.
Proposition 3 For any M < U and β > 0 the optimal contract [q i (α), ti (α)] for the internal
manager (which maximizes (7) subject to the constraints (8) and (9)) is the following:
• For α < α̂, the minimum income constraint (9) binds and the reservation utility constraint
(8) does not. The solution is [q i (α), ti (α)] = [q̃(α), M ].
• For α ∈ [α̂, (1 + β)α̂] both constraints bind. The solution is [q i (α), ti (α)] = [ U −M
β , M ].
• For α > (1 + β)α̂ the reservation utility constraint (8) binds and the minimum income con
α
α
, U − β q̃ 1+β
].
straint does not. The solution is [q i (α), ti (α)] = [q̃ 1+β
Figure 1 illustrates Proposition 3. When the cost parameter is low, or high, the optimal quantities are decreasing, corresponding to the case of binding minimum income and reservation utility
constraints respectively. For medium values of the cost parameter, both constraints bind, and the
optimal remains constant.
The principal’s basic objective is to implement an efficient level of quantity and to extract any
available surplus from the inside manager who derives a rent or satisfaction from quantity (the term
βq i (α)). Naturally, high quantities are desired when the cost parameter α is small. For a given
reservation utility U , the principal’s ability to extract surplus depends on how tight is the minimum
income M relative to U . Proposition 3 summarizes the consequences of both considerations.
Low values of α correspond to high outputs, but the payments cannot be made lower than M .
The minimum income constraint binds, while the reservation utility constraint is slack. The output
will be lower than the first best and independent of the value of β. Having to make the minimum
payment M to the internal manager, the principal finds it suboptimal to implement the first best
α
quantity q̃( 1+β
. As the value of α rises, eventually the quantity falls sufficiently so that both the
minimum income and reservation utility constraints bind. Output becomes constant in this regime.
At high values of α, the first-best quantity becomes small enough so that it is implementable
without violating the minimum income constraint.
14
The principal’s expected profit π i and the manager’s expected rent Ri is
(11)
πi =
Z
α̂
[q̃(α) − αc(q̃(α)) − M ] f (α)dα
0
Z
(1+β)α̂
[q̃(α̂) − αc (q̃(α̂)) − M ] f (α)dα
α
α
(1 + β)q̃
+
− αc q̃
− U f (α)dα,
1+β
1+β
(1+β)α̂
+
Zα̂ᾱ
(12)
R
i
Z
=
α̂
β q̃(α)f (α)dα.
0
The following result is a straighforward implication of Proposition 3, and its proof is omitted.
Corollary 1 If either M converges U , or if β converges to 0, the optimal contract for the internal
manager [q i (α), ti (α)] converges pointwise to [q̃(α), U ].
The special cases where M = U , or where β = 0, turn out to be very simple. If M = U the
participation constraint (8) is implied by the limited liability constraint (9). It is straightforward
to see that the solution then consists in binding the limited liability constraint, ti (α) = M , and
setting the quantity at q i (α, β) = q̃(α) for all α. If β = 0 the limited liability constraint (9) is
implied by constraint (8) instead, but the solution is the same as with U = M . We note that this
solution is first best if and only if β = 0.
4.2
Comparative statics
Ultimately we are interested in whether the principal hires the manager to supervise production,
or instead relies on subcontracting, and so we will be led to compare the expected profits under
both regimes. We would also like to know how this decision is affected by the underlying parameter
values, and among these we will focus on the manager’s marginal benefit from production, β, the
minimum payment M , and the reservation utility U . Thus, before comparing outsourcing with
in-house production we analyze how the solution with in-house production varies as β, M , and U
vary.
15
4.2.1
Comparative statics (M and U )
We note that the quantities described in Proposition 3 do not depend on the absolute level of the
minimum payment M , but on its position relative to the reservation utility U (more precisely, on
U − M ). Thus, changing the reservation utility and the minimum payment by the same amount
would have no effect on the quantities. It would, however, affect the transfers and therefore the
principal’s profit. For instance, increasing both U and M by one unit would decrease the principal’s
expected profit by one unit.
What happens if the reservation utility U is held fixed, and the minimum payment is increased?
