Quality, Shelf Life, and Demand Uncertainty

Proof of Proposition 1

The proof of part (1) is as follows. Consider the maximization problem (2). For the case of corner solution q = 0, the product is not produced. Thus, we consider the case where q > 0. The first order conditions with respect to q and T are

where (11) holds with inequality only if the corner solution T = 1 occurs.

Differentiating (10) with respect to p yields

I next show, evaluated at the optimal solution, ∂ F ∂ p > 0 , ∂ F ∂ q < 0 and ∂ F ∂ T < 0 .

By (10),

Because v′(q) > 0, 0 < δ < 1, 0 ≤ p ≤ 1, and T ≥ 1, ∂ F ∂ p > 0 .

By (10) again,

Because v′′(q) < 0 and c q q ′ ′ ( q , T ) ≥ 0 , ∂ F ∂ q < 0 .

Because v′′(q) < 0 and v(0) = 0,

Similarly, because c Tqq ′ ′ ′ ( q , T ) ≥ 0 and c T ′ ( 0 , T ) = 0 ,

By (11), (14) and (15),

Note that the left side of (16) is the partial derivative of F with respect to T. Thus, ∂ F ∂ T < 0 .

By (12), if d T d p ≤ 0 , d q d p > 0 . This completes the proof of part (1).

The proof of part (2) is as follows. The interior solution to the maximization problem (2) satisfies the first order conditions

Furthermore, by the second order condition, ∀p, the matrix

evaluated at the interior solution, is negative semi-definite. Hence, its determinant is non-negative, i.e. det H ≥ 0.

By the Implicit Function Theorem, if the determinant of the matrix

evaluated at the interior solution, is not zero, i.e. det I ≠ 0, then

Because det H ≥ 0, to determine the sign of d p d T , we need study the sign of det I. By the proof of part (1), F q ′ ( q , T , p ) < 0 , G q ′ ( q , T , p ) < 0 (see (16)), and F p ′ ( q , T , p ) > 0 . I next study the sign of G p ′ ( q , T , p ) .

By (11), it can be verified that

where y ≡ (1 − p).

It can be verified that for y ≤ 1 2 , T − ( 1 − δ ) y ( 1 − y ) ( 1 − δ y ) > 0 and T − ( 1 − δ ) y ( 1 − y ) ( 1 − δ y ) ln ( δ y ) increases in y. Because as y → 0, T − ( 1 − δ ) y ( 1 − y ) ( 1 − δ y ) ln ( δ y ) → − ∞ , then there exists y* such that if y ≤ y*,

and thus G p ′ ( q , T , p ) ≤ 0 . Because y = 1 − p, there exists p*(≡ 1 − y*), such that if p ≥ p*, G p ′ ( q , T , p ) ≤ 0 and det I > 0. By (21), d p d T ≤ 0 . That is, T decreases in p.

For some p, the corner solution T = 1 may occur, the optimal shelf life T in this case is constant in p. Therefore, for p ≥ p*, T decreases or non-increases in p. By part (1) of Proposition 1, the optimal quality increases in p. Thus, quality is negatively correlated with shelf life. □

Proof of Proposition 2

The proof of part (1) is as follows. Let n denote the quantity of products the firm provides for a period. First, I show if v ′ ( 0 ) ∑ i = j m p i > c q ′ ( 0,1 ) , the firm provides no less than n _j units, i.e. n ≥ n _j .

Suppose n _j−1 ≤ n < n _j . Consider producing one more unit with quality q > 0 and shelf life T = 1. The (n + 1)th product yields an expected profit

Because v ′ ( 0 ) ∑ i = j m p i > c q ′ ( 0,1 ) , there exists q* > 0 such that

Because v(0) = 0, v′′(q) < 0, c(0, 1) = 0, and c q q ′ ′ ( q , T ) ≥ 0 , ∀q > 0, T > 0, then

and

Therefore,

Thus, π(q*) > 0. That is, producing one more unit with q = q* and T = 1 is profitable. Hence, n _j−1 ≤ n < n _j is impossible. By similar reasoning, it can be verified that n < n _j−1 is also impossible. Therefore, n ≥ n _j .

Next I show if v ′ ( 0 ) ∑ i = j + 1 m p i ≤ c q ′ ( 0,1 ) , the firm provides no more than n _j units, i.e. n ≤ n _j .

Suppose n _j < n ≤ n _j+1. There must be n − n _j units which have the same probability ∑ i = j + 1 m p i of being sold in a period. Suppose the optimal quality and shelf life for these units are q ̃ > 0 and T ̃ ≥ 1 . Then the expected profit of one unit of these products is

where J ( T ̃ ) ≡ ∑ i = j + 1 m p i 1 − δ T ̃ 1 − ∑ i = j + 1 m p i T ̃ 1 − δ 1 − ∑ i = j + 1 m p i .

