ECON 6555 Homework 5 Due Date: Thursday, March 5
ECON 6555 Homework 5 Due Date: Thursday, March 5 (at The beginning of class)
Construct the normal form of a game where two consumers, A and B, decide whether to adopt a new network technology, with given valuations and payoffs, and determine the pure strategy Nash equilibria. Analyze a market with 50 consumers with valuations uniformly distributed between $1 and $50, and find the equilibria at a price of $600. Examine the software market with consumers valuing technical support, deriving profit-maximizing prices with and without piracy protection. Finally, evaluate a network good with a uniform valuation distribution, form and solve the associated equilibrium and profit functions, and analyze how profits change with a parameter θ.
Paper For Above instruction
The assignment encompasses multiple interconnected economic models focusing on game theory, network effects, pricing strategies, and consumer valuation distributions. Each model requires constructing strategic frameworks, solving for Nash equilibria, deriving optimal pricing strategies, and understanding how market parameters influence outcomes. This integrated approach facilitates a thorough analysis of strategic decision-making in technology adoption, software sales, and network goods, reflecting real-world complexities in digital markets and network externalities.
Strategic Interaction in Technology Adoption
The initial scenario involves a simple simultaneous-move game featuring two consumers, A and B, deciding whether to adopt a new network technology. The consumers have valuations of vA = 100 and vB = 200, and the payoff structure gives each consumer their valuation if both adopt, or zero otherwise, minus an adoption cost of $50. The normal form matrix considers strategies "Adopt" and "Don't Adopt".
In the matrix form, the payoffs when both adopt: Consumer A gets 100 - 50 = 50; Consumer B gets 200 - 50 = 150. If only one adopts, the adopting consumer incurs the $50 cost but gets zero valuation from the network effect unless both adopt. If neither adopts, payoffs are zero.
Analyzing the best responses, it emerges that neither consumer has an incentive to deviate unilaterally once both adopt, meaning that the strategy profile of both adopting (adopt, adopt) is a pure strategy Nash equilibrium with payoffs (50, 150). Other strategy profiles do not satisfy mutual best responses due to the payoff structures. Hence, the sole pure strategy Nash equilibrium is both consumers adopting the network technology.
Extending this analysis to a market with 50 consumers with valuations uniformly distributed from $1 to $50, the network effects imply that the value to a consumer depends on the number of adopters, N. The payoff function is V_N = v + θN, where v is individual valuation, N is the number of adopters, and θ reflects the network externality strength. With a uniform distribution, the valuation for the marginal consumer is finding the smallest v such that the consumer is indifferent between adopting or not, given the number of current adopters.
Graphically, the equilibrium set involves three possible states: no adoption, partial adoption, and full adoption, depending on the price p=600 and the valuation distribution. Algebraically, we find that for a consumer with valuation v, the condition for adoption is v + θN ≥ p. Since v ranges from 1 to 50, and the distribution is uniform, the aggregate behavior depends on the relationship between p, θ, and N. Solving for the equilibrium involves setting v + θN = p and analyzing the feasible N values.
The most probable equilibrium is the one where Facebook-like positive network externalities lead to full adoption if the initial threshold v is sufficiently high and the network effect θ is strong enough to compensate for the high price. Conversely, low θ or high p may lead to partial or no adoption equilibrium.
Market for Software with Network Externalities and Piracy
The second scenario considers a software market with 100 consumers valuing technical support and 100 not valuing it, with specific payoff structures. When piracy is possible, consumers who value support derive a payoff of 2n - p when purchasing, or n if pirating, while consumers who do not value support get n - p when purchasing, or n when pirating, where n is total users. The firm's goal is to set a profit-maximizing price under these conditions.
Without anti-piracy measures, piracy competes directly, and demand includes both legitimate and pirated copies. Because the software costs nothing, and piracy reduces the firm's revenue, the profit maximization involves balancing the price p to maximize n * p, where n depends on demand elasticity. The optimal price in a no-protection scenario often is at a point where marginal revenue equals marginal cost (zero), which suggests setting p=0, but the strategic behavior of consumers, especially those valuing support, complicates this.
With piracy protection, the firm controls access, eliminating pirated copies. The demand then depends solely on consumers willing to pay p, with the highest valuation group setting the maximum p. The firm's profit maximizes at the monopoly price where marginal revenue equals zero, which yields p equal to the maximum consumer willingness to pay. Assuming full valuation, this price approaches their maximum valuation, adjusted for strategic considerations.
Network Goods with Consumer Valuations and Externalities
The final model examines a network good with a continuous valuation distribution from 0 to 1, where each consumer's payoff from purchasing is v + θN. The market size is normalized, and the fraction N depends on the threshold v. The problem involves determining the equilibrium cutoff valuation, profit maximization, and how profits depend on θ.
In part (a), the fraction of adopters N is the proportion of consumers with valuation v ≥ v, where v is the marginal consumer's valuation. Given the uniform distribution, N = 1 - v. Part (b) involves constructing an equation for v by equating the consumer's marginal payoff to zero: v + θN = p. Solving for v yields the critical valuation threshold, which determines N.
In part (c), inserting N into the equilibrium condition gives v as a function of p and θ. The profit function, in part (d), is revenue p N, with N = 1 - v*. The firm chooses p to maximize this profit, leading to a first-order condition that determines the optimal price, as derived from the maximization problem. The maximum profit depends positively on θ because higher network externalities increase the marginal value of additional users, incentivizing higher prices.
Overall, these models reveal complex strategic considerations in technology adoption, pricing, and network effects. They illustrate how valuations, externalities, and market structure influence equilibrium outcomes and profits, providing valuable insights for firms operating in digital and networked markets.