As intuition would suggest, and as shown in the following proposition, this cannot be beneficial for
the principal, but in some cases it does not hurt her either.
Proposition 4 Increasing the minimum payment M by one unit causes a reduction in the principal’s expected profit by one unit if and only if for all cost parameter realizations, only the limited
liability constraint is binding i.e., if ᾱ ≤ α̂. Otherwise it decreases the expected profit by less than
one unit. In particular, if the limited liability constraint does not bind for any cost parameter realization, i.e., if α ≥ (1 + β)α̂, an increase in the minimum payment does not affect the expected
profit.
Conversely, if the minimum payment M is held fixed, and the reservation utility is increased, the
principal is either worse off, or unaffected (this happens if only the limited liability constraint is
binding).
Proposition 5 Increasing the reservation utility U by one unit causes a reduction in the principal’s
expected profit by one unit if and only if for all cost parameter realizations, only the participation
constraint is binding, i.e., if α ≥ (1+β)α̂. Otherwise it decreases the expected profit by less than one
unit. In particular, if the participation constraint does not bind for any cost parameter realization,
i.e., if ᾱ ≤ α̂, an increase in the manager’s reservation utility does not affect the expected profit.
What happens with the manager when the minimum payment is increased? An increase in the
minimum payment means that the principal is less able to substitute the ”in-kind” compensation
16
βq for the transfer t, forcing the principal to leave a larger rent to the manager. An exception to
this rule arises if the limited liability constraint does not bind for any cost parameter realization.
Proposition 6 The manager’s rent increases if the minimum payment M is raised, unless the
limited liability constraint is not binding for any α ∈ [α, ᾱ], in which case the rent is unaffected by
an increase in M .
4.2.2
Comparative statics (β)
Increasing β has two opposing effects on the profit-maximizing quantity. First both threshold values
α̂ and (1 + β)α̂ increase. Therefore, the subset of [α, ᾱ] for which the limited liability constraint
is binding expands, and the quantity offered whenever both constraints bind decreases. On the
other hand, whenever the limited liability constraint is not binding the quantity increases. Figure 2
illustrates this point. It shows the profit-maximizing quantity as a function of α for U = 1, M = 0,
c(q) = q 2 /2, α = .25, and ᾱ = 4.1: the thin line is based on β = 1, and the thicker line on β = 1/2.
For sufficiently small values of α the two curves coincide, so that the value of β is irrelevant. Said
differently, ceteris paribus, for β sufficiently large only the limited liability constraint is binding for
any cost parameter realization: then an increase in β has no impact on the quantity. It should also
be noted that if β is sufficiently small the limited liability is not binding for any α.
Whereas the effect of an increase in β on the threshold values α̂ and (1+β)α̂ affects the expected
profit adversely, the effect on the first-best quantity has a positive impact. A revealed preference
argument shows that the former effect never outweighs the latter: the contract which is optimal
for a given value of β would also be implementable for a higher value of β, so that the principal
cannot lose from an increase in β. Figure 3 displays the expected profit as a function of β, when
α is uniformly distributed on [.25, 1.25], c(q) = q 2 /2, and U = .25, and M = 0. This figure shows
that when β is large the principal does not gain either from a further increase in β. This is because
for large values of β the limited liability constraint prevents the principal from further extracting
the manager’s marginal surplus.
As mentioned earlier if β is small enough the limited liability constraint is not binding for any
value of α, in which case the principal implements the first-best quantity and extracts the full
17
surplus. Figure 4 illustrates this point, by showing the difference between the first-best expected
profit and the in-house expected profit as a function of β: for small values of β this difference is
zero.
The following proposition shows that the properties exhibited in Figures 3 and 4 are general.
Proposition 7 Given [α, ᾱ], U , and M < U , the in-house expected profit π i (β) is non-decreasing
in β. There exists β̄ > 0 such that π i (β) is constant for all β ≥ β̄. There exists β > 0 such that
the in-house expected profit is the first-best expected profit, π i (β) = π ∗ (β), for any β < β.
