When Should a Monopolist Improve Quality in a Network Industry?

A.1 Multiproduct monopolist

Consider first the optimal behavior of the monopolist in the policy where it sells both products, and the market remains uncovered.

A.1.1 The monopolist does not cover the market

We proceed by analyzing the second period problem of the monopolist, given the choice of first period price, . Posit that the monopolist decides to sell goods 1 and 2 in the second period, without covering the market. Since , the market is not covered if the indifferent consumer between buying good 1 and not buying is strictly larger than 0. That is, we must have , which is equivalent to . We will now obtain the optimal prices and then verify if they are consistent with the hypothesis of uncovered market by the multiproduct monopolist.

Lemma 1When the monopolist decides to sell both products in period 2, there exists a positive value of, say, such that the optimal prices are consistent with the option of not covering the market whenever. In this interval of, the initial price of good 1 is positive.

Proof. See Appendix B. ■

Lemma 1 states that for sufficiently small, the monopolist has the option of selling both goods in the second period, leaving the market uncovered. When , the monopolist chooses prices given by

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with corresponding total profits

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Notice that , and thus, if we define introductory pricing by requiring an increasing path of prices over periods, introductory pricing is indeed a candidate optimum. However, and consequently, the installed base is not maximized in period 1. Intuitively, as is low, the consumers do not place enough importance on the network to induce the monopolist to decrease the initial price to zero, foregoing profits in period 1.

A.1.2 The monopolist covers the whole market

Consider now the second period problem when the monopolist decides to cover the market with goods 1 and 2. Given that , the market is covered when .

Lemma 2When the monopolist decides to sell both products in period 2, there exists an interval ofvalues, namelysuch that there exist optimal prices consistent with a covered market. In this interval ofvalues, the optimal pricing for first period is positive if, where, and zero otherwise.¹⁵

Proof. See Appendix B. ■

Lemma 2 states that for some values of the quality differential and the network effect intensity, selling both products and covering the market is a candidate optimum. The corresponding prices are given by

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when , and

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when .

The total profit is given by

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Corollary 2Under the multiproduct innovation policy, the optimal pricing behavior of the monopolist is consistent either with a covered market with initial zero price or with an uncovered market.

The optimal pricing behavior is consistent with an uncovered market when , while it is consistent with a covered market with initial zero price when . In Appendix B, we show that when , thus proving that and are disjoint intervals.

A.2 Strict quality improvement

Now, consider that the monopolist sells good 1 in the first period, while it decides to substitute good 1 for good 2 in the second. The monopolist avoids cannibalizing sales of good 1 in the second period by introducing a high-quality good. Given that network effects are delayed by one period, it disregards the impact of first period sales in period 2 and sets monopoly prices in both periods. The optimum is, thus, given by and , and total profits are .

A.3 No quality improvement

Finally, assume that the monopolist decides to sell only good 1 in the two periods, disregarding completely quality improvement.

Lemma 3When the monopolist decides to disregard the quality improvement, there is a threshold value for, namely, , such that the optimal monopolist solution is given by positive initial pricing whenand zero initial pricing when.

Proof. See Appendix B. ■

The corresponding prices are given by

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for , and

when .

From Lemma 3, we know that, in a solution where the monopolist disregards the quality improvement, the network is maximal only if is not too large with respect to the quality of good 1. Introductory pricing is always chosen at the optimum, however, the initial price is zero only if is high enough.

B.1 Lemma 1

Proof. When the market is uncovered, the monopolist solves the following optimization problem:

The first-order conditions are

The optimal solution is

and the equilibrium demands are

We must guarantee that that is, which occurs when

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Condition [19] depends on and consequently on Its fulfillment can only be assessed after disclosing the optimal choice of The monopolist chooses to maximize overall profit

This function is concave when and convex otherwise. Solving the inequality for we have that whenever the profit function is concave. When the function is concave, the optimum is attained at the root of the first derivative of , namely,

This price, belongs to the admissible interval if and only if . This condition can be written as . If and , the function is concave, however, Finally, when , the function is convex and We must now verify whether condition [19] is verified when both and If , the condition [19], for the existence of an uncovered market optimal solution, is However, since we assume , whenever the optimal solution is the condition is never met. Notice that is zero when Also, there exists an uncovered market solution when if However, so, no uncovered market solution exists with .

