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.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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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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