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Principles of Economics
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Chapter 15 / Exercise 3
Principles of Economics
Mankiw
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Unformatted text preview: Math for Econ I- Practice Problems New York University 1. Supply and demand 1. In a California town, the monthly charge for waste collection is $8 for 32 gallons of waste and $12.32 for 68 gallons of waste. (a) Find a linear formula for the cost, C, of waste collection as a function of the number of gallons of waste, w. (b) What is the slope of the line found in part (a)? Give units and interpret your answer in terms of the cost of waste collection. (c) What is the vertical intercept of the line found in part (a)? Give units and interpret your answer in terms of the cost of waste collection. Solution: (a) We find the slope m and intercept b in the linear equation C = b + mw. To find the slope m, we use m= ∆C 12.32 − 8 = = 0.12 dollars per gallon. ∆w 68 − 32 We substitute to find b: C = b + mw 8 = b + (0.12)(32) b = 4.16 dollars. The linear formula is C = 4.16 + 0.12w. (b) The slope is 0.12 dollars per gallon. Each additional gallon of waste collected costs 12 cents. (c) The intercept is $4.16. The flat monthly fee to subscribe to the waste collection service is $4.16. This is the amount charged even if there is no waste. 2. Production costs for manufacturing running shoes consist of a fixed overhead of $650,000 (“fixed cost”) plus a cost of $20 per pair of shoes (“variable cost”). Each pair of shoes sells for $70. (a) Find the total cost, C(q), the total revenue, R(q), and the total profit, π(q), as a function of the number of pairs of shoes produced, q. (b) How many pairs of shoes must be produced and sold for the company to make a profit? Solution: (a) The cost function is of the form C(q) = b + m · q where m is the variable cost and b is the fixed cost. Since the variable cost is $20 and the fixed cost is $650,000, we get C(q) = 650,000 + 20q. The revenue function is of the form R(q) = pq where p is the price that the company is charging the buyer for one pair. In our case the company charges $70 a pair so we get R(q) = 70q. The profit function is the difference between revenue and cost, so π(q) = R(q) − C(q) = 70q − (650,000 + 20q) = 70q − 650,000 − 20q = 50q − 650,000. (b) We are asked for the number of pairs of shoes that need to be produced and sold so that the profit is larger than zero. That is, we are trying to find q such that π(q) > 0. Solving we get π(q) > 0, 50q 50q − 650,000 > 0 > 650,000 q > 13,000. ! ! Thus, if the company produces and sells more than 13,000 pairs of shoes, it will make a profit. ! 3. Linear supply and demand curves are shown in Figure with price on the vertical axis. ! ! 24 (a) Label the equilibrium price p0 and the equilibrium quantity q0 on the axes. (b) Explain the effect on equilibrium price and quantity if the slope, ∆p/∆q, of the supply curve increases. ANSWER: Illustrate your answer graphically. (a) See Figure ??. (c) Explain the of the on equilibrium price and quantity if the slope, ∆p/∆q, of the demand curve becomes more (b) If the slope effect supply curve increases then the supply curve will intersect the demand negative. Illustrate your answer graphically. curve sooner, resulting in a higher equilibrium price p1 and lower equilibrium quantity q1 . Intuitively, this makes sense since if the slope of the supply curve increases. The amount Solution: produced at a given price decreases. See Figure ??. (a) See Figure below: p p New supply Supply p1 p0 p0 Demand q q0 1-4aq33ansafig Old supply Demand q1 q0 1-4aq33ansbfig q ! upply curve increases then the the slope Figure 1.15 Figure 1.16 (b) If supply curve the supply curve increases then the supply curve will intersect the demand curve sooner, of will intersect the demand ng in a higher equilibrium price p1 and lower equilibrium quantity q1 . p1 and lower equilibrium quantity q1 . Intuitively, this makes sense resulting in a higher equilibrium price es