Industrial Organnization and Firm Strategy (econ 105)ECON ASSIGNMENT
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attachments_2014731.zipProblem Set 2.pdf
Problem Set 2 ECO105 Industrial Organization and Firm Strategy Professor Michael Noel University of California San Diego
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1. A single-product monopolist sells output to two geographically separated markets. Arbitrage is not possible. The inverse demand curve in market 1 is given by p1 = 8 - q1 and in market 2, it is p2 = 4 - q2. Note: if you get stuck in one part, move on to the next.)
a. The monopolist currently produces the output at a single plant (called plant "A") with total cost function C(qA) = 0.5qA
2. Hence qA = q1 + q2 is the total output of the firm (which we can also call Q). Find the quantities sold in each market, q1 and q2, and total output Q. Find the profits of the firm. (Hint: this is the basic third degree price discriminating problem. Make sure to substitute qA out of the problem using the constraint, so that the problem is all in terms of the two “free” variables, q1 and q2.)
b. (From an old exam.) The monopolist is considering adding a second plant, plant "B". The total cost function for plant "B" is C(qB) = qB + 0.25qB
2. Hence, if the monopolist purchases the new plant, her total production is qA + qB = Q, where qA is produced in plant A and qB is produced in plant B. She then sells Q = q1 + q2 in output, where q1 is sold in market 1 and q2 is sold in market 2. Find the output sold in each market, q1 and q2, the amount of good produced at each plant, qA and qB, and the total output, Q. Find the profits of the firm in this case. How much is the firm willing to pay to purchase plant B? (Hint: Do NOT assume that a single plant is matched solely with a single market! You only know that the total produced equals to the total sold, so q1 + q2 = qA + qB ( = Q). There are now four quantities to find, q1, q2, qA, qB, but only three of them are “free variables”, i.e. can be independently chosen. So for example, if you know q1, q2 and qA, it necessarily determines what qB must be. So set up your profit function initially in terms of all four quantities and but then use the constraint q1 + q2 = qA + qB to substitute out one of the four. Which three quantities you keep in the equation is up to you. Then solve for three first order conditions.)
c. (Challenging.) Assume a new law prohibits price discrimination and requires the monopolist to charge the same price in both markets. Repeat part a. under this assumption, i.e. only plant A. Has welfare increased, decreased or remained unchanged relative to part a.? Warning: check to see if both markets are being served in equilibrium! (You will need to “add” up the two demand curves – add the q’s, not the p’s! A diagram will help. Watch out for the kinks, it may be that only one group will be served and it may be that both will be served. At the end it may help to plot your numerical answer on your graph and make sure it makes sense. For example, if you find both groups are served, you should be on the segment of the demand curve in your diagram where both would be served.)
d. Assume instead there is one market with demand given by p1=2 – q1 and two plants. The cost function for plant A is C(qA) = qA and the cost function at plant B is C(qB) = 2qB
2. Solve for the quantity produced at each plant and the total quantity. (Hint: this is known as the “multiplant monopoly” problem. The solution concept is the same as always: the key thing is to think carefully about what the profit function must look like. Note that q1=qA + qB by constraint, so there are only two “free variables” and you should only be solving two first order conditions.)
2. A monopolist producing bicycles faces two types of consumers: light duty users and heavy duty users, indexed by θL and θH respectively, where θL < θH. There are equal numbers of each type of consumer and each consumer has unit demand for bicycles (ie. buys one or zero.) A consumer of type θ has valuation equal to V(θ, q) = 0.5θ(1 – (1-q)2) for a bicycle of quality q. Marginal cost of production is constant and equal to c regardless of quality. a. Find ∂V/∂q, ∂2V/∂q2, ∂V/∂θ, ∂2V/∂q∂θ and verify that the standard assumptions on these values
hold. Explain the importance of each assumption.
b. Assume the monopolist can observe the type of any given individual. Set up the fully non-linear maximization program in which the monopolist offers for sale a high quality version (q = qH) at a price RH and a low quality version (q = qL) at a price RL. (For your practice, do not carry the general “V” functions throughout the problem (or else you are just copying from the board) but instead insert the specific equations for V into your setup of the problem. You can check your answers match what we did in class at the end).
