# In this investigation, I will be modeling the revenue (income) that a firm can expect given it demand curve using my knowledge of linear and quadratic functions

by derp109 (student)

Demand & Revenue Investigation

Reece Chau 11DZBH

Grade 11 Mathematics SL                                                                                             Wednesday 2nd November 2011.

Demand & Revenue Investigation.

Aim: In this investigation, I will be modeling the revenue (income) that a firm can expect given it demand curve using my knowledge of linear and quadratic functions.

Background Information:

The firm we are focusing on is the Very Big Gas Company (VBGC). The VBGC is a government monopoly that supplies natural gas to a national market. The vast majority of its sales involve selling natural gas to consumers who use it for heating of homes and businesses. Market studies have shown that the demand for its product varies each quarter according to seasonal temperatures. Since VBGC is a national monopoly, the price of its product is regulated by a government agency so as to protect consumers from excessively high prices and to maintain a level of consumption that reflects national environmental goals.

For each quarter, data on price and quantity sold has been collected and is recorded in a table below. The price is measured in Euros per cubic metre and the quantity is measured in millions of cubic metres of natural gas.

Table showing the price and quantity of gas sold for each quarter.

Part 1:

From the market data within the tables, we can find linear demand equations for the natural gas sold by the VBGC. Since the VBGC is a national monopoly, consumers can only purchase gas from this firm therefore the demand equation also presents the quantity of natural gas sold by the company. The demand equation shows the relationship between the price of a good (P) and the quantity demanded by consumers (Q), it should be written like this: y = mx + c (straight-line equation). When shown graphically, the price of a good is always shown on the y-axis whereas the quantity demanded always lies on the x-axis.

Using our knowledge of linear functions, we can calculate the gradient for each quarter using the market data providing us with information relating to the price and quantity sold. In order to find the gradient for the linear demand equations we can use the general rule of , which finds the difference of two points hence leaving us with the gradient, which we can use as the basis of our demand curve.

Using the points from quarter 1 and the general rule of  we can substitute the y-values with the price and the x-values with quantity demanded to get the gradient of the line for quarter 1.

• = -0.75

Using my GDC, I wanted to check my answers so I inserted the equation, substituting the x and y values with the price and quantity demanded resulting in the same answer. Now as we have our gradient for quarter 1, we have the first part of our demand curve, which we can substitute it into the straight-line equation, P= -0.75Q + c. We only have to find the value of c in order to complete our linear demand equation for the 1st quarter. To find the value of c, we simply have to substitute the values of the price and quantity demanded into the equation, and then solve it to be left with c.

• P= -0.75Q + c
• 4.5 = -0.75(2) + c
• 4.5 = -1.5 + c
• C = 6

Again, using my GDC I also substituted the values of P and Q to result in the same answer, to make sure it’s accurate and valid. Now as we have our c-value, we can simply insert it into the equation to be left with our linear demand equation for quarter 1. P = -0.75Q + 6.

We can check its validity by inserting the other points from quarter 1 into the equation to ensure that the gradient is correct.

• = -0.75

Now inserting the gradient into the straight- line equation:

• P= -0.75Q + c
• 3 = -0.75(4) + c
• 3 = -3 + c
• C = 6

So now, we are completely sure that the linear demand equation (P = -0.75Q + 6) for quarter 1 is correct.

Moving onto the 2nd, 3rd and 4th quarters, we can apply the same method of finding the gradient, then substituting the values with P and Q in order to get the linear demand equation.

Calculations to get Linear Demand Equations for 2nd, 3rd and 4th quarters:

We can test its validity by ...