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Section 2.2 Demand Function

In Section 2.1 you first made sense of your data by determining what the results from each small test market could tell you about your product's demand in the large nationwide market. The next step, then, is to go beyond your short list of data points to model the overall trend those data points suggest. After all, chances are that the "best" price point to set for your product will not exactly be one of the prices you tested, but something in between. But how do we read in between our data points?

The key quantitative method in this process is regression, the statistical process of varying the parameters in a formula so as to minimize the mean-square error inherent in how that formula's predicted data points differ from your actual data points. In other words, it is the process of finding a "curve of best fit" of a given type for a set of data points. The statistical details of regression analysis are beyond the scope of this course, but the result provides the crucial link between real-world data and idealized mathematical models we can analyze using the tools of algebra and calculus.

Subsection 2.2.1 Motivation 2

Our goal is to determine a demand function, a mathematical mode \(p = D(q)\) which best explains the relationship between the quantity sold in the national market \(q\) and the unit price we'd charge to sell that quantity, \(q\text{.}\)

To get there, take every data scientist's recommendation for a first step in any data project: make a scatter plot. Then, use technology to fit an equation to that data. For our project, we'll choose a quadratic equation (a polynomial of degree 2). That means we'll get a demand function equation that matches the template

\begin{align} y\amp = Ax^2 +Bx+C \amp \text{or, in context,}\tag{2.2.1}\\ p = D(q) \amp = Aq^2 + Bq+C\tag{2.2.2} \end{align}

The question is, what values of the coefficients \(A,B,C\) will give us an equation that best matches the data?

Do it with Excel:

Do it with Desmos:

Subsection 2.2.2 Apply It 2

The \(x\)- and \(y\)-intercepts of an applied function model almost always tell us something meaningful and interesting. Finding them is a snap once we have a model equation, using what we know from precalculus:

  1. To find a \(y\)-intercept, set \(x\) equal to zero and solve for \(y\text{.}\) That is, compute the value of \(y=f(0)\text{.}\) (This usually requires only arithmetic and no algebra.)
  2. To find a \(x\)-intercept, set \(y\) equal to zero and solve for \(x\text{.}\) This often requires considerably more algebraic technique, and is sometimes not feasible without technology.

Here's how that plays out with Decision Report 2's demand function:

Subsection 2.2.3 Decision Report 2

In your previous decision report (Subsection 2.1.2), your team constructed a table of data relating quantities \(q\) (in thousands) and prices \(p\) (in dollars per unit) for your product's projected sales on the national market.

Now, your team's job is to extrapolate from that data: to determine a mathematical function

\begin{equation*} p = D(q) \end{equation*}

which best fits your table of data. This function is known as a demand function, and its job is to capture the relationship between the unit price charged for a product, and the quantity of product that would sell at that price.

Your Vice President of National Sales remains somewhat unconvinced by your previous findings, in which you determined whether a national sales goal of 1.5 million units was realistic. Your job in this decision report is to make an even stronger case for your previous results.

Exercises Exercises

1. Decision Report 2: What are the worst-case scenarios?

Download: Click to download this Decision Report assignment (PDF).

Decide: Reinforce your decision from Exercise by determining both

  • The maximum price that is possible to charge for your product in the national market, and
  • The maximum quantity that is possible to sell in the national market.

Explain your findings, and why they support your previous decision report, in a brief memo to your Vice President.

Deliver: Use a trend line in either Excel or Desmos to fit a quadratic function to your national market data, with quantity \(q\) (in thousands) on the horizontal axis and price \(p = D(q)\) on the vertical.

Report both the equation of this quadratic, using the variables \(q\) and \(p\) in place of \(x\) and \(y\text{,}\) its correlation coefficient, and the computations that inform your decision.