c. Solve this maximization program for the optimal qualities and prices (q L, RL) and (q H, RH). What is the consumer surplus, also known as “rent”, enjoyed by the low type? The high type? Carefully draw the equilibrium in a diagram.
d. Assume instead the monopolist cannot observe the type of any given individual. Set up the fully non-linear maximization program in which the monopolist offers for sale a high quality version and a low quality version. Plug in the specific equations for V right from the start. Solve for (q L, RL) and (q H, RH). What is the consumer surplus, or rent, enjoyed by the low type? The high type? Compare the quantities you find to those in part c.) Carefully draw the equilibrium in a diagram.
e. Assume instead there are exactly two consumers, person H and person L, and let VH(q) = 5q and VL(q) = 4q. (Note that at least one of our usual assumptions on V is violated…which one(s)?) Assume the total cost of producing one high quality good and the low quality good is C(qH) = qH
2/2 and C(qL) = qL 2/2 respectively. Redo part c and d. with these changes. (Note that marginal
cost is no longer constant so the equations from the board…which assumed constant marginal cost…are no longer identical to what you would get here!)
3. A monopolist faces two types of consumers: low demand consumers and high demand consumers. A high demand consumer has valuation equal to VH(q) = 10 + q - q2 for q units of output and a low demand consumer has valuation equal to VL(q) = 10 + q - 2q2 for q units of output. There are equal numbers of each type of consumer. Marginal cost of production is constant and equal to c. The monopolist wishes to offer two packages (q L, RL) and (q H, RH) where qi is the quantity of output in package i and Ri is the total price for package i. a. Assuming the monopolist can observe the type of any given consumer, set up the monopolist’s
fully non-linear maximization program solve this program for the profit maximizing quantities and total package prices (q L, RL) and (q H, RH). (Again, insert your functions for V from the start.)
b. Assuming the monopolist cannot observe the type of any given consumer, set up and solve her maximization program. Compare your answers to part b.)
Problem Set 3.pdf
Problem Set 3 ECO105 Industrial Organization and Firm Strategy Professor Michael Noel University of California San Diego
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1. A monopolist produces a durable good. All production takes place at constant MC= c in the first period and the good lasts until the end of the second period. There is a continuum of consumers uniformly distributed on [0, 1], indexed by θ. A consumer receives utility u = 2 θ - p if she purchases the good in the first period (and enjoys it for two periods) and u = θ - p if she buys in the second period. The monopolist can charge different prices in each period.
a. Assume the monopolist can credibly commit to a specific second period price while still at the
beginning of the first period.
i. Set up the maximization program.
ii. Solve this program for θ1 (the marginal consumer indifferent between buying in the first period and buying in the second), and θ2 (the marginal consumer indifferent between buying in the second and not buying at all.) Warning! – you may be violating an implicit constraint (chances are, you are.) Find out what implicit constraint is being broken, then impose the constraint and solve again.
b. Assuming that the monopolist cannot commit to a second period price, find θ1, θ 2, q1, p1, q2, and p2. Which outcome does the monopolist prefer, that of part a. or part b.?
c. Repeat parts a. and b. but assuming that the utility to a consumer is u = 3θ- p1 if she purchases in period 1, u = θ - p2 if she purchases in period 2, 0 if she does not buy, and MC = 0. (This is someone who gets extra value from owning something brand new.)
2. Two firms compete Cournot. The market demand curve is Q = 16/p and each firm faces a cost function C(qi) = cqi. a. Find the equilibrium firm quantities, industry quantity, price and firm profits.
b. Sketch the best response functions for each firm in a single diagram.
c. Verify that the Lerner Condition (for Cournot competition) holds. d. Repeat a. and b. using the market demand curve Q = 1 – p and a cost function for each firm of
C(qi)=qi/2.
