21 Analytical Models
Marketing models consists of
 Analytical Model: pure mathematicalbased research
 Empirical Model: data analysis.
“A model is a representation of the most important elements of a perceived realworld system”.
Marketing model improves decisionmaking

Econometric models
 Description
 Prediction
 Simulation

Optimization models
 maximize profit using market response model, cost functions, or any constraints.
Quasi and Field experimental analyses
Conjoint Choice Experiments.
“A decision calculus will be defined as a modelbased set of procedures for processing data and judgments to assist a manager in his decision making”(Little 1976):
 simple
 robust
 easy to control
 adaptive
 as complete as possible
 easy to communicate with
Type of game  

Static  Dynamic  
Info Content  Complete  Nash  Subgame perfect 
Incomplete 
Bayesian Nash (Auctions) 
Perfect Bayesian (signaling) 
Mathematical Theoretical Models
Logical Experimentation

An environment as a model, specified by assumptions
Math assumptions for tractability
Substantive assumptions for empirical testing
Decision support modeling describe how things work, and theoretical modeling present how things should work.
Compensation package including salaries and commission is a tradeoff between reduced income risk and motivation to work hard.
Internal and External Validity are questions related to the boundaries conditions of your experiments.
“Theories are tested by their predictions, not by the realism of their super model assumptions.” (Friedman, 1953)
Competition is performed under uncertainty

Competition reveals hidden information
Independentprivatevalues case: selling price = second highest valuation
It’s always better for sellers to reveal information since it reduces chances of cautious bidding that is resulted from the winner’s curse

Competition is better than bargaining
 Competition requires less computation and commitment abilities
Competition creates effort incentives

Types of model:

Predictive model
Sales model: using time series data
Trial rate: using exponential growth.
Product growth model: Bass (1969)

Descriptive model
Purchase incidence and purchase timing : use Poisson process
Brand choice: Markov models or learning models.
Pricing decisions in an oligopolistic market Howard and Morgenroth (1968)

Normative model
 Profit maximization based on price, adverting and quality (Dorfman and Steiner 1976), extended by (H. V. Roberts, Ferber, and Verdoorn 1964; Lambin 1970)

Later, Little (1970) introduced decision calculus and then multinomial logit model (Peter M. Guadagni and Little 1983)
Potential marketing decision automation:
Promotion or pricing programs
Media allocation
Distribution
Product assortment
Direct mail solicitation

Definitions:
Rationality = maximizing subjective expected utility
Intelligence = recognizing other firms are rational.

Rules of the game include
# of firms
feasible set of actions
utilities for each combination of moves
sequence of moves
the structure of info (who knows what and when?)

Incomplete info stems from
unknown motivations
unknown ability (capabilities)
different knowledge of the world.
Pure strategy = plan of action
A mixed strategy = probability dist of pure strategies.
Strategic form representation = sets of possible strategies for every firm and its payoffs.
Equilibrium = a list of strategies in which “no firm would like unilaterally to change its strategy.”
Equilibrium is not outcome of a dynamic process.

Equilibrium Application

Oligopolistic Competition
Cournot (1838): quantities supplied: Cournot equilibrium. Changing quantities is more costly than changing prices
Bertrand (1883): Bertrand equilibrium: pricing.
Perfect competition
Product Competition: Hotelling (1929): Principle of Minimum Differentiation is invalid.

Entry:
first mover advantage
deterrent strategy
optimal for entrants or incumbents
Channels


Perfectness of equilibria
Subgame perfectness
Sequential rationality
Tremblinghand perfectness

Application
Product and price competition in Oligopolies
Strategic Entry Deterrence

Dynamic games
Longterm competition in oligopolies
Implicit Collusion in practice : price match from leader firms

Incomplete Information
Durable goods pricing by a monopolist
predatory pricing and limit pricing
reputation, product quality, and prices
Competitive bidding and auctions
21.1 Building An Analytical Model
Notes by professor Sajeesh Sajeesh
Step 1: Get “good” idea (either from literature or industry)
Step 2: Assess the feasibility of the idea
Is it interesting?
Can you tell a story?
Who is the target audience?
Opportunity cost
Step 3: Don’t look at the literature too soon
 Even when you have an identical model as in the literature, it’s ok (it allows you to think)
Step 4: BUild the model
Simplest model first: 1 period, 2 product , linear utility function for consumers
Write down the model formulation
Everything should be as simple as possible .. but no simpler
Step 5: Generalizing the model
 Adding complexity
Step 6: Searching the literature
 If you find a paper, you can ask yourself why you didn’t do what the author has done.
Step 7: Give a talk /seminar
Step 8: Write the paper
21.2 Hotelling Model
(KIM and SERFES 2006): A location model with preference variety
Stability in competition
Duopoly is inherently unstable

Bertrand disagrees with Cournot, and Edgeworth elaborates on it.
 because Cournot’s assumption of absolutely identical products between firms.
seller try to \(p_2 < p_1 c(lab)\)
the point of indifference
\[ p_1 + cx = p_2 + cy \]
c = cost per unit of time in each unit of line length
p = price
q = quantity
x, y = length from A and B respectively
\[ a + x + y + b = l \]
is the length of the street
Hence, we have
\[ x = 0.5(l  a  b + \frac{p_2 p_1}{c}) \\ y = 0.5(l  a  b + \frac{p_1 p_2}{c}) \]
Profits will be
\[ \pi_1 = p_1 q_1 = p_1 (a+ x) = 0.5 (l + a  b) p_1  \frac{p_1^2}{2c} + \frac{p_1 p_2}{2c} \\ \pi_2 = p_2 q_2 = p_2 (b+ y) = 0.5 (l + a  b) p_2  \frac{p_2^2}{2c} + \frac{p_1 p_2}{2c} \]
To set the price to maximize profit, we have
\[ \frac{\partial \pi_1}{\partial p_1} = 0.5 (l + a  b)  \frac{p_1}{c} + \frac{p_2}{2c} = 0 \\ \frac{\partial \pi_2}{\partial p_2} = 0.5 (l  a + b)  \frac{p_2}{c} + \frac{p_1}{2c} = 0 \]
which equals
\[ p_1 = c(l + \frac{ab}{3}) \\ p_2 = c(l  \frac{ab}{3}) \]
and
\[ q_1 = a + x = 0.5 (l + \frac{a b}{3}) \\ q_2 = b + y = 0.5 (l  \frac{ab}{3}) \]
with the SOC satisfied
In case of deciding locations, socialism works better than capitalism
(d’Aspremont, Gabszewicz, and Thisse 1979)
 Principle of Minimum Differentiation is invalid
\[ \pi_1 (p_1, p_2) = \begin{cases} ap_1 + 0.5(lab) p_1 + \frac{1}{2c}p_1 p_2  \frac{1}{2c}p_1^2 & \text{if } p_1  p_2 \le c(lab) \\ lp_1 & \text{if } p_1 < p_2  c(lab) \\ 0 & \text{if } p_1 > p_2 + c(lab) \end{cases} \]
and
\[ \pi_2 (p_1, p_2) = \begin{cases} bp_2 + 0.5(lab) p_2 + \frac{1}{2c}p_1 p_2  \frac{1}{2c}p_2^2& \text{if } p_1  p_2 \le c(lab) \\ lp_2 & \text{if } p_2 < p_1  c(lab) \\ 0 & \text{if } p_2 > p_1 + c(lab) \end{cases} \]
21.3 Positioning Models
Tabuchi and Thisse (1995)
Relax Hotelling’s model’s assumption of uniform distribution of consumers to nonuniform distribution.

Assumptions:
Equal cost
Consumers distributed over [0,1]
\(F(x)\) = cumulative distribution of consumers where \(F(1) = 1\) = total population

2 distributions:
Traditional uniform density: \(f(x) =1\)
New: triangular density: \(f(x) = 2  22x1\) which represents consumer concentration
Transportation cost = quadratic function of distance.
Hence, marginal consumer is
\[ \bar{x} = (p_2  p_1 + x^2_2x_1^2)/2(x_2x_1) \]
then when \(x_1 < x_2\) the profit function is
\[ \Pi_1 = p_1 F(\bar{x}) \]
and
\[ \Pi_2 = p_2[1F(\bar{x})] \]
and vice versa for \(x_1 >x_2\), and Bertrand game when \(x_1 = x_2\)

If firms pick simultaneously their locations, and then simultaneously their prices, and consumer density function is logconcave, then there is a unique Nash price equilibrium
Under uniform distribution, firms choose to locate as far apart as possible (could be true when observing shopping centers are far away from cities), but then consumers have to buy products that are far away from their ideal.
Under triangular density, no symmetric location can be found, but two asymmetric Nash location equilibrium can still be possible (decrease in equilibrium profits of both firms)

If firms pick sequentially their locations, and pick their prices simultaneously,
 Under both uniform and triangular, first entrant will locate at the market center
Sajeesh and Raju (2010)
Model satiation (varietyseeking) as a relative reduction in the willingness to pay of the previously purchased brand. also known as negative state dependence

