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# 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

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Introduction

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.

 Quarter 1 Jan, Feb March Quarter 2 April, May, June Quarter 3 July, Aug., Sept. Quarter 4 Oct., Nov., Dec. Price Quantity Price Quantity Price Quantity Price Quantity 4.5 2 3.25 1 3 0.5 4.5 1.5 3 4 2 2 2 1 3 3 1.5 6 0.75 3 1 1.5 0.5 5.5

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:

Middle   • = -1.25
• = -2
• = -1

Input value of P & Q, then solve C

• P= -1.25Q + c
• 2= -1.25(2) + c
• 2= -2.5+ c
• c =  4.5
• P= -2Q + c
• 2 = -2(1) + c
• 2= -2 + c
• c = 4
• P= -Q + c
• 3= -1(3) + c
• 3= -3 + c
• c = 6

Final Linear Demand Equations

P = -1.25Q + 4.5

P = -2Q + 4

P= -Q + 6

From these justifications, it is clear that our final linear demand equations for all quarters are accurate. The graph below shows all of the final linear demand equations for all four quarters.

Figure 1: Graph showing Linear Demand Equations for all four quarters.

Part 2: In this section, we will be using the demand equations found in part 1 in order to show the revenue for each of the four quarters. Revenue is the income the company gets, therefore the general equation for revenue is: R= PQ (Revenue= price x quantity). As we know the price (P) already from the linear demand equations, we simply need to multiply everything by Q to get the functions for the revenue.

Calculations:

Quarter One:

• Revenue Equation: R=PQ
• P=-0.75Q+6
• R=(-0.75Q+6)(Q)

Quarter Two:

• Revenue Equation: R=PQ=
• P=-1.25Q+4.5
• R=(-1.25Q+4.5)(Q)

Quarter Three:

• Revenue Equation: R=PQ
• P=-2Q+4
• R=(-2Q+4)(Q)

Quarter Four: P=-Q+6

• Revenue Equation: R=PQ
• P=-Q+6
• R=(-Q+6)(Q)

Now since we have our revenue functions (seen as quadratic functions) we can graph the functions to investigate whether the results make sense and seeing the benefits and limitations to the VBGC.

Figure 2: Graph showing the revenue functions for all four quarters. From looking at the graph, it can be seen that these parabolas are very different implying that the revenue for each quarter is different, fluctuating throughout the year.

Conclusion

• Q=-
• Q=-

From this equation we can always derive our x-value. It simply has to be substituted into the straight-line equation, resulting with the y-value and our final vertex for the location of the maximum revenue.

• P=m()+c
• P=-+ c
• P=
• (Quantity, Price)= (, )

When we substitute the x-value back into the straight-line equation, and we eliminate the terms we are left with as our y value. We can also prove that these work by taking two points from the market table and seeing if the results match the vertex. For example:

• (Quantity, Price)= (, )
• = = 4
• = = 3

(Quantity, Price) (4, 3) This clearly proves the general rule as we derive back to the maximum vertex to find the location on the graph of the maximum revenue.

Result of the elasticity of demand when revenue is maximized.

Referring back to part 4, the elasticity’s for the various demand curves will fluctuate as they depend on seasonal changes and the demand at the time of year. For the maximum revenue, we can derive a general rule that will always prove that the elasticity is always unit elastic. This can be presented in algebra if we substitute the P and Q values in the form of c for the straight-line equation:

As we eliminate the like terms we are left with:

• η== -1

From these calculations, we derive to η being equivalent to -1, which matches the results I got in part 4 justifying my proposition. In relation to the VGBC, with their four quantity and price combinations that maximized the revenue for the firms, it can be seen that as the price changes the expenditure remains constant, always resulting in -1, which is unit elasticity.

In conclusion, it can be seen that with the use of linear and quadratic functions we can relate it to real-life situations such as helping find out the quantity of a product sold, the revenue as well as elasticity for certain firms.

By Reece Chau 11DZBH.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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