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Consider a monopolist who can make a take-it-or-leave-it offer to a single consumer. The monopolist’s cost function is given by C(q) = 4q. The consumer has a budget of m = 100. The consumer’s utility function is given by U(y,q) = y1/2 + 2q, where q is the quantity of the monopolist’s product and y denotes remaining money. What is the optimal offer to the consumer?

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Joshua Stredder
Joshua StredderLv10
29 Apr 2021
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Related questions

1. Consider a monolopy where the inverse demand for its product is given by P = 50-2Q. Total costs for this monopolist are estimnated to be C(Q) = 100 + 2Q + Q^2. At the Profit-maximizing combination of output and price, consumer surplus is?

Answer: $64

2. You are the manager of a firm that sells its product in a competitive market at a price of $48. Your firm's cost function is C = 60+2Q^2. Your firm's maximum profits are

Answer: $228

I already know the answers, please show how it works! Thank you!!

Consider a durable good monopolist whose product is usable for only 2 periods. However, each consumer can benefit from the product only for one period. The quality of the product does not diminish over time and thus those who bought the product in period 1 can sell it in period 2 (if they find buyers). Assume also that the production cost is zero. The inverse demand curve in each period is given by P = 16 - Q where Q denotes the quantity available at that period. Both the consumers and the monopolist use a discount factor d is an element of [0,1] to discount future income or profits.

a) Assuming that the monopolist can commit to future prices and quantities, determine the optimal quantities and price in each period. What is the monopoly's profit? Indicate whether your answer depends on the discount factor d

b) Assume now that the monopolist cannot commit to second-period prices and quantities. Determine the optimal quantities and price in each period. What is the monopoly's profit? Indicate whether your answer depends on the discount factor d

I am only interested in part c,

1. James has a utility function given by U=X^0.5+Y^0.5 , where X is the amount of product 1 consumed per period and Y is the amount of product 2 consumed per period.

a) Derive expressions for James’ price elasticity of demand for good 1, James’ income elasticity of demand for good 1 and James’ cross price elasticity of demand for good 1 with respect to the price of good 2? Calculate the values of these elasticities at and I=$100. Explain the meaning of these elasticity values in words. Show that the sum of these three elasticities must be zero. Comment. Derive James’ indirect utility function and explain its meaning.Px=Py=1

b) What is the minimum income necessary for James to achieve 100 utils when and ? Illustrate your answer with a diagram. Derive James’ compensated demand functions for goods 1 and 2. Suppose now that I=$100, .

c) Use your answers to part (a) to find the optimal values of and . Calculate James’ gain in compensating consumer surplus if declines to $.5. Use James’ compensated demand curve for good 1 to illustrate this. Illustrate the “income” and “substitution” effects of this price reduction using James’ indifference curve map and his budget lines.

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