Measuring the Digital Divide:

Structural Estimation of the Demand for Personal Computers

Jeff Prince

Cornell University

The Digital Divide

• What is it?– The separation between technology

participants and non-participants

• Why study it for PCs?– PCs have a wide range of household

applications:• Education• Information searches• Digital photography• Interactive gaming

Household Heterogeneity Plays a Major Role in the Divide

• The majority of new PC purchases are made by households already owning a PC

• From a revealed preference perspective, the marginal utility of an upgrade for many households is greater than that of crossing the Divide

Frequency of New PC Purchase

No PC entering observation year

.0838

Owned PC entering observation year

.1966

Full Data Set .1720

Separation Between Owners and Non-owners

• The summary statistics above show that upgrades are faster than new purchases

• This implies that the Technological gap is also widening

– Upgraders can do even more with their PC while non-owners still can do nothing

NTIA Statistics Suggest Pertinent Areas of Heterogeneity

• Ownership rates vary greatly across income, education, and age groups

• Ownership rates are:– Strongly increasing with income– Strongly increasing with education– Strongly decreasing with age

• Especially from 35-44 group onward

The Digital Divide and Diffusion Theory

• The PC is a “new” product diffusing through the population

• Non-owners are the late adopters– Generally attributed to heterogeneity – i.e.,

non-owners have a lower valuation for PCs– Adoption takes even longer if there’s a hurtle

such as a set-up or learning cost

The Demand for Personal Computers

This model has three main components:

• Heterogeneity– Stock– Observed– Unobserved

• Dynamics

• Set-up (or learning) costs

Looking at the Divide in a New Way

• The Divide is the result of the interaction of learning costs, persistent heterogeneity, and dynamic technological change

Literature/Contribution

• Goolsbee & Klenow (’00)• Hendel (’99)

• Departure from above papers:– Models the PC as a durable good

• Accounts for dynamic nature of the problem

– Considers replacement• Properly characterizes heterogeneity

• Can address a wider array of questions…

Issues This Model Addresses

• Short-term vs. long-term price elasticity• Price elasticity of owners vs. non-owners• “Technology elasticity”

– I.e., the response of demand to changes in the rate of technological progress

• The marginal value of quality improvements• Set-up costs for first-time buyers• Impacts of long-term and short-term subsidies

for first-time buyers

Findings

• Two demand curves (replacement and first-time)– Replacement price elasticity lower than that of first-time

purchase– Owners more responsive to changes in the rate of quality

improvement

• Dynamics matter– Short-term price elasticity higher than long-term price elasticity

• Fixed costs for first-time purchase are significant

• Subsidizing first-time PC purchases can be effective

Three Main Components: Heterogeneity

• Observed Heterogeneity– Variation in income, education, age, and

family size

• Unobserved Heterogeneity– Techies vs. Non-techies

• Stock Heterogeneity– Variation in households’ current PC holdings

Three Main Components: Dynamics

• The Classic “Buy or Wait” Decision

• Since PCs are durable goods, purchase decisions today affect decisions tomorrow

• Expectations about future available selection affect purchasing decisions today

Three Main Components:Set-up/Learning Costs

• Buying a First PC Involves Learning about Hardware and Software, Suppliers, Set-up in the House, etc.

Outline of Talk

• The Model

• The Data

• Results

• Extensions

• Conclusions

The Model

• Dynamic Stochastic Discrete Choice (DSDC) Model– Good for modeling demand for durables– Accounts for discrete nature of the choice– Accounts for forward-looking consumers

• Follows Rust’s 1987 DSDC Model

Model Details (Conceptual)

• Households Observe the Current State of the World

• They Form Expectations about the Future State of the World

• Based on This Information, They Make the Choice that Maximizes Their Expected Present Value of Utility

What is a Relevant State of the World?

• Household’s Observable Characteristics– Income, education, etc.

• Unobservables (Techie or Non-techie)

• Current PC Ownership– Does the household own a PC, and if so, how

good is it?

• Current Choice Set– PCs available at the time along with “no PC”

What are Relevant Expectations?

• Households Today Form Expectations about PC Choices Available in Upcoming Years

• Households Today Also Form Expectations about Their Future Purchasing Decisions

Model Details (Formal)

• Consumers make PC purchasing decisions to maximize expected discounted lifetime utility:

Expectation Discount Utility Choice Info

• Components:– X’s are vectors of PC characteristics (including price)– Z’s are vectors of household characteristics– θ is a vector of unknown parameters– η is a vector of utility shocks– s(t) is the state of the world at time t

( ( )) sup { ( , , ) ( )| ( ), , }tV s t E U x z d s tt j J tj j

Stochastically Evolving State Space?

