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Moral Hazard, Insurance and Some Collusion

Journal of Economic Behavior&Organization

V ol.50(2003)225–247

Moral hazard,insurance,and some collusion

Ingela Alger a,?,Ching-to Albert Ma a,b,c

a Department of Economics,Boston College,Chestnut Hill,Massachusetts,MA02467,USA

b Department of Economics,Boston University,Boston,Massachusetts,MA02215,USA

c Department of Economics,Hong Kong University of Science an

d Technology,

Clear Water Bay,Kowloon,Hong Kong

Received1July2000;received in revised form12January2001;accepted22January2001

Abstract

We consider a model of insurance and collusion.Ef?cient risk sharing requires the consumer to get a monetary compensation in case of a loss.But this in turn implies consumer–provider collusion incentives to submit false claims to the insurer.We assume,however,that only some providers are collusive while some are honest,and determine the optimal contract specifying the treatment,the insurance policy and the reimbursement policy to the provider.Two cases are analyzed:the?rst allows contract menus that induce self-selection among the providers;the second allows contracts consisting of a single policy.In both cases,deterrence of collusion is optimal only if the probability that the provider is collusive is large.When the contract deters collusion,it is as if the provider was collusive with probability one.The?rst best is achieved only when the provider is honest with certainty.Furthermore,over-consumption of treatment occurs in many cases,and is sometimes used as a substitute for monetary compensation to the consumer.

JEL classi?cation:G22;I11;I12

Keywords:Insurance;Moral hazard;Optimal contracts;Honesty and dishonesty

1.Introduction

Insurance policies cover consumers for the losses they may suffer due to an accident.The moral hazard problem associated with insurance is well known.Because the true state of the loss may be hard to verify,instead of the indemnity approach,insurance policies often cover fully or partially the expenses consumers may have to insure to recover some of the losses. In disability insurance,the problem is compounded by the potential loss of income after an illness or accident.Disability and health insurance policies are very often simultaneously ?Corresponding author.

E-mail addresses:ingela.alger@https://www.doczj.com/doc/9311314352.html,(I.Alger),ma@https://www.doczj.com/doc/9311314352.html,(C.-t.Albert Ma).

PII:S0167-2681(02)00049-5

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offered by employers.Here,the moral hazard problem becomes more extensive:not only is there a potential for consumers to use medical care excessively,consumers and providers may misreport the extent of injury in order to collect disability insurance payments.In this paper,we study the collusion between consumers and providers in the context of health and disability insurance.1

In our model,a risk-averse consumer may become ill,and suffer some health loss as well as income loss due to disability.A provider can partially restore these losses by supplying treatment.Insurance is offered by a risk-neutral insurer by means of contracts with the con-sumer and the provider.These contracts specify payments from the insurer to the provider, from the consumer to the insurer,as well as the amount of treatment.Because treatment only partially recovers the loss,ef?cient insurance requires a monetary compensation to the consumer.

Whether a loss has actually occurred or not is nonveri?able information,and payments and recovery inputs can only be based on the claim that the insurer receives.In considering how claims are?led,we use a new approach and let some economic agents act honestly. While some providers and consumers may collude to defraud the insurer of disability pay-ments,some do not:these provider–consumer coalitions will report truthfully the private information of the loss.2The assumption that some providers behave honestly is motivated by several considerations.First,an economic agent may have made a conscious decision to be honest,concluding that the potential gain from lying may not be worthwhile.This is supported by several theoretical?ndings in the literature.The repeated-game literature has explained“sincere”or“altruistic”behavior as optimal responses in long-term inter-actions.This of course cannot explain all types of cooperation,in particular those that occur in short-term relationships.But honest behavior may also be optimal in short-term relationships.3An agent may have such a distaste for misreporting the truth that he will forgo any potential gain.

Second,a number of papers have suggested that human behavior is driven by factors other than pure sel?shness.Surveying the psychology literature,Rabin(1998)concludes that the standard assumption of sel?sh behavior in economics may be too narrow,and sometimes even misleading.Extensive experimental evidence in bargaining games has demonstrated that a notion of fairness determines behavior(see Rabin,1998,p.21ff).Rabin also quotes the example,originally due to Dawes and Thaler(1988),of farmers leaving fresh produce on a table with a box nearby,expecting customers to pick their purchases and leave money to“complete”the transaction.Similarly,most of us have voluntarily paid for newspapers inside an unlocked box on a sidewalk.In a paper on endogenous preferences,Bowles (1998,p.80)“treat[s]preferences as cultural traits,or learned in?uences of behavior”and includes such examples as never lying and reciprocating dinner invitations.Taken together, 1Clearly,our model applies to any environment in which an accident leads to both injury and loss of income, such as home and automobile insurance.We have chosen to use the health and disability example because it illustrates the situation most succinctly.

2Ma and McGuire(1997)also study claims as a result of collusion between the provider and the consumer,but assumes that there are only collusive provider types.

3In a model where an agent is matched with a new principal each period,Tirole(1996)determines the optimal behavior of opportunistic agents,who co-exist with agents who always cheat and agents who never cheat.He shows that under some conditions,an opportunistic agent maximizes his utility by never cheating the principal.

I.Alger,C.-t.Albert Ma/J.of Economic Behavior&Org.50(2003)225–247227 these?ndings strongly indicate that honest behavior is neither uncommon nor irrational.In the tax compliance and evasion literature,it has been recognized that some taxpayers are always honest;Erard and Feinstein(1994)have studied the impact of this.

Our approach does not assume that economic agents are nonstrategic.Although honest providers always truthfully?le claims with the insurer,they seek to maximize pro?ts, subject to the truthful report of the loss.We have thought about an alternative assumption: that some agents are completely nonstrategic,honestly reporting the health status,as well as always performing any task that is asked of them to do(for example,returning any rent or payoffs above their reservation utility level).We believe that this second assumption is too strong,and hard to justify empirically.Moreover,a consumer’s health status has a certain objective connotation,being a piece of information that is potentially veri?able;in fact,we will exploit this property in setting up the collusion subgame between the consumer and the provider.On the other hand,an agent’s intention is impossible to ascertain objectively; even under the best of circumstances,one’s intention may not be inferred accurately.

We determine the equilibrium insurance and provider payment contracts.As a bench-mark,we?rst assume that the insurer can observe whether the provider is honest or not.If he is honest,the?rst best is implementable.If he is collusive,the contract is second-best: consumers must bear some risk.Interestingly,the provider does not get any rent:collusion is deterred by eliminating the monetary compensation to the consumer when a loss is claimed. In fact,the treatment is used as a substitute for monetary compensation:it is larger than the ?rst-best level.

When the insurer cannot observe whether the provider is honest or not,we consider the general class of deterministic contracts,each of which consists of a menu of insurance and payment policies and recovery inputs.However,the common methodology of solving for the equilibrium by maximizing an objective function under a set of truth-telling constraints is no longer valid.Indeed,a resource-allocation mechanism that exploits the honesty of economic agents must let those who behave strategically succeed in“gaming”against the system.For completeness,we show in Lemma1that some equilibrium allocations cannot be implemented by collusion-proof contracts.4Therefore,we must consider two distinct classes of contracts:those which deter collusion,and those which do not.Interestingly,the optimal collusion-proof contract is independent of the likelihood of the provider’s type,and identical to the equilibrium policy in a model where the provider is known to be collusive always(Proposition2).5

This clearly demonstrates that deterring collusion cannot be always optimal.For example, when the likelihood of a collusive provider is very small,a?rst-best contract,which is not collusion-proof,must perform better.Nevertheless,we show that the?rst-best contract is not an equilibrium contract even when the provider is almost always honest.Indeed,in Propo-sition3,we characterize the optimal noncollusion-proof contract for any given distribution 4An honest provider always forgoes any gain from information manipulation.A contract allowing collusion, therefore,rewards only the collusive type.Therefore,equilibrium allocations that reward the two types of provider differently must imply equilibrium collusion.

5The intuition is easy to understand.When faced with a collusion-proof policy,a collusive provider will not gain from misreporting the loss information.This implies that even if a collusive provider picks a policy that is meant for the honest provider,there will not be any gain from misreporting.So the policy for the honest provider must also be collusion-proof.

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on the provider’s type.Collusion takes the form of the provider–consumer coalition always claiming that a loss has occurred.This leads to a waste in recovery inputs and rents for the provider.The optimal noncollusion-proof policy for the honest provider implements exces-sive recovery inputs as a substitute for monetary compensation;for the collusive provider, insuf?cient inputs are used,because they are wasted with some probability.Risk sharing for the consumer is never perfect.More importantly,when compared to the?rst best,the distortion in the honest provider’s policy varies strictly monotonically with the probability of the honest provider,and tends to zero as this probability tends to one.This contrasts with the optimal collusion-proof contract,which is independent of the https://www.doczj.com/doc/9311314352.html,bining these results,we conclude that if the probability of a collusive provider is small,equilibrium contracts allow collusion.Conversely,if this probability is large,the equilibrium contracts must be collusion-proof.

Equilibria in our model may remain at the usual second best(which obtains when eco-nomic agents are always strategic),or it may be better,but never?rst best.Whenever equilib-ria are better than the second best,the(strategic)provider and the consumer misreport their private information in equilibrium.Equilibria tend to the?rst best as the probability of the provider being honest goes to one,but whenever this probability is within a neighborhood of zero,equilibria do not change with the probability and are second best.

Our analysis of side-contracts follows Tirole’s innovation(Tirole,1986),but adds the assumption that the provider may not always be willing to manipulate information.Hence, the method of considering only collusion-proof equilibria does not apply.Recently,Ma and McGuire(1997)study the effect of collusion on payment and insurance contracts.The model there is quite different,allowing the provider to choose efforts and the consumer to choose quantities.More importantly,in Ma and McGuire(1997)the consumer and provider cannot write any side contract,and collusion may not always maximize their joint surplus. Kofman and Lawarrée(1996)and Tirole(1992)also study models with collusion where some economic agent may not be corrupt.In both papers,it is assumed that the principal offers a single contract;in contrast we allow for a menu of contracts.We?nd that the contracts meant for the honest and the dishonest provider are different whenever collusion is not deterred.Our results can,therefore,be interpreted to imply that the analyses of Kofman and Lawarrée and Tirole involve a loss of generality.

