Unit Price Elasticity of demand and Total Revenue

CFA Level I Economics
#1

Hello All,

Wikipedia (http://en.wikipedia.org/wiki/Price_elasticity_of_demand) says that when price elasticity of demand = unitary, or =-1 for a -ve slope linear demand curve, the change in price won’t affect total revenue.

I am not sure how this is possible. For instance, let’s consider the following example:

Demand Curve: P = A - 0.01*Q, with A=6

Now, given unit elasticity, the price = $3 and Quantity = 300.

If I increase my price by 33.33% from $3 to $4, then the new quantity demanded will be 200.

Hence, total revenue before the price change = $3*300 = $900; while the total revenue after the change = $4*$200 = $800.

Hence, total change in revenue = -11.11%. This is pretty significant.

Can someone please help me ? I am not sure what I am missing here? These basic questions are really tricky. I am really stuck.

Thanks in advance.

#2

Instead of increasing your price by 33%, increase your price by 1%, or 0.1%, or 0.01%.

When you change the price by a large amount, you’re no longer concerned with the elasticity at a particular point; you’re interested in the average elasticity over the entire range of prices. While the elasticity is -1 when P = $3.00, it’s not -1 when P = $3.10, or $3.50, or $3.75, or $4.00.

The slope of the parabola y = _x_² is zero when x = 0, but that doesn’t mean that y won’t change when we change x to, say, 5. It means only that _ at _ x = 0, the slope is zero: if we change x to, say, 0.01, y doesn’t change very much.

#3

Thanks S2000magician for your response. I see your point. However, wouldn’t we get similar effect, i.e. small increase/decrease in revenue, by increasing/decreasing price when elasticity is not equal to -1?

I am still not fully clear. How can we conclusively the direction of the change in total revenue, given a specific elasticity?

#4

Let’s see what happens with the demand curve you gave:

P = 6 – 0.01Q

Then revenue is:

R = PQ = (6 – 0.01Q)Q

= 6Q – 0.01Q²

As you point out, revenue is maximized at the point of unitary elasticity: ($3.00, 300), where:

R = $3.00(300) = $900.00.

Let’s see what happens to revenue when we change the quantity to 299, and to 301:

P(299) = 6 – 0.01(299) = $3.01

P(301) = 6 – 0.01(301) = $2.99

R(299) = $3.01(299) = $899.99

R(301) = $2.99(301) = $899.99.

Thus, at the point of unitary elasticity, any change in quantity (or in price) – up or down – will cause total revenue to decrease.

Let’s see what happens around Q = 200 (where P = $4.00, and R = $800), changing the quantity to 199, and to 201:

P(199) = 6 – 0.01(199) = $4.01

P(201) = 6 – 0.01(201) = $3.99

R(199) = $4.01(199) = $797.99

R(201) = $3.99(201) = $801.99.

So when we decrease the quantity (increase the price) the revenue decreases, but when we increase the quantity (decrease the price) the revenue increases. The percentage change in quantity is greater than the percentage change in price, so this is the elastic portion of the demand curve.

Let’s see what happens around Q = 400 (where P = $2.00, and R = $800), changing the quantity to 399, and to 401:

P(399) = 6 – 0.01(399) = $2.01

P(401) = 6 – 0.01(401) = $1.99

R(399) = $2.01(399) = $801.99

R(401) = $1.99(401) = $797.99.

So when we decrease the quantity (increase the price) the revenue increases, but when we increase the quantity (decrease the price) the revenue decreases. The percentage change in quantity is less than the percentage change in price, so this is the inelastic portion of the demand curve.

#5

Thank you so much S2000magician for your efforts in explaining this to me in detail. I really appreciate it.

Utmost regards,

Allalongthewatchtower