Proposition 8 Given [α, ᾱ], U , and M < U , the manager’s rent Ri is non-decreasing in β. The
R ᾱ
manager’s rent is bounded above by α β q̃(α)f (α)dα.
5
Choice of Contractual Setting
In the previous sections we have studied how profits are affected by a change in the manager’s preference parameter β, and by changes in the minimum payment M . Under both in-house production
and outsourcing a higher β makes the project more profitable. We now investigate which contractual arrangement dominates, and why. We will rely on the comparative statics results derived
above, and on new results to answer this question. We will focus on how varying the minimum
payment M and the manager’s experience rent parameter β affects the principal’s decision.
To begin, let’s consider the special case M = U . Recall from Proposition 1 that if M =
U the principal is not able to extract any of the manager’s experience rent βq. The principal
therefore chooses the quantity q̃(α), which does not depend on β. Hence, the expected profit is also
independent of β. Since there is no information revelation problem with respect to α, the quantity,
as well as the principal’s expected profit, is first-best if and only if β = 0. The expected profit
under in-house production π i is the horizontal line in Figure 5 in the example where c(q) = 21 q 2 , α
is uniformly distributed on [.25, 1.25], and U = M = .25. The other curve is the expected profit
under outsourcing, π o , which is second-best for all values of β, increasing and convex in β. The
shapes of these curves being general (see Propositions 2 and 1), we may state the following result.
18
Corollary 2 Given [α, ᾱ], U , and M = U , there exists β L > 0 such that the principal prefers
in-house production over outsourcing if and only if β ≤ β L .
What would happen if the minimum payment M were reduced slightly away from U ? Recall from
Proposition 4 that, for M = U the limited liability constraint is binding for all cost parameter
realizations. Together with Proposition 4 this implies that if M is reduced the expected profit
under in-house production increases for any value of β (except for β = 0). In Figure 5 there would
be an upward shift in the curve showing π i , except for β = 0 where there would be no change.
Since the expected profit under outsourcing is unaffected the threshold value for β above which
the principal prefers outsourcing to in-house production must increase. Hence, the threshold value
β L identified in Corollary 2 is the lowest possible value for β for which outsourcing may dominate
in-house production from the principal’s viewpoint. Is this threshold unique? We believe that
unless fairly strong restrictions are imposed there is in general not a unique threshold value for
β; for a given value of the minimum payment M as β increases the principal may want to switch
from in-house to outsourcing, back to in-house, etc. However, the convexity of the expected profit
under outsourcing, together with the fact that for β large enough the expected profit under inhouse production is unaffected by further increases in β (see Proposition 7), implies that for β
large enough the principal prefers outsourcing to in-house production. Furthermore, this largest
threshold value, call it β H , increases as M decreases.
We can further establish the following result. The fact that the expected profit under outsourcing is unaffected by a change in M , together with Proposition 7, and Corollary 2, and the fact that
for every value of β the change in the expected profit under in-house production due to a change
in M is continuous, implies:
Corollary 3 For any β > β̂ there exists a threshold value M̄ (β) < U such that π i (M ) ≥ π o for all
M ≤ M̄ (β), and π i (M ) < π o otherwise. Furthermore, M̄ (β) is continuous in β.
6
Welfare
Is the principal’s choice of production organization efficient? Here we address this question while
taking a second-best measure of social welfare where the social surplus achieved in a given institu-
19
tional setting is that corresponding to the constrained profit maximizing production levels rather
than the firs-best levels. Letting S o and S i denote the social surplus evaluated at the principal’s
solution with outsourcing and in-house production, respectively, we have:
(13)
S o = π 0 + Ro
Z ᾱ α + h(α)
α + h(α)
=
(1 + β)q̃
− αc q̃
f (α)dα − U
1+β
1+β
α
and
(14)
S i = π i + Ri
Z α̃
[(1 + β)q̃(α) − αc(q̃(α))] f (α)dα
=
0
Z
(1+β)α̃
[(1 + β)q̃(α̃) − αc (q̃(α̃))] f (α)dα
α
α
+
(1 + β)q̃
− αc q̃
f (α)dα − U.
1+β
1+β
(1+β)α̃
+
α̃
Z +∞
(15)
We can show a series of results which are similar to those found for the profits:
1. S o is increasing in β. A sufficient condition for it to be convex in β is c000 ≤ 0.
2. S i is increasing in β.
3. If M = U there exists a unique threshold value βSL for β such that S i > S o iff β < βSL .
4. We have not been able to show that βSL > β L , i.e., that if M = U the principal would switch
from in-house production to outsourcing for smaller values of β than the social planner would
(which the Matlab figure below may suggest). We indeed suspect that this is not true in general,
although we have not yet produced a counter-example.
5. For β < βSL , we have S i > S o for all M ≤ U .
6. For β ≥ βSL there exists a unique threshold M S such that S i > S o iff M < M S . M S is
continuous in β.
7. For any β we may compare S i to S o at the value of M for which the principal is indifferent.
Since π i = π o this boils down to comparing the rents. If Ri > Ro when the principal switches
from in-house to outsourcing, it means that there is too much outsourcing. Intuition: the surplus
20
being the profit plus the rent, the difference between the social planner and the principal is that
the former takes into account the external effect of the principal’s action on the manager, whereas
the principal does not. Whenever the manager’s rent with in-house is larger than his rent with
outsourcing when the principal switches from in-house to outsourcing, there is too much outsourcing
from the planner’s point of view. And vice versa. So too much outsourcing can occur when the
rent with in-house is large relative to outsourcing, and vice versa. Numerical examples suggest
that there is too much outsourcing when M is close to U and β is not too large. M close to U
corresponds to a situation where, for a given β, the in-house rent is large. This seems indeed like
a good candidate to get too much outsourcing.
For small values of M the switching to outsourcing may only arise for very large values of β. For
such values of β the rent under outsourcing would be very large, especially in the context where the
rent under outsourcing is convex in β. This may explain why the numerical example exhibits too
little outsourcing for such parameter values. We need to investigate more whether this property
may hold in general with a quadratic cost and a uniform distribution.
Proposition 9 Assume that lim inf q→∞
c0 (q)
c00 (q)
> 0. Then for M sufficiently low, there exists β
such that the firm chooses to produce in-house while outsourcing is the socially optimal institutional
arrangement.
Proof: From Equation (??), since
∂q o
∂β (α, β)
=
1+β
(α+h(α))c00 (q(α,β))
and limβ→∞ q 0 (α, β) = ∞, the
assumptions on the cost function imply that the derivative of the manager’s rent under outsourcing
tends to infinity as β tends to infinity. Furthermore, from Proposition 8, the manager’s in-house
rent is bounded above by a linear function of β. Hence, for β large enough, the manager earns a
larger rent under outsourcing independent of the level of M .
Now consider the firm’s choice of institutional arrangement. For a given M , it may achieve the
firs-best profit as long as β is low enough that limited liability does not bind even for the lowest
cost realization α. The critical value of β solves αc0 ( U −M
β ) = 1 + β. As M tends to −∞, β must
tend to ∞ so as to keep the equality satisfied. For β below that critical value, in-house production
is clearly preferred by the firm since it yields the first-best profit.
Hence, for M very small, the lowest β value at which the firm chooses to give up in-house pro-
21
duction in favor of outsourcing must be large so that the manager’s rent is larger with outsourcing.
This implies that for β slightly below the value that makes the firm indifferent, in-house production
is selected, whereas outsourcing would be socially preferable.
Q.E.D.
Note that a sufficient condition in order for Proposition 9 to apply is that c0 be cogconcave,
c0
c00
since it implies that
is increasing. Note that this is also a sufficient condition for convexity
in β of the manager’s expected rent under outsourcing (a picture with the outsourcing rent and
the largest possible in-house rent could illustrate the point that for β large, the manager prefers
outsourcing).
The following lemma establishes various results.
Lemma 1 Letting S o and S i be defined by (13) and (14) we obtain:
1. For any U ≥ 0, β = 0 implies S i > S o .
dS i (U )
dU
2. For any β ≥ 0,
≥
dS o (U )
dU
= 1, with
dS i (U )
dU
= −1 if and only if U ≥ βq ∗ (α).