Previous studies argue that in the presence of variety seeking consumers, firms should enjoy higher prices and profits, but this paper argues that average prices and profits are lower.
 Firms should charge lower prices in the second period to prevent consumers from switching.
Assumptions:
Period 0, choose location simultaneously
Period 1, choose prices simultaneously
Period 2, firms choose prices simultaneously
K. S. Moorthy (1988)
 2 (identical) firms pick product (quality) first, then price.
Tyagi (2000)
Extending Hotelling (1929) Tyagi (1999b) Tabuchi and Thisse (1995)
Two firms enter sequentially, and have different cost structures.
Paper shows second mover advantage
KIM and SERFES (2006)
Consumers can make multiple purchases.
Some consumers are loyal to one brand, and others consume more than one product.
Shreay, Chouinard, and McCluskey (2015)
 Quantity surcharges from different sizes of the same product (i.e., imperfect substitute or differentiated products) can be led by consumer preferences.
21.4 Market Structure and Framework
Basic model utilizing aggregate demand
Bertrand Equilibrium: Firms compete on price
Cournot Market structure: Firm compete on quantity
Stackelberg Market structure: LeaderFollower model
Because we start with the quantity demand function, it is important to know where it’s derived from Richard and Martin (1980)
K. S. Moorthy (1988)
 studied how two firms compete on product quality and price (both simultaneous and sequential)
21.4.1 Cournot  Simultaneous Games
\[ TC_i = c_i q_i \text{ where } i= 1,2 \\ P(Q) = a  bQ \\ Q = q_1 +q_2 \\ \pi_1 = \text{price} \times \text{quantity}  \text{cost} = [a  b(q_1 +q_2)]q_1  c_1 q_1 \\ \pi_2 = \text{price} \times \text{quantity}  \text{cost} = [a  b(q_1 +q_2)]q_1  c_2 q_2 \\ \]
\[\begin{equation} \frac{d \pi_1}{d q_1} = a  2bq_1  bq_2c_1 = 0 (1) \tag{21.1} \end{equation}\] \[\begin{equation} \frac{d \pi_2}{d q_2} = a  2bq_2  bq_1c_2 = 0 \tag{21.2} \end{equation}\]From (21.1)
\[\begin{equation} q_1 = \frac{ac_1}{2b}  \frac{q_2}{2} = R_1 (q_2) \tag{21.3} \end{equation}\]is called reaction function, for best response function
From (21.2)
\[\begin{equation} q_2 = \frac{ac_2}{2b}  \frac{q_1}{2} = \tag{21.4} \end{equation}\]\[ q_1 = \frac{ac_1}{2b}  \frac{ac_2}{4b} + \frac{q_1}{4} \]
Hence,
\[ q_1^* = \frac{a2c_1+ c_2}{3b} \\ q_2^* = \frac{a2c_2 + c_1}{3b} \]
Total quantity is
\[ Q = q_1 + q_2 = \frac{2ac_1 c_2}{3b} \]
Price
\[ abQ = \frac{a+c_1+c_2}{3b} \]
21.4.2 Stackelberg  Sequential games
also known as leaderfollower games
Stage 1: Firm 1 chooses quantity
Stage 2: Firm 2 chooses quantity
\[ c_2 = c_1 = c \]
Stage 2: reaction function of firm 2 given quantity firm 1
\[ R_2(q_1) = \frac{ac}{2b}  \frac{q_1}{2} \]
Stage 1:
\[ \pi_1 = [ab(q_1 + \frac{ac}{2b}  \frac{q_1}{2})]q_1  cq_1 \\ = [ab( \frac{ac}{2b} + \frac{q_1}{2}]q_1 + cq_1 \]
\[ \frac{d \pi_1}{d q_1} = 0 \]
Hence,
\[ \frac{a+c}{2}  b q_1 c =0 \]
The Stackelberg equilibrium is
\[ q_1^* = \frac{ac}{2b} \\ q_2^* = \frac{ac}{4b} \]
Under same price (c), Cournot =
\[ q_1 = q_2 = \frac{ac}{3b} \]
Leader produces more whereas the follower produces less compared to Cournot
\[ \frac{d \pi_W^*}{d \beta} <0 \]
for the entire quantity range \(d < \bar{d}\)
As \(\beta\) increases in \(\pi_W^*\) Firm W wants to reduce \(\beta\).
Low \(\beta\) wants more independent
Firms W want more differentiated product
On the other hand,
\[ \frac{d \pi_S^*}{d \beta} <0 \]
for a range of \(d < \bar{d}\)
Firm S profit increases as \(\beta\) decreases when d is small
Firm S profit increases as \(\beta\) increases when d is large
Firm S profit increases as as product are more substitute when d is large
Firm S profit increases as products are less differentiated when d is large
21.5 More Market Structure
Dixit (1980)
Based on BainSylos postulate: incumbents can build capacity such that entry is unprofitability
Investment in capacity is not a credibility threat if incumbents can change their capacity.
Incumbent cannot deter entry
Tyagi (1999a)
More retailers means greater competition, which leads to lower prices for customers.

Effect of \((n+1)\) st retailer entry
Competition effect (lower prices)
Effect on price (i.e., wholesale price), also known as input cost effect
Manufacturers want to increase wholesale price because now manufacturers have higher bargaining power, which leads other retailers to reduce quantity (bc their choice of quantity is dependent on wholesale price), and increase in prices.
Jerath, Sajeesh, and Zhang (2016)
Organized Retailer enters a market
Inefficient unorganized retailers exit
Remaining unorganized retailers increase their prices. Thus, customers will be worse off.
Amaldoss and Jain (2005)
consider desire for uniqueness and conformism on pricing conspicuous goods

Two routes:
higher desire for uniqueness leads to higher prices and profits
higher desire for conformity leads to lower prices and profits
Under the analytical model and lab text, consumers’ desire for unique is increased from price increases, not the other way around.
Snob:
\[ U_A = V  p_A  \theta t_s  \lambda_s(n_A) \\ U_B = V  p_B  (1\theta) t_s  \lambda_s(n_B) \]
where
\(\lambda_s\) = sensitivity towards externality.
\(\theta\) is the position in the Hotelling’s framework.
\(t_s\) is transportation cost.
Conformist
\[ U_A = V  p_A  \theta t_s + \lambda_c(n_A) \\ U_B = V  p_B  (1\theta) t_s + \lambda_c(n_B) \]
Rational Expectations Equilibrium
If your expectations are rational, then your expectation will be realized in equilibrium
Say, Marginal Snob = \(\theta_s\) and \(\beta\) = number of snob in the market
Snobs
\[ U_A^c \equiv U_B^c = \theta_s \]
Conformists
\[ U_A^c =U_B^c = \theta_c \]
Then, according to rational expectations equilibrium, we have
\[ \beta \theta_s +( 1 \beta) \theta_c = n_A \\ \beta (1\theta_s) +( 1 \beta) (1\theta_c) = n_B \]
where
\(\beta \theta_s\) = Number of snobs who buy from firm A
\((1\beta)\theta_c\) = Number of conformists who buy from firm B
\(\beta(1\theta_s)\) = Number of snobs who buy from firm B
\((1\beta)(1\theta_c)\) = Number of conformists who buy from firm B
which is the rational expectations equilibrium (whatever we expect happens in reality).
In other words, expectation are realized in equilibrium.
The number of people expected to buy the product is endogenous in the model, which will be the actual number of people who will buy it in the market.
We should not think of the expected value here in the same sense as expected value in empirical research (\(E(.)\)) because the expected value here is without any errors (specifically, measurement error).
 The utility function for snobs is such that overall when price increase for one product, snob will like to buy the product more. When price increases, conformist will reduce the purchase.
Balachander and Stock (2009)
Adding a Limited edition product has a positive effect on profits (via increased willingness of consumers to pay for such a product), but negative strategic effect (via increasing price competition between brands)
Under quality differentiation, highquality brand gain from LE products
Under horizontal taste differentiation, negative strategic effects lead to lower equilibrium profits for both brands, but they still have to introduce LE products because of prisoners’ dilemma
Sajeesh, Hada, and Raju (2020)

two consumer segments:
functionalityoriented
exclusivityoriented
Firm increase value enhancements when functionalityoriented consumers perceive greater product differentiation
Firms decrease value enhancements if exclusivityoriented perceive greater product differentiation
21.6 Market Response Model
Marketing Inputs:
 Selling effort
 advertising spending
 promotional spending
Marketing Outputs:
 sales
 share
 profit
 awareness
Give phenomena for a good model:
 P1: Dynamic sales response involves a sales growth rate and a sales decay rate that are different
 P2: Steadystate response can be concave or Sshaped. Positive sales at 0 adverting.
 P3: Competitive effects
 P4: Advertising effectiveness dynamics due to changes in media, copy, and other factors.
 P5: Sales still increase or fall off even as advertising is held constant.
Saunder (1987) phenomena
 P1: Output = 0 when Input = 0
 P2: The relationship between input and output is linear
 P3: Returns decrease as the scale of input increases (i.e., additional unit of input gives less output)
 P4: Output cannot exceed some level (i.e., saturation)
 P5: Returns increase as scale of input increases (i.e., additional unit of input gives more output)
 P6: Returns first increase and then decrease as input increases (i.e., Sshaped return)
 P7: Input must exceed some level before it produces any output (i.e., threshold)
 P8: Beyond some level of input, output declines (i.e., supersaturation point)
Aggregate Response Models

Linear model: \(Y = a + bX\)
Through origin
can only handle constant returns to scale (i.e., can’t handle concave, convex, and Sshape)

The Power Series/Polynomial model: \(Y = a + bX + c X^2 + dX^3 + ...\)
 can’t handle saturation and threshold

Fraction root model/ Power model: \(Y = a+bX^c\) where c is prespecified
c = 1/2, called square root model
c = 1, called reciprocal model
c can be interpreted as elasticity if a = 0.
c = 1, linear
c <1, decreasing return
c>1, increasing returns

Semilog model: \(Y = a + b \ln X\)
 Good when constant percentage increase in marketing effort (X) result in constant absolute increase in sales (Y)

Exponential model: \(Y = ae^{bX}\) where X >0
b > 0, increasing returns and convex
b < 0, decreasing returns and saturation

Modified exponential model: \(Y = a(1e^{bX}) +c\)
Decreasing returns and saturation
upper bound = a + c
lower bound = c
typically used in selling effort

Logistic model: \(Y = \frac{a}{a+ e^{(b+cX)}}+d\)
increasing return followed by decreasing return to scale, Sshape
saturation = a + d
good with saturation and sshape
Gompertz model

ADBUDG model (Little 1970) : \(Y = b + (ab)\frac{X^c}{d + X^c}\)
c > 1, Sshaped

0 < c < 1
Concave
saturation effect
upper bound at a
lower bound at b
typically used in advertising and selling effort.
can handle, through origin, concave, saturation, Sshape
Additive model for handling multiple Instruments: \(Y = af(X_1) + bg(X_2)\)
Multiplicative model for handling multiple instruments: \(Y = aX_1^b X_2^c\) where c and c are elasticities. More generally, \(Y = af(X_1)\times bg(X_2)\)
Multiplicative and additive model: \(Y = af(X_1) + bg(X_2) + cf(X_1) g(X_2)\)
Dynamic response model: \(Y_t = a_0 + a_1 X_t + \lambda Y_{t1}\) where \(a_1\) = current effect, \(\lambda\) = carryover effect
Dynamic Effects

