Chapter 22: Exchange in Capital Markets 22.1: Introduction We are now in a position to examine trade in capital markets. We know that some people borrow and some people save. After a moment s reflection you will also realise that the total amount borrowed must equal the total amount saved if someone wants to borrow some money there must be someone else willing to lend him or her that money. In a capital market which is in equilibrium the price of borrowing and saving must adjust so that borrowing and saving are equal. And what is this price? The rate of return 1+r of course for every 1 borrowed today (1+r) must be repaid in one period s time; for every 1 saved there is a payoff of (1+r) in one period s time. We are assuming there is no inflation 1. This chapter looks at trade through time. We examine a capital market in which individuals can exchange money today for money in one period s time. This may be a bit confusing as what we put on the axes is consumption in the two periods so we should explain a little more what it is that we are assuming. We are assuming that the individuals get utility out of what they consume each period. Consumption is an all-inclusive term but it may simplify things a bit if we think of it as food. So the individuals get utility out of consuming food each period. As we shall assume, they each get an endowment of food each period when they wake up in period 1 there is an endowment of food waiting for them and when they wake up in period 2 there is an endowment of food waiting for them. If they did no trading then they would simply consume the food with which they were endowed and that would be the end of it. However they might prefer to re-arrange their consumption consuming more in one period and less in the other. (For example if they were given lots in one period and very little in the other.) Now suppose that food is perishable so that if it is not consumed in period 1 it goes bad so it can not be consumed in the second period. If that is so then the only way to re-arrange consumption is to trade with someone else A gives some food to B in period 1, for example, and B gives some food to A in period 2. Trade is thus essential if any kind of re-arrangement is to be implemented if the endowment is in perishable food. Now introduce money into the economy and for the moment let us suppose that the price of consumption in both periods is 1 (that is, we are assuming zero inflation). Then money and consumption are synonymous and if we put consumption in period 1 on the horizontal axis it is just the same as if we put money spent in period 1 on the horizontal axis. Similarly, if we put consumption in period 2 on the vertical axis it is just the same as if we put money spent in period 2 on the vertical axis. Now trade on this interpretation is trading money in period 1 for money in period 2. So, for example, A gives some money to B in period 1 and in exchange B gives A some money in period 2. So far so good. The only problem with this interpretation is that money is not like perishable food in a world with no inflation in fact 1 in period 1 remains 1 in period 2. It is therefore clear that no-one will accept a trade which gives them less than 1 in period 2 for each 1 given in period 1. No-one will accept a negative rate of interest as they can always guarantee themselves a zero rate of interest by simply saving the money under the bed 2. So we should be a little careful if we use non-perishable money on the axes rather than perishable food/consumption 1 See later for the differences if there is inflation. 2 If there is inflation say of x% - no-one will accept a rate of interest less than x% because they can once again guarantee a rate of interest equal to x% by saving their money under the bed.
on the axes we should check that the equilibrium rate of interest is positive 3. If it is not then the lenders will choose not to lend and will save their money under the bed. 22.2: Trade in a Capital Market We use the technique for analysing exchange that we introduced in chapter 8 that of the Edgeworth box. We assume a very simple economy consisting of just two people and two goods. The people are individuals A and B; the goods are consumption in period 1 and consumption in period 2. The individuals may or may not differ in their preferences over these two goods and they may or may not differ in their endowments of the two goods. Let us build up a specific example, exactly as we did in chapter 8. We start with individual A and assume that initially he or she has 100 units of consumption in period 1 and 25 units in period 2. This is rather an unbalanced endowment and he or she may well be happy to trade to a more balanced consumption profile. Let us suppose that he or she has Discounted Utility preferences with a square root utility function (as we used in chapter 21) and a discount rate ρ equal to 0.3: A discounts the future at 30%. We draw A s indifference curves and endowment point in figure 22.1. Note that there are lots of points in this figure which A would prefer to his or her endowment point E. Now individual B. I assume that initially he or she has 50 units of consumption in period 1 and 75 units in period 2. He or she too has a somewhat unbalanced initial stock. As for preferences let us assume that B has the same preferences as A Discounted Utility Model preferences with a square root utility function and a discount rate ρ equal to 0.3 so that we can see whether trade is possible when the individuals have the same tastes. B is portrayed in figure 22.2 measured from the usual origin and again in figure 22.3 measured from the top right origin. 3 If there is inflation we should check that the equilibrium real rate of interest is positive.
