Chapter 25: Exchange in Insurance Markets 25.1: Introduction In this chapter we use the techniques that we have been developing in the previous 2 chapters to discuss the trade of risk. Insurance markets exist for that purpose. In particular we look at optimal risk-sharing between individuals and discuss how the trading of risk can make people better off. We use the techniques of chapters 23 and 24 combined with the analysis of chapter 8 (where we discussed trade in general) to understand how the trading of risk works. We use the framework of chapter 8. We assume a very simple economy consisting of two individuals, A and B, and two possible states of the world, state 1 and state 2. Ex ante, when trading may take place, it is not know which of the two states will occur, though their respective probabilities π 1 and π 2 are known. Ex post, one and only one of the two states will occur. The individuals start with some initial endowments of income/consumption, but they may be able to trade away from this initial point. We work throughout with a specific analysis but the story can obviously be generalised. What is important is that you take away from this chapter an understanding of how insurance markets help to allocate risk efficiently. 25.2: An Edgeworth box We start with individual A. We assume that he or she would have with an income of 75 if state 1 occurs and an income of 50 if state. We assume that the two states of the world are equally likely. As for preferences, we assume that A has Expected Utility preferences and we assume here a constant absolute risk aversion utility function with parameter r = 0.01. We picture A s preferences and endowment point in figure 25.1. The straight line is A s certainty line. Now we turn to individual B. We assume exactly the same initial endowment: an income of 75 if state and an income of 50 if state. However we assume that B is more risk averse than A, having a constant absolute risk aversion utility function with parameter r = 0.03. We portray B s preferences and endowment point the normal way up in figure 22.2 and upside down in figure 22.3. The straight line in figure 25.3 is B s certainty line.
Now we do an Edgeworth. Superimpose B s upside down indifference map on top of A s in such a way that the endowment points coincide. We then get an Edgeworth box of dimension 150 (the total income/consumption if state 1 were to occur) times 100 (the total income/consumption if state 2 were to occur), with the endowment point exactly at the middle as their initial endowments are equal. You might like to think of State 2 as some kind of disaster in which the society as a whole loses 2/3 of its income. Inserted into figure 25.4 are the certainty lines of the individuals: the 45 line starting at the bottom left origin is that for A and the 45 line starting at the top right origin is that for B. Note that the slopes of all the indifference curves for both the individuals along their respective certainty lines is 1 the ratio of the probabilities of the two states. If you study this figure you should be able to see where the contract curve is. In this case, where both individuals have constant absolute risk aversion utility functions, the contract curve is in an interesting position and has an interesting shape: it is a 45 line between the two certainty lines, but nearer to the certainty line of the more risk averse individual (B). This is interesting as it indicates the nature of efficient risk sharing. The contract curve is nearer to the certainty line for the
more risk averse individual which means that in any efficient contract B s position is less risky than that of A 1. So A the less risk-averse individual bears more of the risk than the more riskaverse person. (Notice that there is risk to share because the total income is 150 in one state and 100 in the other there is no way to get rid of the risk altogether, though the two individuals can change the way they share it.) Let us look at the competitive equilibrium. It is the point marked C in figure 25.9. Also drawn in the figure are the two price-offer curves and the contract curve. Also joined together are the endowment point and the equilibrium point to indicate the equilibrium relative price (the price of state 1 contingent income relative to the price of state 2 contingent income). Let us summarise what happens. Initial allocation Individual A Individual B Society 75 75 150 50 50 100 Competitive equilibrium allocation Individual A Individual B Society 83 67 150 45 55 100 1 Note that on a particular individual s certainty line there is no risk for that individual. As we move away say at an angle of 90 - the riskiness increases and the further away the more the risk.
