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Rationing by Waiting Lists

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    (Lindsay and Feigenbaum)

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    Background Queues emerge in:

    i) Setting 1: Quantity demanded and quantity supplied

    fluctuate so optimal capacity and equilibrium price isdifficult to determine.

    ii) Setting 2: If prices are below the market-clearinglevel, queues of demanders will form to ration the

    available supply

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    This paper examines Setting 2

    Case where the price is too low.

    Even when demand is perfectly predictable anduniform (opposite of the first setting), lines form andgrow until the

    expected wait = value of goods received

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    Definition of Waiting Queuing using waiting lists

    Does not imply waiting in person

    No cost in terms ofwasted time. One can do whatever he wants with his time except

    enjoy the services of the good sought.

    Question: How does the market clear?

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    Assumptions 1: Delay in receipt of a good can lower its value to

    demanders It is this diminishing value rather than increasing cost of

    obtaining such goods that produces the convergence ofquantity demanded and the quantity supplied.

    2: Individual demand is unpredictable from period toperiod If demand is predictable, the demander can forecast his

    desired quantity in each future period and simply order in

    advance to obtain it. No diminished value for the good because there is no delay in

    between the time the consumer wishes to consume the goodand the time the consumer can actually consume the good.

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    Section I: Theory Individual decision to join the waiting lists is described,

    then aggregated by market

    Sensitivity of the rate of joining to:

    Expected delay in delivery

    Rate at which demand diminishes with delay in delivery

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    Section II: Empirical results Theory is tested on the data from the waiting list for

    admission to British National Health Service (NHS)hospitals.

    Theory is found to be in contrast to currentexplanations of waiting lists in the NHS andelsewhere.

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    Current theory Current theory: Market does not clear due to backlogs Total demand Total supply because total demand includes

    backlog

    But rate at which services are demanded in each period= rate at which services are supplied,

    There is no long-run inadequacy of resources to dealwith demand.

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    Implications Solution to this waiting list problem is through short-

    term efforts

    However, expansion of facilities typically does noteliminate waiting lists, or even substantially reducethem.

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    Section IA. Individual Joining: Join a waiting list when

    PV of the good when delivered > Cost of joining thequeue

    = C

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    When does the individual join the

    queue?

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    Value functionValue of the good/rights:

    i) Depends on price.i) Assume that upon arriving at the top of the waiting

    list, each demander is entitled to purchase a fixedamount of the good at a price (possibly zero) below

    the market-clearing price.

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    ii) Depends on the delay, expressed asgwhich

    incorporates both: A) Discount rate effect. However, the impact ong is

    insignificant as the delays in the empirical tests rarelyexceed several months in duration.

    B) Diminishing demand effect.

    The timing of delivery may affect a goods value due tofashion, circumstance, location, health or whim.

    Major component ofg since the main thrust of the analysis isto predict the influence of differences in decay rates indifferent queues.

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    This gives us:

    wheret = expected date in delivery

    g = decay rate = vector of unknown attributes

    p = delivery price

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    Cost function Cost of joining the list, ci =

    Any costs incurred to qualify for joining other than thepurchase price

    E.g. taking examinations, obtaining approvals andreferrals

    Transactions costs (e.g. expenditures for transportation,legal advice, market information)

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    Market determinant of t

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    How does the market clear? Instead of clearing the market by raising the cost of

    obtaining the good, waiting time clears the market bymaking it less valuable.

    If a good is distributed to a population with varying vandg

    Demanders with high values and low decay factors willcrowd out demanders with lower v and higherg.

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    Rate of joiningAssume that the purchase price and the cost of joining

    as uniform across all persons, so:

    Variation in t is attributable to:

    Decay rateg

    Vector of consumer attributes

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    j(0): Number of people that will join when t=0 (i.e. no

    expected delay in delivery)

    At t=0, the number of people is unaffected by changesing

    At t>0, then an increase ing reduces the rate of joining

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    Elasticity

    Where v= elasticity of joining wrt value placed on the good by the marginal

    joiners

    Substituting (3) into (2) we getFor any given expected wait, the responsiveness of those joiningwaiting lists to changes in this wait will vary positively with the demandelasticity and the decay rate.

    (2)

    (3)

    (1)

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    Number in QueueAt equilibrium, the number in the queue Q is equal to

    the joining ratej(t)*t.

    An increase in supply reduces the equilibrium wait (seeFigure 1).

    A change in expected wait should therefore have thefollowing effect on the number in the queue:

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    Effect of supply shifts on number in

    queue So supply shifts has the following effects on the

    number in the queue:

    Waiting lists will not be shortened because an increasein supply results in a longer waiting lists, due to

    decrease in expected delay. This holds for as long as

    |elasticity of joining wrt to expected wait| >1.

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    Equilibrium Equilibrium is at , where

    Note:

    The joining function allows us to derive an expressionfor the elasticity of joining wrt expected wait in thequeue.

    Increases in service rate is occasionally accompanied byincreases in joining rate (since expected wait decreases),

    assuming elasticity >1.

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    Part II: Empirical Results National Health Service (NHS) in Great Britain.

    It relates the rate at which demanders of hospitalservices join waiting lists to:

    Expected delay, t

    Decay rate,g of demand for these services

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    NHS Features of the NHS (or how to join an NHS

    queue):

    For non-emergency cases, Patient -> GP -> Consultant/Hospitalisation (placed in queue)

    Consultants may only be visited if the patient has a referral.

