## 1.3 Linear Functions and ModelsLinear Demand, Supply, and Time-Change Models

Previous tutorial: Linear Cost, Revenue, and Profit

(This topic is also in Section 1.3 in Finite Mathematics, Applied Calculus and Finite Mathematics and Applied Calculus)

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This is a continuation of a tutorial on linear models. To go back to the start of the tutorial, press here.

Using linear functions to describe or approximate relationships in the real world is called linear modeling. Here, we study several kinds of linear model:

### Linear Demand Functions

It is often observed that, the more you charge for an item, the lower the demand will be: As the price goes up, the demand goes down.

Q How do we measure demand?
A A common way of measuring demand for an item is by the number of items you sell.in some period of time. For instance, you can measure the demand for Starship Troopers T shirts by daily sales.

 Linear Demand Function A demand equation or demand function expresses demand q (the number of items demanded) as a function of the unit price p (the price per item). A linear demand function has the form q = mp + b Note: p plays the role of x and q plays the roles of y Interpretation of m The (usually negative) slope m measures the change in demand per unit change in price. Thus for instance, if p is measured in dollars and q in monthly sales, and m = -400, then each \$1 increase in the price per item will result in a drop in sales of 400 items per month. Interpretation of b The y-intercept b gives the demand if the items were given away. Example If the demand for T-shirts, measured in daily sales, is given by q =-4p + 90, where p is the sale price in dollars, then daily sales drop by four T-shirts for every \$1 increase in price. It the T-shirts were given away, the demand would be 90 T-shirts per day.

Weekly sales of Ludington's Wellington Boots are given by

q = -4.5p + 4000 pairs per week
where p is the price in dollars.

Q The equation tells you that

 Raising the price by \$4.50 results in one less pair sold per week. Lowering the price by \$2 results in 9 more pairs sold per week. Lowering the price by \$90 results in 2 more pairs sold per week. Raising the price by \$4.50 results in 4000 less pairs sold per week. Raising the price by \$1 results in 4000 less pairs sold per week

Ludington's competitor, Lillington, can sell 4500 pairs of boots per week if it charges \$100, and sales drop to 4450 per week if it charges \$110.

Q Lillington's demand equation is

 q =

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### Linear Supply Function, Equilibrium Price

A demand function gives the number of items consumers are willing to buy at a given price, and a higher price generally results in a lower demand. However, as the price rises, suppliers will be more inclined to produce these items (as opposed to spending their time and money on other products), so supply will generally rise. A supply function gives q, the number of items suppliers are willing to make available for sale, as a function of p, the price per item.

 Linear Supply Function, Equilibrium Price A supply equation or supply function expresses q (the number of items suppliers are willing to make available) as a function of the unit price p (the price per item). A linear demand function has the form q = mp + b Interpretation of m The (usually positive) slope m measures the change in supply per unit change in price. Thus for instance, if p is measured in dollars and q in monthly sales, and m = 400, then each \$1 increase in the price per item will result in an increased supply of 400 items per month. Interpretation of b The y-intercept b gives the number of items suppliers would be willing to supply for free. Example The number of T-shirts I am prepared to tie-dye and supply to Campus Creations Inc. per day depends on the price, \$p, I obtain according to q = 2.5p + 5. For every \$2 increase in price, I am willing to supply 5 additional shirts per day. Equilibrium Price Demand and Supply are said to be in equilibrium when demand equals supply. The corresponding values of p and q are called the equilibrium price and equilibrium demand. To obtain the equilibrium price, set demand equal to supply and solve for the unit price p. To obtain the equilibrium demand, evaluate the demand (or supply) function at the equilibrium price.

We saw above that weekly sales of Ludington's Wellington Boots are given by

q = -4.5p + 4000 pairs per week
where p is the price in dollars. On the other hand, Luddington's is not too eager to supply its Wellington Boots at basement bargain rates, and accorindingly controls the supply according to the formula
q = 50p - 1995 pairs per week

Q The supply equation tells you that

 Raising the price by \$50 results in one more pair supplied per week. Raising the price by \$50 results in 1995 more pairs supplied per week. Raising the price by \$50 results in one less pair supplied per week. Raising the price by \$1 results in 50 less pairs supplied per week. Raising the price by \$1 results in 50 more pairs supplied per week.

Again, here are the demand and supply equations:

Demand: q = -4.5p + 4000 pairs per week
Supply: q = 50p - 1995 pairs per week

Q The equilibrium price and demand are given by

 Equilibrium price = \$ Equilibrium demand = pairs

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Q What happens if the price is set lower or higher than the equilibrium price?
A To illustrate what happens at various prices, let us look at the graphs of demand and supply for Luddington's Wellington Boots.

Press the buttons to see the demand and supply at the indicated prices.

• When the price is lower that the equilibrium price, the demand is greater than the supply, resulting in a shortage.
• When the price is set at the equilibrium price, the demand equals supply, so there is no shortage or surplus, and we say that the market clears.
• When the price is greater that the equilibrium price, the supply is greater than the demand, resulting in a surplus.

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### Velocity and Other Time-Change Models

Things all around us change with time. Thus, it is natural to think of many quantities, such as your income or the temperature in Honolulu, as functions of time. We usually use the independent variable t to denote time (in seconds, hours, days, years, etc.). If a quantity q changes with time, then q is a function of t. Here, we are interested in linear functions of t:

q = mt + b       q is a linear function of time t
Let's go through such an example.

Weekly sales of Ludington's Wellington Boots are given by

q = -50t + 4500
where t is the time in weeks from now.

Q Sales this week (t = 0) are

 pairs

Q Sales next week (t = 1) will be

 pairs
 Q Weekly sales of Wellington Boots are increasing decreasing at a rate of pairs per week

In view of the above, we interpret the slope as the rate of change of the quantity q with respect to time t. Here is a summary of what we saw above, together with something new.

 Linear Change Over Time If a quantity q is a linear function of time t, so that q(t) = mt + b, then the slope m measures the rate of change of q, and b is the quantity at time t = 0, the initial quantity. If q represents the position of a moving object, then the rate of change is also called the velocity. Units of m The units of measurement of m are units of q per unit of time; for instance, if q is income in dollars and t is time in years, then the rate of change m is measured in dollars per year. Examples 1/ If the accumulated revenue from sales of your video game software is given by R(t) = 2000t + 500 dollars, where t is time in years from now, then you have earned \$500 in revenue so far, and the accumulated revenue is increasing at a rate of \$2000 per year. 2. You are driving down the Ohio Turnpike such that the number of miles you have traveled after t hours is given by s(t) = 54t + 20. Then your speed is 54 miles per hour, and at time t= 0 you had traveled 20 miles.

All of the preceding examples share the following common theme.

 General Linear Models If y = mx + b is a linear model of changing quantities x and y, then the slope m is the rate at which y is increasing per unit increase in x, while the y-intercept b is the value of y that corresponds to x = 0. The slope m is measured in units of y per unit of x, while the intercept b is measured in units of y.

Now try some of the exercises in Section 1.3 of the textbook, or press "Review Exercises" on the sidebar to see a collection of exercises that covers the whole of Chapter 1.

Last Updated: March, 2006