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Unformatted text preview: ECON 100C Midterm Exam 1 (A00) Name________________________ Prof. Levkoff Seat#_________________________ Spring 2013 PID#_________________________ Version A (Orange) Solutions Time_________________________ Directions: After everyone is seated, the exam will be distributed. You will have the entire lecture to complete the exam. The exam is 8 pages. There are two parts- one short answer/free response section and one problem solving/analysis section. To receive full credit, SHOW ALL WORK and graphs where appropriate and CLEARLY BOX YOUR FINAL ANSWERS for any calculations. Make sure your name, seat #, PID #, and exam time are on the examination sheet before submitting your exam. SCORING BREAKDOWN (for TA use only): Total Score___________________/100 Part I Total:___________________/40 Part II Total:_____________________/60 a) ____________________/10 #1) Total___________________/25 b) ____________________/10 a)___________________/10 c) ____________________/10 b)____________________/5 d) ____________________/10 c)____________________/5 d)____________________/5 #2) Total___________________/35 a)____________________/5 b)____________________/5 c)___________________/10 d)____________________/5 e)___________________/10 1 Part I. Evaluate the validity of each of the following statements. Be sure to clearly accompany your explanation using graphs and equations where appropriate (40 points): a) Price controls tend to cause more damage in terms of surplus reduction in relatively more elastic markets. ANSWER: This is generally true because as agents respond more to price changes (suppliers or demanders), they are more willing to stop participating in the market and we get a reduction in q below equilibrium, award 8 points. OR If students make an argument showing “flatter” (more elastic) curves yielding a larger DWL in comparison to “steeper” (less elastic) curves under any pricing policy (ie: price floors, ceilings, taxes, subsidies), award 8 points. If they do BOTH of the above, I think this is the BEST answer and should receive full credit as it demonstrates understanding of both intuition and analytics. If they get it right in general, but make some mistake on a diagram explaining, give no higher than a score of 7. If they get it wrong, but backwards, (and otherwise made a correct argument), award no more than 5 points. If they only answer True or False, but provide no explanation, a score of zero should be given. b) (10 points) A maximum profit solution to the short-run, uniform pricing monopoly’s output choice problem only exists if the marginal cost of the last unit produced at the optimum is an increasing function of output. ANSWER: This statement is False because we can easily construct a counter example that shows otherwise: We saw in class that it is possible that the MC curve slope downward where it intersects MR, as long as it intersected MR from below (so that MR’<MC’ as per the second order condition), so it doesn’t have to slope upwards for an optimum to exist like it did under perfect competition where MR’=0. For just this answer, award no more than 8 points. If they also argue WHY it is still an optimum with the downward sloping MC curve (small variation argument to show profits decrease), then jointly, the answer should be awarded full credit. If a student answers that this is false, and that it is only true for a P.C. firm, award no more than 8 points as well. While this is correct, I think the above answer is 2 slightly more comprehensive. If they mention this and the above, again award full credit. If students show a diagram with an upward sloping MC curve and argue why it is an optimum (arguing a small variation around q* results in profits decreasing), award no more than 5 points as the student showed only one situation where this is true, but have not considered fully all of the possibilities. Otherwise, students might argue something strange other than “uniform pricing,” and you should be on the lookout for creative answers here provided they are correct. I’ll actually announce that there is a typo and it should say “uniform pricing monopolist” to mitigate this instance. Answers of only True or False with no explanation should receive zero credit. c) If the market inverse demand curve is linear, then a uniform pricing monopolist’s marginal revenue curve is the same function as the inverse demand curve, but with half the slope. ANSWER: They should PROVE that this is false by taking a general form linear demand curve (P=a-bq) and calculating first, the total revenue function (P(q)*q), and then taking the derivative to show that the MR=a-2bq (twice the slope!). This answer is worth full credit. If they answer false, it should have a steeper slope since MR diminishes at a faster rate than P because a monopolist must lower his price to sell additional units, award no more than 7 points. Otherwise, if they answer false, it should have twice the slope, but don’t