Carryover effect: current marketing expenditure influences future sales
 Advertising adstock/ advertising carryover is the same thing: lagged effect of advertising on sales
Delayedresponse effect: delays between when marketing investments and their impact
Customer holdout effects
Hysteresis effect
New trier and wearout effect
Stocking effect
Simple Decayeffect model:
\[ A_t = T_t + \lambda T_{t1}, t = 1,..., \]
where
 \(A_t\) = Adstock at time t
 \(T_t\) = value of advertising spending at time t
 \(\lambda\) = decay/ lag weight parameter
Response Models can be characterized by:
The number of marketing variables
whether they include competition or not

the nature of the relationship between the input variables
 Linear vs. Sshape
whether the situation is static vs. dynamic
whether the models reflect individual or aggregate response

the level of demand analyzed
 sales vs. market share
Market Share Model and Competitive Effects: \(Y = M \times V\) where
Y = Brand sales models
V = product class sales models
M = marketshare models
Market share (attraction) models
\[ M_i = \frac{A_i}{A_1 + ..+ A_n} \]
where \(A_i\) attractiveness of brand i
Individual Response Model:
Multinomial logit model representing the probability of individual i
choosing brand l
is
\[ P_{il} = \frac{e^{A_{il}}}{\sum_j e^{A_{ij}}} \]
where
 \(A_{ij}\) = attractiveness of product j for individual i \(A_{ij} = \sum_k w_k b_{ijk}\)

\(b_{ijk}\) = individual i’s evaluation of product j on product attribute k, where the summation is over all the products that individual
i
is considering to purchase  \(w_k\) = importance weight associated with attribute k in forming product preferences.
21.7 Technology and Marketing Structure and Economics of Compatibility and Standards
Amaldoss and Jain (2005)
Balachander and Stock (2009)
Sajeesh, Hada, and Raju (2020)
21.8 Conjoint Analysis and Augmented Conjoint Analysis
More technical on 27.1
Jedidi and Zhang (2002)
 Augmenting Conjoint Analysis to Estimate Consumer Reservation Price
Using conjoint analysis (coefficients) to derive at consumers’ reservation prices for a product in a category.

Can be applied in the context of
product introduction
calculating customer switching effect
the cannibalization effect
the market expansion effect
\[ Utility(Rating) = \alpha + \beta_i Attribute_i \]
where \(\alpha\)
Netzer and Srinivasan (2011)
Break conjoint analysis down to a sequence of constantsum paired comparison questions.
Can also calculate the standard errors for each attribute importance.
21.9 Distribution Channels
McGuire and Staelin (1983)
Assumptions:
 Two manufacturing (wholesaling) firms differentiated and competing products: Upstream firms (manufacturers) and downstream channel members (retailers)
3 types of structure:
 Both manufacturers with privately owned retailers (4 players: 2 manufacturers, 2 retailers)
 Both vertically integrated (2 manufacturers)
 Mix: one manufacturer with a private retailer, and one manufacturer with vertically integrated company store (3 players)
Each retail outlet has a downward sloping demand curve:
\[ q_i = f_i(p_1,p_2) \]
Under decentralized system (4 players), the Nash equilibrium demand curve is a function of wholesale prices:
\[ q_i^* = g_i (w_1, w_2) \]
More rules:
 Assume 2 retailers respond, but not the competing manufacturer
And unobserved wholesale prices and market is not restrictive, and Nash equilibrium whole prices is still possible.
Under mixed structure, the two retailers compete, and nonintegrated firm account for all responses in the market
Under integrated structure, this is a twoperson game, where each chooses the retail price
Decision variables are prices (not quantities)
Under what conditions a manufacturer want to have intermediaries
Retail demand functions are assumed to be linear in prices
Demand functions are
\[ q_1' = \mu S [ 1  \frac{\beta}{1  \theta} p_1' + \frac{\beta \theta}{1 \theta}p_2'] \]
\[ q_2' = (1 \mu) S [ 1+ \frac{\beta \theta}{1 \theta} p_1'  \frac{\beta}{1 \theta} p_2'] \]
where
\(0 \le \mu , \theta \le 1; \beta, S >0\)
S is a scale factor, which equals industry demand (\(q' \equiv q_1' + q_2'\)) when prices are 0.
\(\mu\) = absolute difference in demand

\(\theta\) = substutability of products (reflected by the cross elasticities), or the ratio of the rate of change of quantity with respect to the competitor’s price to the rate of change of quantity with respect to own price.
\(\theta = 0\) means independent demands (firms are monopolists)
\(\theta \to 1\) means maximally substitutable
3 more conditions:
\[ P = \{ p_1', p_2'  p_i' m'  s' \ge 0, i = 1,2; (1\theta)  \beta p_1' \beta \theta p_2' \ge 0, (1 \theta) + \beta \theta p_1'  \beta p_2' \ge 0 \} \]
where \(m', s'\) are fixed manufacturing and selling costs per unit
To have a set of \(P\), then
\[ \beta \le \frac{1}{m' + s'} \]
and to have industry demand no increase with increases in either price then
\[ \frac{\theta}{1 + \theta} \le \mu \le \frac{1}{1 + \theta} \]
After rescaling, the industry demand is
\[ q = 2 (1 \theta) (p_1+ p_2) \]
Results:
When each manufacturer is a monopolist (\(\theta = 0\)), it’s twice as profitable for each to sell through its own channel
When demand is maximally affected by the actions of the competing retailers (\(\theta \to 1\)), it’s 3 times as profitable to have private dealers.
The breakeven point happens at \(\theta = .708\)
In conclusion, the optimal distribution system depends of the degree of substitubability at the retail level.
Jeuland and Shugan (2008)

Quantity discounts is offered because
Costbased economies of scale
Demand based  large purchases tend to be more price sensitive
Strategic reason single sourcing
Channel Coordination (this is where this paper contributes to the literature
K. S. Moorthy (1987)
 Price discrimination  second degree
Geylani, Dukes, and Srinivasan (2007)
Jerath and Zhang (2010)
21.10 Advertising Models
Three types of advertising:
 Informative Advertising: increase overall demand of your brand
 Persuasive Advertising: demand shifting to your brand
 Comparison: demand shifting away from your competitor (include complementary)
n customers distributed uniformly along the Hotelling’s line (more likely for mature market where demand doesn’t change).
\[ U_A = V  p_A  tx \\ U_B = V  p_B  t(1x) \]
For Persuasive advertising (highlight the value of the product to the consumer):
\[ U_A = A_A V  p_A  tx \]
or increase value (i.e., reservation price).
\[ U_A = \sqrt{Ad_A} V  p_A  tx \]
or more and more customers want the product (i.e., more customers think firm A product closer to what they want)
\[ U_A = V  p_A  \frac{tx}{\sqrt{Ad_A}} \]
Comparison Advertising:
\[ U_A = V  p_A  t\sqrt{Ad_{B}}x \\ U_B = V  p_B  t \sqrt{Ad_A}(1  x) \]
Find marginal consumers
\[ V  p_A  t\sqrt{Ad_{B}}x = V  p_B  t \sqrt{Ad_A}(1  x) \]
\[ x = \frac{1}{t \sqrt{Ad_A} + t \sqrt{Ad_B}} (p_A + p_B + t \sqrt{Ad_A}) \]
then profit functions are (make sure the profit function is concave)
\[ \pi_A = p_A x n  \phi Ad_A \\ \pi_B = p_B (1x) n  \phi Ad_B \]
where
\(\phi\) = per unit cost of advertising (e.g., TV advertising vs. online advertising in this case, TV advertising per unit cost is likely to be higher than online advertising per unit cost)
t can also be thought of as return on advertising (traditional Hotelling’s model considers t as transportation cost)
Equilibrium prices conditioned on advertising
\[ \frac{d \pi_A}{p_A} =  \frac{d}{p_A} () \\ \frac{d \pi_B}{p_B} = \frac{d}{p_B} \]
Then optimal pricing solutions are
\[ p_A = \frac{2}{3} t \sqrt{Ad_A} + \frac{1}{3} t \sqrt{Ad_B} \\ p_B = \frac{1}{3} t \sqrt{Ad_A} + \frac{2}{3} t \sqrt{Ad_B} \]
Prices increase with the intensities of advertising (if you invest more in advertising, then you charge higher prices). Each firm price is directly proportional to their advertising, and you will charger higher price when your competitor advertise as well.
Then, optimal advertising (with the optimal prices) is
\[ \frac{d \pi_A}{d Ad_A} \\ \frac{d \pi_B}{d Ad_B} \]
Hence, Competitive equilibrium is
\[ Ad_A = \frac{25 t^2 n^2}{576 \phi^2} \\ Ad_B = \frac{25t^2 n^2}{576 \phi^2} \\ p_A = p_B = \frac{5 t^2 n }{24 \phi} \]
As cost of advertising (\(\phi\)), firms spend less on advertising
Higher level of return on advertising (\(t\)), firms benefit more from advertising
With advertising in the market, the equilibrium prices are higher than if there were no advertising.
Since colluding on prices are forbidden, and colluding on advertising is hard to notice, firms could potential collude on advertising (e.g., pulsing).
Assumption:
 Advertising decision before pricing decision (reasonable because pricing is earlier to change, while advertising investment is determined at the beginning of each period).
Collusive equilibrium (instead of using \(Ad_A, Ad_B\), use \(Ad\)  set both advertising investment equal):
\[ Ad_A = Ad_B = \frac{t^2 n^2}{16 \phi^2} > \frac{25t^2 n^2}{576 \phi^2} \]
Hence, collusion can be make advertising investment equilibrium higher, which makes firms charge higher prices, and customers will be worse off. (more reference Aluf and Shy  check Modeling Seminar Folder  Advertising).
Combine both Comparison and Persuasive Advertising
\[ U_A = V  p_A  tx \frac{\sqrt{Ad_B}}{\sqrt{Ad_A}} \\ U_B = V  p_B  t(1x) \frac{\sqrt{Ad_A}}{\sqrt{Ad_B}} \]
Informative Advertising
 Increase number of n customers (more likely for new products where the number of potential customers can change)
How do we think about customers, how much to consume. People consume more when they have more availability, and less when they have less in stock (Ailawadi and Neslin 1998)
VillasBoas (1993)
 Under monopoly, firms would be better off to pulse (i.e., alternate advertising between a minimum level and efficient amount of advertising) because of the Sshaped of the advertising response function.
Model assumptions:
 The curve of the advertising response function is Sshaped
 Markov strategies: what firms do in this period depends on what might affect profits today or in the future (independent of the history)
Propositions:
 “If the loss from lowering the consideration level is larger than the efficient advertising expenditures, the unique Markov perfect equilibrium is for firms to advertise, whatever the consideration levels of both firms are.”
Nelson (1974)
Quality of a brand is determined before a purchase of a brand is “search qualities”
Quality that is not determined before a purchase is “experience qualities”
Brand risks credibility if it advertises misleading information, and pays the costs of processing nonbuying customers
There is a reverse association between quality produced and utility adjusted price
Firms that want to sell more advertise more
Firms advertise to their appropriate audience, i.e., “those whose tastes are best served by a given brand are those most likely to see an advertisement for that brand” (p. 734).
Advertising for experience qualities is indirect information while advertising for search qualities is direct information . (p. 734).
Goods are classified based on quality variation (i.e., whether the quality variation was based on searhc of experience).

