We first evaluate the effect of production contracts on information rents, the buyer’s profits, and the total surplus obtained by the farmer and the buyer by comparing the profit-maximizing basic and restrictive contract menus under the standard treatment of asymmetric information. Then, in order to better understand the role of information in driving differences in outcomes under the two contracts, we conduct a marginal analysis by continuously varying the degree of asymmetric information between symmetric information and the standard model. The marginal analysis identifies the factors which influence profits and total surplus as the degree of information asymmetry increases. In the context of production contracts, it identifies the value to the buyer of reducing asymmetric information by imposing additional constraints on production decisions.
Buyer’s profit-maximizing basic and restrictive contracts
This subsection develops the buyer’s profit-maximizing basic and restrictive contracts and then compares output, information rents, production costs, and profits under the two contracts.
Buyer’s profit-maximizing basic contract
Given a menu of basic contracts \((\tilde {{\mathbf {q}}}, \tilde {{\mathbf {t}}})\), the buyer’s profit when the farmer declares a type of \(\hat {\theta }\) is \(p \tilde {{q}} (\hat {\theta }) - \tilde {{t}} (\hat {\theta })\). Thus, the buyer’s problem is to choose the contract menu \((\tilde {{\mathbf {q}}}, \tilde {{\mathbf {t}}})\) that maximizes \(\sum _{i \in \{ {\ell }, {h} \}} \Big \{\phi ^{i} \Big (p \tilde {{q}} (\theta ^{i}) - \tilde {{t}} (\theta ^{i}) \Big)\Big \}\) subject to incentive and participation constraints. Because the principal’s problem when there is a basic contract is the same as one with a single composite input, we can characterize the optimal basic contract menu by drawing on standard results from the mechanism design literature in which production is a function of the farmer’s effort: the constraints that are binding on the buyer are type h’s incentive compatibility constraint and type ℓ’s individual rationality constraint. Consequently, h and ℓ produce, respectively, at and below the symmetric information benchmark solution levels for their types. Moreover, the difference between the transfer offered to ℓ and ℓ’s production cost of delivering the designated output level will just equal ℓ’s reservation utility, which in our model is zero. On the other hand, the transfer offered to h includes a premium, referred to as his information rent, which in the optimal contract will just offset the increment in utility that h would derive by adopting ℓ’s contract rather than the one intended for him. Remark 3 summarizes the textbook treatment of this class of contract (see, e.g., Varian 1992, pp. 457–463).
Remark 3
The optimal basic contract menu has the following properties:
-
1.
farmer ℓ produces less than he does in the symmetric information benchmark, \({\tilde {q}^{\ell }} < q^{\star } (\theta ^{\ell })\), and receives a transfer equal to his production costs:
$$\begin{array}{*{20}l} {{\tilde{t}^{\ell}}} \quad \quad = \quad \quad & \tilde{C}_{\ell}^{P} ({\tilde{q}^{\ell}}). \end{array} $$
(4a)
-
2.
farmer h produces the same quantity as in the symmetric information benchmark, \({{\tilde {q}^{h}}} = q^{\star } (\theta ^{h})\), and receives a transfer greater than his production costs:
$$\begin{array}{*{20}l} {{\tilde{t}^{h}}} \quad \quad = \quad \quad & \tilde{C}_{h}^{P} ({{\tilde{q}^{h}}}) \quad \quad + \quad \quad \left(\tilde{C}_{\ell}^{P} ({\tilde{q}^{\ell}}) \quad - \quad \tilde{C}_{h}^{P} ({\tilde{q}^{\ell}}) \right). \end{array} $$
(4b)
h’s transfer compensates him for his own production costs (the first term) plus pays him the difference between the transfer he would obtain for producing \( {\tilde {q}^{\ell }}\) and his cost of producing it (the difference between the second and third terms).
In what follows, we sometimes use the terminology production costs and information costs to distinguish between costs incurred through production and costs paid to ensure truthful revelation by the buyer.6 The terms “marginal production” and “marginal information” costs will then have the obvious interpretation.
Profit-maximizing restrictive contract
Given a restrictive contract menu \((\bar {{\mathbf {q}}}, \bar {{\mathbf {k}}},\bar {{\mathbf {t}}})\), the buyer’s profit from a farmer declaring a type of \(\hat {\theta }\) is \(p \bar {{q}} (\hat {\theta }) {-} \bar {{t}} (\hat {\theta })\). Thus, her problem is to choose the contract menu \((\bar {{\mathbf {q}}}, \bar {{\mathbf {k}}}, \bar {{\mathbf {t}}})\) that maximizes \(\sum _{i {\in } \{ {\ell }, {h} \}} \left \{\phi ^{i} \left (p \bar {{q}} (\theta ^{i}) {-} \bar {{t}} (\theta ^{i}) \right) \right \}\) subject to incentive and participation constraints.
