22  Project: Taos Bee Association

This project uses linear functions in order to analyze simple models for supply and demand.

In order to make it easier to understand these concepts, we study a hypothetical situation, where the Taos Bee Association is purchasing honey from local farmers and then selling the honey to tourists.

This project involves four tasks:

22.1 Analyzing supply

We start by studying the honey supply. The Taos Bee Association plans to buy honey from a variety of farmers in Taos County. Economic theory tells us that if the Taos Bee Association pays a lower price for honey, then fewer people will want to sell them honey. But when the price is higher, then more people will be willing to sell them honey. The relationship between price and amount of honey are willing to sell is called a supply curve, even though in this case the relationship is actually a straight line.

We use these variables:

  • \(x\) is the amount of honey local farmers are willing to sell, measured in hundreds of pounds. (Thus \(x=3\) means farmers are willing to sell \(300\) pounds of honey.)
  • \(y\) is the price per pound of the honey. (Thus \(y=4\) means that the Taos Bee Association is paying farmers \(\$4\) for a pound of honey.)

For the purposes of this project, we assume that the relationship between the amount of honey sold and the price is given by \[ y = 2 + 3x. \]

The first part of this task is to make a table of values for this formula. Be sure to label what \(x\) and \(y\) mean.

Second, you should make a plot of the equation. Be sure to label what the \(x\) and \(y\) axes mean.

Third, suppose that the Taos Bee Association is willing to buy honey for \(\$20\) per pound. Based on the formula, how many pounds of honey should they expect to buy at this price?

22.2 Analyzing demand

We now analyze the honey demand. We imagine that the Taos Bee Association is selling honey to tourists who visit Taos County. Economic theory tells us that when the price of honey is high, then tourists will only buy a small amount of honey, but when the price of honey is low, then tourists will buy more honey.

We use these variables:

  • \(x\) is the amount of honey tourists are willing to buy, measured in hundreds of pounds. (So \(x = 3\) means that tourists are willing to buy \(300\) pounds of honey.)
  • \(y\) is the price of a pound of honey. (So \(y = 4\) means that honey is sold to the tourists for \(\$4\) per pound.)

The Taos Bee Association did some market research and discovered that

  • In order to sell \(100\) pounds of honey to tourists, the price can be set to \(\$40\) per pound. What values of \(x\) and \(y\) does this correspond to?
  • In order to sell \(400\) pounds of honey to tourists, the price can be set to \(\$28\) per pound. What values of \(x\) and \(y\) does this correspond to?

The first part of this task is to make a table that includes the two data points from the market research. Also put these two points on a graph.

The second part of this task is to construct the linear equation \(y=mx+b\) that connects the two data points. First, find the slope \(m\), and then find the base value \(b\).

The third part of the task is to draw the corresponding line.

Finally, answer these two questions:

  • Suppose that the Taos Bee Association wants to sell \(500\) pounds of honey. What price should they set?
  • Suppose the Taos Bee Association sets the price at \(\$4\) per pound. How many pounds of honey will they be able to sell?

22.3 Analyzing the equilibrium

In an ideal world, the amount of honey that the farmers are willing to sell would be the same as the amount of honey that the tourists are willing to buy. We want to know: what price achieves this balance?

Think about how you might figure out what price achieves this balance. You can use any of our modeling approaches - table of values, plot, equation.

22.4 Summarizing your analysis

Finally, you should create a report that summarizes the results of your analysis. Your summary should include the following components:

  • An introduction that explains the scenario and the questions that you addressed.
  • The plot from your analysis of supply, along with some words explaining what the plot shows.
  • The plot from your analysis of demand, along with some words explaining what the plot shows.
  • The equilibrium value that you found, along with some words explaining how you found that value.

Please use complete sentences throughout your summary.