Mass Market Pricing How to price to markets with large numbers of consumers
Looking forward ... • Concept of mass market demand • Relationship between revenue and demand • Profit maximising price levels • Concept of elasticity • Innovative pricing
Playing games with consumers (High Margin, Low Consumer Surplus) Buy Consumer Not ($0, $0) Firm Choose Price (same for all) (Low Margin, High Consumer Surplus) Buy Consumer Not ($0, $0)
Roll back Few sales with high margins Firm Choose Price (same for all) Lots of sales with low margins
Posted prices versus negotiation • In mass markets, only price-posting may be feasible It’s like making a take-it-or-leave-it offer! • Consumers whose WTP > p purchase from you • If all consumers’ WTPs equal, can extract all the surplus • But, if WTPs are different, same p can’t extract all the surplus
When is price-posting reasonable? • Price haggling too costly relative to value of product • Large numbers of customers bargaining is inefficient • Information requirements too demanding • Do you know buyer’s WTP, in a big anonymous market? • Markets are perfectly competitive • Bargaining not an issue anyway • Price-posting should be efficient • Usually, firms post (set) prices
Puzzle: Luxury Boxes • Among the many decisions made by sports stadium designers is the number of luxury boxes to build • Suppose that, for a particular stadium under construction, luxury boxes will be sold outright to local businesses and can be constructed at a cost of $300,000 apiece. The stadium designer plans to build 25 boxes and expects, at this number, to sell each for $1 million, for a net profit of $700,000 x 25 = $17.5m. • An associate asserts that this is crazy. Since the box can be built for $300,000 and sold for around $1m apiece, building only 25 leaves money on the table, even if a small price reduction is needed if more are built. • Is the associate correct? What if the price to sell 26 is $950,000?
Demand curves • In mass markets, relevant consumer information summarised by a demand curve • A demand curve identifies how many units of product sell at any posted price • The market demand curve is derived from the WTPs of all potential consumers
Numerical Example • 1,000 potential buyers (only interested in a single unit each). Have different WTP ranging from $0 to $999 • Monopoly seller with constant per-unit cost of $200, • Sufficient capacity to supply everyone • Question: what quantity maximises monopolist profit?
Pooled pricing: “one price fits all” • Assume same price charged to everyone • This is called “pooled” pricing • Reasonable in markets with resale and arbitrage • Customers cannot pay different prices • Otherwise, ones with low price can re-sell (at profit) to others • Later, we relax this assumption • The monopolist faces a tradeoff • Prices can be increased, • But only at the expense of lower sales volume (and, visa versa)
What quantity do you choose? • You have a monopoly in this market • But, can you do whatever you want? • For example, can you sell 800 units at a price of $700? • NO • You can sell 800 units at $200 per unit • Or you can price at $700, and sell 300 units • You can choose price OR quantity, but not both! • Quantity choices imply prices • Price choices imply quantities • Pick one or the other(we’ll assume you choose quantity)
Changes in total revenue (TR) • Consider TR as you increase quantity 1 unit at a time • To sell 1 unit, price = $999, TR = $999 • To sell 2 units, price = $998, TR = $998*2 = $1,996 • Notice that, going from 1 unit to 2 units • You have to drop price, which reduces what you get on unit 1 • You were getting $999 • Now you get $998 • But, you get to sell an additional unit at $998 • Dropping price, you lose $1 TR on unit 1 but gain $998 on unit 2 • The net effect is an increase in TR of $1,996 – $999 = +$997 • As quantity increases 1 unit at a time • You lose more and more TR from lower prices on previous units • You gain less and less on the additional unit sold • At some point, the losses exceed the gains!
Graphically: TR from 4 to 5 units The change in TR going from 4 to 5 units is $991 = 995 – 4 To sell 4 units, price = $996 To sell 5 units, price = $995 P = $996 P = $995 Lose $1 per unit on 4 units TR = area of rectangle = 996*4 = 3984 Gain$995on5th unit
Marginal revenue • The marginal revenue (MR) of the nth unit is the change in total revenue induced by going from the (n – 1)th unit • In our example Here, WTP is low ($600) and the lost revenue on previous units required to get the sale is large ($399) – net effect $201
MR = $1 MR = – $1 MR = – $3 MR = $3 Graphically MR = change in TR • The marginal revenue of the 500th unit is $1 • The marginal revenue of the 502th unit is – $3
Total cost (TC) versus marginal cost (MC) • Costs can be analysed in exactly the same way • The MC of the nth unit is the change in TC induced by moving production from (n – 1) units to n units • If you supply 100 units, MC = additional cost of supplying 100 instead of 99 • In this example, each additional unit costs $200 MC = $200 for any level of production
Decision rule: MR = MC • Monopolist should produce as long as doing so is profitable • The monopolist wants to continue to expand sales up to the point where the last unit sold adds just enough TR to offset its effect on TC • Obviously, if the overall effect on TR does not offset the overall effect on TC, don’t produce it! • Profit is maximised by producing 1 unit less than the first unit at which MR – MC < 0
Maximising Profit • Marginal Profit = (Revenue from another unit) - (cost of another unit) = Marginal Revenue - Marginal Cost • When should monopolist supply one more unit? • Whenever the marginal profit is positive • That is, until marginal profit just-above or equal to 0 Produce so long as MR MC!
