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CHAPTER 2

OPTIMAL DECISIONS USING

MARGINAL ANALYSIS

The following table shows the five topic sections of this chapter and the associated study guide problems that pertain to each topic section.

Section Topic 1 A Simple Model of the Firm

Problems M1-M5. 2 Marginal Analysis

Problems M6-M7, S1, L1. 3 Marginal Revenue and Marginal Cost

Problems M8-M13, S2-S4, L2-L5. 4 Sensitivity Analysis

Problems M14-M19, L6-L9. 5 Appendix: Calculus and Optimization Techniques

Problems M20, S5-S8, L10-L11.

Multiple Choice

M1 According to the model of the firm, management’s main objective is to maximizea. Revenue.b. Return on Investment.c. Profit.d. Market share in targeted segments.e. Earnings growth.

M2 According to the simple model of the firm, management can predict a. Both revenues and costs with certainty.b. Costs with certainty, but revenues imprecisely.c. Its output level with certainty, but the market-clearing price imprecisely.d. Neither revenues nor costs very precisely.e. Revenues and costs with certainty only in the short run (next 3 months).

M3 A firm’s total cost function is given by: C = 1,000 + 15Q. At an output of 200 units,a. Total cost is $4,000 and marginal cost is $3,000.b. Total cost is $3,000 and fixed cost is $1,000.c. Total cost is $4,000 and marginal cost is 15.d. Average total cost is constant.e. Answers c and d are both correct.

M4 A firm’s demand curve is described by the equation: Q = 200 – 4P. The inverse demand curve is:a. P = 200 – 4Q.b. P = 50 - .25Q.c. P = 50 - .5Q.d. P = 200 - .25Q.e. None of the above answers is correct.

M5 A firm’s price and cost equations are given by P = 200 - 2Q and C = 1,000 + 40Q, respectively. Therefore, a. The firm’s revenue is: R = 200 – 2Q2.b. The firm’s profit is: = -1000 + 160Q – 2Q2

c. The firm’s revenue is: R = 200Q – 2Q2.d. At Q = 50 units, the firm’s total profit is 2,500.e. Answers b and c are both correct.

M6 At its current output level, a firm’s marginal profit is negative. Therefore, it shoulda. Increase output until M = 0b. Reduce output because MR > MC.c. Reduce output because MR < MC.d. Shut down production to avoid continuing losses.e. Reduce both its output and its price.

M7 A firm’s profit equation is given by: = -100 + 160Q – 20Q2. Therefore,a. The firm’s fixed cost is 100.b. M = 160 – 20Q.c. The firm’s profit-maximizing output is Q = 4.d. M = 160 – 40Q.e. Answers a, c, and d are all correct.

M8 For a downward sloping demand curve, the associated marginal revenue curvea. Coincides with the demand curve.b. Lies below and parallel to the demand curve.c. Has the same price intercept but a steeper slope than the demand curve.d. Is positive for all levels of sales.e. None of the above answers is correct.

M9 A firm can sell as much output as it wishes at the fixed price, P = $10 per unit. Then,a. The revenue curve has the usual inverted U-shape.b. To maximize profit, the firm will produce at full capacity.c. MR = 10 - 2Q.d. Marginal revenue is constant and equal to $10.e. Answers a and c are both correct.

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M10 In order to maximize profit, management should set output such thata. Marginal profit is zero.b. Marginal profit is maximized.c. Marginal revenue is zero.d. Marginal revenue equals marginal cost.e. Answers a and d are both correct.

M11 A firm’s price and cost equations are given by P = 200 - .2Q and C = 1,000 + 40Q, respectively. Therefore, its profit maximizing level of output isa. Q = 500 units.b. Q = 400 units.c. Q = 300 unitsd. Q = 600 units.e. None of the above answers is correct.