3 types of goods
experience durable
experience nondurable
search goods
Experience goods are advertised more than search goods because advertisers increase sales via increasing the reputability of the sellers.
The marginal revenue of advertisement is greater for search goods than for experience goods (p. 745). Moreover, search goods will concentrate in newspapers and magazines while experience goods are seen on other media.
For experience goods, WOM is better source of info than advertising (p. 747).
Frequency of purchase moderates the differential effect of WOM and advertising (e.g., for low frequency purchases, we prefer WOM) (p. 747).
When laws are moderately enforced, deceptive advertising will happen (too little law, people would not trust, too much enforcement, advertisers aren’t incentivized to deceive, but moderate amount can cause consumers to believe, and advertisers to cheat) (p. 749). And usually experience goods have more deceptive advertising (because laws are concentrated here).
Iyer, Soberman, and VillasBoas (2005)
Firms advertise to their targeted market (those who have a strong preference for their products) than competitor loyalists, which endogenously increase differentiation in the market, and increases equilibrium profits
Targeted advertising is more valuable than target pricing. Target advertising leads to higher profits regardless whether firms have target pricing. Target pricing increased competition for comparison shoppers (no improvement in equilibrium profits). (p. 462  463).
Comparison shoppers size:
\[ s = 1  2h \]
where \(h\) is the market size of each firm’s consumers (those who prefer to buy product from that firm). Hence, \(h\) also represents the differentiation between the two firms
See table 1 (p. 469).
\(A\) is the cost for advertising the entire market
\(r\) is the reservation price
Yuxin Chen et al. (2009)
Combative vs. constructive advertising

Informative complementary and persuasive advertising
Informative: increase awareness, reduce search costs, increase product differentiation
Complementary (under comparison): increase utility by signaling social prestige
Persuasive: decrease price sensitivity (include combative)

Consumer response moderates the effect of combative adverting on price competition:
It decreases price competition
It increases price competition when (1) consumers preferences are biased (firms that advertise have their products favored by the consumers), (2) disfavor firms can’t advertise and only respond with price. because advertising war leads to a price war (when firms want to increase their own profitability while collective outcome is worse off).
21.11 Product Differentiation
Horizontal differentiation: different consumers prefer different products
Vertical differentiation: where you can say one good is “better” than the other.
Characteristics approach: products are the aggregate of their characteristics.
21.12 Product Quality, Durability, Warranties
Horizontal Differentiation
\[ U = V p  t (\theta  a)^2 \]
Vertical Differentiation
\[ U_B = \theta s_B  p_B \\ U_A = \theta s_A  p_A \]
Assume that product B has a higher quality
\(\theta\) is the position of any consumer on the vertical differentiation line.
When \(U_A < 0\) then customers would not buy
Point of indifference along the vertical quality line
\[ \theta s_B  p_B = \theta s_A  p_A \\ \theta(s_B  s_A) = p_B  p_A \\ \bar{\theta} = \frac{p_B  p_A}{s_B  s_A} \]
If \(p_B = p_A\) for every \(\theta\), \(s_B\) is preferred to \(s_A\)
 K. S. Moorthy (1988)
\[ \pi_A = (p_A  c s_A^2) (Mktshare_A) \\ \pi_B = (p_B  cs_B^2) (Mktshare_B) \\ U_A = \theta s_A  p_A = 0 \\ \bar{\theta}_2 = \frac{p_A}{s_A} \]
 Wauthy (1996)
\(\frac{b}{a}\) = such that market is covered, then
\[ 2 \le \frac{b}{a} \le \frac{2s_2 + s_1}{s_2  s_1} \]
for the market to be covered
In vertical differentiation model, you can’t have both \(\theta \in [0,1]\) and full market coverage.
Alternatively, you can also specify \(\theta \in [1,2]; [1,4]\)
\[ \theta \in \begin{cases} [1,4] & \frac{b}{a} = 4 \\ [1,2] & \frac{b}{a} = 2 \end{cases} \]
Under Asymmetric Information
Adverse Selection: Before contract: Information is uncertain
Moral Hazard: After contract, intentions are unknown to at least one of the parties.
Alternative setup of Akerlof’s (1970) paper
Used cars quality \(\theta \in [0,1]\)
Seller  car of type \(\theta\)
Buyer = WTP = \(\frac{3}{2} \theta\)
Both of them can be better if the transaction occurs because buyer’s WTP for the car is greater than utility received by seller.
 Assume quality is observable (both sellers and buyers do know the quality of the cars):
Price as a function of quality \(p(\theta)\) where \(p(\theta) \in [\theta, 3/2 \theta]\) both parties can be better off
 Assume quality is unobservable (since \(\theta\) is uniformly distributed) (sellers and buyers do not know the quality of the used cars):
\[ E(\theta) = \frac{1}{2} \]
then \(E(\theta)\) for sellers is \(1/2\)
\(E(\theta)\) for buyer = \(3/2 \times 1/2\) = 3/4
then market happens when \(p \in [1/2,3/4]\)
 Asymmetric info (if only the sellers know the quality)
Seller knows \(\theta\)
Buyer knows \(\theta \sim [0,1]\)
From seller perspective, he must sell at price \(p \ge \theta\) and
From buyer perspective, quality of cars on sale is between \([0, p]\). Then, you will have a smaller distribution than \([0,1]\)
If \(E[(\theta)  \theta \le p] = 0.5 p\)
Buyers’ utility is \(3/4 p\) but the price he has to pay is \(p\) (then market would not happen)
21.12.1 Akerlof (1970)
 This paper is on adverse selection
 The relationship between quality and uncertainty (in automobiles market)
 2 x 2 (used vs. new, good vs. bad)
\(q\) = probability of getting a good car = probability of good cars produced
and \((1q)\) is the probability of getting a lemon
Used car sellers have knowledge about the probability of the car being bad, but buyers don’t. And buyers pay the same price for a lemon as for a good car (info asymmetry).
Gresham’s law for good and bad money is not transferable (because the reason why bad money drives out goo d money because of even exchange rate, while buyers of a car cannot tell if it is good or bad).
21.12.1.1 Asymmetrical Info
Demand for used automobiles depends on price quality:
\[ Q^d = D(p, \mu) \]
Supply for used cars depends on price
\[ S = S(p) \]
and average quality depends on price
\[ \mu = \mu(p) \]
In equilibrium
\[ S(p) = D(p, \mu(p)) \]
At no price will any trade happen
Example:
Assume 2 groups of graders:

