Notes on Quantity Setting Oligopoly Model

Oligopoly: A market structure characterized by a small number of firms, each holding significant market share, which leads to interdependent pricing and output decisions. In oligopolistic markets, actions taken by one firm directly influence others, necessitating strategic planning and competitive behavior among the firms.
Quantity Setting Model: In this model, firms simultaneously decide the quantities of their products to produce, which subsequently affects market prices based on the total output.

Firms (n firms) decide production quantities, denoted as ( q1, q2, …, q_n ). This denotes that each firm makes independent output decisions while considering the output levels of competitors.
- Total quantity ( Q ) is the sum of individual outputs: ( Q = q1 + q2 + … + q_n ). A crucial aspect of the model is acknowledging how the collective output influences market prices and individual firms' profits.
Industry Demand Equation: The demand faced by the industry can be represented by the equation ( P(Q) = a - bQ ), where ( a ) represents the maximum price consumers are willing to pay when the quantity is zero, and ( b ) is the slope of the demand curve indicating how much the price decreases as more quantity is produced.
Cost Structure: For simplicity in evaluating the model, firms are often assumed to have identical marginal costs (MC), expressed as ( MC_i = c ) for all firms. This assumption allows for a clearer analysis of competitive outcomes without the complications introduced by varying cost structures among firms.

The profit functions for firm ( i ) can be expressed as: ( \Pii(q1, …, qn) = (a - bqi - bQ{-i} - c)qi ). This implies that each firm's profits are determined not only by its output ( q*i ) but also by the total output of competing firms ( Q_{-i} ). The consideration of competitors’ outputs is fundamental in oligopoly settings, where firms must anticipate the actions and reactions of each other.

The equilibrium condition is achieved by setting the marginal revenue ( MRi ) equal to marginal cost ( MCi ). This leads to the firm’s best response function given by: ( q*i(Q{-i}) = \frac{a - c}{2b} - \frac{Q{-i}}{2} ). Here, ( Q_{-i} ) denotes the output of all other firms, reflecting the interconnectedness of firms in determining output strategies.
At equilibrium, it is assumed that all firms produce the same quantity: ( q* = \frac{a - c}{(n + 1)b} ). The total output in the market thus becomes: ( Q* = n \cdot q* = \frac{n(a - c)}{(n + 1)b} ).
Equilibrium Price: The market price at equilibrium can be derived as: ( P* = a + \frac{nc}{n + 1} ), which suggests that the price setting is sensitive to both the number of firms and their cost structures. Per-firm Profit at equilibrium can then be expressed as: ( \Pi*_i = \frac{(a - c)^2}{(n + 1)^2b} ), highlighting the profits achievable by firms given the prevailing market conditions.

As the number of firms ( n \to \infty ) in the industry increases, the per-firm output approaches ( 0 ) (( q* \to 0 )) and the market price converges toward marginal cost (( P* \to c )). This phenomenon illustrates the competitive nature of markets, wherein an overabundance of firms leads to diminished profits and prices, reaching a point of perfect competition where per-firm profit approaches ( 0 ).

Market Power: The ability of a firm to influence the price of the product it sells in contrast to being a price taker typical in perfectly competitive markets. Market power is crucial for firms operating in an oligopoly as it allows them to retain substantial margins over their costs.
Lerner Index (LI) quantitatively defines market power as: ( LI = \frac{P - MC}{P} = s^2 ), where ( s ) is the firm’s market share. A higher Lerner index indicates more market power and reflects the firm’s ability to set prices above marginal cost, thus capturing greater profit per unit.

In scenarios where firms offer differentiated products rather than identical goods, strategic interactions and competitive outcomes differ significantly. The demand functions for such differentiated products can be expressed as: ( P^1(q1, q2) = a - q*1 - g \cdot q_2 ), with ( 0 <= g < 1 ) denoting the degree of differentiation between the products offered by the firms. This aspect is crucial as differences in products can lead to varied demand curves.
The presence of product differentiation influences firms’ profits considerably, as each firm must optimize its output considering the product preferences of consumers and the outputs of competitors. Profit functions become more complex under differentiation but follow similar maximization principles as in homogeneous product scenarios.

Increased product differentiation generally results in:
- Higher equilibrium outputs ( q* ) for each firm, as firms can capture distinct segments of the market.
- Elevated market prices for goods, owing to the perceived value added through differentiation, allowing firms to maintain pricing power.
- Greater per-firm equilibrium profits, as differentiation often insulates firms from direct price competition and allows for enhanced margins.
Furthermore, product differentiation alters competitive dynamics under Cournot duopoly, leading to distinct market outcomes compared to situations with homogeneous goods, fostering varied price and output phenomena.

Understanding the dynamics of oligopoly, particularly within quantity-setting contexts, facilitates a deeper comprehension of market behaviors and strategic interactions among firms, aiding in the analysis of both homogeneous and differentiated product environments. The implications of these dynamics extend to economic policies and regulatory considerations, impacting how industries and markets operate under varying competitive pressures.