M12 A firm’s price equation is given by: P = 200 - 20Q. Then, the firm’sa. Revenue-maximizing output is 6 units.b. Revenue-maximizing price is $100.c. Profit-maximizing price is $100.d The firm’s profit-maximizing output is 3 units.e. Answers a and d are both correct.

M13 A firm’s cost equation is given by C = 200 + 10Q + 2Q2. At Q = 20, marginal cost isa. $1,200b. 1,200/20 = $6c. $1,000d. 1,000/20 = $5e. $90

M14 If fixed costs increase, what will happen to marginal revenue and marginal cost?a. Both will increase.b. Both will decrease.c. Marginal cost will increase; marginal revenue will be unchanged.d. Both will be unchanged.e. Marginal cost will increase; marginal revenue will decrease.

M15 Suppose that the demand for a product increases. What is the most likely effect on the firm’s price and quantity?a. Both will increase.b. Both will be unchanged.c. Both will decrease.d. Management can choose to hold one of them constant and increase the other.e. Price will decrease, quantity will increase.

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M16 Suppose that the firm’s marginal cost decreases. What is the effect on the firm’s optimal price and quantity?a. Both will increase.b. Both will be unchanged.c. Price will decrease, quantity will increase.d. There is not enough information to determine how price and quantity will change.e. Price will increase, quantity will decrease.

M17 Suppose that demand decreases. What is the most likely effect on the marginal revenue and marginal cost curves?a. Marginal revenue will decrease, marginal cost will not change.b. Marginal revenue and marginal cost will both decrease.c. Neither will change.d. Marginal revenue will not change, marginal cost will decrease.e. Marginal revenue will increase, marginal cost will decrease.

M18 According to the text, what was the nature of the conflict between Burger King and the local franchisees?a. Burger King wanted the franchisees to maximize profit, but the franchisees

wanted to maximize revenue.b. Burger King wanted to maximize revenue for itself, but wanted the franchisees to

maximize profit.c. Burger King wanted the franchisees to maximize revenue, but the franchisees

wanted to maximize profit.d. Burger King wanted the franchisees to minimize cost of sales, but the franchisees

wanted to maximize revenue.e. Burger King wanted the franchisees to maximize output, but the franchisees

wanted to maximize revenue.

M19 The fact that Amazon profits by selling a predictable number of e-books for each Kindle it sells implies thata. It earns additional MR for each Kindle sold.b. It should raise the price of Kindle so as to avoid cannibalizing e-book sales.c. It can raise the price of the Kindle and still increase the quantity of Kindles it

sells. d. It should discount the price of the Kindle.e. Answers a and d are both correct.

M20 If a firm’s profit is given by: = -Q3 + 18Q2 - 60Q - 100, then its optimal output isa. Q = 2b. Q = 10c. Q = 14d. A maximum does not exist. Profit is unbounded.e. None of these answers is correct.

Short Problems and Questions


S1 Suppose that the firm’s marginal cost is $500 per unit and that demand is as given in the following situations. In each case, find the profit-maximizing level of output.

a. P = 1,500 - 25Q.b. P = 2,000 - 50Q.c. Q = 3,000 - 5P.

S2 Suppose that a firm faces the demand curve, P = 30 - 1.5Q, where P denotes price in dollars and Q denotes total unit sales. The cost equation is C = 28 + 12Q.a. Determine the firm’s profit-maximizing output and price.b. Suppose that there is a change in the labor contract so that the cost equation

becomes C = 15 + 16Q + .25Q2. Determine (and explain) the resulting effect on the firm’s output and price.

c. Suppose that the firm sells in a competitive market and faces the fixed price, P = $21. For the cost function in part b, find the firm’s new optimal quantity.

S3 Suppose that a farmer produces grain that sells in a highly competitive market where the price is $4 per unit. The cost equation is given by C = 360 + .01Q2.a. Determine the optimal quantity for the farmer to produce.b. Now suppose that the market price falls to $3.60 per unit. What is the new

optimum? Verify the result that a drop in price leads to a reduction in output.