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    Rationing by waiting lists (1) Over time, some who are in the queue would

    have recovered/moved away/died while awaitingtreatment.

    (2) Expected wait itself reduce the attractiveness ofjoining in the first place

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    Implications of Theory Rate of joining is positively related to

    1) The value of the services provided, v

    Rate of joining is inversely related to: 1) Expected delay, t

    2) The decay rate,g

    3) The cost of joining, c

    Elasticity: Membership in the queue is positively related tothe rate of service where |t| > 1 and negatively related tomembership in the queue where |t| < 1

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    Data Base 14 administration regions

    Data on mean waiting time and the number ofdischarges:

    Reported annually for each ICDA disease category

    Reported by region

    Data of for the year 1974

    22 conditions observed in 14 regions for a total of 308observations

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    Data shortcomings Data aggregated to regional level masks the

    intraregional variability in waiting times and theother variables

    However, the demanders are not restricted to a singlehospital, but may shop among alternatives in theirregion for hospitals offering the shortest wait.

    Therefore the differences in hospital waiting lists withinthe region is assumed to be small relative tointerregional variation

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    Separate queues for admission are not explicitlymaintained in each hospital for separateconditions

    Beds can be assigned for a variety of conditions.

    However, since the waiting times for different conditionsvary greatly, it suggests that separate queues areimplicitly maintained.

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    Assigning separate decay rates The decay rate,g, for individual hospitalizable conditions is

    not objectively measured.

    Hence, decay rates are assigned by grouping them.

    Categories are given separate decay rates

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    Category I conditions (high decay rate): Nonemergency cases, typically susceptible to drug therapy for

    which alternatives to hospital care were available Cases that respond to treatment and are controlled within a

    reasonable time in most instances, even if hospitalization is notprovided

    Category II conditions (low decay rate): Nonemergency cases such as hernia or cataracts that do not grow

    worse with delays in treatment but for which no alternative tohospital based therapy is available

    Category III conditions (negative decay rate): Conditions that rapidly grow more serious over time Negative decay rate as demand increases rather than subsides over

    time

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    Hypothesisj= j (t,g,v)

    j/t < 0, j/g < 0, j/v>0

    Estimate j = a0 +a1t +a2g.t +a3v +u

    where

    a1 < 0, a2 < 0, a3 > 0

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    Dummy variables:

    Dummy variable 1 indicates inclusion in the high decayrate, Category I

    Dummy variable 0 indicates inclusion in the low decayrate, Category II

    Category III conditions removed from sample These emergencies are moved to the head of the queue and do not

    follow the process outlined in the papers theory

    No influence on the results predicted for the remaining categories

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    Shortcomings of tests No data on numbers actually joining each queue in

    each period

    Model suggests that in equilibrium the rate of joining =rate of output s

    So data on service rates per period are used instead

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    Since rate of supply may also be influenced by

    delay (not independent oft), OLS estimates ofthese coefficients may be inconsistent.

    To reduce this inconsistency: Used predictors oft that are uncorrelated with the

    disturbance term u

    Predicted values oft are used in the joining equation to

    obtain unbiased estimates of its structure Delay is structured in weeks

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    Different queues will have difference in the

    number of potential joiners, j(0). This affects both the estimated constant term and the

    slope of the joining function

    Separately estimating regressions for each queue and

    identifying these constants Deflate the dependent variable of one queue by the ratio of

    the constant terms

    This makes the two different queues comparable

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    Empirical results Demand

    Supply

    Elasticities Decay rate, g

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    Demand i) Demand (joining rate)

    Proven that j/t < 0 and j/g < 0

    Proven that there is a difference in decay rates betweenCategory I and II conditions

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    Supply ii) Supply (cases treated per 1000 population)

    The longer the expected wait per condition, the higher isthe rate of output, s/t > 0

    Beds available per capita and doctors per capita have apositive effect on supply of services

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    Elasticities iii) Elasticities (computed from joining and supply

    equation coefficients)

    Elasticity of joining wrt expected wait is lower for lowdecay rate conditions than higher decay rate conditions(both are negative)

    Elasticity of supply wrt delay is positive.

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    Decay rate, g iv) Estimates of demand decay rates

    Mean expected delay for Category I was lower than forCategory II conditions

    According to American and Canadian studies, priceelasticity of demand is very inelastic at low prices

    The money price of hospital care is zero under the NHS so thisconclusion is relevant

    The decay rate for Category I conditions is greater than forCategory II conditions

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    ConclusionWaiting list queues function as rationing devices

    Membership in such queue itself imposes no cost,

    so the waiting lists may ration only through theinfluence of delay on the value of the servicedelivered Rates at which demand decays over time were found to be

    positive for both categories Category I decay rate > Category II decay rate

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    Homogenous households Comparative static predictions about the response

    of such queues to changing market conditions ispossible.

    j/t < 0

    j/g < 0, and

    s/t > 0.

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    Non-homogenous householdsWhere markets serve households for whom

    demand diminishes at different rates, rationingwill occur on the basis ofdecay rates as well asvalue

    People who value the good less might obtain the goodbecause others (who value it more) are discouraged bythe waiting time and do not join the waiting lists


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