Lecture 1: Optimal Pricing for Monopoly with Multiple Goods Jacob LaRiviere Justin Rao 1 maxπ₯1,π₯2,β¦,π₯π π π₯1 , π₯2 , β¦ , π₯π Composite Commodity Theory Assume there are n goods (e.g., apples, bananas, carrots, etcβ¦) but we really only care about one of them (e.g., apples). How do we handle this problem as economists? Lets start with the very general consumerβs problem: constrained maximization. Needed assumptions are completeness, reflexivity and transitivity. π . π‘. π1 β π₯1 + β― + ππ β π₯π β€ πΌ Suggestion: Say that we only solved this optimization problem for goods 2, β¦, n. Take the first good as a parameter [e.g., like βmβ in y(x)=mx+b] max π₯2,β¦,π₯π π π₯2 , β¦ , π₯π ; π₯1 π . π‘. π2 β π₯2 + β― + ππ β π₯π β€ πΌα This leads to a bunch of solution functions for all of the parametersβ¦ π₯2β π2 , β¦ , ππ , πΌα , β¦ , π₯πβ π2 , β¦ , ππ , πΌα α weβre netting out expenditures on goods 2, 3, β¦, n. NOTE: I now is πΌ; π₯2β π2 , β¦ , ππ , πΌα , β¦ , π₯πβ π2 , β¦ , ππ , πΌα -> π π₯1 , π₯2 , β¦ , π₯π Composite Commodity Theory Idea: with these βsolution functionsβ for the goods we donβt care about, lets plug them back in to the original utility functionβ¦ Call this new function βVβ and let x1 vary again: π π₯1 , π₯2 , β¦ , π₯π = π π₯1 , π₯2β π2 , β¦ , ππ , πΌα , β¦ , π₯πβ π2 , β¦ , ππ , πΌα α π2 , β¦ , ππ = π π₯1 , πΌ; This reduces the problem to a function of a bunch of parameters (e.g., m and b) rather than variables (e.g., y and x). This is useful; parameters arenβt complicated but variables are! Noting that the prices of goods 2, 3, β¦, n are fixed, consider maximizing V α π2 , β¦ , ππ maxπ₯1,πΌα π π₯1 , πΌ; s.t. π1 β π₯1 + πΌα β€ πΌ maxπ₯1,π₯2 π π₯1 , π₯2 Composite Commodity Theory We can call πΌα the composite commodity and evaluate βeverything we donβt care aboutβ as it and now call it x2. NOTE: Effectively weβve normalized the cost of the composite commodity to $1 s.t. π1 β π₯1 + π₯2 β€ πΌ This problem has a solution where for different levels of p1 and I there are different solutions for π₯1 and π₯2 . π₯1β π1 , πΌ , π₯2β π1 , πΌ Finally, plot π₯1β π1 , πΌ against π1 and voila! You have a demand curveβ¦ β¦sum them over all consumers and you have a market demand curve! How do monopolists price goods? Monopolists, like all firms, should price to maximize profits. As a result, the demand curve and costs matter Monopolists, like all firms, should price to maximize profits. As a result, the demand curve and costs matter Perfect Comp: MB = MC = P (other firms can undercut price otherwise) Monopolists, like all firms, should price to maximize profits. As a result, the demand curve and costs matter Monopolist doesnβt have to worry about competitors -> set Q such that MC = MR Monopolists, like all firms, should price to maximize profits. As a result, the demand curve and costs matter Monopolist doesnβt have to worry about competitors -> set Q such that MC = MR How to construct MR? To sell another unit, must lower the price so firm loses money on units they were selling (intensive margin) and gains money from additional units sold (extensive margin) MR = extensive gains β intensive losses maxπ π π = π π π β ππΆ(π) Monopolistβs math Maximizes profits (TR β TC) by setting a quantity and charging needed price to have market clear e.g., price is a function of quantity: P(q) and note the TR = P(q)*q f.o.c.: πβ² π β π β + π π β β ππΆ β² π β = 0 πβ² π β π β + π π β β ππΆ π β = 0 πβ² π β π β + π π β = ππΆ π β NOTE: Pβ(q) < 0 since demand slopes downward πβ² π β π β Intensive margin loss as q increases π πβ Extensive margin gain as q increases πβ² π β π β + π π β = ππΆ π β Monopolistβs math ππ(πβ ) β π ππ + π(π β ) = π Note that Pβ(q) is the slope of the demand curve and that economists think about elasticities rather than slopes. ππ(π β ) β π(π ) β π = β π ππ Assume MC(q) = c for simplicity (e.g., constant MC) Divide both sides by P π(π β ) β π ππ(π β ) π =β π(π β ) ππ π(π β ) Lerner Equation: If demand curve is relatively inelastic, charge a high markup. NOTE 1: Assume that elasticity is constant along all portions of the demand curve for convenience. NOTE 2: βConstant Elasticityβ makes demand curve nonlinear but a perfectly valid assumption. NOTE 3: π refers to own price elasticity unless otherwise noted. β ππ(π β ) π(π β ) β π % Ξπ 1 1 π(π β ) =β =β =β =β % Ξπ ππ π(π β ) % Ξπ π % Ξπ π π·(πβ ) β π π = β π·(πβ ) π p MC Dead Weight Loss pm Monopoly Profits D q qm qc MR 12 Choosing Quantity β’ Marginal Revenue, the increment to revenue from a increase in quantity sold MR ο½ p(q) ο« qpο’(q) β’ Elasticity of demand: tells you the % change in quantity for a 1% change in price ππ %Ξπ ππ π 1 π π π=β =β =β =β ππ %Ξπ ππ π πβ²(π) π π 13 Examining the elasticity function 1 π(π) π=β β² π π π First derivative of the demand curve. At any point π0 it gives the slope at that point. The demand curve. Gives the price as function of q. Relative levels of price and quantity 14 Special case: linear demand 1 π(π) π=β π ππππ π First derivative is constant At high prices, q is low. A 1% change in p is relatively large, especially compared to quantity. Elasticities will be relatively large. At low prices, p will be small and q is large. Elasticities will mechanically be low. 15 Some general facts about elasticity β’ At high prices, a fixed % change in price is larger in level terms. Since q will tend to small, a given level change in q, will be a larger in percentage terms. β’ This creates a relationship such that elasticities will tend to be high at the βtop of the demand curveβ and low at the bottom. β’ When we think of βsmall changesβ in price, the impact of raising and lowering will be symmetric. However, for larger changes, e.g. 10%, an increase and decrease need to have the same magnitude impact 16 Elasticity and total revenue Total revenue (TR) = p(q)*q Product Rule Algebra Substituting in οTR d ο¨ pq ο© qdp ο« pdq dp ο¦ pdq οΆ ο§ο§1 ο« ο·ο· ο½ %οpο¨1 ο ο₯ ο© %οTR ο½ ο½ ο½ ο½ TR pq pq p ο¨ qdp οΈ Translating to calculus Negative of this is elasticity 17 Elasticity and total revenue Total revenue (TR) = p(q)*q οTR d ο¨ pq ο© qdp ο« pdq dp ο¦ pdq οΆ ο§ο§1 ο« ο·ο· ο½ %οpο¨1 ο ο₯ ο© %οTR ο½ ο½ ο½ ο½ TR pq pq p ο¨ qdp οΈ β’ One minus the elasticity translates a price increase in percent to a revenue increase. β’ For example, if the elasticity is 3.5, a 1% price increase causes a -2.5% impact on revenue (a loss). 18 Elasticity and total revenue %οTR ο½ %οpο¨1 ο ο₯ ο© β’ Elasticity = 1 ο small changes in price do not impact revenue β’ Elasticity < 1 ο price drops lower revenue, price increases raise revenue β’ Elasticity>1 ο price drops raise revenue 19 Inverse Elasticity Rule β’ Profit Max (MR=MC) MR MC οΆο° 0ο½ ο½ p ο« qpο’(q ) ο c' (q ) οΆq π β π β² π = βππβ² π π β πβ²(π) π β² 1 =β π π = π π π 20 MR Inverse Elasticity Rule β’ Profit Max (MR=MC) MC οΆο° 0ο½ ο½ p ο« qpο’(q ) ο c' (q ) οΆq β’ Price-cost margin (Lerner index) = 1 over elasticity p ο cο’(q ) ο qpο’ 1 ο½ ο½ . p p ο₯ β’ Price minus marginal costs divided by price is referred to as gross margin. 21 Inverse Elasticity Rule p ο cο’(q ) ο qpο’ 1 ο½ ο½ . p p ο₯ β’ Suppose MC=0. Then quantity is chosen so that elasticity is 1. p ο qpο’ 1 ο½1ο½ ο½ p p ο₯ β’ Intuition: if marginal costs are zero, then optimize for revenue. Total revenue grows until elasticity = 1. β’ If MC>0, one will βstopβ before reaching elasticity = 1. 