3. There are 9 firms in an industry. The demand curve is given by Q = 55 - p.
a. Assume each firm has cost function C(qi) = 5qi. (That is, MCi = 5, for all i = 1..9). Find the total
industry output Q, the price, each firm’s profits and the Herfindahl index. b. Assume instead each firm i has cost function C(qi) = iqi, i = 1..9 (That is, MC1 = 1, MC2 = 2,
MC3 = 3,…, MC9 = 9.) Find the total industry output and the industry price.
c. Is the Herfindahl index higher in part a. or part b.? Is welfare higher in part a. or part b.? Do not attempt to do any math here (which is cumbersome) to find welfare explicitly – use your answers for Q and P to infer which outcome is more efficient. What does the asymmetry in costs reveal about the Herfindahl?
d. Let C(qi) = iqi as in part b. but assume that demand has fallen to Q = 10 - P. Find total industry output and industry price. Hint: sum up the first order conditions. This is tricky -- look carefully at your answer…are you violating any implicit constraints? If you are (and you probably are), impose the constraints and solve again.
4. (From an old exam.) Polar Parkas are sold are two independent and very distant markets. Market N is controlled by a profit-maximizing monopolist. In Market S, there are two Cournot oligopolists: oligopolist #1 is currently producing twice as much as the oligopolist #2 in the equilibrium. Also assume that the marginal cost of oligopolist #1 at its equilibrium quantity is equal to the marginal cost of the Market N monopolist at its optimal quantity. Also, at the equilibrium prices and quantities, the elasticity of demand in market N is 6 in absolute value, and in market S the elasticity of demand is 2 in absolute value. a. Show that the marginal cost of oligopolist #1 is 80% of the marginal cost of oligopolist #2, at the
equilibrium price and quantities.
b. Show that the price in market N is 80% of the price in market S in equilibrium. Since market N is dominated by a monopolist, and market S has competing firms, explain in a sentence how this price ordering can be possible?
5. Consider two firms facing a market demand curve p = 2 - Q, and each having a cost function C(qi) = qi. (That is, constant MC=1 and no fixed costs.)
a. Assuming the two firms compete Stackelberg where firm 1 plays first, find equilibrium firm quantities, industry quantity, price and firm profits. Calculate total welfare (the sum of consumer surplus and total industry profits). Calculate the market shares of the two firms.
b. Now assume the cost function for firm 2 has decreased to C(q2) = 0. The cost function for firm 1 remains C(q1) = q1. Assuming the two firms compete Stackelberg, find equilibrium firm quantities, industry quantity, price and firm profits. Calculate total welfare. Compare your results from this part with those of part a. If you were a Department of Justice economist, would you be concerned about the increased market share of firm 2?
c. Assume instead the market demand curve is P = a – bQ and costs are zero to each of the two firms. Assuming Stackelberg competition, solve for the equilibrium price, firm quantities, industry quantity, firm profits, consumer surplus, and welfare.
6. Two firms each produce a differentiated good at a constant marginal cost equal to 1. Firm i faces demand of qi = 4 + pj - pi, where pi is the price of its own good and pj is the price of the other firm's good. a. If prices are chosen simultaneously, find the best response functions, equilibrium prices,
quantities and profits.
b. Sketch the best response functions. Label the equilibrium point E1.
c. If firm 1 is able to credibly commit to choosing its price first (ie. a Stackelberg leader in this price game), find the equilibrium prices, quantities and profits. Compare your results to those of part a. Who benefits most, the leader or the follower? On the same diagram, label this equilibrium as point E2. (Notice that E2 is not on firm 1’s best response function. Why not?)
d. Repeat parts a. and b. using the demand curve qi = 1 + pj – pi and assuming costs to each firm are zero.
7. (From an old exam.) Consider two firms (firm i and firm j) each selling a differentiated good at zero marginal cost in the country of Kanada. The demand for the product sold by firm i is qi = 1 – pi – pj,
with i ≠ j.
a. Are these two goods substitutes or complements?
b. Assuming firms compete by simultaneously choosing its prices. What are the equilibrium prices?
c. Now assume that firm 1 and firm 2 enter into a contract in which they agree fix prices. Although illegal in the U.S., price fixing contracts are legally binding in the country of Kanada. Assuming prices are chosen to maximize industry profits (and are then split equally between them), what are the prices that the colluding firms choose? What’s going on here?
d. In no more than a few sentences, discuss the welfare implications of this price-fixing agreement. (You may calculate welfare in each case if you are unsure.) Price fixing agreements are almost always antitcompetitive. But should Kanada outlaw price fixing agreements generally? Use your answer to part a. in your answer here.
