Demand Modeling

UCSD MGT 100 Week 3

Kenneth C. Wilbur and Dan Yavorsky

Demand Curves

  • Theory
  • Challenges
  • What firms do

Inverse Demand Curve

Demand Curves: Theory

  • Useful theoretical concept that summarizes market response to price

      - Why is it useful?

  • Often taught with perfect competition

      - Typically assumes stable, competitive, frictionless markets w free entry, full information, no differentiation 
  - Model predicts zero LR economic profits
  - Any evidence?

  • Also, taught with monopoly i.e. market power

      - What is market power? How would we measure it?

Demand Curves: Challenges

  • Idealized demand curves are hard to estimate (why?)

  • Where do product attributes come in?

      - What if we don't observe all attributes? "Price endogeneity"

  • Many things can predict demand

      - preferences, information, advertising, quality, match value, quality, complements, substitutes, competitor prices, entry, taxes and other policies, retail distribution, nature of equilibrium, stockpiling, consumer income, ...

  • What do we need to estimate Demand?

      - Observable, exogenous variation in costs or price
  - Otherwise, "price endogeneity" will bias demand estimates

How firms learn demand

  • Market research

    - Conjoint analysis, customer interviews, simulated purchase environments

  • Expert judgments, e.g. salesforce input

  • Cost-driven price adjustments

      - Often one-sided

  • Demand modeling with archival data

  • Price experiments

      - Market tests, digital experiments, bandits, digital coupons

  • Best practice: triangulation

Price Experiments

  • Ideally, the best way to learn demand, because you create exogenous price variation; but…

    • Competitors & consumers can observe price variation

    • May change purchase timing, stockpiling or
      reference prices

    • Competitors, distribution partners or suppliers may react

  • Hence, experimenting to learn demand may change future demand

Demand Modeling: Pros

  • Relatively inexpensive for large organizations

  • Fully compatible with price experiments

      - Either/or framing would be a false dichotomy: "Yes-and"

  • Confidential, fast

  • Depends on real consumer choices, i.e. revealed preferences

      - As opposed to stated preferences

  • Enables demand predictions at counterfactual prices

  • Enables predictions of competitor pricing response

  • Prediction accuracy: evaluable after price changes

Demand Modeling: Cons

  • Requires data, exogenous price variation, time, effort, training, commitment, trust, organizational buy-in

  • Always subject to untestable modeling assumptions

  • Requires the near future to resemble the recent past

        - To be fair, all predictive analytic techniques require these 3

Do Demand Models Work?

  • Evidence is supportive, but not thick

    - Informally, I know several people who maintain demand models in large orgs
- Formal evidence requires researchers and firm to collaboratively (a) estimate demand, (b) act on demand estimates, (c) observe how actions affect outcomes, & (d) report the results publicly. Hard to do & incentives conflict
- Demand modeling can also go badly, e.g. due to price endogeneity

Multinomial Logit

  • history, math, properties, discussion

Multinomial Logit (MNL)

  • Time tested, famous, popular demand model
  • Ported to econ from stats & psych by McFadden (1974, 1978, 1981; “Logistic regression’’)

Multinomial Logit (MNL)

  • Let \(i\) index consumers, \(j=1,...,J\) products, and \(t\) index choice occasions

  • Assume each \(i\) gets indirect utility \(u_{ijt}\) from product \(j\) in market \(t\):

\[u_{ijt}=x_{jt}\beta-\alpha p_{jt}+\epsilon_{ijt}\]

  • Then, assuming each \(i\) picks the \(j\) that maximizes \(u_{ijt}\), has unit demand, and that \(\epsilon_{ijt}\sim\)i.i.d.\(EV_1(0,1)\), market share is

\[Prob.\{u_{ijt}>u_{ikt}\forall{k\ne j}\}\equiv s_{jt}=\frac{e^{x_{jt}\beta-\alpha p_{jt}}}{\sum_{k=1}^J e^{x_{kt}\beta-\alpha p_{kt}}}\]

With \(N_t\) consumers, \(q_{jt}(\vec{x}; \vec{p})=N_t s_{jt}\). See Train 2009 sec 3.10 for proof

  • Estimating \(\alpha\) and \(\beta\) enables us to predict every product’s quantity response to a change in any product’s attributes \(x_{jt}\) or price \(p_{jt}\)