Alger and Renault(1998,2000)consider a principal-agent model(without collusion) in which the agent has private information about the surplus generated by the economic exchange with principal(the circumstances).The agent either discloses the information to the principal honestly or he must be induced to do so.Alger and Renault(1998)is closest to the present paper in that two periods are considered:the agent knows only his ethics in the?rst period and learns the circumstances in the second period.6In the?rst period,the principal offers a menu of contracts that specify allocations as a function of the circumstances announced by the agent in the second period.Some of our results are similar to theirs:in both papers,the standard second-best contract is optimal when dishonest behavior is suf?ciently likely,and ethics are truthfully revealed in the?rst period;moreover,it is optimal to let 6Alger and Renault(2000)allow for contracts in which the agent announces his ethics and the circumstances simultaneously;they also consider three alternative de?nitions of honesty,depending on whether the honest agent honestly reveals his ethics or not.

I.Alger,C.-t.Albert Ma/J.of Economic Behavior&Org.50(2003)225–247229 the dishonest type lie in equilibrium when honest behavior is suf?ciently likely.This is comforting since it indicates that these results are robust.However,our?ndings differ for the contracts offered when honest behavior is suf?ciently likely.They?nd that the?rst-best contract is offered to the honest agent(and to the dishonest agent if the principal’s utility does not directly depend on the agent’s type,as is the case here).In contrast,we?nd that the honest and dishonest providers are offered different contracts;moreover,these contracts are never?rst best.The reason behind this difference will be explained in the text.

The next section presents the model and the?rst best;we also de?ne the side-contract and the extensive form.The analysis is presented in Section3.First,we de?ne a collusion-proof policy,and show that generally both collusion-proof and noncollusion-proof policies must be considered.The following two subsections,respectively,consider contracts that deter and permit collusion.Our main results are also presented there.For completeness,in Section 4,we derive the optimal contract when menus are unavailable.We show that this involves a loss of generality.Finally,concluding remarks are made in the last section.Proofs are contained in the appendix unless otherwise stated.

2.The model

We study the design of contracts between an insurer,a provider and a consumer.We begin by presenting the basic setting and the?rst best.Then we de?ne the side-contracting subgame played by the provider and the consumer,as well as the extensive form of the game played by the three parties.Our model is quite general,but because its application to health and automobile insurance is straightforward,often we illustrate our model by these markets.7 The consumer suffers a loss with probability p.The loss can be due to an illness(as in the health insurance case)or an accident(the automobile insurance case).We express the loss in monetary terms and denote it by .There are,thus,two possible states of nature.8We use the index i=h,s to denote them:i=s is the state when the consumer’s loss is ,otherwise,i= h.In the health-care market,a consumer may suffer from some symptoms.If those symptoms are actually insigni?cant,the true state of nature is h(the“healthy”state);otherwise,the state is s(the“sick”state).In the automobile insurance market,the states of nature refer to whether an accident has actually led to a loss.The consumer is risk averse with an increasing and strictly concave utility function U,de?ned in terms of money.Her initial wealth is W. The provider has a technology that recovers some of the consumer’s loss if that has oc-curred.The recovery technology is,however,completely unproductive when the consumer has not experienced a loss.Let m denote the input.In the health market,m denotes the quantity of treatment;in the automobile market,the repair work.Each unit of the input m costs the provider c.The output is measured in monetary units through the function f,where f is increasing and strictly concave.We assume that f is bounded from above,and that max f(m)< ;in other words,the technology cannot completely recover the consumer’s total loss .Many situations?t this assumption.This is certainly true for illnesses that may 7See Lu(1997)for an example of evidence of misreporting in the health care sector.

8We will introduce another set of states later;they concern the strategic interaction between a consumer and a provider.

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lead to disability and loss of income.Moreover,consumers may not be able to recover fully from some illnesses.For the automobile market,it re?ects the inconvenience caused by a repair of the car after an accident,the time and effort required to?nd a replacement, or the unavoidable risk that a repair might have been done improperly.9We assume that the recovery input is veri?able information,and that it can be speci?ed in a contract.The provider is risk neutral,but has limited liability;we normalize his reservation pro?t to0. An insurer offers a policy comprising insurance for the consumer and a reimbursement scheme for the provider.The policy speci?es a transfer t i from the consumer to the in-surer,a transferαi from the insurer to the provider,and the amount of recovery input m. An insurance-reimbursement policy can,thus,be denoted by{(αh,t h);(αs,t s,m)}.If the consumer suffers a loss,and pays an amount t in order to obtain m units of recovery,her utility is U(W? +f(m)?t);if there is no loss and the consumer pays t,her utility is

U(W?t).If the provider supplies m units of recovery in return for a remunerationα,his utility isα?mc.We suppose that the provider can always refuse to serve the consumer,in which case he obtains his reservation utility.The risk-neutral insurer is assumed to operate in a competitive market,and to set its policy to maximize the consumer’s expected utility. In the?rst best,whether the consumer has suffered the loss is veri?able information,

and a contract can be based on that.The?rst best is denoted by{(α?

h ,t?h);(α?s,t?s,m?)},and

is the solution to the following program:chooseαh,αs,t h,t s and m to maximize (1?p)U(W?t h)+pU(W? +f(m)?t s),

subject to

αh≥0,

αs?mc≥0,

(1?p)(t h?αh)+p(t s?αs)≥0.

The?rst two constraints are the participation constraints for the provider,and the third one is the insurer’s budget constraint.Trivially,the insurer setsαh=0andαs=mc.Then, straightforward calculation from the?rst-order conditions yields:

t?h=t?s+ ?f(m?),(1)

f (m?)=c.(2) The?rst condition says that all risks are absorbed by the insurer.The second is the productive ef?ciency condition:the marginal bene?t of recovery f (m?)is equal to the marginal cost c.Because >f(m?),the consumer’s payment in the event of a loss is lower:t?h>t?s.10 Havin

g established the?rst best,we proceed to study the interaction between the provider and the consumer when they can manipulate the information about the consumer’s loss. While the information about the true state of nature may be obtained at a cost,frequently the insurer relies on the information reported by the provider or the insured.This is clearly the 9Alternatively,we can assume that the marginal cost of recovery,c,is suf?ciently high.

10Alternatively,one can interpret this lesser payment in the event of a loss as a refund of the premium.That is, the consumer pays the premium t h upfront,and if there is a loss,t h?t s part of the payment will be refunded.

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case in the health-care market;although medical records are available and potentially can be used to settle disputes and law suits,almost always a provider simply submits a claim on behalf of the consumer to complete the?nancial transactions due to the treatments.

As we noted in the previous section,the economic relationship between a provider and a consumer may develop in an uncertain way:while some providers may collude with consumers,others may not.For our purpose of analyzing the effects of collusion,it is unimportant to model explicitly the reasons behind this,although one can easily think of a number of reasons.For example,there may be a chance that collusion results in a prosecution and penalty,and due to differences in wealth or opportunity costs,some providers?nd collusion unattractive.Alternatively,some providers may have developed a long-term relationship with the insurer and care more about their reputation.

The provider is,therefore,either truthful or collusive;denote these types by the index j=τandσ,respectively.The type of the provider is not observable by the insurer.The probability that a provider is collusive isθ.For simplicity,we assume that the consumer colludes whenever the provider is willing to.A truthful provider always submits a truthful report about the consumer’s loss.By contrast,a collusive provider may agree to submit a false claim through a side-contract with the consumer,which we now de?ne:

De?nition1(The side-contract subgame).A side-contract is an offer(x,k)made by the provider to the consumer,where x is a transfer from the consumer to the provider,and k, k=h,s,is the report.If the consumer accepts the side-contract,report k is made,and the transfer x is paid by the consumer to the provider.Otherwise,the true state of nature is reported.

Observe that any party in the side-contract subgame can unilaterally enforce truthful reporting.If the true state is i,the provider can simply offer the consumer a transfer of 0and a report of i;likewise,a consumer can always reject a side-contract offer from the provider,and the true state will be reported.Hence,for the consumer and provider to agree on a(nontrivial)side-contract,both of them must be at least as well off as when reporting the true state.11

The transfer allows the provider–consumer coalition to maximize their joint surplus.To illustrate,suppose that the true state is h.If they report the true state,the joint surplus is αh?t h;if they report s,it becomesαs?mc?t s.Note that since treatment is veri?able,it is carried out although it has no value to the consumer.Suppose that

αs?mc?t s>αh?t h.(3) Then there must exist an x such that

t h?t s≥x≥αh?αs+mc,

one of the inequalities being strict.Both parties can be made better off by the following side-contract:they report k=s and the consumer pays x to the provider,where?t s?x≥?t h, andαs?mc+x≥αh.Since the provider makes the side-contract offer,he can extract the 11As a tie-breaking rule,we assume that if neither the provider nor the consumer gains strictly by misreporting the state,the true state is reported.

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whole surplus generated by the misreporting.Thus,if(3)holds,the provider offers a side-contract with the transfer x=t h?t s,implying a net payoffαs?mc+t h?t s to the provider. To complete the description of the setting,we de?ne the extensive form of the game played by the insurer,the consumer,and the provider.In stage1,the consumer is randomly matched with a provider and“nature”determines the type of the provider;with probabilityθthe provider is a collusive type.The insurer offers an insurance-payment contract to the provider. We will consider two cases:?rst,the contract contains two policies for the provider to choose between;second,it consists of one single policy.In stage2,the provider picks a policy(we will describe this stage in more detail later).In stage3,the consumer suffers the loss with probability p.Whether the consumer has suffered this loss becomes the consumer’s and the provider’s private information.In stage4,the provider and the consumer play the side-contract subgame if the provider is collusive;at the end of this stage,a report on whether the consumer has suffered the loss is made.If the provider is collusive,this report is the result of the side-contract agreement.Otherwise,the report is h if and only if the consumer has not suffered a loss.Finally,payments to the provider,transfers from the consumer,and any recovery inputs are executed according to the selected policy and the report.

