Proof of Proposition 3

We consider the two cases, where the monopoly sells to a single vendor and two vendors. In both cases, we substitute the equilibrium markup, obtained from maximizing the monopoly’s profits, into the firms’ equilibrium decision rules. We then substitute these rules into the inverse demand functions to obtain the equilibrium generic and the new-product prices when the monopoly sells to one or two firms. These prices are

For both generic and new-product prices, we subtract the equilibrium price with one firm from the equilibrium price with two firms:

[20]

[21]

Inequalities [10] imply that the sign of is the same as the sign of .

Proof of Proposition 7

We first state a lemma that is also used in a proof in “Consumer welfare” in Appendix.

Lemma 1For all values of m, regardless of whether the monopoly uses one or two vendors, sales in the generic market and the new market satisfy the relation:

[22]

Proof. When the upstream monopoly sells to both firms, we obtain the equilibrium conditions for aggregate sales in the two markets in a symmetric equilibrium, as functions of m using the linear model. We invert the formula for sales of the new product () to obtain an expression for m as a function of aggregate sales, :

[23]

We, then, substitute this equation into the equilibrium condition for aggregate generic sales, , as a function of m to obtain aggregate generic sales as a linear function of aggregate sales of the new product. The resulting relation is eq. [22]. Given this constraint, a choice of, for example, determines the value of and also m. The values of these variables determine the monopoly’s profits.

By its choice of m, the upstream monopoly selects a point on this line. The monopoly solves a maximization problem subject to two constraints, eqs [22] and [23].

When the monopoly sells to a single firm, its maximization is subject to the three equilibrium conditions: the two first-order conditions for generic sales and Firm 1’s first-order condition for new-product sales, which can be written as functions of m. We invert the equation for Firm 1’s new-product sales to write m as a function of . The result is

[24]

We use this equation to eliminate m from the remaining two equations (generic sales of the two firms) to obtain expressions for and as functions of . By adding the resulting two equations, we obtain the expression for aggregate generic sales as a function of aggregate new-product sales, again leading to eq. [22]. ■

The equilibrium level of new-product sales when the upstream monopoly sells to a single firm, conditional on m, is

and the monopoly’s optimal level of m is

[25]

Conditional on the optimal level of m, the equilibrium level of new-product sales with one vendor is

The equilibrium level of new-product sales when the monopoly sells to two firms, conditional on m, is

and the optimal level of m is

[26]

Eq. [25] shows that when the monopoly sells to a single firm, it sets (a unit subsidy) if and (a positive markup) if . Eq. [26] shows that the markup is always positive, when the monopoly sells to two firms. In view of these two results, for . To show that the markup with a single firm is larger than the markup with two firms when C is close to its upper bound, b, we need to compare the markup for large C. We have

Evaluating the term in square brackets at (the supremum of C), we have

Because , we conclude that for C close to (but smaller than) , .

Consumer welfare

Proposition 11If, then a representative consumer has higher utility when the monopoly sells to a single downstream firm. Ifandis “sufficiently small” (in a sense made precise in the proof), then consumers have higher welfare when the monopoly sells to two downstream firms. The consumer is indifferent between the two alternatives if.

The proof of this proposition relies on Lemma 1 and the following lemma that collects several properties of the indirect utility function that stem from the linear relationship described by Lemma 1. Let be the indirect utility function for a representative agent, y be income, and (a function of prices and income) be the marginal utility of income.

Lemma 2(i) Holding y fixed, as aggregate sales ofincreases andadjusts as specified by eq. [22], the change in utility is

[27]

(ii) For, at every point on the line given by eq. [22], so utility reaches its maximum at the intercept of eq. (22)

[28]

(iii) For, for small, and V reaches a minimum at an interior point on the line where

[29]

The maximum of V might be at either intercept of eq. [22].

Proof. (Lemma 2) (i) Totally differentiating the indirect utility function, holding y constant, dividing the result by , and using Roy’s identity implies

where the last line uses the total derivatives of the inverse demand equations [8]. Divide both sides of the final equation by to obtain

Simplify this expression using eq. [22] to eliminate and noting that along the line in eq. [22], .

(ii) Because the sign of the right-hand side of eq. [27] is the sign of the change in indirect utility due to an increase in , evaluated on the line given by eq. [22]. By inequalities [10], the coefficient of on the right-hand side of eq. [27] is positive, so for , V is maximized at the corner given by eq. [28].

(iii) For , V is decreasing on this line in the neighborhood of the corner . Setting implies eq. [29]. Finally, we need to show that this value of is less than the value at the intercept, . Subtracting these two values, we have

where the inequality follows from inequalities [10]. ■

We now provide the proof of Proposition 11.

Proof. Conditional on the optimal level of m (see “Proof of Proposition 7” in Appendix), the equilibrium level of new-product sales with two vendors is

The difference between new-product sales in the two cases is

Inequalities [10] imply that the sign of is the same as the sign of .

If so that , consumer welfare is higher when the monopoly sells to a single firm, because utility is increasing in by Lemma 2(ii).

If so that , by Lemma 2(iii), V is increasing in for . Therefore, for , a sufficient condition for consumers to be better off with two downstream firms selling the new product is . Using the definition of in eq. [29], we have

[30]

where

[31]

Because the denominator in the last line of eq. [30] is positive, a necessary and sufficient condition for is . Define and write in terms of :

This expression shows that for small , . A sufficient condition for is that is smaller than the smallest positive root of .

If , then , so sales of both the new and the generic products are the same regardless of whether the monopoly sells to one firm or two firms. Consequently, consumer welfare is also the same in the two cases.

A more general linear demand

The monopoly still prefers to sell to two firms, if and only if the cost F is sufficiently small. Its increase in profits from selling to two rather than to one firm (exclusive of the additional cost F) is

which is independent of .

Calculations parallel to those described in “Proof of Proposition 3” Appendix show that the increase in the generic price if Apple sells to two rather than to one firm is

The equation simplifies to eq. [20] for . A sufficient condition for the generic price to be higher when the monopoly sells to two rather than to one firm is . For and sufficiently large in absolute value, the generic price is lower when the monopoly sells to two firms. For , define

which has the same sign as .

The necessary and sufficient condition for is

[32]

The empirically relevant case is where both (the reservation price for the new product, net of production costs, exceeds the reservation price for the generic), and where (the income elasticity of the new product exceeds that of the generic). This situation corresponds to the second line of inequality [32]; here, the generic price is lower when the monopoly sells to two firms rather than to one firm, if .

The difference between the new-product price when the monopoly sells to two firms rather than to a single firm is

which simplifies to eq. [21] for . The necessary and sufficient condition for is

[33]

As noted above, the empirically relevant setting is and , corresponding to the second line of inequality [33]. In this case, the new-product price is higher when the monopoly sells to two firms rather than to one firm, if and only if is not greater than .

Consequently, if and , both the new and generic prices are lower when the monopoly sells to two vendors.

The markup when the monopoly sells to a single firm is

The necessary and sufficient condition for this markup to be positive is

If Apple sells to both firms, its markup is

which is positive provided that , a negative number.