Do
Multiproduct Firms with Market Power
Cross-Subsidize?
by
Dale S. Bremmer
Professor of
Economics
HSS Department
Rose-Hulman
Institute of Technology
5500 Wabash Avenue
Terre Haute , IN 47803
Email:
dale.Bremmer@rose-hulman.edu
Presented at the “Market
Structure” Session of the
76th Annual WEA
International Conference
Do Multiproduct Firms with
Market Power Cross-Subsidize?
I. Introduction
Firms
with multiple products and market power are often alleged to charge higher
prices on some of their products to cover the high cost they incur in producing
other products. There are numerous
anecdotal examples such as the argument that colleges allegedly cross-subsidize
the cost of graduate education with the monies from undergraduate
education. Paper producers claim
mandatory recycling of their products is used to cross-subsidize the less
profitable recycling of cans and bottles. The high stamp prices in
Using a simple model with linear demand curves and constant marginal costs, this paper demonstrates that a two-good monopolist facing demand curves that are known with certainty will not increase the price of one good because of an increase in the cost of producing another, unrelated good. With certain demand, a monopolist will not cross-subsidize the increase in the cost of one good by charging a higher price on another good. However, once uncertainty is introduced, the paper shows that a monopolist that maximizes expected utility will increase the price of the good facing uncertain demand given an increase in the cost of an unrelated good which faces a certain demand curve. So it is in the face of uncertainty that the monopolist will charge higher prices for some goods in response to higher cost for another good. However, if the cost of producing the good with uncertain demand increases, the price of the good with certain demand is not affected.
Following these introductory comments is a brief review of the literature upon which the cross-subsidization model developed in this paper is based. The paper’s third section presents two models - - one where both of the monopolist’s products faces certain demand and another where one of the products faces uncertain demand. Conclusions and thoughts about future research are discussed in the fourth and final section of the paper.
II. Review of the Literature
This paper’s genesis has roots in three branches of economic literature: the issues concerning price theory and uncertainty, the output-price decisions of a multiproduct monopolist, and the cross-subsidization literature. These different areas of the literature are discussed in that order.
Price theory and uncertainty
Mills (1959) analyzed the effect of uncertainty on a monopolist’s pricing decisions, finding the results depends on the shape of the monopolist’s marginal cost curve. For example, in the case of constant marginal cost, Mills determined that the monopolist’s optimal price will be lower with uncertainty than without. In a dynamic theory that emphasizes the role of inventories of the finished product, Zabel (1970) also examined the pricing behavior of a monopoly with uncertain demand. He demonstrates that the monopolist’s optimal level of inventory will fall as the holding cost of inventory increases. Given a uniform distribution of demand and constant marginal cost, Zabel finds a monopoly will increase its price as the holding cost of inventory also increases.
In an often-cited work, Sandmo (1971) investigates the output decisions of a competitive firm under price uncertainty.[1] He finds two key facts. First, under price uncertainty, the output of the competitive firm is less than the output that would occur with price certainty. Secondly, if decreasing absolute Arrow-Pratt risk aversion is assumed,[2] then a competitive firm with price uncertainty will reduce output as its fixed cost increases. Leland (1972) extends Sandmo’s result to the theory of monopoly under uncertainty. He finds that under uncertainty a monopolist’s price and output decision are not invariant to changes in fixed cost. If the demand curve exhibits what Leland calls the “principle of increasing uncertainty,” then the quantity-setting monopolist will produce a smaller output than the certainty amount where a known marginal revenue curve intersects the marginal cost curve.
Harris and Raviv (1981) use a model of demand uncertainty, and they find that endogenously derived pricing schemes for a monopolist depend on capacity constraints. In their results, an optimal single price exists only if capacity constraints are not binding. Using a capital asset pricing model that explicitly includes risk, Brick and Jagpal (1981) examine a monopoly’s decisions regarding price and advertising under uncertainty. Not surprising, they find increases in demand leads to increases in the monopoly price. The optimal level of advertising, however, depends on how responsive the risk-adjusted price elasticity of demand is to changes in advertising.
Multiproduct firms
One of the first papers to study the implications of uncertain demand on a multiproduct monopoly was the work done by Dhrymes (1964). He decomposed this problem into two components. First, the monopolist determines the optimal output mix by maximizing expected utility. Next, given this optimal output mix, the monopoly determines the optimal combination of inputs by minimizing cost. Dhrymes concludes that the qualitative results of his model are similar to those of a uniproduct firm; but, the multiproduct firm’s response to changes in the state of the uncertainty is more complex than that of a uniproduct firm. Exogenous shocks to the state of uncertainty include both changes in the firm’s attitude toward risk and changes in the underlying probability distribution function that characterizes the firm’s risk.
