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If somebody is risk averse then they are willing to pay a risk premium in order to avoid the risk – usually in order for someone else to bear the risk in the form of insurance.

Consider the case where a person has a utility function of $U=10W^{0.3}$ where U signifies utility and W signifies wealth in £. The more wealth they have, the more utility they have so the happier they are.

Say they start with wealth of £2000. They own a machine that has a 10% chance of breaking down in a particular time period. If it breaks down it will cost £800 to repair it.

There are two possible outcomes here, a good outcome where the machine doesn’t break down and the person has wealth of £2000, or a bad outcome where the machine breaks down and they have £1200. We can use the utility function to calculate the amount of utility in the respective states:
Good outcome: U = 97.793

The machine has a 10% chance of breaking down, so that means a 90% chance of us having the good outcome and 10% chance of the bad outcome. We can use this to calculate the expected utility of this whole situation:

EU = (0.9 x 97.793) + (0.1 x 83.899) = 96.404

Now if we went back into the original utility function to find what wealth corresponds to a utility level of 96.404, we get $96.404=10W^{0.3} \Rightarrow W = 1906.838$.

In other words the expected utility of this situation is equivalent to having wealth of £1906.84. The difference between this and the starting wealth of £2000, £93.16, is the maximum risk premium the machine owner is willing to pay to remove the risk. If an insurer offers to sell insurance that will fully cover the costs of machine repair if it breaks down, then the machine owner is willing to pay up to £93.16 in order to buy that insurance.

Say the insurer offers to provide full insurance for a cost of £90. Then the machine owner has a choice – either he takes the insurance, which means he has guaranteed wealth of £1910, and a corresponding guaranteed utility of 96.452, or he doesn’t take insurance, which will mean he either has £2000 (and utility of 97.793) if it doesn’t break down, or £1200 (and utility of 83.899) if it does. In terms of expected utility, the expected utility from taking insurance is 96.452, and the expected utility from not taking the insurance is 96.404. As the expected utility from taking the insurance is higher than the expected utility from not taking insurance, then it is rational to take the insurance. If the insurance had cost more than £93.16, then the expected utility from taking the insurance would have been lower than the expected utility from not taking the insurance, so it would have been rational not to take out the insurance.

## Risk aversion and utility

We can think of risk aversion in the context of utility. The most intuitive way is to consider wealth. Most of us would probably have a utility function that increases with diminishing returns as wealth increases. That is, the more wealth we have, the more utility we get (the happier we are), but the increases in our utility are greatest when we have less wealth. If you have wealth of nothing, and somebody offers you £10000, you are probably going to get a great deal of additional happiness from that £10000. However if you are already a multi-millionaire, then an extra £10000 might increase your happiness but it won’t have the same effect as it would if you had nothing.

A risk averse person will have a utility function relative to wealth, that looks similar to this:

A risk neutral person will have a utility function that looks like this:

A risk loving person will have a utility function that looks like this:

The shapes of the utility functions reflect the extent to which utility increases with wealth. The one of most interest to us is the risk averse one, because people are more likely to be risk-averse than anything else. This brings us to the concept of risk premium and insurance.

## Income and substitution effects

Optimal choice when faced with a budget constraint involves moving to an indifference curve that is tangential to the budget line.

But if the prices a good changes, it will change the budget line. Here we will take the opportunity to use the “two good” model to specifically focus on the effects of the price change of one good, by considering good X on the horizontal axis and “AOG” or “all other goods” on the vertical axis. This allows us to see how much someone can spend on all other goods, once they have finished spending on good X.

If the price of good X rises, then the budget line will pivot inwards around the point where it crosses the vertical axis. At the vertical intercept, the consumer is spending all his money on “all other goods” and not buying any of good X, so a price change in good X won’t change anything. But the horizontal intercept will be different, as the consumer is then spending all his money on good X, which is now more expensive, so obviously he can afford less given his budget.

Here at the original budget constraint of B1, the optimal bundle is at A, where the consumer has aX of good X and aAOG of all other goods. When the price of X increases, the budget line shifts to B2. Here the optimal bundle is at B, where the consumer is consuming less of good X and more of other goods.

