Profit Maximization Under Natural Monopoly (transcript)

Natural monopoly with decreasing average total cost can still make profit by equating marginal revenue with marginal cost while achieving economic efficiency through price discrimination.

Profit is maximized when marginal revenue (MR) from selling the product is equal to marginal cost (MC) of producing it.

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Profit maximization

MR = MC

But marginal revenue (MR) depends on how pricing is done.

We will compare two modes of pricing, namely single pricing vs perfect price discrimination.

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Single pricing vs perfect price discrimination

And marginal cost (MC) depends on whether the firm is producing under increasing returns or diminishing returns.

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Increasing returns vs diminishing returns.

Specifically, we will talk about the cost curves of natural monopoly where one firm is big enough to dominate the whole market.

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Natural monopoly

TR

L

Single price (Psp) applies to all units sold.

Therefore Psp is an average price.

i.e., every buyer pays the same price even though some are willing to pay more.

R

Discriminating price (Pdp) applies only to the marginal unit sold.

Therefore Pdp price is a marginal price.

i.e., every unit is sold at a different highest possible price.

L

When a single-pricing firm faces a downward sloping demand curve, it must search for the maximum profit price. e.g., a firm in monopoly or monopolistic competition is such a price searcher

As the price searcher lowers its price, total revenue will first increase when sales increases faster than price decreases.

i.e., elasticity of demand > | 1 |

TR (total revenue) would reach a maximum when the sales increases no faster than price decreases.

i.e., elasticity of demand = | 1 |

Then TR decreases as sales increases slower than price decreases.

i.e., elasticity of demand < | 1 |

R

If the firm can sell each unit of output according to the buyer's willingness to pay (i.e., reservation price), the total revenue generated will be much larger and reaches its maximum when price charged is equal to zero.

This total revenue generated through perfect price discrimination can be called total willingness to pay (TWP)

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Observation #1: Because single price is an average price, single-pricing seller's TR is affected by the elasticity of demand. TR maxs out when elasticity = 1.

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Observation #2: Because discriminating price is a marginal price, price-discriminating seller's TR (i.e., TWP) maxs out only when price = 0.

MR

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From the single-pricer's TR curve, we can derive the marginal revenue (MRsp) at any output level from the slope of TR.

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From the discriminating pricer's TR curve (TWP), we can derive the marginal revenue (MRdp) at any output level from the slope of TWP.

Observation #1: MRdp is the same as the marginal willingness to pay (MWP).

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Observation #2: TR reaches its maximum when MRsp = 0. This happens mid-point down the straight-line demand curve.

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Observation #3: TWP reaches its maximum when MRdp = 0. This happens at the end of the demand (=MWP) curve.

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Observation #4: Under single pricing, MR < P because price must be lowered for all units in order to sell one more unit.

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Observation #5: Under discriminating pricing, MR (= MWP) = P because each unit is sold at a different price.

ATC

The U-shaped ATC and MC curves are typical of firms under diminishing returns where the optimal capacity of the fixed input is soon reached, leading to high variable costs.

In some cases, higher fixed cost can be traded off for lower variable costs. Think computer software where the high fixed cost of development leads to very low variable cost of duplicating the codes.

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L

When fixed cost is very high and variable cost is very low, ATC will not reach its minimum until a very high output level. A natural monopoly results since the market is only big enough for one firm.

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Natural monopoly

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R

Let's keep the low fixed cost and high variable cost model on the right panel for comparison with the natural-monopoly cost curves on the left panel.

L and R

Observation #1: Fixed cost is very high with the natural-monopoly compared to the low fixed cost model.

Observation #2: Variable cost increases very slowly with the natural monopoly compared to the low fixed cost model.

L

Observation #3: MC is below ATC persistently under natural monopoly.

L and R

Observation #4: For the same high output level, natural-monopoly ATC is lower than the low fixed-cost model.

Profit Maximization

With these 2 sets of cost curves, we can now compare profit maximization under single pricing vs perfect price discrimination.

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The single-pricing monopolist will maximize profit by setting MRsp = MC.

At MRsp = MC, output is too low to take advantage of the scale economy and profit (TR=TC) is too low even though price is quite high.

L and R

But, a price-discriminating monopolist with the same cost curves would be able to produce much higher output level (Qdp) with a much higher profit by setting MRdp (= Pdp) = MC.

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MRdp (= Pdp) = MC.

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Observation #1: The price-discriminating firm can achieve higher output and higher profit because its TR (=TWP) reflects full marginal price for every unit...

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MRdp = Pdp

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while the TR of the single-pricing firm reflects the same average price for every unit.

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P > MRsp

Regulation

Observation #2: The maximum profit output (Qsp) of the single pricing monopoly is too low to exploit the maximum-efficiency output (P=MC) of a declining ATC at Qsp2.

Observation #3: The single-pricing firm cannot produce at Qsp2 without incurring a substantial loss.

Therefore, regulator cannot require natural monopoly to charge a price equal to MC.

Observation #4: The most the regulator can do is to require the natural monopoly to set a price equal to ATC.

But profit is zero when P = ATC.

Observation #5: At zero-profit output Qsp3, the output level is still below the maximum-efficiency output at Qsp2 where P = MC.

L and R

Observation #6: Given the same cost curves, the price-discriminating monopoly would naturally produce at the maximum-efficiency output (P = MC) because its MRdp = Pdp.

Observation #7: Even if ATC is entirely above the demand curve where not even a single-pricing monopoly would want to produce, ...

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ATC > P

R

perfect price discrimination could still ensure profit for the natural monopoly without any government regulation.