Profit Maximization Under Single Pricing (transcript)

Single-price searchers maximize profit by setting a uniform price where marginal revenue is equal to marginal cost.

Single pricing and total revenue

Most sellers of differentiated products are faced with a downward-sloping demand curve. Because their competitors do not sell perfect substitute products, they have some power to search for the profit-maximizing price.

Price searchers can set different prices for different buyers if they have sufficient market power. But most fixed-price sellers resort to single pricing.

Single pricing means the same price is applied to all units even though some buyers are willing to pay more. Therefore, a lower price needed to sell just one more unit must be offered for all units.

Because the same price applies to all units sold at any given level, total revenue (TR) is equal to price times quantity sold (TR = P * Q). Here TR is represented by the blue area.

Let's plot the total revenue curve (TR). The vertical distance in the upper panel simply maps the blue area below.

When demand is price elastic (i.e., when the percentage change in quantity demanded is larger than the percentage change in price), lowering price will generate larger total revenue.

In a straight-line downward sloping demand curve, the mid-point locates the price that maximizes total revenue.

After maximum total revenue is reached, further decrease in price would lead to a decline in total revenue because the percentage change in quantity demanded is smaller than the percentage change in price.

P > MR

Since a single-price searcher will not generally lower its price beyond the maximum revenue point, we can concentrate on the rising TR segment to see how lowering price affects the marginal revenue (MR). MR is the additional revenue generated when one additional unit is sold.

Because TR is increasing at a decreasing rate, the slope of its tangents (which represents MR) decreases when more is sold at lower prices.

MR < 0 after maximum total revenue point is reached and will not be shown here.

Because a lower price needed to sell just one more unit must be offered for all units,

the marginal revenue (MR) gained by the seller by lowering price is always less than the price (P) paid by the buyers (i.e., P > MR).

MR = MC

When a firm must choose its price to maximize profit, how does it determine the maximum profit output?

Assuming that we have some fixed inputs in the short run and that diminishing returns set in as more variable inputs are added, we will have an inverted S-shaped total cost curve TC.

Let's bring back ATC... and MC

The firm identifies the maximum profit output level where TR - TC is the greatest. This output level occurs where the slopes of the tangents to TC and TR are equal.

Since the slope of tangent to TR = MR and slope of the tangent to TC = MC, when slope of these two tangents are equal, MR = MC.

That means, we can directly determine the maximum profit output by the intersection of MR and MC.

Positive Profit

Maximum profit output occurs when MR = MC.

At Qπ, the firm will be able to sell all it produces at as indicated by the demand curve. the average profit is indicated by the gap between P and ATC i.e., P-ATC. Total profit is indicated by the rectangular area (P-ATC) * Q, i.e. average profit per unit times the number of units.

Observation #1

Maximum profit is represented by an area in the bottom panel because the vertical axis is based on per unit value.

Observation #2

Price for the maximum profit output Qπ cannot be set equal to MC (=MR) because the low price would create excess demand because more will be demanded than supplied.

And in this case, this low price leads to economic loss.

P = MC

At maximum profit output Qπ, the market clearing price is Pπ.

Total profit is indicated by the rectangular area i.e., (P-ATC) * Q.

At maximum profit output Qπ, Pπ > MC. If buyers are willing to pay more than the MC of producing Qπ, why not charge less or produce more?

The firm could expand output from Qπ to Q2 and charge a price P2 = MC, but the total profit would be smaller than when MR = MC.

No Profit

Because P > MR under single pricing with a downward-sloping demand curve

and MR = MC is the condition for profit maximization,

Therefore, P > MC when profit is maximized.

But P > MC does not guarantee positive profit.

Positive profit depends on the relative position of ATC to the demand curve.

If ATC is low compared to P, there could be positive profit.

When ATC is high compared to P due to higher fixed cost for example, the same demand condition can result in zero profit or negative profit.

Observation:

Since the change in TFC does not affect TVC, MC stays the same. Therefore, the maximum profit output can stay unchanged as long as the demand condition stays the same, (i.e., MR stays unchanged).