Two things have gone on here. There is a substitution effect whereby the consumer is deciding to substitute some of good X for some other goods because X is now more expensive relative to other goods. There is also an income effect whereby the fact that X has got more expensive whilst other goods are unchanged in price, means that the consumer is now relatively poorer than he was before, his money doesn’t go as far.

To show how much of the overall change in spending on the different goods is due to the substitution effect, we consider what would happen if we were ‘compensating’ the consumer by giving him enough of a raise in income to allow him to stay at the same level of utility as he was on at bundle A, now that good X has become more expensive.

We can break down the substitution and income effects like this:

Here we have drawn a new budget line (the black dashed line) with the same slope as the final budget line (B2), indicating the same relative prices between X and other goods as the final relative prices after the price change. This has been moved to the point that is tangential to the indifference curve that the original bundle, A, was on. Now this budget line effectively represents an increased budget, because it is a parallel shift outwards from B2. It shows us the budget that the consumer would have needed, to stay at the original level of utility, the same indifference curve as he was on at bundle A. However A would not be the optimal choice bundle here, it would be bundle C. Bundle C gives the same level of utility as bundle A, but it would satisfy the new relative prices after the price change. This shows us the substitution effect. If the consumer was compensated by being given enough raise in income to allow him to stay at the same level of utility as he was on at bundle A, he would have responded to the change in relative prices by shifting his spending to buy less good X and more other goods, hence the move from A to C.

The rest of the effect is the income effect. The shift from C to B, where the consumer has less money to spend on good X and also less money to spend on other goods, is as a result of the income effect of the price change making him poorer overall.

## Optimal choice with a budget constraint

One of the basic premises of ‘normal’ preferences in microeconomics is that we assume that more is better. In other words, if we are considering two goods, X and Y, and we have a choice between a bundle of 3X and 4Y, or 10X and 25Y, we are going to prefer the second because we get more of both.

However most of the time there is a cost involved with both of the goods, and we don’t have unlimited resources. The more we spend on good X the less we spend on good Y. So we will have a budget constraint depending on the resources available to us. We could either spend all our budget on good X, or all on good Y, or on some combination of the two.

We usually illustrate the budget constraint on a diagram along with indifference curves indicating our preferences.

Here we have two goods, X and Y, and a budget constraint indicated by the black line. All combinations of bundles between the origin and the budget line are affordable. Anything on the budget line is just affordable, it means we are using up all our budget on the combination of goods X and Y chosen. Anything inside the budget line (ie closer to the origin) is affordable and we will have some of the budget to spare. Anything outside the budget line (ie further away from the origin than the line) is not affordable.

There are four bundles shown, A, B, C and D, each on a different indifference curve. The most preferable bundle is D, because this is on the highest indifference curve, furthest away from the origin, but this bundle lies outside our budget constraint so it is unaffordable.

Bundle A is inside our budget constraint so this is not using our full resources and we can easily move to a more preferable bundle (like B or C) simply by spending more. B and C are both bundles that use up the full budget, but C is preferable to B as it is on a higher indifference curve.

C is the optimal choice when faced with the budget constraint given here, as it is the indifference curve that is tangential to the budget constraint. This means that at point C, the slope of the indifference curve, or the marginal rate of substitution, is equal to the slope of the budget line.

## Marginal utilities and the marginal rate of substitution

The marginal rate of substitution is the rate at which the consumer is willing to substitute one good for another in order to retain the same level of utility.

Lets say our goods are are X and Y, and the total utility derived from having a bundle that is a combination of some X and some Y is U.

We will have a utility function of the form $U(X,Y)$.

The marginal utility we get by adding a unit more of X will be $\frac{\partial U}{\partial X} = MU_X$.

The marginal utility we get by adding a unit more of Y will be $\frac{\partial U}{\partial Y} = MU_Y$.

The marginal utility of X is also the change in total utility we get divided by the change in X, $MU_X =\frac{\Delta U}{\Delta X} = \frac{U(X+\Delta X, Y) - U(X,Y)}{\Delta X}$.

When the change in X is small then we can simply approximate to $\Delta U = MU_X \Delta X$, and by the same logic when the change in Y is small we can approximate to $\Delta U = MU_Y \Delta Y$.

