Chapter 2 Handouts: Basic Economic Relations



Chapter 2 Handouts: Basic Economic Relations

Optimal decision: choice alternative that produces a result most consistent with managerial objectives. Such objectives include:

• Maximizing profits

• Minimizing losses

• Minimizing costs

• Maximizing revenues

• Minimizing errors

Managers must weigh the appropriate costs and benefits—and evaluate the present value of those costs and benefits in making optimal decisions.

Functional Relationships:

• Commonly used in optimization analysis to indicate which factors are important in the decision making process.

• They can be very broad

Ex: TR=f(Q): Total revenue is a function of the Quantity

Total revenue is the dependent variable

Quantity is the independent variable.

• They can be more precise

Ex: TR=P x Q

Equations give a more precise expression of the relationship

Ex: TR=$5.00 x Q

Total, Average, and Marginal Relationships:

• Most frequently used in optimization analysis

• Definitions:

o Totals: the comprehensive sum of the dependent variable

o Marginals: the change in the dependent variable caused by a one unit change in an independent variable.

▪ Ex: producing one extra unit of a good creates more total cost. The marginal cost (MC) is only those specific costs associated with producing that one additional unit. (ΔTC/ΔQ).

▪ Ex: selling one more unit of your product increases your total profits. The marginal profit (Mп) is the extra profit earned because of that one extra unit sold. (ΔTп/ΔQ).

▪ Ex: selling one more unit provides more revenue. The extra revenue from selling one more unit is the marginal revenue (MR). (ΔTR/ΔQ).

o Averages: the per unit value.

▪ Ex: 3 of us can make 12 cakes in a day. Our average production level is 12/3=4. On average, each of us can make 4 cakes. All total, we make 12.

Calculating and Graphing these relationships

|QUANTITY (Q) |TOTAL PROFIT (Tп) |MARGINAL PROFIT (Mп) |AVERAGE PROFIT (Aп) |

|0 |0 | | |

|1 |200 | | |

|2 |500 | | |

|3 |600 | | |

|4 |600 | | |

|5 |550 | | |

1. Fill in the columns for marginal profit and average profit.

2. Create two graphs one above the other.

a. Graph Total profit against Q in the upper graph

b. Graph Marginal and average profit against Q in the lower graph

Geometric Relationships:

1. At any point along the total curve, the corresponding average figure is given by the slope of a straight line from the origin to that point.

a.

2. At any point along the total curve, the corresponding marginal figure is given by the slope of a line drawn tangent to the total curve at that point. This means that the marginal figure is the derivative of the total figure.

a.

b.

c.

*Relationship between marginal and average.

a. When the marginal is above the average value, the average will increase (When Mп>Aп, then Aп↑).

b. When the marginal is below the average value, the average will decrease (When Mп< Aп, then Aп↓).

c. It is not necessarily true that when Mп↑ then Aп↑ or that when Mп↓ then Aп↓.

Ex: think of course grades. Each test is your marginal performance. Your tests combined give you a sense of your average.

These relationships are important because they are used over and over in business decisions. We will use spreadsheets and graphs, but we will also use equations to represent the functional relationships.

Three primary managerial rules that use the marginal/average relationship:

1. Maximize Revenues

2. Maximize Profit

3. Minimize Cost

Using marginal relationships to maximize Revenue or maximizing sales income

• Firms may choose to maximize revenue (particularly as a short run motive) in order to increase consumer awareness of products (sell more product than might maximize profit), or to limit entry of new competition by keeping prices lower and sales quantities higher.