Changes between the two allocations Individual A Individual B Society +8-8 0-5 +5 0 Individual B is easy to understand. He or she starts out with an ex ante risky income (75, 50) by which we mean 75 if state and 50 if state. The expected income is 62.5 and the riskiness is ± 12.5 by which we mean that the two values are 12.5 either side of the mean. After trading to the equilibrium point, B has an ex ante risky income of (67, 55). This has an expected value of 61 which is lower than the expected income before trade but the riskiness is now ± 6. So individual B has reduced the riskiness of his or her ex ante income, at the cost of having to accept a reduction in the expected income. But because he or she is rather risk-averse, he or she is happy to accept this reduction in the expected income in exchange for a reduction in the risk. Note that at C, rather obviously, individual B is on a higher indifference curve than he or she was at E. A, on the other hand, starts out with the same ex ante risky income (75, 50) - which has an expected value of 62.5 and a riskiness of ± 12.5. After the trade at the competitive equilibrium A has an ex ante risky income of (83, 45). Note that this has an expected income of 64 and a riskiness of ± 19. We see that A has accepted an increase in the riskiness of his or her position but he or she has also benefited with an increase in his or her expected income. Despite the fact that A is moderately risk-averse he or she is prepared to take a little bit of extra risk in return for a little bit of extra expected income. Note that at C, rather obviously, individual A is on a higher indifference curve than he or she was at E. We end up with more efficient risk-sharing than we had originally: in the original position they were both exposed to the same amount of risk; in the competitive equilibrium A (the relatively less risk-averse) has more of the risk than B (the relatively more risk-averse person) though A is rewarded with a higher expected income, while B has to pay by accepting a lower expected income. [Incidentally this is not a fair insurance market because the relative price is clearly not equal to 1 the ratio of the probabilities. In fact the relative price is less than 1. You might like to ask yourself why.] 25.4: Same Risk Attitudes but Different Endowments Let us explore some other cases. Let us begin with a case in which the individuals have the same risk attitude but different endowments. We take constant absolute risk aversion functions with r = 0.02 for both individuals. You can probably guess where the contract curve is yes half way between the certainty line of A and that of B. We assume different endowments in terms of riskiness but assume the same expected values. Specifically A starts with (100, 25) and B starts with (50, 75). Both have an expected value of 62.5; A s riskiness is ± 37.5 while B s is ± 12.5. The Edgeworth box is in figure 25.10.
The various allocations are in the table below. Initial allocation Individual A Individual B Society 100 50 150 25 75 100 Competitive equilibrium allocation Changes between the two allocations Individual A Individual B Society 68 82 150 44 56 100 Individual A Individual B Society -32 +32 0 +19-19 0 The competitive equilibrium is obviously on the contract curve but it is closer to A s origin than to B s. In fact as you will see individual B does rather well out of the deal: starting with a risky income with expected value 62.5 and riskiness ± 12.5, he or she ends up in the competitive equilibrium with (82, 56) which has expected value 69 and riskiness ± 13. He or she accepts a little more riskiness and gets in exchange quite a big increase in the expected value. A, on the other hand, starts out with expected value 62.5 and riskiness ± 37.5 and ends up with (68, 44) which has
expected value 56 and riskiness ± 12. He or she accepts a big reduction in the riskiness in exchange for giving up quite a bit of expected value. But then A was initially expose to a lot of risk. 25.5: Similar Endowments but Different Preferences An interesting case is when each individual gets income in just one state of the world: A in state 1 and B in state 2. What do they do? Consider the case when A s ex ante income is 100 in state 1 and 0 in state 2, while B s is 0 in state 1 and 100 in state 2. Between the two of them there is no uncertainty: their joint income is 100 whichever state occurs. So the Edgeworth box is of size 100 by 100 and the endowment point is at the bottom right hand corner. Let us suppose different preferences: A is constant absolute risk averse with parameter r = 0.01 while B is constant absolute risk averse with parameter r = 0.03. As before the contract curve lies between the two certainty lines but as it happens in this case the certainty lines coincide (with the diagonal from the bottom left to the top right corners), and therefore so does the contract curve. Figure 25.12 illustrates. The two curves through the endowment point E are the price-offer curves. They are found in the usual way (see Chapter 8) by finding, for each possible budget line through the initial endowment point, the optimal choice of the individual. They intersect at the point C on the contract curve at the middle of the box. At the competitive equilibrium they are both at the point (50, 50) whichever state of the world happens both have an income 50. This is an interesting case as between the two of them they manage to completely remove the risk that they both had originally. Note also that the slope of the equilibrium budget constraint is 1. Note also that we get this equilibrium, given the endowment, irrespective of the risk aversion of the two individuals. The contract curve is bound to be the main diagonal since this coincides with the certainty line for each of them and so we know that the slopes of both their indifference curves are equal to 1 and hence are equal. Only at point C on the contract curve is the slope of the line joining E and that point on the contract curve also equal to 1. This is quite a reassuring result.