• For Many DSDC Models, the State Space Evolves Stochastically

• For PCs, Only the Choice Set Could be Stochastic– Personal Characteristics and the PC Owned

Won’t Change (in general)– The Choices Available Next Year Can be

Considered Stochastic– But, Perfect Foresight is Plausible

The Choice Set

• Accounting for All Specs (MHz, RAM, ROM, Brand, etc.) is Unfeasible for This Model

• Choices Fit Nicely into Three Categories:– High-end – the “souped-up” PC– Median – the “standard” PC– Low-end – the “cheap” PC

• The Choice Set for each household is: {H,M,L,Q}– Q is “status quo”, or “No New Purchase”

Key Assumptions

• The Utility Function is Additively Separable

• Markov Process– Knowing this year’s state is enough to optimally

predict next year’s state

• Conditional Independence– Given the observables this period, the error term this

period is independent of the error term last period

( , , ) ( , )U x z u x zt tjt jt

The Recursive Problem

• Households solve the same infinite-horizon problem every period

• The value function can be written as:

• Households make decisions to maximize current utility plus expected future utility

• So, for given θ, we can solve for V or E[V] using Bellman’s equation

( ( )) max [ ( , , ) ( ) [ ( ( ), ( ))]]{ ( )} {0,1}

1V s t U x z d t E V s t d tj J tjt j jJd tj j

Specific Example

• Consider the following formula for U– measures quality, is price, and z measures income

• Unobserved heterogeneity is measured through

– Random coefficient taking on two possible values• High with probability p• Zero with probability 1-p

( , , ) ( , )U x z u x zt tjt jt

1 2 1 3 4 1 5 2 6 1( , ) ( 0," " ( ))jt j t j t j t j t

u x z x z zx x I x noPC s t

7 1( " ")j t

I x topPC

1x

2x

7

Derivation of Probabilities

• Error Term Assumed Type I Extreme Value

• Probability of Choosing Option A:

• P(H) =

• P(L) =

• P(A) = P(H) + P(L)

exp{ ( , ) [ ( ( ), )]}*

exp{ ( , ) [ ( ( ), )]}( )

at HH

jt Ht H

u x E V s t ap

u x E V s t jj C x

exp{ ( , ) [ ( ( ), )]}(1 )*

exp{ ( , ) [ ( ( ), )]}( )

at LL

jt Lt L

u x E V s t ap

u x E V s t jj C x

Identification

• As Usual, Parameters for Factors Common to All Choices Aren’t Identified– Includes Coefficients on z’s and 1

• How Can We Identify Dynamic Preferences With Only Cross-Sectional Data?– Variation in Holdings Identifies This– Looking for Marginal Value of Quality

• We Have Large Variation in Quality “Jumps”

Solving the Model

• Maximum Likelihood

• Likelihood function built from probabilities above

• Maximization over the parameter space requires numerical methods– Outer Loop

• Sequentially make guesses for the optimal θ• Amoeba method

– Inner Loop

• Each guess for θ requires a solution of Bellman’s equation

Data

• Forrester Research– Surveys on technology purchases and preferences each year

• Data includes:– Demographic information (Age, income, education)– Details on last PC purchased by the household

• Questions include:– Please indicate in what year you or someone in your household

purchased the computer that was bought most recently.– How much did you pay (in dollars) for your last computer,

including the new monitor?

Acquiring PC Stock and Quality

• The PC Stock entering an observation year– Not directly provided in the Forrester Data– Requires an overlap of two surveys for each

household• Approx. 30,000 overlapping surveys for ’98 – ’99 and 20,000

for ’00 – ’01

• PC quality is inferred– Use price paid and year purchased along with yearly

price lists from PC World Magazines

Summary Statistic: Demand Differences between Owners and Non-owners

Persists across Income

2001Category Subgroup % of

subgroup owning PC

entering 2001

% of subgroup

buying new PC in 2001

% of PC owners in subgroup

buying new PC in 2001

% of non-PC owners in subgroup

buying new PC in 2001

Income < $20k 44.54 9.00 14.85 4.30

$20k - $35k 68.54 12.53 14.76 7.67

$35k - $60k 82.25 15.71 16.90 10.22

$60k - $100k 88.10 20.09 21.17 12.10

$100k+ 90.10 24.31 25.20 16.18

Summary Statistic:Propensity to Replace by Quality of PC Owned

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

PC 1(noPC)

PC 2 PC 3 PC 4 PC 5 PC 6 PC 7 PC 8 PC 9 PC10

PC11

PC level

Pro

ba

bili

ty o

f re

pla

ce

me

nt

1999

2001

Results from DSDC ModelCovariate 1999 Estimate 2001 Estimate

MHz .4059 (.043)’’’ .0933 (.018)’’’

