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Elastic demand is a type of demand that will rise or fall depending on the price of the good. For example, candy bars are an elastic demand. If the price of candy is around 5AED, most people will buy the candy and it will be high in demand. However, if that same candy bars price rose up to 8AED, most people would not buy the candy.
Income elasticity of demand measures the responsiveness of the demand for a good to a change in the income of the people demanding the good, ceteris paribus. It is calculated as the ratio of the percentage change in demand to the percentage change in income. For example, if, in response to a 10% increase in income, the demand for a good increased by 30%, the income elasticity of demand would be 30%/10% = 3.
Cross-price elasticity measures whether goods are substitutes or complements. It looks at the response of people in buying one product when the price of another product changes.
If goods are complements, cross-price elasticity will be negative. For example, if the price of gasoline rises, the sales of large cars will decline. The positive change in the denominator (bottom) is matched with a negative change in the numerator (top) of the equation. The result is therefore negative. If cross-price elasticity is positive, B is a substitute for A. For example, sales of Coke will fall if the price of Pepsi falls because some Coke drinkers will switch from Coke to Pepsi.
Inelastic demand is the opposite. People will buy goods with an inelastic demand no matter what the price is. A good example of this would be life-saving medications. Even if they are expensive, people will still buy them or else they could possibly die.
So basically the elastic demand for a good will rise or fall depending on the price where as an inelastic demand for a good wont.