• Many government agencies or not for profit organizations don’t sell products—they provide services in which case they may want to maximize their outreach services in order to help more people or secure higher budgets in subsequent years

Suppose you have the following information about sales at different price points:

|Quantity Sold |Price |Total Revenue (TR) |Marginal Revenue (MR) |

|0 |$50 | |------ |

|1 |$45 | | |

|2 |$40 | | |

|3 |$35 | | |

|4 |$30 | | |

|5 |$25 | | |

|6 |$20 | | |

• Fill in the table above

• Sketch out the demand curve and find the inverse demand equation p=f(Q) or P=a +bQ where a is the intercept of the price axis (Y) and b is the slope coefficient ∆y/∆x or ∆P/∆Q

• Use any two data points to find coefficient “b”;

• Plug into functional relationship: P=a - 5Q

• Find value for “a” by substituting in any P,Q combo from table

o

o

o

o _______. (can double-check table to see that this is true. However, if you are NOT given the “a” value in the table use this method to determine it!

o

• Determine the Total Revenue (TR) equation:

o

o

o

Now you can find TR for any quantity value and are no longer limited by the table:

Confirm your TR equation is correct by choosing a value from the table such as Q=3 and determining TR:

TR= 50Q -5Q2

TR= 50(3) -5(3)2

TR=150-45=$105. Confirm in table.

Graph TR numbers:

• Determine Marginal Revenue (MR) equation:

o

o

o

o Note: your MR numbers may be slightly different in the table relative to numbers calculated using the derivative. This is OK.

o Plot MR numbers from table on the same graph as TR above.

• Revenue maximization occurs where TR is maximized or where MR=0.

o Find this point algebraically using MR equation

o

o

o

o TR is maximized at Q=5. You will see that for quantities up to five units, MR is positive indicating that as you produce and sell another unit the total revenue increases. After Q=5, MR is negative indicating that each additional unit produced and sold results in a reduction of total revenue. Therefore, the max TR is where MR=0.

Conclusions:

•

•

•

• This rule is different from profit maximization which also considers costs!

Using Marginal Relationships to Maximize Profit

Selling Golf Carts—given demand equation and total cost equation only.

Demand Equation: P=$7,500 - $3.75Q

Total Cost Equation: TC=$1,012,500 + $1500Q + 1.25Q2

Find on your own:

Total Revenue Equation:

Marginal Revenue Equation:

Marginal Cost (∆TC/∆Q):

A. Determine Profit Maximizing Quantity

Profit π= total revenue (TR) – total cost (TC)

Marginal profit (mπ) = MR-MC

To maximize the total you need to set the marginal equal to zero or mπ=0

o From 232: you may recall that this also means that MR-MC=0 or MR=MC (you could set MR=MC and get the same answer as follows)

Mπ=MR-MC

Mπ=

Mπ=

Set Mπ=0

0=

Q=600 units.

B. Determine the selling price, P, using the given demand equation:

C. Determine the Total Revenue earned at Q=600 units:

D. Determine total cost at Q=600 units

Given: TC=$1,012,500 + $1500Q + 1.25Q2

E. Determine Total Profit:

F. Determine Profit per unit: (П/Q)

П/Q=$787,500/600=$1312.50

Try at home: You should note that maximizing profit and maximizing revenue are two different strategies. For this problem you could go through and max revenue to find this would occur at Q=1000 units and profits would be $-12,500.

Using marginal analysis to minimize cost.

The relationship between marginal and average is such that the average is either maximized or minimized where the two cross (they maximize in the case of revenues or profits; and minimize in the case of costs).

Given total cost (TC):

o Find Marginal cost (MC) as the derivative as done previously

o Find Average Cost (AC or sometimes called ATC) as TC/Q. Hence, take your TC equation and divide it by Q

▪ Ex: TC= 1500 +10Q+2Q2

▪

▪

▪

In the practice questions recommended you will work through an example on your own.

CH. 2 PRACTICE QUESTIONS pp. 43-52

Self-Test Problems and Solutions

ST 2.1

ST 2.2

Problems:

P2.2

P2.6A (omit part B)

These are for you to practice. You will have a quiz in a day or so (Wednesday or Thursday depending on when we finish Ch. 2 material) that will be very similar in nature. The answers to most of the problems are either directly underneath the problem or posted on the class website (faculty.wiu.edu/tn-westerhold)

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