Low/Med. Wealth*MHz .2133 (.036)’’’ .0141 (.010)

Med. Wealth*MHz .3373 (.033)’’’ .0273 (.011)’’’

Med./High Wealth*MHz .4201 (.034)’’’ .0904 (.014)’’’

High Wealth*MHz .6041 (.040)’’’ .2022 (.018)’’’

H.S. Education*MHz .0737 (.032)’’ .0063 (.010)

Some College*MHz .1868 (.034)’’’ .0368 (.013)’’’

College/Post College*MHz .2263 (.034)’’’ .0807 (.015)’’’

Age(35-49)*MHz -.0289 (.027) -.0053 (.015)

Age(50-59)*MHz -.1015 (.029)’’’ -.0392 (.016)’’

Age(60+)*MHz -.2898 (.030)’’’ -.0947 (.015)’’’

FamSize(2)*MHz .0622 (.026)’’ .0126 (.009)

FamSize(3+)*MHz .0863 (.027)’’’ .0531 (.012)’’’

Top PC 5.3537 (.089)’’’ 5.5477 (.370)’’’

Probability of techie .1154 (.006)’’’ .0362 (.007)’’’

Marginal Utility of Money .3374 (.003)’’’ .3030 (.004)’’’

PC Learning Cost 9.9124 (.106)’’’ 6.7701 (.248)’’’

Heterogeneity in Marginal Values

• Marginal Value (for an extra 200 MHz) varies greatly across demographic groups– Increasing in Income, Education, Family Size– Decreasing in Age

• Highest marginal value is $392 for 1999 and $142 for 2001– Household of 3 with head under 35 making $100,000 with

college degree

• Lowest marginal value is $34 for 1999 and $0 for 2001– Household of 1 with head over 60 making less than $20,000 with

less than a high school degree

Learning Costs

• Estimated as $2938 in 1999 and $2234 in 2001

• If a Current Owner Requires an Increase in Value of K to make New Purchase, New Owner Requires K+LC

• Estimates for learning costs are expected to increase over time – Estimates for the average non-owner– Increases in user friendliness and exposure to PCs in

general may explain the decrease

Price Elasticity

• Short-term vs. Long-term and Owners vs. Non-owners

• Short-Term Price Elasticity– Measured by considering a one-year-only price decline

• 1999: 3.6 for non-owners vs. 2.9 for owners• 2001: 2.6 for non-owners vs. 2.1 for owners

• Long-Term Price Elasticity– Measured by considering a long-term price decline

• 1999: 3.2 for non-owners vs. 2.1 for owners• 2001: 2.7 for non-owners vs. 1.7 for owners

Technology Elasticity

• Demand’s Response to Expected Acceleration (Deceleration) in Quality Improvements

• Consider Acceleration in Quality Improvements from Doubling Every 2 Years to Every 1.5 Starting in the Subsequent Year– 1999: Demand falls 4.2% for non-owners; falls 6.4%

for owners– 2001: Demand falls .5% for non-owners; falls 3.9% for

owners

Policy Issues

1999 2001

% change in demand for non-owners

% change in demand for non-owners

Short-term Subsidy

$100 29.38% 27.94%

$200 65.04% 61.76%

Long-term Subsidy

$100 4.98% 5.83%

$200 10.15% 11.93%

Model Comparisons

• Dynamic vs. Static– Dynamic performs significantly better in both years

(likelihood ratio test)– Static over-emphasizes observable differences in

marginal value and under-emphasizes learning cost

• Stock vs. No Stock– Can’t test directly– However, results for no stock are non-sensical

• Expect this for a model inconsistent with the data

Robustness Analysis

• Discount Rate– Technically can solve for it, but practically unlikely– Discount Rate of .9 was better fit than lower ones (.8, .7, etc.)

• Horizon Length– Assumed 7 years– Results for cap at 6 and 8 years yielded trivial differences

• Technological Evolution– Evolution assumed to continue roughly as it has for the last

decade– Small fluctuations yielded trivial differences in the results

Results Recap

• Two demand curves (replacement and first-time)– Replacement price elasticity lower than that of first-time

purchase– Owners more responsive to changes in the rate of quality

improvement

• Dynamics matter– Short-term price elasticity higher than long-term price elasticity

• Fixed costs for first-time purchase are significant

• Subsidizing first-time PC purchases can be effective– Impact depends on time structure

Possible Extensions

• Nested Logit

• Random Coefficients?

• Alternative considerations of unobserved heterogeneity

• Time inconsistency

Conclusions

• Heterogeneity is important– Variation in value of PC quality across demographic groups is

large

• Dynamics matter– For price elasticity– For technology elasticity

• Stock effects matter

• Overall, models like this one incorporate the major factors behind purchases of durable goods