22 p MC Dead Weight Loss pm Monopoly Profits D q qm qc MR 23 Digression on Margin Formula β’ Perfect competition says price=MC, or zero markup, which implies elasticity of infinity. In other words, by deviating from market price I can sell all my units. What are some examples where this is approximately true in practice? β’ In general MC will depend on price. Cannot in general say βwhat are my marginal costsβ to get optimal mark up 24 Markup formula cont. π π= πβ² π πβ1 β’ Seems to say βfixed markup on marginal costsβ, but elasticity will depend on the demand function, so will not be fixed across a firmβs products and across customers. β’ Optimal pricing is much more complicated than a fixed markup 25 Markup formula cont. π π= πβ² π πβ1 β’ Markup > 1 β’ Elasticity will generally depend on q, costs depend on q β’ With constant elasticity, firm passes on more than 100% of cost or tax (a tax is like a marginal cost, firm βmarks up the taxβ) β’ Works at firm level, with elasticity measured at firm, not industry β’ Does not capture competitor reactions 26 Monopoly Pricing Formula β’ Prices depends on elasticity, which depends on the product, customer characteristics β’ Offer discounts to elastic (price sensitive) customers β’ Discounts offered on basis of factors correlated with price sensitivity β’ Pricing can often be understood by how decision variables correlate with price sensitivity 27 Pricing Multiple Goods 28 Basic idea β’ One firm selling multiple goods β’ A firmβs goods will, in general, βcompete with each otherβ to some degree β’ A rational firm takes this into account when setting price. Optimal price will depend on own price elasticity (what we just learned about) and cross price elasticities 29 Pricing Related Goods Review: substitutes and complements β’ Complements and Substitute Products (Relationship) ο§ ο§ Sales of a good rise when the price of a complement falls β’ Console and games β’ Drinks and food at a restaurant (e.g. happy hour to attract customers) Sales of a good fall when the price of a substitute falls β’ Games vs. other games β’ Food at a restaurant ο§ Lower price of substitute cannibalizes demand from other product ο§ Lower price of complement promotes sales of other product ο§ Prices of substitutes (complements) are higher (lower) than standalone profit-maximizing prices 30 Inverse Elasticity Rule 2 β’ Suppose we sell n goods indexed i=1,β¦,n β’ Demands xi(p) β’ Profit n ο° ο½ ο₯i ο½1 ο¨ pi ο mc(i ) ο©xi (p). β’ If we assume constant marginal cost, this simplification is an example of selling the same good in multiple markets or to multiple customer βtypesβ p j dxi β’ Cross-price elasticity ο₯ij ο½ xi dp j . β’ Note no minus sign. β’ Positive ο substitutes; Negative ο complements 31 Representative Consumer Assumption β’ If there is a representative consumer maximizing utility: max u(x)-px, so u ο’(x) ο½ p and u ο’ο’(x)dx ο½ dp β’ Thus there are symmetric cross-derivatives οΆxi οΆx j ο½ οΆp j οΆpi Recall this rule from multivariate calculus From the total derivative of FOC This rule need not hold in practice, but is a commonly made assumption 32 In Matrix Notation pi ο mc(i ) , Price cost margin: Li ο½ pi οΆx j οΆxi οΆο° n n 0ο½ ο½ xi ο« ο₯ j ο½1 ( p j ο mc(i )) ο½ xi ο« ο₯ j ο½1 ( p j ο mc(i )) οΆpi οΆpi οΆp j ο¦ οΆ n ( p j ο mc (i )) ο· ο½ xi ο§1 ο« ο₯ j ο½1 ο₯ ij ο§ ο· p j ο¨ οΈ 0 = 1 + E L, and thus L = - E-1 1 33 Rule for inverting a 2x2 matrix Two Good Formula β’ L = - E-1 1 yields π³π π³π πππ = π ππ πππ πππ πππ β πππ π³π = β πππ πππ β πππ πππ =β πππ Factor out πππ βπ π π Divide top and bottom by πππ πππ πππ = πππ πππ πππ β πππ πβ πππ πππ πβ πβ π πππ πππ =β πππ πππ πππ πππ π ππ πβ πβ πππ πππ πππ πππ Multiply top and bottom by πππ 34 Two Good Formula β’ L = - E-1 1 yields π Lπ = β πππ πππ πππ πππ πππ πππ πβ πππ πππ πππ πβ πππ πππ will be between 0 and 1 because πππ πππ < πππ πππ This is because cross price elasticities have to be smaller than the relevant own price elasticities. 