On the Microfoundation of Linear Oligopoly Demand

2.1 Issue 1: The Objective

Both the Bowley (1924) and the Shubik and Levitan (1980) demand specifications are derived from a representative consumer gross utility function that takes the following general form:

with α, β, γ > 0.^[8] Notice that the way in which we present this objective function, it effectively consists of three distinct parts. Starting with the third, γ ⋅ ( x 2 − x 1 ) 2 , this part captures the complementarity between products x ₁ and x ₂. Consistent with classic consumer theory, utility is ceteris paribus higher with a more balanced consumption plan. In fact, the representative agent is induced to buy both products in equal amounts (x ₁ = x ₂) so that the third-term disutility is minimized and de facto disappears.

The first two parts, α ⋅ ( x 1 + x 2 ) − β ⋅ ( x 1 + x 2 ) 2 , capture utility coming from total rather than relative consumption and express the extent to which the goods are considered substitutes. These components indicate that the consumer derives utility from consuming more products, but only up to a certain amount. That is, utility increases at low levels of total consumption through the first term, but at higher levels of total consumption the second term starts to dominate the first. This implies that there is a point at which the consumer is satiated. Notice that this is true also when the representative agent does not face a budget constraint and holds even when the goods would be offered, in the broadest sense, for free. Contrary to the third term favoring balance in consumption, this therefore is at odds with traditional consumer theory. Indeed, the fact that the objective function has a unique maximum makes that the common assumption of nonsatiation is violated.^[9]

2.2 Issue 2: The Solution

To solve the representative consumer problem, one naturally needs to take account of the cost of consumption. In the following, we point out that the linear oligopoly demand structure will only result from the representative agent’s maximization problem under fairly specific, and arguably strong, assumptions.

Towards that end, let F be a set of vectors ( x , m ) ∈ ℝ n × ℝ and consider the utility function V : F ↦ ℝ . For (x, m) ∈ F, therefore, utility V(x, m) is obtained from consuming an amount of x ∈ ℝ n goods as well as from the unspent money m.^[10] Moreover, let the vector of prices be given by p ∈ ℝ n . If the available income is I, then the representative consumer faces the following general maximization problem:

where p ⋅ x is total expenditure.

It can be easily verified, however, that the linear oligopoly demand system is the solution to:

Thus, in light of the general maximization problem, V(x, m) = U(x) + m and the constraint m ≥ 0 is either ignored or income is (implicitly) assumed large enough. Observe that this specification effectively treats expenditures as a disutility, which is linearly subtracted from the gross utility function. At first sight this may seem natural and innocuous, but it does imply two strong assumptions:

1. The representative consumer’s utility function is quasi-linear in money;

2. The representative consumer can borrow unlimited amounts of money for free.

The first condition means the absence of a wealth effect as utility is linearly increasing in money. Irrespective of whether utility is high or low, the marginal gain of an extra dollar is the same.^[11] The second states that the budget restriction has no bite and is effectively non-existent. Given the quadratic nature of the objective function, this holds trivially when the representative consumer has a sufficiently large income.^[12] Note, however, that this approach makes it irrelevant how much budget is available. Indeed, even when the representative agent loses all his income (I = 0) or would have severe debts (I < 0), demand for both products stays the same.

2.3 Issue 3: The Products

Following the textbook approach and thus assuming A1 and A2, the representative consumer picks x ₁ and x ₂ to maximize:

Goods are substitutes in utility when for every η 1 > 0 there is an η 2 > 0 such that:

That is, every decrease in utility resulting from a reduction in the consumption of good 1 can be compensated by an increase in the consumption of good 2.^[13] If we consider x ₂ an implicit function X 2 ( x 1 ) along a level curve of V, then X 2 ′ < 0 . Implicit differentiation yields:

Rearranging gives:

Thus, X 2 ′ < 0 requires

and

or the reverse.

Note that the above inequalities do not hold for all consumption bundles (x ₁, x ₂). For instance, x ₁ = 0, x ₂ = 1 and β − γ < α 2 < β + γ violates the inequalities. In this model, therefore, utility can remain constant when the representative agent receives more or less of both goods.^[14] One therefore may need a rather peculiar quadratic aggregate utility function to obtain a relatively straightforward demand system. This is arguably problematic since the representative consumer’s utility function is ultimately intended to serve as a microfoundation for linear oligopoly demand.

3.1 Duopoly

Consider the linear duopoly demand system as described above:

In the following, we will show how this demand structure can be derived directly from a population of heterogeneous consumers.

Consider a price-setting duopoly where both firms are located on the boundary of an interval [0, 2]. In particular, and without loss of generality, firm 1 and firm 2 are respectively situated at 0 and 2. There are three types of consumers, each with uniform population density λ i on [0, 2], i = 1, 2, 3. The total number of type i buyers is thus given by 2 ⋅ λ i . Type 1 customers are assumed to obtain positive gross utility when buying from firm 1, s > 0, and no utility when buying from firm 2. By contrast, Type 2 customers attach no value to the products of firm 1 and derive positive gross utility from buying at firm 2, v > 0. Finally, Type 3 customers value both equally and have a willingness to pay of 4 for each.^[15] Consumers either buy one unit of the product or do not buy and are characterized by their location. In the spirit of spatial IO settings, there are costs associated with distance between buyer and seller and these are assumed to be linearly increasing.