Problem Set 4.pdf
Problem Set 4 ECO105 Industrial Organization and Firm Strategy Professor Michael Noel University of California San Diego
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1. Two firms, located at either end of a linear city, compete in prices. Each firm has constant marginal cost equal to c. Each consumer i has unit demand, and receives utility of uij = v - pj - tzij if she buys from firm j located a distance of zij away at price pj. She receives utility equal to zero if she does not buy.
a. If not the whole market is served in equilibrium, find the equilibrium price and profits for each
firm. Do profits rise or fall with an increase in t?
b. If the whole market is served in equilibrium, find the equilibrium price and profits for each firm. Do profits rise or fall when t increases?
c. Give the economic intuition why the sign of dπ/dt is different in part a. and part
d. Repeat a. and b. using the utility function uij=1-pj-zij and assuming marginal costs are zero to each firm. Ignore the questions pertaining to t.
2. Two firms compete in prices along a linear city (as always, of distance one). One firm is located at the west end at point 0, the other is located in the center at point ½. Each firm has constant marginal cost equal to c. Each consumer i has unit demand, and receives utility of uij = v - pj - tzij if she buys from firm j located a distance of zij away at price pj. She receives utility equal to zero if she does not buy. Assume that v is large enough that all consumers buy in equilibrium. Find the equilibrium price and profits for each firm.
3. Five infinitely lived firms produce homogeneous goods and operate in a market with demand curve P = 1 - Q and no marginal or fixed costs of production. Assume the firms have a cartel agreement in which they agree to split monopoly profits. Assume each firm plays a grim trigger strategy: that is, if any firm cheats, all firms revert back to the Nash equilibrium strategy of the static game and stay
there forever. For each of the following oligopoly paradigms, find the range of the discount factor δ necessary to support the collusive agreement in equilibrium.
a. Firms are Bertrand competitors.
b. Firms are Cournot competitors.
4. Consider two identical, infinitely-lived firms competing in two markets with identical demand. However, in market A, each firm can observe in period t the price its opponent set in period t-1, ie. cheating can be detected in the next period. In market B, each firm in period t can only observe past prices up to (and including) period t-2, ie. a two-period detection lag. (Thus each firm cannot learn its opponents price in t-1 until period t+1.) Assume firms compete in prices. Consider the trigger strategy in which each firm sets the monopoly price in each market as long as each has done so in every past period, but once cheating in either market has been detected by the opponent, they set price equal to marginal cost in both markets forever. a. Warmup: if only market A existed, show that collusion is sustainable if δ≥1/2. b. Warmup: if only market B existed, show that collusion is sustainable if δ≥1/√2. c. Now assume both markets exist. If a firm were to cheat, what is the most profitable way to do so?
(Hint: it will cheat as much as possible before being detected.) Show that collusion is sustainable if δ = (1 + √17)/8 = 0.64.
5. Assume an incumbent who can choose advertising level K in the first period. In the second period, the entrant may enter by paying a fixed cost F. If it enters, the two firms play Cournot, otherwise the incumbent is a monopoly. Demand is given by P = K – q1 – q2. There are no marginal costs. The cost to the incumbent of K units of advertising is C(K) = K3. (Don’t forget to include this at the end of the incumbent’s profit function you write down.)
a. What K does the incumbent choose if entry is blockaded? For what F is entry blockaded?
b. What K does the incumbent choose if it wishes to deter entry? What are its profits (as a function of F) when it deters entry?
c. Does investing in K make the incumbent soft or tough? Does the incumbent over or underinvest in K when deterring?
d. What K does the incumbent choose if it accommodates entry? What are its profits if it accommodates entry?
e. For which F will the incumbent deter? For which F will the incumbent accommodate?
f. Does the incumbent over or underinvest when accommodating? To find out, calculate KOL, the K that it would choose if the entrant were unable to observe and react to the incumbents choice of K. Then your answers to parts d. and e.
g. Let us try to answer e. another way. Calculate the strategic effect of K on the incumbent’s profits. Is it positive or negative? Should the incumbent over or underinvest?
h. Verify that the total derivative of K on the incumbent’s profits is equal to the direct effect of K on the incumbent’s profits plus the strategic effect of K on firm 1’s profits via q2.
i. Let us try to answer e. yet a third way. Can you use the taxonomy approach (ie. think furry animals) to decide whether or not the incumbent should over or underinvest? If not, why not? If so, which strategy does this game call for and does it call for over or underinvestment?
