Price-Cap Regulation of Firms That Supply Their Rivals

Proof. Assume that α _E /α _I  > β. Arya, Mittendorf, and Sappington (2008) show that entry occurs. Denote the incumbent’s downstream price under duopoly by p I D . Solving the game by backward induction, we find that

On the other hand, when the incumbent operates as a monopolist, its downstream price is given by p I M = ( α + c I ) / 2 . Moreover,

so that the incumbent sets a higher downstream price under duopoly than under monopoly.

1. Solving the game by backward induction shows that consumer surplus under duopoly is given by

   C S D = 64 − 23 β 4 − 5 β 6 α I 2 + 4 α E 2 4 + 5 β 2 − 4 α I α E 16 β + 3 β 3 − β 5 8 1 − β 2 8 + β 2 2 .

   On the other hand, consumer surplus under monopoly (and hence under foreclosure) is given by C S M = α I 2 / 8 . Thus, the consumer-welfare gain from entry is

   (17) C S D − C S M = α E − β α I 4 α E − 12 α I β + 5 α E β 2 + 2 α I β 3 + α I β 5 2 1 − β 2 8 + β 2 2 .

   Because α _E /α _I  > β, the sign of CS ^D  − CS ^M is the same as that of the second parenthesized expression in the numerator of the term on the right-hand side of Equation (17): 4α _E  − 12α _I β + 5α _E β ² + 2α _I β ³ + α _I β ⁵. For fixed values of α _I and β, let φ(⋅; α _I , β) denote the function (of α _E ) that this expression defines. Note that φ(⋅; α _I , β) is continuous and increasing. Furthermore,

   lim α E ↓ α I β φ ( α E ; α I , β ) = − α E 1 − β 2 8 + β 2 < 0 .

   Thus, entry of a relatively inefficient provider decreases consumer surplus relative to the case of monopoly. In particular,

   sign C S D − C S M = sign α E − α I 12 β − 2 β 3 − β 5 4 + 5 β 2 .

   Furthermore, when 12 β − 2 β 3 − β 5 / 4 + 5 β 2 ≥ 1 (which holds for β > 0.42), CS ^D  > CS ^M only if α _E  > α _I .

2. Under duopoly, the firms’ equilibrium profits are

   π I D = 8 − 3 β 2 − β 4 α I 2 + 4 α E 2 − 8 β α I α E 4 1 − β 2 8 + β 2 ,

   (18) π E D = 2 + β 2 2 α E − β α I 2 1 − β 2 8 + β 2 2 .

   Under monopoly, the incumbent’s equilibrium profit is π I M = α I 2 / 4 . The difference in welfare levels between duopoly and monopoly is given by

   (19) W D − W M = α E − β α I 28 α E − 36 α I β + 15 α E β 2 − 8 α I β 3 + 2 α E β 4 − α I β 5 2 1 − β 2 8 + β 2 2 .

   Because α _E /α _I  > β, the sign of W ^D  − W ^M is the same as that of the second parenthesized expression in the numerator of the term on the right-hand side of Equation (19): 28α _E  − 36α _I β + 15α _E β ² − 8α _I β ³ + 2α _E β ⁴ − α _I β ⁵. For fixed values of α _I and β, let ϕ(⋅; α _I , β) denote the function (of α _E ) that this expression defines. Note that ϕ(⋅; α _I , β) is continuous and increasing. Furthermore,

   lim α E ↓ α I β ϕ ( α E ; α I , β ) = − α E 1 − β 2 8 + β 2 < 0 .

   Thus, entry of a relatively inefficient provider decreases total welfare relative to the case of monopoly. In particular,

   sign W D − W M = sign α E − α I 36 β + 8 β 3 + β 5 28 + 15 β 2 + 2 β 4 .

   Because 36 β + 8 β 3 + β 5 / 28 + 15 β 2 + 2 β 4 < 1 for all β ∈ (0, 1), entry unambiguously increases total welfare if the entrant is more efficient than the incumbent. □

Proof

It is clear from Equation (18) that π E D is increasing in α _E . Observe that

Hence, both firms’ profits increase with α _E . Finally, we have

which implies that

For given values of α _I and β, let ξ(α _E ; α _I , β) ≡ 8α _E  − 16α _I β + 10α _E β ² − 3α _I β ³ + α _I β ⁵. Note that ξ(⋅; α _I , β) is continuous and increasing in α _E , and that

However, ξ(α; α _I , β) > 0 for

The expression on the right-hand side of Inequality (20) is strictly increasing in β and bounded above by 1. □

Proof

It is clear from Equation (18) that π E D is decreasing in α _I . Note that

Because the denominator in Equation (21) is negative for β ∈ (0, 1),

Thus, ∂ π I D / ∂ α I > 0 if and only if

The function of β that is defined by the expression on the right-hand side of Inequality (22) is unbounded above and decreasing in β. It approaches 1 as β approaches 1.

