Chapter 2: Optimization Techniques and New Management ToolsInstructor: Maharouf Oyolola

As Discussed in the previous chapter, the objective of a business firm is to maximize profits or the value of the firm or minimize cost, subject to some constraints. In this chapter, we present optimization techniques, or methods for maximizing or minimizing the objective function of a firm.

Methods of expressing Economic relationships Economic relationships can be expressed in the form of equations, tables, or graphs. When the relationship is simple, a table and/or graph may be sufficient. However, when the relationship is complex, expressing the relationship in equational form may be necessary.

Examples Example of simple demand equation: Qd= a-b*P Example of a complex equation: TR=100Q-10Q2

Total, Average, and Marginal RelationshipsThe relationship between total, average, and marginal concepts and measures is crucial in optimization. This relationship is basically the same whether we deal with revenue, product, cost, and marginal cost

Total Revenue schedule of the firmQ 0 1 2 3 4 5 6 7 TR=100Q-10Q2 TR 100(0)-10(0)2 100(1)-10(1)2 100(2)-10(2)2 100(3)-10(3)2 100(4)-10(4)2 100(5)-10(5)2 100(6) -10(6)2 100(7)-10(7)2 0 90 160 210 240 250 240 210

the total revenue curve of the firm300 total revenue(TR) 200 100 0 0 2 4 output (Q) 6 8 TR

Total, Average, and Marginal Costs of a firmQ 0 1 2 3 4 TC $20 140 160 180 240 AC 140 80 60 60 MC 120 20 20 60

Optimization with calculus Determining a Maximum or a Minimum by calculus Optimization often requires finding the maximum or the minimum value of a function. For example, a firm might want to maximize its revenue, minimize the cost of producing a given output, or, more likely, maximize its profits.

Multivariate Optimization The Unconstrained Optimization Assume a car dealer for FORD and HONDA would like to maximize his/her profit. As a manager you want to know how many Honda and Ford should I sell to maximize my profit? The unconstrained optimization does not take into account factors that might impede the freedom of the firm such as the legal environment, the space, the personnel.

The Constrained Optimization.It is unrealistic to assume that the manager of the firm faces no constraints. Most of the time, however, managers face some constraints in their optimization decisions. As a manager, you want to know how many cars (FORD and HONDA) should I order from the manufacturers to maximize my profit?

As the manager of the car dealing company, here are some of the constraints you might be dealing with 1) Do I have enough space for the cars I order? 2) Do I have enough personnel to take care of the cars? As we discussed last week, the objective of the firm is to maximize its profit. Therefore, minimize its costs. In the optimization with constraint, you take the aforementioned factors into consideration while deciding how many cars you should order to maximize your profit.

Example 2 If you are the manager of a coffee shop. Your objective is to maximize the firms profit. Assume your order, milk, which is one of your inputs directly from the manufacturer. Milk is perishable good. How many milks should I order to maximize my profit?

Problems In-class Problems Problem #4 page 77