Functions and Models

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Demand and Supply

The demand for a commodity usually goes down as its price goes up. On the other hand, the amount a manufacturer is willing to bring to the market, the supply, generally goes up as the price goes up.
Demand and Supply Models

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 supply equation or supply function expresses supply q (the number of items a supplier is willing to bring to the market) as a function of the unit price p (the price per item). It is usually the case that demand decreases and supply increases as the unit price increases.

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 find the equilibrium price, determine the unit price p where the demand and supply curves cross (sometimes we can determine this value analytically by setting demand equal to supply and solving for p). To find the equilibrium demand, evaluate the demand (or supply) function at the equilibrium price.

Examples

If the demand for Ludington's Wellington Boots is q = -4.5p + 4000 pairs sold per day and the supply is q = 50p - 1995 pairs per week (see the graph below), then the equilibrium point is obtained when demand = supply:

    -4.5p+4000 = 50p-1995
    54.5p = 5995
giving p = 5995/54.5 = $110. The equilibrium price is therefore $110 and the equilibrium demand is q = −4.5(110) + 4000 = 3505 pairs. per week. What happens at prices other than the equilibrium price can be seen in the following figure:




  • 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.

The next quiz is similar to Exercise 24 in Section 1.2 of Note: You will need to enter algebraic expressions using proper graphing calculator format. Press the button to see some examples.

The demand for monorail service in the three urbynes (or districts) of Utarek, Mars can be modeled by

where p is the fare the Utarek Monorial Cooperative charges in zonars* (). Assume the cooperative is prepared to provide service for
* The zonar is the official currency on Mars.
The demand and supply graphs will appear when you correctly enter them.
 
In December 2085 the Utarek Monorial Cooperative was charging per ride.
The fare resulted in a shortage
surplus
of million rides per day.
Use the graph to estimate the above answer accurate to the nearest whole number.

Modeling Change over Time

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 we can regard q as a function of t.

In the next example (similar to Example 5 in Section 1.2 of ) we are asked to select from among several curve-fitting models for given data.

The following table shows monthly sales s(t) of Ludington's Wellington Boots (t = 0 represents January):

t
  s(t)  

The graph below shows a plot of these data. After successfully answering the first question below you can enter the equation of any curve to plot if you like, but you won't see anything unless graph happens to go through the window shown, like 500+50t for example.

        Monthly sales of Wellington Boots
s(t) =
Which of the following types of model would be appropriate for the data? [In some cases, there may be more than one correct answer, but you should not choose a nonlinear model if a linear model appears to fit well. ]
Which of the following specific models best fits the data? [There is only one correct answer. Hint: Try plotting some of them before selecting.]

An important analytic model of change over time comes from the compound interest formula in finance.
Compound interest

If an amount (present value) P is invested for t years at an annual rate of r, and if the interest is compounded (reinvested) m times per year, then the future value A is

    A(t) = P\Bigleft(1+\frac{r}{m}\Bigright)^{mt}.      An exponential function of t
A special case is interest compounded once a year:
    A(t) = P(1 + r)^t         Put m = 1 in the above formula.
Example

If $2,000 is invested for two and a half years in a mutual fund with an annual yield of 12.6% and the earnings are reinvested each month, then P = 2,000, r = 0.126, m = 12, and t = 2.5, which gives

    A(2.5) = 2,000\Bigleft(1+\frac{0.126}{12}\Bigright)^{12×2.5} ≈ $2,736.02         Tech formula: 2000(1+0.126/12)^(12*2.5)

One for you:

You invest in a mutual fund with an annual yield of and the interest is reinvested

    Answer must be a whole number.
    Use the Function Evaluator and Grapher or a table in your graphing calculator to obtain the last answer.

You now have several options:

Last Updated: November, 2009
Copyright © 2009 Stefan Waner