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 good.
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...