We now do Edgeworth s clever trick. We put the figure of B upside down on top of A the right way up making sure that the endowment points coincide - and we get the Edgeworth box of figure 22.4. Note very carefully that the width of the box is the total initial stock of consumption in period 1: A has 100 units and B has 50, giving a total of 150 units. The height of the box is the total initial stock of consumption in period 2: A has 25 units and B has 75 units, giving a total of 100 units. We ask whether trade is possible. In figure 22.4 we have not drawn the contract curve but it is drawn in figure 22.9 below. Its position, however, is obvious it is the straight line joining the two origins (because the preferences are identical 4 ). The contract curve has the same interpretation as in chapter 8: it is the locus of efficient points in the space. Any point off the curve is inefficient in the sense that there is always a direction to move in which both individuals are better off than before. However, once on the contract curve any movement is bound to make at least one of the two worse off. We might therefore expect that any contract entered into would be on the contract curve. From the picture it will be seen that the initial point is off the contract curve. In this example the contract curve is the straight line joining the two origins the reason for this is that they have identical preferences. But note that they do not have identical endowments the endowment point E is not at the middle of the box and indeed is off the contract curve. Some trade should be possible. As before, let us investigate where competitive trading takes the two individuals. As we have already noted the price of consumption in period 1 relative to the price of consumption in period 2 is 1+r - where r is the rate of interest - for the simple reason that every pound consumed in period 1 costs 1+r in terms of period 2 consumption - as that is what could be consumed in period 4 And, strictly speaking, because the preferences we have assumed are homothetic which is a mathematical concept outwith the scope of this course.
2 if the pound was not consumed in period 1. So the slope of a budget constraint is (1+r). This, of course, we already knew from the material in chapter 20. Now for each rate of interest r we can draw the budget constraint a line with slope (1+r) passing through the point E and hence find the optimal point for each individual. In this way we can find the price-offer (or the interest-rate offer ) curve for each individual. These are illustrated in figure 22.9. In this figure the convex curve passing through the endowment point E is the interest-rate offer curve for A and the concave curve passing through E is the interest-rate offer curve for B. The straight line joining the two origins is the contract curve which passes through the point where the two interest-rate offer curves intersect. This is the competitive equilibrium and the line joining it and the endowment point E is the equilibrium budget constraint. Let us study carefully this competitive equilibrium which is almost but not quite at the centre of the box. It is at the point (76.24, 50.83) as measured from the bottom left origin and at the point (73.76, 49.17) as measured from the top right origin. We thus get the following analysis. Initial Allocation Individual A Individual B Society Money/Consumption 100 50 150 in period 1 Money/Consumption in period 2 25 75 100 Competitive Individual A Individual B Society Equilibrium allocation Money/Consumption 76.24 73.76 150 in period 1 Money/Consumption 50.83 49.17 100
in period 2 Changes between the two Money/Consumption in period 1 Money/Consumption in period 2 Individual A Individual B Society -23.76 +23.76 0 +25.83-25.83 0 So we get the following exchange: A gives to B 23.76 units of consumption in period 1 and B gives to A 25.83 units of consumption in period 2. Note very carefully that A forgoes 23.76 units of period 1 consumption but gets in return 25.83 units of period 2 consumption. Looking at it in terms of borrowing and saving, A lends 23.76 to B in period 1 and gets back from B 25.83 in period 2. For every unit saved (lent to B) in period 1 A gets back 25.83/23.76 = 1.087 units in period 2 a rate of return 1.087 and hence a rate of interest 8.7%. Note that this is positive (which means that A is better off lending the money to A rather than keeping the money under the bed). Note also that the slope of the equilibrium budget constraint the line joining E and the competitive equilibrium is 1.087. In this equilibrium both individuals re-arrange their consumption stream A ends up consuming less than his first period income (which was big) and in exchange gets more period 2 consumption. B increases his first period consumption but has to pay (interest) for the privilege in terms of having a more than lower second period consumption. The competitive equilibrium rate of interest is positive. You might like to ask yourself whether this is always going to be the case. 22.3: A Different Scenario In this section we look at a different scenario. In this Scenario 2 we assume identical endowments but different preferences. The first of these means the endowment point is at the centre of the box and the second of these means that the contract curve is not the straight line joining the two origins. To be specific we assume identical square root utility functions but different discount factors the ρ for A is equal to 0.1 and that for B equal to 0.5. So both discount the future though B more heavily than A. For this reason the contract curve is above and to the left of the line joining two origins for any given division of period 2 consumption B gets rather more than the equal share of consumption in period 1.