First group: \(U_1 = M = \sum_{i=1}^n x_i\) where
\(M\) is the consumption of goods other than cars
\(x_i\) is the quality of the ith car
n is the number of cars
Second group: \(U_2 = M + \sum_{i=1}^n \frac{3}{2} x_i\)
Group 1’s income is \(Y_1\)
Group 2’s income is \(Y_2\)
Demand for first group is
\[ \begin{cases} D_1 = \frac{Y_1}{p} & \frac{\mu}{p}>1 \\ D_1 = 0 & \frac{\mu}{p}<1 \end{cases} \]
Assume we have uniform distribution of automobile quality.
Supply offered by first group is
\[ S_2 = \frac{pN}{2} ; p \le 2 \]
with average quality \(\mu = p/2\)
Demand for second group is
\[ \begin{cases} D_2 = \frac{Y_2}{p} & \frac{3 \mu}{2} >p \\ D_2 = 0 & \frac{3 \mu}{2} < p \end{cases} \]
and supply by second group is \(S_2 = 0\)
Thus, total demand \(D(p, \mu)\) is
\[ \begin{cases} D(p, \mu) = (Y_2 + Y_1) / p & \text{ if } p < \mu \\ D(p, \mu) = (Y_2)/p & \text{ if } \mu < p < 3\mu /2 \\ D(p, \mu) = 0 & \text{ if } p > 3 \mu/2 \end{cases} \]
With price \(p\), average quality is \(p/2\), and thus at no price will any trade happen
21.12.1.2 Symmetric Info
Car quality is uniformly distributed \(0 \le x \le 2\)
Supply
\[ \begin{cases} S(p) = N & p >1 \\ S(p) = 0 \end{cases} \]
Demand
\[ \begin{cases} D(p) = (Y_2 + Y_1) / p & p < 1 \\ D(p) = Y_2/p & 1 < p < 3/2 \\ D(p) = 0 & p > 3/2 \end{cases} \]
In equilibrium
\[ \begin{cases} p = 1 & \text{ if } Y_2< N \\ p = Y_2/N & \text{ if } 2Y_2/3 < N < Y_2 \\ p = 3/2 & \text{ if } N < 2 Y_2 <3 \end{cases} \]
This model also applies to (1) insurance case for elders (over 65), (2) the employment of minorities, (3) the costs of dishonesty, (4) credit markets in underdeveloped countries
To counteract the effects of quality uncertainty, we can have
 Guarantees
 Brandname good
 Chains
 Licensing practices
21.12.2 Spence (1973)
Built on (Akerlof 1970) model
Consider 2 employees:
Employee 1: produces 1 unit of production
Employee 2: produces 2 units of production
We have \(\alpha\) people of type 1, and \(1\alpha\) people of type 2
Average productivity
\[ E(P) = \alpha + 2( 1 \alpha) = 2 \alpha \]
You can signal via education.
To model cost of education,
Let E to be the cost of education for type 1
E/2 to be the cost education for type 2
If type 1 signals they are highquality worker, then they have to go through the education and cost is E, and net utility of type 1 worker
\[ 2  E < 1 \\ E >1 \]
If type 2 signals they are highquality worker, then they also have to go through the education and cost is E/2 and net utility of type 2 worker is
\[ 2  E/2 > 1 \\ E< 2 \]
If we keep \(1 < E < 2\), then we have separating equilibrium (to have signal credible enough of education )
21.12.3 S. Moorthy and Srinivasan (1995)
Moneyback guarantee signals quality
Transaction cost are those the seller or buyer has to pay when redeeming a moneyback guarantee
Moneyback guarantee does not include product return (buyers have to incur expense), but guarantee a full refund of the purchase price.
If signals are costless, there is no difference between moneyback guarantees and price

But signal are costly,
Under homogeneous buyers, lowquality sellers cannot mimic highquality sellers’ strategy (i.e., moneyback guarantee)

Under heterogeneous buyers,
when transaction costs are too high, the seller chooses either not to use moneyback guarantee strategy or signal through price.

When transaction costs are moderate, there is a critical value of seller transaction costs where
below this point, the highquality sellers’ profits increase with transaction costs
above this point, the highquality sellers’ profits decrease with transaction costs
Uninformative advertising (“moneyburning”) is defined as expenditures that do not affect demand directly. is never needed

Moral hazard:
 Consumers might exhaust consumption within the moneyback guarantee period
Model setup
Highquality sellers (\(h\))  Lowquality sellers (\(l\))  

Cost \(c_h > c_l\) 
\(c_h\)  \(c_l\) 
21.13 Bargaining
Books:
Abhinay Muthoo  Bargaining Theory with Applications (1999) (check books folder)
Josh Nash  Nash Bargaining (1950)
Allocation of scare resources
Buyers & Sellers  Type 

Many buyers & many sellers  Traditional markets 
Many buyers & one seller  Auctions 
One buyer & one seller  Bargaining 
Allocations of

Determining the share before gametheoretic bargaining
Use a judge/arbitrator
Meetinthemiddle
Forced Final: If an agreement is not reached, one party will use take it or leave it
Art: Negotiation
Science: Bargaining
Game theory’s contribution: to the rules for the encounter
Area that is still fertile for research
21.13.1 Noncooperative
Outline for noncooperative bargaining
 The rules
Takeitorleaveit Offers
Bargain over a cake
If you accept, we trade
If you reject, no one eats
Under perfect info, there is a simple rollback equilibrium
In general, bargaining takes on a “takeitorcounteroffer” procedure

If time has value, both parties prefer to trade earlier to trade later
 E.g., labor negotiations  later agreements come at a price of strikes, work stoppages
Delays imply less surplus left to be shared among the parties
Twostage bargaining
I offer a proportion, \(p\) , of the cake to you
If rejected, you may counteroffer (and \(\delta\) of the cake melts)

Payoffs:
In the first period: 1p, p
In second period: \((1\delta) (1p),(1\delta)p\)

Since period 2 is the final period, this is just like a takeitorleaveit offer
 You will offer me the smallest piece that I will accept, leaving you with all of \(1\delta\) and leaving me with almost 0
Rollback: then in the first period: I am better off by giving player B more than what he would have in period 2 (i.e., give you at least as much surplus)
You surplus if you accept in the first period is \(p\)
Accept if: your surplus in first period greater than your surplus in second period \(p \ge 1  \delta\)
IF there is a second stage, you get \(1  \delta\) and I get 0
You will reject any offer in the first stage that does not offer you at least \(1  \delta\)
In the first period, I offer you \(1  \delta\)
Note: the more patient you are (the slower the cake melts) the more you receive now

Whether first or second mover has the advantage depends on \(\delta\).
If \(\delta\) is high (melting fast), then first mover is better.
If \(\delta\) is low (melting slower), then second mover is better.
Either way  if both players think, agreement would be reached in the first period
In any bargaining setting, strike a deal as early as possible.
Why doesn’t this happen in reality?
reputation building
lack of information
Why bargaining doesn’t happen quickly? Information asymmetry
 Likelihood of success (e.g., uncertainty in civil lawsuits)
Lessons:
Rules of the bargaining game uniquely determine the bargain outcome
which rules are better for you depends on patience, info
What is the smallest acceptable piece? Trust your intuition
delays are always less profitable: Someone must be wrong
Nonmonetary Utility

each side has a reservation price
 LIke in civil suit: expectation of wining
The reservation price is unknown

One must:
probabilistically determine best offer
but  probability implies a chance that non bargain will take place
Example:
Company negotiates with a union

Two types of bargaining:
Union makes a takeitorleaveit offer

Union makes a n offer today. If it’s rejected, the Union strikes, then makes another offer
 A strike costs the company 10% of annual profits.
Probability that the company is “highly profitable”, ie., 200k is \(p\)

If offer wage of $150k
Definitely accepted
Expected wage = $150K

If offer wage of $200K
Accepted with probability \(p\)
Expected wage = $200k(p)

\(p = .9\) (90% chance company is highly profitable
best offer: ask for $200K wage
Expected value of offer: \(.9 *200= 180\)

\(p = .1\) (10% chance company is highly profitable
best offer: ask for $200K wage
Expected value of offer: \(.1 *200= 20\)
If ask for $10k, get $150k
not worth the risk to ask for more
If firstperiod offer is rejected: A strike costs the company 10% of annual profits
Strike costs a highvalue company more than a low value company
Use this fact to screen

What if the union asks for $170k in the first period?
Low profit firms ($150k) rejects  as can’t afford to take

HIgh profit firm must guess what will happen if it rejects
Best case: union strikes and then asks for only $140k (willing to pay for some cost of strike), but not all)
In the mean time: strike cost the company $20K
Highprofit firm accepts
Separating equilibrium
only highprofit firms accept the first period
If offer is rejected, Union knows that it is facing a lowprofit firm
Ask for $140k
What’s happening

Union lowers price after a rejection
Looks like giving in
looks like bargaining

Actually, the union is screening its bargaining partner
Different “types” of firms have different values for the future
Use these different values to screen
Time is used as a screening device
21.13.2 Cooperative
two people diving cash
If they do not agree, they each get nothing
They cant divide up more than the whole thing
21.13.3 Nash (1950)
Bargaining, bilateral monopoly (nonzerosum two person game).
Non action taken by one individual (without the consent of the other) can affect the other’s gain.

Assumptions:
Rational individuals (maximize gain)
Full knowledge: tastes and preferences are known
Transitive Ordering: \(A>C\) when \(A>B\), \(B>C\). Also related to substitutability if two events are of equal probability
Continuity assumption

Properties:
\(u(A) > u(B)\) means A is more desirable than B where \(u\) is a utility function

Linearity property: If \(0 \le p \le 1\), then \(u(pA + (1p)B) = pu(A) + (1p)u(B)\)
 For two person: \(p[A,B] + (1p)[C,D] = [pA  (1p)C, pB + (1p)D]\)

Anticipation = \(p A  (1p) B\) where
\(p\) is the prob of getting A
A and B are two events.
\(u_1, u_2\) are utility function
\(c(s)\) is the solution point in a set S (compact, convex, with 0)
Assumptions:

If \(\alpha \in S\) s.t there is \(\beta \in S\) where \(u_1(\beta) > u_2(\alpha)\) and \(u_2(\beta) > u_2(\alpha)\) then \(\alpha \neq c(S)\)
 People try to maximize utility
If \(S \in T\) , c(T) is in S then \(c(T) = c(S)\)

If S is symmetric with respect to the line \(u_1 = u_2\), then \(c(S)\) is on the line \(u_1 = u_2\)
 Equality of bargaining
21.13.4 Iyer and VillasBoas (2003)
 Presence of a powerful retailer (e.g., Walmart) might be beneficial to all channel members.
21.13.5 Desai and Purohit (2004)
2 customers segment: hagglers, nonhagglers.
When the proportion of nonhagglers is sufficient high, a haggling policy can be more profitable than a fixedprice policy
21.14 Pricing and Search Theory
21.14.1 Varian and Purohit (1980)

From Stigler’s seminar paper (Stiglitz and Salop 1982; Salop and Stiglitz 1977), model of equilibrium price dispersion is born

Spatial price dispersion: assume uninformed and informed consumers
 Since consumers can learn from experience, the result does not hold over time
Temporal price dispersion: sales


