MIDTERM RECAP TUTORIAL TA Andrea Venegoni

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1 MIDTERM RECAP TUTORIAL TA Andrea Venegoni 1- Monopoly A monopolist has the following cost function and faces the following demand curve for its products: C(Q) = 20Q Demand : P = 100 Q 1. Explain why in monopoly a deadweight loss results with respect to the perfectly competitive outcome. 2. Calculate and show in a graph the deadweight loss 3. Calculate firms profits SOLUTIONS a) A deadweight loss occurs since under monopoly fewer quantity is exchanged at a higher price compared to the perfect competitive outcome. b) In order to compute the deadweight loss is necessary to calculate equilibrium price and quantity both under perfect competition and under monopoly. Starting from monopoly we have to recall that in equilibrium the monopolist, in order to maximize its profits, set the price in order for the marginal revenues to be equal to the marginal costs (Monopoly equilibrium condition : MR = MC). Then, we have to compute marginal costs and marginal revenues. Starting from marginal costs we know that: MC(Q) = dc(q) =20. dq For what concerns marginal revenues, we have first to compute total revenues, that are given by the multiplication between price and quantity. Knowing from the demand function that P=100-Q we can write the total revenues (TR) expression as follows: TR(Q) = P (Q) Q = (100 Q) Q = 100Q Q 2. Once having derived the expression of TR we can compute marginal revenues (MR): MR(Q) == dtr(q) =100-2Q dq 1

2 Now, having both MC and R, we can equate them to get the quantity that maximizes monopolist profits: MR=MC 100-2Q=20 80=2Q Q = 40 Substituting Q in the demand equation P = = 60 The efficient quantity, i.e. the quantity exchanged in perfect competition, is given by: P = MC 100 Q = 20 Q = 80 To compute the deadweight loss must be multiplied the difference between the perfect competition and the monopolistic quantity by the difference between the monopolistic and the perfect competition price: DWL= (P M P PC ) (Q PC Q M ) 2 = 800 2

3 d) Compute the profits of the monopolist To compute the profits of the monopolist total revenues and total costs associated to monopolist price and quantity must be computed: Π = TR TC = = Monopoly and Perfect Competition Suppose that a monopolist s market demand is given by D : P = Q and the marginal cost is given by: MC(Q) = 10Q. a) Now suppose that the market is perfectly competitive. Calculate the resulting competitive equilibrium price and quantity b) Calculate the profit-maximizing price and quantity for the monopolist 3

4 SOLUTIONS c) Compare the consumer surplus and producer surplus in part a) with that in part b). What is the deadweight loss resulting from the monopoly? Cfr the previous exercise. 3. Supply and Demand Suppose the demand and supply of tennis rackets in the local sportwear store season is given by the following functions: Qd = 20 4P Qs = 2P 4 1. What is the market-clearing price of tennis rackets? How many tennis rackets are sold by the local sportswear store? 2. Suppose that an Italian tennis player wins the Roland Garros tournament, making young people be willing to play tennis. As a result the demand curve shifts outwards to Qd = 50 4P. What will be the new market clearing price and quantity sold? SOLUTIONS 1) To find the equilibrium price and quantity, knowing that the market equilibrium condition is given by Q S =Q D we have to equate the supply and demand expressions as follows: 2P 4 = 20 4P Solving for P, and hence taking all the terms with it on the right side we obtain: 2P + 4P = P = 24 P = 24 6 = 4 P=4 is the equilibrium price. To find the equilibrium quantity lets substitute this value into either of the two equations, the supply and demand ones. Lets choose this time the demand equation, we obtain: So the equilibrium quantity will be 4 Q D = = = 4 4

5 2) If the demand shifts we have to repeat what we have done at the previous point with the new demand function that now is Qd=50-4P. To find the equilibrium price and quantity, knowing that the market equilibrium condition is given by Q S =Q D we have to equate the supply and demand expressions as follows: 2P 4 = 50 4P Solving for P, and hence taking all the terms with it on the right side we obtain: 2P + 4P = P = 54 P = 54 6 = 9 P=9 is the equilibrium price. To find the equilibrium quantity lets substitute this value into either of the two equations, the supply and demand ones. Lets choose this time the demand equation, we obtain: Q D = = = 14 So the new equilibrium quantity will be Elasticity The manager of a sports center decides to increase the per hour price of the soccer fields from 10 to 11 euros. As a consequence, the number of hours booked per week decreases from 60 to 45. a. Calculate the elasticity of demand using the point formula (at the original price and quantity). b. Calculate the impact of the price increase on sports center revenues. Do they go up or down? c. Based on the estimate of elasticity do you think that it will be more convenient for the manager to cut down the price? d. Using the point elasticity calculated in (a) above, estimate the total amount of hours that will be demanded and total revenues per week if the price were to be reduced from 10 to 8. Answer: a. POINT FORMULA Price: Increases from 10 to 11 Quantity: Declines from 60 to 45 Ed = (45 60)/45 (11 10)/10 = 15/60 1/10 = = 2.5 5

6 b. Revenues before the price increase: 60 * 10 = 600 Revenues after the price increase: 45 * 11 = 495 So, the price increase results in a decline in Sports center s revenues. c. Since demand is elastic, a decrease in price will result in an increase in revenue (because elastic demand implies that the % increase in quantity demanded will be greater than the % decline in price). d. Recall that the elasticity formula can be rewritten as: % Q Q / Q Q E d * % P P / P P P Q Substituting the values from the problem into the formula yields: Ed = -2.5 ΔP = - 2 (from 10 to 8) P = 10 Q = 60 Thus: -2.5 = (ΔQ/-$2) * 10/60 Solving for ΔQ yields: -2.5 = ΔQ * 10/(-2 *60) ΔQ = -2.5* -(60/10) ΔQ = + 30 Therefore, as a result of the price cut, total soccer fields hour sold will be = 90 per week. Revenues with the price cut will be 90* 8 = 720 Note: Quicker way to calculate the change in quantity as a result of the change in price: 6