provide any other reasoning, no more than 5 points should be awarded. An answer of only True or False with no explanation should be scored a zero. d) (10 points) The Lerner index for an unsubsidized profit maximizing, uniform pricing monopolist facing a downward sloping, linear demand curve is between zero and one. ANSWER: This is true. Use the first order condition to find the Lerner index for a profit max monopolist (2 points) Write that the Lerner index is equal to (P-MC)/P=-1/elasticity (3 points) argue that the lowest price a monopolist charges corresponds to elasticity = -1, 3 making the highest value the Lerner index equal to 1 (since the elasticity can’t be any smaller in magnitude). (3 points) Otherwise, the elasticity can approach negative infinity, in which case the index is driven to zero, implying P=MC. So definitely, for the monopolist, the index is on (0,1). (2 points) in which case the index is driven to zero, implying P=MC. So definitely, for the monopolist, the index is on (0,1). (5 points) Part II. Problems/Analysis (60 points) #1) (25 points) Consider a market with a demand curve given by qd 1000 10 P and supply curve given by q S 10 P . a) (10 points) Calculate the equilibrium price and quantity and show that the quantity you found solves where the marginal benefit and marginal cost functions are the inverse demand and supply curves, respectively. Explain the implications of your result. ANSWER: Set the quantities equal and solve for p to find p*=$50 and q*=500. (1 point) Find the inverse demand/supply curves correctly (1 points) Compute the integral correctly (2 points) Optimize the result correctly to find q*=500 (2 points) Explanation: the equilibrium allocation is the one that maximizes total surplus (the invisible hand!) (4 points for some explanation along these lines). b) (5 points) How many units are traded if the government imposes a price floor of $40? What about $60? Calculate the deadweight loss in both situations. ANSWER: At a price floor of $40, the market can still reach equilibrium @ p*=$50. Thus, there is no deadweight loss, and q*=500 units are supplied, demanded, and traded. (1 point) However, if the price floor is set at $60: -then the quantity demanded is restricted to 400 units 4 -producers would like to supply 600 units (inducing a surplus of 200 units) -so only 400 units get traded. (2 points) This means that output is reduced by 100 units below equilibrium. The DWL here is (1/2)(100)(20)= $1000. (2 points) If the word is done correctly, but the answers are wrong due to a miscalculation of q* from a previous part, take only 1 point off. c) (5 points) If, instead of a price floor, the government were to impose a specific tax of $10 on the sale of each unit, how many units would be traded? What prices do the sellers receive? What prices do the buyers pay? ANSWER: if a specific tax is levied on the sale of each good of $10 per unit, then q will be reduced to 450 units being traded (found by solving where the difference between p_b-p_s=$10 and substituting the inverse demand curves in). (3 points) At 450 units, buyers are willing to pay $55 and sellers are willing to accept $45 from the inverse demand and supply curves. (2 points) d) (5 points) For the specific tax in c), calculate the deadweight loss and show ∗ that it is equal to under the tax. where is the quantity traded ANSWER: Diagrammatically calculate the DWL as (1/2)(10)(50)=$250. (1 point) Correctly calculate the MB and MC functions as the inverse demand/supply curves (1 point) Correctly compute the integral assuming q* and q under the tax are correct and that it gives the same value as computing diagrammatically. (3 points) If the answer is correct given some error in the previous part, take off 1 point provided all of the work subsequently is correct. #2) (35 points) Consider a market with inverse demand curve given by P( q) 900 q . Suppose technology is such a way that the total cost of production is given by 400. 5 a) (5 points) Set up the perfectly competitive firm’s short-run profit maximization problem and derive the supply curve for a perfectly competitive firm in this model. ANSWER: Solve Max_{q} Profit = p*q-TC(q) where p is constant. (2 points for correct profit function) First order conditions (taking derivative and setting equal to zero yield) (1 point) p=MC => q=p/2 for one firm. (2 points) b) (5 points) Calculate the price at which the profit maximizing, perfectly competitive firms will break even. ANSWER: This occurs at the minimum of the ATC curve where profits are driven to zero. This can be found three ways: 1) The way I did it in the first problem set in question #2 2) Calculating ATC, and setting the derivative of it to zero and solving for q 3) Recognizing that at the Min of ATC, ATC=MC and solving for q. All three of these methods will yield a solution of zero profits when the market price is equal to $40 at 20 units of output. (5 points) c) (10 points) Set up the monopolist’s profit maximization problem as a function