Under a restrictive contract, the input mix is no longer exogenous to the buyer’s decision. Consequently, the textbook single-input model can no longer be used to characterize the optimal restrictive contract menu. Instead, Lemma 1 below establishes that the principal’s constrained profit maximization problem is equivalent to an unconstrained profit maximization problem which substitutes into the principal’s expression for profits the two binding contraints: the low ability farmer’s reservation utility constraint and the high ability farmer’s incentive compatibility constraint.
Lemma 1
The problem of choosing the optimal restrictive contract menu is equivalent to the following problem:
$$\begin{array}{*{20}l} \max_{({\bar{\mathbf{q}}},{\bar{\mathbf{k}}})} \; \sum_{i \in \{{\ell},{h}\}} & \left\{\phi^{i} \left(p {\bar{{q}}} (\theta^{i}) - {\bar{{t}}} (\theta^{i})\right)\right\} \\ \text{where} \quad {{{\bar{t}}^{h}}} \quad \quad = \quad \quad & \bar{C}_{h}^{P} ({\bar{q}}^{h}, {\bar{k}}^{h}) \quad \quad + \quad \quad \left(\bar{C}_{\ell}^{P} ({\bar{q}}^{\ell}, {\bar{k}}^{\ell}) \quad - \quad \bar{C}_{h}^{P} ({\bar{q}}^{\ell}, {\bar{k}}^{\ell}) \right) \notag \\ {\bar{t}}^{\ell} \quad \quad = \quad \quad & \bar{C}_{\ell}^{P} ({\bar{q}}^{\ell}, {\bar{k}}^{\ell}). \notag \end{array} $$
(5)
That is, any solution to (5) is a solution to the buyer’s restrictive problem and vice versa.
The proofs, and all subsequent ones, are deferred until the “Appendix”. Equipped with Lemma 1, we establish that the restrictive contract problem shares many of the properties of the single-input problem. Formally,
Proposition 1
The optimal restrictive contract menu has the following properties:
-
1.
farmer ℓ produces less than he does in the symmetric information benchmark, \({\bar {q}^{\ell }} < q^{\star } (\theta ^{\ell })\), with capital level \(\bar {k}^{\ell }\) and receives a transfer equal to his production costs:
$$\begin{array}{*{20}l} \bar{t}^{\ell} \quad \quad = \quad \quad & \bar{C}_{\ell}^{P} \left({\bar{q}^{\ell}},{\bar{k}^{\ell}}\right). ` \end{array} $$
(6a)
-
2.
farmer h produces the same quantity as in the symmetric information benchmark, \({\bar {q}^{h}} = q^{\star } (\theta ^{h})\), using the neo-classical input vector (e
⋆(θ
h),k
⋆(θ
h)), and receives a transfer greater than his production costs:
$$\begin{array}{*{20}l} {{\bar{t}^{h}}} \quad \quad = \quad \quad & \bar{C}_{h}^{P} \left(\bar{q}^{h}\right) \quad \quad + \quad \quad \left(\bar{C}_{\ell}^{P} \left({\bar{q}^{\ell}},\bar{k}^{\ell}\right) \quad - \quad \bar{C}_{h}^{P} \left({\bar{q}^{\ell}},\bar{k}^{\ell}\right) \right). \end{array} $$
(6b)
h’s transfer compensates him for his own production costs (the first term) plus pays him the difference between the transfer he would obtain for producing \({\bar {q}^{\ell }}\) and his cost of producing it (the difference between the second and third terms).
-
3.
farmer ℓ’s capital-effort ratio exceeds the neo-classical ratio \(\tilde {\beta }\).
Let \(\bar {C}^{I} (q,k) = \bar {C}_{\ell }^{P} (q,k) - \bar {C}_{h}^{P} (q,k)\) denote the information cost of contracting with type ℓ to produce q with capital level k under a restrictive contract. It follows from Lemma 1 that the task of choosing the optimal restrictive contract menu can be reformulated as the following (unconstrained) maximization problem:
$$\begin{array}{*{20}l} \max_{(\bar{\mathbf{q}},\bar{\mathbf{k}})} \; \sum_{i \in \{{\ell},{h}\}} & \phi^{i} \left(p \bar{{q}} (\theta^{i}) - \bar{C}^{P}(\bar{{q}} (\theta^{i}), \bar{{k}}(\theta^{i}), \theta^{i}) \right) \quad + \quad \phi^{h} \bar{C}^{I}(\bar{{q}}(\theta^{\ell}), \bar{{k}}(\theta^{\ell})). \end{array} $$
(7a)
Similarly, from Proposition 3, the task of choosing the optimal basic contract can be reformulated as
$$\begin{array}{*{20}l} \max_{\tilde{\mathbf{q}}} \; \sum_{i\in\{{\ell},{h}\}} & \phi^{i} \left(p \tilde{{q}} \left(\theta^{i}\right) - \tilde{C}^{P}\left(\tilde{{q}} \left(\theta^{i}\right), \theta^{i}\right) \right) \quad + \quad \phi^{h} \tilde{C^{I}} \left(\tilde{{q}} \left(\theta^{\ell}\right)\right) \end{array} $$
(7b)
where \(\tilde {C^{I}} (q) = \tilde {C}_{\ell }^{P} (q) - \tilde {C}_{h}^{P} (q)\) denotes the information cost of contracting with type ℓ to produce q under a basic contract.
