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Unformatted text preview: Problem Set 1 Solutions: The Partial Equilibrium Competitive Model in the Short Run Econ 100B James Rauch Problem 1 There are 200,000 identical consumers, each with utility function U i = x 2 i y i and income = $40. The price of y is fixed at $1. There are 20,000 identical perfectly competitive firms producing good x . Each firm has the production function x j = (1 / 3) K 1 / 2 L 1 / 2 and capital stock fixed at 12 in the short run. The wage is $4 and the rental rate on capital is $9. (a) Derive the demand function for each consumer. (b) Derive the market demand function. (c) Derive the supply function for each firm. (d) Derive the market supply function. (e) Find the competitive market equilibrium price and quantity for good x in the short run. (f) How much profit will each firm make in the short run? Solution (a) First set MRS =- p x / p y MRS =- u/x u/y =- 2 x i y i x 2 i =- 2 y i x i- p x p y =- p x 1 yielding: p x = 2 y i x i (1) 1 Now use the budget constraint: p x x i + p y y i = m i p x x i + y i = 40 (2) Combining 1 and 2, x * i = 80 3 p x (b) Each consumers demand is x * i = 80 / (3 p x ), and there are 200,000 identical consumers. Hence, market demand is simply: x D = n X i...
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- Fall '07