If , condition [19] can be rewritten as

and it is satisfied when It is easy to show that . Hence, an uncovered market solution exists in the domain , and it is equal to

B.2 Lemma 2

Proof. The monopolist solves the following problem:

Writing the Lagrangian, we have

and the Kuhn–Tucker conditions are

The unique finite solution for this problem occurs, when , and We must check whether the demands are positive and smaller than 1 at this solution. The demand of good 1 evaluated at the optimal prices is , and we must guarantee that

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In the first stage, the monopolist chooses to maximize overall profit given by

This function is concave when , or equivalently when , and convex otherwise. Whenever the function is concave, it attains its maximum at This value of is strictly lower than . However, if , or equivalently, , it can assume negative values, in which case the optimum is attained in the boundary of the admissible set, namely, at . When the global profit is convex, it is easily seen that the optimum is attained at . In summary, for , the function is convex, and ; for , the profit function is concave, and the optimum is again attained at ; and for , the profit function is concave, and the optimal is positive.¹⁶

We must now check whether the solutions , and , allow to maintain consistency with the initial condition for the existence of the second period covered market equilibrium, condition [20]. When the solution is valid if and only if . As is a solution if and only if , the first and second period solutions are valid in the domain that can be shown to be non-empty (for ).

When condition [20] implies that

which is equivalent to This expression is true whenever , which happens for , the domain where the solution is valid. So, it is possible to have , inducing a covered market solution in the second stage. ■

B.3 Corollary 2

Proof. Here, we show that . Note that

The RHS is positive for (it is increasing in for , and 0 for ). Taking squares,

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B.4 Lemma 3

Proof. The demand of the monopolist is given by the mass of consumers whose valuation is above price, that is,

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The demand at period 2 is given by

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In the second period, the monopolist solves the following problem:

The Lagrangian of the problem is the following:

and the Kuhn–Tucker conditions are

This system has two solutions. We consider first the case where the restriction is not binding, i.e. and The demand must be positive and less than 1, and this occurs if and only if The condition that must be satisfied is

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If the solution is valid, the monopolist chooses to maximize:

This function is concave when in which case the profit is maximized for So, when , the profit function is convex, and the optimum is obtained at . When , the function is concave, but the optimum is attained in the lower boundary of the admissible set for namely When the function is concave and We must now check whether or are consistent with condition [23]. For we must have however, the solution occurs only when , and thus this solution is not consistent with the second period optimal solution given by As for the condition [23] boils down to , which is true whenever situation where is effectively optimal.

We analyze now the possibility that the restriction is binding. In that case, and . The condition for this solution to be valid is

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If this solution is valid, the monopolist maximizes in the first period, the following overall profit function:

This function is overall concave, and the optimum is attained at If , if . We must now check whether the condition [24] works. Consider first , then we must have , which coincides with the area of parameters where is effectively optimal. Now, if , the condition is . However, for , is not optimal. The condition is not met. ■

B.5 Theorem 1

Proof. We start by identifying the monopolist’s options as follows:

1. Selling both goods, leaving a part of the market unserved, and not practicing introductory pricing. The profit is

2. Selling only good 1 in the second period and not practising introductory pricing (no improvement option). Profit is

3. Selling only good 2 in the second period and obtaining profit

4. Selling only good 1 in the second period and practising introductory pricing (no improvement option). Profit

5. Selling both goods, covering the market, and practising introductory pricing. The profit is

6. Selling both goods, covering the market, and not practising introductory pricing. The profit is .

Lemma A.1Wheneverwe have

Proof. First we show that

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Whenever , we obtain From Corollary 2, it follows that We must just show that

Whenever , we obtain ■

The monopolist faces the following options for the different subsets of the parameter space,

• Case 1: the monopolist chooses between options (a), (b), (c) and (f).

• Case 2: the monopolist chooses between options (a), (c), (d), and (f).

• Case 3: the monopolist chooses between options (c), (d), and (f).

• Case 4: the monopolist chooses between options (c), (d), and (e).

• Case 5: the monopolist chooses between options (c) and (d).

Case 1: we show first that (a) dominates (b), then that (a) dominates (c), and finally that (a) dominates (f)

Also,

Finally,

It is easy to see that the last two conditions hold for .