of output produced and show that the price/quantity solution to this problem is the same as if the monopolist’s profit maximization problem was instead, written as a function of the price charged. ANSWER: Write the optimization problem as Max_{q} Profit=p(q)*q-TC(q) where p(q) is the inverse market demand (2 points) Take the first order conditions to find MR=MC or 900-2q=2q (1 point) so that q=225 => p*=$675 (2 points) AND 6 Write the optimization problem as function of p as Max_{p} Profit=p*q(p)-TC(q(p)) where q(p) is the direct market demand (2 points) Take the first order conditions w.r.t p (1 point) Solve to find p*=$675 => q*=225 (2 points) d) (5 points) If the monopolist can discriminate the market a second time, (assuming that the good is durable and those that purchased it in c) will no longer want to purchase it again), what price will the monopolist charge on the additional units and how many additional units will he/she sell? ANSWER: In the second market, the residual demand is just $675-q. (1 point) Setting MR for the residual demand = MC in the second market (accounting for the units sold already, which is $450 on the last unit) gives us 7 675-2q=2q+450 (OR from setting up and deriving the optimization problem) (2 points) => q*=56.25 additional units at a price of p*=$618.75 from the residual demand curve (2 points) e) (10 points) If the monopolist, instead, could implement first degree (perfect) price discrimination right away, show that the output level that corresponds to maximal profits solves and calculate the monopolist’s maximum feasible profit. ANSWER: Computing the integral using the inverse demand P(q) (2 points) Optimizing it with respect to q, to find q*=300. (3 points) Substituting for q* into the maximal profit expression should yield a profit under first degree of (1/2)(900)(300) -400 fixed cost= $134,600. (5 points) (if they forgot to account for the $400 fixed cost, award 3 points instead of 5) 8 ECON 100C Midterm Exam 2 Name____________________________ Prof. Levkoff Time: 6:30pm / 8:00pm (circle one) Spring 2013 Seat #____________________________ Version A Solutions PID#_____________________________ Directions: You will have the entire lecture to complete the exam. There are two parts- one short answer/free response section and one problem solving/analysis section. To receive full credit, SHOW ALL WORK and graphs where appropriate and CLEARLY BOX YOUR FINAL ANSWERS for any calculations. Make sure your name, section #, seat #, and PID are on the examination sheet before submitting your exam. Part I. Evaluate the validity of each of the following statements. Be sure to clearly accompany your explanation using graphs and equations where appropriate (10 points/question): a) (10 points) It is always optimal to charge different prices to different individuals according to their maximum willingness to pay because this allows the seller to extract the maximal surplus from consumers. ANSWER: Not necessarily. If the seller cannot prevent resale, then trying to charge different prices to different individuals may not be optimal if low paying individuals can resell the good to higher paying individuals, stealing profit from the seller in the resale market. If, in addition, the seller cannot distinguish between consumer types, then every consumer, when faced with different prices has an incentive to misrepresent themselves as the lowest paying type. 7 points - if they mention cannot prevent resale 3 points – if they mention cannot distinguish consumer types b) (10 points) Since the Cournot (as well as the Bertrand!) static duopoly model(s) predicts a breakdown from the cartel arrangement due to the incentive to deviate, we should never expect to see cartels operational in practice. ANSWER: False. Just because the static model doesn’t predict cartel stability doesn’t mean we shouldn’t see them in practice because we do! The fact that cartels do exist is evidence that they are in fact not playing a static game, but rather a repeated infinite horizon game where the threat of future punishment exists. 7 points – mention that cartels are not playing a static, but rather an infinitely repeated dynamic game 3 points – mentions that we do see them exist in reality c) (10 points) It is advantageous to be the first mover in games since the ability to precommit to a particular action can help determine the most beneficial path of play. ANSWER: Not always. Some games have second mover advantages (ie: matching pennies, price setting games, etc.). 7 points – mention 2nd mover vs. 1st mover advantage 3 points – show/discuss an example Part II: Problems / Analysis (70 points) #1) (25 points total) Consider the following duopoly game below between two firms selling the same (homogeneous) product that are competing by setting prices simultaneously. Each firm can set a price of $8, $9, or $10 for its product. The profits to either firm in thousands of dollars are shown in the following normal-form payoff matrix below: a) (5 points) If the firms set prices simultaneously, calculate the equilibrium prices to set for either firm. What type of equilibrium did you find? Is it efficient? Explain. ANSWER: {$8, $8} is the dominant strategy equilibrium of the game. It is not efficient. Both players could be made better off by jointly charging $9 or, even better, $10 (the most efficient outcome). 