If the provider always reports truthfully(θ=0),the insurer can rely on the provider’s information and implement the?rst best.In contrast,if the provider is willing to write side-contracts with a consumer with some probability(θ>0),the?rst best is not imple-mentable.Consider the?rst-best allocation:under state h,there is no recovery input,and the

consumer pays t?

h ;in state s,the consumer receives treatment m?,pays t?s and the provider

is reimbursed m?c.From(1)and(2),we know that t?h>t?s.The provider’s net payoff is in-dependent of the state and is equal to0.This creates an incentive for the consumer–provider coalition to avoid the higher payment in state h by reporting state s when the true state is h. They will waste the recovery input,but this is not costly,since the provider is completely reimbursed.Explicit attention must,therefore,be paid to the possibility of the formation of side-contracts.As a benchmark,we begin our analysis with the standard assumption, namely,the provider is collusive with probability1.Results under this assumption will then be compared with those in the more general case(0<θ<1),which we analyze subsequently.

When a provider is always collusive,the insurer offers a single policy:{(ασ

h ,tσh);

(ασs,tσs,mσ)}.In this case,it can easily be proved that there is no loss of generality in re-stricting attention to contracts which deter equilibrium side-contracts between the provider and the consumer(see Tirole(1986)).

De?nition2(Collusion-proof policies).A policy{(ασ

h ,tσh);(ασs,tσs,mσ)}is said to be

collusion-proof if the following conditions are satis?ed:

ασh?tσh≥ασs?mσc?tσs,(4)ασs?mσc+f(mσ)?tσs≥ασh?tσh.(5)

Inequalities(4)and(5)ensure that the joint surplus from making a truthful report exceeds the joint surplus from lying about the state of nature,in states h and s,respectively.For later use,it is useful to note that both constraints cannot be violated simultaneously:if(4) is violated,then(5)is satis?ed,and vice versa.

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The insurer maximizes the expected utility of the consumer:

(1?p)U(W?tσh)+pU(W? +f(mσ)?tσs),(6) subject to the collusion-proofness constraints(4)and(5),as well as the participation con-straints for the provider and the insurer’s break-even constraint:

ασh≥0,(7)ασs?mσc≥0,(8) (1?p)(tσh?ασh)+p(tσs?ασs)≥0.(9)

Proposition1.When the provider is always collusive,the optimal collusion-proof policy {(ασh,tσh);(ασs,tσs,mσ)}has the following properties:

1.The provider obtains zero pro?t:ασ

h =ασs?mσc=0.

2.The consumer pays the same transfer in states h and s:tσs=tσh.

3.The consumer is imperfectly insured:W? +f(mσ)?tσs

4.Recovery input is excessive relative to the?rst best:mσ>m?,or f (mσ)

The policy in Proposition1will be referred to as the standard second-best policy.It has two main features:insurance is imperfect,and the recovery input is excessive compared to the?rst best.The intuition is as follows.As we pointed out above,given the?rst best, the consumer and the provider have an incentive to report s when the true state is h.The relevant(and binding)collusion-proofness constraint is,the one which ensures that the consumer–provider coalition prefers reporting the truth in state h(constraint(4)).Two options are available to make this constraint hold:either increase the payment to the provider when he reports h,or reduce the difference between the transfers of the consumer to the

insurer tσ

h ?tσs.The?rst option is suboptimal:if the payment to the provider is positive,

the expected utility of the consumer can be raised simply by decreasing this payment and decreasing the consumer’s transfer to the insurer by the same amount.Hence,the provider

must get zero pro?t,and the optimal way to deter collusion is to reduce tσ

h ?tσs to zero.In

other words,risk sharing through differences in monetary transfers across states h and s is not optimal,and the recovery input is used as a substitute:it is increased from the?rst-best level,mσ>m?.Now we turn to the case where the provider colludes with the consumer with probabilityθ,0<θ<1.

3.Self-selection among providers

In this section,we assume that the insurer offers a contract consisting of a menu of two

policies:[{(ατ

,tτh);(ατs,tτs,mτ)},{(ασh,tσh);(ασs,tσs,mσ)}].In stage2,with the knowledge

of whether he is truthful or collusive,the provider picks a policy{(αj

h ,t j h);(αj s,t j s,m j)},

j=τ,σ.We assume that,independently of his type,the provider picks the policy that max-imizes his expected utility,which is calculated according to what he anticipates will happen in equilibrium in stage4.That is,if he is a collusive provider,he will have to consider the

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possibility of side-contracts for the reporting of the information about the consumer’s loss.In contrast,if he is an honest provider,he anticipates that he will reveal that information truth-fully.The interpretation in the health insurance case is straightforward:although a physician may have a high ethical standard while working(i.e.report the truth and avoid prescribing treatment when unnecessary),he chooses the highest expected payment for doing so. The two policies may be distinct and provide incentives for different types to self-select in stage2of the game.But of course,the policies may also be identical.So there is no loss of generality to consider only those equilibria of subgames starting at stage2in which the j type provider picks{(αj h,t j h);(αj s,t j s,m j)},j=τ,σ.

A?rst and crucial observation is that when the provider colludes with the consumer with probabilityθ,it is no longer possible to consider only collusion-proof policies.Con-

sider a contract[{(ατ

h ,tτh);(ατs,tτs,mτ)},{(ασh,tσh);(ασs,tσs,mσ)}]that de?nes a subgame

at stage2,and a continuation equilibrium,and suppose that{(ασ

h ,tσh);(ασs,tσs,mσ)}is not

collusion-proof.Then the continuation equilibrium of the subgame may not be replicated by a collusion-proof policy for the collusive type.

If{(ασ

h ,tσh);(ασs,tσs,mσ)}is not collusion-proof,the collusive provider and the consumer

will?nd it optimal to lie in some state.At the beginning of stage2,the collusive provider then anticipates that his expected utility,denoted x,is strictly greater than his expected

utility if he was to report truthfully:x>y≡(1?p)ασ

h +p(ασs?mσc).Because a type

τprovider does not write a side-contract,his expected utility is just y if he picks theσpolicy.Of course theτpolicy will have to give him at least y in expected utility;say it gives him exactly y.Now consider a collusion-proof contract that mimics the outcome of the noncollusion-proof contract and,therefore,gives the same expected utilities to typesσand τof the provider.Since the new policy is collusion-proof,the typeσgets expected utility x by reporting truthfully.So the typeτprovider will also get expected utility x in equilibrium (he can always pick the policy meant for typeσin stage2).Therefore,the collusion-proof policy cannot mimic the equilibrium of the subgame of the noncollusion-proof policy.We have,thus,proved:12

Lemma1.Forθ∈(0,1),a(continuation)equilibrium of a contract may not be a(con-tinuation)equilibrium of a contract consisting only of collusion-proof policies.

The need to consider policies that permit the provider and consumer to lie about the consumer’s loss stems from the existence of the honest provider.An honest provider will not exploit a policy which is not collusion-proof.In other words,a policy that can be manipulated by the collusive provider may give a higher pro?t to the collusive than to the honest provider.The possibility of“differentially”rewarding different types of the provider through a side-contract by the collusive type is the key to our analysis.13

Lemma1implies that we must consider contracts consisting of policies that may not be collusion-proof.In most models in information economics,an incentive problem typically

12Alger and Renault(1998)contains a similar argument in a different model.

13Lemma1does not depend on the assumption that the provider gets all the potential gain from misreporting. Indeed,if the provider earns any strictly positive share of the gain from a side-contract,the lemma continues to hold.

I.Alger,C.-t.Albert Ma /J.of Economic Behavior &Org.50(2003)225–247235

results in deterrence of misreporting of information.In contrast,our analysis must consider allocations that are generated by the misrepresentation of information.Again,this is due to our assumption that a player in our model may not want to exploit the superior information he has in some subgame.In the next two subsections,we analyze two classes of contracts:those which prevent collusion,and those which do not.

3.1.Optimal contract with collusion deterrence

Here we derive the optimal contract [{(ατh ,t τh );(ατs ,t τs ,m τ)},{(ασh ,t σh );(ασs ,t σs ,m σ)}]that deters collusion.In stage 4,the provider of type σmust not ?nd it pro?table to write a

side-contract,so constraints (4)and (5)must hold.Since the policies must allow the provider to obtain his reservation utility in both states,we have the following participation constraints:

αj h ≥0,

j =σ,τ,(10)αj s ?m j c ≥0,

j =σ,τ.(11)Next,we state the incentive constraints that guarantee self-selection for the two types of the provider.In stage 2,the type j provider must ?nd it optimal to select policy {(αj h ,t j h );(αj s ,t j s ,m j )},j =τ,σ,anticipating the equilibrium moves in stage 4.Type τprovider always reveals the true state in stage 4,regardless of the policy he has chosen.Therefore,the incentive constraint for the truthful provider type is the following:

(1?p)ατh +p(ατs ?m τc)≥(1?p)ασh +p(ασs ?m σc),(12)

The condition for a collusive provider to choose the σpolicy is slightly more involved.Although the policy indexed by σis collusion-proof,the policy indexed by τmay not be.When a collusive provider selects the τpolicy,he will consider any gain from a side-contract at stage 4.Thus,if the state turns out to be h,he can either forgo the side-contract to obtain ατh or use a side-contract to report s and get ατs ?m τc +t τh ?t τs .Similarly,if the state is s,the provider reports k =s (respectively,k =h)when ατs ?m τc is greater (respectively,smaller)than ατh ?f (m τ)+t τs ?t τh .To summarize,the incentive constraint that ensures that type σprovider picks the σpolicy is:

(1?p)ασh +p(ασs ?m σc)≥(1?p)max[ατh ,ατs ?m τc +t τh ?t τs ]

+p max[ατs ?m τc,ατh ?f (m τ)+t τs ?t τh ].(13)

Finally,the insurer’s budget constraint is:

θ[(1?p)(t σh ?ασh )+p(t σs ?ασs )]+(1?θ)[(1?p)(t τh ?ατh )+p(t τs ?ατs )]≥0,

(14)

and the objective function is:

θ[(1?p)U(W ?t σh )+pU (W ? +f (m σ)?t σs )]

+(1?θ)[(1?p)U(W ?t τh )+pU (W ? +f (m τ)?t τs )].(15)

An optimal contract deterring collusion maximizes (15)subject to (4),(5),(10)–(14).

236I.Alger,C.-t.Albert Ma /J.of Economic Behavior &Org.50(2003)225–247

Proposition 2.Suppose 0<θ<1.The optimal contract deterring equilibrium collusion is independent of θand offers the same policy to the truthful and collusive types of provider.In fact ,this policy is the standard second-best policy of Proposition 1.

