Unformatted text preview: Econ 121 – Fall 2010 UC Berkeley Professor Cristian Santesteban Problem Set 8 Solutions Due: Thursday, December 2 by 11:15am Problem 1 (35 points) Suppose demand is p = 80 – q and marginal cost for each firm is 20. There are two firms competing. a) Find the Cournot equilibrium price and quantity. Π1 = (80 – q1 – q2)q1 – 20q1 FOC: 80 – 2q1 – q2 – 20 = 0 Firm 1 reaction function: q1 = 30 – ½ q2 By symmetry, qi = 30 – ½ qi qi = 20, Qc = q1 + q2 = 40, Pc = 40 b) Find the equilibrium if there is a hypothetical monopolist. By how much is price raised? Π1 = (80 – Qm) Qm – 20∙Qm FOC: 60 – 2Qm = 0 Qm = 30, Pm = 50, ΔP% = 25% c) Show that for linear demand p = a – bq that εii(p0) = p0/(a – p0). Rewrite inverse demand function as a demand function: q = a/b – p/b Take the derivative of this demand function with respect to p: dq/dp = 1/b The definition of demand (or price) elasticity is: ε = dq/dp ∙ (p/q) Since the elasticity varies with p, we need to evaluate the elasticity at a particular price, Po: ε(Po) = (Po / b ) ∙ b / (a – Po) = Po / (a – Po) d) Find the critical elasticity of demand for an SSNIP equal to 5%. How does this compare to the prevailing (Cournot) elasticity of demand? Determine on the basis of the critical elasticity of demand whether this market is an antitrust market. Critical Elasticity (for linear demand): εcrit = 1/ (m + 2t), where m = prevailing margin = (40 – 20)/40 = 1/2 and t = 1/20 So, εcrit = 20/ 12 = 1.67 The prevailing elasticity is evaluated at the Cournot equilibrium price: ε(40) = 40 / (80 – 40) = 1 Since εcrit > ε(40), this is an antitrust market. e) Suppose now that marginal cost equals 65. Redo (a), (b), and (d). a’) q1 = q2 = 5, Q = 10, Pc = 70 b’) Qm = 7.5, Pm=72.5, ΔP% = 3.6% d’) εcrit = 1/(5/70 + 1/10) = 70/12 < ε(70) = 70/(8070) = 7; this is not an antitrust market anymore. Problem 2 (30 points) Consider a market with constant elasticity of demand (ε) and constant marginal cost equal to c. The gross increase in profit from increasing price by dp is dp(q + dq). The gross decrease in profit from increasing price by dp is −(m)(p)(dq) where m is the prevailing price‐cost margin (p − c)/p. [Hint: dq is negative.] a) Show that the gross increase in profits from raising price by dp equals pqt (1 − tε), where t = (dp/p). Plug in t=dp/p pqdp (1 – dp∙ε/p)/p qdp(1dp∙ε/p) dp(q q∙dp∙ε/p) Plug in ε = (dq/dp)∙p/q dp(q+dq), which is what we want to show. b) Show that the gross decrease in profits from raising price by dp equals mptqε. mptqε Plug in ε = (dq/dp)∙p/q and t=dp/p mpdpq(dq/dp)∙p/(q∙p) −(m)(p)(dq), which is what we want to show c) Use (a) and (b) to determine the maximum margin that makes a price increase of 100t% unprofitable. pqt (1 − tε) > mptqε (1 − tε) > m ε m < (1/ε – t) = mmax Problem 3 (35 points) Assume for the purposes of this problem that, contrary to its protests, Microsoft has a monopoly in providing operating systems, called “Windows,” for personal computers. Assume also that the marginal cost to Microsoft of supplying its OS for more than one computer is zero. Denote by w the price charged by Microsoft for its operating system. (Assume that Microsoft sets a single price per computer, that is, does not employ two‐part tariffs, quantity discounts, or other forms of price discrimination.) Computer Original Equipment Manufacturers (OEMs) assemble computers. Suppose that the “bill of materials” for a computer, that is, the cost to the OEM of all the parts necessary to build a computer, adds up to $900 per machine, and that assembly costs another $100 per machine. Finally, assume (contrary to the efforts of Dell, IBM, and HP) that computers are a homogeneous good, and the annual demand for computers is given by Q = 50,000,000 – 10,000 p. Suppose that the OEM business is perfectly competitive. a) For any given price, w, of operating systems, what will be the price and sales of computers? The downstream industry is competitive so the equilibrium price will equal marginal cost. P = 1000 + w. Qr = 40M – 10K∙w b) What price w should Microsoft set for its operating system? How much money will Microsoft make? How much money will OEMs make? What will be the price of a computer? Πm = w∙Qr = w(40M – 10K∙w) FOC w: 40M – 20K∙w = 0 Πm = $40B ($40 billion) Πr = $0 P = $3,000 c) How much money would a vertically integrated firm controlling both the supply of Windows and the assembly of computers make? What price would such a firm charge for computers? Πv = (P1000)Q = (P1000)(50M – 10K∙P) FOC P: 60M – 20KP = 0 Πv = $40B d) Could Microsoft make more money by integrating downstream into computer assembly? Why or why not? No, Microsoft could not make more money by integrating downstream because the downstream firm is competitive. Microsoft can extract the maximum monopoly rents by charging the monopoly price for w and then having the downstream firms pass it on 1 for 1. Suppose now (definitely contrary to reality) that a single firm, HP, has a monopoly over the assembly of computers. e) For a given price, w, for Windows, what price p would HP set for computers, and how many computers would be sold? Πhp = (P – 1000 – w)∙(50M – 10K∙P) FOC: 60M + 10K∙w – 20K∙P = 0 P = 3000 + 0.5w P = $3,000 w* = 2,000 Qhp = 50M – 10K∙(3K + 0.5w) = 20M – 5K∙w f) What price, w, should Microsoft set for its OS? How much money will Microsoft make? How much money will HP make? What will be the price of a computer? Πm = w∙Qhp = w∙(20M – 5K∙w) FOC: 20M – 10K∙w = 0 w* = $2,000 Πm = 2000 (20M10M) = $20B Πhp = $10B P = $4000 g) Could Microsoft and HP make more money by merging? If so, how much? Would such a merger benefit or harm computer users? By how much? Microsoft would do better by merging. It would earn $20B more. Such a merger would help consumers because the price of computers would fall from $4000 to $3000. ...
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- Fall '07
- Cournot equilibrium price, critical elasticity