Because information costs are independent of type h’s contractual variables, the presence of information asymmetry has no impact on h’s choice of inputs or output, and hence cost of production, under either type of contract. Hence, all these values coincide with their symmetric information benchmark values. For this reason, we shall ignore this aspect of the buyer’s problem for the remainder of the paper and focus our attention on the contract targeted for type ℓ. Because information costs depend on the difference in productivity between the two types, h will continue to be in the discussion.
To streamline notation,7 we divide by ϕ
ℓ and write ϕ
h/ϕ
ℓ as Φ.
$$ \begin{array}{ll}\mathsf{Basic}:\kern2em & \underset{\overset{\sim }{q}}{ \max}\kern2.77626pt \left(p\overset{\sim }{q}-{\overset{\sim }{C}}^P\left(\overset{\sim }{q},{\theta}^{\ell}\right)\right)\kern1em -\kern1em \varPhi \kern0.3em \overset{\sim }{C^I}\left(\overset{\sim }{q}\right).\kern2em \end{array} $$
(7c)
$$ \begin{array}{ll}\mathsf{Restrictive}:\kern2em & \underset{\left(\bar{q},\bar{k}\right)}{ \max}\kern2.77626pt \left(p\bar{q}-{\bar{C}}^P\left(\bar{q},\bar{k},{\theta}^{\ell}\right)\right)\kern1em -\kern1em \varPhi \kern0.3em {\bar{C}}^I\left(\bar{q}\right).\kern2em \end{array} $$
(7d)
Given q, let \( \tilde {C^{I}} (q) = \tilde {C}_{\ell }^{P} (q) - \tilde {C}_{h}^{P} (q)\) denote the information cost of having q produced under a basic contract, when both types use the neo-classical input mix. For given k, let \(\bar {C}^{I} (q,k) = \bar {C}_{\ell }^{P} (q,k) - \bar {C}_{h}^{P} (q,k)\) denote the information cost of having q produced under a restrictive contract requiring the use of the capital level k. The results which follow are consequences of the following inequality:8
$$\begin{array}{*{20}l} \text{for all positive } q, \text{and all } k \ge \tilde{k} \left(q,\theta^{\ell}\right),\quad \tilde{C^{I}} (q) > \bar{C}^{I} (q,k). \end{array} $$
(8)
That is, the information cost associated with the profit-maximizing basic contract is always greater than the information cost associated with the profit-maximizing restrictive contract (which requires type ℓ to utilize a super-optimal level of capital).
Figure 1 illustrates (8) by comparing a basic contract in which the farmer is required to produce an arbitrary output level q against a restrictive contract in which the farmer is required to produce q using the neo-classical input mix for his type. The purpose of the figure is to show that the latter contract will be more profitable than the former. It is not, however, the most profitable way for the buyer to obtain q: from part 3 of Proposition 1, the buyer can do even better than the illustrated restrictive contract if she requires ℓ to choose a super-optimal capital-labor mix.
The two isoquants in each panel of the figure, labeled IQ
ℓ
(q) and IQ
h
(q), indicate the input combinations with which ℓ and h can produce q. The parallel lines represent isocost curves. The brace in the left panel indicates the cost differential when farmer type i produces q using the neo-classical input mix for his type (including k
i). The brace in the right panel indicates the reduced cost differential when the contract designed for type ℓ specifically requires that q must be produced using the capital level k
ℓ that is optimal for type ℓ; because of this restriction, if type h chose the contract for type ℓ, he would be obliged to use an excessive amount of capital. (A sufficient condition for the cost differential to be smaller in the right panel than the left is that effort and capital are not perfectly substitutable.) The braces in the two panel also represent the respective information rents that the buyer must pay farmer h to produce q, under either a basic and restrictive contract: the input-mix penalty built into the restrictive contract lowers the incentive for h to “cheat” and pretend to be ℓ and hence also the incentive payment required to induce truthful revelation.
Figure 1 also demonstrates how the buyer can construct a restrictive contract which exactly mimics any basic contract, except for the added restriction on the input mix that h must use if she deviates from truthful behavior. The buyer’s revenues are the same under both contracts because outputs are the same. Production costs are also the same because, provided the farmer of type i chooses the contract designed for type i, the input mix he selects will be identical under the two contracts. But information rents are lower under the restrictive contract, and so profits associated with q are higher. Since q and k were chosen arbitrarily, this argument applies in particular to \(\tilde {q}(\theta ^{\ell })\), the output assigned to ℓ in the optimal basic contract. It follows that profits under this contract must be strictly less than profits he obtains by “mimicking” ℓ and choosing ℓ’s restrictive contract described above, and hence lower by an even greater margin than profits under the optimal restrictive contract. The preceding remarks are summarized in Proposition 2.