Proposition 2says that if the contract must deter collusion,then the existence of the truthful type of provider is inconsequential:the equilibrium contract is the same as if the provider was collusive with certainty.Despite the fact that the τtype provider always reports truthfully,he gets the optimal collusion-proof policy.

The key to understanding this result lies in the pair of inequalities (12)and (13).Given that the contract deters collusion,the σtype provider cannot bene?t from writing a side-contract after choosing the σpolicy,{(ασh ,t σh );(ασs ,t σs ,m σ)}.Also,the collusive provider must not ?nd it attractive to pick the τpolicy ({(ατh ,t τh );(ατs ,t τs ,m τ)})—even if he can write a side-contract on it:see inequality (13).Now,the truthful provider prefers the τpolicy to the σpolicy—see inequality (12).But this must mean that when the truthful provider picks the τpolicy,even if he could write side-contracts,he would be unable to bene?t.Indeed,combining inequalities (13)and (12)yields:

(1?p)ατh +p(ατs ?m τc)

≥(1?p)max[ατh ,ατs ?m τc +t τh ?t τs ]+p max[ατs ?m τc,ατh ?f (m τ)+t τs ?t τh ],

which says that the τpolicy is collusion proof:the right-hand side is the expected utility for a provider when side-contracts can be written.Therefore,requiring the σpolicy to be collusion-proof implies that the τpolicy must also be collusion-proof.Hence,the optimal collusion-proof contract simply consists of the policy in Proposition 1to both provider types.

3.2.Optimal contract without collusion deterrence

In this subsection,we consider contracts [{(ατh ,t τh );(ατs ,t τs ,m τ)},{(ασh ,t σh );(ασs ,t σs ,m σ)}]for which the σpolicy is not collusion-proof.As we noted in the discussion of De?nition 2,under a noncollusion-proof policy the provider–consumer coalition either

always reports state h or always reports state s.We will consider only those policies in which the coalition always reports s,and verify later that this is optimal.So consider those σpolicies for which the joint surplus from reporting state s is greater than reporting state h truthfully:

ασh ?t σh <ασs ?m σc ?t σs .(16)

In state h the collusive provider offers a side-contract with the transfer x =t σh ?t σs from the consumer to the provider.Therefore,the provider’s utility is ασs ?m σc +t σh ?t σs ,and the consumer’s utility remains at U(W ?t σh ).In state s,the information is reported truthfully;hence,the provider’s and the consumer’s utilities are ασs ?m σc and U(W ? +f (m σ)?t σs ),respectively.The objective function is the same as in the previous section:

θ[(1?p)U(W ?t σh )+pU (W ? +f (m σ)?t σs )]

+(1?θ)[(1?p)U(W ?t τh )+pU (W ? +f (m τ)?t τs )].(17)

I.Alger,C.-t.Albert Ma /J.of Economic Behavior &Org.50(2003)225–247237

As before,the provider must obtain his reservation utility:

ατh ≥0,(18)

αj

s ?m j c ≥0j =σ,τ.(19)Notice that since in equilibrium ασh is not chosen by type σprovider in state h,we do not impose a lower bound on it.The budget constraint for the insurer is modi?ed:

θ(t σs ?ασs )+(1?θ)[(1?p)(t τh ?ατh )+p(t τs ?ατs )]≥0.(20)When the provider’s type is σ,the recovery input will be used with probability 1;this explains the ?rst term of (20),which says that the transfer collected from the consumer is always t σs while the payment to the provider is always ασs .Finally,the self-selection constraints for the provider are as follows.For the truthful provider:

(1?p)ατh +p(ατs ?m τc)≥(1?p)max[ασh ,0]+p(ασs ?m σc).(21)

Because we have not required ασh ≥0,we allow for the possibility that the truthful provider refuses the transfer in state h if he has picked the σpolicy.Next,using the infor-

mation on the collusive provider’s equilibrium utility from the optimal side-contract,we can determine his expected utility if he selects the σpolicy:

(1?p)(ασs

?m σc ?t σs +t σh )+p(ασs ?m σc).The incentive constraint for type σprovider is:

ασs

?m σc +(1?p)(t σh ?t σs )≥(1?p)max[ατh ,ατs ?m τc +t τh ?t τs ]+p max[ατs ?m τc,ατh ?f (m τ)+t τs ?t τh ].

(22)

Similar to (13),the right-hand side takes into account the possibility of collusion when the collusive provider picks the policy that is meant for the truthful provider.The contract [{(ατh ,t τh );(ατs ,t τs ,m τ)},{(ασh ,t σh );(ασs ,t σs ,m σ)}]which maximizes (17)subject to constraints (18)–(22)is described in the following proposition.

Proposition 3.Given that the policy {(ασh ,t σh );(ασs ,t σs ,m σ)}is not collusion-proof ,the optimal contract has the following properties :

1.The truthful provider obtains zero pro?t :ατh =ατs ?m τc =0.

2.In state s,the collusive provider reports k =s and obtains zero pro?t :ασs ?m σc =0;in state h,he reports k =s and obtains pro?t t σh ?t σs ≥0through a side-payment from the consumer .

3.The consumer is imperfectly insured,whether matched with a truthful or a collusive provider :W ? +f (m j )?t j s

4.When the provider is truthful,recovery input is excessive relative to the ?rst best :m τ>m ?;when the provider is collusive ,the recovery input is smaller than in the ?rst best :

m σ

238I.Alger,C.-t.Albert Ma/J.of Economic Behavior&Org.50(2003)225–247

Given that the collusive provider always reports s,14recovery inputs are used even when they have no value.For this reason,the level of recovery input must be reduced from the ?rst best when the consumer is matched with a collusive provider.And maintaining full insurance under a reduced recovery input is too costly,so a consumer who is matched with a collusive provider must face some risk.The truthful provider does not exploit the mis-reporting incentive.Nevertheless,?rst-best risk-sharing is still not optimal for a consumer who is matched with a truthful provider.This is due to the fact that the collusive provider

earns a rent tσ

h ?tσs which is equal to tτh?tτs by the binding incentive constraint(22).Hence,

to limit the rent to the collusive provider,tτ

h ?tτs is reduced from the?rst best.Given the

lack of full insurance,it is optimal to raise the recovery input from the?rst-best level to reduce the amount of risk faced by the consumer matched with a truthful provider.This is reminiscent of the regular second-best policy,where treatment is also used as a substitute for monetary compensation(see Proposition1).

As mentioned in the introduction,the result in the above proposition differs from that in Alger and Renault(1998):given that the principal’s utility does not depend directly on the agent’s type,they?nd that the principal offers the?rst-best contract to both the honest and the dishonest agents when the dishonest agent’s type is not revealed in equilibrium. In their model,when the dishonest agent lies in equilibrium,it is as if the dishonest agent was of one particular type with probability1.As a result,the principal sets the decision variable to its?rst-best level conditional on that type:offering the?rst-best contract also to the honest agent then does not violate the dishonest agent’s incentive constraint.In our model it is not optimal to do so,because of the insurance motive:even though the dishonest provider always claims that the patient is in the sick state,the consumer’s utility is not the same in the healthy and the sick states.Moreover the treatment is unproductive when the consumer is healthy.Therefore,it is not optimal to set the treatment level and the payments to their?rst-best levels when the consumer is matched with a collusive provider;in order to satisfy the incentive constraint(22)the contract for the honest provider then also has to be distorted.

3.3.The equilibrium contract

To determine the equilibrium contract,we compare the consumer’s expected utilities from the optimal collusion-proof and noncollusion-proof contracts.In contrast with the collusion-proof contract of Proposition2,the values of the variables in theτ-policy in Proposition3vary withθ(this is readily seen by examining the?rst-order conditions).In fact,asθgoes to zero from above,the optimal policy for the truthful type tends towards the ?rst-best policy.

14Note that condition(16)must hold for this to be true.For small values ofθ(the likelihood of the collusive type

being small),we can show that tτ

h ?tτs=tσh?tσs>0,so that(16)indeed is satis?ed.Note further that a simple

revealed preference argument can be used to verify that the insurer could not do better by offering some contract

[{(ατ

h ,tτh);(ατs,tτs,mτ)},{(ασh,tσh);(ασs,tσs,mσ)}]such that the collusive provider always reports h.Note that for

such a contract,no recovery input is used when the consumer is in state s and is matched with a collusive provider. Hence,it must be dominated by the one considered above:indeed,in the contract above,mσ=0could have been chosen,but was not.

I.Alger,C.-t.Albert Ma/J.of Economic Behavior&Org.50(2003)225–247239 Corollary1.Asθtends to0,the optimal policy for the truthful type provider tends to the?rst-best policy:limθ→0{(ατh,tτh);(ατs,tτs,mτ)}={(α?h,t?h);(α?s,t?s,m?)}.Moreover, tτh>tτs(so that(16)is satis?ed).

Corollary1and Proposition3together say that the distortion of theτpolicy changes in a strictly monotonic way asθ,the probability for a collusive provider,begins to increase from 0.Whenθ=0,the insurer offers the?rst-best policy to the truthful provider,the insurer bears all the risks,and production is ef?cient.Asθincreases from0,it is optimal to depart from the?rst best in order to reduce the collusive provider’s rent and production inef?ciency that result from the side-contract between the collusive provider and the consumer.This follows from the Envelope Theorem.Because risk sharing and production are ef?cient in

the?rst-best policy,reducing the difference between tτs and tτ

h and increasing the recovery

input mτslightly leads to a second-order loss,but this results in a?rst-order gain because the collusive provider’s incentive constraint is relaxed.Whenθbegins to increase from0, the equilibrium contract must begin to adjust.Therefore,although the?rst-best policy is feasible for the truthful provider,it is not offered in equilibrium.However,the policy for the truthful provider is approximately?rst-best whenθis in the neighborhood of zero.Note that this implies that allowing collusion must outperform deterring collusion whenθis small. Next we turn to the case whenθis close to1.Here,the rent to the collusive provider becomes large,as does the waste of the recovery input due to the misreporting in state h.Furthermore,the expected bene?t to the consumer of the truthful provider’s behavior becomes small.The overall bene?t from allowing collusion,thus,becomes small.Not surprisingly,whenθis large enough,it is better to deter collusion than to allow it.

Corollary2.For allθsuf?ciently close to1,the contract in Proposition3has tσ

h ?tσs=

tτh?tτs=0.That is,it satis?es the collusion-proofness constraints in De?nition2. Corollary2implies that forθsuf?ciently close to1,the optimal noncollusion-proof policy

must give the consumer a lower expected utility than the optimal collusion-proof policy.

