Delegation in Vertical Relationships: The Role of Reciprocity

Proof of Lemma 1

If firm U does not produce the appropriate input in a given state of the world, his payoff equals p − c i + μ ( p − E ( c | I ̂ ) ) r . The reciprocal part of this payoff is constant as long as U is not producing the appropriate input. Accordingly, he maximizes his overall utility by producing the cheapest input among those that are not appropriate.

Next, suppose it is optimal for firm U to produce the appropriate input in a state s ̇ < n . From what written above, the best alternative to producing the appropriate input is for firm U to produce q _n . Since we are assuming that q s ̇ is optimal in state s ̇ , the following condition holds:

As by construction c s ̇ > c s , ∀ s > s ̇ , if the above condition holds for s ̇ , it holds a fortiori ∀ s > s ̇ . Note, however, that if the state of the world is n, the relevant condition is:

It is immediate to verify that the left-hand side (LHS) of condition (4) is strictly smaller than the LHS of condition (5), ∀ s ̇ < n and that the right-hand side (RHS) of condition (4) is strictly larger the RHS of condition (5). Accordingly, if condition (4) holds, so does condition (5). We conclude that if firm U produces the appropriate input in a state of the world s ̇ , he produces the appropriate input in any “larger” state s > s ̇ .

For the last part of the proof we require beliefs to be in equilibrium. After substituting I ̂ = I ̄ ( n − s ′ ) in K ( p ; I ̂ ) , individual rationality simplifies to:

As 1 + μ E ( Π D | I ̄ ( n − s ′ ) ) > 0 by construction, individual rationality implies that p ≥ E c | I ̄ ( n − s ′ ) ; accordingly, any unkind contract is rejected. If this is the case, firm U always produces the appropriate input in the state n (cfr. condition (5) above after substituting I ̂ = I ̄ ( n − s ′ ) ). In particular, for firm U to never produce the appropriate input, I ̄ ( 0 ) , the proposed contract must be perceived as unkind. However, any such contract would be rejected in first place. In conclusion, I ̄ ( n − s ′ ) must be such that ∀s ∈ [0, …, n − 1], firm U produces the appropriate input for any state s > s′ and q _n otherwise. □

Proof of Proposition 1

First, we require beliefs to be in equilibrium: I ̂ = I ̄ ( n − s ′ ) . Let s′ + 1 ≠ n. By the arguments in the proof of Lemma 1, U produces the appropriate input in any state s > s′ if the following condition holds:

After some algebra, the above expression simplifies to p ≥ p n − s ′ * = E ( c | I ̄ ( n − s ′ ) ) + c s ′ + 1 − c n μ ρ . In any other state firm U produces q _n if the following condition holds:

After some algebra, the above expression simplifies to p < E ( c | I ̄ ( n − s ′ ) ) + c s ′ − c n μ ρ . It is immediate to verify that p n − s ′ * satisfies this condition. Further, as K ( p n − s ′ * ; I ̄ ( n − s ′ ) ) ≥ 0 , p n − s ′ * satisfies individual rationality. Next, if s′ + 1 = n, by substituting s′ = n − 1 in the expression for p n − s ′ * one gets exactly p 1 * = E ( c | I ̄ ( 1 ) ) = c n . It is immediate to verify that p 1 * solves the problem for s′ + 1 = n. □

Proof of Corollary 1

Consider the following input-contingent payments to be paid after the delivery of the input: p(q _i ) = c _i . As U is then indifferent between producing any input, I ̄ ( n − s ′ ) describes a possible behavior of firm U. The expected price for this contract is 1 n ∑ s = s ′ + 1 n c i + s ′ c n . This price equals the limit for μ → ∞ of p n − s ′ * :

□

Proof of Corollary 2

Consider the price identified in Proposition 1:

The first term, the expected production cost of firm U is decreasing in s′: as s′ increases, firm U produces the cheapest input instead of a more expensive appropriate input in more states of the world, thus his expected costs decrease. The second term, c s ′ + 1 − c n μ ρ , is strictly decreasing in s′ by assumption. We can conclude that p n − s ′ * is strictly decreasing in s′. □

