Relationship between Average and Marginal Revenue Curves
Introduction
Cost and revenue are just like two different faces of the same coin. The costs and revenues of a firm determine its nature and the levels of profit. Cost refers to the expenses incurred by a producer for the production of a commodity. Revenue denotes the amount of income, which a firm receives by the sale of its output. The revenue concepts commonly used in economic are total revenue, average revenue and marginal revenue.
Total Revenue
Total revenue refers to the total sale proceeds of a firm by selling its total output at a given price. Mathematically TR = PQ, where TR = Total Revenue, P = Price, Q = Quantity sold. Suppose a firm sells 100 units of a product at the price of $5 each, the total revenue will be 100 × $5 = $500.
Average Revenue
Average revenue is the revenue per unit of the commodity sold. It is obtained by dividing the total revenue by the number of units sold. Mathematically AR = TR/Q; where AR = Average revenue, TR = Total revenue and Q = Quantity sold. In our example, average revenue is = 500/100 = $5. Thus, average revenue means price.
Marginal Revenue
Marginal revenue is the addition to total revenue by selling one more unit of the commodity.
Algebraically it is the total revenue earned by selling ‘n’ units of the commodity instead of n1. Thus,
MR_{n} = TR_{n} – TR_{n1}; where MR_{n} = Marginal revenue of the n^{th} unit
TR_{n} = Total revenue of n units
TR_{n1} = Total revenue of n1 units
N = Any given number of units sold.
Suppose 5 units of a product are sold at a revenue of $50 and 6 units are sold at a total revenue of $60. The marginal revenue will be $60  $50 = $10. It implies that the 6^{th} unit earns an additional income of $10.
Relationship between AR and MR curve
Let us consider the relationship between marginal, average and total revenue under pure completion and under imperfect competition.
1. Under Pure competition
Under pure (or perfect) competition, a very large number of firms are assumed to be present. The supply of each seller is just like a drop of water in a mighty ocean so that any increase or decrease in production by any one firm exerts no perceptible influence on the total supply and on the price in the market. The collective forces of demand and supply determine the price in the market so that only one price tends to prevail for the whole industry. Each firm has to take the market price as given and sell its quantity at the ruling market price. In simple terms, the firm is a ‘pricetaker’ and the firm’s demand curve is infinitely elastic. As the firm sells more and more at the given price, its total revenue will increase but the rate of increase in the total revenue will be constant, since AR = MR.
Table 1: Pure Competition
Q
 AR (P)
 TR
 MR


1
 10
 10
 10

2
 10
 20
 10

3
 10
 30
 10

4
 10
 40
 10

5
 10
 50
 10

6
 10
 60
 10

7
 10
 70
 10

In figure 1, OX – axis represents the number of units sold and OY axis represents the price per unit. The price of the unit remains constant at P_{1}. Consequently AR and MR curves coincide with each other.
2. Under Imperfect Competition (Monopoly)
Unlike under perfect competition, a firm under imperfect competition such as under monopoly can sell more only by lowering its price. Therefore, the average revenue curve is downward sloping and its corresponding marginal revenue curve lies below it.
Table 2: Monopoly
Q
 AR (P)
 TR
 MR


