Revenue, Relationship of Revenues with Price Elasticity of Demand

Revenue (Too incomplete to underatand)

Three types of revenue exist:

• Total Revenue: Total revenue, often known as gross revenue, is the sum of all a company's sales-related receipts. There are two methods for estimating total revenue:

  • It is obtained by multiplying total sales quantity (or output) with the per unit price. Thus, TR = Q × P where, TR = Total Revenue, Q = Total Sales Quantity or Output and P = Per unit price.
  • It is the aggregate of marginal revenues. Thus, TR = ∑MR = ∑ (MR1 + MR2 + …………. + MRn) Where TR = Total Revenue, ∑ = Summation, MR = Marginal Revenue, MR1 = Marginal revenue of 1st unit of output, MR2 = Marginal revenue of 2nd unit of output and MRn = Marginal revenue of nth unit of output.

• Average Revenue: It is the outcome of the total revenue divided by the total output. In other words, it is the per unit revenue. Thus, AR = TR/Q or AR = (P × Q) / Q = P. Therefore, AR = Price where, AR = Average Revenue, TR = Total Revenue, Q = Total output, P = Price 

• Marginal Revenue: Marginal Revenue is defined as the addition made to the total revenue by selling one more unit of the output. In other words, it is the ratio of change in the total revenue with a change in the total sales quantity (by selling one more unit of output). It reflects the rate of change in total revenue.

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MR = ΔTR /ΔQ or MR = TR(n) – TR(n-1)

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where, MR = Marginal Revenue, ΔTR = change in total revenue, ΔQ = change in output,

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TR(n) = Total revenue of ‘n’ units and TR(n-1) = Total revenue from (n-1) units,

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Suppose, a producer sells 900 kg at a per unit price of Rs.10/-

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Then, TR = P × Q = 10 × 900 = 9000,

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AR = TR / Q = 9000/900 = 10

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Again,

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Then, MR = ΔTR / ΔQ = 10/1 = 10
Or, MR = TRn – TRn-1 = 9010 – 9000 = 10

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Relationship of Revenues with Price elasticity of demand

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There is a very useful relationship between price elasticity of demand, average revenue and marginal revenue at any level of output. It is stated above that the average revenue of a firm is a really same thing as the demand curve of consumers for the firm’s product. Therefore, the price elasticity of demand on a consumer’s demand curve is the same thing as the elasticity of demand on the given point on the firm’s average revenue curve. It is seen in the figure that price elasticity of demand at point C on the average revenue curve AB = CB / CA. To study the relationship between average revenue, marginal revenue and price elasticity of demand at any level of output, we have to compute the price elasticity.

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Fig 1: Revenue and Price elasticity of demand

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Price elasticity of demand at point C on the average revenue ( or demand ) curve = CB / CA.

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Following conclusions have been drawn in this regard:
a. When price elasticity of demand is greater than one, MR is positive and TR is increasing.
b. When price elasticity of demand is less than one, MR is negative and TR is decreasing.
c. When price elasticity of demand is equal to one, MR is equal to zero and TR is maximum and constant.

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The relationship between TR, AR, MR and price elasticity of demand can be proved with the help of the following schedule.

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According to the above schedule,

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• At the output range of 1 to 5 units, the price elasticity of demand is greater than one according to total outlay method. Hence, TR is increasing and MR is positive.
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• At the output range of 5 to 6 units, the price elasticity of demand is equal to one. Hence, TR is maximum and MR equals to zero.At the output range of 6 units to 10 units,
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• At the output range of 6 units to 10 units, the price elasticity of demand is less than unity. Hence, TR is decreasing and MR is negative.
   
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Based on point elasticity,

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• At the output range of 1 to 5 units, point price elasticity on linear demand curve or AR curve is greater than unity. TR curve slopes upwards to the right and bends to the X–axis indicating that it increases at a decreasing rate. MR curve slopes downwards to the right indicating that MR decreases but remains positive.
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• At output equal to 6 units, point price elasticity is equal to unity on the midpoint of a linear demand curve. TR curve is parallel to X – axis. It implies that TR is maximum and constant. MR curve touches X–axis. It implies that MR becomes zero.
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• At the output range of 6 to 10 units, point price elasticity on linear demand curve is less than unity. TR curve slopes downwards to the right indicating that it is decreasing. MR curve slopes negative (or lies on negative Y–axis) indicating that MR is decreasing but becomes negative.
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Fig 2: Revenue and Price elasticity of demand

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A seller never produces (or sells) output in the inelastic range of its demand curve. It is because when price elasticity < 1, TR decreases and MR becomes negative (i.e. firm has to bear loss).

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Example:
1. Let, AR = 10, MR = 5. Compute price elasticity of demand
Since, price elasticity = AR / (AR – MR)

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Therefore, Price elasticity = 10 / (10 – 5) = 2
Price elasticity of demand is greater than one.

2. Let, AR = 30, price elasticity of demand = 3. Compute MR.
Since, MR = AR [(e-1) / e]
Therefore, MR = 30 [(3-1) / 3] = 5

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3. Let, price elasticity = 0.5, MR = -10. Compute price.
Since, price = AR = MR [e/(e-1)]
Therefore price = -10 [0.5/(0.5-1)] = 10

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Reference

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Koutosoyianis, A (1979), Modern Microeconomics, London Macmillan