Notice where the competitive equilibrium is. The equilibrium interest rate is 5% (the slope of the line joining the endowment point and the competitive equilibrium has slope 1.05). Although the two individuals start out with the same endowment, A is induced by a positive rate of interest to lend some money to individual B the reason being that, relative to A, B prefers to consume in the first period and is willing to pay interest for the privilege. As a consequence A ends up consuming more over the two periods combined. 22.4: Comments You will realise that the analysis of this chapter is very similar to that of chapter 8. In fact it is identical except for nomenclature: instead of two general goods we have consumption in period 1 and consumption in period 2 and trade takes place in the capital market where the individuals exchange money/consumption in period 1 for money/consumption in period 2. The price of period 1 consumption in terms of period 2 consumption is one plus the rate of interest. It follows that all the results of chapter 8 remain relevant including: (1) the general possibility of trade except when the initial point is on the contract curve (which happens if the endowments are identical and the preferences are identical); (2) the efficiency of trade along the contract curve; and (3) the efficiency of the competitive equilibrium. Also remaining relevant is the fact that the competitive equilibrium, and hence the equilibrium rate of interest, depends upon the initial endowments and the preferences. 22.5: Summary
We have shown that some kind of intertemporal trade is usually possible, though in a world with no inflation and with non-perishable money the competitive equilibrium may not be implementable because one of the agents prefers to store his or her money under the bed rather than accept a negative rate of interest. Otherwise all the results from chapter 8 (which considered trade in general) remain valid. In a world with inflation we simply correct the rate of interest for the inflation and hence get a real rate of interest - equal to the nominal rate of interest minus the rate of inflation. We get the same kind of result: the equilibrium real rate of interest must be positive to induce both agents to indulge in trade. We have shown that the equilibrium (real) rate of interest depends upon the endowments and the preferences of the individuals. 22.6: How can real rates of interest be negative? You may have noticed that, in certain periods of history, real interest rates have been negative. What do we mean by this? Simply that the rate of inflation exceeds the money rate of interest. Consider the following table, taken from statistics in the UK publication Economic Trends Annual Supplement: Year Retail Price index (1985=100) Implied Rate of Inflation between current year and following year (%) Interest Rate on Treasury Bills (%) 1973 25.1 9.1 12.52 1974 29.1 15.9 11.30 1975 36.1 24.1 10.93 1976 42.1 16.2 14.09 1977 48.8 - - Take 1974 for example. Between then and 1975, the retail price index rose 15.9% whilst the rate of interest on Treasury Bills was 11.30%. This means that 100 invested in Treasury Bills in 1974 became worth 111.30 in 1975. However, in 1975 111.30 could buy 100/36.1 = 3.08 in goods whereas in 1974 100 could buy 100/29.1 = 3.44 in goods; that is, the 100 invested in Treasury Bills was worth 10.4% less in 1975, in terms of its buying power over goods, than in 1974 because of the effect of inflation. Prices rose 15.9% while the interest rate was only 11.30% - implying a negative real rate of interest. Money invested in the capital markets was worth less in 1975 than in 1974 in terms of its buying power over goods. You might well ask: how can this happen?
