Chapter 17: Aggregation


 Verity Gregory
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1 Chapter 17: Aggregation 17.1: Introduction This is a technical chapter in the sense that we need the results contained in it for future work. It contains very little new economics and perhaps contains nothing that is not reasonably obvious after a little thought, but it is nevertheless very important. The basic point concerns aggregation of supply and demand. To date most of the book has been concerned with individuals with individual demand and individual supply. To apply this material in the real world, where we are dealing with lots of individuals, we need to be able to move from the individual level to the aggregate level we need to be able to aggregate the results we have obtained. In essence there are two parts to this one obvious and one perhaps not so obvious. The first is the simple aggregation of the demand and supply functions with which we have been working. The second is the aggregation of the important concept of surplus that we have been using. With regard to the first of these we want to ask how we do the aggregation, and perhaps we are also interested in whether the aggregate functions retain the properties of the individual functions. With regard to the second of these we want to know whether the surplus as measured with respect to the aggregate function is in fact the aggregate of the surpluses as measured with respect to the individual functions. Depending on your technical sophistication you may regard this latter question as either obvious or meaningless. To make this chapter easy I use just two individuals, individual A and individual B: if we can aggregate two individuals we can aggregate more than two. 17.2: Aggregating Demand We start with a familiar example one for the demand for a discrete good. We have in fact already done this. You will recall that the demand function for a good that is discrete takes the form of a step function, with a step at every reservation price. You might then be able to anticipate the results that follow. We work with a simple example. Let us ssume that individual A would buy up to 3 units of the good and has reservation prices 10, 5 and 1 for the 3 units. So his or her demand function is as in figure 17.1.
2 For B let us assume that he or she would buy up to 2 units with reservation prices of 8 and 4 as in figure The aggregation process is simple if we remind ourselves that we should add up horizontally because the quantity demanded is on the horizontal axis and we want to find the aggregate demand at any given price. So the aggregation process is the following: for each price we find the demand of A and the demand of B and then add them together. For example, at a price of 5 A s demand is 2 and B s demand is 1 so the aggregate demand is 3. Proceeding in this way we get figure Another way of thinking about this is through reservation prices. We know that the demand curve for a discrete good is in the form of a step function with a step at every reservation price. Now we already know that A has reservation prices of 10, 5 and 1 and that B has reservation prices of 8 and 4. Taken together then (and put in order) the two individuals have reservation prices of 10, 8, 5, 4 and 1. Hence the aggregate function is as in figure You will notice that it has the same form as the individual functions in that they are step functions and so is the aggregate. This is not necessarily the case, though there are other examples where the aggregate function has the same form as the individual functions. One example is when the individual demand functions all have the same intercept on the vertical axis. We will present this algebraically. Suppose A s demand function is q A = 10 p so that A s demand becomes zero at a price of 10. Suppose that B s demand function is q B = 20 2p so that B s demand becomes zero also at a price of 10. Now find the aggregate demand. If Q denotes the aggregate demand then Q = q A + q B = (10 p) + (20 2p) = 30 3p. The aggregate demand function is Q = 30 3p. You will see that the aggregate demand becomes zero also a price of 10. This is a second example where the aggregate function has the same form as the individual functions.
3 But this is not always the case. Consider the more general case of linear individual demand functions. We present this case graphically as the algebra is messy while it is clear from the graphs what is happening. We start with A. His or her demand function is linear. The same is true for B. When we aggregate we have to be careful. Remember that we are adding up horizontally. Note that A does not start buying until the price is below 10 whereas B does not start buying until the price is below 7. Therefore when looking at the aggregate, when the price is between 7 and 10, only A is in the market and therefore the aggregate function in this price range is simply A s demand. For prices below 7 however both A and B are in the market. We thus get the aggregate demand of figure Note that below a price of 7 A s demand is linear in price and so is B s. It follows that the sum of their demands must also be linear in price.
