Question 1: Consumer Theory
In both the Marshallian and Hicksian consumer optimisation problems, it is assumed that consumers are supposed to be rational. The main focus of these problems are cost minimisation and utility maximisation, which play a huge part in consumer demand, but in real life, these are not the only problems that are considered. Also, it is assumed that every consumer’s indifference curve for two goods would be the same – they are very generalised models, and do not take into account other factors. For example, not many consumers would spend their entire budget on said goods – one thing to consider would be a consumer’s marginal propensity to consume and save. Though both of the problems provide a framework and model of consumer decisions, they are not plausible when applying them to real-life terms, because we have imperfect knowledge.
The expression given in the question, is the rearranged derivative of the Hicksian demand being equal to the Marshallian demand, when income from the budget constraint is equal to minimised expenditure, whereby m=ep, μ. This is given by: dDdp= dHdp- dDdm . dedp
using m = e.
Shephard’s Lemma provides us an alternative way of deriving Hicksian demand functions, using e. It is given by: dedp= x*
It is important to note that e is strictly increasing in p, due to Shephard’s Lemma, and x* >0,by assumption. Substituting this into the above expression gives: dDdp= dHdp- dDdm x* This expression now represents a complete law of demand, as it has combined both Marshallian and Hicksian demand, whereby income from the budget constraint of Marshallian demand, is equal to minimised expenditure of Hicksian demand. Therefore, it has maximised utility and minimised cost simultaneously, to create an optimal quantity of demand in x*. The first term, dDdp, means that Marshallian demand (maximising utility) increases, relative to the price of the good. dHdp represents the Hicksian part of the expression, whereby expenditure is minimised, relative to the price of the
Question 3: Adverse Selection, Moral Hazard and Insurance
Insurance markets are needed when risk is present. Risk occurs when there is uncertainty about the state of the world. For example, car drivers do not know if they will crash their car in future, and suffer a loss of wealth – so they would purchase insurance to eliminate this risk of loss, and protect them if they were to ever crash their car. Agents (buyers of insurance) will use insurance markets to transfer their income between different states of the world. This allows insurance markets to trade risk between high-risk and low-risk agents/states. These can be described as Pareto movements. A Pareto improvement is the allocation, or reallocation of resources to make one individual better off, without making another individual worse off. Another term for this is multi-criteria optimisation, where variables and parameters are manipulated to result in an optimal situation, where no further improvements can be made. When the situation occurs that no more improvements can be made, it is Pareto efficient.
A condition for efficiency is the least risk-averse agent bears all the risk in an insurance market. If a risk-averse agent bears risk, they would be willing to pay to remove it. A risk-averse agent has a diminishing marginal utility of income; whereby his marginal utility is different across states, if his income is different across states. The agent would give up income in high-income states, in which his marginal utility is low, to have more income in low-income states (e.g. bad state of the world causing a loss of wealth), where his marginal utility would be high. If the insurance market is risk neutral, they will sell insurance to the customer, as long as the payment received is higher than the expected value of pay-outs that the insurer is contracted to give to the customer in different states of the world.
Whenever the agent bears some risk, unexploited gains from trade exist. Absence of unexploited gains from trade is a requirement in an efficient insurance market, therefore the situation must arise, whereby the agent’s income is equalised across the states of the world. A risk neutral insurance company can charge a premium to equalise the agent’s income across states of the world, in the best interests of the risk-averse agent. Also, for an insurance market to be efficient, a tangency condition is implied. The tangency of the indifference curves of a risk-averse agent, and a risk-neutral agent, is where efficiency occurs. At this point, one cannot be made better off, without the other being made worse off (Pareto efficiency).
However, an insurance company will never be completely efficient in real life, as information asymmetry exists. The first type of information asymmetry to arise in an insurance market is moral hazard, whereby the actions that an agent may take after signing the contract cannot be observed. This gives the company a trade-off decision between giving full insurance or offering incentives for the agent. Full insurance is first-best in the absence of asymmetric information, when the insurance company is risk-neutral and the agent is risk-averse. However, if the agent is fully insured by the company, they have no reason to prevent a bad state of the world from happening. To solve this problem, the insurance company will not offer full insurance, in order to provide the agent with an incentive to avoid losses.
The second type of information asymmetry to occur in an insurance market, is adverse selection. This is when the agent has private information about his risk type and characteristics, and agents in the market are heterogenous. As the insurer doesn’t know which agents are high-risk or low risk, the company will not offer different types of full insurance to match risk-types, as high-risk agents will prefer contracts that are designed for low-risk agents. To solve this, the insurer will offer low-risk agents less insurance – this ensures that high-risk types do not have the incentive to choose a contract for low-risk customers, as they will want more insurance, because they know they will need to claim more.