S4 Compute average total cost (C/Q) and marginal cost for the following cost functions. a. C = 80Q.b. C = 20 + 40Q.c. C = 100 + 30Q + 5Q2.

S5 What is the second derivative and why is it important for optimization problems?

S6 In each case below, find the profit-maximizing level of output. Verify that each is a maximum by checking the second derivative.a. = -50 + 100Q - 10Q2.b. = -20 + 150Q - 2Q3.c. = -40 - 180Q + 39Q2 - 2Q3.

S7 Compute the derivatives of the following functions:a. y = 4x1.5. b. y = 1/x.

S8 Compute the appropriate partial derivatives for the following functions.a. Q = 12K.5L.5.b. Q = PaYb.


Longer Problems and Discussion Questions

L1 Define marginal analysis and explain how management uses it in decision making.

L2 A small liberal arts college is trying to predict enrollment for the next academic year. The vice president for business states that enrollment has tended to follow a pattern described by E = 18,000 – .5P, where E denotes total enrollment and P is yearly tuition.a. If the school sets tuition at $20,000, how many students can it expect to enroll?b. If the school seeks to enroll 6,000 students, what tuition should it charge?c. If the school wants to maximize total tuition revenue, what tuition should it

charge?d. As the vice president for business, what tuition would you recommend? Why?

L3 Land Grant College has a successful football team, and sells tickets to students, alumni, and the public. Experience has shown that attendance has followed the demand relationship: Q = 80,000 - 5,000P where Q is attendance per game and P is the ticket price in dollars. Current capacity of the football stadium is 50,000 people.a. If Land Grant charges $7 per ticket, predict attendance.b. The athletic director wants to maximize revenue in order to pay for other athletic

programs at Land Grant. What is the appropriate ticket charge?c. Because the stadium is to be substantially repaired, capacity next year will be

limited to 35,000 seats. Given this limitation, what ticket price should Land Grant set?

L4 A firm faces the inverse demand curve, P = 20 - .5Q and cost function, C = 25 + 2Q + Q2.

Prepare a table listing quantities from 0 to 12 and corresponding prices, revenues, costs, and profits. In the final two columns, list marginal revenues and marginal costs. Verify that the MR = MC rule results in maximum profit.

L5 State the basic rule for maximizing profit in production. Carefully explain why this is an appropriate rule to follow.

L6 John Deere sells tractors and other agricultural equipment in competition U.S. and international manufacturers. Over a one-year period, the U.S. dollar as depreciated about 25% vis-à-vis other foreign currencies. How would this affect Deere’s demand curve? In turn, what would be the ultimate effect on Deere’s output and average sales price?

L7 The Dutch Peddler is a small bicycle repair shop, specializing in minor repairs and tune-ups. The direct, out-of-pocket cost of a spring tune-up is $18, and the price is currently set at $30. Dutch Peddler has noticed that its number of tune-ups has increased in each of the past four years, while its tune-up price has remained unchanged. The owner of Dutch Peddler is an old college friend of yours, and asks you for some free advice on how to improve earnings at the shop. In particular, she is considering a price increase. What would you advise?

L8 Keith Johnson, manager of the Campus BookStore, has to make a decision. Recently, a sales rep from a clothing manufacturer offered to produce sweatshirts with the school’s


logo. The quoted cost is $10 per sweat (including shipping) plus $1,600 to prepare the patterns for producing the logo. Keith has heard from other bookstore managers at a recent convention that school-labeled products sell very well in college bookstores, so he is seriously considering adding this line of clothing to boost earnings.a. Demand for sweats is: P = 40 - .02Q. Determine the profit-maximizing price and

quantity for Campus BookStore.b. Due to limited shelf space, Johnson must limit the number of sweats that he

stocks to 80% of the profit-maximizing quantity found in part a. What is the “marginal cost” associated with this constraint? Explain.

c. Now suppose demand falls to: P = 20 - .02Q. (Costs remain the same as before.) Is it still possible to earn a profit on sweats? Explain.

