We've updated our

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Putting It Together

Summary

Back to our original problem at the beginning of this section. This discussion focused on whether to raise or lower the price of Product X to increase sales. What we are basically looking at is the Elasticity of this produce (i.e. how responsive are the sales of this product based on the current price). Our formula for elasticity is:

$\displaystyle{E} = {-}\frac{{p}}{{q}}\cdot\frac{{dq}}{{dp}}$

Examples

In this case, p is our independent variable. We know

$\displaystyle {q} = 400 - {2p}$

Which tells us:

$\displaystyle\frac{{dq}}{{dp}} = {-2}$

$\displaystyle{E} = {-}\frac{{p}}{{400 - 2p}} \cdot{-2}$

$\displaystyle{E} = \frac{{2 p}}{{400 - 2p}}$

Evaluating at $\displaystyle{p} = {\100}$

$\displaystyle{E} = \frac{{{2}\cdot {100}}}{{400 - {2}\cdot{100}}} = \frac{{200}}{{200}} = 1$

Since $\displaystyle{E} = 1$, this tells us to do nothing to the price. Neither increasing it nor decreasing it will increase our overall profits.