Title: Product Differentiation and Oligopoly A Network Approach
Domain: economics

Question: design a general equilibrium model of oligopoly where firms compete in a network of differentiated products. Start by defining a hedonic demand system derived from a representative agent's quadratic utility function over product characteristics. The similarity between products, represented as cosine similarity of their characteristic vectors, should endogenously determine the structure of competition. Proceed to solve for the Cournot-Nash equilibrium. The central task is to decompose each firm's equilibrium markup into two fundamental, orthogonal components: a measure of "hedonic-adjusted productivity" reflecting quality and cost efficiency, and a measure of "product market centrality" reflecting the firm's position in the competitive network. Finally, demonstrate mathematically how this centrality measure dictates the distribution of surplus between producers (as monopoly rents) and consumers.

Background: This problem is situated in the field of industrial organization, specifically concerning models of imperfect competition and product differentiation. Classic approaches to modeling differentiation include spatial models (like Hotelling's line), representative agent models with a preference for variety (like Dixit-Stiglitz), and discrete choice models (like Logit or Probit). A parallel tradition is the hedonic approach, which posits that consumers derive utility not from products themselves, but from the underlying characteristics they possess. This allows for a more continuous and nuanced view of product substitutability. The current problem merges the hedonic approach with modern network theory. By representing products as nodes in a "product space," where the links between them are weighted by their similarity, one can apply tools from network science. Concepts like network centrality, traditionally used to measure the importance of individuals in a social network, can be adapted to quantify a firm's structural market power based on its product's position relative to its competitors.

Definitions: Product Space: Each product i is represented as a strictly positive, unit-length vector a_i in an m-dimensional space of "common characteristics." The matrix of these vectors is A. | Competition Matrix (Σ): The intensity of competition between any two products i and j is defined by σ_ij = α(a_i'a_j), where α is a parameter governing horizontal differentiation. The off-diagonal entries of the competition matrix are Σ_ij = σ_ij. | Hedonic-Adjusted Productivity (ω_i): A novel, unit-free measure of a firm's efficiency. It is defined as the ratio of the marginal utility of the first unit produced (a measure of quality, b_i) to its baseline marginal cost (c_i^0). Formula: ω_i = b_i / c_i^0. | Product Market Centrality (χ_i): An exogenous measure of a firm's structural market power. It captures how "isolated" a firm is in the network of product characteristics, summarizing the entire matrix of cross-price derivatives into a single scalar for each firm. It ranges from 0 (monopolist) to 1 (perfectly competitive). | Monopolistic Markup (μ̄_i): The markup a firm would charge if it had zero degree centrality (i.e., no competitors). It is determined solely by the firm's hedonic-adjusted productivity. Formula: μ̄_i = (1 + ω_i) / 2.

Methodology: To find the Cournot-Nash equilibrium, formulate the Cournot Potential function Φ(q). The equilibrium quantity vector q^Φ is the unique maximizer of this function. This approach elegantly transforms a complex multi-agent game into a single-variable optimization problem. | The equilibrium conditions, particularly the relationship between price-cost margin and quantity, are derived by analyzing the first-order conditions (FOCs) of each firm's profit maximization problem (∂π_i/∂q_i = 0). | The core decomposition of the markup and the link between centrality and surplus sharing are established through a series of direct algebraic substitutions. The key is to use the definitions of χ_i, M_i, and the equilibrium conditions (p_i-c_i=q_i and q_i = f(d_i)) to express endogenous variables in terms of exogenous ones. | The entire system can be solved concisely using matrix algebra. The equilibrium quantity vector q^Φ is found by computing the inverse of the matrix (2I + Σ). This inverse matrix is known as the Bonacich centrality matrix and is fundamental to deriving the firm-level centrality measures.