L9 Suppose that a firm employs a sales force that is compensated on a commission system. The firm pays a 10% commission on the first $300,000 of gross sales booked by a seller and a 15% commission on sales over $300,000.a. Is there a conflict in objectives between the sales force and management of the

firm? Explain.b. The use of an increasing commission rate provides a strong incentive for agents to

increase gross sales. When would a company not want to encourage increased sales by its agents? Explain.

L10 New Car Motors sells cars and trucks in a rapidly growing suburb. Based on past trends, the sales manager estimates that the inverse demand equation for cars is: PC = 40 -.05QC where QC is the number of cars sold per year and PC is the price per car (in thousands of dollars). The price equation for trucks is: PT = 30 - .1QT. In turn, NCM pays $10 thousand for each car from the manufacturer and $8 thousand for each truck.a. Determine the profit-maximizing outputs of cars and trucks for NCM.b. Answer the questions in part a, supposing that capacity is limited to a total of 260

vehicles per year.

L11 Suppose that Q (output) depends upon the use of inputs K (capital) and L (labor) according to the production function:

Q = K.5L.5

The firm has allocated $1,000 per week to buy capital and labor, where K costs $4 per unit and L costs $5 per unit. The firm seeks to maximize output subject to its $1,000 budget. How much of each input should the firm buy? What is its maximum level of output? Use the method of Lagrangian multipliers to address these questions.

SOLUTIONS

Multiple Choice

M1 cM2 a This is a simplifying assumption.M3 c Total cost is C = 1,000 + (15)(200) = $4,000 and MC = dC/dQ = $15 per unit.M4 bM5 e


M6 cM7 eM8 cM9 d In perfect competition, the price equation is simply P = $10, implying MR = $10.M10 e The conditions MR = MC and M = 0 are equivalent.M11 b Setting MR = MC implies 200 - .4Q = 40, or Q = 400.M12 b To maximize revenue, set MR = 0. Thus, 200 - 40Q = 0, implying Q = 5 and

P = $100.M13 e MC = dC/dQ = 10 + 4Q. Thus, at Q = 20, MC = $90.M14 d A change in fixed cost has no effect on marginal cost or marginal revenue. M15 aM16 cM17 aM18 cM19 eM20 b M = -3Q2 + 36Q – 60 = -3(Q - 2)(Q – 10). The maximum is at Q = 10.

Short Problems and Questions

S1 a. To maximize profit, set MR = MC. We know that revenue is: R = PQ soR = (1,500 - 25Q)(Q) = 1,500Q - 25Q2, implying: MR = 1,500 - 50Q.Setting MR = MC = 500 implies 1,500 - 50Q = 500 or 1,000 = 50Q. Thus, the optimal quantity is Q* = 20.

b. P = 2,000 - 50Q implies MR = 2,000 - 100Q. Again, setting MR = MC, we have: 2,000 - 100Q = 500 or 1,500 = 100Q. Thus, Q* = 15.

c. Rearranging Q = 3,000 - 5P, we have: 5P = 3,000 – Q, or P = 600 -.2Q.Setting MR = MC, we have 600 - .4Q = 500. Thus, Q* = 250.

S2 a. = R - C = (30 - 1.5Q)(Q) – [28 + 12Q]. After simplifying, this becomes: = 18Q - 1.5Q2 - 28. Therefore, M = d/dQ = 18 - 3Q. Setting M = 0 implies: 18 - 3Q = 0, or Q* = 6. In turn, P* = 30 - (1.5)(6) = $21.Equivalently, if we set MR = MC, we have MR = 30 – 3Q = 12, so Q* = 6.

b. From the new cost equation, we have: MC = dC/dQ = 16 + .5Q. Again, settingMR = MC implies: 30 - 3Q = 16 + .5Q. Thus, 14 = 3.5Q or Q* = 4. Because MC has increased (relative to part a), the firm’s optimal quantity declines. In turn, P* = 30 - (1.5)(4) = $24.

c. With the price fixed at $21, it follows that R = 21Q and MR = dR/dQ = 21. Setting MR = MC implies 21 = 16 + .5Q, or Q* = 10. With MR constant (rather than downward sloping), it pays for the firm to increase output.