For consumer surplus,

so that

Therefore, ∂CS ^D /∂α _I  > 0 if and only if

Just as in Inequality (22), the function of β that is defined by the expression on the right-hand side of Inequality (23) is unbounded above and decreasing and approaches 1 as β approaches 1. □

Proof

The proof is by contradiction. Suppose that w < 0. If p _I(w) < c _I , the incumbent loses money on both upstream sales of the input and downstream sales of the product and can do strictly better by setting w = 0 and p _I (w) = c _I .

Now assume that p _I (w) = c _I . Consider a deviation to a higher input price, w′ ∈ (w, 0), while keeping the incumbent’s downstream price fixed. The deviation has three effects:

1. The incumbent incurs a smaller loss on each inframarginal sale of the input than at w.

2. The entrant’s marginal cost rises. In response, the entrant raises its downstream price, which lowers the quantity demanded for its product. As a result, the entrant purchases less of the input. This effect further reduces the incumbent’s upstream loss.

3. The rise in the entrant’s downstream price increases the incumbent’s downstream sales.

Finally, consider the case in which p _I (w) > c _I , which can hold only if the incumbent does not face a price cap. As in the previous case, consider a deviation to w′ ∈ (w, 0). The three effects identified above also arise here, but the third effect now results in a strict increase in the incumbent’s profit. □

Proof

A price-capped incumbent achieves positive downstream sales at input price w if and only if

In view of Lemma 2, we can substitute c _I in for p ̄ I ( w ) into Inequality (24), which can then be rewritten as

Because 0 < β < 1, Inequality (25) holds if and only if the numerator in the left-hand side term is positive. That condition can be rewritten as

The result now follows by observing that the expression on the right-hand side of Inequality (26) is negative (and hence Inequality (26) holds for every w ≥ 0) if and only if α _E /α _I  < 2/β − β. In particular, this latter condition holds whenever α _I  > α _E , and, for fixed values of α _E and α _I , it holds for β sufficiently close to zero (i.e. when the retail products are sufficiently differentiated). □

Lemma 4

Let f 1 : [ 0,1 ] → R and f 2 : [ 0,1 ] → R be two affine functions with f ₁(0) < f ₂(0) and f ₁(1) < f ₂(1). Let g : [ 0,1 ) → R be a differentiable, convex function that satisfies g(0) < f ₁(0) and lim_t↑1 g(t) > f ₂(1). There exist unique numbers t ₁, t ₂ ∈ (0, 1) that satisfy f ₁(t ₁) = g(t ₁) and f ₂(t ₂) = g(t ₂). Furthermore, t ₁ < t ₂.

Proof

For each i ∈ {1, 2}, let h i : [ 0,1 ) → R be defined by h _i  ≡ g − f _i . Because f ₁(0) > g(0) and lim_t↑1 f ₁(t) = f ₁(1) < lim_t↑1 g(t), we have h ₁(0) < 0 < lim_t↑1 h ₁(t). By the Intermediate Value Theorem, there exists at least one t ₁ ∈ (0, 1) such that h ₁(t ₁) = 0. We now prove by contradiction that there exists exactly one such t ₁. To this end, suppose that there exist t ₁ and t ̃ 1 that satisfy 0 < t 1 < t ̃ 1 < 1 and h 1 ( t 1 ) = h 1 ( t ̃ 1 ) = 0 . By Rolle’s Theorem, there exists t 0 ∈ ( t 1 , t ̃ 1 ) such that h 1 ′ ( t 0 ) = 0 . Furthermore, the Mean Value Theorem implies that there exists t* ∈ (0, t ₁) such that h 1 ′ ( t * ) = − h 1 ( 0 ) / t 1 > 0 = h 1 ′ ( t 0 ) . On the other hand, since f ₁ is affine and g is convex, h 1 ′ is nondecreasing on (0,1), which contradicts the finding that h 1 ′ ( t * ) > h 1 ′ ( t 0 ) . Thus, there exists exactly one t ₁ ∈ (0, 1) for which h ₁(t ₁) = 0 (or, equivalently, f ₁(t ₁) = g(t ₁)). An analogous argument shows that there exists exactly one t ₂ ∈ (0, 1) that satisfies f ₂(t ₂) = g(t ₂).

It remains to show that t ₁ < t ₂. First, note that f ₁ < f ₂. To see why, suppose that there exists t ₀ ∈ (0, 1) for which f ₁(t ₀) ≥ f ₂(t ₀). Let h ≡ f ₁ − f ₂. Since h is affine, it is monotonic, which contradicts the fact that max{h(0), h(1)} < 0 ≤ h(t ₀). So f ₁ < f ₂. Now note that g(t ₁) = f ₁(t ₁) < f ₂(t ₁), and g(t ₂) = f ₂(t ₂), so that h ₂(t ₁) < 0 = h ₂(t ₂). By the Mean Value Theorem, there exists t ̄ ∈ ( 0 , t 2 ) such that h 2 ′ ( t ̄ ) = − h 2 ( 0 ) / t 2 > 0 . By the convexity of h ₂, h 2 ′ ( t ) ≥ h 2 ′ ( t ̄ ) > 0 for all t ∈ [ t ̄ , 1 ) ; that is, h ₂ is increasing on [ t ̄ , 1 ] . In particular, h ₂(t) ≥ 0 for all t ∈ [t ₂, 1]. From h ₂(t ₁) < 0, it therefore follows that t ₁ < t ₂. □