Let us use the analysis of the chapter. The key question is whether the equilibrium rate of interest can be negative. We have, so far, assumed zero inflation, and we will continue to do so for the moment, and add inflation in later. In the chapter we assumed that both individuals put more weight on present consumption than on future consumption. It can be shown that, if this is true, then the equilibrium rate of interest must be positive, whatever the initial endowments. Hence, in order to get a case in which we have a negative rate of interest, we must suppose that one of the two individuals puts more weight on future consumption. Consider the following graph. To produce this figure we have assumed that the rate of discount for Individual A (whose consumption we measure from the bottom left-hand origin) is 0.4 and the rate of discount for Individual B (whose consumption we measure from the top right-hand origin) is 0.4. Note carefully the minus sign. So A puts relative weight 0.714 (=1/(1+0.4)) on second period consumption, while B puts relative weight 1.667 (= 1/(1-0.4)) on second period consumption. A cares more about period 1 consumption as we have assumed throughout our analysis - but B cares more about period 2 consumption than period 1 consumption. There is nothing to say that this cannot be the case; perhaps you know individuals who feel that way? The figure shows the consequences. We have assumed in this figure that we have a perfectly symmetric endowment point both individuals have income of 50 of each good in each period. The endowment point is indicated with the letter E, and the competitive equilibrium is at the point where the two price-offer curves and the contract curve intersect at the point labelled C, approximately at (71,31). So in the equilibrium exchange, B gives to A 21 units of consumption in period 1 and A gives in exchange 19 units of consumption in period 2. The slope of the equilibrium budget constraint is thus -19/21 = -0.904. This, as we know, is equal to (1+r) where r is the rate of interest. We therefore have an equilibrium rate of interest 0.096, that is, - 9.6%. A is happy with this exchange as he or she prefers period 1 consumption, and B is happy with this as he or she prefers period 2 consumption. (Recall that the initial endowment point has equal consumption in both periods for both individuals.)
At this stage you could retort that B would not enter into this exchange because rather than give up 21 units of period 1 consumption for just 19 of period 2 consumption, he or she would prefer to put the 21 units of period 1 consumption under the bed and consume it in period 2. If the good is perishable this is clearly not possible if it is perishable food, for example. But if it is money, could he or she not simply put the money under the bed? This might work if money did not change value during the period. But consider the possibility of inflation. Let the price of food in period 1 be p 1 and that in period 2 p 2. Suppose the incomes are in food both individuals have an endowment of 50 units of food in each period, and suppose that food is perishable. Then individual B, in order to carry money over to next period, has to sell some of his or her endowment of food in period 1, and then buy some more food in period 2. How does this change the analysis? Note that money per se has no value and cannot be consumed directly. We repeat the analysis under this scenario and what do we find? There is an equilibrium price ratio between food in the two periods which is clearly such that p 1 /p 2 = 19/21 because the slope of the equilibrium budget line is 19/21. It follows therefore that the price in period 2, p 2, is 21/19 times the price in period 1: there is inflation in the price of food between the two periods. Therefore, putting the money under the bed makes no difference, as it is worth less in period 2. Indeed sufficiently less that if B puts 21 in money under the bed in period 1, it will only buy 19 units of food in period 2 exactly the same solution as if trade takes place between A and B! Of course, if there was some other asset, which kept its value between the two periods (or which fell in value less than money), then B should buy that asset. Otherwise, he or she has to accept a negative rate of interest between the two periods precisely because he or she values consumption more in period 2 than in period 1. This perhaps explains why we observe negative real rates of interest from time to time.