Appunti Managerial Economics - Modulo 1

Optimization in business decisions

Making optimal decisions about the levels of various business activities is an essential skill for all managers. It requires managers to analyze benefits and costs to make the best possible decision under a given set of circumstances. A manager's decision is considered optimal if it leads to the best outcome. This involves applying the fundamental principles of optimization theory.

Components of an optimization problem

An optimization problem involves the specification of three things:

• Objective function (usually profit for managers and satisfaction for consumers) to be maximized or minimized;
• Activities or choice variables that determine the value of the objective function (e.g., the value of profit depends on the number of units of output produced and sold);
• Constraints that may restrict the value of the choice variables.

The constraint can or cannot depend on the origin of the constraint. We have a plateau of information of variables, and if we have some constraints, we have fewer degrees of freedom to move within our content. As managers, we can control our profit by changing the level of activity. We have the possibility of controlling the things we want to maximize, but we need to be able to manage constraints, costs, and other aspects.

Problems with optimization

• Maximization problem: An optimization problem that involves maximizing the objective function.
• Minimization problem: An optimization problem that involves minimizing the objective function.

Optimization problems are further categorized as unconstrained or constrained, depending on whether the decision-maker can choose the values of the choice variables in the objective function from an unconstrained or constrained set of values.

• Unconstrained optimization: The decision-maker can choose the level of activity from an unrestricted set of values. This is the easiest scenario, allowing movement without restriction.
• Constrained optimization: The decision-maker chooses values for the choice variables from a restricted set of values.

For instance, in a public context, we have some restrictions and are not free to choose, thus having less freedom. For example, if the total cost of the activity must equal a specific cost, we don't have a lot of degrees of freedom. If a price is decided by the government, we must consider it and constrain our activity.

Types of choice variables

• → Discrete choice variables: Can only take specific integer values.
• → Continuous choice variables: Can take any value between two endpoints.

We have one choice to maximize our profit or minimize our costs. The constrained maximization and minimization problems follow a simple rule for the solution: we must conduct our choice to marginal analysis.

Marginal analysis

Marginal analysis is an analytical technique for solving optimization problems by changing values of choice variables by small amounts to see if the objective function can be further improved. We have our constraint and our variables, and what we can do to increase our profit? We can start moving a little bit our variables until we understand that, at a certain point, we reach the peak.

The idea behind marginal analysis is this: When a manager contemplates whether a particular business activity needs adjusting, either more or less, to reach the best value, the manager needs to estimate how changing the activity will affect both the benefits the firm receives from engaging in the activity and the costs the firm incurs. If changing the activity level causes benefits to rise by more than costs rise, or, alternatively, costs to fall by more than benefits fall, then the net benefit the firm receives from the activity will rise. The manager should continue adjusting the activity level until no further net gains are possible, which means the activity has reached its optimal value or level.

Unconstrained maximization

Any activity that decision-makers might wish to undertake will generate both benefits and costs. Consequently, decision-makers will want to choose the level of activity to obtain the maximum possible net benefit from the activity, where the Net Benefit (NB) associated with a specific amount of activity (A) is the difference between total benefit (TB) and total cost (TC) for the activity: NB = TB - TC

The optimal level of the activity (A*) is the level that maximizes net benefit and represents the choice variable. If I want to make a decision in a specific setting, I have to consider the cost and benefit of taking this action. This is the main aim of the maximization problem. We want to find the level of activity that maximizes the profit, and this level is the optimal level. The amount of activity is our variable that managers continue to adjust to reach the optimal level of activity.

If we are in an unconstraint setting, the level of activity could be chosen, and we are free to move between zero and infinity. Within a setting with a constraint, we are not free to move (it is not the case of any level of activity can be chosen).

If deciding to maximize and reach the optimal level of activity, we have to consider one important thing: Keep this A* is the one that maximizes our total benefit curve. If we decide to go further and use more, for example at point F or G, our benefit increases, but what we really want to do is to maximize the net benefit. So if we move from B to F, which has a higher total benefit, this is associated with a net benefit equal to zero, because costs increase.

Net benefit at any particular level of activity is measured by the vertical distance between the total benefit and total cost curves. At 200 units of activity, for example, net benefit equals the length of line segment CC9, which happens to be $1,000 as shown in Panel B at point Panel B of Figure 3.1 shows the net benefit curve associated with the TB and TC curves in Panel A. As you can see from examining the net benefit curve in Panel B, the optimal level of activity, A*, is 350 units, where NB reaches its maximum value. At 350 units in Panel A, the vertical distance between TB and TC is maximized, and this maximum distance is $1,225.

Note

1. At the optimal level of activity in an unconstrained maximization problem, the total benefit is still rising (see point B → G).
2. The optimal level of activity in an unconstrained maximization problem does not result in minimization of total cost (this happens at zero units of activity).

Marginal benefit & Marginal cost

• Marginal benefit (MB): Change in total benefit (TB) caused by an incremental change in the level of the activity.
• Marginal cost (MC): Change in total cost (TC) caused by an incremental change in the level of the activity.

A little increase or a little decrease in the margin, so for example, if it moves from x to x1, can cause a change in the marginal benefit or cost, so we have to calculate the difference between the two values. Marginal variables measure rates of change in corresponding total variables. Marginal benefit (marginal cost) of a unit of activity can be measured by the slope of the line tangent to the total benefit (total cost) curve at that point of activity.

Panel A in Figure 3.2 illustrates the procedure for measuring slopes of total curves at various points or levels of activity. Marginal benefit (marginal cost) is the change in total benefit (total cost) per unit change in the level of activity. The marginal benefit (marginal cost) of a particular unit of activity can be measured by the slope of the line tangent to the total benefit (total cost) curve at that point of activity.

Using marginal analysis to find optimal activity levels

• → If marginal benefit > marginal cost: Activity should be increased to reach the highest net benefit. We assure that the total benefit we obtain is greater than the marginal costs, we have to go on with our activity to further produce in order to reach the optimal level of activity and the highest net benefit.
• → If marginal cost > marginal benefit: Activity should be decreased to reach the highest net benefit. When we have the marginal cost that is greater than the marginal benefit, it is no more convenient, so we have to stop there and decrease the level of activity otherwise we lose benefit.
• → Optimal level of activity: When no further increases in net benefit are possible. Occurs when MB = MC. Any point in time we re-evaluate the situation of the margin in order to adjust the activity and produce the net benefit.

If we are in c'' we have to adjust the level of activity because we can do better. If we are in d'' we have to adjust the level of activity and reduce it because we are producing too much.

What can I do if I have a plan that produces different things? If I have to move an input that is connected to more than an output (so it is not so easy and linear, because if I increase the net benefit in one sense, in the other could decrease). The simultaneous decision is really hard.

If, at a given level of activity, a small increase or decrease in activity causes net benefit to increase, then this level of the 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—the level that maximizes net benefit—is attained when no further increases in net benefit are possible for any changes in the activity, which occurs at the activity level for which marginal benefit equals marginal cost: MB = MC.

Unconstrained maximization with discrete choice variables

• Increase activity if MB > MC;
• Decrease activity if MB < MC;
• → Optimal level of activity: Last level for which MB exceeds MC (MB = MC or the nearest level to this equation).

Irrelevance of sunk, fixed, and average costs

These costs are irrelevant in decision-making because you can't recover them, and we cannot adjust our activity with respect to them.

• → Sunk costs: Previously paid and cannot be recovered;
• → Fixed costs: Constant and must be paid no matter the level of activity;
• → Average (or unit) costs: Computed by dividing total cost by the number of units of activity.