1
 10
 10
 10

2
 9
 18
 8

3
 8
 24
 6

4
 7
 28
 4

5
 6
 30
 2

6
 5
 30
 0

In table 2, AR falls by $1 at a time whereas MR falls by $2 at each stage.
In figure 2, OX – axis represents the number of units of the commodity sold. OY represents the price. The AR curve as well as the MR curve slope downwards. However, the rate of all in marginal revenue is double that of the fall of the average revenue.
How much is MR below AR?
(i) When MR and AR are straight lines and slope downwards
When AR and MR are straight lines, sloping downwards, the marginal revenue falls twice as much as the fall in the average revenue. In other words, the marginal revenue will cut any line perpendicular to the y – axis at halfway to the average revenue curve. This can be proved mathematically. In figure 3, AB = BC.
Total Revenue = Average Revenue × Output
CM × OM = Area of Rectangle ACMO
Also Total Revenue = Area under the marginal revenue curve = RDMO
Also ACMO = ABDMO + BCD and RDMO = ABDMO + RAB
Therefore, ABDMO + BCD = ABDMO + RAB
Or BCD = RAB
But RAB = BCD, being right angles
And RBA = CBD, being vertically opposite angles.
Thus, the two triangles are equal in area and BCD = RAB
Therefore, AB = BC
Hence, it is proved that marginal revenue curve will cut any line perpendicular to the Yaxis at halfway to the average revenue curve.
(ii) When the average revenue curve is convex to the origin
When the average revenue curve is convex to the origin, the marginal revenue curve cuts any perpendicular line to the Yaxis at more than halfway from the average revenue curve. This is shown in figure 4.
(iii) When the average revenue curve is concave to the origin
When the average revenue curve is concave to the origin, the marginal revenue curve cuts any perpendicular line to the Yaxis at less than halfway from the average revenue curve. This is shown in figure 5.
Price Elasticity, Average Revenue and Marginal Revenue
Mrs. Joan Robinson in her book ‘The Economics of Imperfect Competition’ has shown the empirical relationship between price elasticity, average revenue and marginal revenue.
The relationship is expressed in the formula.
AR = MR or MR = AR (e/(e1)); where, AR = Average Revenue, MR = Marginal Revenue and ‘e’ = price elasticity of demand.
In figure 6, AR and MR are the average revenue and the marginal revenue curves. Elasticity of demand at point R on the average revenue curve = RT/RS
In triangles PSR and MRT,
∟SPR = ∟RMT (right angles)
∟SRP = ∟RTM (corresponding angle)
Therefore, ∟PSR = ∟MRT
Therefore, triangles PSR and MRT are similar.
Hence, RT/RS = RM/SP (1)
Now in triangle PSK and KRQ,
PK = KR
∟PKS = ∟RKQ (vertically opposite angles)
∟SPK = ∟KRQ (right angles)
Therefore, triangles PSK and RQK are congruent.
Hence, PS = RQ (2)
From (1) and (2), we get,
Elasticity at R = (RT/RS) = (RM/SP) = (RM/RQ)
But RM/RQ = RM/(RMRQ)
But RM = Average revenue = price
QM = Marginal revenue
Elasticity at R = Average revenue/(Average revenue – Marginal revenue)
= AR/(ARMR)
If A stands for Average revenue, M stands for Marginal revenue and ‘e’ stands for elasticity on the average revenue curve, then e = A/(AM).
Therefore, e(AR) – e(MR) = AR
e(AR) – AR = e(MR)
AR = e(MR)/(e1)
AR = MR(e/(e1))
MR = AR((e1)/e)
A few examples:
 Suppose the price of a product is $6 and the elasticity of demand is 2. Marginal revenue will be MR = AR((e1)/e) = $6 × (21)/2 = $6 × (1/2) = $3.
 When the price of the product is $6 and price elasticity of demand is 1, marginal revenue will be MR = AR((e1)/e) = $6 × (11)/1 = $6 × 0 = 0.
If MR = 0, it is a case in which the MR curve coincides with the Xaxis.
Some Special Cases of Revenue Curves
Mrs. Joan Robinson has also pointed out many special cases of Marginal and average revenue curves.
Rectangular Hyperbola: If the demand for the firm’s product is unitary elastic (e = 1), then the average revenue will assume the form of a rectangular hyperbola. This limiting case is possible under pure monopoly where the monopoly product has no substitutes at all.
According to the formula MR = AR ((e1)/e)
Putting e=1, we have MR = AR ((11)/1) = AR × 0 = 0
Thus, when the elasticity of demand is equal to one or unity, though not the average revenue curve, the marginal revenue curve will be zero. Therefore, the marginal revenue curve coincides with the Xaxis.
Under Oligopoly: Oligopoly is a market where there are only few sellers. The demand curve of a firm under oligopoly is not supposed to be smooth. The demand curve has a kink at point P on the demand curve indicating the price policy of the firm. If the firm raises the price above this price (kink P), his rivals will not follow suit. Consequently, his sales and profit will suffer. On the contrary, if it lowers the price, the rival firms will retaliate by following the same action. Therefore, the firm cannot gain more by lowering the price. When there is a kink in the average revenue curve, the marginal revenue is discontinuous at the point of the kink. The gap in the marginal revenue depends upon the nature of the elasticity on the upper and lower portions of the kinked demand curve. This is shown in figure 7.
Significance of the Concept of Revenue
(a) In determining the nature of profit
The concepts of MR and AR both together constitute a powerful analytical tool in economic analysis. Average revenue is the price per unit of output. To find out whether the firm earns super normal profits or only normal profits or losses the following rule is followed. At the point of equilibrium –
 If AR is tangent to AC there will be normal profit
 If AR is above AC there will be super normal profits
 If AR is below AC there will be loss
 Helpful in decisionmaking
The concept is also vital in determining the equilibrium of a firm. The aim of every firm is to obtain maximum profits. The rule for profit maximization is MC = MR.
If MR > MC expansion in output will be profitable
If MR < MC expansion incurs loss
If MC = MR equilibrium output is attained
The intersection of MC = MR determines price, output, and the profit or loss of a firm.
(b) Concept of excess capacity
This concept is helpful to indicate to the entrepreneur whether the firm possesses excess capacity or not. Under perfect competition, production will be carried on up to the minimum point of the LAC. Therefore, excess capacity is not possible.
However, under imperfect competition (monopoly or monopolistic competition) the firm can earn more by reducing its output. So production will not be carried on up to the minimum point of the longrun average cost curve. Thus, imperfect competition leads to idle capacity. It is a wastage from the society’s point of view.
(c) FactorPricing
In fixing the prices of factors in the factor markets, AR and MR concepts are very useful. In factor pricing, the average revenue curve becomes the average revenue productivity curve, and marginal revenue curve becomes the marginal revenue productivity curve, ARP and MRP are inverted ‘U’ (bell Shaped) curves. The point of intersection of MFP and MFC (Marginal Factor Cost) determines the equilibrium level of price, output and profit for a firm under various cost conditions.
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