7 A point elasticity of demand of 2.5 implies that a 1% change in price will result in a 2.5% change in quantity demanded. In this case, price declines from 10 to 8. That corresponds to an 20% decline in price ( 2/10). Therefore, quantity should increase by 20 * 2.5= 50% from the original level of 60, i.e. 60 *.50 = 30. Thus, with the price cuts, total soccer hours increase by 30 to ( = 90 meals/week Total revenues increase to (90 * 8) = Taxation The Spanish government decides to impose a luxury tax (over and above the regular value added tax) of 4,500 euros on the buyers of luxury large boats and yachts. The management at Astondoa (a Spanish luxury yacht producer) is very concerned about this new development and decides to hire a consultant to evaluate the effect of this tax. Astondoa provides the consultant with estimates of demand and supply functions for luxury Yachts in Spain: Qs = P Qd = P Where Q is the number of Yachts (in thousands) sold per year, and P is the average price (in thousands of euros) for its most popular models. Astondoa s management asks the consultant to estimate the following: a. The effect of the tax on Astondoa s overall sales and revenues. That is, calculate the effect of the tax on the average price and quantity sold compared to price and quantity prior to the imposition of the tax and calculate the resulting % decline in Astondoa s monthly revenues. b. The amount of monthly tax revenue raised by this tax for the Spanish government. c. Who bears the larger burden of the tax, buyers or sellers. Please show how you obtained your answer. d. How could Astondoa use the results provided by the answers to (a) (c) above to develop arguments against the imposition of the tax? Be specific. Think strategically. Answer: a. The market equilibrium price absent the tax is given by finding the P that equates Qd and Qs: P = P 88= 2P P* = 44 The equilibrium quantity is given by plugging the equilibrium price into the Qd or Qs equation: Qd = *44 = 80-66=14 7

8 The tax of 4,500$ creates a wedge between the price buyers pay (Pb) and the price sellers receive (Ps). Note that it does not matter to the market equilibrium whether the tax is imposed on buyers or sellers of cigarettes. Pb = Ps The demand and supply equations become: Qd = Pb = (Ps + 4.5) = Ps 6.75 Qs = Ps The new market equilibrium will occur where Qd = Qs Ps 6.75= Ps = 2Ps Ps = This is the price sellers receive. Pb = Ps = = This is the price buyers pay. The new equilibrium quantity Q* is determined by substituting the new price paid by buyers Pb of 11.6 in the Qd equation or the new price received by sellers Ps= 9.6 in the Qs equation: Qd = * = To compute the pre-tax and post-tax total revenues of Astondoa is necessary to multiply the price and quantity of equilibrium in the two situations: TR pre tax = P pre tax Q pre tax = = 616 TR post tax = PS Q post tax = = b. To compute the tax revenues raised by the government is sufficient to multiply the amount of the tax times the post-tax quantity: Tax Revenues = Tax Q post tax = = c. To calculate the proportion of the tax borne by buyers, we have to compare the price paid by buyers with and without the tax. As a result of the tax, the market price went up by 1.125, from 44 to Therefore consumers bear of the 4.5 of the tax or 25% of the tax. To calculate the proportion of the tax paid by sellers, we have to compare the price received by sellers before and after the tax. As a result of the tax, the price received by sellers fell from 44 to Therefore, sellers bear the remaining dollars of the tax or 75% of it. From calculations above, sellers bear a significantly larger proportion of the tax than buyers. This implies that the supply of sellers is less elastic (with regard to price) than the demand of buyers. d. The main complain that Astondoa s manager can make to the Spanish government is that if, as appears, their intention was to levy a luxury tax on rich people doing this they haven t quite reach their goal, because, as demonstrated by the calculations, the biggest portion of the tax burden does not fall on buyers but on producers. 6. Production Theory Please compute the following given this total cost function: TC(Q)= Q 2 8

9 a. average total cost b. marginal cost c. fixed cost d. average fixed cost e. average variable cost Calculate which is the quantity that minimizes ATC. If the firm produces a quantity equal to 8 is it exploiting economies of scale or suffering from diseconomies of scale? Please explain. SOLUTIONS A) Average total costs = TC = 36+4Q2 Q Q B) Marginal costs = δtc = 8Q δq = 36 Q + 4Q C) Fixed costs = the component of the cost function that does not depend on quantity = 36 D) Average fixed costs = FC = 36 Q Q E) Average variable costs = VC = 4Q Q = 4Q2 Q 7.Production Theory The Carter Enterprise that produces cutlery wants to assess which is the right level of production that will allow it to minimize its average production costs in order to make its business more efficient. The use of the plant and the machineries costs $ while for each unit produced the cost to be substained is 50Q + 100Q 2 a- Which is the cost function for the Carter Enterprise? b- Calculate fixed costs, average fixed costs, variable costs and marginal costs for the production function of Carter enterprise. c- How can we determine the quantity to be produced in order to minimize costs? d - Calculate the quantity that minimizes the average total costs. a) The cost function for Carter is: Q + 100Q 2 b) Fixed Costs = Variable Costs = 50Q + 100Q 2 Average Total Costs = ( Q + 100Q 2 )/Q Marginal 9

10 Costs= Q c) To determine the quantity to be produced in order to minimize the average total costs we have to calculate the quantity that makes marginal costs equal average total costs. d) ( Q +100Q 2 ) /Q = Q Q +100Q 2 = 50Q + 200Q = 100Q = Q 2 Q = 50 So, ATC is minimized at 50 units of output. 10