MNL: Common factors

  • Suppose \(\gamma_{t}\) indicates how popular the category is in market \(t\), so utility is \(u_{ijt}=\gamma_{t}+x_{jt}\beta-\alpha p_{jt}+\epsilon_{ijt}\). Then market share becomes

\[s_{jt}=\frac{e^{\gamma_{t}+x_{jt}\beta-\alpha p_{jt}}}{\sum_{k=1}^J e^{\gamma_{t}+x_{kt}\beta-\alpha p_{kt}}}\]

\[s_{jt}=\frac{e^{\gamma_{t}} e^{x_{jt}\beta-\alpha p_{jt}}}{e^{\gamma_{t}}\sum_{k=1}^J e^{x_{kt}\beta-\alpha p_{kt}}}\]

\[s_{jt}=\frac{e^{x_{jt}\beta-\alpha p_{jt}}}{\sum_{k=1}^J e^{x_{kt}\beta-\alpha p_{kt}}}\]

  • Similar exercise applies to individual-specific intercepts \(\gamma_i\), or individual-time interactions \(\gamma_{it}\). Only differences across products affect predicted market shares

MNL: Share differences

  • We set product \(j=1\) utility to \(u_{i1t}=\epsilon_{i1t}\) to normalize utility, i.e.

\[s_{1t}=\frac{1}{\sum_{k=1}^J e^{x_{kt}\beta-\alpha p_{kt}}}\]

  • Hence for every \(j\ne 1\), consider an affine transformation of market shares:

\[ln(s_{jt})-ln(s_{1t})=x_{jt}\beta-\alpha p_{jt}\]

  • Looks like a regression equation… we sometimes add an error \(\xi_{jt}\) to represent unobserved product attributes:

\[ln(s_{jt})-ln(s_{1t})=x_{jt}\beta-\alpha p_{jt}+\xi_{jt}\]

  • We use the \((J-1)T\) differences in market shares to estimate demand parameters

MNL Estimation

Define \(y_{ijt} \equiv 1\{i \text{ chose } j \text{ in } t\}\). I.e., \(y_{ijt}=1\) iff \(i\) choose \(j\) at \(t\); otherwise \(y_{ijt}=0\).

  1. Maximum likelihood: Assume the probability of observation \(\{i,j,t\}\) is \(s_{jt}^{y_{ijt}}\);
    then lik. is \(L(\beta )=\prod\limits_{\forall i,j,t} s_{jt}^{y_{ijt}}\); & choose parameters to maximize log lik:

\[\sum_{\forall i,j,t} y_{ijt}\ln s_{jt}\]

  1. Method of Moments: Choose \(\alpha\) and \(\beta\) to solve some vector of “moments,” i.e. interactions between exogenous observables \(z\) and error terms, e.g. 

\[ \sum_{\forall j,t}z_{jt} (\frac{\sum_{\forall i}y_{ijt}}{N_t} - s_{jt})=0\]

  1. Linearize the model, choose parameters to minimize the sum of square errors

MNL Goodness-of-fit Statistics

  • Discrete choice models predict choice probabilities rather than choices, because utility is always unobserved; hence nonstandard fit statistics

  • Predicted outcomes are inherently stochastic, so limited predictive ability

  1. Likelihood Ratio Test: (not R-sq)

\[\rho=1-\frac{ln L(\hat\beta)}{ln L(0)}\]

  • As \(L(\hat\beta)\to 1\), \(ln L(\hat\beta)\to 0\), \(\rho\to 1\)

  • As \(ln L(\hat\beta)\to ln L(0)\), \(\rho\to 0\)

        -   Heuristic: 0.2-0.4 is pretty good

  1. Hit Rate: % of individuals for whom most-probable choice was actually chosen

  2. R-sq using prediction errors at the \(jt\) level

MNL Pros

  1. Microfounded, i.e. behavioral predictions are consistent with a clearly specified theory of consumer choice

       - Theory is utility maximization
   - Economists widely believe that microfounded models are more generalizable than purely statistical models*

  2. Extensible to accommodate preference heterogeneity

       - We'll cover 3 types of extensions in heterogeneous demand modeling

  3. Likelihood function is globally concave in the parameters, ensuring fast and reliable estimation

       - Remember our local vs. global optimum discussion?

MNL Cons

  1. Assuming \(\epsilon_{ijt}\sim\)i.i.d.\(EV_1(0,1)\) is convenient but unrealistic

     - More likely, more similar products would experience more similar demand shocks
 - Alternatives exist but can be computationally expensive

  2. Analyst selects the choice set \(j=1,...,J\), market size \(N_t\), attributes \(x_{jt}\), and price structure \(p_{jt}\).

     - What's a j? What's a t? What's in x? How do we measure p? Who's in N?
 - "Tuning factors" or "Analyst degrees of freedom"

  3. Market share derivatives depend on market shares alone (IIA; see Train Sec. 3.6)

  4. Price Endogeneity

       - Affects all demand models, not just MNL

IIA: Deeper dive

  • Famous example from McFadden (1974)
  • Suppose you estimate demand for transportation with three options: {Blue Bus, Red Bus, Car}, each with 33% market share
  • Now suppose you painted the red buses blue and want to predict market shares in the new choice set: {Blue Bus, Car}
  • MNL will predict Blue Bus and Car shares of 50%, not 67% and 33% (why?)