Because v′′(q) < 0 and c q q ′ ′ ( q , T ) ≥ 0 , ∀q > 0, T > 0,

and

Because v ′ ( 0 ) ∑ i = j + 1 m p i ≤ c q ′ ( 0,1 ) ,

It can be verified that J′′(T) < 0, ∀T ≥ 0. Thus,

Similarly, because c T T ′ ′ ( q , T ) ≥ 0 , ∀q > 0, T > 0,

By (24), (25) and (26),

Because J′′(T) < 0, c T T ′ ′ ( q , T ) ≥ 0 again,

By (24) and (27), the right side of (28) is less than that of (29). Thus,

That is, π ( q ̃ , T ̃ ) < 0 . Hence, n _j < n ≤ n _j+1 is impossible. By similar reasoning, n > n _j+1 is also impossible. Thus, n ≤ n _j .

Therefore, n = n _j . Because n _j < n _m , the market is partially covered.

If n = n _j , the firm will divide the n _j units into j groups. Each group has a different probability of being sold in a period from another. The products between groups are thus different. Otherwise, (10) will be violated. Hence, the number of varieties is j. This completes the proof of part (1).

Analogous to the proof of part (1), it can be verified that if v ′ ( 0 ) p m > c q ′ ( 0,1 ) , the firm provides no less than n _m units. Because n _m is the potential market size, the firm won’t provide more than n _m units. Hence, n = n _m and the number of varieties is m. The market is fully covered.

If v ′ ( 0 ) ∑ i = 1 m p i ≤ c q ′ ( 0,1 ) , the firm provides no more than n ₀ units. Because n ₀ = 0, the firm produces nothing and the market completely collapses. □

Proof of Proposition 3

Suppose under L ̃ , the firm provides n _j > 0 units of products in a period, then

Otherwise, if v ′ ( 0 ) ∑ i = j m p i ̃ ≤ c q ′ ( 0,1 ) , by the Proof of Proposition 2, the firm provides at most n _j−1 units in a period.

Because L first-order stochastically dominates L ̃ ,

Thus,

By the Proof of Proposition 2 again, (32) implies the quantity of products the firm provides in a period under L is no less than n _j . Hence, the number of product varieties under L is no less than that under L ̃ . □

Proof of Proposition 4

The proof of part (1) is as follows. Let F(⋅) and F ̃ ( ⋅ ) be the respective cumulative distribution functions of L and L ̃ . F(⋅) is the same as (1). F ̃ ( ⋅ ) is obtained by replacing p _j in (1) with p ̃ j . Because L second-order stochastically dominates L ̃ , for all x ≥ n ₀,

Then it can be verified that, for all k ∈ {0, 1, …, m − 1},

I now show

It can be verified that the left side of (34) is equal to n m − ∑ i = 0 m p i n i and the right side is equal to n m − ∑ i = 0 m p ̃ i n i . Because the two distributions have the same mean, ∑ i = 0 m p i n i = ∑ i = 0 m p ̃ i n i . Therefore, (34) holds.

I next show ∑ i = 0 m − 1 p i ≥ ∑ i = 0 m − 1 p ̃ i . Otherwise, if ∑ i = 0 m − 1 p i < ∑ i = 0 m − 1 p ̃ i , then

By (34) and (35),

This contradicts (33). Therefore, ∑ i = 0 m − 1 p i ≥ ∑ i = 0 m − 1 p ̃ i . Because ∑ i = 0 m p i = ∑ i = 0 m p ̃ i = 1 , p m ≤ p ̃ m .

If L induces full coverage of the market, by Proposition 2, v ′ ( 0 ) p m > c q ′ ( 0,1 ) . Then, v ′ ( 0 ) p ̃ m > c q ′ ( 0,1 ) . By Proposition 2 again, the market is also fully covered under L ̃ .

If the market completely collapses under L, by Proposition 2, v ′ ( 0 ) ( 1 − p 0 ) ≤ c q ′ ( 0,1 ) . By (33),

Thus, p 0 ≤ p ̃ 0 . Then v ′ ( 0 ) ( 1 − p ̃ 0 ) ≤ c q ′ ( 0,1 ) . By Proposition 2 again, the market completely collapses under L ̃ .