This paper is based on
Stiglitz: assume informed (choose lowest price store) and uninformed consumers (choose stores at random)
Shilony (Shilony 1977): randomized pricing strategies
\(I >0\) is the number of informed consumers
\(M >0\) is the number of uninformed consumers
\(n\) is the number of stores
\(U = M/n\) is the number of uninformed consumers per store
Each store has a density function \(f(p)\) indicating the prob it charges price \(p\)
Stores choose a price based on \(f(p)\)
Succeeds if it has the lowest price among n prices, then it has \(I + U\) customers
Fails then only has \(U\) customers
Stores charge the same lowest price will share equal size of informed customers
\(c(q)\) is the cost curve
\(p^* = \frac{c(I+U)}{(I+U}\) is the average cost with the maximum number of customers a store can get
Prop 1: \(f(p) = 0\) for \(p >r\) or \(p < p^*\)
Prop 2: No symmetric equilibrium when stores charge the same price
Prop 3: No point masses in the equilibrium pricing strategies
Prop 4: If \(f(p) >0\), then
\[ \pi_s(p) (1F(p))^{n1} + \pi_f (p) [1(1F(p))^{n1}] =0 \]
Prop 5: \(\pi_f (p) (\pi_f(p)  \pi_s (p))\) is strictly decreasing in \(p\)
Prop 6: \(F(p^* _ \epsilon) >0\) for any \(\epsilon> 0\)
Prop 7: \(F(r \epsilon) <1\) for any \(\epsilon > 0\)
Prop 8: No gap \((p_1, p_2)\) where \(f(p) \equiv 0\)
Decision to be informed can be endogenous, and depends on the “full price” (search costs + fixed cost)
21.14.2 Lazear (1984)
Retail pricing and clearance sales
Goods’ characteristics affect pricing behaviors
Market’s thinness can affect price volatility
Relationship between uniqueness of a goods and its price
Price reduction policies as a function of shelf time
Set up
Single period model
\(V\) = the price of the only buyer who is willing to purchase the product
\(f(V)\) is the density of V (firm’s prior)
\(F(V)^2\) is its distribution function
Firms try to
\[ \underset{R}{\operatorname{max}} R[1  F(R)] \]
where \(R\) is the price
\(1  F(R)\) is the prob that \(V > R\)
Assume \(V\) is uniform \([0,1]\) then
\(F(R) = R\) so that the optimum is \(R = 0.5\) with expected profits of \(0.25\)
Twoperiod model
Failures in period 1 implies \(V<R_1\).
Hence, based on Bayes’ theorem, the posterior distribution in period 2 is \([0, R_1]\)
\(F_2(V) = V/R_1\) (posterior distribution)
\(R_1\) affect (1) sales in period 1, (2) info in period 2
Then, firms want to choose \(R_1, R_2\) .Firms try to
\[ \underset{R_1, R_2}{\operatorname{max}} R_1[1  F(R_1)] + R_2 [1F_2(R_2)]F(R_1) \]
Then, in period 2, the firms try to
\[ \underset{R_2}{\operatorname{max}} R_2[1  F_2(R_2)] \]
Based on Bayes’ Theorem
\[ F_2(R_2) = \begin{cases} F(R_2)/ F(R_1) & \text{for } R_2 < R_1 \\ 1 & \text{otherwise} \end{cases} \]
Due to FOC, second period price is always lower than first price price
Expected profits are higher than that of oneperiod due to higher expected probability of a sale in the twoperiod problem.
But this model assume
no brand recognition
no contagion or network effects
In thin markets and heterogeneous consumers
we have \(N\) customers examine the good with the prior probability \(P\) of being shoppers, and \(1P\) being buyers who are willing to buy at \(V\)
There are 3 types of people
 customers = all those who inspect the good
 buyers = those whose value equal \(V\)
 shoppers = those who value equal \(0\)
An individual does not know if he or she is a buyer or shopper until he or she is a customer (i.e., inspect the goods)
Then, firms try to
\[ \begin{aligned} \underset{R_1, R_2}{\operatorname{max}} & R_1(\text{prob sale in 1}) + R_2 (\text{Posterior prob sale in 2})\times (\text{Prob no sale in 1}) \\ & R_1 \times ( F(R_1))(1P^N) + R_2 \{ (1F_2(R_2))(1 P^N) \} \times \{ 1  [(1  F(R_1))(1P^N)] \} \end{aligned} \]
Based on Bayes’ Theorem, the density for period 2 is
\[ f_2(V) = \begin{cases} \frac{1}{R_1 (1 P^N) + P^N} \text{ for } V \le R_1 \\ \frac{P^N}{R_1 (1 P^N) + P^N} \text{ for } V > R_1 \end{cases} \]
Conclusion:
As \(P^N \to 1\) (almost all customers are shoppers), there is not much info to be gained. Hence, 2period is no different than 2 independent oneperiod problems. Hence, the solution in this case is identical to that of oneperiod problem.
When \(P^N\) is small, prices start higher and fall more rapid as time unsold increases
When \(P^N \to 1\), prices tend to be constant.
\(P^N\) can also be thought of as search cost and info.
Observable Time patterns of price and quantity
Pricing is a function of
The number of customers \(N\)
The proportion of shoppers \(P\)
The firm’s beliefs about the market (parameterized through the prior on \(V\))
Markets where prices fall rapidly as time passes, the probability that the good will go unsold is low.
Goods with high initial price are likely to sell because high initial price reflects low \(P^N\)  low shoppers
Heterogeneity among goods
The more disperse prior leads to a higher expected price for a given mean. And because of longer time on shelf, expected revenues for such a product can be lower.
Fashion, Obsolescence, and discounting the future
The more obsolete, the more anxious is the seller
Goods that are “classic”, have a higher initial price, and its price is less sensitive to inventory (compared to fashion goods)
Discounting is irrelevant to the pricing condition due to constant discount rate (not like increasing obsolescence rate)
For nonunique good, the solution is identical to that of the oneperiod problem.
Simple model
Customer’s Valuation \(\in [0,1]\)
Firm’s decision is to choose a price \(p\) (label  \(R_1\))
Oneperiod model
Buy if \(V >R_1\) prob = \(1R_1\)
Not buy if \(V<R_1\) probability = \(R_1\)
\(\underset{R_1}{\operatorname{max}} [R_1][1R_1]\) hence, FOC \(R_1 = 1/2\), then total \(\pi = 1/2(1/2)^2 = 1/4\)
Twoperiod model
Two prices \(R_1, R_2\)
\(R_1 \in [0,1]\)
\(R_2 \in [0, R_1]\)
\[ \underset{R_1}{\operatorname{max}} [R_1][1R_1] + R_2 (1  R_2)(R_1) \]
\[ \underset{R_1}{\operatorname{max}} [R_2][\frac{R_1  R_2}{R_1}] \]
FOC \(R_2 = R_1/2\)
\[ \underset{R_1}{\operatorname{max}} R_1(1R_1) + \frac{R_1}{2}(1  \frac{R_1}{2}) (R_1) \]
FOC: \(R_1 = 2/3\) then \(R_2 = 1/3\)
\(N\) customers
Each customer could be a
shopper with probability p with \(V <0\)
buyer with probability \(1p\) with \(V > \text{price}\)
Modify equation 1 to incorporate types of consumers
\[ R_1(1  R_1)(1 p^N) + R_2 (1 R_2) R_1 (1p^N) [ 1  (1R_1)(1p^N)] \]
Reduce costs by
Economy of scale \(c(\text{number of units})\)
Economy of scope \(c(\text{number of types of products})\) (typically, due to the transfer of knowledge)
Experience effect \(c(\text{time})\) (is a superset of economy of scale)
Lal and Sarvary (1999)
Conventional idea: lower search cost (e.g., Internet) will increase price competition.

Assumptions:

Demand side: info about product attributes:
digital attributes (those can be communicated via the Internet)
nondigital attributes (those can’t)
Supply side: firms have both traditional and Internet stores.


Monopoly pricing can happen when
high proportion of Internet users
Not overwhelming nondigital attributes
Favor familiar brands
destination shopping
Monopoly pricing can lead to higher prices and discourage consumer from searching
Stores serve as acquiring tools, while Internet maintain loyal customers.
Kuksov (2004)
For products that cannot be changed easily (design),lower search costs lead to higher price competition
For those that can be easily changed, lower search costs lead to higher product differentiation, which in turn decreases price competition, lower social welfare, higher industry profits.
21.15 Pricing and Promotions
Extensively studied
Issue of Everyday Low price vs Hi/Lo pricing

Shortterm price discounts
offering tradedeals

consumer promotions
shelfprice discounts (used by everybody)
centsoff coupons (some consumers whose value of time is relatively low)
Loyalty is similar to inform under analytic modeling:
Uninformed = loyal
Informed = nonloyal
30 years back, few companies use price promotions
Effects of Shortterm price discounts

measured effects (84%) (Gupta 1988)
Brand switching (14%)
purchase acceleration (2%)
quantity purchased
elasticity of ST price changes is an order of magnitude higher

Other effects:
general trial (traditional reason)
encourages consumers to carry inventory hence increase consumption
higher sales of complementary products
small effect on store switching

Asymmetric effect (based on brand strength) (bigger firms always benefit more)
 expect of store brands
Negative Effects
Expectations of future promotions
Lowering of Reference Price
Increase in price sensitivity
Postpromotional dip
Trade Discounts
Shortterm discounts offered to the trade:

Advantages
Incentivize the trade to push our product
gets attention of sales force

Disadvantages
might not be passed onto the consumer
trade forward buys (hurts production plans)
hard to forecast demand
trade expects discounts in the future (cost of doing business)
Scanback can help increase retail pass through (i.e., encourage retailers to have consumer discounts)
Determinants of pass through

Higher when
Consumer elasticity is higher
promoting brand is stronger
shape of the demand function
lower frequency of promotions
(Online) Shelfprice discounts (Raju, Srinivasan, and Lal 1990)
 If you are a stronger brand you can discount infrequently because the weaker brands cannot predict when the stronger brands will promote. Hence, it has to promote more frequently
Coupons
Little over 1% get redeemed each year
The ability of centsoff coupons to price distribution has reduced considerably because of their very wide availability
Sales increases required to make freestandinginsert coupons profitable are not attainable
Coupon Design

Expiration dates
 Long vs short expiration dates: Stronger brands should have shorter windows (because a lot more of your loyalty customer base will utilize the coupons).