Therefore,for these values ofθ,the equilibrium contract must be the collusion-proof contract

in Proposition2.We have been unable to show that the value of the objective function at

the solution of the program for the optimal noncollusion-proof contract is monotonic in θ.So for intermediate values ofθ(those not in the neighborhood of0or1),we cannot characterize conditions for which the equilibrium contract is the collusion-proof contract in

Proposition2,or the noncollusion-proof contract in Proposition3.Nevertheless,we suspect

that the noncollusion-proof contract is the equilibrium contract if and only ifθis below a

certain threshold.To summarize our results,we have:

Proposition4.When the probability of the provider acting collusively is suf?ciently small,

the equilibrium contract consists of policies that do not deter collusion;in equilibrium,

the collusive provider writes a side-contract with the consumer,while the truthful provider

is given a policy that is approximately?rst best.When the probability of the provider

acting collusively is suf?ciently high,the equilibrium contract consists of the equilibrium

collusion-proof policy(independent ofθ)as if the provider were always collusive,namely

the contract in Proposition1.

240I.Alger,C.-t.Albert Ma/J.of Economic Behavior&Org.50(2003)225–247

The intuition for why the equilibrium contract is independent ofθonce it is close to1 is this.Supposeθis equal to1,clearly,the equilibrium contract is collusion-proof.Now, letθdecrease from1slightly.If the contract is now changed slightly,so that it is no longer collusion-proof,a side-contract will be written and the collusive type will always report state s.This results in a waste of the treatment in state h.Put differently,changing from the collusion-proof to a noncollusion-proof policy forθclose to1results in a discrete decrease in the payoff.So departing from the collusion-proof contract forθclose to1is suboptimal.

4.Two types of provider,one single policy

For completeness and comparison,we now consider a regime in which the insurer offers one single policy,{(αh,t h);(αs,t s,m)}.Screening menus are ruled out in this section.The analysis turns out to be much simpler.Clearly,the policy must be either collusion-proof or not.If it is collusion-proof,the optimal policy is the standard second-best policy.We only need to determine the optimal noncollusion-proof policy,and again,we only need to consider those policies where the consumer–provider coalition always reports s.So condition(16) must hold.The consumer’s expected utility is independent of the provider type and it is: (1?p)U(W?t h)+pU(W? +f(m)?t s).(23) The participation constraints for the provider and the insurer’s budget constraint are:αh≥0,(24)αs?mc≥0,(25) (1?θ)(1?p)(t h?αh)+[θ+(1?θ)p](t s?αs)≥0.(26) All three constraints are binding,and they can be solved to replaceαh,αs and t h in the objective function.The?rst-order conditions with respect to m and t s are,respectively:

1?θ

U (W?t h)c=pU (W? +f(m)?t s)f (m),(27)

1?θ

U (W?t h)=pU (W? +f(m)?t s).(28)

Simpli?cation shows that these conditions imply:

Proposition5.If a single noncollusion-proof policy is offered,the optimal policy satis?es:

1.Unlessθ=0the consumer is imperfectly insured:W? +f(m)?t s

2.The recovery input is at the?rst-best level:m=m?.

3.The truthful provider obtains zero pro?t:αh=αs?mc=0.

4.In state s,the collusive provider reports k=s and obtains zero pro?t:αs?mc=0;in state h,he reports k=s and obtains pro?t t h?t s≥0through a side-payment from the consumer.

I.Alger,C.-t.Albert Ma/J.of Economic Behavior&Org.50(2003)225–247241

In the previous section,we found that with a noncollusion-proof menu of policies,there was under-treatment if the provider was collusive and over-treatment if he was honest. Here,it turns out that the level of treatment is always?rst best.It can be explained by two differences:here,the consumer’s utility does not depend on whether the provider is honest or dishonest;furthermore,there is no leeway in adjusting the amount of treatment according to the type of provider.However,in order to limit the payment to the collusive provider, the difference between the payments t h and t s is reduced compared to the?rst-best policy, resulting in imperfect insurance.15

To determine the equilibrium contract,we compare the consumer’s expected utilities from the optimal collusion-proof and noncollusion-proof contracts.With the former,the expected utility is independent ofθ.Furthermore:

Corollary3.If the policy is not collusion-proof,the expected utility of the consumer is strictly decreasing inθ.

Proof.Since m=m?for allθ,the expected transfer from the consumer to the insurer increases withθ.Furthermore,condition(28)implies that the difference between W?t h and W? +f(m)?t s increases asθincreases. Moreover,whenθis close to one,the standard second-best policy dominates the optimal noncollusion-proof policy.Thus,we have proved:

Proposition6.If a single policy is offered,there exists?θsuch that:

(i)ifθ≤?θ,the equilibrium policy is the optimal noncollusion-proof policy in Proposition

(ii)ifθ>?θ,the equilibrium policy is the standard second-best collusion-proof policy in Proposition1.

5.Conclusion

In this paper,we have characterized the equilibrium insurance-payment contracts when the consumer–provider coalition may sometimes?le false claims with the insurer.Our focus has been on the risk sharing and treatment ef?ciency properties.We have explored two possibilities:either the insurer offers a menu of policies to screen the prospective providers,or it offers a single policy.Restricting attention to single policies leads to a loss of generality,and lower welfare.In both cases,however,allowing collusion turns out to be optimal when the coalition will falsify claims with small probabilities.When menus are used by the insurer,the equilibrium treatment level is higher than the?rst best when it is supplied by the honest provider;lower,when supplied by the dishonest provider.Insurance is never perfect,and the consumer must face some risks.The use of monetary transfers to 15An interesting implication of this result is that production ef?ciency obtains when insurance is provided with positive loading:indeed,the problem can be interpreted as a standard insurance problem with positive loading.To see this,chooseλ≡(θ(1?p)/p)as the loading factor.

242I.Alger,C.-t.Albert Ma/J.of Economic Behavior&Org.50(2003)225–247 compensate the consumer’s loss not recovered by treatment only occurs when collusion is tolerated.When an insurance and payment contract is designed,it is important that the interaction between consumers and providers be understood properly.As we have shown, the nature of this interaction—collusive or honest—determines the optimal design critically. When the insurer is unable to screen the providers by offering a menu of policies,a simple monotonicity results obtains:collusion is tolerated if and only if the provider is dishonest with a suf?ciently low probability(see the very last proposition).We have been unsuccessful in obtaining a similar monotonicity result when the insurer is able to use menus of policies.There remains the possibility that there are multiple equilibrium switches between collusion-proof and noncollusion-proof policies as the probability of the provider being dishonest varies,although we have been unable to construct such an example.

Our paper is one of the few in the insurance and health economics literature that explicitly models agents who are not completely self-interested.It seems likely that this framework will shed light on many institutions and phenomena,and future research in this direction may prove fruitful.

Acknowledgements

The research was conducted while the?rst author was visiting the Department of Eco-nomics at Boston University and supported by the Sweden–America Foundation.We thank Michael Manove for discussing various issues with us.For their comments,we thank Thomas McGuire and seminar participants at Boston College,Boston,Queen’s,Rice,Van-derbilt and Washington Universities,CIDE,University College London,and the Universities of Dundee,Munich,Namur,and Southampton.

Appendix A

A.1.Proof of Proposition1

Obviously,(9)is binding at the optimum;otherwise,tσs and tσ

h could be reduced.Also,(7)

binds;otherwise,ασ

h and tσ

could be reduced by the same amount.An identical argument

establishes that(8)binds.We have proved1in Proposition1.Withασ

h =0=ασs?mσc,

and with(9)binding,the problem becomes:choose tσ

h ,tσs,and mσto maximize(6)subject

to:

(1?p)tσh+p(tσs?mσc)=0,(A.1)?tσh≥?tσs,(A.2) f(mσ)?tσs≥?tσh.(A.3)

Constraint(A.2)binds.If it did not,tσ

h could be increased and tσs decreased(in proportions

such that(A.1)is unaffected).Such a change would not violate(A.3),but would decrease the difference between W?tσ

and W? +f(mσ)?tσs,since f(mσ)< .This proves

I.Alger,C.-t.Albert Ma/J.of Economic Behavior&Org.50(2003)225–247243

2of the proposition.Now when tσ

h =tσs,3of the proposition follows from the assumption

of f(m)< .

Further,the binding constraint(A.2)implies that(A.3)is satis?ed as a strict inequality.

Substituting the binding constraint(A.2)into(A.1),we have tσ

h =pmσc.Putting this into

the objective function,we have reduced the problem to the maximization of (1?p)U(W?tσh)+pU(W? +f(mσ)?tσh),

subject to

tσh=pmσc.

Letλbe the Lagrange multiplier.The?rst-order conditions for tσ

h and mσare,respec-

tively:

(1?p)U (W?tσh)+pU (W? +f(mσ)?tσh)=λ,

pU (W? +f(mσ)?tσh)f (mσ)=λpc.

Combining the above,we have

U (W? +f(mσ)?tσh)

(1?p)U (W?tσh)+pU (W? +f(mσ)?tσh)

f (mσ)=c.

Since 1.Therefore, f (mσ)

A.2.Proof of Proposition2

First,constraints(12)and(13)together imply:

ατh?tτh≥ατs?mτc?tτs,(A.4)ατs?mτc+f(mτ)?tτs≥ατh?tτh,(A.5) which are equivalent to collusion-proofness constraints for the truthful provider.So adding these two constraints into the program is inconsequential.

Next,we relax the program by dropping(12)and(13);we will show that they are satis?ed at the solution of the relaxed program.So the relaxed program is the maximization of(15) subject to(4),(5),(A.4),(A.5),(10)and(11)and the budget constraint(14).

