Elasticity is the percent change in one ("independent") variable divided by the percent change in another ("dependent") variable.  It's that simple.

For example, if a 10% increase in the price of waffles causes consumption of waffles to fall by 20%, then the (own-price) elasticity of demand for waffles is (-20%) / (+10%) = -2.

Elasticity doesn't have any units (the percents cancel out); it's just a measure of the responsiveness of one variable to another.

You should keep track of the signs on elasticity measures: they tell you the direction of the change.

If a demand elasticity measure is greater than +1 or less than -1, then economists say the demand is "elastic," meaning that demand is pretty price-responsive.   If the elasticity measure is between -1 and +1, then economists say the demand is "inelastic," meaning that it is not very price-responsive.

When talking about elasticity, try to be specific--elasticity with respect to what?  As we know, there are lots of variables that can influence the demand for a particular good.

Cross-Price and Income Elasticities

If the price of waffles increases by 25% and consumption of pancakes rises by 20%, then the cross-price elasticity of demand for waffles with respect to the price of pancakes is +20% / +25% = +0.8.  Pancakes and waffles are substitutes for each other, the cross-price demand schedule for pancakes with respect to waffle prices is positively-sloped, and the cross-price elasticity is positive too.

If the price of waffles increases by 20% and consumption of syrup falls by 10%, then the cross-price elasticity of demand for waffles with respect to the price of syrup is -10% / +20% = -0.5.  Waffles and syrup are complements to each other, the cross-price demand schedule for syrup with respect to waffle prices is negatively sloped, and the cross-price elasticity is negative too..  The sign on the cross-price elasticity measure tells you if the goods are substitutes (positive sign) or complements (negative sign).

If household incomes rise 10% and waffle consumption rises 2%, then the income elasticity of demand for waffles is +2% / +10% = +0.2.  Waffles are a normal good; the Engel curve (demand with respect to income) for waffles is positively-sloped, and the income elasticity is positive too.  If incomes fall 10% and lottery ticket sales rise by 2%, then the income elasticity of demand for lottery tickets is +2% / -10% = -0.2.  Any inferior good has a negative income elasticity.  The sign on the income elasticity measure tells you if the goods are normal (positive sign) or inferior (negative sign).

If household incomes rise 10% and health care expenditures rise 15%, then the income elasticity of demand for health care is +15% / +10% = +1.5.  A luxury good is defined as any normal good with an income elasticity greater than +1, so health care would be a luxury good.

Demand Elasticity and Total Revenue

The total revenue a firm or industry earns is the amount of money they collect in sales.  Total profit is calculated by subtracting the total cost of making and selling the good from total revenue.  The total revenue earned in a market is the quantity sold (Q) times the price per unit (P): TR = P x Q.

Graphically, total industry revenue is represented by the area of the rectangle P x Q (base x height).  Here the area of the rectangle P₀ x Q₀ is larger than the area of the rectangle P₁ x Q₁.

The equation TR = P x Q seems to suggest that a firm can increase its revenues by either selling more (increasing Q), or by charging more per unit (increasing P), or doing both.  Unfortunately, that's not the case.  The firm can't simply force people to buy more and pay more too.  Markets are voluntary, and customers only buy what they want.

If the firm is facing a downward-sloping demand schedule, the only way it's going to sell more is by reducing the price.  And the firm has to reduce it's price for everybody, since it can't get away with price discrimination (charging different prices to different customers).  Customers talk to each other, and they compare prices!  So markets have pretty uniform prices for each good.

If the firm has to reduce its price in order to sell more, how will this affect its total revenue?  This depends on the elasticity of demand.

If demand is price-elastic (%change in Q divided by the %change in P is greater than 1 or less than -1), then a lower price will attract enough extra customers to increase total revenue.  "Price-elastic" means the percent increase in quantity sold will be larger than the percent decrease in price.  So the extra quantity sold more than makes up for the lower price per unit, and total revenues rise.

Example:  Suppose the elasticity of demand for widgets is -2.0, and at a price of $2.00 each, US demand for widgets is 400 million per year, so total industry revenue is $2 x 400 million = $800 million per year.  Now if the widget industry raises prices 10% to $2.20, demand for widgets falls 20% to 320 million units, so total industry revenue is $2.20 x 320 million = only $682 million.  Since it is selling to an elastic demand schedule, the widget industry would have to lower its prices to increase its total revenues.

On the other hand, if demand is price-inelastic (meaning that % change in Q will be smaller than % change in P), then a lower price will not attract enough extra customers, so a price cut will reduce total revenues.  In this case, the way to increase total revenues is to increase price.

Inelastic demand for food and US farm incomes

Demands for agricultural commodities like corn and wheat are quite inelastic (around -0.2).  People consume pretty much the same number of food calories each day, and lower food prices don't get people eating much more.

US farmers have become much more efficient at growing food over the past century, and crop yields per acre have risen dramatically.  But this creates a big problem for farmers: they keep growing too much and driving crop prices way down.  Since food demand is so inelastic, the more farmers grow, the less total revenue they earn!

Farmers are always complaining about this.  Small farms keep going out of business, but their cropland doesn't disappear.  It mainly gets bought by bigger farm operations, and stays in production.

The government has various programs to help farmers maintain their incomes.  One way the government keeps farm prices from going too low is by paying some farmers not to farm some of their land!

Terms to know:
elasticity = (% change in Q) / (% change in P)
    ("elastic:" |E| >1; "inelastic:" |E| <1.)
cross-price elasticity = (% change in Q) / (% change in P of another good)
    (complements: E_cp > 0; subsitutes: E_cp < 0)
income-elasticity = (% change in Q) / (% change in income)
    (normal goods: E_I > 0; inferior goods: E_I < 0; luxury goods: E_I > +1)
how to calculate %change:  %change in Q = 100 x (Q₁ - Q₀) / Q₀
another method:  %change in Q = 100 x (Q₁ - Q₀) / [(Q₁+Q₀)/2]
total revenue TR = P x Q (price per unit times total number of units sold)
profit = TR - total cost.
if demand is elastic, reducing price to sell more increases TR.
if demand is inelastic, increasing price (and selling less) increases TR.