Market_For_Lemons - Atlas of Economic Models
 

Market For Lemons

Introduction

The main assumption in the 'Market for Lemons' model is that buyers are unable to perfectly observe the quality of the good, instead using some market statistic to calculate the expected quality of that good. This gives sellers an incentive to market low-quality goods since buyers cannot know for sure the quality of a good and any change to the expected quality affects the market as a whole rather than individual sellers. As more low-quality goods are introduced to the market (or as high-quality goods are withdrawn) the expected quality declines, and with it the size of the market, eventually reducing to zero. Market failure is caused by the 'bad' goods (lemons) driving out the 'good' since sellers of low-quality goods have no incentive to reveal their type. This model is taken from Akerlof's 1970 paper entitled 'The market for 'lemons': Quality uncertainty and the market mechanism' (akerlof_1970).

Model

There is a market for a particular good $$ x $$ composed of $$ N_1 $$ sellers and $$ N_2 $$ buyers. The supply of $$ x $$ depends on its price $$ p $$ and a vector of all other relevant factors $$ y $$. There is a market statistic $$ \mu $$ which summarises the expected quality of $$ x $$ and depends on $$ p $$ such that as $$ p $$ falls, so does $$ \mu $$. The demand for $$ x $$ depends on $$ p $$, $$ \mu $$ and a vector of all other relevant factors $$ z $$. Buyers are unable to directly observe the quality of $$ x $$ , $$ q $$ , but know that it lies within a certain range $$ q \in [a,b] $$, necessarily implying that $$ \mu \in [a,b] $$. At the equilibrium, supply equals demand for the given quality.

  • $$ S = S(p,y) $$
  • $$ Q^d = D(p,\mu(p),z) $$
  • $$ \mu = \mu (p) $$ such that $$ \frac{\partial \mu}{\partial p} > 0 $$

  • $$ S(p,y) = D(p,\mu(p),z) $$ in equilibrium

The demand and supply functions are derived from the utility functions of B and S, where $$ n $$ is the units of good $$ x $$ already owned by the individual and $$ m $$ is a vector of all other goods. B and S have reservation utility levels of $$ \bar {U_B} $$ and $$ \bar{U_S} $$ respectively.

  • $$ U_B = U_B (p,q,n,m) $$ where $$ \frac{\partial U_B}{\partial p} < 0 $$ and $$ \frac{\partial U_B}{\partial q} > 0 $$ $$ \Rightarrow $$ B's utility is decreasing in price $$ p $$ and increasing in quality bought $$ q $$.

  • $$ U_S = U_S (p,q,n,m) $$ where $$ \frac{\partial U_S}{\partial p} > 0 $$ and $$ \frac{\partial U_S}{\partial q} < 0 $$ $$ \Rightarrow $$ S's utility is increasing in $$ p $$ and decreasing in quality supplied $$ q $$.

Solving

Symmetric information

If both B and S can observe $$ q $$ (or equally if neither can observe $$ q $$ and both parties know that the other cannot) then the market for $$ x $$ will clear such that $$ S(p,y) = D(p,\mu(p),z) $$ . The range of prices over which trades take place $$ p \in [p_{L}*,p_{H}*] $$ is determined by the particular forms of $$ U_B $$ and $$ U_S $$ and the restriction $$ q \in [a,b] $$ .

Asymmetric information

  1. If $$ q $$ is observable to S but B only knows the distribution of $$ q $$ then B must use a market statistic $$ \mu $$ to calculate the expected value of $$ q $$ . $$ U_B $$ now becomes a random variable, but B's expected utility at a particular price $$ E[U_{B} (p,\mu,n,m) | p] $$ can be found by replacing $$ q $$ with its expected value $$ \mu $$. The maximum price B would be willing to pay for $$ x $$ given their expected utility, $$ p_{1}^{max} $$ , is found by setting $$ E[U_{B} (p,\mu,n,m) | p] = \bar{U_B} $$ and solving for $$ p $$ . $$ p_{1}^{max} < p_{H}* $$ (i.e. the upper bound price for trades to occur under symmetric information) as quality uncertainty lowers the price that B are willing to pay.

  2. However, this now gives an additional supply constraint of $$ p \le p_{1}^{max} $$ . For some S it may be the case that $$ U_S(p_{1}^{max}) < \bar{U_S} $$ so they withdraw from the market and now $$ N_{1}' < N_1 $$ . These S who withdraw are those that supply $$ x $$ of a high $$ q $$ , since they do not receive sufficient compensation for their high $$ q $$ in the form of high $$ p $$ . This causes B to revise their expectations $$ \mu $$ as they know that the average quality of $$ x $$ has fallen and now $$ q \in [a,b'] $$ where $$ b' < b $$ and $$ b'$$ is such that $$ U_S (q=b' , p=p_{1}^{max}) = \bar{U_S} $$ (i.e. given $$ p_{1}^{max} $$ buyers know that S will only supply quality $$ x $$ up to the point where reservation utility is obtained).