Proposition 2
The buyer’s profits under the optimal restrictive contract strictly exceed her profits under the optimal basic contract.
Marginal analysis of the basic and restrictive contracts
We use marginal analysis to study the relationship between the two kinds of contracts. Specifically, we vary continuously the amount of uncertainty faced by buyers in a situation of asymmetric information, starting with symmetric information and ending with the incomplete information model described in the “Methods” section. Let Φ =ϕ
h/ϕ
ℓ denote the ratio of the probabilities that the farmer is of either type. We will now, for each γ∈[0,Φ ], solve for the optimal basic and restrictive contracts, when Φ in each problem is replaced with γ. When γ=0, the asymmetric information component of the buyer’s problem is eliminated, since the farmer is known to be of type ℓ. Now, we can use calculus techniques to examine the impact of a small “increase in buyer uncertainty,” d
γ. For each agent type, the symmetric information benchmark solution (p. 18) is independent of the type-probability ratio γ; consequently, the rates at which, conditional on each type, output, the buyer’s profits, etc. decline as γ increases are pure measures of the marginal impacts of buyer uncertainty. We then integrate these marginal impacts over the interval [0,Φ ] to recover and compare the total impacts of buyer uncertainty on the solutions to the buyer’s original problems (7c) and (7d).
Marginal analysis of the restrictive contract
We begin by determining the minimum cost to the buyer of having type ℓ produce at least q under a restrictive contract for a given γ, while ensuring that type h does not have an incentive to pretend to be type ℓ. This cost minimization problem requires picking the nonnegative vector (e,k), e=(e
ℓ,e
h), which minimizes the production cost (we
ℓ+rk) plus the expected information cost
γ
w(e
ℓ−e
h) of producing q under the restrictive contract, subject to the constraints that (a) farmer ℓ produces q using (k,e
ℓ) and (b) if farmer h selects the contract designed for ℓ, he produces q using (k,e
h). Summarizing, the principal’s problem is
$$\begin{array}{*{20}l} \min_{(\textbf{e},k)}\: \underbrace{\left\{w \left((1 + \gamma) {e}^{\ell} - \gamma {e}^{h}\right) + r k\right\}}_{\text{Term A}} & \; \text{s.t.} \; f^{{\ell}} ({e}^{\ell}, k) = q, \text{} f^{{h}} ({e}^{h}, k) = q~ \text{and } ({\textbf{e}},k) \ge 0. \end{array} $$
(9)
As a consequence of a restriction, we shall later impose (see (15) below), the solution values (e,k) for (9) are necessarily positive. Because of this, we will omit the nonnegativity constraints from our specification of the Lagrangian, which is
$$\begin{array}{*{20}l} \bar{L}({\textbf{e}},k, {\boldsymbol{\lambda}};q, \gamma) = w \left((1 + \gamma) {e}^{\ell} - \gamma {e}^{h}\right) + r k + \lambda^{{\ell}} (q - f^{\ell} ({e}^{\ell},k)) + \lambda^{h} (f^{{h}}({e}^{h},k)- q)). \end{array} $$
(10)
where λ=(λ
ℓ,λ
h) is the vector of multipliers for the restricted problem. Let \(\left ({\bar {\vec {e}}}(q,\gamma),, \bar {k} q,\gamma), {\bar {\boldsymbol {\lambda }}}q,\gamma)\right)\) denote the solution to (9). Inserting these values into Term A of (9), we obtain the restrictive cost function, \(\bar {C} (q,\gamma) = \bar {C}^{P}(q,\gamma) + \bar {C}^{i}(q,\gamma)\), where \(\bar {C}^{P}(q,\gamma) = (w \bar {{e}}^{{\ell }} (q,\gamma) + r \bar {k}(q,\gamma))\) is the production cost, and \(\bar {C}^{i}(q,\gamma) = \gamma w (\bar {{e}}^{{\ell }} (q,\gamma) - \bar {{e}}^{h} (q,\gamma))\) is the information cost of producing q under the restrictive contract.
The first-order condition for \(\bar {L}\) has five equations in five unknowns:
$$\begin{array}{*{20}l} \bigtriangledown \bar{L} \quad = \quad \left[ \begin{array}{c} \bar{L}_{{e}^{\ell}} \\ \bar{L}_{{e}^{h}} \\ \bar{L}_{k} \\ \bar{L}_{\bar{\lambda}^{{\ell} }} \\ \bar{L}_{\bar{\lambda}^{h}} \end{array} \right] \quad = \quad & \left[ \begin{array}{l} (1 + \gamma) w - \bar{\lambda}^{\ell} f_{{e}^{\ell}} \\ - \gamma w + \bar{\lambda}^{h} f_{{e}^{h}} \\ r - \bar{\lambda}^{\ell} f_{k}^{\ell} + \bar{\lambda}^{h} f_{k}^{{h}} \\ q - f^{\ell} ({e}^{\ell},k)\\ f^{h} ({e}^{h},k) - q \\ \end{array} \right] \quad = \quad 0. \end{array} $$
(11)
At the solution to (11), \(\left (\bar {\textbf {e}}(q,\gamma),\bar {k}(q,\gamma), {\bar {\boldsymbol {\lambda }}}(q,\gamma)\right)\), the constraints are identically zero so that the restrictive cost function
\(\bar {C} (q,\gamma)\)—defined as the minimum attainable value of total cost under the restrictive contract for each (q,γ) pair—is identically equal to the minimized value of \(\bar {L}\), henceforth denoted by \(\bar {L} (\cdot \; ; q, \gamma)\).
