Please assist with answering the attached questions.

Chapter 03 - Marginal Analysis for Optimal Decisions Chapter 3: MARGINAL ANALYSIS FOR OPTIMAL DECISIONS Net Benefit = Total Benefit - Total Cost 1. Formulating an optimization problem involves specifying: a. the objective function to be either maximized or minimized, b. the activities or choice variables that determine the value of the objective function, and c. 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 vertical distance between total benefit and total cost curves) The net benefit function is the objective function to be maximized in unconstrained maximization problems. o Unconstrained means that decision makers get to choose ANY level of activity they wish. The optimal level of the activity (A*) is the level of activity that maximizes net benefit. o The point at which Total Benefit exceeds Total Costs the most: NB* = $1,225. This corresponds with the optimal level of activity A* = 350 which maximizes Net Benefit. (SEE FIGURES BELOW) Note: A* (our choice variable) does not maximize TB nor TC. (TB still growing after B and TC can be minimized at A=0) 4. The choice variables determine the value of the objective function. Choice variables can be either o Continuous: Uninterrupted span (or continuum) of values o Discrete. A span of values that is interrupted by gaps. Chapter 03 - Marginal Analysis for Optimal Decisions Chapter 03 - Marginal Analysis for Optimal Decisions Marginal Analysis 5. Marginal benefit (MB): The change in total benefit caused by an incremental change in the level of activity. 6. Marginal cost (MC): 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. 7. Marginal benefit and marginal cost can be expressed mathematically as MB = change in total benefit DTB = change in activity DA MC = change in total cost DTC = change in activity DA The symbol D means the change in and A denotes the level of activity. o Marginal benefit and marginal cost are also slopes of total benefit and total cost curves, respectively. Marginal benefit of a particular unit of activity is measured by the slope of the line tangent to the total benefit curve at that point of activity. Marginal cost of a particular unit of activity is measured by the slope of the line tangent to the total cost curve at that point of activity. Slopes have to be equal! 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. o The activity must then be increased (if marginal benefit exceeds marginal cost) or o Activity must be decreased (if marginal cost exceeds marginal benefit) to reach the highest net benefit. o 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. ACTIVITY TOTAL BENEFIT A Slope =Rise/Run $ MB = MB of: B +100 520 A* = 350 520/100 $5.2 350th unit of activity C +100 640 200 640/100 $6.4 200th D +100 820 320 320/100 $3.2 600th ACTIVITY TOTAL COST A Slope =Rise/Run $ MC = MC of: B +100 520 A* = 350 520/100 $5.2 350th unit of activity C +100 340 200 640/100 $3.4 200th D +100 820 600 820/100 $8.2 600th $ NET BENEFIT $ NET COST Chapter 03 - Marginal Analysis for Optimal Decisions Chapter 03 - Marginal Analysis for Optimal Decisions Optimal Activity Levels with Marginal Analysis 8. 9. When a manager faces an unconstrained maximization problem and must choose among discrete levels of an activity, the manager should: o Increase the activity if MB > MC and o Decrease the activity if MB