3 points – correct equilibrium 1 point – correctly identify it as a DSE 1 point – correct efficiency explanation (joint deviations to $9 OR $10 suffices to show that the equilibrium isn’t efficient) b) (10 points) If firm A sets its price in the first period, followed by firm B in the second period, who is capable of observing firm A’s first period choice, show that the equilibrium outcome that you found in a) is also sub game perfect (survives elimination) in this sequential game (hint: you should draw the game as an extensive form tree). ANSWER: Using backwards induction, the DSE from part a) indeed survives elimination and is the sub game perfect equilibrium outcome. 5 points - specifying the correct tree diagram 5 points - identifying the backwards induction SPE Outcome (don’t need to write the fully contingent strategy here). c) (10 points) If the firms are playing the simultaneous game repeatedly over an infinite horizon, for what values of each firm’s discount factor δ is it possible to sustain prices of $10 under the threat of “grim trigger” punishment? (hint: the game is not symmetric so you’ll need to consider this for both players). ANSWER: Need to consider δ for both players since the game isn’t perfectly symmetric. Firm A: … … … … ∗ Firm B: … … … … ∗ Thus, in order to guarantee cooperation, we need both firms to have a discount rate of at least 4/7 so that cooperation is optimal relative to deviation. 5 points – firm A’s problem correct 5 points – firm B’s problem correct No need to mention that both would need to have a discount factor greater than 4/7 to sustain collusion since the question asks you to find the delta for each firm. #2) (45 points total) Consider a monopolist selling to two types of consumer exhibiting demand 150 and 110 . Suppose the marginal cost of curves given by production is constant and equal to the average cost per unit so that $10. a) (5 points) Calculate the optimal 3rd degree prices to charge each type. ANSWER: Either write out the multivariate profit max problem, or use the condition that marginal revenue must equal marginal cost across both markets to find that ∗ & ∗ . 1 point – correct work (should show either MR=MC across all markets OR set up a profit maximization problem). 2 points – correct price to type 1 2 points – correct price to type 2 (give points if carry through mistake, but everything else was done correctly) b) (15 points) Calculate the optimal uniform two-part tariff to implement as a monopolist (same price and entry fee to both types). ANSWER: Case I: Exclude type 2. Charge with entry fee ∗ $ , ∗ and extract all surplus from type 1 . Case II: Sell to both types. Charge a marginal price per unit of and charge both types the fame entry fee according to the low type’s (type 2’s) consumer surplus effectively extracting all of the low type’s surplus (but the high type 1 will have some surplus left). Direct demand curves for each type are given by and so that total demand when selling to both types is . The entry fee is given by . Then the firm solves: → Since ∗ → ∗ $ , and ∗ $ , , the pricing plan in Case II is the optimal solution. 5 points – correctly considered case I 3 points – correctly set up the profit function 2 points – correctly differentiated the profit function (give points if carry through mistake, but everything else was done correctly) 3 points – correct optimal p and F (give points if carry through mistake, but everything else was done correctly) 2 points – identify optimal pricing plan as case II (give points if carry through mistake, but everything else was done correctly) . Suppose now, that there is only the type 1 consumer so that total market demand is just 150 . However, there is now a second firm with cost function in addition to the first firm with $10. Both firms produce the same identical product. c) (15 points) If the firms in the duopoly compete for the type 1 consumer by setting quantities simultaneously, calculate the Cournot-Nash equilibrium outputs for each firm to set as well as the market clearing price that prevails. ANSWER: Firm 1 Solves: , , . . , . . , ⇒ Firm 2 Solves: ≡ , , ⇒ ≡ Solving the system of best responses yields the Nash equilibrium outputs and market price: ∗ . & ∗ . → ∗ $ . 3 points – correctly specify firm 1’s problem 3 points – correctly identify firm 1’s first order condition OR best response function (give points if carry through mistake, but everything else was done correctly) 3 points – correctly specify firm 2’s problem 3 points – correctly identify firm 2’s first order condition OR best response function (give points if ca...
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