Unformatted text preview: Problem Set 1 #4) When a per-unit sales tax is added, either the Demand curve or the Supply curve shifts. Tax on Demand curve: left-shift (people will buy less if materials are taxed). Tax on Supply curve: left-shift (sellers need to increase price to make up for the tax). When the Supply curve is shifted, the point moves along the Demand curve. Thus, the new Price Equilibrium at this point is the price that consumers pay. When the Demand curve is shifted, the point moves along the Supply curve. Thus, the new Price Equilibrium at this point is the price that sellers sell at. Then, if you go straight down (in a Supply curve shift) or straight up (in a Demand curve shift) from the new Price Equilibrium, you will calculate what the other is paying. The difference is the tax that goes to the government. Problem Set 2 #4) Let p and q=2p be the prices of apples and bananas respectively. If you spend your 1st dollar on apples, you can buy 1/p pounds of apples which gives you an extra utility of 1 util x 1/p. If you spend it on bananas you can buy 1/2p pounds of bananas which gives you an extra utility of 3 utils x 1/2p. Hence, you should spend this dollar on bananas. i.e. 1 (1/p) utils vs. 1.5 (1/p) utils. #5) The marginal utility for apples and bananas is 1 and 3 respectively. To see this, notice that an extra pound of apples (resp. bananas) increases your utility by 1 (resp. 3) and this is independent of how many apples and bananas you are consuming. Therefore, we have: 1/p < 3/2p. When we have MU(a)/ Pa < MU(b)/Pb, it means that you should decrease your consumption of good A and buy more of good B. But, in this instance you are already consuming zero units of good A and you cannot further decrease your consumption. Problem Set 3 #1) Need to solve for PE (price of exercise equipment) by plugging in given values for PV (price of exercise videos) and QE (quantity of exercise equipment). PV is given as 2, and QE will be zero to calculate the choke price. Then, using the own price-price elasticity formula ( E = P/(Pchoke P) ). Now you plug in PE (also given) and the Pchoke (which you have found) to get the elasticity. #2) Complement goods >>> Cross-price elasticity = negative (+/-) or (-/+) Substitute goods >>> Cross-price elasticity = positive(+/+) or (-/-) Cross-price elasticity = (% change in Qe) / (% change in Pv) ... i.e. the change in quantity of one good over the change in price of the other good. #4) For linear demands, we know that Elasticity = P / (Pchoke P). The first demand curve has a choke price of 100, whereas the second has a choke price 50. Therefore, the second one, i.e. the one with the lowest choke price, is more elastic at any price. NOTE: that E = (Q/P) x (P/Q) ... thus, when comparing 2 linear demand curves (SEE 9/15 PACKET, p 11), the "P/Q" ratio differs greatly at any one price. i.e. the two points on the 2 lines have the same price, but one's quantity is significantly larger than the other's...which messes up the entire Elasticity equation. Problem Set 4 #1) Average Cost = TC / q ... Av. Variable Cost = VC / q ... Marginal Cost = TC / q ... Total Cost = Fixed Cost + VC ............... find TC with q = 0 ... Then, set the given TC(q) curve to FC + VC (find TC w/ q = 0) ... Solve for VC in terms of q. Divide by q to obtain Av. Variable Cost ... Set MC (given) = AC (found) and solve for q ... Then, set MC = Av. VC (found) and solve for q. Problem Set 5 #1) Find Inverse Demand. Marginal Revenue = same intercept, 2x the slope. Price that maximizes Profits = (Pchoke + Marginal Cost) / 2 ... Pmax = (Pchoke + MC) / 2 Profits = (Max Price Marginal Cost)(Quantity) = (Pmax MC)(Q) #2) MR (found) = 30 Q ... MC (given) = 6 ... MC = 6 + TAX Set MC = MR ... solve for Q in terms of t. Tax Revenue = Q x t Set TR = desired tax (in this case 23). Solve. Problem Set 6 #1) Find MRs for both segments. Set equal to MC = 5. Find max Q, P for both segments. MC = Price (1 (1/E) ). Solve for Elasticity in both segments. #3) MRa [Qa] = MRb [Qb] = MC [Qa + Qb] First, find MRa and MRb. Set them = to each other. Solve for Qb in terms of Qa. Then set MRa [Qa] = MC [Qa + Qb] (formula). Substitute in for Qb what you found earlier. Solve for Qa. Then you can find Pa, as well as Qb and Pb. #4) MR = MC = P ... find P, plug into Cost Function to get Q. Revenue = P x Q Profit = Rev. Cost Producer Surplus = area of under the MC curve Problem Set 8 #1) Inverse Demand. Marginal Revenue. MR=MC. Find Pchoke by plugging in 0 for Q. Use formula: P = (Pchoke + MC) / 2. You will have P1 in terms of P2 and P2 in terms of P1...set equal to each other and solve. #2) Now that you have P1 in terms of P2 and P2 in terms of P1, take the original P1 Inverse Demand, and plug in new P2 value (in terms of P1). Solve for P1 in terms of Q1. Note: now you have a new