The optimal choice of the monopolist is to sell both products, uncovering the market and not practising introductory pricing.

Case 2: the monopolist may choose between options (a), (c), (d), and (f). As shown in Case 1, (the proof was independent of the interval of values for the parameter ). Now, we show that The strategy that we will use is the following: First, we show that whenever and then we use the result in Case 1 to conclude that .

We can easily show that whenever Therefore, the above inequality can be rewritten as:

Then, . Finally, we must show that . We have that , whenever . It can be shown, after some lengthy computations, that this is the case in the domain and .¹⁷

Case 4: the monopolist chooses between (c), (d), and (e). We have that

and .

Then, the optimal solution is to cover the market selling both products.

Case 5: the monopolist chooses between options (c) and (d).

If this inequality holds for the smallest value of , it holds everywhere. For , we have

And so, we can conclude that holds.

Finally, for Case 3, we must check now how compares to and to The sign of the expression depends on the sign of under concavity of the profit function. This polynomial has a single root which is always negative. At , the polynomial is positive, which means that it is positive for all values of So, we have that . Now, we compute

The denominator is positive. The numerator is a polynomial of the second degree with two roots, one negative and one positive. We obtain that, within the interval that we are analyzing, the function is positive, i.e. in the interval . If , we know that in the interval , and thus . The monopolist would then choose to sell only good 2 in period 2. However, we can guarantee that if and only if (this is the reason why we have assumed that ). Likewise, if , in the interval . So, the monopolist prefers to sell both goods and cover the market, rather than selling only one of the qualities. ■

B.6 Corollary 1

Proof. The proof of the first part of Corollary 1 follows from the proof of Theorem 1. Here, we prove the second part of Corollary 1. When , we have to solve the problem of a multiproduct vertically differentiated monopolist. As mentioned before, a monopolist will never cover the market if network externalities are not present. The question, which variety or varieties will it offer. Offering both varieties, its profit function is given by

Demands are positive if and only if Also, we assume that at least the agent with the highest valuation is willing to buy either good, i.e. and . Given these restrictions the profit is maximized for

This implies that

So, in practice, prices are set such that no consumer buys good 1. There is strict improvement of the quality. ■

B.7 Theorem 2

Proof. Introductory pricing occurs when the optimal price in period 1 is lower than the optimal price in period 2. We will now compare and (obtained in Appendix A) for the optimal innovation policies in each of the intervals of .

We can compute

Notice that the numerator of the expression is positive, whereas the denominator is positive under the condition that the profit function for the multiproduct uncovered market case is concave, as specified in the proof of Theorem 1. So, and introductory pricing is optimal. Moreover, , since (see proof of Theorem 1).

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We can compute

For , the numerator is positive. The denominator is also positive under the condition that the profit function for the multiproduct covered market case is concave, as specified in the proof of Theorem 1. So, and introductory pricing is optimal. Moreover, , since (see proof of Theorem 1).

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Then,

B.8 Theorem 3

Proof. First, we compute the welfare of each alternative. In the uncovered market case, it is easy to obtain

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and

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For the covered market solution with

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For the covered market solution with ,

and

Also, if strict improvement is the optimal solution, the consumer surplus is

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and the overall welfare is

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Finally, for the no improvement solution, the consumer surplus is given by

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and

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Total welfare is

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Now, we perform the comparison of the welfare in the different subsets of parameter’s space.

Case 1: , first of all we compare and . Both these functions are convex and increasing, and at , and at , . Given this, there exists a level of such that for , and for . Concerning the other options we have seen that . Now, we show that also It is easy to see that can be written as where K is positive and , such that

is always verified. We can thus conclude that . Also, and , so, So, in this case, the optimal policy in terms of welfare is to choose an uncovered market solution, if is lower than , and a covered market with positive prices in period 1, otherwise.

Case 2: As before, . And now, also , because and . Monotonicity and convexity of and imply that results from Case 1 are also valid in this domain of parameters and, as such . In this case, the best policy is to uncover the market.

Case 3: follows from Case 2. .

Case 4: first we show that ,

Now, we show that

Case 5: We know that for , . Given that , whenever the result holds.

Therefore, as long as , the solutions chosen through welfare maximization coincide with the solutions chosen by the monopolist. ■