This process is iterative, as since now $$ q \in [a,b'] $$ buyers form a new expectation of $$ q $$ , $$ \mu ' $$ , where $$ \mu ' < \mu $$ . Step (1) is repeated to find $$ p_{2}^{max} $$ where $$ p_{2}^{max} < p_{1}^{max} $$. $$ p_{2}^{max} $$ becomes the new supply constraint, so step (2) is then repeated to give $$ q \in [a,b"] $$ where $$ b" < b' $$ . After $$ t $$ iterations the market collapses such that $$ \mu = a $$ and $$ p_{t}^{max} $$ is such that $$ E[U_{B}(\mu = a) | p] = \bar{U_B} $$ .

Under asymmetric information, the market collapses because sellers of low-quality goods have no incentive to reveal this information as buyers cannot observe quality. In order to overcome this problem, the buyer needs to propose a set of contracts such that it is in the interests of sellers to voluntarily reveal the quality of their goods by accepting the contract that is designed for them. See Monopolist_Screening_Model.

Examples

The Used Car Market

The example in Akerlof's paper is that of the used car market (in the USA 'bad' cars are known as 'lemons'). Let the quality of used cars in the market $$ q $$ be a continuous random variable which is uniformly distributed $$ q \in [0,1] $$. Buyers B and sellers S have the following utility functions:

  • $$ U_{B} (p,q) = (3/2)q - p $$
  • $$ U_{S} (p,q) = p - q $$
  • $$ \bar{U_B} = \bar{U_S} = 0 $$

Symmetric information

If $$ q $$ is observable to both B and S then it is sufficient for a sale to occur that $$ p \in [q, (3/2)q] $$:

  • If $$ q = 0 $$ then $$ U_B = -p $$ and $$ U_S = p $$
    • If $$ p = q = (3/2)q = 0 $$ then $$ U_B = \bar{U_B} = U_S = \bar{U_S} = 0 $$ $$ \Rightarrow $$ both B and S are indifferent between trading and not.
  • If $$ q = 1 $$ then $$ U_B = (3/2) - p $$ and $$ U_S = p - 1 $$
    • If $$ p = q = 1 $$ then $$ U_B = (1/2) $$ and $$ U_S = 0 $$ $$ \Rightarrow $$ S are indifferent between trading and not, B get positive utility.
    • If $$ p = (3/2)q = (3/2) $$ then $$ U_B = 0 $$ and $$ U_S = (1/2) $$ $$ \Rightarrow $$ B are indifferent between trading and not, S get positive utility.

Asymmetric information

If $$ q $$ is observable to S but not to B then B must calculate the expected quality of a car. In this case the market statistic used is the expected value of the uniform random value $$ q \in [0,1] $$, i.e. $$ \mu = \hat{q} = (1/2) $$. At price $$ p $$ the expected utility of B is $$ E[U_{B} | p] = (3/2)\hat{q} - p = (3/4) - p $$, so the maximum that B are willing to pay is $$ p^{max} = (3/4) $$.

However, given that B will not pay $$ p > (3/4) $$, S who own cars of $$ q > (3/4) $$ will not sell since this would result in $$ U_S < \bar{U_S} $$. Thus the new upper bound for $$ q $$ becomes $$ (3/4) $$ giving $$ q \in [0,(3/4)] $$.

B now revise their beliefs about the expected quality of cars, with $$ \hat{q}' = (3/8) $$ and $$ E[U_B | p ] = (9/16) - p $$, giving a new maximum price of $$ p^{max} = (9/16) $$. As a result the market again reduces in size, since S who own cars of $$ q > (9/16) $$ withdraw from the used car market.

This process of B revising expectations and S withdrawing from the market continues until the market has collapsed. After $$ n $$ iterations the maximum price B is willing to pay is $$ p_{n}^{max} = (3/4)^n $$, so as $$ n \to \infty $$ , $$ p^{max} \to 0 $$ and $$ \hat{q} \to 0 $$ , thus the market collapses. The only equilibrium is $$ p = \hat{q} = 0 $$ and no trades take place.

Applications and Extensions

Health Insurance

The medical condition of an individual is known to the buyer (applicant) but cannot be perfectly observed by the seller (insurance company). Instead the insurance company observes the average medical condition of applicants, and as the price of cover increases the average medical condition of applicants decreases since only those with greater-than-average need for health insurance (i.e. those with a lower-than-average medical condition) will take out cover at the average price. This is a form of adverse selection problem, see *Stiglitz Model of Insurance*. Since they cannot discriminate between applicants, to avoid making a loss insurance companies set their prices high such that only those with a poor medical condition would wish to purchase cover. However, this means that 'high quality' individuals (i.e. those in good medical condition) will be priced out of the market and withdraw since they would pay more in premiums than they expect to get in payouts, and so the average 'quality' of this smaller pool of applicants falls. Again this is an iterative process, with each round prices rising, applicants withdrawing and the average 'quality' of applicants falling until eventually the market collapses and no insurance is offered.

An example is that medical insurance is rarely offered to people aged 65 and over. The over-65s are likely to have a lower-than-average medical condition and as such greater-than-average need for insurance, although obviously some will be in better health than others. However, insurance companies cannot differentiate between individuals in this age group with varying medical conditions, so even those who are in good health will not be offered insurance.

One way for insurance companies to overcome the asymmetric information problem is to engage in cream-skimming. A screening method is used to eliminate 'low-quality' applicants so that insurance can be offered to 'high-quality' applicants at lower prices. An example of this is offering health insurance only to the employed, since individuals must have at least a certain standard of health to be employed in the first place so those who are in poor health are excluded.

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