Note that because \(\bar {L}_{{e}^{\ell }}\), \(\bar {L}_{{e}^{h}}\), and \(\bar {L}_{k}\) are all zero, we have
$$\begin{array}{*{20}l} \bar{\lambda}^{\ell} = \frac{(1 + \gamma) w} {{f_{{e}}^{\ell}}} > \bar{\lambda}^{h} = \frac{\gamma w} {f_{{e}}^{h}}. \end{array} $$
(12)
Substituting the expressions for the λ’s (Eq. (12)) into the expression for \(\bar {L}_{k}\) (eq. (11)) yields:
$$\begin{array}{*{20}l} \frac{r}{w} = \frac{\bar{f}_{k}^{\ell}}{\bar{f}_{e}^{\ell}} + \gamma \left(\frac{{\bar{f}_{k}^{\ell}}}{{\bar{f}_{e}^{\ell}}} - \frac{{\bar{f}_{k}^{h}}}{{\bar{f}_{e}^{h}}} \right). \end{array} $$
(13)
Because h is more efficient than ℓ and both use the same level of capital to produce q, h’s effort level under the restrictive contract must be less than ℓ’s. That is, \( \frac {\bar {k}}{\bar {{e}}^{{\ell }}} > \frac {\bar {k}}{\bar {{e}}^{h}}\) which in turn implies \( \frac {{\bar {f}_{k}^{\ell }}}{{\bar {f}_{e}^{\ell }}} > \frac {{\bar {f}_{k}^{h}}}{{\bar {f}_{e}^{h}}}\). Hence, the term in parentheses in (13) is positive, implying \(\frac {{\bar {f}_{k}^{\ell }}}{{\bar {f}_{e}^{\ell }}} < \frac {r}{w }\). Proposition 3 follows immediately.
Proposition 3
In a restrictive contract for a given (q,γ)≫0, the prescribed capital-effort ratio for the low ability farmer is greater than the neoclassical ratio \(\tilde {\beta }\).
(For a vector \({\mathbf {x}} \in \mathbb {R}^{n}\), we write x≫0 if x
i
>0, for i=1,...n). Note that Proposition 3 is more general than, and hence implies, property 3 of Proposition 1. Figure 2 provides intuition for Proposition 3. Its top panel reproduces Fig. 1 above. Consider the effect on the buyer’s problem of increasing γ from zero, for the moment holding the output level constant at an arbitrary output level q. When γ=0, the type ℓ farmer is required to use the neo-classical input mix. By the envelope theorem, a small increase in capital intensity above the neoclassical level has only a second-order impact on the production costs of farmer ℓ (see the bottom panel of Fig. 2). On the other hand, because the neo-classical k level for farmer ℓ is super-optimal for farmer h, the given increase in capital intensity would result in a first-order increase in farmer h’s production cost if he chose the contract designed for ℓ. Thus, a small increase in capital intensity beyond the neoclassical level for ℓ results in a first-order reduction in information costs, and a second-order increase in production costs. It follows that whenever γ>0, the prescribed level of capital for farmer ℓ will exceed the neoclassical level for her prescribed level of output.9
The restrictive marginal cost function, denoted by \(\overline {{MC}}\), is identically equal to \(\frac {d \bar {L}(\cdot \; ; q, \gamma)}{d q}\) which, by the envelope theorem, equals \(\frac {\partial \bar {L}(\cdot \; ; q, \gamma)}{\partial q}\). This partial derivative in turn equals the difference between the two Lagrangians, \(\bar {\lambda }^{{\ell }}\) and \(\bar {\lambda }^{h}\), so that \(\overline {{MC}} (q,\gamma) = \bar {\lambda }^{{\ell }} (q,\gamma) - \bar {\lambda }^{h} (q,\gamma)\). Moreover, at the buyer’s optimum, \(\overline {{MC}} (\bar {{q}}(\gamma),\gamma) = p\), where \(\bar {{q}}(\gamma)\) is the profit maximizing level of output produced by the farmer of type ℓ at price p under the restrictive contract.
Our strategy for studying the properties of the restrictive contract is to apply the implicit function theorem to the first-order conditions (11), along the path \(\{(\bar {q}(\gamma),\gamma) : \gamma \in [0,{\Phi \,}]\}\). This requires, of course, that the determinant of the Hessian of \(\bar {L}(\cdot \; ;\bar {q}(\gamma),\gamma)\), denoted \({\Delta ^{\overline {{HL}}}}(\gamma)\), is non-zero along this path. It is easy to verify that \({\Delta ^{\overline {{HL}}}}(0)\) is positive. By continuity, the previous requirement is then equivalent to requiring that \({\Delta ^{\overline {{HL}}}}(\cdot)\) is positive on [0,Φ].