Let us now specify the utility function of a Type 1 consumer located at z ∈ [ 0,2 ] . This customer has basically three options: (1) buy from firm 1 (value s − z − p ₁), (2) buy from firm 2 (value z − 2 − p ₂) or (3) buy nothing (value 0). Notice that the third choice dominates the second, because z − 2 − p ₂ ≤ 0 for positive prices. Thus, the utility function of a Type 1 buyer located at z ∈ [ 0,2 ] effectively is

The Type 1 customer who is indifferent between option (1) and option (3) is located at z = s − p ₁.

The utility function of a Type 2 customer located at z ∈ [ 0,2 ] can be determined in a similar fashion and is given by

The Type 2 customer who is indifferent between buying and not buying is thus located at z = p ₂ − v + 2. Finally, the utility function of a Type 3 customer at z ∈ [ 0,2 ] is

Under the assumption that prices are sufficiently low, the indifferent Type 3 buyer is located at z = 1 + 1 2 ( p 2 − p 1 ) .^[16]

On the basis of these utility specifications, we can now derive the corresponding demand functions. For a given combination of prices (p ₁, p ₂) in the relevant range, demand for firm 1 is given by the sum of consuming Type 1 buyers and the part of Type 3 buyers preferring the product of firm 1.^[17]

Demand for the products of firm 2 can be derived in a similar way.

The above approach therefore allows one to derive any linear duopoly demand system of the form:

where λ 1 = b 1 − c , λ 2 = b 2 − c , λ 3 = 2 c , s = a 1 − 2 c b 1 − c and v = a 2 − 2 c b 2 − c .

3.2 Oligopoly

The above duopoly demand structure has been generalized to any number of firms by Häckner (2000). If the set of firms is N, then demand for each firm k ∈ N takes the following form:

with a _k , b _k , c > 0. We will now use a variation on the ‘spokes-model’ introduced by Chen and Riordan (2007) to show how this demand system can be derived with the approach laid out in the previous subsection.^[18]

Consider a price-setting oligopoly where each firm is ‘located’ at an end node of a star-shaped network. The edge from the center to firm i is denoted by E _i . Let the distance to the center (i.e., the length of E _i ) be equal to 1 so that the distance between each pair of firms is 2.

For each firm i ∈ N , there is a Type i consumer with uniform population density λ i on E _i . The total number of type i buyers is thus given by λ i . Type i customers are assumed captive in the sense that they obtain positive gross utility v _i  > 0 when buying from firm i and no utility when buying from any other firm j ∈ N , j ≠ i.

Next, for each pair of firms (i, j), type ij customers value the goods from both firm i and firm j equally and have a willingness to pay of 4 for each. Type ij customers are distributed with uniform population density λ i j along the path E i ∪ ​ E j from firm i to firm j. Consumers either buy one unit of the product or do not buy and are characterized by their location. As before, there are costs associated with distance between buyer and seller and these are assumed to be linearly increasing.

Let us now specify the utility function of a Type i consumer located at z ∈ E i , where z = 0 indicates the location of firm i at the endpoint of E _i and z = 1 is the location of the center. Such a customer has basically three options: (1) buy from firm i (value v _i  − z − p _i ), (2) buy from another firm j, j ≠ i, (value z − 2 − p _j ) or (3) buy nothing (value 0). Notice that the third choice dominates the second since z − 2 − p _j  ≤ 0 whenever p _j is positive. Thus, the utility function of a Type i buyer located at z ∈ E i is effectively given by:

The Type i customer who is indifferent between option (1) and option (3) is located at z = v _i  − p _i .

The utility function of a Type ij customer located at z ∈ E i ∪ ​ E j can be determined in a similar fashion and is given by

Under the assumption that prices are sufficiently low, the indifferent Type ij buyer is located at z = 1 + 1 2 ( p j − p i ) .^[19]

On the basis of these utility specifications, we can now derive the corresponding demand functions. For a given combination of prices p = (p ₁,…, p _n ) in the relevant range, demand for firm i is given by the sum of consuming Type i buyers and the part of Type ij buyers preferring the product of firm i.

The above approach therefore allows one to derive any linear oligopoly demand structure of the form

where λ _k  = b _k  − c, λ k j = 2 c n − 1 , and v k = a k − 2 c b k − c .

This result contrasts with Jaffe and Weyl (2010) which shows that this type of linear demand structure cannot be derived from a discrete choice setting when the number of goods exceeds 2.^[20] In Jaffe and Weyl (2010), the distribution of customers over (realizations of) valuations is given by a smooth probability density function. As a result, the probability of observing a customer who does not value a particular product is zero. The above model does not fit within this framework due to the presence of captive consumers.^[21] We thus conclude that linear oligopoly demand can be consistent with a discrete choice foundation for any number of products when each firm has a loyal customer base.