For the relaxed program,constraints(10)and(11)bind;this follows an argument similar to the one in the proof of Proposition1.So we can substitute(10)and(11)as equalities to the other constraints and eliminate theαvariables.Furthermore,the missing constraints (12)and(13)are satis?ed once the constraints(10)and(11)have been shown to be binding. So we know that the solution to the relaxed program is the solution to the original problem. After having substituted for theαvariables,we rewrite the constraints:

?tσh≥?tσs,(A.6) f(mσ)?tσs≥?tσh,(A.7)

244I.Alger,C.-t.Albert Ma /J.of Economic Behavior &Org.50(2003)225–247

?t τh ≥?t τs ,

(A.8)f (m τ)?t τs

≥?t τh ,(A.9)θ[(1?p)t σh +p(t σs ?m σc)]+(1?θ)[(1?p)t τh +p(t τs ?m τc)]≥0.(A.10)

Following a now familiar argument,constraints (A.6)and (A.8)are binding;as a result,constraints (A.7)and (A.9)are satis?ed with slack.Now use (A.6)and (A.8)as equalities to substitute into (A.10)and the objective function.So we simplify the program into choosing t σh ,m σ,t τh

,and m τto maximize θ[(1?p)U(W ?t σh

)+pU (W ? +f (m σ)?t σh )]+(1?θ)[(1?p)U(W ?t τh )+pU (W ? +f (m τ)?t τh )],

subject to

θ(t σh

?pm σc)+(1?θ)(t τh ?pm τc)≥0.The proposition follows from comparing the ?rst-order conditions of this program to those in the proof of Proposition 1.

A.3.Proof of Proposition 3

We begin by assuming that ατh ?t τh <ατs ?m τc ?t τs .We will later consider the other possibility.Then,constraint (22)reduces to:

ασs ?m σc +(1?p)(t σh ?t σs )≥ατs ?m τc +(1?p)(t τh ?t τs ).(A.11)

First note that we can set ασh =0.We next show that (19)for j =τis binding.Suppose that it does not bind.Then we can reduce ατs ,while increasing ατh so as to leave the left-hand sides of (20)and (21)unaffected.Constraint (A.11)has been relaxed by the decrease of ατs .Hence,t σh can be decreased.Since t σh is absent from the other constraints,these are not affected.This results in an increase of the expected utility.Next,we show that for j =σ

(19)is binding.Suppose that it is not binding.Then,we can increase m σ.For instance,increase m σc by >0and suf?ciently small.This relaxes constraint (21),so that ατh can be decreased by (p/(1?p)) .By (20),t τh can then also be decreased by (p/(1?p)) .This in turn implies that the right-hand side of constraint (A.11)decreases by .Since the left-hand side was also decreased by ,this constraint remains unaffected.But the expected utility has increased,since m σhas been increased,and t τh has been decreased.Given that the constraints (19)for j =τand for σare binding,constraint (21)is https://www.doczj.com/doc/9311314352.html,st,we show that constraint (18)must bind.Indeed,if it did not,ατh could be decreased.By (20),t τh could then be decreased,thus,raising the expected utility.We have,therefore,proved 1and https://www.doczj.com/doc/9311314352.html,ing (18)and (19)as equalities to substitute for the αvariables,we simplify the budget constraint (20)and the incentive constraint for the collusive provider (A.11)to:

θ(t σs ?m σc)+(1?θ)[(1?p)t τh +p(t τs ?m τc)]≥0,

(A.12)t σh ?t σs ≥t τh ?t τs .(A.13)

some和any的用法与练习题

some和 any 的用法及练习题( 一) 一、用法： some意思为：一些。可用来修饰可数名词和不可数名词，常常用于肯定句 . any 意思为：任何一些。它可以修饰可数名词和不可数名词，当修饰可数名词时要用复数形式。常用于否定句和疑问句。注意： 1、在表示请求和邀请时，some也可以用在疑问句中。 2、表示“任何”或“任何一个”时，也可以用在肯定句中。 3、和后没有名词时，用作代词，也可用作副词。二、练习题： 1.There are ()newspapers on the table. 2.Is there ( )bread on the plate. 3.Are there () boats on the river? 4.---Do you have () brothers ?---Yes ,I have two brothers. 5.---Is there () tea in the cup? --- Yes,there is () tea in it ,but there isn’t milk. 6.I want to ask you() questions. 7.My little boy wants ()water to drink. 8.There are () tables in the room ,but there aren’t ( )chairs. 9.Would you like () milk? 10.Will you give me () paper? 复合不定代词的用法及练习一.定义: 由 some，any，no，every 加上 -body,-one,-thing,-where构成的不定代词，叫做复合不定代词 . 二. 分类： 1.指人：含 -body 或 -one 的复合不定代词指人 . 2.含-thing 的复合不定代词指物。 3.含-where 的复合不定代词指地点。三：复合不定代词： somebody =someone某人 something 某物，某事，某东西 somewhere在某处，到某处 anybody= anyone 任何人，无论谁 anything任何事物，无论何事，任何东西 anywhere 在任何地方 nobody=no one 无一人 nothing 无一物，没有任何东西 everybody =everyone每人，大家，人人 everything每一个事物，一切 everywhere 到处 , 处处 , 每一处

初中英语名词练习题与详解

名词判断对错 1、［误］ Please give me a paper. ［正］ Please give me a piece of paper. ［析］不要认为可以数的名词就是可数名词，这种原因是对英语中可数与不可数名词的概念与中文中的能数与不能数相混淆了，所以造成了这样的错误，因paper 在英语中是属于物质名词一类，是不可数名词。而不可数名词要表达数量时，要用与之相关的量词来表达，如： two pieces of paper. 2、［误］ Please give me two letter papers. ［正］ Please give me two pieces of letter paper. ［析］ paper 作为纸讲是不可数名词，而作为报纸、考卷、文章讲时则是可数名词，如：Each student should write a paper on what he has learnt. 3、［误］ My glasses is broken. ［正］ My glasses are broken. 4、［误］ I want to buy two shoes. ［正］ I want to buy two pairs of shoes. ［析］英语中glasses—眼镜， shoes—鞋， trousers—裤子等由两部分组成的名词一般要用复数形式。如果要表示一副眼镜应用 a pair of glasses 而这时的谓语动词应与量词相一致。如：5、This pair of glasses is very good. ［误］ May I borrow two radioes? ［正］ May I borrow two radios? ［析］以o 结尾的名词大都是用加es 来表示其复数形式，但如果 o 前面是一个元音字母或外来语时则只加s 就可以了。这样的词有zoo— zoos,piano—pianos. 6、［误］ This is a Mary's dictionary. ［正］ This is Mary's dictionary. ［析］如名词前有指示代词this, that, these those, 及其他修饰词our,some, every, which,或所有格时，则不要再加冠词。 7、［误］ There are much people in the garden. ［正］ There are many people in the garden. ［析］可数名词前应用 many, few, a few, a lot of 来修饰，而 people 是可数名词，而且是复数名词，如： The people are planting trees here. 8、［误］ I want a few water. ［正］ I want a little water. ［析］不可数名词前可以用 a little, little, a lot of, some来修饰，但不可用many,few 来修饰。 9、［误］ Thank you very much. Y our family is very kind to me. ［正］ Thank you very much. Y our family are very kind to me. 10、［误］ Tom's and Mary's family are waiting for us. ［正］ Tom's and Mary's families are waiting for us. 11、［误］ I'm sorry . I have to go. Tom's families are waiting for me. ［正］ I'm sorry. I have to go. Tom's family are waiting for me. ［析］集合名词如果指某个集合的整体，则应视为单数，如指某个集合体中的个体则应视为复数。如 :My family is a big family. When I came in, Tom's family were watching TV. 即汤姆一家人正在看电视。这样的集合名词有：family class, team 等。

some和any的用法

some和any的用法：（1）两者修饰可数单数名词，表某一个；任何一个；修饰可数复数名词和不可数名词，表一些；有些。〔2）一般的用法：some用于肯定句；any用于疑问句，否定句或条件句。 I am looking for some matches. Do you have any matches? I do not have any matches. （3）特殊的用法： (A) 在期望对方肯定的回答时，问句也用some。 Will you lend me some money? (=Please lend me some money.) (B) any表任何或任何一个时，也可用于肯定句。 Come any day you like. （4）some和any后没有名词时，当做代名词，此外两者也可做副词。 Some of them are my students.〔代名词） Is your mother any better?（副词） 3. many和much的用法：（1）many修饰复数可数名词，表许多； much修饰不可数名词，表量或程度。 He has many friends, but few true ones. There hasn't been much good weather recently. （2）many a: many a和many同义，但语气比较强，并且要与单数名词及单数形动词连用。 Many a prisoner has been set free. (=Many prisoners have been set free.) （3）as many和so many均等于the same number of。前有as, like时, 只用so many。 These are not all the books I have. These are as many more upstairs.

some与any的用法区别教案资料

s o m e与a n y的用法区别

some与any的用法区别一、一般说来，some用于肯定句，any用于否定句和疑问句。例如： She wants some chalk. She doesn’t want any chalk. Here are some beautiful flowers for you. Here aren’t any beautiful flowers. 二、any可与not以外其他有否定含义的词连用，表达否定概念。例如： He never had any regular schooling. In no case should any such idea be allowed to spread unchecked. The young accountant seldom (rarely, hardly, scarcely) makes any error in his books. I can answer your questions without any hesitation. 三、any可以用于表达疑问概念的条件句中。例如： If you are looking for any stamps, you can find them in my drawer. If there are any good apples in the shop, bring me two pounds of them. If you have any trouble, please let me know. 四、在下列场合，some也可用于疑问句。 1、说话人认为对方的答复将是肯定的。例如： Are you expecting some visitors this afternoon?（说话人认为下午有人要求，所以用some）