Consider the case in which the production function g is CES in addition to satisfying A1–A3. In this case, the expression for the determinant reduces to
$$\begin{array}{*{20}l} {\Delta^{\overline{{HL}}}}(\gamma) = -\tau \times \left\{ \frac{\gamma}{f_{{e}}^{h}({e}^{h},k)}\;\frac{{e}^{\ell}{f_{{e}}^{\ell}}({e}^{\ell},k)} {k{f_{k}^{\ell}}({e}^{\ell},k)} - \frac{1+\gamma}{{f_{{e}}^{\ell}}({e}^{\ell},k)}\;\frac{{e}^{h}{{f_{{e}}^{h}}}({e}^{h},k)} {k{f_{k}^{h}}({e}^{h},k)}\right\}. \end{array} $$
(14)
where τ>0 depends only on parameters of the model. Clearly, expression (14) will be positive in a neighborhood of γ=0. For large γ, however, positivity is difficult to guarantee when inputs are close substitutes and h is much more efficient than ℓ. Figure 3 illustrates the problem. When isoquants have minimal curvature, the difference between \({f_{{e}}^{\ell }}({e}^{\ell },k)\) and \({{f_{{e}}^{h}}}({e}^{h},k)\) depends on the efficiency gap, but only minimally on the input ratio, while the ratios \(\frac {{e}^{\ell }{f_{{e}}^{\ell }}({e}^{\ell },k)}{k{f_{k}^{\ell }}({e}^{\ell },k)}\) and \(\frac {{e}^{h}{{f_{{e}}^{h}}}({e}^{h},k)}{k{f_{k}^{h}}({e}^{h},k)}\) are very similar. Given all other parameters, therefore, we can construct an example in which e
h is arbitrarily close to zero (as in Fig. 3), ensuring that (14) will be positive except when γ is very small. Lemma 2 establishes that this problem does not arise when the elasticity of substitution between effort and labor is bounded above by unity. 10
Lemma 2
If g is CES in effort and capital, with constant elasticity of substitution parameter \(\bar {\sigma }_{k {e}} \le 1\), then \({\Delta ^{\overline {HL}}}(\cdot)\) will be positive on [0,Φ].
Because the sufficient condition in Lemma 2 is far from necessary for the property we need, we will hold the condition in reserve for the moment and, in Propositions 4 and 6 below, simply assume that \({\Delta ^{\overline {HL}}}(\cdot) > 0\). A convenient implication of this assumption—which we invoked when we specified the Lagragian (10)—is
$$\begin{array}{*{20}l} \text{For all } \gamma > 0, \text{if } {\Delta^{\overline{HL}}}(\gamma) \text{is positive then } \qquad\qquad \notag \\ {e}^{h} (\bar{q}(\gamma),\gamma)~ \text{and hence } {e}^{\ell} (\bar{q}(\gamma),\gamma)~ \text{and } \bar{{q}} (\gamma)~ \text{are also positive.} \end{array} $$
(15)
To see this, note that if \({e}^{h} (\bar {q}(\gamma),\gamma){=}0\), the first term in (14) would be positive and the second term zero.
Proposition 4
A sufficient condition for the restrictive marginal cost function \(\overline {{MC}} (\cdot,\gamma)\) to be increasing in q, and for \(\bar {{q}} (\cdot)\)to be a continuously differentiable function of γ, for all γ∈[0,Φ ], is that the determinant of the Hessian of the Lagrangian (10) is positive on [0,Φ ]. Moreover,
$$\begin{array}{*{20}l} \bar{q}^{\prime} (\cdot)= & \frac{w \left(\bar{{e}}^{{\ell}} - \bar{{e}}^{h}\right)}{(\alpha-1)(\bar{\lambda}^{\ell}-\bar{\lambda}^{h})}<0. \end{array} $$
(16)
It is straightforward to verify, from (12) and (29), that if the ratio, γ, of high types to low types is sufficiently small, the determinant of the Hessian of the Lagrangian will indeed be positive.
Marginal analysis of the basic contract
In order to compare outcomes under the two contracts, we must obtain an expression for \(\frac {d \tilde {{q}}(\gamma)}{d \gamma }\). Remark 2, established that it is sufficient to analyze the equivalent single-input formulation of the buyer’s problem under the basic contract.