Meyer (1975, 1976) extends the analysis of a monopoly under uncertainty to a monopoly with multiple outputs and multiple inputs. In his 1975 paper dealing with simultaneous pricing and capacity decisions under uncertainty, Meyer found that the optimal investment decision usually entailed some excess capacity. His 1976 paper applied components of the capital asset pricing theory to directly incorporate the market price of risk. In this paper, Meyer found that the optimal pricing structure depended on the marginal risk associated with each distinct group of customers. One interesting result was his finding that optimal pricing may involve selling output to several groups of customers at a price below the marginal production cost.
The cross-subsidization
literature
When
the economic literature refers to cross-subsidization, often it is in reference
to the behavior of a regulated monopolist.
Church and Ware (1999, p.797) state cross-subsidization exists “if the
revenues from a product are less than its costs of production.” An example would be ATT’s cross-subsidization
of low local rates with higher long-distance rates that occurred before its
breakup in the 1980s. A current example
of a regulated firm using cross-subsidization would be Palmer’s (1992)
conclusion that local businesses in
In these studies, cross-subsidization usually referred to the use of profits from one activity to cover the losses in another activity. This paper uses an expanded definition of cross-subsidization. The question here is whether multiproduct firms will increase the price of one product given an increase in the per unit cost of another, unrelated product.
III. The Model
Suppose a profit-maximizing, quantity-setting monopolist sells two products, good 1 and good 2, in separate markets.[3] Symbolically, q1 is the output of good 1, q2 is the output of good 2, p1 is the market price of good 1, and p2 is the market price of good 2. The monopolist faces a linear demand curve in each market; so p1 = a0 – bq1 and p2 = g0 – hq2, where a0 and g0 are the positive vertical intercepts of the demand curves, and b and h are the absolute values of the slopes of the two demand curves. Each product is produced with constant per unit cost. The constant marginal cost of good 1 is c1, while c2 is the constant marginal cost of good 2.
Goods 1 and 2 are assumed to be unrelated. Changes in the price of one good does not affect the demand for the other good.[4] In addition, there are no synergies in production as c1 and c2 are unrelated.[5] First, these assumptions are used in a model with complete certainty about the demand for both products. The results from this simple model with certainty are later compared to the outcome of another model where the demand for good 1 is uncertain.
The case of certain demand
The monopolist chooses q1 and q2
to maximize profits, p, which equals
Differentiating equation (1) with respect to q1 and q2 results in two first-order conditions:
and
The
profit-maximizing amount of each good, 
and 
, is found by solving equations (2) and (3) to find
and
The key result from the model with
certainty can be seen from equations (4) and (5). Given these optimal solutions, note that 
. In the case of
certainty, the monopolist’s profit-maximizing output in one market is invariant
to the monopolist’s per unit cost in another, unrelated market. A change in c2 will not affect the
monopolist’s profit-maximizing price for p1 just as the firm’s
optimal choice of p2 is invariant to a change in c2. This
result is intuitive. If a monopolist
is charging the profit-maximizing price for one product, it is common sense
that the firm will not change this price given a change in the production cost
of an unrelated product. Any deviation
in the price of the good whose cost did not change from its initial
profit-maximizing level will decrease total profits more than the initial
increase in the other good’s per unit cost.
Thus, in this static model with certainty, a profit-maximizing,
multiproduct monopolist does not cross-subsidize increases in the per unit cost
of one good with increases in the price of an unrelated good that it also
sells.
The case of uncertain demand
To introduce uncertainty in the model, good 1 is assumed to have uncertain demand. Suppose there are two states of the world. In the first state, State 1, which occurs with probability z1, the demand curve for good 1 is p1 = a0 – bq1. State 2, the second state, occurs with probability z2; however, the demand for good 1 is p1 = a1 – bq1 where a1 > a0. Thus, uncertainty in the demand for good 1 is captured by an intercept shift. Profits in State 1, p1, equal the sum of the certain profits from the sale of good 2 and the profits from selling good 1 at the lower demand or
Likewise, the profits the monopoly earns in State 2, p2, equal the sum of the certain profits generated by the sale of good 2 and the profits from the sale of good 1 with the increased demand or
Given the uncertainty about the
demand for good 1, the monopolist determines the optimal values of q1
and q2 by maximizing its expected utility function, 
, where U(x) is the utility of income.[7]
Differentiating E(U) with respect to q1 and q2 results in two first-order conditions or
and
Given that both z1 and z2
are positive fractions, and the marginal utility of income, 
and 
, are positive, then equation (9) implies that (g0
– 2hq2 – c2) equals 0.