Answer: I. Detailed Model Formulation and Equilibrium Derivation | Demand Side Derivation: | The model begins with a representative agent's quadratic utility over common characteristics (x) and idiosyncratic characteristics (y), with linear disutility for labor (H): | U(x,y,H) = α(x'b^x - (1/2)x'x) + (1-α)(y'b^y - (1/2)y'y) - H | Product quantities q are mapped to characteristics via x = Aq and y = q, where A is the m x n matrix of unit-length characteristic vectors a_i. Substituting this into the utility function and rearranging terms yields a utility function in terms of goods q: | U(q, H) = q'b - (1/2)q'(I + Σ)q - H | where b is a vector of quality/demand intercepts and Σ is the competition matrix with off-diagonal elements Σ_ij = α(a_i'a_j), capturing product similarity. | The consumer chooses q to maximize their surplus, U(q, H) - (p'q - H). The first-order condition (FOC) yields the linear inverse demand system: | p(q) = b - (I + Σ)q. | Supply Side and Cournot-Nash Equilibrium: | Firm i's profit is π_i(q) = p_i(q)q_i - h_i(q_i), with a linear marginal cost c_i. The FOC for firm i maximizing its profit is p_i + (∂p_i/∂q_i)q_i - c_i = 0. From the inverse demand, ∂p_i/∂q_i = -1. Substituting this gives a fundamental equilibrium condition: the price-cost margin equals the quantity produced. | p_i - q_i - c_i = 0. | II. The Detailed Markup Decomposition | Relating Quantity to Centrality: | Define the (endogenous) output-weighted degree centrality of firm i as d_i = Σ_{j≠i}σ_{ij}q_j. Substitute the demand function into the firm's FOC: | (b_i - q_i - Σ_{j≠i}σ_{ij}q_j) - q_i - c_i = 0 | Solving for the equilibrium quantity q_i gives: | q_i = (b_i - d_i - c_i) / 2 | Deriving the Markup Decomposition: | Define the (exogenous) Product Market Centrality χ_i as the normalized degree centrality: χ_i = d_i / (b_i - c_i). This implies d_i = χ_i(b_i - c_i). Substitute this back into the equilibrium quantity equation. Since p_i - c_i = q_i, we get: | p_i - c_i = (1 - χ_i)(b_i - c_i) / 2 | Now, express the markup μ_i = p_i / c_i. | μ_i = 1 + (1 - χ_i)(b_i - c_i) / (2c_i) | Define the Monopolistic Markup μ̄_i as the markup when χ_i=0 (no competition), which can be written in terms of Hedonic-Adjusted Productivity (ω_i = b_i/c_i) as μ̄_i = (ω_i + 1)/2. | Substituting μ̄_i into the general markup equation and simplifying gives the final decomposition: | μ_i = χ_i * 1 + (1 - χ_i) * μ̄_i | This result shows that the equilibrium markup is a convex combination of the competitive markup (1) and the monopolistic markup (μ̄_i), with the weight determined by the firm's structural centrality (χ_i). | III. Centrality and the Division of Surplus | Surplus Definitions: | Monopoly Rents (Producer Surplus): r_i = (p_i - c_i)q_i. From the FOC p_i-c_i=q_i, this simplifies to r_i = q_i^2. | Consumer Surplus generated by firm i: The total consumer surplus can be decomposed, and the portion attributable to the consumption of good i is given by s_i(q) = (1/2)(q_i^2 + q_i d_i). | Ratio of Rents to Consumer Surplus: | The ratio is r_i / s_i = q_i^2 / [ (1/2)(q_i^2 + q_i d_i) ] = 2q_i / (q_i + d_i). | Define the Weighted Market Share as M_i = q_i / (q_i + d_i). Thus, the rent-to-surplus ratio is directly proportional to this weighted market share: r_i / s_i = 2M_i. | Linking Centrality to Market Share: | The goal is to express M_i solely in terms of the exogenous centrality χ_i. We use the two relations d_i = χ_i(b_i - c_i) and b_i - c_i = 2q_i + d_i. Substituting the second into the first gives d_i = χ_i(2q_i + d_i). Solving for d_i in terms of q_i and χ_i yields: | d_i = (2χ_i / (1 - χ_i)) * q_i | Substitute this expression for d_i back into the definition of weighted market share M_i: | M_i = q_i / (q_i + (2χ_i / (1 - χ_i))q_i) | Canceling q_i and simplifying gives the final relationship: M_i = (1 - χ_i) / (1 + χ_i). | Final Result: | By combining these results, we can express the division of surplus as a direct function of a firm's structural network position: | r_i / s_i = 2 * ( (1 - χ_i) / (1 + χ_i) ) | This equation demonstrates that a firm's ability to capture surplus is entirely determined by its product market centrality, a measure derived from the underlying topology of the product space.

Checklist: 1. Correctly derives the GHL inverse demand system p(q) = b - (I + Σ)q from the specified quadratic utility function.  
2. Identifies the game as a potential game and correctly states the Cournot Potential function Φ(q) whose maximization yields the Cournot-Nash equilibrium.  
3. Correctly defines the two core components: Hedonic-Adjusted Productivity (ω_i) and Product Market Centrality (χ_i).  
4. Successfully derives and states the main markup decomposition theorem: μ_i = χ_i * 1 + (1 - χ_i) * μ̄_i.  
5. Correctly defines the weighted market share M_i and derives its relationship with monopoly rents and consumer surplus (r_i/s_i = 2M_i).  
6. Demonstrates the mathematical derivation linking the exogenous Product Market Centrality (χ_i) to the endogenous Weighted Market Share (M_i), yielding M_i = (1 - χ_i) / (1 + χ_i).  
7. Combines the previous steps to show the final relationship between a firm's structural network position (χ_i) and its ability to capture surplus, i.e., r_i/s_i = 2 * ((1-χ_i)/(1+χ_i)).