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Here is an elaborated discussion on the relationship between price, marginal revenue and price elasticity demand.

The monopolist follows the same basic principle of profit maximisation that the competition firm uses- produce that output where marginal cost and marginal revenue are equal. In fact, the major difference between the monopolist and the competitive firm lies in the difference between their revenue functions.

We know that marginal revenue and price are identical for the competitive firm. It is supply and demand that together determine market price and, as a price taker, a competitive firm faces a perfectly elastic demand at that market price. Since its output increases total revenue by a constant amount, that is equal to the price. Thus, a competitive firm’s marginal revenue is its price. This, however, is not true for the monopolist.

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Since the monopolist is the only producer, the industry demand curve and the firm demand curve are one and the same. In fact, a monopolist’s output decision will have a large impact upon price, and the monopolist is well aware of that fact. The consequences of this can be seen in Fig. 22.1.

At a quantity of three units, consumers are willing to pay Rs. 14 per unit. In order for the monopolist to expand output and sale by one unit he must lower Z price of all units from Rs. 14 to Rs. 12. Clearly, the effect of expanding output is to reduce the price as the monopolist moves down along the demand curve.

Since marginal revenue is defined to be the change in total revenue resulting from a one unit change in output, this means that marginal revenue will be less than the price. To see this, we shall consider what happens to total revenue when the firm changes output from 3 to 4 units. At Rs. 14 per unit, the sale of three units of output generated a total revenue of Rs. 42.

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When the manager of a monopoly firm expands his output to 4 units, price falls to Rs. 12 and total revenue is Rs. 48. Marginal revenue is the difference in total revenue at 3 units of output and at 4 units of output, which is Rs. 48 – 42 = Rs. 6. It is less than the price which is Rs. 12. Fig. 22.1 shows that the sale of the additional output increase total revenue by the new price, which is Rs. 12.

But the necessary Rs. 2 reduction in price causes a loss in revenue of Rs. 2 times the original quantity of three units, which is Rs 6. On balance, total revenue increases by only Rs. 6, and this is less than the price of Rs. 12.

**An Expression of Marginal Revenue: **

We can derive a more general expression for the monopolist’s marginal revenue. Total revenue equals price times quantity, and marginal revenue equals the change in total revenue that accompanies a one unit change in quantity, or;

MR = Δ (PQ)/ΔQ

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= P (ΔQ/ΔQ) + Q (ΔP/ΔQ)

= P + Q (ΔP/ΔQ) … … (1)

Here ΔP/ΔQ is the slope of the demand curve and, therefore, is a negative number. Thus, marginal revenue is less than price. We can use this expression to identify the revenue gains and losses in Fig. 22.1: P represents the revenue gain of Rs. 12, whereas Q ΔP/ΔQ represents the revenue loss of Rs. 6. In this case, Q = 3 and ΔP/ΔQ = – Rs. 2.

The relationship between the monopolist’s marginal revenue and price (i.e., average revenue) is reflected in the price elasticity of the industry demand curve. Since P/P = 1, we can write equation (1) as;

Clearly marginal revenue equals zero if the price elasticity equals one. The increase in total revenue resulting from an increase in quantity is exactly offset by the reduction in total revenue resulting from the accompanying fall in price.

Selling more output increases total revenue only if marginal revenue is positive, which occurs when the industry price elasticity exceeds one. As the price elasticity rises, marginal revenue gets closer to price.

**Significance of Elasticity of Demand at Equilibrium under Monopoly:**

It may be noted that a profit-making monopolist always operates on the elastic part of the demand curve. The reason is that if it is on the elastic part of its demand (AR) curve, price cut will lead to an increase in its total revenue and marginal revenue will be positive. But if it operates on the inelastic part price cut will lead to a fall in total revenue and marginal revenue will be negative. Such a situation cannot be considered economically feasible.

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No firms wants to lose revenue by selling more. Thus the monopolist will not operate on the inelastic part of its demand curve. Furthermore, profit-seeking monopolist, like any profit seeking firm, always attains equilibrium by equating MR with MC. We know that MC is always positive.

Therefore, MR has also to be positive. Otherwise the equality between the two cannot be ensured. This means that the monopolist has also to operate on the elastic part of its demand curve as Fig. 22.2 shows. Moreover, a monopolist cannot operate at the point of its demand curve at which demand is unitary price elastic (E_{p} = 1).

At this point, AR is constant and MR is zero. So this possibility is also ruled out. It, therefore, follows that a profit-seeking monopolist will always operate on the elastic part of its demand curve. In this context we may refer to the inverse elasticity rule.

**Inverse Elasticity Rule****: **

**If the monopolist knows his marginal cost (MC) and price elasticity of demand (E _{p}), it should set price (P) such that: **

(P – MC)/P = 1/E_{p}

The left hand side is the mark-up of price over marginal cost expressed as percentage of price. The expression shows that to maximise profit, the price mark-up should equal the inverse of the demand elasticity.

**The smaller the price elasticity of demand, the greater the price mark-up: **

P = MC/ [1 – (1 /E_{p})]

If the monopolist knows his demand elasticity and marginal cost, the foregoing expression can be used to calculate its profit-maximising price.

We know that the change in total revenue (ΔTR) associated with a change in quantity sold (ΔQ) is equal to area B minus area A. Area B equals P (ΔQ) and area A equals Q (ΔP). Thus;

ΔTR = P (ΔP) + Q (ΔP) … … (4)

**Since ΔTR/ΔQ is marginal revenue, dividing (9.4) by ΔQ yields: **

MR = P + (ΔP/ΔQ) Q … … (5)

We know that the elasticity of demand E_{P} equals (when it is expressed as a positive number) (ΔQ/Q)/ (ΔP/P), ΔP/ΔQ equals (-1/E_{P})(P/Q).

**Substituting (-1 IE _{p}) (P/Q) for ΔP/ΔQ in (6) produces: **

MR = P + Q [(-1/E_{p}) (P/Q) = P – (P/E_{P}) = P [1 – (1/E_{p}) … … (6)

At the profit-maximising output, MC = MR, so

MC = P [1 – (1/E_{p})] … … (7)

**Subtracting P from both sides of equation (7) and then multiplying through by (1/P) yields: **

(P – MC)/P = 1/E_{P}

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