Stopping rule for our example? • Each additional unit yields less incremental revenue • But each additional unit costs $200 • Each additional unit increases your total costs by $200 • Monopolist has constant marginal cost of $200 • If you earn more than $200 in revenue from increasing output by one more unit, do it! • If you earn less than $200, don’t! Stop when Marginal Revenue is $200
A Snapshot MR – MC goes negative here Stop here –1 +1
This can also be solved graphically Profit maximising price Point where MR = MC
Restricting supply • Motive for restricting supply, under bargaining: = reducing the bargaining power of buyers • Motive for restricting supply, under posted pricing = capturing more value from buyers with high WTP
Marginal Thinking Very powerful: right way to approach any decision in which • You can make choices in small increments • Each increment brings less benefit Look at the last increment: is it worth it? • Should I study 8 or 9 hours, today? • Should I eat another spoonful of ice cream? Marginal thinking, for the firm: Should I produce one more unit?
Calculus “shortcut” • Working through these calculations using the brute-force approach (i.e., unit-by-unit) is tedious, time-consuming and prone to error • We can get very good approximations using calculus • Cost: learn appropriate calculus rules • Benefit: huge reduction in time spent working unit-by-unit cases • For those who remember their calculus, cost = 0 • To proceed, we need to make some simplifying assumptions
A necessary assumption • To use calculus, must assume price-quantity relationship is “smooth” • That is, must assume the relationship • P = 1000 – Q • holds for all quantities – even non-integer amounts • The equation above, called the inversedemand curve, describes price as a function of quantity • Demand is actually “lumpy” • At P = $999.00, sell 1 units • At P = $998.50, sell 1 unit! P = 1000 – Q • In large markets, it is simpler (and close enough)to assume smooth demand • At P = $999.00, sell 1 units • At P = $998.50, sell 1.5 units
Demand & inverse demand curves • The equation describing price as a function of quantityis called the inversedemand curve • The sister equation, describing quantity as a function of price, is called the demand curve • If we wish to treat quantity as the choice variable (we do), use inverse demand curve • Key difference between lumpy and smooth demand: • Tiny change in quantity tiny change in price • Indeed, changes can be infinitesimal • When quantities are in the millions, increasing demand one unit is close to an infinitesimal change • It may help to imagine the product is something divisible into arbitrary quantities, like petrol P = 1000 – Q
$250,000 Total Revenue(smooth version) Smooth TR curve Once we assume price is a smooth function of quantity, TR is also a smooth function of quantity • TR = Price x Quantity = PQ • So, substitute (1000 – Q) in for price (from inverse demand equation) to get TR as a function of quantity only • TR = (1000 – Q)Q = 1000Q – Q2 1000Q – Q2
$250,000 $1 $249,999 1 unit Marginal revenue of 500th unit MR of 500th unit = change in TR, 499 to 500 units = $1 NOTE: 1 = slope of the line through these points on the TR curve (slope = “rise over run”) But, with smooth TR, quantity increments can be much smaller than 1 unit … … and, this is useful! TotalRevenue
Maxima of smooth functions The slope of a curve at a point is its instantaneous rate of change at that point(e.g., the change in TR for a minuscule change in quantity) $250,000 Notice anything special about this line? $249,999 FACT: The maximum of a function occurs at the point where its slope = 0 FACT: The instantaneous rate of change in TR at a given quantity is the slope of the line tangent to the TR curve at that quantity TotalRevenue
Calculus: the management summary • The derivative of a curve at a particular point is the slope of the curve at that point • To use calculus, you need to know the rule for finding the derivative • Calculus knowledge for Man Ec • If y is a function of x, the “derivative of y with respect to x,” denoted y/ x, is the slope of the function – the rate at which y changes with x • The only calculus rule you ever need (in this class) is that curves of the form where a, b and c are constants (may be positive, zero, or negative) • Have derivatives of the form • So, plug a number in for x to determine the slope of y(x) at that value of x
Why calculus is useful TR = PQ = 1000Q - Q 2 • To find the slope at any Q, write down the derivative using the “only rule you will ever need” • This tells you the slope of the TR curve for any value of Q • Total revenue hits its max when the slope = 0
MR and MC with smooth functions • MR at a specific Q is equal to the value of the derivative at the quantity • From before, • E.g., MR at Q = 300 is 1000 – 2*300 = 400 • The same idea applies to costs • TC = 200Q, so, using the “only rule you will ever need” • MC at Q = 300 is 200 • In this case, marginal costs are constant (the same at any level of production)
Firm objective: maximise profit • Now, let’s apply the procedure to find maximum profit (what the monopolist really cares about!) • Monopolist wishes to choose optimum Q • Profit = TR – TC = (1000Q – Q2) – 200Q = 800Q – Q2 • Apply the calculus rule to get • Calculate where the slope equals 0 Notice: same as the answerobtained using brute-force
Graphically Same principle as before Profit = 1000Q – Q2 Q*
This can be stated as a “marginal rule” • Marginal profit = 800 – 2Q = (1000 – 2Q) – 200 • In other words, marginal profit = MR – MC • MR = 1000 – 2Q • MC = 200 • To maximise profit, set MR = MC • At the ideal output, Q*: 1000 - 2Q* = 200 or Q* = 400 • To find ideal price, substitute Q* into the inverse demand function: P* = 1000 – Q* = $600.