25.6: One Risk Neutral Individual When one of the individuals is risk neutral and the other is risk averse, we get a rather nice result. Let A be risk neutral and B risk averse. We get figure 25.13. In this figure the contract curve coincides with the certainty line of individual B so the competitive equilibrium is on B s certainty line. (It is at the point C given the endowment point E.) So, in fact, wherever is the endowment point, the competitive equilibrium is a position of certainty for individual B moreover, given that the slope of A s price-offer curve is 1 (equal to the slope of his or her indifference curves) it follows that the expected income of B is the same at the competitive equilibrium as at the endowment point B simply loses the risk. A gets all the risk but he or she does not care as he or she is risk neutral. This example is instructive as it shows that the presence of a risk-neutral individual removes the risk from elsewhere. 25.7: Summary We showed that we could use the general exchange apparatus of chapter 8 to examine the exchange of risk between two individuals. We saw that in general some exchange of risk is possible. In some cases we saw the individuals managing to completely eliminate the risk they faced. In others we saw that this was not possible but that some kind of efficient risk sharing was possible usually with the less risk-averse person taking more of the risk. Insurance markets help to achieve an efficient sharing of the risk in society. In general, with an insurance market, the less risk-averse ends up taking more of the risk.
In particular, if one individual is risk-neutral and the other risk-averse then the former ends up taking all of the risk. 25.8: What if people are not different? Throughout this book, we have argued that, in general, if people are different, then there will be an opportunity for mutually advantage exchange between them. Indeed, we have shown that if we have a simple society of two individuals, who are different in either their preferences or in their endowments, then mutually advantageous trade is possible as long as their initial position is not on the contract curve. In the examples given in the text, if the preferences are identical then the contract curve is the straight line joining the two origins, which passes through the mid-point of the box. It follows, therefore, that if the endowments are identical, we start at the mid-point of the box on the contract curve - and mutually advantageous trade is not possible. Note, however, that in the examples considered so far, we have assumed convex indifference curves. With such curves the proposition is true: if the individuals are identical in both their preferences and their endowments, then mutually advantageous trade is not possible. But, in the context of risky choice, concave indifference curves are possible they simply mean that the individual likes risk. Clearly there are such people in society: you know someone who plays the National Lottery? Consider now two identical, risk-loving, individuals. Possible indifference curves are illustrated in the following figure. The income/consumption of individual A is measured from the bottom lefthand origin his or her indifference curves are those that are concave (with respect to that origin). The income/consumption of individual B is measured from the top right-hand origin his or her indifference curves are those that are concave (with respect to that origin), and hence are those convex with respect to the bottom left-hand origin. Note that both goods are still good, so moving upwards and rightwards in the box makes A better off and B worse off.
Where is the contract curve? We have to be careful. At first glance it would appear to be the line joining A s origin with B s origin, but a moment s reflection will make you realise that points along this line are not efficient. Indeed we can find a direction to move away from this diagonal and increase the welfare of both A and B. For example, moving from the mid-point to either of the corners C or D makes both individuals better off that is, puts them both on higher indifference curves. In fact it can be shown that in this case, the contract curve is the whole of the perimeter of the box: once we are on it we cannot make one individual better off without making the other worse off; if we are off it, we can always find a direction to move and make both individuals better off. You should be aware that the contract curve is in an odd place in this example precisely because both individuals are risk-lovers. Around the perimeter they are both exposed to risk which is what they like. Now suppose they start with identical endowments the endowment point is that marked E at the centre of the box. Do they want to trade? Can we find a competitive equilibrium? Can we find the price-offer curves of the two individuals? These are the points to which they would move at given prices. That for A can be shown to be parts of the perimeter of the box: specifically the part from (100,0) to (100,50) and the part from (0,100) to (50,100). These parts are coloured blue in the above figure. Similarly we can find the price offer curve for B: it can be shown to be parts of the perimeter of the box: specifically the part from (50,0) to (100,0) and the part from (0,50) to (0,100). These parts are coloured red in the above figure. Notice that both individuals would choose to move away from the certainty of the mid-point of the box to a risky point on the perimeter 2. These two price-offer curves intersect at the points C and D the upper left-hand and the lower right-hand corners of the box. Note that C and D are on the contract curve. We therefore have two competitive equilibria, at points C and D, with the same equilibrium budget constraint the line joining C and D. 2 Though not any old point.
So we have trade, even though the two individuals have identical preferences and endowments. Note the nature of the trade they start off in a completely safe situation, each getting income 50 in whatever state of the world happens. But they are risk-lovers, preferring risk to certainty. So they trade to a position in which one gets income 100 in one state of the world and 0 in the other, while it is the other way round for the other individual. It is just as if they agree to a bet: A pays to B 100 if something happens and B pays 100 to A if it does not happen. They are both happier than not having this bet (because points C and D are on higher indifference curves than point E for both of them). They are happier with the bet than without it. Do you not know people like that?