Proof of Proposition 2

To begin with, we need to identify the value of s′ that maximizes the expected profits of firm D. Formally we are looking for s ′ * = arg max s ′ ∈ [ 0 , … , n − 1 ] ⊂ N E ( Π D | I ̄ ( n − s ′ ) ) − p n − s ′ * . To solve this integer program we first consider the relaxed problem for s ′ ∈ R which is a concave parabola. We then let s ̃ * denote the abscissa of its vertex. As this parabola is symmetric about the vertical line trough the vertex, we identify s ^′* as the nearest integer to s ̃ * that belongs to [0, …, n − 1].

First, we rewrite the expected profits of firm D as follows:

We then apply Assumption 1 and substitute c _i = c ₁ − (i − 1)Δc in the above expression. After some algebra we obtain:

When s ′ ∈ R , the above expression describes a concave parabola, whose equation can be expressed as α(s′)² + βs′ + γ, where

The abscissa of its vertex is:

Let ⌊ s ̃ * ⌉ be the nearest integer to s ̃ * .^[15] Given that the axis of symmetry of this parabola is the vertical line trough the vertex we have that ⌊ s ̃ * ⌉ = arg max s ′ ∈ N E ( Π D | I ̄ ( n − s ′ ) ) − p n − s ′ * . Let s′^* be the value of s′ that maximizes D’s revenues, that we henceforth call the optimal value of s′. If s ̃ * < 1 2 or s ̃ * ≥ n − 3 2 , s ^′* equals 0 or n − 1, respectively. If 0 ≤ s ̃ * < n − 3 2 , s ′ * = ⌊ s ̃ * ⌉ . To sum up we have that:

where s ̃ * = n − 1 2 + n μ ρ − ρ Δ c .

The condition for s ^′* = n − 1 can be rearranged as follows:

This inequality holds for ρ ∈ 1 − 1 + 4 n μ Δ c Δ c 2 , 1 + 1 + 4 n μ Δ c Δ c 2 . As 1 − 1 + 4 n μ Δ c Δ c 2 < 0 and ρ > 0 by construction, we conclude that s ^′* = n − 1 for ρ ≤ ρ ̲ = 1 + 1 + 4 n μ Δ c Δ c 2 > 0 .

The condition for s ^′* = 0 can be rearranged as follows:

This inequality holds for ρ > n − 1 + ( n − 1 ) 2 + 4 n μ Δ c Δ c 2 and for ρ < n − 1 − ( n − 1 ) 2 + 4 n μ Δ c Δ c 2 . As n − 1 − ( n − 1 ) 2 + 4 n μ Δ c 2 < 0 and ρ > 0 by construction, we conclude that s ^′* = 0 for ρ > ρ ̄ = n − 1 + ( n − 1 ) 2 + 4 n μ Δ c Δ c 2 > 0 .

It is immediate to verify that for ρ ∈ ρ ̲ , ρ ̄ , s ^′* ∈ {1, …, n − 2}.

As n ≥ 2, we have that ρ ̄ ≥ ρ ̲ , with strict inequality for n > 2. If n = 2, ρ ̄ = ρ ̲ and s ^′* = 0 if ρ > ρ ̄ , while s ^′* = 1 if ρ ≤ ρ ̄ . □

Proof of Corollary 3

The corollary follows directly from the Proof of Proposition 2, and in particular from (6) ad (7). □

Proof of Corollary 4

To begin with, note that lim ρ → 0 s ̃ * = lim ρ → 0 n − 1 2 + n μ ρ − ρ Δ c = + ∞ . By (7) s ^′* = n − 1. The relevant condition for the contract ( p 1 * , I ̄ ( 1 ) ) to be preferred to vertical integration is then condition (1) from Section 4: ρ ≥ n(c _n − c _n+1). As n(c _n − c _n+1) > 0 by construction, the condition does not hold when ρ → 0 and firm D integrates vertically. □

Proof of Proposition 3

To begin with, consider the case s ^′* = n − 1. By (7) this is the case when s ̃ * ≥ n − 3 2 . Substituting for s ̃ * and rearranging leads to: n ρ ≥ μ ρ Δ c − 1 . The term in braces is strictly positive by Assumption 2 and μ n − 1 = n ρ ρ Δ c − 1 − 1 .

