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MICROECONOMICS II FIRST PART: Aggregate Demand and Supply Exercise 1: Consider the following demand functions: si 0 y1 ( p ) = 20 − 2 p si p > 10 p ≤ 10 y2(p)=3/p y3(p)=1/pα a) Find the price elasticity (ε1 (p), ε2 (p), ε3 (p)) of each of the three functions. What is the value of the elasticities when p = 6? Explain the meaning of these numbers. b) Obtain the aggregate market demand, y(p), assuming that there are 4 consumers with the first type of consumer demand function and 3 with the second, and no consumer with the third. c) Find the price elasticity of aggregate demand in question b) and evaluate it at p= 6. Interpret its value. Exercise 2: The demand for a consumer (A) of good X is equal to xA (p) = max {200-p, 0}.The demand for another consumer (B) of good X is equal to xB (p) = max {90-4p, 0}. a) What is the price elasticity of demand for each of the goods? b) At what price is the price elasticity of A unitary? And at what price is the price elasticity of B unitary? c) Draw the demand curves of consumers A and B and the aggregate demand curve for good X. d) Find a price other than zero (0) at which there is a nonlinear and positive aggregate demand for good X. What is the aggregate demand for prices below the point where the slope changes? What is the aggregate demand for higher prices? e) At what point of the aggregate demand curve is the price elasticity unitary? At what price will the income from the sale of good X be maximized? f) If the purpose of the sellers of good X is to maximize their income, to what consumer/s will they sell the good? Exercise 3: The typical demand function of match tickets of FC Barcelona is q (p) = 200.00010.000p. The goal of managers is to maximize their income. The total capacity of Camp Nou is 100.000 viewers. a) Write the inverse demand function. b) Write expressions for the Total Income (I) and the Marginal Income (Img) as functions of the total number of tickets sold. c) What price will generate the most income? What amount of tickets will be sold at this price? d) Selling the amount of the previous section, what is marginal income? What is the price elasticity of demand for tickets? Is the stadium full? Due to the current good luck of the team, the demand for tickets has increased and has become q (p) = 300.000-10.000p. e) What is the new inverse demand function? f) Write an expression for marginal income as a function of the number of entries. g) Ignoring stadium capacity, what price will generate the most income? How many tickets would be sold at this price? h) Taking now into account the capacity of the stadium, how many tickets will go on sale to maximize revenue? At what price? i) When the tickets are sold at the price of the previous section, what would be the marginal income from selling an extra ticket? What is the elasticity of demand for tickets to this combination of price and quantity? Exercise 4: Josep has quasi-linear preferences and loves sweets. Its inverse demand function for sweets is p (x) = 49 - 6x where x is the number of sweets he eats. He currently consumes 8 sweets at the price of 1 € per sweet. If the price of candy goes off at 7 € per sweet, calculate the change in consumer surplus (Josep’s). Exercise 5: The bicycle industry in Barcelona is composed of 100 companies with the same cost curve c (y) = 2 + (y2 / 2) and 80 companies with the same cost curve c (y) = y2/ 6. What is the supply curve for this industry? Exercise 6: In a perfectly competitive market there are two types of firms. Type 1 firms have total costs represented analytically by the function C (q) = 2q³-2q²+6q and type 2 firms by the function C (q) = 8q³-4q²+2q. Determine the aggregate supply of market generated by 8 companies of type 1 and 10 of type 2. ADDITIONAL EXERCISES Exercise 7 Assume C(y) = y3 – 6y2 + 18y is the long term cost function of a competitive firm. Market demand is given by D(p) = Max{147 - 3p, 0}. a) Find the long term supply function of this firm qi LP (p). Take into account corner solutions (q might be 0). Draw it. b) Assume all existent and potential firms are equal. Find the equilibrium of this market in the long term (price, number of firms, total production and each firm’s production). Exercise 8: Assume C(y) = y3 – 8y2 + 22y is the long term cost function of a competitive firm. Market demand is given by D(p) = Max{ 118 - 3p, 0}. a) Find the long term supply function of this firm qi LP (p). Take into account corner solutions (q might be 0). Draw it. b) Assume all existent and potential firms are equal. Find the equilibrium of this market in the long term (price, number of firms, total production and each firm’s production). Exercise 9: Assume C(y) = y3 – 4y2 + 18y is the long term cost function of a competitive firm. Market demand is given by D(p) = Max{142 - 2p, 0}. a) Find the long term supply function of this firm qi LP (p). Take into account corner solutions (q might be 0). Draw it. b) Assume all existent and potential firms are equal. Find the equilibrium of this market in the long term (price, number of firms, total production and each firm’s production). Exercise 10: Assume C(y) = y3 – 10y2 + 35y is the long term cost function of a competitive firm. Market demand is given by D(p) = Max{130 - 3p, 0}. a) Find the long term supply function of this firm qi LP (p). Take into account corner solutions (q might be 0). Draw it. b) Assume all existent and potential firms are equal. Find the equilibrium of this market in the long term (price, number of firms, total production and each firm’s production). SOLUTION TO ADDITIONAL EXERCISES Exercise 7 min Cme: y = 3, CMe=9 min CMg: y = 2, CMg=6 CMe = Cmg: y=0; y=3 Equilibrio: p=9, n=40, y=3, Y=120 Exercise 8 min Cme: y = 4, CMe=6 min CMg: y = 2,67, CMg=0,67 CMe = Cmg: y=0; y=4 Equilibrio: p=6, n=25, y=4, Y=100 Exercise 9 min Cme: y = 2, CMe=14 min CMg: y = 1,33, CMg=12,67 CMe = Cmg: y=0; y=2 Equilibrio: p=14, n=50, y=2, Y=100 Exercise 10 min Cme: y = 5, CMe=10 min CMg: y = 3,33, CMg=1,67 CMe = Cmg: y=0; y=5 Equilibrio: p=10, n=20, y=5, Y=130