From the envelope theorem
dπ o (β)
=
dβ
Z
ᾱ
q o (α, β)f (α)dα,
α
which is positive and increasing in β. Holding β fixed, we have:
π
o
Z
ᾱ =
α
F (α)
o
(1 + β)q (α) − α +
c (q (α)) − U0 f (α)dα,
f (α)
o
implying
dπ o (U0 )
= −1.
dU0
Proof:
πi =
Z
α̃(U )
[q ∗ (α) − αc(q ∗ (α))] f (α)dα
α
Z
(1+β)α̃(U ) +
α̃(U )
Z ᾱ
+
U
− αc
β
(1 + β)q
(1+β)α̃(U )
∗
U
β
f (α)dα
α
1+β
22
α
∗
− αc q
− U f (α)dα.
1+β
Using the fact that q ∗ (α̃) =
U
β
dπ i (β)
dU
we get
(1+β)α̃(U ) Z
=
α̃(U )
Z ᾱ
−
1
1
− αc0
β β
U
β
f (α)dα
f (α)dα.
(1+β)α̃(U )
The term within the square brackets in the first integrand is decreasing in α, and it takes the value
0 for α = α̃, and the value -1 for α = (1 + β)α̃. Hence if α < (1 + β)α̃(U ) the first integral is strictly
greater than:
Z
(1+β)α̃(U )
−
f (α)dα.
α̃(U )
As a result we obtain
dπ i (β)
dU
Z
ᾱ
≥ −
f (α)dα ≥ −1,
α̃(U )
where the two inequalities hold as equalities if and only if (1 + β)α̃(U ) ≤ α.
Q.E.D.
Whichever institutional arrangement will dominate from a social welfare view point depends on
the extent of the distortions in production that the principal introduces so as to limit the manager’s
rent. Since the principal chooses to deteriorate efficiency in order to reduce the manager’s rent,
the principal’s incentives in her choice of production organization are somewhat aligned with the
social optimum: she likes situations where she does not have to give up much rent so that she may
introduce only limited inefficiencies. For instance, when the manager’s reservation utility U0 is
large enough, the principal may achieve the first-best social outcome when production is in-house
so that this is the optimal organizational form both for the principal and for social welfare. More
generally, the social benefits from resorting to in-house production rather than outsourcing decrease
as the manager’s minimum payment increases so that outsourcing may only be optimal if M is large
enough (as in Lemma 3).
Still, as the next example illustrates, the principal’s organizational choice will not systematically
coincide with the social optimum because the principal does not entirely capture social surplus.
The principal’s choice between the two organizational forms will be socially suboptimal whenever
the discrepancy between the manager’s rent in-house and under outsourcing is wide enough. Our
23
numerical example below shows that there is no systematic bias in the principal’s decision: she may
choose outsourcing when it is socially dominated or choose in-house production when outsourcing
would have been socially optimal.
Figure 6, produced with Matlab, shows, for c(q) = 21 q 2 and U = 1, and for different values of β
and M , whether or not the principal chooses the socially optimal organizational form (conditional
on the quantities and transfers being chosen by the principal). In this figure a value of 0 on the
vertical axis means that the retailer’s decision is socially optimal; a value of 1 means that the
retailer chooses to outsource whereas the planner would choose in-house production, and a value
of -1 means the opposite.
24
Appendix
Proof of Proposition 3
Consider first a relaxed program which omits the minimum income constraint (9). The reservation
utility constraint (8) must bind in this relaxed program, so t = U − βq. We can substitute t in the
principal’s payoff by this value, and the quantity that maximizes this payoff is
α
q(α) = arg max(1 + β)q − αc(q) − U = q̃
.
q
1+β
This is the solution if the omitted minimum income constraint is satisfied; that is, if
(16)
U − β q̃(
α
) ≥ M,
1+β
or
U −M
α
≥ q̃(
).
β
1+β
By the definition of α̂, we have
U −M
= q̃(α̂).
β
(17)
The function q̃ is strictly decreasing. Therefore, for
α
1+β
≥ α̂, or equivalently α ≥ (1 + β)α̂, the
minimum income constraint is satisfied by q(α) defined above. This proves the third part of the
proposition.