Exercise: marginal cost • Same demand curve as before, but now the cost of production is increasing: Total Cost = 200Q + Q2. • What is the cost of producing 30 units? 31 units? What is the marginal cost of the 31st unit? • Calculate Marginal Cost, using derivatives. Is your answer for the marginal cost of the 31st unit approximately correct? • How much does the monopolist choose to produce? • Now suppose that you can sell as much as you want to on the US market, for $900 apiece, but demand in the Australian market is (1000 – P). What do you do?
Why split profits into MR and MC? • Marginal analysis is a powerful tool, beyond the case of one factory producing for one market • If selling in two markets: • Equalise the MRs • Why? Imagine you sell a fixed quantity (say 180) • If MR higher in AU, do better by moving one unit from US to AU • This gradually pushes MR down in Australia. • Now choose total quantity: set MR = MC • Double-check that you want to sell in both markets!
Similar: plant production levels • Assume 2 factories, 1 in Melbourne and 1 in Geelong: Total Cost in Factory M = 200QM Total Cost in Factory G = 150QG + QG2 • Same demand curve as before (can only sell in AU) • How much do you produce in each factory? • How about if the total cost in Factory G = 300QG + QG2?
Solution: same idea as before The marginal revenue and marginal cost equations (for each plant) are: Set MC = MR in both plants, solve for optimal quantities (2 eqs, 2 unknowns): When the marginal cost in factory G is 300 + 2QG
Elasticity A convenient measure of sensitivity for use in pricing
Your demand curve in ...the REAL world! • Algebraic analysis is useful for building general intuition • But not for setting prices in the real world • Usually, you don’t know the whole of your demand curve • You do know at least one point on your demand curve = demand at the current price • You can experiment with slightly higher prices and slightly lower prices, to see how demand changes • If you also know your marginal cost, that’s enough to figure out if you should move your price up or down
Elasticity • In determining Q & P, it is important to know how sensitive demand is to price changes. • If it is relatively insensitive, then by raising price the monopolist does not exclude many buyers • If it is relatively sensitive, raising price can exclude many buyers • The measure of how sensitive a demand function is to a change in price = “elasticity” • Prices are higher in markets with less sensitive demand (“less elastic”) • In our example: AU market more inelastic than US • Price higher in Australia
Perfectly Elastic Demand Price Demand Quantity
Perfectly Inelastic Demand Price Demand Quantity
Calculating Elasticity • Price elasticity of demand is the percentage change in quantity demanded divided by a given percentage change in price • More often we use the point elasticity of demand = elasticity for a “minuscule” percentage change in price (derivative of quantity with respect to price, times price divided by quantity)
Some Properties of Elasticity • e is a negative number: e.g., if 10% increase in price of oil decreases quantity by 20%, e = – 2 • “More elastic” means “Bigger in absolute value” e.g., if eUS= – 2 and eAU= – 10, AU demand is more elastic • Unit-Free Measure • you can compare elasticities among different goods • Is oil more price sensitive than butter, at their current prices? • Elasticity vs. Slope • These are not the same thing • Slope is P/Q
Estimated Price Elasticities Elasticities calculated at current market prices
Accounting for Differences • Degree of Substitutability • Temporary vs. Permanent Price Changes • Long-run vs. Short-run elasticity