IIA: Remedies

IIA is testable & usually rejected by data

Common remedies:

  1. Extend the model to impose structure on choice set,
    e.g. Nested Logit or Ordered Logit

  2. Change \(\epsilon_{ijt}\sim\)i.i.d.\(EV_1(0,1)\), e.g. Multivariate Probit with correlated errors

  3. Change model structure so IIA property does not obtain, e.g. heterogeneous logit

Price endogeneity:
35 INSANE explanations

  • #4 will SHOCK you. Like, comment, subscribe
  • Covered on the exam…ask lots of questions

2. Fundamental issue

  • Demand model is a causal price-quantity relationship
    Yet observed prices may correlate with other demand and supply determinants
    Exogenous price variation req’d to distinguish correlation from causation (“identification”)

    - Price endogeneity is a "data problem" not a "model problem"
- Can be hard to verify empirically--needed data is missing--but widely believed important
- Implies wrong demand slope, biased demand predictions

  • Sign of bias depends on unobserved correlation

    - If corr(price,unobs)<0 --> estimated demand is "too flat" or too elastic
- If corr(price,unobs)>0 --> "too steep" or too inelastic
- Affects all demand models, not just MNL

3. E-Commerce Example

  • Imagine an unobserved demand shock, such as a viral Instagram post, increases Amazon product awareness and sales

  • Sales spike, inventory drops, automated pricing system increases price to monetize remaining inventory

  • What do data show? Corr(sales, price) > 0 !!

      - Common enough to be unsurprising when this happens

4. An Analogy

  • Foot size data significantly predict reading comprehension among children!

      - In fact, age causes both foot size 
  - And age causes reading comprehension 
  - Without age, you cannot accurately estimate the causal relationship between foot size and reading comprehension. You can only measure the correlation

  • Also, crib purchases significantly predict childbirth!

5. Sports example

  • You have data on Lakers ticket prices and sales before and after the Luka trade

  • Prices and sales are both higher after the trade

  • Does this mean that higher price caused higher sales?

6. Example: Digital systems

  • Uber surge pricing:
    Positive Demand shocks increase price
    Negative Supply shocks increase price

  • System adjusts the price without knowing the causes

      Many digital inventory-based pricing systems are similar

7. Shrinkflation

  Shrink the package, maintain the price
  Also, "Skimpflation" : reduce ingredient intensity

8. Multiple determinants

  • Price causes quantity demanded; but so do other things

9. Oft-unobserved price correlates

  1. Changes in consumer preferences, income, market size

  2. Retail distribution, prominence, stocking

  3. Digital marketing, including search ads, display ads, affiliates, influencers, coupons

  4. Competitor prices, preference shocks, retail, dig mktg

Any may correlate with equilibrium prices, leading to endogeneity biases if left uncontrolled

10. General model interpretation

  • Posit a model \(q=f(x,p,e)\) for \(p=\)price, \(q=\)quantity,
    \(e=\)error reflecting all relevant unobservables; estimate \(\frac{dq}{dp}\)

  • What does \(\frac{dq}{dp}\) mean exactly? 2 possibilities:

    1. Correlation: Empirical tendency of q to change with p, holding other observable attributes x constant
2. Causality: Causal effect of 1-unit change of p on q
1 is descriptive analytics; 2 is diagnostic analytics

  • Standard econometric assumptions only admit #2 when \(corr(p,e)==0\) (“exogenous”)

    Hence, what the estimates can teach us depends on what we cannot see
This is a tricky situation: When can we trust our demand model?
Answer 1: When we have exogenous price variation, hence corr(p,e)==0; OR
Answer 2: When we observe all demand drivers; but this is difficult to verify

11a. Simulation

  • Suppose 2 firms, correlated cost shocks and correlated prices, MNL demand

  • Suppose true demand is \(q_1=f(p_1, p_2, \epsilon_1, \epsilon_2)\)

  • Suppose we use OLS to estimate \(q_1=\alpha + p_1\beta + \epsilon\)

  • We mistakenly believe that \(corr(p,\epsilon)==0\)

        Incorrect: corr(p_1,p_2)>0, and p2 omitted, hence p2 is in epsilon

  • \(\hat{\beta}\) is biased to fit the model’s assumption that \(\sum p_1\epsilon=0\)

  • Wrong \(\hat{\beta}\) means wrong demand curve slope
    …implies wrong demand predictions in response to price
    …recommended price will be wrong, may reduce profit