Next consider a relaxed program which omits the participation constraint (8). The minimum
income constraint (9) must bind, so t = M . Replacing t by this expression in the objective implies
(18)
q(α) = arg max q − αc(q) − M = q̃(α).,
q
This is the solution if the omitted reservation utility constraint is satisfied; that is, if β q̃(α)+M ≥ U .
Again, from the definition of α̂ and the fact that q̃ is strictly decreasing, the reservation utility
constraint (8) is satisfied for any α < α̂. This proves the first part of the proposition.
The preceding arguments imply that for any α ∈ [α̂, (1 + β)α̂] both reservation utility and
minimum income constraints bind, and we have
(19)
q(α) =
U −M
,
β
and
This proves the second part of the proposition.
t(α) = M.
Q.E.D.
25
Proof of Proposition 4
For given values of α, ᾱ, U , and β, the expected profit under in-house production may be written
as a function of M :
(20)
α̃(M )
Z
i
[q̃(α) − αc(q̃(α)) − M ] f (α)dα
π (M ) =
0
Z
(1+β)α̃(M )
[q̃(α̃) − αc (q̃(α̃)) − M ] f (α)dα
+
α̃(M )
Z +∞
(1 + β)q̃
+
(1+β)α̃(M )
α
1+β
α
− αc q̃
− U f (α)dα.
1+β
After simplification we get
(21)
dπ i (β)
dM
Z
(1+β)α̃
= −
0
1
f (α)dα +
β
Z
(1+β)α̃ αc0 (q̃(α̃)) − 1 f (α)dα.
α̃
If ᾱ ≤ α̃ this expression reduces to
Z
ᾱ
−
f (α)dα = −1,
α
which is thus a lower bound for
(22)
dπ i (β)
dM
since the second term in (21) is positive. Indeed,
α̃c0 (q̃(α̃)) = 1,
which implies
(23)
αc0 (q̃(α̃)) > 1
for α > α̃.
Finally we note that if α ≥ (1 + β)α̃ the expression in (21) reduces to zero.
Q.E.D.
Proof of Proposition 5
For given values of α, ᾱ, M , and β, the expected profit under in-house production may be written
as a function of U as follows (compared to the expression in (20) we have replaced M by U − β q̃(α̃)
26
for α between α̃ and (1 + β)α̃):
Z α̃(U )
i
[q̃(α) − αc(q̃(α)) − M ] f (α)dα
π (U ) =
0
Z
(1+β)α̃(U )
[(1 + β)q̃(α̃) − αc (q̃(α̃)) − U ] f (α)dα
+
α̃(U )
Z +∞
α
α
(1 + β)q̃
+
− αc q̃
− U f (α)dα.
1+β
1+β
(1+β)α̃(U )
After simplification we get
(24)
dπ i (β)
dU
Z
+∞
= −
f (α)dα +
α̃
1
β
Z
(1+β)α̃ 1 + β − αc0 (q̃(α̃)) f (α)dα.
α̃
If α ≥ (1 + β)α̃ this reduces to
Z
ᾱ
−
f (α)dα = −1.
α
The second term in (24) is positive since
(25)
(1 + β)α̃c0 (q̃(α̃)) = 1 + β,
which implies
(26)
1 + β − αc0 (q̃(α̃)) > 0
for α < (1 + β)α̃.
Finally we note that if ᾱ ≤ α̃ the expression in (24) reduces to zero.
Q.E.D.
Proof of Proposition 6
For given values of α, ᾱ, U , and β, the manager’s expected rent under in-house production may be
written as a function of M :
(27)
Ri (M ) =
Z
α̃(M )
β q̃(α)f (α)dα.
0
We immediately get
(28)
dRi (M )
dM
= β q̃(α̃)f (α̃)
27
dα̃(M )
,
dM
which is strictly positive as long as f (α̃) > 0, since
dα̃(M )
dM
> 0.
Q.E.D.