35 Two Good Formula β’ L = - E-1 1 yields π Lπ = β πππ π =β πππ πππ πβ πππ πππ πππ πβ πππ πππ + πβ β =β π >π ++ πππ π β π πβ ββ 36 Two Good Formula for Substitutes β’ L = - E-1 1 yields π Lπ = β πππ π =β πππ πππ πβ πππ πππ πππ πβ πππ πππ + πβ β =β π >π ++ πππ π β π πβ ββ 37 Two Good Formula for Substitutes β’ L = - E-1 1 yields π Lπ = β πππ πππ πβ πππ πππ πππ πβ πππ πππ 38 Two Good Formula for Complements β’ L = - E-1 1 yields πππ π πβ β Lπ = β πππ π β π 39 Two Good Formula Review β’ L = - E-1 1 yields π =β πππ πππ πβ πππ π π π β ππ ππ πππ πππ πππ > π, goods are substitutes. A price decrease on product 2 decreases sales on product 1 (go in same direction) πππ < π, goods are complements. A price decrease on product 2 increases sales on product 1 (go in opposite directions) πππ & πππ will be negative due to law of demand (note before we βembedded the negative sign) 40 Bundling β’ Pure bundling: only sell the bundle β’ Cars & tires β’ Cable TV β’ Cars + feature βmodelsβ β’ Mixed bundling: sell separately with a discount for bundle β’ Video games w/ console β’ Sports passes (clubs, ski resorts) 41 Enormous Bundle 42 Utilities with Independent Values Action Utility Buy Nothing 0 Buy Good 1 v 1 β p1 Buy Good 2 v 2 β p2 Buy Both v1+v2 β pB 43 v2 Buy neither, but would buy a bundle. Starting from the top right of the square, there is always some bundle I want to offer pB Buy Good 2 Buy Both p2 Buy Nothing Buy Good 1 p1 v1 44 v2 If p2 optimal, this price reduction doesnβt affect profits β sales gains just balance price cut pB Buy Good 2 Buy Both p2 Buy Nothing Buy Good 1 p1 v1 45 v2 pB Buy Good 2 Buy Both p2 If p1 optimal, this price reduction doesnβt affect profits β sales gains just balance price cut Buy Nothing Buy Good 1 p1 v1 46 v2 pB Buy Good 2 Buy Both p2 Reducing bundle price gives the additional sales of both goods with a single price cut Buy Nothing Buy Good 1 p1 v1 47 Conceptualizing bundles β’ A bundle can be thought of as a βconditional discountβ. E.g. If you buy good 1, Iβll give you a discount on good 2. β’ This lets me give βtargeted offersβ β’ Especially powerful when my valuation of good 2 is much lower if I already have good 1. E.g. gym memberships bundle many partner locations for a small increase in price because otherwise people would rarely buy more than 1. 48 Bundling as a part of corporate strategy β’ Rethink Product β’ Jack Walsh noticed GE made more profit on engine service than aircraft engines β’ Redefined product: sell engines in order to sell service β’ Rather than selling service to make engines more attractive β’ Very profitable β’ IBM pivoted from providing software and services to sell hardware β’ IBM Global Services β’ Hardware margins typically low except Apple 49 Bundling Insights β’ Mixed bundling is always more profitable than no bundling β’ With independent or negatively correlated goods β’ Better for consumers as well! β’ Often a βgrand bundleβ does well for firms, but can be bad for consumers β’ Skims out the most willing-to-pay with a βsuper goodβ β’ Bundles can be used to help customers 50
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