Chapter 3: MARGINAL ANALYSIS FOR OPTIMAL DECISIONS. Essential Concepts

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1 Solutions Manual Managerial Economics Foundations of Business Analysis and Strategy 12th Edition Thomas Maurice Completed downloadable package SOLUTIONS MANUAL for Managerial Economics 12th Edition by Christopher Thomas, S. Charles Maurice Instant download: Chapter 3: MARGINAL ANALYSIS FOR OPTIMAL DECISIONS Essential Concepts 1. Formulating an optimization problem involves specifying three things: (1) the objective function to be either maximized or minimized, (2) the activities or choice variables that determine the value of the objective function, and (3) any constraints that may restrict the range of values that the choice variables may take. 2. Marginal analysis involves changing the value of a choice variable by a small amount to see if the objective function can be further increased (in the case of maximization problems) or further decreased (in the case of minimization problems). 3. Net benefit from an activity (NB) is the difference between total benefit (TB) and total cost (TC ) for the activity: NB = TB TC. The net benefit function is the objective function to be maximized in unconstrained maximization problems. The optimal level of the activity (A*) is the level of activity that maximizes net benefit. 4. The choice variables determine the value of the objective function. Choice variables can be either continuous or discrete. A choice variable is continuous if the decision maker can choose from an uninterrupted span (or continuum) of values. A discrete choice variable is one for which the decision maker chooses from a span of values that is interrupted by gaps. 5. Marginal benefit (MB) is the change in total benefit caused by an incremental change in the level of activity. Marginal cost (MC) is the change in total cost caused by an incremental change in the level of activity. An incremental change in activity is a small positive or negative change in activity, usually a one-unit increase or decrease in activity. Marginal benefit and marginal cost can be expressed mathematically as change in total benefit MB = = DTB change in activity DA MC = change in total cost change in activity = DTC DA where the symbol D means the change in and A denotes the level of activity. 6. Because marginal variables measure rates of change in corresponding total variables, marginal benefit and marginal cost are also slopes of total benefit and total cost curves, respectively. Marginal benefit (cost) of a particular unit of activity is measured by the slope of the line tangent to the total benefit (total cost) curve at that point of activity.

2 7. If, at a given level of activity, a small increase or decrease in activity causes net benefit to increase, then this level of activity is not optimal. The activity must then be increased (if marginal benefit exceeds marginal cost) or decreased (if marginal cost exceeds marginal benefit) to reach the highest net benefit. The optimal level of the activity is attained when no further increases in net benefit are possible for any changes in the activity. This point occurs at the activity level for which marginal benefit equals marginal cost: MB = MC. 8. When a manager faces an unconstrained maximization problem and must choose among discrete levels of an activity, the manager should increase the activity if. The optimal level of activity is the last level for which MB exceeds MC. MB < MC MB > MC and decrease the activity if 9. Sunk costs are costs that have previously been paid and cannot be recovered. Fixed costs are costs that are constant and must be paid no matter what level of activity is chosen. Average (or unit) cost is the cost per unit of activity, which is computed by dividing total cost by the number of units of activity. Decision makers wishing to maximize net benefit should ignore any sunk costs, any fixed costs, and the average costs associated with the activity because none of these costs affect the marginal cost of the activity, and so are irrelevant for making optimal decisions. 10. The ratio of marginal benefit divided by the price of an activity (MB/P) tells the decision maker the additional benefit of that activity per additional dollar spent on that activity, sometimes referred to informally as bang per buck. In constrained optimization problems, the ratios of marginal benefits to prices of the various activities are used by managers to determine how to allocate a fixed number of dollars among activities. 11. To maximize or minimize an objective function subject to a constraint, the ratios of the marginal benefit to price must be equal for all activities, MB A P A = MB B P B = = MB Z P Z and the values of the choice variables must meet the constraint. Answers to Applied Problems 1. a. One way of reducing traffic deaths is to reduce speed. While it may be possible to eliminate all traffic deaths by allowing motorists to drive no faster than 15 MPH in cars equipped with driver and passenger air bags, most American drivers would not view a 15 MPH speed limit as optimal. Most drivers seem willing to accept some additional probability of death in return for faster speeds. The 70 MPH speed limit would be optimal if the marginal benefit of reducing speed limits equals the marginal cost of reducing speed limits. Just because reducing speed limits to 65 MPH would save even more lives does not, by itself, mean that further reduction in speed limits should be undertaken. The marginal benefit of speed reduction must be compared to the marginal cost. b. If it costs nothing to eliminate pollution (i.e. MC = 0), then the optimal level of pollution would indeed be zero. When the marginal cost of pollution abatement is greater than zero, as it is for virtually every type of pollution, the optimal level of pollution occurs at that level of pollution for which the marginal benefit to society of eliminating more pollution just equals the marginal cost of eliminating more pollution. In fact, it is possible to have too little pollution if pollution abatement activities have been undertaken such that the marginal cost of abatement exceeds the marginal benefit. c. To maximize net benefit, troops should be left in Iraq if the marginal benefit exceeds the marginal cost. Since marginal benefit and marginal cost are measures of additional (or extra) benefits and