Consider now the case s ^′* = 0. By (7) this is the case when s ̃ * < 1 2 . Substituting for s ̃ * and rearranging leads to n ρ < μ ρ Δ c − n + 1 . The term in braces is strictly positive by Assumption 2 and μ ₀ = ∞.

By (7), s ^′* = s, s ∈ [1, …, n − 2] if s − 1 2 ≤ s ̃ * < s + 1 2 . Substituting for s ̃ * and introducing σ ( s ) = s − n + ρ Δ c , the condition becomes: σ ( s ) ≤ n μ ρ < σ ( s + 1 ) . From Assumption 2 we know that σ(s) > 0, ∀s ∈ [1, …, n − 2] and μ s = n ρ σ − 1 ( s ) .

Summing up we have that μ ₀ = ∞, while for any other s ∈ [1, …, n − 1], μ _s is finite and can be expressed as μ s = n ρ σ − 1 ( s ) . Turning now to Δμ _s , we have that Δμ ₀ = ∞, while any other Δμ _s is finite and simplifies to Δ μ s = n ρ σ ( s ) σ ( s + 1 ) − 1 > 0 . Clearly, Δμ ₀ > Δμ ₁, while for any other difference we have that Δ μ s − Δ μ s + 1 = 2 n ρ σ ( s ) σ ( s + 1 ) σ ( s + 2 ) − 1 > 0 . □

Proof of Remark 1

Let p n − s ′ * * be the optimal price such that U behaves as described in I ̄ ( n − s ′ ) when he is liquidity constrained. Liquidity constraints imply that p n − s ′ * * must be non-smaller than the largest production cost faced by U. Accordingly, if the price identified in Proposition 1, p n − s ′ * , is such that p n − s ′ * > c s ′ + 1 , then p n − s ′ * * = p n − s ′ * and the analyses from the previous sections continue to apply, otherwise p n − s ′ * * = c s ′ + 1 . To see this is the case consider that by liquidity constraints U can only produce inputs q _i for which p ≥ c _i . Let L(p) bet the set of these inputs. Incentive compatibility becomes:

As we are assuming that p n − s ′ * * > p n − s ′ * , firm U produces the appropriate input ∀s > s′ by arguments analogous to those in the Proof of Proposition 1. In any other s ≤ s′, firm U produces q _n either because doing so is preferred to producing the appropriate input, or because the appropriate input for such states does not belong to L p n − s ′ * * and by Lemma 1 U produces q _n . □

Proof of Corollary 5

Consider two subsequent prices p n − s ′ * * and p n − ( s ′ + 1 ) * * , determined as in Remark 1. We want to show that p n − s ′ * * > p n − ( s ′ + 1 ) * * . To do so, we consider four possible cases. First, if p n − s ′ * * = c s ′ + 1 and p n − ( s ′ + 1 ) * * = c s ′ + 2 , the claim is true by construction. Next, if both prices are determined as in Proposition 1 the claim holds by Corollary 2. If p n − s ′ * * = c s ′ + 1 and p n − ( s ′ + 1 ) * * = p n − ( s ′ + 1 ) * , by Remark 1 it must be that c s ′ + 1 ≥ p n − s ′ * ; and by Corollary 2 p n − s ′ * > p n − ( s ′ + 1 ) * = p n − ( s ′ + 1 ) * * . Lastly, if p n − s ′ * * = p n − s ′ * and p n − ( s ′ + 1 ) * * = c s ′ + 2 , we have that by Remark 1 p n − s ′ * * = p n − s ′ * ≥ c s ′ + 1 and, by construction, c s ′ + 1 > c s ′ + 2 = p n − ( s ′ + 1 ) * * . □