Method of distribution
Instore (is better)
Through the package
Targeted promotions
Package Coupons acquisition and retention tradeoffs
3 types of package coupons:
Peeloff (lots more use the coupons) lowest profits for the firm
inpacks (fewer customers will buy the products in the first period)
onpacks (customers buy the products and they redeem in the next period) best approach
Summary
Trade and consumer promotion are necessary
Consumer promotion (avoid shelf price discount/news paper coupons, use package coupons
strong interaction between advertising and promotion (area for more research)
3 degrees price discrimination
 Firstdegree: based on willingness to pay
 Seconddegree: based on quantity
 Thirddegree: based on memberships
 Fourthdegree: based on cost to serve
21.15.1 Narasimhan (1988)

Marketing tools to promote products:
Advertising
Trade promotions
Consumer promotions

Pricing promotions:
Price deals
Centsoff labels
coupons
Rebates
Brand loyalty can help explain the variation in prices (in competitive markets)

Firms try to make optimal tradeoff between
attracting brand switchers
loss of profits from loyal customers.
Deviation from the maximum price = promotion
Assumptions:
Firms with identical products, and cost structures (constant or declining). Noncooperative game.
Same reservation price

Three consumer segments:
Loyal to firm 1 with size \(\alpha_1 (0<\alpha_1<1)\)
Loyal to firm 2 with size \(\alpha_2(0 < \alpha_2 < \alpha_1)\) (asymmetric firm)
Switchers with size \(\beta (0 < \beta = 1  \alpha_1  \alpha_2)\)
Costless price change, no intertemporal effects (in quantity or loyalty)
To model \(\beta\) either
 \(d \in (b, a)\) is switch cost (individual parameter)
\[ \begin{cases} \text{buy brand 1} & \text{if } P_1 \le P_2  d \\ \text{buy brand 2} & \text{if } P_1 > P_2  d \end{cases} \]
 Identical switchers (same d)
 \(d = 0\) (extremely price sensitive)
For case 1, there is a pure strategy, while case 2 and 3 have no pure strategies, only mixed strategies
Details for case 3:
Profit function
\[ \Pi_i (P_i, P_j) = \alpha_i P_i + \delta_{ij} \beta P_i \]
where
\[ \delta_{ij} = \begin{cases} 1 & \text{ if } P_i < P_j \\ 1/2 & \text{ if } P_i = P_j \\ 0 & \text{ if } P_i > P_j \end{cases} \]
and \(i = 1,2, i \neq j\)
Prop 1: no pure Nash equilibrium
Mixed Strategy profit function
\[ \Pi_i (P_i) = \alpha_i P_i + Prob(P_j > P_i) \beta P_i + Prob (P_j = P_i) \frac{\beta}{2} P_i \]
where \(P_i \in S_i^*, i \neq j; i , j = 1, 2\)
Then the expected profit functions of the twoplayer game is
\[ \underset{F_i}{\operatorname{max}} E(\Pi_i) = \int \Pi_i (P_i) d F_i (P_i) \]
\(P_i \in S_i^*\)
such that
\[ \Pi_i \ge \alpha_i r \\ \int dF_i (P_i) = 1 \\ P_i \in S_i^* \]
21.15.2 Balachander, Ghosh, and Stock (2010)
 Bundle discounts can be more profitable than price promotions (in a competitive market) due to increased loyalty (which will reduce promotional competition intensity).
21.15.3 Goić, Jerath, and Srinivasan (2011)

Crossmarket discounts, purchases in a source market can get you a price discounts redeemable in a target market.
 Increase prices and sales in the source market.
21.16 Market Entry Decisions and Diffusion
Peter N. Golder and Tellis (1993)
Peter N. Golder and Tellis (2004)
Boulding and Christen (2003)
Van den Bulte and Joshi (2007)
21.17 Principalagent Models and Salesforce Compensation
21.17.1 Gerstner and Hess (1987)
21.17.2 Basu et al. (1985)
21.17.3 Raju and Srinivasan (1996)
Compare to (Basu et al. 1985), basic quota plan is superior in terms of implementation
Different from (Basu et al. 1985), basic quota plan has
 Shapeinduced nonoptimality: not a general curvilinear form
 Heterogeneityinduced nonoptimality: common rate across salesforce
However, only 1% of cases in simulation shows up with nonoptimality. Hence, minimal loss in optimality
Basic quota plan is a also robust against changes in
salesperson switching territory
territorial changes (e.g., business condition)
Heterogeneity stems from
Salesperson: effectiveness, risk level, disutility for effort, and alternative opportunity
Territory: Sales potential and volatility
Adjusting quotas can accommodate the heterogeneity
To assess nonoptimality, following Basu and Kalyanaram (1990)
Moral hazard: cannot assess salesperson’s true effort.
Assumptions:
The salesperson reacts to the compensation scheme by deciding on an effort level that maximizes his overall utility, i.e., the expected utility from the (stochastic) compensation minus the effort distuility.
Firm wants to maximize its profit
compensation is greater than saleperson’s alternative.
Dollar sales \(x_i \sim Gamma\) (because sales are nonnegative and standard deviation getting proportionately larger as the mean increases) with density \(f_i(x_it_i)\)
Expected sales per period
\[ E[x_i t_i] = h_i + k_i t_i , (h_i > 0, k_i >0) \]
where
 \(h_i\) = base sales level
 \(k_i\) = effectiveness of effort
and \(1/\sqrt{c}\) = uncertainty in sales (coefficient of variation) = standard deviation / mean where \(c \to \infty\) means perfect certainty
salesperson’s overall utility
\[ U_i[s_i(x_i)]  V_i(t_i) = \frac{1}{\delta_i}[s_i (x_i)]^{\delta_i}  d_i t_i^{\gamma_i} \] where
 \(0 < \delta_i <1\) (greater \(\delta\) means less riskaverse salesperson)
 \(\gamma_i >1\) (greater \(\gamma\) means more effort)
 \(V_i(t_i) = d_i t_i^{\gamma_i}\) is the increasing disutility function (convex)
21.17.4 Lal and Staelin (1986)
A menu of compensation plans (salesperson can select, which depends on their own perspective)
Proposes conditions when it’s optimal to offer a menu

Under (Basu et al. 1985), they assume
Salespeople have identical risk characteristics
identical reservation utility
identical information about the environment
When this paper relaxes these assumptions, menu of contract makes sense
If you cannot distinguish (or have a selection mechanisms) between high performer and low performer, a menu is recommended. but if you can, you only need 1 contract like (Basu et al. 1985)
21.17.5 Simester and Zhang (2010)
21.19 Marketing Resource Allocation Models
This section is based on (Mantrala, Sinha, and Zoltners 1992)
21.19.1 Case study 1
Concave sales response function
 Optimal vs. proportional at different investment levels
 Profit maximization perspective of aggregate function
\[ s_i = k_i (1 e^{b_i x_i}) \]
where
 \(s_i\) = currentperiod sales response (dollars / period)
 \(x_i\) = amount of resource allocated to submarket i
 \(b_i\) = rate at which sales approach saturation
 \(k_i\) = sales potential
Allocation functions

Fixed proportion
\(R_i\) = Investment level (dollars/period)
\(w_i\) = fixed proportion or weights
\[ \hat{x}_i = w_i R; \\ \sum_{t=1}^2 w_t = 1; 0 < w_t < 1 \]

Informed allocator
 optimal allocations via marginal analysis (maximize profits)
\[ max C = m \sum_{i = 1}^2 k_i (1 e^{b_i x_i}) \\ x_1 + x_2 \le R; x_i \ge 0 \text{ for } i = 1,2 \\ x_1 = \frac{1}{(b_1 + b_2)(b_2 R + \ln(\frac{k_1b_1}{k_2b_2})} \\ x_2 = \frac{1}{(b_1 + b_2)(b_2 R + \ln(\frac{k_2b_2}{k_1b_1})} \]
21.20 Mixed Strategies
Games with finite number, and finite strategy for each player, there will always be a Nash equilibrium (might not be pure Nash, but always mixed)
Extended game
Suppose we allow each player to choose randomizing strategies
For example, the server might serve left half of the time, and right half of the time
In general, suppose the server serves left a fraction \(p\) of the time
What is the receiver’s best response?
Best Responses
If \(p = 1\), the receiver should defend to the left
\(p = 0\), the receiver should defend to the right
The expected payoff to the receiver is
\(p \times 3/4 + (1p) \times 1/4\) if defending left
\(p \times 1/4 + (1p) \times 3/4\) if defending right
Hence, she should defend left if
\(p \times 3/4  (1p)\times 1/4 > p \times 1/4 + (1p) \times 3/4\)
We said to defend left whenever
\[ p \times 3/4  (1p)\times 1/4 > p \times 1/4 + (1p) \times 3/4 \]
Server’s Best response
Suppose that the receiver goes left with probability \(q\)
if \(q = 1\), the server should serve right
If \(q = 0\), the server should server left
Hence, serve left if \(1/4 \times q + 3/4 \times (1q) > 3/4\times q + 1/4 \times (1q)\)
Simplifying, he should serve left if \(q < 1/2\)
Mixed strategy equilibrium:
A mixed strategy equilibrium is a pair of mixed strategies that are mutual best responses
In the tennis example, this occurred when each play chose a 5050 mixture of left and right
Your best strategy is when you make the option given to your opponent is obsolete.
A player chooses his strategy to make his rival indifferent
A player earns the same expected payoff for each pure strategy chosen with positive probability
Important property: When player’s own payoff form a pure strategy goes up (or down), his mixture does not change.
21.21 Bundling
Equipment  Installation  

Customer Type 1  $8,000  $2,000 
Customer Type 2  $5,000  $3,000 
Say we have equal numbers of type 1 and type 2, then you would like to charge $5,000 for the equipment and $2,000 for installation when considering equipment and installment equally . If you price it separately, then your total profit is 14,000.
But you bundle, you get $16,000.
If we know that bundles work. But we don’t see every company does ti?
Because it depends on the number of type 1 and 2 customers, and negative correlation in willingness to pay.
For example:

Information Products
margin cost is close to 0.
Bundling of info products is very easy
hence always bundle
21.22 Market Entry and Diffusion
Product Life Cycle model
Bass (1969)

