Informal lecture notes.

Total revenue is determined by both the price and the quantity sold. If we sell a 100 units at $4, we have a total revenue of $400. We can create a simple mathematical model of this fact:

\[ Total\, Revenue = Price\, \times\, Quantity \]

or

\[ TR(Q) = P \times Q \]

Remember from the law of demand that prices and quantity have a negative relationship. This means that as we increase price, we would expect the quantity sold to be smaller. How much smaller? The sensitivity of the market to changes in price is captured by the shape of the demand curve.

Assuming you are a profit maximizer, it would be important to know how quantity changes when we increase price. If the change in quantity associated with a change in price is **relatively** small, we would expect Total Revenue to increase. On the other hand, if a small increase in price brings about a **relatively** large decrease in quantity sold, we would expect Total Revenue to decrease. In the second case, we would increase revenue by lowering our price.

0 | 12 | 0 |

1 | 10 | 10 |

2 | 8 | 16 |

3 | 6 | 18 |

4 | 4 | 16 |

5 | 2 | 10 |

6 | 0 | 0 |

We use \(\eta\) to denote elastisity, |x| means the absolute value of x, and \(\Rightarrow\) means *implies*.

If \(|\eta| > 1 \Rightarrow\) demand is elastic and \(TR\) increases with quantity.

If \(|\eta| < 1 \Rightarrow\) demand is inelastic and \(TR\) decreases with quantity.

Some real world estimated elasticity.

Eggs | 0.1 |

Cigarettes | 0.4 |

Rice | 0.5 |

Hotel room | 1.0 |

Beef | 1.6 |

Coke | 4.4 |

\[percent\, change = \frac{New\, value-Old\, value}{Old\, value}\times 100\]

We are interested in how prices affect revenue. This depends on how much quantity will vary with a change in price. When you increase price:

We increase our revenues per unit sold

We lose customers because of the price increase

If the percentage change in price (+) is larger than the corresponding percentage change in quantity (-), total revenue will increase with the change in price:

\[|\%\Delta P|>|\%\Delta Q| \iff \left|\frac{\%\Delta Q}{\%\Delta P}\right| < 1\]

When the above inequaliy holds we say that demand is inelastic, because the response to a change in price is weak.

If the percentage change in price (+) is smaller than the corresponding percentage change in quantity (-), total revenue will decrease with the change in price:

\[|\%\Delta P|<|\%\Delta Q| \iff \left|{\frac{\%\Delta Q}{\%\Delta P}}\right| > 1\]

When the above inequaliy holds we say that demand is elastic, because the response to a change in price is strong.

\[\frac{percentage\, change\, in\, quantity}{percentage\, change\, in\, price}\]

which can also be written as

\[\frac{(Q_1-Q_0)/Q_{avg}}{(P_1-P_0)/P_{avg}}= \frac{Q_1-Q_0}{P_1-P_0}\frac{P_1+P_0}{Q_1+Q_0} \]

Note how with elasticity we use the mean of both point instead of the â€˜old valueâ€™ in the percentage change formula.

The letter â€˜dâ€™ is called delta in greek, itâ€™s lower case is \(\delta\), and itâ€™s upper case is \(\Delta\). Scientists use the \(\Delta\) to mean difference. For example, if an economist write \(\Delta GDP\), she would mean \(GDP_{today}-GDP_{yesterday}\). We can use this notation to write the formula for elasticity.

\[\frac{\%\Delta Q}{\%\Delta P}\]

1 | 1.00 | 1 |

2 | 0.50 | 1 |

3 | 0.33 | 1 |

4 | 0.25 | 1 |

5 | 0.20 | 1 |

6 | 0.17 | 1 |

When the price increases from $4 to $6 quantity decreases from 120 to 80. What is the arc elasticity between these two points?

The slope:

\[\frac{Q_1-Q_0}{P_1-P_0}=\frac{80-120}{6-4}=20\]

and the ratio of averages:

\[\frac{P_1+P_0}{Q_1+Q_0}=\frac{6+4}{80+120}=.05\]

The elasticity is \(\eta= 20 * .05 = 1\)

Small \(\Delta\) in price \(\Rightarrow\) \(-\infty \Delta\) in quantity; \(\eta=-\infty\).

\(\infty \Delta\) in price \(\Rightarrow\) zero \(\Delta\) in quantity; \(\eta=0\).

What is the percentage change in quantity demanded due a 1% change in income?

\[\eta_I=\frac{\%\Delta Q}{\%\Delta I}\]

\(\eta_I > 0\) for normal goods.

and

\(\eta_I < 0\) for inferior goods.

What is the percentage change in quantity demanded due a 1% change in the price of another good?

\[\eta_{x,y}=\frac{\%\Delta Q_x}{\%\Delta P_y}\]

\(\eta_{x,y} > 0\) for substitutes.

and

\(\eta_{x,y} < 0\) for complements.