S3 a. The farmer’s profit is: = R - C = 4Q - .01Q2 – 360. Thus, M = d/dQ = 4 - .02Q. Setting M = 0 implies: 4 - .02Q = 0, or Q* = 200.

b. At P = $3.60, profit is: = 3.6Q - .01Q2 – 360. Thus, M = d/dQ = 3.6 - .02Q. Setting M = 0 implies: 3.6 - .02Q = 0, or Q* = 180. This verifies that a drop in price leads to a reduction in output.

S4 a. With C = 80Q, AC = 80Q/Q = 80 and MC = dC/dQ = 80.


b. With C = 20 + 40Q, AC = 20/Q + 40 and MC = 40.c. With C = 100 + 30Q + 5Q2, AC = 100/Q + 30 + 5Q and MC = 30 + 10Q.

S5 The second derivative is found by twice differentiating a given function, that is, by taking the first derivative then taking the first derivative of that. For example, marginal profit (the rate of change of profit) is found by taking the derivative of the profit function. By taking the derivative of marginal profit, we find the second derivative of the profit function. For instance, if = -50 + 20Q - 2Q2, then we have: M = d/dQ = 20 - 4Q and d2/dQ2 = d(M)/dQ = -4.

The sign of the second derivative determines whether a function is at a maximum (if the sign is negative) or a minimum (if the sign is positive). In this example, setting M = 0 implies Q* = 5. Since the second derivative is negative (-4) at this output, we know that this is a maximum.

S6. a. M = d/dQ = 100 - 20Q = 0 implies Q* = 5.In turn, d2/dQ2 = -20, so we have found a maximum.

b. M = d/dQ = 150 - 6Q2 = 0 implies Q* = 5. In turn, d/dQ2 = -12Q, which is negative at Q* = 5, so we have found a maximum.

c. M = d/dQ = -180 + 78Q - 6Q2 = 0. Dividing each side by 6 and rearranging, we get : Q2 – 13Q + 30 = 0, which factors into (Q - 3)(Q – 10) = 0. Thus, the solution is Q* = 3 or Q* = 10. In turn, d2 /dQ2 = 78 - 12Q. This is positive at Q* = 3 and negative at Q* = 10. The first output is a minimum and the latter is a maximum.

S7 a. dy/dx = 6x.5 = 6√ x .b. y = 1/x = x-1. Therefore, dy/dx = -x-2 = -1/x2.

S8 a. For Q = 12K.5L.5, Q/K = 6K-.5L.5 and Q/L =6K.5L-.5. b. For Q = PaYb, Q/P = aPa-1Yb and Q/Y= bPaYb-1.

Longer Problems and Discussion Questions

L1 Marginal analysis is the process of considering a small change in a decision and determining whether the given change will improve the ultimate objective. Management uses marginal analysis to identify an optimal decision – that is, a decision that maximizes a given objective (usually profit).For instance, a firm might try charging a slightly different price and monitoring the effect on unit sales, revenues, costs, and profits. Alternatively, it might change its mix of inputs (capital, labor, and so on) in seeking to minimize the cost of producing a given level of output.

L2 a. From the demand equation, E = 18,000 - (.5)(20,000) = 8,000 students.b. Rearranging the demand equation (.5P = 18,000 – E) give the price equation

P = 36,000 – 2E. For E = 6,000, we find: P = $24,000. c. Tuition revenue is maximized when MR = 0. Revenue = PE = (36,000 - 2E)E.

