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1. A monopoly firm sells a product to two types of consumers, low valuation con- sumers whose valuations are 2q and high valuation consumers whose valuations are 3q. Consumers are equally likely to have high or low valuations, and the cost of producing a product of quality q is q², that is c(q) = q². (a) What quality product, qı and q2, would a social planner sell to each con- sumer? (b) Assume that the monopolist sells the same two products that a social planner would choose (answer to part a). What prices would the monop- olist charge the consumers if it could not observe the consumers' types? That is, solve the monopolist's pricing problem subject to individually rational and incentive compatibility constraints. Let t₁ and t2 denote the two price, so profit is t₁ -q+t₂-92. (c) What quality products, and at what prices, would the monopolist sell to the consumers if i) it could not observe the consumers' types and ii) it chose the qualities to maximize profits. You should write out the firm's profit function and the constraints, explain which constraints bind and why, substitute the binding constraints into the objective function, and finally, solve the resulting unconstrained optimization. (d) Explain intuitively how changing the product quality in the way you de- rived in part (c) increases the firm's profits.

1. A monopoly firm sells a product to two types of consumers, low valuation con- sumers whose valuations are 2q and high valuation consumers whose valuations are 3q. Consumers are equally likely to have high or low valuations, and the cost of producing a product of quality q is q², that is c(q) = q². (a) What quality product, qı and q2, would a social planner sell to each con- sumer? (b) Assume that the monopolist sells the same two products that a social planner would choose (answer to part a). What prices would the monop- olist charge the consumers if it could not observe the consumers' types? That is, solve the monopolist's pricing problem subject to individually rational and incentive compatibility constraints. Let t₁ and t2 denote the two price, so profit is t₁ -q+t₂-92. (c) What quality products, and at what prices, would the monopolist sell to the consumers if i) it could not observe the consumers' types and ii) it chose the qualities to maximize profits. You should write out the firm's profit function and the constraints, explain which constraints bind and why, substitute the binding constraints into the objective function, and finally, solve the resulting unconstrained optimization. (d) Explain intuitively how changing the product quality in the way you de- rived in part (c) increases the firm's profits.

Managerial Economics: A Problem Solving Approach
Managerial Economics: A Problem Solving Approach
5th Edition
ISBN:9781337106665
Author:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Chapter16: Bargaining
Section: Chapter Questions
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1. A monopoly firm sells a product to two types of consumers, low valuation con- sumers whose valuations are 2q and high valuation consumers whose valuations are 3q. Consumers are equally likely to have high or low valuations, and the cost of producing a product of quality q is q?, that is c(9) = q°. (a) What quality product, q1 and q2, would a social planner sell to each con- sumer? (b) Assume that the monopolist sells the same two products that a social planner would choose (answer to part a). What prices would the monop- olist charge the consumers if it could not observe the consumers' types? That is, solve the monopolist's pricing problem subject to individually rational and incentive compatibility constraints. Let t and t2 denote the two price, so profit is t, - af + t2 – 3. (c) What quality products, and at what prices, would the monopolist sell to the consumers if i) it could not observe the consumers' types and ii) it chose the qualities to maximize profits. You should write out the firm's profit function and the constraints, explain which constraints bind and why, substitute the binding constraints into the objective function, and finally, solve the resulting unconstrained optimization. (d) Explain intuitively how changing the product quality in the way you de- rived in part (c) increases the firm's profits.
Transcribed Image Text:1. A monopoly firm sells a product to two types of consumers, low valuation con- sumers whose valuations are 2q and high valuation consumers whose valuations are 3q. Consumers are equally likely to have high or low valuations, and the cost of producing a product of quality q is q?, that is c(9) = q°. (a) What quality product, q1 and q2, would a social planner sell to each con- sumer? (b) Assume that the monopolist sells the same two products that a social planner would choose (answer to part a). What prices would the monop- olist charge the consumers if it could not observe the consumers' types? That is, solve the monopolist's pricing problem subject to individually rational and incentive compatibility constraints. Let t and t2 denote the two price, so profit is t, - af + t2 – 3. (c) What quality products, and at what prices, would the monopolist sell to the consumers if i) it could not observe the consumers' types and ii) it chose the qualities to maximize profits. You should write out the firm's profit function and the constraints, explain which constraints bind and why, substitute the binding constraints into the objective function, and finally, solve the resulting unconstrained optimization. (d) Explain intuitively how changing the product quality in the way you de- rived in part (c) increases the firm's profits.
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