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### ps 4 solutions

Course: ECON econ 100a, Summer 2007
School: UCSC
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Microeconomics Intermediate Summer 2007 Monika Thomas Solutions: Problem Set 4 10.2 With a \$3 tax, setting Q d = Q s implies 10 - .5( P s + 3) = -2 + P s Ps = 7 Substituting into the equation for P d implies P d = 10 . Substituting this price into the equation for quantity demanded implies Q = 5 million. At these prices and quantities, consumer surplus is \$25 million, producer surplus is \$12.5 million, and...

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Microeconomics Intermediate Summer 2007 Monika Thomas Solutions: Problem Set 4 10.2 With a \$3 tax, setting Q d = Q s implies 10 - .5( P s + 3) = -2 + P s Ps = 7 Substituting into the equation for P d implies P d = 10 . Substituting this price into the equation for quantity demanded implies Q = 5 million. At these prices and quantities, consumer surplus is \$25 million, producer surplus is \$12.5 million, and government tax receipts are \$15 million. The deadweight loss is \$1.5 million. The deadweight loss measures the difference between potential net benefits (\$54 million) and the net benefits that are actually achieved (\$25 + \$12.5 + \$15 = \$52.5 million). 1 10.5 Using the supply and demand curves from Example 10.1 and an excise tax of \$0.40 implies 165 - 50( P s + 0.40) = 66 + 55 P s P s = 0.75 Substituting into the equation for P d implies P d = 1.15 . Substituting P s into the supply equation implies Q = 66 + 55(0.75) = 107.40 . Finally, the government tax receipts will be tQ = 0.40(107.40) = 42.95 . These values correspond with those in Table 10.1. Graphically, the solution is 3.50 3.00 2.50 2.00 1.50 1.00 0.50 0.00 0 Demand Supply + 0.40 Supply Price 50 100 150 200 250 Quantity 2 10.7 a) With perfectly elastic supply, a \$4 excise tax shifts the supply curve up as pictured below. Consumers will bear the entire burden of the tax. The end result is that the equilibrium quantity falls from Q1 to Q2, the price producers receive remains the same (\$60), and the price consumers pay rises from \$60 to \$64. P \$64 \$60 S + \$4 S D Q2 Q1 Q b) When supply is perfectly inelastic, producers will bear the entire burden of the tax. Since supply is inelastic, it is easiest to think about this problem graphically as shifting down the demand curve. [Convince yourself you'd get the same result had we done this in part (a).] The \$4 tax has no effect on the equilibrium quantity. However, the price sellers receive falls from \$60 to \$56. The price consumers pay remains constant at \$60. P \$60 \$56 D D \$4 Q1 Q S 3 16.1 a) In equilibrium we must have quantity supplied equal to quantity demanded in both the butter and margarine markets. This implies in equilibrium we will have d s QM = QM d s QB = QB Substituting in the given curves implies 20 - 2 PM + PB = 2 PM 60 - 6 PB + 4 PM = 3PB Solving for PB in the first equation and substituting into the second equation implies 60 + 4 PM = 9(4 PM - 20) 60 + 4 PM = 36 PM - 180 PM = 7.5 When PM = 7.5 , PB = 10 . At these prices, QM = 15 and QB = 30 . s When the supply curve for margarine shifts to QM = PM , we have b) 20 - 2 PM + PB = PM 60 - 6 PB + 4 PM = 3PB Solving the first equation for PB and substituting into the second equation implies 60 + 4 PM = 9(3PM - 20) 60 + 4 PM = 27 PM - 180 PM = 10.43 When PM = 10.43 , PB = 11.30 . At these prices, QM = 10.43 and QB = 33.91 . The increase in the price of vegetable oil increases the price of margarine and decreases the quantity of margarine consumed. As consumers switch to butter, the price of butter rises and the quantity of butter consumed goes up. 4 The price of butter rises when the price of vegetable oil rises because butter and margarine are substitutes. The effects can be seen in the following graphs. Market for Margarine 40 30 Price 20 10 0 0 5 10 15 20 25 Quantity D 30 35 S' S Market for Butter 25 20 15 10 5 0 0 10 20 30 40 50 Quantity S Price D' D 60 70 Because the goods are substitutes, when the supply of margarine shifts inward from S to S', raising the price of margarine, consumers substitute butter for margarine, shifting demand for butter outward from D to D'. This raises both the equilibrium price and quantity of butter. 