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Using Data to Determine Parameters

One task that economists face all the time is determining the values of parameters appearing in key economic equations. For example, suppose we know that a particular firm faces a demand curve for its product that looks like the following:

Q = A - B*P

where Q is the quantity people buy, P is the price of the product, and A and B are unknown constants. For this to be of any use to the firm, however, we need to figure out the numerical values of A and B.

In order to do that, we need to collect and use data. In this case, we could look at the prices the firm has charged in the past and the quantities it has sold. Suppose that firm has the following information:

Year Price Charged Sales
1995 10.00 800.00
1996 20.00 600.00

Since the demand curve would have held in both years, we can put these numbers into it to get the following two equations:

1995: 800 = A - B*10

1996: 600 = A - B*20

Graphically, the two data points look like this:

We now have two equations and two unknowns, A and B. To find A and B we can use ordinary algebra. Rewriting the 1995 equation:

A = 800 + B*10

Substituting this into the second equation:

600 = (800 + B*10) - B*20 = 800 - B*10

We can solve this for B, which shows that B=20. Substituting this back into the equation above for A shows that A=1000. We now know that the firm's demand curve is the following:

Q = 1000 - 20*P

Graphically, the finished demand curve looks like this:

When economists do this for real problems, they usually do the calculations using a branch of statistics called econometrics. For example, they might run a linear regression of Q on a constant term and P. The basic idea is the same, however.

Something to think about:

A very important thing to think about is that we couldn't have done this calculation if the firm had only one year of data, or had charged the same price in both years. Think about it and see if you can figure out why.

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Peter J Wilcoxen, The Maxwell School, Syracuse University
Revised 11/29/2003