6. A new firm (the "entrant", firm 2) is about to enter an industry currently dominated by a monopolist (the "incumbent", firm 1.) The entrant does not have to pay a start up fee so entry cannot be deterred. When the entrant enters in the second period, the two firms will produce differentiated products and compete in prices (chosen simultaneously.) MC = 0 for both firms. In the first period, before the entrant gets set up, the incumbent can strategically choose its advertising level. If the incumbent puts ads in k newspapers in the first period, its total profits are Π1 = (3 + 3k - p1 + p2)p1 - 4k3/3, where the last term represents the cost of advertising. Profits to the entrant is given by Π2 = (3 - p2 + p1)p2. (Note the lack of fixed costs.) The world ends at the end of the second period. a. Show that, given the incumbent’s advertising level k, equilibrium prices in the second period are
p1 = 3 + 2k and p2 = 3 + k. Solve for the profits of each firm in the second period as a function of k.
b. What value of k will the incumbent choose in the first period, knowing the entrant will observe and react to it in the second period? In other words, what is kAC?
c. What value of k would the incumbent have chosen in the first period if the entrant had not been able to observe and strategically react to the choice of k? That is, what is kOL?
d. Comparing kAC and kOL from parts b. in c., does the incumbent overinvest or underinvest in advertising?
e. Let us answer part d. another way. Calculate the strategic effect of k on firm 1’s profits. Is it positive or negative? Hence, does the incumbent overinvest or underinvest in advertising?
f. Let us answer part d. a third way, using the "taxonomy of strategies" approach. (Note that direct effect of k on firm 2’s profits is zero.) Does more advertising make the incumbent tough or soft? Are prices strategic substitutes or strategic complements? Given these answers, what strategy should the firm use? (Think furry animals.) Does this strategy call for overinvestment or underinvestment in advertising?
Problem Set 5.pdf
Problem Set 5 ECO105 Industrial Organization and Firm Strategy Professor Michael Noel University of California San Diego
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1. Consider three equally spaced firms competing in prices (chosen simultaneously) around a circular city. There are no fixed costs of production and zero marginal cost. A consumer receives utility u = v - p - tz if she purchases from a store a distance of z away, and you may assume that the parameters are such that all consumers buy exactly one good in equilibrium (even after the merger discussed below). a. Find the prices and profits of each firm, and total industry profits in equilibrium.
b. Now assume firms 1 and 2 merge. Locations remain the same and all consumers continue to be served following the merger.
Find prices, profits to each firm and total industry profits. Compare to part a. In particular, did prices rise or fall?
c. If you were a government advisor, would you recommend preventing this merger if your objective was low prices?
2. Consider two firms competing in prices (chosen simultaneously) at opposite ends of a linear city. There are no fixed costs of production and marginal cost to each firm is equal to c. A consumer receives utility u = v - p - tz if she purchases from a store a distance of z away, and v is so large that all consumers will buy exactly one good in equilibrium (even after the merger discussed below). a. Find prices and the profits to each firm.
b. Show that it would be profitable for the two firms to merge.
c. Show that prices would increase as a result.
d. Imagine the marginal cost of the new firm would be cPOST < c. Is there a cPOST low enough that would cause prices to fall after the merger?
3. A common practice for authorities in evaluating mergers is to observe the stock price responses of rival firms to the proposed merged firm. What information does this provide? Is there an incentive for the rival firms to hide their response or mislead? Discuss why stock price responses are more informative to authorities than, for example, lobbying efforts by the rival firm itself for or against the merger?