sense since if the slope of the if the slope ofincreases. The amount (c) since supplyslope of the demand curve increases. Thenegative, produced atfunction will decreases. See Figure When the curve the supply curve becomes more amount the demand a given price price decreases. See Figure below: more rapidly and will intersect the supply curve at a lower value of q . This will ??. decrease 1 also result in a lower value of p1 and so the equilibrium price p1 and equilibrium quantity p q1 will decrease. This follows our intuition, since if demand for a product lessens, the price New supply Supply Old supply and quantity purchased of the product will go down. See Figure ??. 15 p Supply p1 p0 Demand q Demand q1 q0 1-4aq33ansbfig ! p0 q p1 Old demand Figure 1.16 1-4aq33anscfig q1 q0 New demand the demand curve becomes more negative, the demand function will ly and will intersect the supply curve at a lower value of q1 . ThisFigure 1.17 will r value of p1 and so the equilibrium price p1 and equilibrium quantity is follows our intuition, since SHORT ANSWER: if demand for a product lessens, the price sed of the product will go down. See Figure ??. p (a) q Figure 1.15 Figure 1.16 (c) When the slope of the demand curve becomes more negative, the demand function will (c) When the slope of the demand curve becomes more negative, the demand function will decrease more rapidly decrease more rapidly and will intersect the supply curve at a lower value of q1 . This will and will intersect the supply curve at a lower value of q1 . This will also result in a lower value of p1 and also result in a lower value of p1 and so the equilibrium price p1 and equilibrium quantity so the equilibrium price p1 and equilibrium quantity q1 will decrease. This follows our intuition, since if q1 will decrease. This follows our intuition, since if demand for a product lessens, the price demand for a product lessens, the price and quantity purchased of the product will go down. See Figure and quantity purchased of the product will go down. See Figure ??. below: (a) p Supply p0 p1 Old demand q q1 q0 New demand 1-4aq33anscfig ! Figure 1.17 Job: 1-4aq34-main Sheet: 1 Page: 1 (September 8, 2012 11 : 21) [1-4aq34-main] SHORT ANSWER: 4. A demand curvephas equation q = 100 − 5p, where p is price in dollars. A $2 tax is imposed on consumers. Find 1 the equation of the new demand curve. Sketch both curves. Supply 1-4aq34 1. A demand curve has equation q = 100 − 5p, where p is price in dollars. A $2 tax is imposed on consumers. Find theSolution: new demand curve. Sketch both curves. equation of the ANSWER: p0 The original demand equation, q = The original demand equation, q = 100 − 5p, tells us that 100 − 5p, tells us that Demand Quantity demanded = 100 − 5demanded = 100 − . Quantity q Amount per unit paid by consumers q0 5 (Amount per unitpaid by consumers ) . The consumers consumers payunit + 2 dollars the price p plusbecause they pay the price p plus $2 tax. Thus, the new demand The pay p + 2 dollars per p because they pay per unit $2 tax. Thus, the new demand equation is b) Equilibrium price will increase; equation is q = 100 − 5(p + 2) = 90 − 5p. equilibrium quantity will decrease q = 100 − 5(p + 2) = 90 − 5p. See Figure 1. (c) Equilibrium price and quantity will decrease Shifts in Supply and pDemand Curves -Taxes 20 18 Demand without tax: q = 100 − 5p Demand with tax: q = 90 − 5p 1-4aq34ansfig See Figure above. ! SHORT ANSWER: q = 90 − 5p 90 100 Figure 1 q 5. A tax of $8 per unit is imposed on the supplier of an item. The original supply curve is q = 0.5p − 25 and the demand curve is q = 165 − 0.5p, where p is price in dollars. Find the equilibrium price and quantity before and after taxes. Solution: Before the tax is imposed, the equilibrium is found by solving the equations q = 0.5p − 25 and q = 165 − 0.5p. Setting the values of q equal, we have 0.5p − 25 p = 165 − 0.5p = 190 dollars Substituting into one of the equation for q, we find q = 0.5(190) − 25 = 70 units. Thus, the pre-tax equilibrium is p = $190, q = 70 units. The original supply equation, q = 0.5p − 25, tells us that Quantity supplied = 0.5 (Amount per unitreceived by suppliers ) − 25. When the tax is imposed, the suppliers receive only p − 8 dollars per unit because $8 goes to the government as taxes. Thus, the new supply curve is q = 0.5(p − 8) − 25 = 0.5p − 29. The demand curve is still q = 165 − 0.5p. To find the equilibrium, we solve the equations q = 0.5p − 29 and q = 165 − 0.5p. Setting the values of q equal, we have 0.5q − 29 = 165 − 0.5p q = 194 dollars Substituting into one of the equations for q, we find that q = 0.5(194) − 29 = 68 units. Thus, the post-tax equilibrium is p = $194, q = 68 units. 6. Consider the supply and demand functions S = 3p − 50 D = 100 − 2p. (a) Sketh the graphs of the supply and demand functions. (b) Find the equilibrium price and quantity. (c) Suppose that a specific tax of $5 per unit is imposed upon suppliers. What are the new equilibrium price and quantity. Justify your answer graphically. What is the effect of the tax on the demander, what is it on the supplier. Solution: (a) To find the equilibrium price and quantity, we find the point at which Supply = Demand 3p − 50 = 100 − 2p 5p = 150 p = 30. The equilibrium price is $30. To find the equilibrium quantity, we use either the demand curve or the supply curve. At a price of $30, the quantity produced is 100 − 2 · 30 = 40 items. The equilibrium quantity is 40 items. In Figure below the demand and supply curves intersect at p∗ = 30 and q ∗ = 40. The equilibrium price is $30. To find the equilibrium quantity, we use either the demand curve or the supply curve. At a price of $30, the quantity produced is 100 − 2 · 30 = 40 items. The equilibrium quantity is 40 items. In Figure 1.50, the demand and supply curves intersect at p∗ = 30 and q ∗ = 40. p (price) S(p) p∗ = 30 D(p) ∗ fig1aq49 (b) q = 40 ! q (quantity) Figure 1.50: Equilibrium: p∗ = 30, q ∗ = 40 (c) The consumers pay p dollars per unit, but the suppliers receive only p − 5 dollars per unit because $5 goes to the government as taxes. Since Quantity supplied = 3(Amount per unit received by suppliers) − 50, the new supply equation is Quantity supplied = 3(p − 5) − 50 = 3p − 65; the demand equation is unchanged: Quantity demanded = 100 − 2p. At the equilibrium price, we have Demand = Supply 100 − 2p = 3p − 65 165 = 5p p = 33. The equilibrium price is $33. The equilibrium quantity is 34 units, since the quantity demanded is q = 100 − 2 · 33 = 34. Notes: In this example, the equilibrium price was $30; with the imposition of a $5 tax in Example ??, the equilibrium price is $33. Thus the equilibrium price increases by $3 as a result of the tax. Notice that this is less than the amount of the tax. The consumer ends up paying $3 more than if the tax did not exist. However the government receives $5 per item. The producer pays the other $2 of the tax, retaining $28 of b: bchap1-temp Sheet: 37 Page: 37 (August 15, 2012 10 : 14) [ex-1-4] the tax was imposed on the producer, some of the tax is passed on to the price paid per item. Although the consumer in terms of higher prices. The tax has increased the price and reduced the number of items sold. See Figure below. Notice that the taxes have the effect of moving the supply curve up by $5 because suppliers have to be paid $5 more to produce the same quantity. 1.4 APPLICATIONS OF FUNCTIONS TO ECONOMICS 37 p (price paid by consumers) Supply: With tax Supply: Without tax 33 30 28 Demand fig1aq50 34 40 q (quantity) Figure 1.51: Specific tax shifts the supply curve, altering the equilibrium price and quantity ! Budget Constraint An ongoing debate in the federal government concerns the allocation of money between defense and social programs. In general, the more that is spent on defense, the less that is available for social programs, and vice versa. Let’s simplify the example to guns and butter. Assuming a constant budget, we show that the relationship between the number of guns and the quantity of butter is linear. Suppose that there is $12,000 to be spent and that it is to be divided between guns, costing nstraint