Proof of Proposition 7
For given values of α, ᾱ, U , and M , the expected profit under in-house production may be written
as a function of β:
i
Z
α̃(β)
[q̃(α) − αc(q̃(α)) − M ] f (α)dα
π (β) =
0
Z
(1+β)α̃(β)
[q̃(α̃) − αc (q̃(α̃)) − M ] f (α)dα
+
α̃(β)
Z +∞
α
α
+
(1 + β)q̃
− αc q̃
− U f (α)dα
1+β
1+β
(1+β)α̃(β)
The first statement in the proposition, namely that the expected profit is non-decreasing in β
follows from a simple revealed preference argument. Consider some β, and the optimal contract
associated with this β. The principal may offer the same contract for any higher value of β: the
transfer being the same the limited liability would be satisfied, and the participation constraint
would continue to hold since the manager’s utility would increase due to the increase in β.
To prove the second statement, namely that for β large enough the expected profit is constant,
we note that α̃, where
(29)
1
α̃(β) =
c0
U −M
β
(see (10)) is strictly increasing in β. Hence, there exists β̄ such that α̃(β̄) = ᾱ, and α̃(β) ≥ ᾱ for
all β > β̄. For any β ≥ β̄ the expected profit equals
Z
ᾱ
[q̃(α) − αc(q̃(α)) − M ] f (α)dα,
α
which is independent of β.
We further note that as β tends to zero, monotonicity of c implies that α̃ tends to 0. Together
with the monotonicity of α̃, and the assumption α > 0, this implies that there exists β > 0 such
28
that (1 + β)α̃(β) = α, and (1 + β)α̃(β) ≤ α for all β < β. Thus, for any β ≤ β the expected profit
equals
i
ᾱ Z
π (β) =
(1 + β)q̃
α
α
1+β
α
− αc q̃
− U f (α)dα.
1+β
which is the first-best expected profit. This proves the third statement in the proposition. Q.E.D.
Proof of Proposition 8
For given values of α, ᾱ, U , and M , the manager’s expected rent under in-house production may
be written as a function of β:
Z
i
(30)
α̃(β)
R (β) =
β q̃(α)f (α)dα.
0
We immediately get
(31)
dRi (β)
dβ
Z
=
α̃
q̃(α)f (α)dα + β q̃(α̃)f (α̃)
0
dα̃(β)
,
dβ
which is strictly positive.
Q.E.D.
29
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q
5
3 .7 5
2 .5
1 .2 5
0
0
2 .5
5
i
1 2
2q ,
7 .5
10
al p h a
Figure 1. q for c(q) =
and
= U0 = 1
Figure 1: q i as a function of α, for c(q) = 21 q 2 , β = U = 1, and M = 0.
q
5
3 .7 5
2 .5
1 .2 5
0
0
1
2
1
3
4
al p h a
Figure 2. q i for c (q) = 12 q 2 ,
= U0 = 1 (thin line), and 2 = U0 = 1 (thick
line)
Figure 2: q i as a function of α, for c(q) = 12 q 2 , β = U = M + 1 = 1 (thin line), and 2β = U =
M + 1 = 1 (thick line).
1
33
p ro fit
2
1 .5
1
0 .5
0
0 .2 5
0 .5
0 .7 5
1
1 .2 5
1 .5
b eta
Figure 3.
i
for c (q) =
1 2
2q ,
= :25,
= 1:25, and U0 = :25
Figure 3: π i as a function of β, for c(q) = 21 q 2 , α = .25, ᾱ = 1.25, and U = M + .25 = .25.
p ro fit d ifferen
ce
1
1
0 .7 5
0 .5
0 .2 5
0
0 .2 5
0 .5
0 .7 5
1
1 .2 5
1 .5
b eta
-0 . 2 5
Figure 4.
i
for c (q) = 12 q 2 , and
= :25,
= 1:25
Figure 4: π ∗ − π i as a function of β, for c(q) = 12 q 2 , α = .25, ᾱ = 1.25, and M = 0.
34
profit
3
2
1
0
0
0.25
0.5
0.75
1
1.25
1.5
beta
-1
Figure 5. ^ i and ^ o for cÝqÞ =
1
2
q 2 , J =. 25, J = 1. 25, and U
Figure 5:
35
Figure 6: Social efficiency of the retailer’s organizational choice.
36