Proof

Suppose that w*(k) < c _I for all k ∈ [0, 1]. Then, if we show that the difference between the derivatives of regulated and unregulated incumbent profits with respect to k, π ̃ I D ′ ( k ) − π I D ′ ( k ) is positive when k = 0 and k = 1, then because both π ̃ I D ′ ( k ) and π I D ′ ( k ) are affine functions of k, from Lemma 4, it follows that the unique equilibrium level of k is higher under regulation than without.

Equation (27) is increasing in c _E and α, so that if it is positive for c _E  = 0 and the lowest value of α that satisfies our assumptions, then it holds for all c _E and α. Note that c _E  = 0 implies that Inequality (8) holds, so that foreclosure does not occur. Moreover, Inequality (12) stipulates an infimum for α. Substituting c _E  = 0 into Inequality (12) yields the following infimum for α

Substituting c _E  = 0 and Equation (28) into the right-hand side of Equation (27) yields

which proves that π ̃ I D ′ ( 0 ) − π I D ′ ( 0 ) > 0 when w*(k) < c _I .

Equation (30) is also increasing in c _E and α, so that we may proceed as above, by substituting c _E  = 0 and Equation (28) into the right-hand side of Equation (30). This yields

completing the proof for the case of w*(k) < c _I for all k ∈ [0, 1].

Now suppose that w*(k) = c _I for all k ∈ [0, 1]. The proof of this case is analogous. In particular, the first-order condition analog to Equation (14) in this case is

It is readily shown that both π ̄ I D ′ ( 0 ) − π I D ′ ( 0 ) and π ̄ I D ′ ( 1 ) − π I D ′ ( 1 ) are increasing in c _E and α. Substituting c _E  = 0 and Equation (28) into these expressions yields respectively

completing the proof when w*(k) = c _I for all k ∈ [0, 1].

Finally, recalling that w*(k) is increasing in k up to c _I , suppose that w*(k) binds above some k ̂ ∈ ( 0 ,  1 ) , but not below it. From the left-hand side of Inequality (15), recall that π ̃ I D ′ ( k ) is increasing in k, whereas from the right-hand side of Equation (32), it follows that π ̄ I D ′ ( k ) is constant in k, so that the curve consisting of π ̃ I D ′ ( k ) for k ∈ [ 0 ,  k ̂ ] and π ̄ I D ′ ( k ) for k ∈ ( k ̂ ,  1 ] represents the relevant set of profit derivatives over all possible values of k. But then Inequality (29) and the rightmost inequality in Expression (33) again imply that the unique equilibrium level of k is higher under regulation, completing the proof. □

Proof

Here, we complete the proof for the case in which the upstream price binds under price-cap regulation by comparing Z and X, the differences in the entrant’s equilibrium profit assuming w = 0 as opposed to the equilibrium value of w that prevails with and without price-cap regulation, respectively. Like, Y − X, Z − X can be broken down into a quadratic equation that can be plotted over parameter space α _E /α _I  × β. However, in this case, to make this decomposition tractable, we first define λ ≡ c _I /α _I . By definition, λ ∈ [0, 1). We can then write,

where A ̄ = − 768 + 896 β 2 − 64 β 4 − 64 β 6 , B ̄ = β ( 512 − 512 β 2 − 16 β 4 + 24 β 6 − 8 β 8 ) + λ ( 2048 − 512 β 2 − 96 β 4 + 16 β 6 + 2 β 8 ) , C ̄ = β 4 ( 64 − 52 β 2 − 16 β 4 + 4 β 6 ) − β λ ( 2048 − 512 β 3 − 96 β 4 + 16 β 6 + 2 β 8 ) − λ 2 ( 1024 − 256 β 2 − 48 β 4 + 8 β 6 + β 8 ) , and D ̄ = 4 ( 1 − β 2 ) ( 32 − 4 β 2 − β 4 ) 2 . For each λ ∈ [0, 1), we can now plot Equation (34) over parameter space α _E /α _I  × β, as we do for λ = 0.375 in Figure 7.^[29]

Recall that under price-cap regulation, the equilibrium input price binds only if α E − α I β / 2 > c I , which may be rewritten as α _E /α _I  > 2λ + β. Graphically, α _E /α _I  = 2λ + β forms a line parallel to and above the line denoting the foreclosure condition in Figure 7. Because the equilibrium input price does not bind below the line, only the space above the line is relevant for the comparison in Figure 7. Additionally, by varying λ, it is easy to see that as was the case with Y relative to X, an entrant that is less efficient (downstream) than the incumbent, will never choose to self-provision under regulation if it would not be worthwhile to do so in an equilibrium without price-caps. □