n ongoing debate in the federal governmentin the federalallocation of money betweenallocation of money between defense and social pro7. An ongoing debate concerns the government concerns the defense d social programs. In grams. In general, the more on defense, theon defense, available that is available for social programs, and vice general, the more that is spent that is spent less that is the less for cial programs, and vice versa. Let’s simplify the example to guns andand butter. Suppose that there is 12,000 dollars to be spent and that versa. Let’s simplify the example to guns butter. Assuming a constant dget, we show that the relationship between guns, costing $400 and the quantity of costingis$2000 a ton. Therefore, the equation of the it is divided between the number of guns each, and butter, butter ear. Suppose that there is $12,000budget constraint is 400g + 2000b = between guns, costing company’s to be spent and that it is to be divided 12, 000. 00 each, and butter, costing $2000 a ton. Suppose guns as a functionbought is g, and the number (I) Write the number of the number of guns of the tons of butter. tons of butter is b. Then the amount offunction from part (I) (which and the amountb, is on the horizontal axis?) (II) Graph the money spent on guns is $400g, variable, g or spent on tter is $2000b. Assuming all the money is spent, Solution: (I) As stated in the problem, Amount spent on guns + Amount spent on butter =guns + Amount spent on butter = $12,000 Amount spent on $12,000 or 400g + 2000b = 12,000. 400g + 2000b = 12,000. Thus, dividing both sides by 400, hus, dividing both sides by 400, g + 5b = 30. g + 5b = 30. This equation is the budget constraint. Since the budget constraint can be written as his equation is the budget constraint. Since the budget constraint can be written as g = 30 − 5b, g = 30 − 5b, the graph of the budget constraint is a line. e graph of the budget constraint is a line. See Figure 1.52. (II) g (number of guns) 30 g + 5b = 30 fig1aq51 Section 1.4 6 ! b (tons of butter) Figure 1.52: Budget constraint 8. A person has m dollars to spend on the purchase of two commodities. The prices for commodities are p and q dollars per unit. Suppose x units of the first and y units of the second commodity are bought. Write a formula to describe the budget of the buyer. Answer: m = px + qy 9. If a firm sells Q tons of a product, the price P in dollars, received per ton is P = 1000 − Q. The price it has to pay per ton is P = 800 + Q. In addition, it has transportation costs of 100 dollars per ton. (a) Express the firm’s profit π as a function of Q, the number of tons sold, and find the profit maximizing quantity. (b) Suppose the government imposes a tax on the firm’s product of 10 dollars per ton. Find the new expression for the firms profits π and the new profit maximizing quantity. ˆ Answer: (a) The profit function is π(Q) = Q(1000 − Q) − (Q(800 + Q)) − 100Q = −2Q2 + 100Q. To find the profit maximizing quantity we complete the square: −2(Q2 − 50Q) = −2((Q − 25)2 − 252 ) = −2(Q − 25)2 + 1250. Hence the profit maximizing quantity is 25 tons and the maximum profit is 1250 dollars. (b) The new profit function is π(Q) = Q(1000 − Q) − (Q(800 + Q)) − 110Q = −2Q2 + 90Q. To find the profit maximizing quantity we complete the square: −2(Q2 − 45Q) = −2((Q − 22.5)2 − 22.52 ) = −2(Q − 22.5)2 + 1012.5. Hence the profit maximizing quantity is 22.5 tons and the maximum profit is 1012.5 dollars. 10. (Textbook Section 4.6 Exercise 8-Harder Problem-Same ideas as in the previous problem, instead of numbers you have constants. I want you feel comfortable with these.) If a cocoa shipping firm sells Q tons of cocoa in England, the price received is given by P = α1 − 1 Q. On the 3 1 other hand, if it buys Q tons from its only source in Ghana, the price it has to pay is given by P = α2 + 6 Q. In addition it costs γ per ton to ship cocoa from its supplier in Ghana to its customers in England (its only market). The numbers α1 , α2 and γ are all positive. (a) Express the cocoa shipper’s profit as a function of