50套初中英语数词

50套初中英语数词一、初中英语数词 1．We throw rubbish every year. A. ton of B. tons C. tons of D. a ton of 【答案】 C 【解析】【分析】句意：我们每年扔大量的垃圾。ton，吨，前面没有具体数字，因此用tons of，大量的，故选C。【点评】考查固定搭配，注意平时识记。 2．Two students to the opening ceremony last Friday. A. hundreds; were invited B. hundred; were invited C. hundreds of; invited 【答案】 B 【解析】【分析】句意：上周五有200名学生被邀请参加开幕式。根据题干中的two与选项中的hundred可知此题考查确切数量的表达方式，hundred要用单数形式；students与invite存在动宾关系，此处要用被动语态，由last Friday，可知要用一般过去时，故选B。【点评】考查数量的表达方式以及被动语态。注意确切数量与不确切数量在表达上的不同。 3．—Do you know the boy is sitting next to Peter？ —Yes. He is Peter's friend. They are celebrating his birthday. A. who; ninth B. that; nine C. which; ninth 【答案】 A 【解析】【分析】句意：——你知道那个坐在彼得旁边的男孩吗?——是的。他是彼得的朋友。他们正在庆祝他的九岁生日。分析句子结构可知，第一空所在句子是定语从句，先行词是人，连接词在从句中作主语，所以应该用who/that引导，which连接定语从句时先行词应该是物，故排除C；nine九，基数词；ninth第九，序数词；第二空根据空后的birthday为名词单数可知，此处需要序数词，表示某人几岁生日应该用序数词表示第几个生日，故选A。【点评】考查定语从句的连接词的辨析和序数词。注意区别定语从句的连接词的使用原则，理解单词词义。 4．There are ________________ months in a year. My birthday is in the ________________ month. A. twelve; twelve B. twelfth; twelfth C. twelve; twelfth D. twelfth; twelve 【答案】 C 【解析】【分析】句意：一年有12个月，我的生日在第12个月。名词复数months前是基数词，twelve是基数词，the定冠词后是序数词，twelfth是序数词，故选C。【点评】考查数词，注意名词复数前是基数词，定冠词后是序数词的用法。

some和any的用法及练习题

some和any的用法及练习题 (一) 一、用法： some意思为：一些。可用来修饰可数名词和不可数名词，常常用于肯定句. any意思为：任何一些。它可以修饰可数名词和不可数名词，当修饰可数名词时要用复数形式。常用于否定句和疑问句。注意：1、在表示请求和邀请时，some也可以用在疑问句中。 2、表示“任何”或“任何一个”时，也可以用在肯定句中。 3、和后没有名词时，用作代词，也可用作副词。二、练习题： 1.There are ( )newspapers on the table. 2.Is there ( )bread on the plate. 3.Are there ( ) boats on the river? 4.---Do you have ( ) brothers ?---Yes ,I have two brothers. 5.---Is there ( ) tea in the cup? ---Yes,there is ( ) tea in it ,but there isn’t milk. 6.I want to ask you ( ) questions. 7.My little boy wants ( )water to drink. 8.There are ( ) tables in the room ,but there aren’t ( )chairs. 9.Would you like ( ) milk? 10.Will you give me ( ) paper? 复合不定代词的用法及练习一.定义: 由some，any，no，every加上-body,-one,-thing,-where构成的不定代词，叫做复合不定代词. 二.分类： 1.指人：含-body或-one的复合不定代词指人. 2.含-thing的复合不定代词指物。 3.含-where的复合不定代词指地点。三：复合不定代词： somebody =someone某人 something某物，某事，某东西 somewhere在某处，到某处 anybody= anyone任何人，无论谁 anything任何事物，无论何事，任何东西 anywhere在任何地方 nobody=no one无一人 nothing无一物，没有任何东西 everybody =everyone每人，大家，人人 everything每一个事物，一切 everywhere到处,处处,每一处

some和any的用法

some和any的用法 1.some adj.一些；某些；某个pron. 某些；若干；某些人 a.adj. some可以修饰可数名词或不可数名词，意为“某些”。 Some people are playing football. (some+可数名词) I ate some bread. (some+不可数名词) b.adj. some后面可以修饰可数名词的单数，意为“某(个)”。 Some day you will know. (some+可数名词的单数) 有一天你会知道的。 Some student cheated in the exam.(some+可数名词的单数) 有个学生考试作弊。对比：Some students cheated in the exam.有些学生考试作弊。 c.pron. some此时作代词，后面不需要再加名词就可以表示“有些(人)”的意思。 All students are in the classroom, and some are doing their homework. d.pron. some作代词，意为“若干(…)”。 There are 10 apples on the table. You can take some. 桌上10个苹果，你可以拿走一些。 2.any adj.任何的；所有的pron.任何一个；任何 a.adj. any可以修饰可数名词和不可数名词，意为“任何的，所有的任何一（…）”。（用于否定意义的陈述句、疑问句、条件状语从句if中） Do you have any ideas?(any+可数名词复数)(疑问句) 你有什么想法吗？ I don’t have any bread.(any+不可数名词)(否定意义的陈述句) Please tell me if you have any problem.(if引导的条件状语从句) b.any后面可以加可数名词的单数，意为“任何一（…）”。 Any error would lead to failure.(any+可数名词单数) 任何（一个）错误都会导致失败。 c.pron. any此时作代词，与some里面c点的用法相似，只是表示这个意义的时候，any多用于否定句和疑问句中。比较：There are 10 apples on the table. You can take some. 桌子上有10个苹果，你可以拿走一些。 There are 10 apples on the table, but you can’t take any. 桌子上有10个苹果，但是你不能拿。 There are some apples on the table.桌上有些苹果。 There aren’t any apples on the table.桌上没有苹果。由此，把陈述句变为否定句/一般疑问句的时候，要把some改成any。思考：some只用于肯定句，any只用于否定句和疑问句中吗吗？不一定，要看句子本身想表达的意思。 1.some可以用于肯定句和疑问句中。在表示请求、邀请、提建议等带有委婉语气的疑问句中，用some表示说话人希望得到肯定的回答。例如： Would you like some coffee?你想喝咖啡吗? 这里用some而不用any，是因为说话人期待得到对方肯定的回答。（因此Would you like…?你想要…吗?这个句型中多用some而不用any）比较： Do you have any books?这里用any而不用some，说明这只是因为这只是纯粹的疑问。

Some和any的用法

Some和any的用法 some和any 既可以修饰可数名词又可以修饰不可数名词，some常用在肯定句中，而any则常用在否定和疑问句中。因此some和any 的用法主要是考虑用在肯定句、疑问句还是否定句中，与名词的可数与否无关。 some意为“一些”，可作形容词和代词。它常修饰可数名词复数。如：some books一些书，some boys一些男孩，也可修饰不可数名词，如：some water一些水，some tea一些茶叶，some常用在肯定句中。any意为“任何一些”，它也可修饰可数名词复数或不可数名词，常用于疑问句和否定句。如： --I have some tea here. 我这儿有些茶叶。--I can’t see any tea. 我没看见茶叶。 --Do you have any friends at school? 你在学校有些朋友吗? 但在表示建议，反问，请求的疑问句中，或期望得到肯定回答时，多用some而不用any。如： Would you like some coffee? 你要不要来点咖啡? What about some fruit juice? 来点水果汁如何? 当any表示“任何”的意义，起强调作用时，它可以用在肯定句中； Any student can answer this question.任何学生都可以回答这个问题。辨析some和any的不同用法：some 常用在肯定句中，而any 则常用在否定和疑问句中。在表示建议，反问，请求的疑问句中，或期望得到肯定回答时，多用some而不用any。 Enough的用法英语中enough表示“足够”，它可以用作名词、形容词和副词，它的具体用法有：一、用作形容词，意思是“充足的; 足够的”；在句中可作定语和表语。作定语时它的位置较灵活，既可放在所修饰的词前，也可放在所修饰的词后。例如： We have enough seats（seats enough）for everyone.我们有足够的座位让大家都能坐。二、用作副词，意思是“充分地；足够地”，修饰形容词、副词或动词，位于所修饰词的后面。例如： 1. He is not strong enough. 他不够强壮。 2. She is old enough to understand this. 她年纪已足够大了能了解这事。 3 ．I didn't know her well enough. 我对她不够了解。4．He did not work hard enough. 他不够用功。［提示］enough常用在“be+形容词+enough +for+sb.+to do.”结构中，注意do后面不接宾语。例如： The book is easy enough for you to read.这本书很简单，你们可以看懂。三、作名词，意思是“足够; 充分”。例如： 1. —Would you like another cup of tea? 你想再来一杯茶吗？ —I have had enough. Thank you very much. 我已经喝够了。非常感谢。 2. He couldn't earn enough money to keep a family. 他赚得钱不够养活家人。我们在使用enough时还要注意以下几点： 1）enough不能与no连用。如不可以说：I have no enough money to buy a car.而应说：I don't have enough money to buy a car. 2）enough用作形容词时不可被very修饰，但可用quite修饰。例如： We have quite enough time. 我们有足够的时间。 3）can't / can never... enough表示“越……越好；无论怎样……也不过分”。例如： You can never be careful enough. 你越细心越好。下面的“口诀”可能对大家理解和掌握enough的用法有所帮助：

人教版初中英语短语大全最全

人教版初中英语短语大全 1)be back/in/out 回来/在家/外出 2)be at home/work 在家/上班 3)be good at 善于，擅长于 4)be careful of 当心，注意，仔细 5)be covered with 被……复盖 6)be ready for 为……作好准备 7)be surprised (at) 对……感到惊讶 8)be interested in 对……感到举 9)be born 出生 10)be on 在进行，在上演，（灯）亮着 11)be able to do sth. 能够做…… 12)be afraid of (to do sth. that…)害怕……(不敢做……，恐怕……) 13)be angry with sb. 生(某人)的气 14)be pleased (with) 对……感到高兴(满意) 15)be famous for 以……而著名 16)be strict in (with) (对工作、对人)严格要求 17)be from 来自……，什么地方人 18)be hungry/thirsty/tired 饿了/渴了/累了 19)be worried 担忧 20)be (well) worth doing (非常)值得做…… 21)be covered with 被……所覆盖…… 22)be in (great) need of (很)需要 23)be in trouble 处于困境中 24)be glad to do sth. 很高兴做…… 25)be late for ……迟到 26)be made of (from) 由……制成 27)be satisfied with 对……感到满意 28)be free 空闲的，有空 29)be (ill) in bed 卧病在床 30)be busy doing (with) 忙于做……(忙于……) (二)由come、do、get、give、go、have、help、keep、make、looke、put、set、send、take、turn、play等动词构成的词组 1)come back 回来 2)come down 下来 3)come in 进入，进来 4)come on 快，走吧，跟我来 5)come out出来 6)come out of 从……出来 7)come up 上来 8)come from 来自…… 9)do one's lessons/homework 做功课/回家作业 10)do more speaking/reading 多做口头练习/朗读 11)do one's best 尽力 12)do some shopping (cooking reading, clean ing)买东西(做饭菜，读点书，大扫除) 13)do a good deed (good deeds)做一件好事(做好事) 14)do morning exercises 做早操 15)do eye exercises 做眼保健操 16)do well in 在……某方面干得好 17)get up 起身 18)get everything ready 把一切都准备好 19)get ready for (=be ready for) 为……作好准备 20)get on (well) with 与……相处(融洽) 21)get back 返回 22)get rid of 除掉，去除 23)get in 进入，收集 24)get on/off 上/下车 25)get to 到达 26)get there 到达那里 27)give sb. a call 给……打电话 28)give a talk 作报告 29)give a lecture (a piano concert)作讲座(举行钢琴音乐会) 30)give back 归还，送回 31)give……some advice on 给……一些忠告 32)give lessons to 给……上课 33)give in 屈服 34)give up 放弃 35)give sb. a chance 给……一次机会 36)give a message to……给……一个口信 37)go ahead 先走，向前走，去吧，干吧 38)go to the cinema 看电影 39)go go bed 睡觉(make the bed 整理床铺) 40)go to school (college) 上学(上大学) 41)go to (the) hospital 去医院看病

