Exercise: Price elasticity

If you know that the market demand for good X corresponds to the following equation:

$P=-2Q+450$

and the supply curve can be model by the equation:

$P=3Q+200$

Calculate the price elasticity for each curve at the equilibrium point. Furthermore explain which curve is relatively more elastic.

The first step consists of solving for the equilibrium price and quantity:

$\\ (1.) -2Q+450=3Q+200 \\ (2.) Q_e=50 \\ (3.) -2(50)+450=350=P_e$

Remember the formula for the price elasticity is

$E_d,E_s=\frac{\%\Delta Q}{\%\Delta P}$

Therefore pick the two closest points and calculate the price elasticity, in this case let's pick a point that is one quantity less than the equilibrium point on both curves.

$\\(49,352)\; \textit{and }(50,350) \rightarrow E_d=|\frac{\frac{50-49}{49}}{\frac{350-352}{352}}|\approx 3.5 \\ (49,347)\; \textit{and }(50,350)\rightarrow E_s=\frac{\frac{50-49}{49}}{\frac{350-347}{347}}\approx 2.3$

Since we are dealing with linear equations we can use another formula which is specific for these types of curves and is more precise than the method we used so far. The formula is the fraction of the price over the quantity multiplied by the inverse of the slope.

$\\ E_d=|\frac{350}{50} \cdot \frac{1}{2}|=3.5 \\ E_s= \frac{350}{50} \cdot \frac{1}{3}=2.3\bar{33}$

Since Ed>Es, the demand curve at the equilibrium point is relatively more elastic than the supply curve. Note that both curves are elastic since both have an elasticity greater than 1. Furthermore, one could have made an educated guess by just looking at the absolute value of the slope for each curve.

The State wants to impose a price floor on the market for good X where the price elasticity of demand is equal to 6. Derive the equation for the price floor.

To answer this question, let's remember that one way to derive the elasticity for a linear function is by using a formula that combines the price, quantity and slope of a linear curve. Therefore, if we know the elasticity, we can create a linear function that intersects the demand curve at the price floor.

$\\(1.)E_d=6=(\frac{P}{Q}) \cdot \frac{1}{2} \\ (2.)P=12Q \\(3.)12Q=-2Q+450 \\(4.)Q_{pf}\approx 32.13, P_{pf}= 12(32.13)\approx 385.6$

Thus, since the equation of a price floor is only bounded by the price, its equation is equal to P=385.6, at this point the price elasticity of the demand curve is about 6.

Let's visualize the problem:

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