Q, the number of tons shipped. (b) Assuming that α1 − α2 − γ > 0, find the profit maximizing shipment of cocoa. (c) Suppose the government of Ghana imposes an export tax on cocoa of t per ton. Find the new expression for the shipper’s profit and the new quantity shipped? (d) Calculate the government’s export tax revenue as a function of t, and advise it on how to obtain as much tax revenue as possible. Solution: (a) The profit function is π(Q) = Q(α1 − 1 Q) − (Q(α2 + 1 Q)) − γQ = − 1 Q2 + (α1 − α2 − γ)Q. 3 6 2 (b) To find the profit maximizing quantity we complete the square: 1 1 1 1 − (Q2 −2(α1 −α2 −γ)Q) = − ((Q−(α1 −α2 −γ))2 −(α1 −α2 −γ)2 ) = − (Q−(α1 −α2 −γ))2 + (α1 −α2 −γ)2 2 2 2 2 1 Hence the profit maximizing quantity is (α1 − α2 − γ) and maximum profit is 2 (α1 − α2 − γ)2 . 1 (c) The new profit function is π(Q) = Q(α1 − 3 Q) − (Q(α2 + 1 Q)) − γQ − tQ = − 1 Q2 + (α1 − α2 − γ − t)Q. 6 2 To find the profit maximizing quantity we complete the square: 1 1 − (Q2 − 2(α1 − α2 − γ − t)Q) = − ((Q − (α1 − α2 − γ − t))2 − (α1 − α2 − γ − t)2 ). 2 2 Hence the profit maximizing quantity is (α1 − α2 − γ − t). (d) The governments export tax revenue on the profit maximizing quantity is R = t ∗ Q∗ = t(α1 − α2 − γ − t) = −t2 + (α1 − α2 − γ)t. To find t that maximizes tax revenue we need to complete the square. 1 1 R = −(t2 − (α1 − α2 − γ)t) = − (t − (α1 − α2 − γ))2 − (α1 − α2 − γ)2 2 2 Hence revenue will be maximized if the government charges t = 1 (α1 − α2 − γ) dollars per ton. 2 11. (Bertrand Model of Price Competition) If there is more than one producer/seller in a market but not so many as to make the perfectly competitive model applicable, we say that the market structure is oligopolistic. One model that describes behavior of firms in this setting is the so-called Bertrand model. To make the matters simple, we assume that there are two firms in the market. We assume that two firms compete in prices. Each firm sets a price and then meets whatever demand exists for its product at that price. We assume that they produce identical commodities, if one firm charges a lower price than the other, then all the consumers will purchase from that producer. If the two firms charge the same price, then we assume that consumers’ purchases will be split evenly between two producers. Thus we need to think about how revenue for each firm changes at prices that are altered. Let’s suppose that the demand function is y = 20 − 2p and the cost function is C = 4y. Analyse firm 1’s profit and revenue functions for alternative prices given that a specific price has been set by firm 2. For simplicity first assume that firm 2 sets a price of p = 7 dollars. Then generalize this to any price. Solution: Suppose firm 2 charges p2 = 7 dollars. If p1 > 7 then the revenue of firm 1, R1 = 0. If p1 = 7 then they share the market. Since for p = 7 the demand is y = 20 − 2(7) = 6 units firm 1 will get only 3 of it hence its revenue is then 3 · 7 = 21 dollars. Also since the cost of producing 3 units is C = 4 · 3 = 12, firm 2’s profit for p1 = 7 will be 21 − 12 = 9 dollars. If p1 < 7 dollars then firm 1 will get everybody, hence it’s revenue is R1 = p1 (20 − 2p1 ). So for this special case when we assume that p2 = 7 dollars we get the following revenue, and profit functions for company 1: p1 (20 − 2p1 ) R1 (p1 ) = 21 0 if p1 < 7 if p1 = 7 if p1 > 7 p1 (20 − 2p1 ) − 4(20 − 2p1 ) π1 (p1 ) = 9 0 if p1 < 7 if p1 = 7 if p1 > 7 Now this can be easily generalized to a case where the set price for firm 2 is p2 . p1 (20 − 2p1 ) 1 R1 (p1 ) = 2 p1 (20 − 2p1 ) 0 if p1 < p2 if p1 = p2 if p1 < 7 p1 (20 − 2p1 ) − 4(20 − 2p1 ) π1 (p1 ) = 1 (p1 (20 − 2p1 ) − 4(20 − 2p1 )) 2 0 if p1 < p2 if p1 = p2 if p1 > p2 2-main Sheet: 1 Page: 1 (August 25, 2014 10 : 57) [1-4aq2-main] Job: bchap1-temp Sheet: 38 Page: 38 (August 15, 2012 10 : 14) [ex-1-4] 2. Exponential growth, present values, completing the square 1 1. In Figure below, which shows the cost and revenue functions for a product, label each of the following: 38 Chapte...
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