This ensures that the insurance company maintains non-negative profit, as high-risk individuals cost more to insure. However, these solutions carry agency costs, because the result is less efficient than if symmetric information was present. I believe that risk neutrality of an insurance company is a sufficient condition for insurance to take place. Insurance companies are risk-neutral to maximise expected profits, therefore as the principal, will design contracts to achieve this, as well as making certain that the agent picks the desired effort (i.e to prevent a bad state of the world) for that contract, and to make sure that the agent even picks the contract in the first place. Making sure incentives are compatible, and ensuring participation by the correct risk types, are constraints on maximising expected profits.
If an insurance company was risk-averse, without the availability of symmetric information, they cannot differentiate between different risk-types, and therefore would not want to take on the risk of possible high-risk agents buying low-risk contracts. They would charge a higher premium to offset this, which would discourage low-risk customers to sign a contract with the company, as it would not be maximising their own utility. This would lead to a missing market, where trade would be prevented, because other risk-neutral companies would offer better contracts, and they would be able to steal all the low-risk customers. The magnitude of this would depend on the number of low- and high-risk people in the population. This leads me to believe that risk neutrality is also a necessary condition for insurance to take place.
An insurance company will sell a policy, c, r, if it makes non-negative profits, then: → r-pic ≥0, where c = payout, pi = probability of the loss state, r = premium. Competition in the market drives profit down to zero, therefore r-pic = 0 in equilibrium. For the contract to be at equilibrium, it must satisfy two conditions: the break-even condition, whereby no contract makes negative profits; and absence of unexploited opportunities for profit, because if there was a contract outside of the offered set, with non-negative profit, would mean the offered set is not in equilibrium. If all agents are homogenous, if all agents face the same probability of loss, pi=p, insurance companies would know each buyer’s pi. The firm must maximise each agent’s utility subject to the firm breaking even. This would be at the point of tangency of the agent’s indifference curve and zero-profit constraint. This would be in equilibrium as another profit-making policy could not be offered.
Therefore, as they can observe agent’s risk types, they can offer different policies, to different types: θi= ri, ci. It follows that each is offered full and fair insurance. In real life, heterogeneity is usually the case. This is when pi varies with all individuals. Assuming that there are two types: high-risk types, H, and low-risk types, L, where the probability of loss for H is higher than for L. Individuals know their own probability of loss i=H, L, but insurance companies are unable to observe this. In this case, there are two different kinds of equilibria that insurance companies could opt with: the candidate pooling equilibrium and the candidate separating equilibrium. The pooling equilibrium is where all risk types buy the same policy. In contrary, the separating equilibrium is based on each risk type buying a different policy. In the pooling equilibrium, if both H and L risk-types choose the same policy, the probability of loss is p and the probability of no loss is 1- p.
Therefore, the slope of the ‘aggregate fair-odds line is -1-pp. The pooling contract must lie on this line to be in equilibrium, to ensure the firm breaks even exactly. The contract must also ensure both types want to buy it – it must take both L and H to higher indifference curve than the indifference curve they would be on if they stayed uninsured. Agent L ends up below his fair odds line, and H above his, which means L pays more than expected costs, and H pays less – both pay the fair pooled premium, but H claims on the policy more. So if L prefers to buy the contract, so will H. This leads me to believe both L and H will be able to get full insurance, though it’s not completely fair, as the firm does not need H to choose a different policy to remain breaking even. However, this brings to mind the notion that if full insurance is offered, the agent will not have the incentive to prevent a loss state.
Therefore, less insurance will probably be offered, and as both risk types are paying the same premium of the same policy, neither will receive full insurance, as it impossible to differentiate between the two – they will both choose the same policy offered. In the separating equilibrium, one contract would be offered to L, and another to H. Each risk type must prefer the contract designed for that type (i.e. the incentives must be compatible). The contracts offered should give each type the highest possible utility, subject to the firm breaking even. If full insurance contracts were offered to both L and H, where their respective indifference curves are tangent with their respective zero-profit constraints/fair-odds lines, low risk customers would prefer the policy designed for them, but high-risk customers would also prefer the same policy, not the policy designed for them.
So they would not both be offered full insurance, as this gives rise to the problem of preventing H from imitating L – low-risk agents are cheaper to insure for the firm (claim less often) so they get a better rate. Therefore, instead of offering L full insurance, they are offered C, which is still on their fair odds line, but on a lower indifference curve, still ensuring the zero-profit constraint.
Now, if the high-risk agents were to choose between the policy designed for them, and C, they will choose the policy designed for them, because they prefer to have more insurance for less money. So, in conclusion, in the separating equilibrium, high-risk (H) customers receive full insurance, and low-risk (L) customers only receive partial insurance – they pay the price to prevent H from imitating them. L is worse off than if there was symmetric information in the market, but no difference to H.