If we were increasing both X and Y then the total change in utility would be $\Delta U = MU_X \Delta X + MU_Y \Delta Y$.

The concept of marginal rate of substitution is that it tells us how much we are willing to substitute of one good in order to get more of another, whilst keeping our overall utility constant. So the key thing here is that overall utility is being unchanged. This means that if for instance we are adding a unit more of X, then we are having to give up some of Y to make up for it.

So if overall utility is unchanged, $\Delta U = 0$ so $0 = MU_X \Delta X + MU_Y \Delta Y$. Hence $- MU_Y \Delta Y = MU_X \Delta X \Rightarrow \frac{\Delta Y}{\Delta X} = \frac{- MU_X}{MU_Y}$. $\frac{\Delta Y}{\Delta X}$ is $\frac{dY}{dX}$ so this gives us an expression for the marginal rate of substitution: it is simply the ratio of the marginal utilities. In the context of an indifference curve, this is the slope of the indifference curve, which makes sense as it is $\frac{dY}{dX}$.

## The Marginal Rate of Substitution and gains from trade

The slope of the indifference curve at a particular point shows us the rate at which the consumer is willing to substitute one good for another in order to retain the same level of utility. This is the marginal rate of substitution, and it is the centrepiece behind ideas of trade and exchange.

To understand the idea of being ‘indifferent’ here, consider the possibility that there are two consumers, me and you, and we both have goods X and Y. I decide that I want some more of good Y, and I am willing to offer you some of my X in order to get it. How much more X you want in order to give up one unit of Y, will depend on a couple of factors – how much you like X compared to Y, and how much X and Y you already have. If you already have loads of X and not much Y, you are likely to be less keen on my offer of trading X for Y than you would be if you had loads of Y and not much X. People generally prefer exchanging the ‘good they have more of’, for the ‘good they have less of’ (apologies for the poor English). This is also known as diminishing marginal rate of substitution and is a property held by most normal convex shaped indifference curves.

The marginal rate of substitution is the amount of Y you would be willing to give up for a unit of X, in other words the change in Y over the change in X. As you will see, this changes as you move along the indifference curve, in other words as you have different combinations of goods.

At point A, you have a lot of Y and not much X, so here the MRS is very steep, you are willing to give up Y for not much X in return. But down at point B, you have a lot of X and not much Y, and here if you are to give up one of your precious units of Y, you are asking for a lot more X in compensation.

Now you can see what would happen if you were offered trade on fixed terms (ie a certain amount of Y in exchange for a certain amount of X). This is a ‘rate of exchange’. In the diagram below, the black dotted line illustrates a fixed rate of exchange which the consumer is being offered. We will consider that he starts off at bundle A.

Here the terms of trade are very good for the consumer. He can exchange some of good Y for some good X and move to bundle C, and make a simple gain, moving to a higher indifference curve representing a gain of utility. But that would not be the full extent of the possibilities of his gain from trade. He could carry on trading at that rate of exchange all the way up to point D, where he has reached the highest indifference curve possible at this rate of exchange, and the highest level of utility possible at this rate of exchange.

What is special about point D? It is the point at which the MRS for the consumer is the same as the MRS of the rate of exchange offered, the indifference curve is tangential to the rate of exchange.

This shows an important result. When a consumer holds a bundle of goods and is offered trade at any rate of exchange different to his MRS, then he can gain utility by trading at that exchange rate. Only when the rate of exchange offered is exactly equal to his MRS, will he not be able to make any gains by trading at that rate of exchange.

## Shapes of indifference curves

Most indifference curves that follow normal preferences have a convex shape:

However you will come across some different shapes:

Perfect substitutes are like this:

Here the consumer only cares about the total number of X + Y that they have, not whether they are getting more of X or Y, so the indifference curves are straight lines.

Perfect complements are like this:

This is a bit like the case where X is a left shoe and Y is a right shoe. If you have 3 left shoes and 4 right shoes, then you are no better off than having 3 left shoes and 3 right shoes as you need the pair together. In fact you are no better off even if you have 3 left shoes and 100 right shoes. So having more Y does not increase your utility unless you increase X as well.