Solving for quantity demanded using price elasticity of demand - FreeEconHelp.com, Learning Economics... Solved!

## 3/4/12

The price elasticity of demand measure can be used for predicting consumer response to price changes.  One of the most powerful tools in economics is using knowledge of consumer behavior to predict what will happen before the change actually takes place.  The following question considers the consumer response to a price increase in gasoline.  It is always a good thing to study behavior before making a change, otherwise you could potentially not only lose customers, but also go out of business.

At the local gas station, the price elasticity of demand for gasoline is 0.5 at the current price of \$3 per gallon and the current quantity demand of 5,000 gallons. The gas station is about to increase the price to \$3.15 per gallon. How many gallons of gas will be sold at this higher price?

This question requires us to deal with elasticities again, so review the concepts if necessary.

Now, we should begin by writing down our basic price elasticity of demand equation:

Or:
% change in quantity / % change in price = elasticity

Remember that the percent change in QUANTITY goes ON TOP, and that the percent change in PRICE goes ON THE BOTTOM.  Think of quarter pound, or some other trick if that helps.

Using the equation above we can plug in -0.5 as our elasticity measure, and we know the before and after price is \$3, and \$3.15.  This gives us a percent change in price of:

.15/3.075 or .049 which is about 1/20.  We can then multiply this amount by our elasticity measure to get the predicted percent change in quantity.  So 1/20*-0.5 is -1/40.  So we would expect the percent change in quantity to be about -0.025.  So what would the after price increase quantity of gallons need to be to get a percent change of -0.025, let’s set up the equation:

(X-5,000) / ((X+5,000)/2) = -0.025

This is where the algebra may get a little tricky for some, first let’s multiply both sides by ((X+5,000)/2), then multiply both sides by 2, to get:

2*(X-5,000) = -0.025 (X+5,000)

We can multiply everything out to get:

2X – 10,000 = -0.025X -125

Now add 0.025 X to both sides, and add 10,000 to both sides to get:

2.025X = 9,875 and divide both sides by 2.025 to get:

X = 4,876.5 (after rounding)

Now if we plug this number back into our elasticity measure, you will see that it does indeed give us a percent change in quantity value of -0.025 which is what we want.

So the confirmed amount of gasoline to be sold after the price increase is going to be 4,876.5 gallons, and keep in mind that this value was found using the midpoint formula, so it is valid for either a price increase or decrease.  If you do not use the midpoint formula, then the value will be somewhat different because you are using a different method to calculate percent change values.

So what does this mean for the seller of gasoline, does his revenue go up/down/ or stay the same???  Well, we can predict that his revenue would go up because he is operating in the inelastic range of price and quantity but let’s confirm that.

What is his revenue before the price increase?  5,000*3.00 = 15,000
What is his revenue after the price increase? 4,876.5*3.15 = 15,360

So his total revenue has increase by \$360 because he raised his price.  His revenue has increased, even though he has sold less product.