Appunti Managerial Economics - Modulo 1

OPTIMIZATION

Making optimal decisions about the levels of various business activities is an essential skill for all

managers, one that requires managers to analyze benefits and costs to make the best possible decision

under a given set of circumstances. A manager’s decision is optimal if it leads to the best outcome

under a given set of circumstances. Finding the best solution involves applying the fundamental

principles of optimization theory.

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 (For example,

- the value of profit depends on the number of units of output produced and sold. The

production of units of the good is the activity that determines the value of the objective

function, which in this case is profit);

Any 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 less degrees of freedom to move within our

content. As manager we can control our profit changing the level of activity. We have the possibility

of controlling 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.

In addition to being categorized as either maximization or minimization problems, optimization

problems are also categorized according to 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: an optimization problem in which the decision maker can choose

the level of activity from an unrestricted set of values. Easiest, when we have the possibility to

choose any level of activity, we are free to move in our context without restriction.

• Constrained optimization: an optimization problem in which the decision maker chooses values

for the choice variables from a restricted set of values.

For instance, when we are in a public context, we have some restriction and we are not free to choose,

we have less freedom.

For example, if the total cost of the activity must be equal to the specific cost, we don't have a lot of

degrees of freedom. If we have a price the price that is decided from the government, we have to

consider it and constraint our activity.

Activities or choice variables determine the value of the objective function. We have two kinds of

choices:

• →

Discrete choice variables Can only take specific integer values

• →

Continuous choice variables Can take any value between two end points

We have only one choice to maximize our profit or minimize our costs. the constrained maximization

and the constrained minimization problems have one simple rule for the solution: we have to conduct

our choice to marginal analysis. lOMoAR cPSD| 10171683

Marginal analysis is an analytical technique for solving optimization problems that involves

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 since that 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 from engaging in the activity. 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

Optimal level of the activity (A*) is the level that maximizes net benefit and represents the choice

variable. If I want to take a decision in a specific setting, I have to consider:

The cost of taking the action;

- The benefit of taking this action.

-

This is the main aim of the maximization problem. We want to find the level of activity that maximize

the profit, and this level is the optimal level. The amount of activity is our variable that manager

continues to adjust in order to reach the optimal level of activity. If we are in an unconstraint setting,

the level of activity could be chosen, we are free to move between zero and infinity. Within a setting

with a constraint, we are not free to move (is not the case of any level of activity can be chosen).

If decide to maximize and reach the optimal

level of activity, we have to consider one

important thing:

Keep this A* is the one that maximize our total

benefit curve. If we decide to go further and use

more, for example at point F or G, out 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 to 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

c”.

$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 (5 NB*). lOMoAR cPSD| 10171683

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 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 highest net benefit.

- When we have the marginal cost that are 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.

lOMoAR cPSD| 10171683

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 produce different things? If I have to move an input that is

connected to more than an output (so is not so easy and linear, because if i increase the net benefit

in one sense, in the other could decreases). 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

Irrelevant in decision making because you can't recover them, and we cannot adjust our activity

with respect to them.

• →

Sunk costs Previously paid & cannot be recovered;

• →

Fixed costs Constant & 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.

The very important part in our decision making is the increm