5 16.5 a) b) & 4 3 Mary Oranges 2 1 0 Josh 0 2 4 6 8 10 Apples c) To be economically efficient, the two consumers must have identical marginal rates of substitution at the allocation. While we are not given the MRS for each consumer, we are told that each has an identical utility function. This implies that at an efficient allocation where the MRS for each consumer is the same, the ratio of apples to oranges must be the same. Since at the current allocation Josh has a ratio of apples to oranges equal to 5 and Mary has a ratio of 1.67, this allocation cannot be efficient. The contract curve in this case will be a straight line between the origins for each consumer. 6 11.2 a) Since the demand curve is written in inverse form and is linear, the MR curve has the same vertical intercept and twice the slop as the demand curve. Thus, MR = 40 4Q. Total revenue will be maximized when MR = 0, or when Q = 10. At that quantity, the price will be P = 40 2Q = 20. Total revenue is PQ = 20(10) = 200. b) 11.7 a) If demand is given by P = 300 - Q then MR = 300 - 2Q . To find the optimum set MR = MC . 300 - 2Q = Q Q = 100 At Q = 100 price will be P = 300 - 100 = 200 . At this price and quantity total revenue will be TR = 200(100) = 20, 000 and total cost will be TC = 1200 + .5(100)2 = 6, 200 . Therefore, the firm will earn a profit of = TR - TC = 13,800 . b) The price elasticity at the profit-maximizing price is Q,P = Q P P Q With the demand curve Q = 300 - P , price Q P = -1 . Therefore, at the profit-maximizing Q , P = -1 Q , P = -2 200 100 The marginal cost at the profit-maximizing output is MC = Q = 100. The inverse elasticity pricing rule states that at the profit-maximizing price P - MC 1 =- P Q,P 7 In this case we have 200 - 100 1 =- 200 -2 1 1 = 2 2 Thus, the IEPR holds for this monopolist. 11.14 Remember that the demand elasticity in a constant elasticity demand function is the exponent on P when the demand function is written in the regular form, i.e. Q = f ( P ). We can manipulate the inverse demand function to get the regular demand function, Q = 10,000 P -2 . This implies that the demand elasticity is 2. P - MC 1 Therefore, using the IEPR, = . So the optimal percentage mark-up of P 2 price over marginal cost is , or 50 percent. 11.20 a) For this monopsonist w L MEL = 4 L + L(4) MEL = w + L MEL = 8 L 8 b) The monopsonist will maximize profit at the point where MRPL = MEL , where MRPL = P Q L In this example, Q L = 0.5 , so MRPL = 0.5P . Since P = 32 , MRPL = 16 . Now setting MRPL = MEL implies 16 = 8 L L=2 At this quantity of labor, w = 4 L = 8 . c) In a competitive labor market, w = MRPL. So the competitive supply of labor satisfies 4L = 16 or L = 4, with w = 4L = 16. The deadweight loss due to monopsony is equal to area A in the graph below, or 0.5(16 8)(4 2) = 8. w MEL = 8L w = 4L 16 A 8 2 4 MRPL = 16 L 9 13.1 a) b) P = MC implies 70 2Q = 10, or Q = 30 and P = 10. A monopolist produces until MR = MC yielding 70 4Q = 10 so Qm = 15 and Pm = 40. Thus m = (40 10)*15 = 450. For Amy, MRA = MC implies 70 4qA 2qB = 10. We could either calculate Beau's profit-maximization condition (and solve two equations in two unknowns), or, inferring that the equilibrium will be symmetric since each seller has identical costs, we can exploit the fact that qA = qB in equilibrium. (Note: You can only do this after calculating marginal revenue for one Cournot firm, not before.) Thus 70 6qA = 10 or qA = 10. Similarly, qB = 10. Total market output under Cournot duopoly is Qd = qA + qB = 20, and the market price is Pd = 70 2*20 = 30. Each duopolist earns d = (30 10)*10 = 200. c) 10
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