Thus, R = 36,000E - 2E2, implying MR = dR/dE = 36,000 - 4E = 0. Consequently, E = 9,000 students and tuition, P =36,000 – (2)(9,000) = $18,000 per year.


d. The recommended tuition depends on several factors. One is the cost of operating the school, especially the marginal cost of an additional student. Another is the objective of the school. Colleges are interested in attracting high-quality students and faculty and in enhancing their academic reputations. Typically, liberal arts colleges do not try to maximize profit, but they do pay attention to enrollment and revenue. By offering financial aid, they also offer different tuition charges to different students.

L3 a. Q = 80,000 - (5,000)($7) = 80,000 - 35,000 = 45,000 people.b. Start by rearranging the demand equation: Q = 80,000 - 5,000P becomes

5,000P = 80,000 – Q, or P = 16 – Q/5,000. To maximize revenue, the athletic director sets MR = 0. From the price equation, it follows that MR = 16 – 2Q/5,000 or MR = 16 – Q/2,500. Setting MR = 0 implies 16 – Q/2,500 = 0, or Q = 40,000. In turn, the appropriate ticket price is P = 16 – 40,000/5,000 = $8. Maximum revenue is R = ($8)(40,000) = $320,000.

c. To fill the reduced stadium capacity (35,000 seats), the appropriate price is:P = 16 - 35,000/5,000 = 16 - 7 = $9. This generates less revenue ($315,000) than in part b

L4 Revenues are computed using: R = 20Q - .5Q2. In turn, MR = dR/dQ = 20 – Q. From the cost equation, C = 25 + 2Q + Q2, we have: MC = dC/dQ = 2 + 2Q.

After completing the table, we find the profit-maximizing output to be Q* = 6. At this output, profit is 29 and MR = MC = 14. The same result can be found algebraically by setting MR = MC implying: 20 - Q = 2 + 2Q. Therefore, Q = 6.

Q P R C MR MC1 19.5 19.5 28 -8.5 19 42 19 38 33 5 18 63 18.5 55.5 40 15.5 17 84 18 72 49 23 16 105 17.5 87.5 60 27.5 15 126 17 102 73 29 14 147 16.5 115.5 88 27.5 13 168 16 128 105 23 12 189 15.5 139.5 124 15.5 11 2010 15 150 145 5 10 2211 14.5 159.5 168 -8.5 9 2412 14 168 193 -25 8 26

L5 The firm’s profit maximizing level of output occurs when the additional revenue from selling an extra unit just equals the extra cost of producing it, that is, when MR = MC.This rule is appropriate because it focuses attention on the effects of a change in current activities, in particular, the effects of expanding or contracting production. If marginal revenue exceeds marginal cost, this means the additional revenue from producing and selling a unit exceeds the cost of doing so. This adds net profit to the firm (that is, it increases the firm’s total profit or it reduces its loss). If marginal revenue is less than marginal cost, then increasing the activity implies a net reduction in profit.


In a multi-products firm, if the products are assumed to be independent of one another, the firm should follow this rule for each of the products. In this case, maximum total profit is achieved by maximizing profit product by product.

L6 This dollar depreciation makes imports of foreign goods to the U.S. more expensive in dollar terms, while making Deere’s exports less expensive to the rest of the world in terms of their foreign currencies. Both effects increase demand for Deere’s goods causing its demand curve to shift outward. In responsive to the positive demand shift, Deere’s new optimal decision (at the intersection of a shifted MR and an unchanged MC) calls for increased Q* and P*. Deere should partially increase its prices and also increase its planned sales.