Discussion of sales has 2 types
Innovators
Imitators
\(p\) = coefficient of innovation (fraction of innovators of the untapped market who buy in that period)
\(q\) = coefficient of imitation (fraction of the interaction which lead to sales in that period)
\(M\) = market potential
\(N(t)\) = cumulative sales till time \(t\)
\(M  N(t)\) = the untapped market
Sales in any time is People buying because of the pure benefits of the product, plus people buy the product after interacting with people who owned the product.
\[ S(t) = p(M N(t)) + q \frac{N(t)}{M} [MN(t)] \\ = pM + (qp) N(t)  \frac{q}{M} [N(t)]^2 \]
one can estimate \(p,q,M\) from data
\(q > p\) (coefficient of imitation > coefficient of innovation) means that you have life cycle (bellshaped curve)
Peter N. Golder and Tellis (1993)
Previous use
limited databases (PIM and ASSESOR) (Urban et al. 1986)
exclusion of nonsurvivors
singleinformant selfreport
New dataset overcomes these limitations and show 50% of the market pioneers fail, and their mean share is much lower
Early market leaders have greater longterm success and enter on average 13 years after pioneers.
Definitions (p. 159)
 Inventor: firms that develop patent or important technologies in a new product category
 Product pioneer: the first firm to develop a working model or sample in a new product category
 Market pioneer is the first firm to sell in a new product category
 Product category: a group of close substitutes
Boulding and Christen (2003)
At the business level, being the leader can give you longterm profit disadvantage from the samples of consumer and industrial goods.
Firsttomarket leads to an initial profit advantage, which last about 12 to 14 years before becoming longterm disadvantage.
Consumer learning (education), market position (strong vs. weak) and patent protection can moderate the effect of firstmover on profit.
Peter N. Golder and Tellis (2004)
Research on product life cycle (PLC)
Consumer durables typically grow 45 per year over 8 years, then slowdown when sales decline by 15%, and stay below those of the previous peak for 5 years.
Slowdown typically happens when the product penetrates 3550% of the market
large sales increases (at first) will have larger sales declines (at slowdown).
Leisureenhancing products tend to have higher growth rate and shorter growth stages than non leisureenhancing products
Timesaving products have lower growth rates and longer growth stages than non timesaving products
Lower likelihood of slowdown correlates with steeper price reduction, lower penetration, and higher economic growth
A hazard model gives reasonable prediction of the slowdown and takeoff.
Van den Bulte and Joshi (2007)

Innovations market have two segments:
Influentials: aware of new developments an affect imitators
Imitators: model after influentials.
This market structure is reasonable because it exhibits consistent evidence with the prior research and market (e.g., dip between the early and later parts fo the diffusion curve).
” Erroneously specifying a mixedinfluence model to a mixture process where influentials act independently from each other can generate systematic changes in the parameter values reported in earlier research.”
Twosegments model performs better than the standard mixedinfluence, the Gamma/Shifted Gompertz, the WeibullGamma models, and similar to the KarmeshuGoswami mixed influence model.
21.23 PrincipalAgent Models and Salesforce Compensation
Key Question:
 Ensuring agents exert effort
 Design compensation plans such that workers exert high effort?
Designing contracts:
Effort can be monitored
Monitoring costs are too high
Timing
 Manger designs the construct
 manager offers the construct and worker chooses to accept
 Worker decides the extent of effort
 Outcome is observed and wage is given to the worker
Scenario 1: Certainty
e = effort put in by worker
2 levels of e
 2 if he works hard
 0 if he shirks
Reservation utility = 10 (other alternative: can work somewhere else, or private money allows them not to work)
Agent’s Utility
\[ U = \begin{cases} w  e & \text{if he exerts effort e} \\ 10 & \text{if he works somewhere else} \end{cases} \]
Revenue is a function of effort
\[ R(e) = \begin{cases} H & \text{if } e = 2 \\ L & \text{if } e = 0 \end{cases} \]
Contract
\(w^H\) = wage if \(R(e) = H\)
\(w^L\) = wage if \(R(e) = L\)
Constraints:
Worker has to participate in this labor market  participation constraint \(w^H  2 \ge 10\)
Incentive compatibility constraint (ensure that the works always put in the effort and the manager always pay for the higher wage): \(w^H  2 \ge w^L 0\)
Hence,
\[ w^H = 12 \\ w^L = 10 \]
Thus, contract is simple because of monitoring
Scenario 2: Under uncertainty
\[ R(2) = \begin{cases} H & \text{w/ prob 0.8} \\ L & \text{w/ prob 0.2} \end{cases} \\ R(0) = \begin{cases} H & \text{w/ prob 0.4} \\ L & \text{w/ prob 0.6} \end{cases} \]
Agent Utility
\[ U = \begin{cases} E(w)  e & \text{if effort e is put} \\ 10 & \text{if they choose outside option} \end{cases} \]
Constraints:
Participation Constraint: \(0.8w^H + 0.2w^L 2 \ge 10\)
Incentive compatibility constraint: \(0.8w^H + 0.2w^L  2 \ge 0.4 w^H + 0.6w^L  0\)
Thus,
\[ w^H = 13 \\ w^L = 8 \]
Expected wage bill that the manager has to pay:
\[ 13\times 0.8 + 8 \times 0.2 = 12 \]
Hence, the expected money the manager has to pay is the same for both cases (certainty vs. uncertainty)
Scenario 3: Asymmetric Information
Degrees of risk aversion
Manger perspective
\[ R(2) = \begin{cases} H & \text{w/ prob 0.8} \\ L & \text{w/ prob 0.2} \end{cases} \]
Worker perspective (the number for worker is always lower, because they are more risk averse, managers are more risk neural) (the manager also knows this).
\[ R(2) = \begin{cases} H & \text{w/ prob 0.7} \\ L & \text{w/ prob 0.3} \end{cases} \]
Participation Constraint
\[ 0.7w^H + 0.3w^L  2 \ge 10 \]
Incentive Compatibility Constraint
\[ 0.6 w^H + 0.3 w^L  2 \ge 0.4 w^H + 0.6 w^L  0 \]
(take R(0) from scenario 2)
\[ 0.7 w^H + 0.3 w^L = 12 \\ 0.3w^H  0.3w^L = 2 \]
Hence,
\[ w^H = 14 \\ w^L = 22/3 \]
Expected wage bill for the manager is
\[ 14 * 0.8 + 22/3*0.2 = 12.66 \]
Hence, expected wage bill is higher than scenario 2
Risk aversion from the worker forces the manager to pay higher wage
Salesperson  

Risk neutral  Risk averse  
Effort  Observable 
Any mix desired effort 
All salary Desired effort 
Not observable 
All commission Desired effort 
Specific mix (S+C) Salesperson shirks 
Grossman and Hart (1986)
 landmark paper for principal agent model
21.23.1 Basu et al. (1985)
Types of compensation plan:
Independent of salesperson’s performance (e.g., salary only)
Partly dependent on output (e.g., salary with commissions)
In comparison to others (e.g., sales contests)
Options for salesperson to choose the compensation plan
In the first 2 categories, the 3 major schemes:
 Straight salary
 Straight commissions
 Combination of base salary and commission
Compensation Type  Best when  Limitation 

Straight salary 
Longterm objective Hard to measure performance 
Less effort 
Straight commission  Easytomeasure performance 
Effortreward ratio is emphasized High risk (uncertainty) to the salesperson 
Combination 
Dimensions that affect the proportion of salary tot total pay (p. 270, table 1)
Previous research assumes deterministic relationship between sales and effort, but this study says otherwise (stochastic relationship between sales and effort).
Assumptions:
Firm: Risk neutral: maximize expected profits
Salesperson: Risk averse . Hence, diminishing marginal utility for income \(U(s) \ge 0; U'(s) >0, U''(s) <0\)
Expected utility of the salesperson for this job > alternative
Utility function of the salesperson: additively separable: \(U(s)  V(t)\) where \(s\) = salary, and \(t\) = effort (time)
Marginal disutility for effort increases with effort \(V(t) \ge 0, V'(t)>0, V''(t) >0\)
Constant marginal cost of production and distribution \(c\)
Known utility function and saleseffort response function (both principal and agent)
dollar sales \(x \sim Gamma, Binom\)
Expected profit for the firm
\[ \pi = \int[(1c)x  s(x)]f(xt)dx \]
Objective of the firm is to
\[ \underset{s(x)}{\operatorname{max}} \int[(1c)x  s(x)]f(xt)dx \]
subject to (agent’s best alternative e.g., other job offer  \(m\))
\[ \int [U(s(x))]f(xt) dx  V(t) \ge m \]
and the agent wants to maximize the the utility
\[ \underset{t}{\operatorname{max}} \int [U(s(x))]f(xt)dx  V(t) \]
21.23.2 Lal and Staelin (1986)
21.23.3 Raju and Srinivasan (1996)
Compare quotabased compensation with (Basu et al. 1985) curvilinear compensation, the basic quota plan is simpler, and only in specical cases (about 1% in simulation) that differs from (Basu et al. 1985). And it’s easier to adapt to changes in moving salesperson and changing territory, unlike (Basu et al. 1985)’s plan where the whole commission rate structure needs to be changed.
Heterogeneity stems from:
Salesperson: disutility effort level, risk level, effectiveness, alternative opportunity
Territory: Sales potential an volatility
Adjusting the quota (per territory) can accommodate the heterogeneity
Quotabased < BLSS (in terms of profits)
Constraints:
 quotabased from curve (between total compensation and sales) (i.e., shapeinduced nonoptimality)
 common salary and commission rate across salesforce (i.e., heterogeneityinduced nonoptimality)
To assess the shapeinduced nonoptimality following
21.23.4 Joseph and Thevaranjan (1998)
21.23.5 Simester and Zhang (2010)
 Tradeoff: Motivating manager effort and info sharing.
21.26 Marketing Mix Optimization Models
Check this post for implementation in Python
21.28 Twosided Platform Marketing Models
Example of Marketing Mix Model in practice: link