some和any地用法

(1)some和any 的用法： some一般用于肯定句中,意思是“几个”、“一些”、“某个”作定语时可修饰可数名词或不可数名词。如：I have some work to do today. (今天我有些事情要做)/ They will go there some day.(他们有朝一日会去那儿) some 用于疑问句时,表示建议、请求或希望得到肯定回答。如：Would you like some coffee with sugar?(你要加糖的咖啡吗？) any 一般用于疑问句或否定句中,意思是“任何一些”、“任何一个”,作定语时可修饰可数或不可数名词。如：They didn’t have any friends here. (他们在这里没有朋友)/ Have you got any questions to ask?(你有问题要问吗？) any 用于肯定句时,意思是“任何的”。Come here with any friend.(随便带什么朋友来吧。) (2)no和none的用法： no是形容词,只能作定语表示,意思是“没有”,修饰可数名词(单数或复数)或不可数名词。如：There is no time left. Please hurry up.(没有时间了,请快点) / They had no reading books to lend.(他们没有阅读用书可以出借) none只能独立使用,在句子中可作主语、宾语和表语,意思是“没有一个人(或事物)”,表示复数或单数。如：None of them is/are in the

classroom.(他们当中没有一个在教室里) / I have many books, but none is interesting.(我有很多的书,但没有一本是有趣的) (3)all和both的用法： all指三者或三者以上的人或物,用来代替或修饰可数名词；也可用来代替或修饰不可数名词。 both指两个人或物,用来代替或修饰可数名词。all和both在句子中作主语、宾语、表语、定语等。如：I know all of the four British students in their school.(他们学校里四个英国学生我全认识) / --Would you like this one or that one? –Both.(你要这个还是那个？两个都要。) all和both既可以修饰名词(all/both+(the)+名词),也可以独立使用,采用“all/both + of the +名词(复数)”的形式,其中的of 可以省略。如：All (of) (the) boys are naughty.(是男孩都调皮) (4)every和each用法： every是形容词,只能作定语修饰单数名词,意思是“每一个”,表示整体概念； each是形容词、代词,可用作主语、宾语、定语等,意思是“每个”或者“各个”,表示单个概念；each可以放在名词前,可以后跟of 短语,与动词同时出现时要放在“be动词、助动词、情态动词”之后或者行为动词之前 every和each都用作单数理解,但是下文中既可以用单数的代词(如he/him/his)也可以用复数的代词(如they/them/their)替代。如：Every one of the students in his class studies very hard.(他班上每个学生学

some,any,one ones those that的区别和用法

some和any的区别和用法要表示"一些"的意思，可用some, any。 some 是肯定词，常用于肯定句；any是非肯定词，常用于否定句或疑问句。例如： There are some letters for me. There aren''t any letters for me. Are there any letters for me? I seldom get any sleep these days. any也常用于条件分句以及带有否定含义的句子中： If you have any trouble, please let me know. 如果你有任何麻烦，请让我知道。 I forgot to ask for any change. 我忘了要一些零钱。当说话人期待肯定回答时，some也可用于疑问句, 比如当说话人期待来信时，他可以问道：Are there some letters for me? 当购物时向售货员提问或者主人向客人表示款待时，也可在疑问句中用some: Could I have some of these apples? some和any 既可以修饰可数名词又可以修饰不可数名词，some常用在肯定句中，而any则常用在否定和疑问句中。因此 some和any 的用法主要是考虑用在肯定句、疑问句还是否定句中，与名词的可数与否无关。 some意为“一些”，可作形容词和代词。它常修饰可数名词复数。如：some books一些书，some boys一些男孩，也可修饰不可数名词，如：some water一些水，some tea一些茶叶，some常用在肯定句中。any意为“任何一些”，它也可修饰可数名词复数或不可数名词，常用于疑问句和否定句。如： --I have some tea here. 我这儿有些茶叶。 --I can’t see any tea. 我没看见茶叶。 --Do you have any friends at school? 你在学校有些朋友吗? --I have some English books, they are my best friends. 我有英语书，它们是我最好的朋友。但在表示建议，反问，请求的疑问句中，或期望得到肯定回答时，多用some而不用any。如：Would you like some coffee? 你要不要来点咖啡? What about some fruit juice? 来点水果汁如何? 当any表示“任何”的意义，起强调作用时，它可以用在肯定句中； Any student can answer this question.任何学生都可以回答这个问题。选题角度：辨析some和any的不同用法：some 常用在肯定句中，而any 则常用在否定和疑问句中。在表示建议，反问，请求的疑问句中，或期望得到肯定回答时，多用some而不用any。一般用于疑问句或否定句中，用于never, hardly, without等词之后，用于if / whether 之后。而some则用于肯定句中，用于建议或请求的疑问句中，用于预料会作肯定回答的疑问句中，用于表示反问的否定的疑句中。如： 1. I’d been expecting ________ letters the whole morning, but there weren’t ________ for me. (全国卷)卷 A. some; any B. many; a few C. some; one D. a few; none

英语量词用法大全

英语量词用法大全常用语法场景1：当和外国友人走在公园，突然看到一群鸽子。这时候你是不是就会说:There are many pigeons 或者a lot of呢？想再高大上一点吗？Sam, look! There is a flock of pigeons over there. 场景2:当说三张纸的时候，是不是很习惯说：“three papers”？错了哦。paper 是不可数名词。正确说法是：three pieces of paper。 ◆◆量词用法整理+讲解◆◆ 量词可以用于描述可数名词比如a herd of elephants(一群大象）, 也可以描述不可数名词；比如three pieces of paper（三张纸） 1、描述一群...量词+群+of+名词一群人a/an crowd／group／army／team／of people；一群牛、象、马、天鹅a herd of cattle／elephants／horses／swans

一群鸟、鹅、母鸡、羊、燕子a flock of birds／geese／hens／goats／swallows 一群猎狗、狼a pack of hounds／wolves 2、描述一丝／点／层一丝怀疑a shadow of doubt 一线未来之光a glimpse of future 一缕月光a streak of moonlight 一层霜／雪／糖霜a layer of frost／snow／cream 3. piece块；片；段；项；件；篇；首；幅；张 a piece of bread／paper／wood／furniture／land／advice／news／meat ／cloth／music... 4、英译“一阵” 一阵哭泣／喝彩／炮击／雷声

some与any的用法区别

初中英语名词用法讲解

一、名词的分类名词可分为普通名词和专有名词两大类。 1. 普通名词又可分为：（1）个体名词。如：cup，desk，student等。一般可数，有单复数形式。（2）集体名词。如：class，team，family等。一般可数，有单复数形式。（3）物质名词。如：rice，water，cotton等。一般不可数，没有单复数之分。（4）抽象名词。如：love，work，life等。一般不可数，没有单复数之分。 2. 专有名词：如：China，Newton，London等。二、名词的数（一）可数名词的复数形式的构成规则 1. 一般情况下在名词的词尾加s，如：book books，pencil pencils. 2. 以-s，-x，-ch，-sh结尾的名词加-es，其读音为［iz］。如：bus buses，box boxes，watch watches，dish dishes等。 3. 以-y结尾的名词：（1）以“辅音字母+y”结尾的名词，把y改为i再加es，读音为［iz］，如：factory factories，company companies等。（2）以“元音字母+y”结尾的名词，或专有名词以y结尾，直接在词尾加-s，读音为［z］。如：key keys，Henry Henrys等。 4. 以-f和-fe结尾的名词：（1）变-f或-fe为v再加-es，读音为［vz］。如：thief thieves，wife wives，half halves等。（2）直接在词尾加-s，如：roof roofs，gulf gulfs，chief chiefs，proof proofs等。（3）两者均可。如：handkerchief handkerchiefs或handkerchieves. 5. 以-o结尾的名词：

关于some和any的用法和区别

关于some和any的用法和区别 some和any都是常见词汇，他们有共同点也有很多不一样的地方，你们知道他们饿区别在哪里吗?接下来小编在这里给大家带来some和any的用法和区别，我们一起来看看吧! some和any的用法和区别一、some和any作为形容词或代词，可以用来说明或代替复数名词或不可数名词，表示不定量，意为一些，其区别是：对其所说明或代替的名词持肯定态度时，用some;持非肯定(否定或疑问)态度时，用any。在以下句子中使用some： 1.肯定句(包括肯定的陈述句和祈使句以及反意疑问句中肯定的陈述部分)。如： There are some new books on the teachers desk. We have a lot of sugar. Take some with you, please. He bought some bread, didnt he? 2.持肯定态度的一般疑问句。如： Are there some stamps in that drawer? Didnt she give you some money? 3.表示请求或建议的一般疑问句，通常都希望得到对方肯定的答复，所以也用some。如： May I ask you some questions?

Would you like some tea? 4.特殊疑问句及选择疑问句。因为特殊疑问句和选择疑问句并不对some所说明或代替的名词表示疑问。如： Where can I get some buttons? Do you have some pens or pencils? 在以下句子中使用ANY： 1.否定句(包括否定的陈述句和祈使句以及反意疑问句中否定的陈述部分)。如： I cant give you any help now. Do not make any noise. There werent any trees here, were there? 2.含有除not以外的其他否定词或否定结构的句子。如： Jim hardly makes any mistakes in his homework. He went to London without any money in his pocket. She was too poor to buy any new clothes. 3.一般疑问句(持肯定态度的除外)。如： Did she buy any tomatoes yesterday? I want some paper. Do you have any? 4.条件状语从句。如： If you are looking for any ink, you can find it on my desk. If there are any good apples there, get me two kilos, please.