L7 This problem is a bit subtle, because demand is not known with precision. However, we

know some things that can help us make a recommendation about a price change.(1) The goal of the manager is to increase profits, which gives an adviser a clearly stated goal for the decision-maker.(2) Though price has been constant, quantity has steadily increased over the years. Thus, demand is increasing (the demand curve is shifting to the right). If the $30 price was once appropriate, a higher price in response to increased demand is probably appropriate now. (An increase in MC over time would also warrant an increase in price.)(3) Overall, it seems that a small price increase (say, to $35 or $40) will likely increase total revenue and profit (even though it will slow the growth in volume). If the manager pursues this course, she should carefully track unit sales to measure the willingness of customers to pay the increased price.

L8 a. Johnson finds the optimal quantity and price of sweats by setting MR = MC.From the price equation, P = 40 - .02Q, it follows that MR = 40 - .04Q. The cost of an additional sweatshirt is $10. Thus, 40 - .04Q = 10 or Q* = 750 sweats.The optimal price is: P = 40 - (.02)(750) = $25.

b. Suppose Johnson is limited to ordering (.8)(750) = 600 sweats. At this quantity, marginal revenue is: MR = 40 - (.04)(600) = $16, while MC = $10. Thus, the store would earn a marginal profit of $6 by ordering and selling an extra sweatshirt. This lost profit (for unsold shirts 601 up to 750) is the “cost” of the limitation to 600 sweats.

c. If demand falls to P = 20 - .02Q then, MR becomes 20 - .04Q. Setting MR = MC, we find 20 - .04Q = 10, or Q = 250 shirts. In turn, P* = $15. The store’s profit on the sweats falls to: = (15 - 10)(250) - 1,600 = -$350. With the fall in demand, ordering the sweats becomes unprofitable.

L9 a. No, there is not necessarily a conflict. Clearly, because of the commission structure, the sales force has a strong incentive to increase sales Top management also wants to increase sales, as long as the additional revenue of the sales exceeds the additional cost of production and sales commission.


b. Beyond some level, a company might want to discourage additional sales in two

cases: 1) When the increased (net) revenue is insufficient to cover the increased cost of commissions. This is possible and suggests that management should examine the commission scheme and perhaps lower commission rates or increase prices. 2) When the firm faces a constraint in production, making an increase in production very costly or impossible. In this case, any increase in sales is either infeasible (leaving disappointed customers) or not worth the cost.

L10 a. The profit from cars is: C = R - C = 40QC - .05Q2C - 10QC.

Setting M = 0 implies: 30 - .1QC = 0, or QC = 300 cars.The profit from trucks is: T = R - C = 30QT - .1Q2

T - 8QT.Setting M = 0 implies: 22 - .2QT = 0, or QT = 110 trucks.

b. NCM is limited to 260 vehicles (instead of the desired 410 vehicles in part a). The relevant constraint is: QC + QT = 260. Before, NCM attained maximum profit by setting MC = 0 and MT = 0. Now, the best it can do is to set MC = MT. Equating marginal profits implies: 30 - .1QC = 22 - .2QT. Solving this equation and QC + QT = 260 simultaneously, we find: QC = 200 cars and QT = 60 trucks.

L11 The objective is to maximize Q, subject to the constraint of spending no more than $1,000 for inputs. The Lagrangian is: £ = K.5L.5 + z(1,000 - 4K - 5L).Compute the partial derivatives, and set them equal to zero:

£/K = .5K-.5L.5 - 4z = 0£/dL = .5K.5L-.5 - 5z = 0£/z = 100 - 4K - 5L = 0

1) We multiply the first partial by 5 and the second partial by 4 so as to cancel the z term: 2.5K-.5L.5 - 20z = 0 and 2K.5L-.5 - 20z = 0. Therefore, 2.5K-.5L.5 = 2K.5L-5. 2) Next, we multiply both sides by K.5L.5. Therefore, 2.5L = 2K so that K = 1.25L. 3) Finally, substitute this into the constraint: 4K + 5L = 1,000, implying 4(1.25L) + 5L = 1,000 or 10L = 1,000. Thus, L* = 100 and K* = (1.25)(100) = 125.Total output is (125).5(100).5 = 111.8 units.

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