Let \(\ddot {g}\) denote the single-input production function corresponding to f
ℓ. Let \(\ddot {{{e}}}^{i} (q)\) denote the level of composite input required for type i to produce q.11 From Remark 1, \(\ddot {e}^{h} (\cdot)= {\Theta \,}^{1/\alpha } \ddot {e}^{\ell } (\cdot)\) (recall from page 15 that \({\Theta \,} = \frac {\theta ^{\ell }}{{\theta ^{h}}}\)), so that the information cost of producing q under a basic contract is \(\ddot {C}^{i} (q) = \ddot {v} \left (1 - {\Theta \,}^{1/\alpha } \right) \ddot {{{e}}}^{\ell } (q)\). Thus, the basic cost function is
$$\begin{array}{*{20}l} \tilde{C}(q, \gamma) \quad = \quad \ddot{C}^{P} \left(q,\theta^{\ell}\right) \quad + \quad \gamma \ddot{C}^{i} (q) \quad \quad = \quad \quad \ddot{v} \left\{1 + \gamma \left(1 - {\Theta\,}^{1/\alpha}\right)\right\} \ddot{{e}}^{\ell} (q). \end{array} $$
(17)
Because \(\frac {d \ddot {{{e}}}^{\ell } (q)}{d q} = \left (\ddot {g}\,'(\ddot {{{e}}}^{\ell } (q)) \right)^{-1}\), the basic marginal cost function is
$$ \widetilde{MC}(q,\gamma) \quad = \quad \ddot{v} \left\{1 + \gamma \left(1 - {\Theta\,}^{1/\alpha}\right)\right\} \left(\ddot{g}\,'(\ddot{{{e}}}^{\ell} (q))\right)^{-1}. $$
(18)
The profit maximizing level of output, \(\tilde {{q}}(\gamma)\), produced by the farmer ℓ at price p under the basic contract is defined by the condition \(\widetilde {MC} (\tilde {{q}}(\gamma),\gamma) = p\). Applying the implicit function theorem,
$$\begin{array}{*{20}l} \frac{d \tilde{q}(\gamma)}{d \gamma} \quad \quad = \quad \quad & -\left.{\frac{\partial \widetilde{MC} (\tilde{{q}}(\gamma),\gamma)}{\partial \gamma}}\right/ {\frac{\partial \widetilde{MC} (\tilde{{q}}(\gamma),\gamma)}{\partial q}} \notag\\ \quad \quad = \quad \quad & {\ddot{v} \left(1 - {\Theta\,}^{1/\alpha} \right) /\ddot{g}\,'} {\frac{\ddot{v} \left\{1+\gamma \left(1-{\Theta\,}^{1/\alpha} \right)\right\}} {(\ddot{g}\,')^{2}}\frac{\ddot{g}\,{~}^{\prime\prime}}{\ddot{g}\,'} }. \notag \end{array} $$
Euler’s theorem implies that \(\ddot {g}\,{~}^{\prime \prime } = (\alpha -1)\ddot {g}\,'/\ddot {{{e}}}^{\ell } (q)\). because \(\ddot {g}\,'\) is homogeneous of degree α−1. By profit maximization, \(\left \{ 1 + \gamma \left (1 - {\Theta \,}^{1/\alpha } \right) \right \} = p \ddot {g}\,'/\ddot {v}\). Using these two equalities, the expression becomes
$$\begin{array}{*{20}l} \frac{d \tilde{{q}}(\gamma)}{d \gamma} \quad = \quad & \frac{\ddot{v} \ddot{{{e}}}^{\ell}(q) \left(1 - {\Theta\,}^{1/\alpha} \right)} {(\alpha- 1)p} \quad = \quad \frac{(w {{\tilde{{e}}^{\ell}}} (q) + r {{\tilde{k}^{\ell}}}(q)) \left(1 - {\Theta\,}^{1/\alpha} \right)} {(\alpha- 1)p} \quad <\quad 0. \\ \notag \end{array} $$
(19)
Comparing the marginal effects of the restrictive and basic contracts
This subsection establishes two factors that contribute to the dominance, from the buyer’s perspective, of the restrictive over the basic contract. Inequality (8) above established that information costs are lower under the the optimal restrictive contract than under the optimal basic contract; Proposition 3 established that production costs are higher. Proposition 5 demonstrates that the former inequality dominates.
Proposition 5
For any given (q,γ)≫0, the buyer’s total cost of optimally obtaining q under a restrictive contract is less than the corresponding costs under a basic contract. That is,
$$\begin{array}{*{20}l} \text{for all} q \text{and all} \gamma > 0, \quad \bar{C}(q, \gamma) \quad < \quad \tilde{C}(q, \gamma). \end{array} $$
(20)
The proof of Proposition 5 is immediate: the neoclassical input mix is feasible under the restrictive contract, but, by Proposition 3, violates the first-order condition (13). The proposition reflects the fact, illustrated in Fig. 2, that the first-order reduction in information costs obtained by moving away from the neoclassical input mix necessarily offsets the resulting, second-order increase in production costs.
Our next result is less immediate. It establishes that relative to the symmetric information benchmark, output is less distorted under the restrictive contract than under the basic contract.
Proposition 6
If the determinant of the Hessian of the Lagrangian (10) is positive on [0,Φ], then output produced by farmer ℓ is higher under the optimal restrictive contract than under the optimal basic contract.