3 costs, benefits and costs already incurred do not matter (i.e. do not affect MB and MC). Sunk costs or benefits do not enter the decision making process, only incremental benefits and costs matter. d. See answer to part c. e. Insurance premiums are fixed costs. The optimal level has nothing to do with how high or how low fixed costs go. 2. Appalachian Coal Mining should minimize net cost by choosing that level of pollution (P) where the marginal benefit of pollution reduction equals the marginal cost of pollution reduction: 1,000 10P = 40P P * = 20 units of pollution. 3. The second partner is basing his objection to the move on costs that are sunk. The money spent on office stationary, business cards, and a sign that cannot be moved to the new office are not marginal costs in the decision to move and should thus be ignored. In other words, the cost of the old cards, old stationary, and old sign are sunk costs to be ignored in making the decision to move the office, while any costs of purchasing new business cards, new stationary, or a new sign are part of the marginal cost of making the move. If, as the first partner seems to believe, MB exceeds MC for making the move, then net benefit rises even though new cards, stationary, and a sign must be purchased. 4. a. 2 b. $500 (= $25 20 radios not stolen due to hiring 1 guard) c a. The following graph illustrates such a situation. Clean-up activity is plotted along the horizontal axis and marginal benefits and costs along the vertical. For any amount of clean up greater than A*, MC exceed MB, and that amount of clean-up activity is too much. b. The following graph illustrates such a situation. Notice that the shape of MB reflects Breyer s assertion that most of the benefit of clean-up comes at relatively low levels of clean up activity. The shape of MC puts most of the clean-up cost at relatively high levels of clean-up.

4 6. Never give up : You should give up an activity when MC > MB for extra units of the activity. Anything worth doing is worth doing well : How well you choose to do something should be determined by weighing costs and benefits at the margin. Waste not, want not : If saving a unit of a resource costs more than the value of the resource (usually measured by the price of a unit of the resource), it is NOT optimal to save the resource. For example, if a manufacturer must spend $100 to prevent $15 worth of a raw material from being wasted, then it is optimal to waste the raw material. 7. a. With a payroll of $160,000, the manager should hire five people with high school diplomas and two people with bachelor's degrees. This choice maximizes the number of customers served because the last dollar spent on each type of employee yields the same addition to the number of customers served; MB HS / P HS (60 / $20,000) = MB B / P B (90 / $30,000) = b. No, she is not making the correct decision. If the manager hires three employees of each type, the marginal contributions of the last person hired are the same for both types of employees (80), but the marginal contribution per dollar is higher for a high school diploma (80/$20,000) than for a bachelor's degree (80/$30,000). By hiring more employees with a high school diploma and fewer with a bachelor's degree, the manager can spend the same amount on payroll and increase the number of customers served. c. With a budget of $240,000, she should hire six people with high school diplomas and four people with bachelor's degrees. 8. a. Q* = 6,000 wine decanters MR 6,000 = $70 b. TR 6,000 = 70 6,000 = $420,000 TC 6,000 = 10, (6,000) (6,000) 2 = $340,000 Profit = $420,000 $340,000 = $80,000 c. The 6,001 st decanter adds $70 to total revenue and slightly more than $70 to total cost. Therefore the 6,001 st unit would (slightly) reduce profit.

5 9. a. 5 b. 4 c. 3 Answers to Mathematical Exercises 1. a. MB = 170 2x; MC = x b. NB = 170x x x 2x 2 dnb dx x* = = 170 2x x = 180 6x = 0; = 30 c. NB = 170(30) (30) (30) 2(30) 2 = 2, a. MB = 100 4x; MC = x 2 12x + 52 b. NB = 100x 2x 2 (1/3)x 3 + 6x 2 52x 80 dnb dx = 100 4x x x 52 = x x 2 = 0 This factors to (x + 4)( x + 12) = 0 x = 4, x = 12 x* = 12 maximizes NB c. NB = 100(12) 2(12) 2 (1/3)(12) 3 + 6(12) 2 52(12) 80 = Z = 3x + xy + y + λ(70 4x 2y) x = 3 + y λ4 = 0 y = x + 1 λ2 = 0 l = 70 4x 2y = 0 Dividing the first equation by the second, 3 + y x + 1 = 4 ; 3 + y = 2x + 2; y = 2x 1 2 Substituting this solution into the third equation: 70 4x 2(2x 1) = 0 and solving for x = 9; y = 2(9) 1 = 17 B max = 3(9) + 9(17) + 17 = 197

6 4. Z = 6x + 3y + λ(288 xy) x y = 6 λy = 0 = 3 λx = 0 = 288 xy = 0 l y 2 = 2; y = 2x; substituting into the third equation, 2x(x) = 2x2 = 288; x = Min c = 6(12) + 3(24) = $ = 12; y = 2(12) = a. Marginal benefit is the derivative of TB with respect to A: MB = dtb = A. At point C, da MB = (200) = $6.40. The rest of the points can be similarly verified. b. Marginal cost is the derivative of TC with respect to A: MC = dtc = A. At point C, MC da = (200) = $3.40. The rest of the points can be similarly verified. c. NB = TB TC = 8A A 0.004A A 2 = 7A 0.010A 2. The derivative of NB equals MB MC. d. Yes. Completed download links: managerial economics 12th edition thomas test bank pdf managerial economics foundations of business analysis and strategy answers managerial economics applied problems answers managerial economics foundations of business analysis and strategy 11th edition pdf managerial economics foundations of business analysis and strategy pdf managerial economics thomas maurice solutions managerial economics by thomas and maurice free download managerial economics thomas maurice pdf