11b. Simulated data generating process

# 1. Simulation parameters and correlated cost shocks
set.seed(14)                   # for reproducibility
n_periods    <- 100            # number of periods
market_size  <- 100            # total market size (e.g. number of customers)
rho          <- 0.9            # influence of costshock1 on costshock2

# Demand model parameters
alpha       <- 0.2             # price sensitivity (common across products)
intercept1  <- 9               # baseline utility for product 1
intercept2  <- 9               # baseline utility for product 2
# (Outside option utility is normalized to 0)

# Simulate cost shocks for the two firms (correlated)
shock1 <- rnorm(n_periods)                          
shock2 <- rho * shock1 + (1 - rho) * rnorm(n_periods)

# Derive prices from costs (higher cost shock -> higher price)
base_cost <- 1
price1 <- 3 * base_cost + shock1  
price2 <- 3 * base_cost + shock2
cor(price1, price2)

# 2. Compute market shares and quantities using multinomial logit demand
data <- tibble(
  period = 1:n_periods,
  shock1 = shock1,
  shock2 = shock2,
  price1 = price1,
  price2 = price2
) %>% 
  mutate(
    # Indirect utilities for each product and outside option:
    U1 = intercept1 - alpha * price1,
    U2 = intercept2 - alpha * price2,
    U0 = 0,  # outside option utility (baseline 0)
    # Convert utilities to choice probabilities (logit formula):
    expU1 = exp(U1),
    expU2 = exp(U2),
    expU0 = exp(U0),
    share1 = expU1 / (expU1 + expU2 + expU0),
    share2 = expU2 / (expU1 + expU2 + expU0),
    Q1 = market_size * share1,
    Q2 = market_size * share2,
    Q0 = market_size * (1 - share1 - share2)
  )

# Create decile variable for Firm 2's price
data <- data %>%
  mutate(p2_decile = ntile(price2, 10))

# 3. OLS regressions for Firm 1's demand
model_naive <- lm(Q1 ~ price1, data = data)
summary(model_naive)

model_full  <- lm(Q1 ~ price1 + price2, data = data)
summary(model_full)

> # 3. OLS regressions for Firm 1's demand
> model_naive <- lm(Q1 ~ price1, data = data)
> summary(model_naive)

Call:
lm(formula = Q1 ~ price1, data = data)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.31867 -0.34908 -0.00637  0.32919  1.19378 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 51.78942    0.17660  293.26   <2e-16 ***
price1      -0.59737    0.05565  -10.73   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.5011 on 98 degrees of freedom
Multiple R-squared:  0.5404,    Adjusted R-squared:  0.5357 
F-statistic: 115.2 on 1 and 98 DF,  p-value: < 2.2e-16

> model_full  <- lm(Q1 ~ price1 + price2, data = data)
> summary(model_full)

Call:
lm(formula = Q1 ~ price1 + price2, data = data)

Residuals:
       Min         1Q     Median         3Q        Max 
-4.177e-04 -6.103e-05  7.340e-06  7.187e-05  7.270e-04 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  5.000e+01  7.456e-05  670528   <2e-16 ***
price1      -4.999e+00  1.321e-04  -37832   <2e-16 ***
price2       4.998e+00  1.489e-04   33571   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.0001478 on 97 degrees of freedom
Multiple R-squared:      1, Adjusted R-squared:      1 
F-statistic: 1.226e+09 on 2 and 97 DF,  p-value: < 2.2e-16

Common solutions

  • Experiments

    Randomizing price eliminates confounding with unobservables
Gold standard, but has drawbacks mentioned earlier

  • Quasi-experiments using archival data

    1. Instrumental variables
2. Regression discontinuities
3. Natural experiments
4. Difference-in-differences
5. Synthetic controls
6. Double/debiased machine learning
These approaches are beyond the scope of this class

  • Model the price-setting process

      But without exogenous price variation, this is difficult to evaluate

  • Best practice: Triangulate

Exogenous smartphone price variation

  • Smartphone discounts are randomly assigned,
    thus we have exogenous price variation to identify \(\beta\)

      - This class focuses on how we can use demand models

  - Endogeneity remedies: Future metrics classes or graduate study

  - Treat the topic as a demand modeling risk to be understood

  • Describe price endogeneity in your own words

      Generate a novel example related to price and demand
  Explain how to resolve it

Class script

  • Wrangle data
  • Estimate MNL
  • Interpret parameters and SEs
  • Assess model fit

Wrapping up

Recap

  • Demand modeling enables data-driven sales predictions for counterfactual prices and attributes, facilitating profit maximization
  • MNL is popular bc it is powerful, microfounded and tractable
  • MNL has limitations (eg, IIA), but is extesnible by modeling heterogeneity
  • Price endogeneity is a data problem that biases demand parameter estimates when price is correlated with unobserved supply or demand shifters. Usually resolved with exogenous price variation

Going further