The demand for a product varies depending on the price charged. For some products, demand changes only slightly with a large increase in price. For other products, demand changes significantly with only a small increase in price. The *Price Elasticity of Demand* is defined as follows:

Price Elasticity of Demand = (∂Q / ∂P) * (P / Q)

Where Q is quantity demanded, P is price, ?Q is change in quantity demanded, and ?P is change in price.

So, supposed that a family demands 20 gallons of gas per week when the price is $1.99 per gallon. But, if the price is $2.02 per gallon, the family only demands 19.5 gallons of gas. The price elasticity is:

Price Elasticity of Demand for Gas = (0.5 / 0.03) * (1.99 / 20) = 1.66

The meaning of this value is that a 1% increase in price of gasoline reduces the family’s demand for gasoline by 1.66%.

Of course, price elasticity will depend on what kind of product is sold (i.e., whether its a necessity or a luxury item), the income level of the consumers, and will also depend on the locality. For example, in developing countries, the price elasticity of demand for gasoline will probably be much larger than it is in the developed world. Also, a product with many close substitutes is more likely to have a larger elasticity.

We can see from the above equation that a price elasticity of demand close to zero means that demand will not change significantly with price. But, a large value indicates that the demand changes significantly with price. Such a large price elasticity is often called a “flat demand curve.” If the demand curve is completely flat, an unlimited quantity can be sold at a given price; but, nothing will be sold if the price is raised only slightly.

Its important to note that the price elasticity of demand is a snapshot at a certain point on the demand curve. The 1.66% reduction in demand for a 1% increase in price of gasoline may be true for the family when gasoline is priced around $2 per gallon, but will probably not be true when the price of gasoline is $4 per gallon. Perhaps, at $4 per gallon, the family will reduce its demand for gasoline by 3% for every 1% increase in the price of gas. Keep this in mind when calculating the price elasticity of demand with data with large differences in price. If such varying data is the only data available, a more accurate way to calculate the elasticity may be to average the price and demand over the range resulting in the following equation:

Price Elasticity of Demand = (ΔQ * [P

_{1}+ P_{2}]) / ( ΔP * [Q_{1}+ Q_{2}])

**Price Elasticity of Demand from the Demand Curve**

Suppose that it is know that a product has the following demand *linear *curve:

P = a – bQ

Where a and b are constants. It can be shown using the above equations that:

Price Elasticity of Demand = (b*P)/Q

With this linear demand curve, the price elasticity of demand approaches zero as the price approaches 0. This makes sense if we again consider gasoline as an example. A family will likely acquire the same amount of gasoline if the price is 1 cent per gallon as it would acquire if the price were 2 cents per gallon.

Likewise, as Q approaches 0, the price elasticity of demand approaches infinity. Using the gasoline example: If a family demands one gallon of gas per week if the price is $20 per gallon, a 10% increase of price to $22 per gallon might cause it to cut its consumption to half gallon per week.

**Price Elasticity and Revenue Maximization**

For a firm, the important question to ask is: What price should be charged to maximize revenue?

The total amount spent by customers on a product is P*Q. This is the firm’s **total revenue.** In the simple case where price is = a – bQ:

TR = P*Q = (a-bQ)Q = aQ – bQ

^{2}

A price is considered *elastic* if the price elasticity of demand is > 1. The total amount spent by customers decreases when price rises.

A price is considered *inelastic* if the price elasticity of demand is < 1. So, the total amount spent by customers *increases *when price rises.

The equilibrium point occurs when the price elasticity of demand equals 1. This is the price point which maximizes total revenue.

**Marginal Revenue**

We know that total revenue is described by the equation:

TR = P*Q = (a-bQ)Q = aQ – bQ

^{2}

The total revenue increases as the quantity increases up to a certain point and then it begins to fall. The amount of revenue (or cost) of the final item produced is called the *Marginal Revenue.* The Marginal Revenue is the first derivative of the Total Revenue:

MR = a – 2bQ

Note that the marginal revenue curve has a slope that is exactly twice the slope of the demand curve.

If marginal revenue is positive, then making extra units of the product will increase the total revenue. If it is negative, then making extra units will decrease the total revenue. Obviously, once marginal revenue comes equal to zero, no more units should be produced. This is the point where price elasticity is equal to 1.

Another way to write the equation for marginal revenue is:

MR = P * (1 -[1 / Price Elasticity])

**Maximizing Profit**

Maximizing profit and maximizing revenue are not the same thing. If the marginal cost of making one extra unit of a new product is $2 and the marginal revenue obtained by selling that product is $1, producing it does not make sense. The profit maximizing point is where marginal cost equals marginal revenue:

MC = MR

MC = P * (1 -[1 / Price Elasticity])

For example, if the marginal cost to product a product is $10 and the price elasticity of demand is 2,

10 = P * (1 – [1/2])

P = $20 should be the selling price

Note that, as the price elasticity increases, the optimal selling price decreases. So, if it costs $10 to produce a product and the price elasticity of demand is 4:

10 = P * (1 – [1/4])

P = $13.33 should be the selling price

Obviously, determining the price elasticity of demand is critical for a firm when setting the price of a product.

why is the price function price = a – bQ

couldnt it be anything? why chose a function with two paramters?

why is this simple formulation reasonable? how is it derived