Thus, the multiproduct monopolist will produce that level of q2
that equates the marginal revenue in the second market to the marginal cost in
the second market. The monopolist
produces the certainty output in the market where demand is certain. In other words, the optimal resulting from
equation (9) is the same level of q2 resulting from equation (3)
when the demand for both products is known with certainty, namely 
. This result
anticipates the comparative statics below that show the optimal level of q2
is invariant to changes in c1.
Given the
assumptions about the zi and the
, 
equation (8) implies that the two terms - - (a0 –
2bq1 – c1) and (a1 – 2bq1 – c1)
- - have opposite signs. Since a1
> a0, it follows that (a0 – 2bq1 – c1)
< 0 and (a1 – 2bq1 – c1) > 0. Given the uncertain demand for good 1, the
optimal level of q1,, must satisfy two conditions. First, atthe marginal revenue associated with the lesser demand for
good 1 is less than the marginal cost of producing good 1. Conversely, the marginal revenue associated
with the greater demand for good 1 is greater than the marginal cost of good 1
at. If q1†
is the profit-maximizing level of output of good 1 with the lesser demand and q1‡
is the profit-maximizing level of output of good 1 with the increased demand,
then as Figure 1 shows, q1† < q1*
< q1‡.[8] Thus, given uncertain demand, the firm’s
optimal output of q1 exceeds the profit-maximizing amount in the
case of the lesser demand but is less than the profit-maximizing amount in the
case of the greater demand.
The sufficient, second-order conditions of this optimization problem involve the second partial derivatives of the expected utility function, E(U). In matrix form, these second partial derivatives are
where
and
Maximizing E(U) requires H11
< 0, H22 < 0, and the determinant of H, 
, which equals H11H22, must be
positive. The off-diagonal terms of
matrix H are zero because
and (g0 - 2hq2 – c2) = 0 because of the first order-condition in equation (9).
Evaluating
equations (8) and (9) at the solutions, 
and 
, and differentiating both of these equations with respect to
c2, a standard comparative statics exercise finds
where
and
Based on the assumptions of the model, θ2 is unambiguously positive, as indicated, and the sign of θ1 is indeterminate. However, as will be shown below, if decreasing absolute Arrow-Pratt risk aversion is assumed, θ1 will be positive. Applying Cramer’s rule to equation (13) obtains
and
In the two equations above, the +
signs appearing above or below certain terms indicates terms that are
unambiguously positive, just as the – signs appearing above other terms denote
terms are that are unambiguously negative; however, the ? sign above a term indicates a term
whose sign is indeterminate. This same
convention in notation is also followed below in equations (20) and (22). The economic interpretation of equation (17)
is straightforward. Given the demand for
a good is known with certainty, if its per unit cost increases, an expected
utility-maximizing, multiproduct monopolist will produce less of it. Since the demand curve for q2 is
downward sloping, then 
. If c2
increases, then a utility-maximizing, multiproduct monopolist will decrease and increase 
.
As
mentioned above, the sign of the expression in equation (16) cannot be
determined unless additional restrictions are placed on the utility
function. θ1 will be
positive if, like Sandmo (1971, p. 68), it is assumed that the utility function
exhibits decreasing absolute Arrow-Pratt risk aversion. If RA(π) is the measure of
absolute risk aversion, then 
, where it is assumed that 
Since 
, the first-order condition in equation (8) can rearranged to
obtain
Since RA(π2) < RA(π1), then
Given RA(π2) > 0, equation (19) ensures that θ1 is positive. With the additional assumption of decreasing absolute Arrow-Pratt risk aversion, then equation (16) becomes
According to equation (20), if the
per unit cost of good 2 - - the good whose demand is known with certainty - -
increases, the monopolist will produce less of good 1, the good with uncertain
demand. Since the demand curve for good
1 is downward sloping, a rise in c2, implies a fall in 
, and an increase in 
, or 
. In the case of
uncertain demand, the multiproduct monopolist will cross-subsidize - - it will
increase the price of an unrelated good given an increase in the per unit cost
of another good. Again this outcome is
intuitive as it expands Sandmo’s (1971) result.
In terms of determining the optimal amount of good 1, the per unit cost
of good 2 acts like fixed cost, and in the presence of uncertainty, an increase
in fixed cost should lead to a decrease in q1.
To
find the effect of a change in c1 on the optimal quantities of q1
and q2, the first-order conditions in equations (8) and (9) are once
more evaluated at the solutions, 
and 
, and differentiated
with respect to c1. This
comparative statics exercise results in the following two-equation system
Solving for 
, equation (22) shows that even in the case of uncertain
demand, if the unit cost of good 1 increases, the monopolist will produce less
of q1, or
Since the demand curve for good 1
is downward sloping, less q1 means a higher p1 or 
. Conversely, if the
demand for q2 is known with certainty, then the firm’s optimal level
of q2 is invariant to the value of c1 or
This result also coincides with
intuition. If the demand for good 2 is
known with certainty, why vary its output given a change in c1? To do so would deviate from a known amount of
output that maximizes profits in market 2, and would reduce expected utility by
more than the initial increase in the cost of good 1. Given no change in q2, 
remains constant, and 
.