An interpretation of Proposition 6 is that the restrictive marginal cost curve (including both production and information rent costs) is strictly lower under the restrictive contract than under the basic contract. Intuitively, this holds because under the restrictive contract, the buyer can limit the extent to which farmer h could substitute between effort and capital if he were to select the contract designed for ℓ.
The social cost of information asymmetry
As we have seen, the buyer’s profits are higher under the optimal restrictive contract than under the optimal basic contract. This does not imply, however, that restrictive contracts are preferable to basic contracts from a social perspective. While the buyer’s objective is to minimize the sum of production and information costs, only production costs matter for total surplus. Information costs are simply a transfer from the buyer to farmer h. Our task in this section is to compare total surplus under the two types of contracts.
In the present model with perfectly elastic demand, total surplus is equal to the buyer’s total revenue minus production costs because the information rent obtained by h is a cost to the buyer. Although information rents are lower under the optimal restrictive contract, average production costs are higher because the input mix is sub-optimal from a pure production standpoint. We refer to this distortion as the input mix effect. The second factor which affects total surplus is the level of production. Proposition 6 establishes that production is always higher under the optimal restrictive contract. We refer to this difference as the output effect. The difference between total surplus under the two contracts is the sum of the two effects. Whether the positive output effect or the negative input mix effect dominates depends on a number of considerations, including the elasticity of substition, the relative importance of the two inputs in the production process, and the productive efficiency gap between the two types.
We find that if the elasticity of substitution is sufficiently small, then total surplus is higher under the restrictive contract than under the basic contract. To obtain this result, we need certain parameters to be bounded away from their natural boundaries. Specifically, the ratio of types’ efficiency factors (Θ), the probability of type ℓ (ϕ
ℓ), and the homogeneity factor (α) must all belong to compact subsets of (0,1), and the neoclassical input ratio (\(\tilde {\beta }\)) and the level of output associated with input vector \((1,\tilde {\beta })\) must belong to compact subsets of (0,∞). Since any compact sets will do, we define them all in terms of an arbitrarily small scalar, \(\check {\omega } \in (0, 0.5)\). Let
$$\begin{array}{*{20}l} G = \left\{g\text{~satisfying A1--A3}: \Theta, \phi^{\ell},\alpha\in [\check{\omega}, 1 - \check{\omega}], \text{\& } \tilde{\beta}, g(1, \tilde{\beta}) \in [\check{\omega}, \left.1\right/ {\check{\omega}}]\right\}. \end{array} $$
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Given a production function g, let σ(e,k|g) denote the elasticity of substitution of capital for effort for the function g at (k,e) and let \(\bar {\sigma }(g) = \text {sup} \left \{\sigma (e,k|g) : (e,k) \in {\mathbb {R}}_{+}^{2}\right \}\).
Obviously, for the special case in which g is CES, \(\bar {\sigma } (g)\) is just the familiar constant measure. Significantly, the fact that g belongs to G does not impose any restriction on \(\bar {\sigma } (g)\).
Proposition 7
There exists \(n\in {\mathbb {N}}\) such that for all g∈G with \(\bar {\sigma } (g) \le 1/n\), social surplus with technology g is higher under the restrictive contract than under the basic contract.
The intuition for this result is straightforward.
As the elasticity of substitution declines, the buyer can obtain a larger and larger “bang for the buck” in terms of information rents. That is, a small distortion in the capital-effort ratio away from the optimal ratio results in a larger and larger decrease in information rents. Figure 4 illustrates the effect of the elasticitiy of substitution. The two panels display the effect of a given increase in ℓ’s k away from its neoclassical value for a given quantity. The top panel illustrates that the reduction in information rents is relatively small when the elasticity of substitution is large. In the bottom panel, the reduction in information rents is large when the elasticity of substitution is small. As a consequence of this relationship, as the elasticity of substitution approaches zero, the quantity produced under the optimal restrictive contract approaches the first-best quantity, while the distortion in the capital-effort ratio goes to zero. It follows that for n sufficiently large, the optimal restrictive contract will generate greater total surplus than the optimal basic contract.
Matters are less straightforward when the two inputs are close substitutes. First (as in Fig. 3), an interior solution may exist but the determinant of the Hessian of the Lagrangian may be negative for sufficiently large γ (see (14)). In this case, none of the machinery on which Proposition 7 is based can be applied. Second, suppose, for a sequence (g
n) satisfying A1–A3 with \(\bar {\sigma } (g^{n})\) increasing without bound, that an interior solution does exist and the determinant of the Hessian of the Lagrangian is positive on [0,Φ ]. In this case, both the input mix and output effects will converge to zero and there is no guarantee that the latter effect will dominate the former. Hence, it is possible that when the two inputs are close substitutes, the optimal basic contract yields a higher level of total surplus than the optimal restrictive contract. Finally, an interior solution for the restrictive contract may not exist. Specifically, if θ
h was slightly higher than the value represented in Fig. 3, then type h would be able to imitate ℓ without utilizing any effort at all.