The proof of part (2) is as follows. Suppose it is optimal for the firm to provide n _j units for a period under L, for any j ∈ {1, 2, …, m − 1}. By Proposition 2, v ′ ( 0 ) ∑ i = j m p i > c q ′ ( 0,1 ) . If ∑ i = j m p ̃ i ≥ ∑ i = j m p i , then v ′ ( 0 ) ∑ i = j m p ̃ i > c q ′ ( 0,1 ) and the firm thus provides no less than n _j units for a period under L ̃ . If ∑ i = j m p ̃ i < ∑ i = j m p i and v ′ ( 0 ) ∑ i = j m p ̃ i ≤ c q ′ ( 0,1 ) , the firm provides less than n _j units for a period under L ̃ . Both cases are possible, if L second-order stochastically dominates L ̃ . □

Proof of Proposition 5

By part (1) of Proposition 2, if v ′ ( 0 ) ∑ i = j m p i > c q ′ ( 0,1 ) and v ′ ( 0 ) ⋅ ∑ i = j + 1 m p i ≤ c q ′ ( 0,1 ) , the firm provides n _j units in a period. For all k ∈ {1, 2, …, j}, n _k − n _k−1 units of them will be sold with a probability of P ( k ) = ∑ i = k m p i in a period. I next show all of these n _j units have the same optimal shelf life T = 1.

Consider the n _k − n _k−1 units which will be sold with a probability of P(k) in a period, ∀k ∈ {1, 2, …, j}. Let the optimal quality for these units is q _k > 0. Because v′′(q) < 0 and c Tqq ′ ′ ( q , T ) ≥ 0 , then v(q _k ) < v′(0)q _k and c T ′ ( q k , 1 ) ≥ c q T ′ ′ ( 0,1 ) q k . Because ln(δ(1 − P(k))) < 0, and

then

Because c T T ′ ′ ( q , T ) ≥ 0 , c T ′ ( q k , 1 ) ≤ c T ′ ( q k , T ) . Hence, ∀T ≥ 1,

Therefore, the optimal shelf life for these units is the corner solution T = 1.

By the first order condition of the firm’s maximization problem with respect to q (see (10)),

Because P(1) > P(2) > ⋯ > P(j), v′′(q) < 0, and c q q ′ ′ ( q , T ) ≥ 0 , ∀q > 0, T > 0, then

That is, there are j different quality levels. □

Proof of Proposition 6

Consider two products which have the same shelf life T* but different quality levels q ₁ and q ₂, where q ₁ > q ₂ > 0. Denote their respective absolute profit margins by AM ₁ and AM ₂. Then,

By (10), v ′ ( q 1 ) − c q ′ q 1 , T * > 0 . Because v′′(q) < 0 and c q q ′ ′ ( q , T ) ≥ 0 ∀ q > 0 , T > 0 , then for all q < q ₁, v ′ ( q ) − c q ′ ( q , T * ) > 0 . Hence, ∫ q 2 q 1 v ′ ( q ) − c q ′ ( q , T * ) d q > 0 , that is, AM ₁ > AM ₂. Thus, a higher quality product has a higher absolute profit margin.

Denote the two products’ respective percentage margins by PM ₁ and PM ₂. Then,

Consider the function K ( q ) ≡ v ( q ) c ( q , T * ) .

Because v′′(q) < 0 and c q q ′ ′ ( q , T ) ≥ 0 , ∀q > 0, T > 0, then v(q) > v′(q)q and c ( q , T * ) ≤ c q ′ ( q , T * ) q , ∀q > 0. By (37), K′(q) < 0. Therefore, PM ₁ < PM ₂, that is, a higher quality product has a lower percentage margin. □

Proof of Proposition 7

The proof of part (1) is as follows. The incentive compatibility implies

and

By (38),

By (39),

Hence,

Because θ _h > θ _l > 0, q ̲ h ≥ q ̄ l .

The proof of part (2) is as follows. First, I show the incentive compatibility constraint (6) is nonbinding for all q l < q l ̄ . Because in equilibrium each consumer is indifferent among the products intended for his type, then for all q _l ∈ Q _l and the associated F _l ,

Because θ _h > θ _l > 0,

The incentive compatibility implies, for all q _h ∈ Q _h and the associated F _h ,

Thus, for all q l < q l ̄ ,

Analogously, it can be verified that the incentive compatibility constraint (7) is nonbinding for all q h > q h ̲ .

Next consider the incentive compatibility constraints (38) and (39). I first show (38) is binding. Suppose (38) is nonbinding, then the firm can increase its profit by raising F ̲ h (and all other F _h as well) slightly while still keeping (38), (39) and all other incentive compatibility constraints valid. Thus, (38) is binding, namely,

If q ̲ h = q ̄ l , F ̲ h = F ̄ l . Thus, (39) is also binding.

If q ̲ h > q ̄ l , (39) is nonbinding. The reason is as follows. Suppose (39) is binding, then θ l v ( q ̄ l ) − F ̄ l = θ l v ( q ̲ h ) − F ̲ h . Because θ _h > θ _l > 0, and q ̲ h > q ̄ l ,

That is, (38) is nonbinding. This is a contradiction.

Because in equilibrium each consumer is indifferent among the products intended for his type, then if q ̲ h = q ̄ l , (8) and (9) are binding; if q ̲ h > q ̄ l , only (8) is binding. □