IV. Conclusions
This paper investigates how a change in per unit cost of one good affects the prices of a multiproduct monopolist. Using a simple model of a firm producing two goods with linear demand curves and constant per unit costs, two cases are investigated - - one where the demand for both the goods is known with certainty and the other where one of the goods has uncertain demand. Demand uncertainty is captured by assuming two possible states of the world where the demand curve for good 1 shifts parallel to the left or right. When the demand curves are known with certainty, changes in the per unit cost of one good does not affect the price of the other unrelated good. In this case, the profit-maximizing monopolist does not cross-subsidize increases in the cost of one good by charging a higher price on the other good. When the demand for good 1 is uncertain, an increase in the cost of good 2 will lead the expected utility-maximizing monopolist to charge higher prices for both good 1 and good 2. So in the case of uncertainty, a monopolist will cross-subsidize higher cost for one good with a higher price for the other unrelated good, demonstrating a key point of the paper. The next key result of the paper shows that an increase in the cost of the good with uncertain demand will result in less of it being produced, but this increase in cost has no effect on the output and price of the good whose demand is known with certainty.
This simple model opens several avenues for future research. Instead of a simple discrete probability approach to model uncertainty, the model should be expanded to continuous probability to verify that its results are robust. Other forms of demand uncertainty rather than parallel shifts in the demand curve need to be investigated to also ensure the robustness of the results. Additionally, other types of attitudes toward risk could be considered such as risk neutral or a risk lover. The model could be expanded to included economies of scope to get a fuller measure of how changes in cost affect prices of a multiproduct, imperfectly competitive firm. In this case, it may not be necessary to assume demand uncertainty in order to show cross-subsidization. Another extension of the model would be to disaggregate per unit cost into specific input prices and to examine how a change in input prices affect the prices charged by a multiproduct firm. These inputs could be common or unique to each product. Finally, the model may be expanded to show loss-leader behavior and examine the impact of changes in cost on such behavior. This future work would expand the use of the model and add to the literature of multiproduct firms.
References
Bernhardt, Irwin, “’Fixed Costs’ and the Competitive Firm Under Price Uncertainty: Comment,” The American Economic Review, vol. 62, no. 1 / 2, 1972, 193.
Brick, Ivan E. and Harsharanjeet Jagpal, “Monopoly Price-Advertising Decision-Making Under Uncertainty,” Journal of Industrial Economics, vol. 29, no. 3, Mar. 1981, 279-285.
Church, Jeffrey
and Roger Ware, Industrial Organization: A Strategic
Approach. Irwin McGraw-Hill:
Dhrymes, Phoebus, “On the Theory of the Monopolistic Multiproduct Firm Under Uncertainty,” International Economic Review, vol. 5, no. 3, Sept. 1964, 239-256.
Faulhaber, Gerald R., “Cross-Subsidization: Pricing in Public Enterprises,” The American Economic Review, vol. 65, no. 5, Dec. 1975, 966-977.
Harris,
Meyer, Robert A., “Monopoly Pricing and Capacity Choice Under Uncertainty,” The American Economic Review, vol. 65, no. 3, Jun. 1975, 326-337.
Meyer, Robert A., “Risk-Efficient Monopoly Pricing for the Multiproduct Firm,” The American Economic Review, vol. 90, no. 3, Aug. 1976, 461-474.
Mills, Edwin S., “Uncertainty and Price Theory,” Quarterly Journal of Economics, vol. 73, no. 1, Feb. 1959, 116 – 130.
Palmer, K., “A Test for Cross-Subsidies in Local Telephone Rates: Do Business Customers Subsidize Residential Customers,” RAND Journal of Economics, vol. 23, 1992, 415-431.
Sandmo, Agnar, “On the Theory of the Competitive Firm Under Price Uncertainty,” The American Economic Review, vol. 61, no. 1, Mar. 1971, 65 – 73.
Sandmo, Agnar, “’Fixed Costs’ and the Competitive Firm Under Price Uncertainty: Reply,” The American Economic Review, vol. 62, no. 1 / 2, 1972, 194-195.
Shaffer, Sherrill, “Cross-Subsidization In Checking Accounts: Note,” Journal of Money, Credit, and Banking, vol. 16, no. 1, Feb. 1984, 100-109.
Zabel, E., “Monopoly and Uncertainty,” The Review of Economic Studies, vol. 37, no. 2, Apr. 1970, 205 – 219.
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