Pchoke. Use formula: P = (Pchoke + MC) / 2. Find P1. Then plug new P1 value into the equation you found in the problem before. #3) P = 100 Q1 Q2 Q3. (x = Q1). SO: P = 100 x Q2 Q3. Take MR. MR = 100 x Q3 2(Q2) = MC = 10. Solve for Q2 in terms of Q3. Set Q2 = Q3. Find Q = (90 x) / 3. Plug back into: P = 100 x Q2 Q3. (Q2=Q3). Take MR again. Solve for x. Problem Set 9 #2) First, set the 2 inverse demand curves = to each other. Solve for Q. Then, add marginal external cost to supply curve (a.k.a. the marginal private cost curve). Set this new supply curve equal to the original demand curve. Solve for Q. Calculate the difference between the first Q and the second Q. This is your deadweight loss. Problem Set 10 #3) Suppose I pay x for insurance. Then, the lotteries I face (with and without insurance respectively) are: L1 = (10000-x,2/3; 10060-x,1/3) L2 = (10000, 1/3 ; 9880, 1/3; 10060, 1/3) If you buy insurance, but the price falls by 10%, you will not exercise the insurance. MIDTERM I #4) Amount consumed of the services of domestic servants declined during the first half of the 20th century, while per- capita income was increasing. Does this mean that domestic servants are an inferior good? No. An inferior good is defined as a good whose quantity goes down us income goes up, when all else is held fixed. During the 20th century tastes have changed (demand shifted). #7) When looking at 2 demand curves (not linear), and they have the same price, one will have a lower quantity. This quantity affects the "P/Q" part of the Elasticty formula. Then; you must also look at the Q/P part of the Elasticity formula. Analyze the slop to determine which outweighs the other: the change in Q or the change in P. For one case, you will be able to tell because both components of the equation will go up. For the other, one will go up and the other will go down; thus the result is ambiguous. #8) Option 1: do not buy card. Buy up to the point where MU = P. This is at 9 DVDs. Thus, you spend 9x5=45. Your Total Utility = all the MUs added up to that point. Calculate Total Utility total expenditures = 36...Option 2: buy the card. Buy up to the point where MU = P. This is at 11 DVDs. Thus, you spend 11x3=33 + card. So it's: 20 + 11x3 = 53. Your Total Utility is all the MUs added up to that point. Calculate Total Utility total expenditures = 88. #10) Set MPlabor = MPmachines. Find that M = 4L. Plug this into L^.5 + 2(4L)^.5. Set equal to 10. Solve. MIDTERM 2 #4) Prisoner's Dilemma: there is one Nash Equilibrium, but there is 1 other possibility that makes both players better off. There are also Dominated Strategies...Coordination Game: there are 2 Nash Equilibria. #6-7) Take equation P=140-Q and make it P=140 - Q1 Q2. Find MR. Set MR=MC. Find both reaction curves: You will have an equation with both Q1 and 2(Q2). Solve for Q2. This will be the "same" reaction curve for Q1. (Find the Qne by setting both Q1 and Q2 = Qne in either of the reaction curves). To find Monopoly Price, use Pmax=(Pchoke + MC) / 2. Plug this back into P=140-Q. Now you have the Monopoly prices and quantities and the Nash prices and quantities. NOW, use your reaction curves. Plug in the respective Nash and Mon. quantities to find out what the other firms will produce on Tuesday, Wednesday, etc. Repeat. #10) Let x and y be the strategies of players 1 and 2 respectively. Any pair (x, y) such that x + y = 100 constitutes a NE. SAMPLES I #7) AC = TC / 60 = 25 AC = TC / 60 = 22 ... BUT: if TC were higher, then AC = 26. This means that the AC curve would attain its minimum elsewhere, because MC does not = AC at AC's minimum. SAMPLES II #4-5) Take market inverse demand (P = 140 Q) and make it P = 140 Q1 Q2. Then find MR. Set MR = MC. Set Q1=Q2=Qne. Find Qne. Find Pne...Then, find the Monopoly Q and Monopoly P. To do this, use formula: Pmax = (Pchoke +MC) / 2. Then, plug this P back into the original equation (P = 140 Q). Now you have the Prices and Quantities for Nash and Monopoly...Then, to find the profits for when one chooses Nash and the other chooses Monopoly. Take Qne and Qmon and plug them back into the equation P = 140 Q1 Q2. Then you have a price. To find the profits for this, simply multiply this price by Qne and Qmon respectively. #7) Derive the reaction curve for firm 1 as a function of the prices of firms 2 and 3. To do this, use the mid-point pricing rule (i.e. set Q1=0). Then, solve for the NE prices, which you know they have to be equal, because the game is symmetric (i.e. set P1=P2=P3="P") FINAL I Exercise: Consider a perfectly competitive market. Each firm has the following total and marginal cost curves: TC = 400 + Q^2 ... MC = 2Q The market demand is Q = 2800 30P. Calculate the long run equilibrium price, the quantity that each firm produces and the number of firms. Step1. Each firm produces where MC = AC...Step 2. AC = 400/Q + Q ...Step 3. AC = MC occurs when Q = 20...Step 4. The long run equilibrium quantity is at the minimum AC. The AC attains its minimum when AC = MC. This happens when Q = 20 and for that quantity AC = 40...Step 5. Q = 2800 30(40) = 1600...Step 6. Number of firms = 1600 / 20 = 80 NOTE: Maximizing revenue is equivalent to maximizing profit when MC = 0. NOTE: The higher the choke price, the more inelastic (lower elasticity) for any price. NOTE: If the elasticity of demand is 0.8, can you conclude that at least one car firm is not maximizing profits? It is true that any firm that maximizes profits will produce at the elastic (greater than 1) part of the demand. However, the estimate of 0.8 refers to the industry demand and is therefore not informative about the pricing behavior of individual firms. NOTE: Positive Network Externalities: If other consumers are more likely to buy, I am also more likely to buy, because with positive externalities my valuation increases with the number of people that use a good or a service. FINAL 2 #5) Profits = Revenue FC TVC...Notice that TVC is the area under MC. #6) NOTE: A tax on the supplier, monopolist, etc. shifts the MC curve up by the tax. Now, set MC + tax = MR and solve for Q. Take Q x tax and find government profits. #7) Long-run (zero profit) equilibrium price = minimum AC. Set AC=MC. Find Q. Plug Q into AC. #10) Firm A and Firm B are in a Cournot competition game. They have to decide how many widgets they want to produce. The market inverse demand is: P = 220 - 2Q, where Q = QA + QB is the total widget capacity. MC=20. Find the Stackelberg equilibrium quantities (capacities) for each firm, when British Airways is the leader. First we find the reaction curve for the follower. Then, plug in the reaction curve into the inverse demand. Take MR=MC again, and find the quantity. Then plug back into inverse demand. #20) NOTE: The law of diminishing returns states that as you increase the use of one factor of production (labor), keeping all other factors (machines) constant, the extra output (marginal product) you get decreases. The original PPF (solid) has the property that as you move down the curve to increase the production of ice cream, you have to sacrifice less and less sweets. This is inconsistent with the law of diminishing returns. According to the law, as you move down the curve, you need more and more workers to increase ice-cream output, and hence you need to sacrifice more and more sweets. FINAL III #2) NOTE: The Stackelberg outcome has to be on the follower's reaction curve. Since the game has positive externalities and strategic complements (reaction curves have positive slope) we know that in the Stackelberg game both the leader and the follower increase their actions. #4) NOTE: Firms in competitive market will produce where P=MC. Remember: MC=AC, at AC's minimum. #10) Consider a two-player game where each player has a dominant strategy. Let that strategy be A B for player 2. Then, (A, B) is an equilibrium in dominant strategies. Is (A, B) also a nash equilibrium?...Yes. The profile (A, B) is a Nash equilibrium, because any equilibrium in dominant strategies in also a NE. Given that player 2 plays B, A is a best response for player 1, and given that 1 plays A, B is a best response for 2. #11) t or f: There is probably someone whose marginal valuation of going to the beach (net of whatever costs she actually bears) is lower than the marginal cost she exerts on others...True. Beachgoers do not internalise the externality and they "over-consume", i.e. too many people go to the beach, so that the MV for the last person going to the beach is lower than the MC. #17) Consider a profit maximizing monopoly with increasing marginal cost of production. Its production workers have won a wage increase, causing the firm's marginal cost of production to rise. Answer the following: 1. Can you tell whether revenue will necessarily rise or fall, as a consequence? 1. Revenue falls, because we are in the elastic portion of the demand. 2. Can you tell whether cost will necessarily rise or fall? 2. Cost may increase or decrease. On one hand, less is produced, but what we produce is now produced at a higher cost. 3. Can you tell whether profits will necessarily rise or fall? 3. This is harder to see, but profits will fall. Here is why: Let (P1, Q1) be the profit maximizing price and quantity before the wage increase. Let (P2, Q2) be the profit maximizing price and quantity after the wage increase. Before the wage increase (P2, Q2) brings less profit than (P1, Q1), since the latter brings the maximum profit. After the wage increase (P2, Q2) brings even less profit since the marginal cost has increased. ...
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- Spring '06
- Supply And Demand, inverse demand