4 Note the kink at the price of 7. This is the price at which B enters the market. You will note that, whereas the individual demand curves are linear the aggregate demand curve is piecewise linear it has a different form from the functions from which it was aggregated. This is almost certainly going to be the case. Note moreover that while the aggregate demand is a function of the price of the good it will, in general, also be a function of all the individual incomes. Only in very exceptional circumstances will aggregate demand be a function of the price of the good and aggregate income. To show this let us do a little algebra. Suppose there are two goods, good 1 and good 2, with prices p 1 and p 2, and two individuals A and B with incomes m A and m B, and we denote the demand for individual A for good 1 as q 1 A and so on, then, in general we have that the demands are given by q 1 A = f 1 A (p 1, p 2, m A ) and q 1 B = f 1 B (p 1, p 2, m B ) for good 1 and similarly for good 2, where f 1 (.) and f 2 (.) are two functions whose form depends upon the preferences of the two individuals. The aggregate demand for good 1 is therefore given by: Q 1 = q 1 A + q 1 B = f 1 (p 1, p 2, m A ) + f 2 (p 1, p 2, m B ) Only in very special circumstances can we write this as Q 1 = f 1 (p 1, p 2, m) where m = m A + m B is the aggregate income of the two individuals. One circumstance in which we can do this is when the individual functions are linear in the variables and the coefficient on income is the same for both individuals. We then have and so the aggregate demand is q 1 A = a 0 + a 1 p 1 +a 2 p 2 + cm A and q 1 B = b 0 + b 1 p 1 +b 2 p 2 + cm B Q 1 = (a 0 + b 0 ) + (a 1 + b 1 ) p 1 + (a 2 + b 2 ) p 2 + cm So the aggregate demand is dependent only on the prices and the aggregate income. In other words the distribution of income between the two individuals does not affect aggregate demand. The reason is simple: if we take money away from A and give it to B then A reduces his or her demand for the good while B increases it by exactly the same amount. Note that if the coefficients on income in the individual equations are different then the aggregate demand depends not only on the aggregate income but also on its distribution. This latter point is important as it shows that, except in rather special cases, changing the distribution of income in society, while keeping the aggregate income of society fixed, will change the aggregate demand for a good or a service. This means that changing the distribution of income in society will change the demand. An obvious example is the demand for food if we take money away from the rich and give it to the poor, the demand for food will rise because the rich will reduce less their consumption of food than the poor will increase their consumption. 17.3: Aggregating Buyer Surplus
5 You may be wondering whether our result about the buyer s surplus survives aggregation. If you recall well you will realise that we have already answered this question as far as a discrete good is concerned. Let us recall this point using the demand function above. We take a particular example. Suppose that A and B both can buy any amount that they wish at a price of 4. What are their individual demands and surpluses? Well A demands 2 units and has a surplus on the first unit of 6 and a surplus on the second of 1 (recall this his or her reservation prices are 10, 5 and 1) giving a total surplus of 7 at a price of 4. B demands either 1 or 2 units and his or her surplus is 4 on the first unit and 0 on the second (if bought, recall this his or her reservation prices are 8 and 4) giving a total surplus of 4 at a price of 4. The total surplus of the two individuals is 11. Let us now look at the surplus as measured with respect to the aggregate demand curve. Again take a price of 4. The aggregate demand curve is that of figure 17.3 and the surplus measured in the usual way the area between the price paid and the aggregate demand curve is illustrated in figure If we add up the areas it is equal to = 11. This is exactly equal to the aggregate surplus as measured above. Indeed when we think about it clearly must be so: we have a step at every reservation price and the difference between the reservation price and the price paid is the surplus or profit on that unit. If you know some mathematics you will realise that this must always be the case irrespective of the form of the demand curve. We are adding horizontally to get the aggregate demand curve and it must be the case that the areas also aggregate. Consider the linear demand curve example we presented graphically above. Take a price paid of 4. What is A s surplus? The area between the price paid of 4 and A s demand curve as given in figure This area is a triangle with base 6 and height 6 the surplus therefore is 18. As for B his or her surplus is the area between the price paid of 4 and B s demand curve as given in figure This area is a triangle with base 6 and height 3 the surplus is therefore 9. The total surplus is (18 + 9) = 27. Now consider the aggregate demand curve figure and work out the area between 4 and this demand curve as in figure
6 What is this area? Well, we can chop it up into two triangles formed by extending the first (upper) linear segment of the demand curve until it reaches a price of 4. The left hand triangle has a base of 6 and a height of 6 (is this familiar?) and the right hand triangle a base of 6 and height 3 (is this familiar?). The total area is thus ½ x 6 x 6 + ½ x 6 x 3 = 27 exactly as before. Indeed for the reasons that we have already stated this must always be the case. So we have the important result that: The surplus as measured from the aggregate demand curve in the usual way (by measuring the area between the price paid and the aggregate demand curve) is always equal to the aggregate surplus found by aggregating the individual surpluses. Or more simply, in the following shorthand: The aggregate of the surpluses is always equal to the surplus of the aggregate. This result is very important. What it means is that if all we are interested in is the aggregate surplus (and not in its distribution) then all we need to know is the aggregate demand curve we do not need to know the individual demand curves. 17.4: Aggregating Supply This section does for supply what section 17.2 did for demand. The methods are the same and the conclusions are the same. Again these conclusions are important. We start as in section 17.2 with a familiar example one for the supply for a discrete good. We have in fact already done this. You will recall that the supply function for a good that is discrete takes the form of a step function, with a step at every reservation price. You should be able to anticipate the results that follow. We work with a simple example. Assume that individual A would sell up to 3 units of the good and has reservation price 3, 4 and 12 for the 3 units. So his or her supply function is as in figure
7 For B assume that he or she would sell up to 2 units with reservation prices of 6 and 9 as in figure The aggregation process is simple if we remind ourselves that we should add up horizontally because the quantity supplied is on the horizontal axis and we want to find the aggregate supply at any given price. So the aggregation process is the following: for each price we find the supply of A and the supply of B and then add them together. For example, at a price of 7 A s supply is 2 and B s supply is 1 so the aggregate supply is 3. Proceeding in this way we get figure Another way of thinking about this is through reservation prices. We know that the supply curve for a discrete good is in the form of a step function with a step at every reservation price. Now we already know that A has reservation prices of 3, 4 and 12 and that B has reservation prices of 6 and 9. Taken together then (and put in order) the two individuals have reservation prices of 3, 4, 6, 9 and 12. Hence the aggregate function is as in figure You will notice that it has the same form as the individual functions in that they are step functions and so is the aggregate.
8 This is not necessarily the case, though there are other examples where the aggregate function has the same form as the individual functions. One example is when the individual supply functions all have the same intercept on the vertical axis. We will present this algebraically. Suppose A s supply function is q A = p 2 so that A s supply becomes zero at a price of 2. Suppose that B s supply function is q B = 2p 4 so that B s supply becomes zero also at a price of 2. Now find the aggregate. If Q denotes the aggregate supply then Q = q A + q B = (p 2) + (2p 4) = 3p 6. The aggregate supply function is Q = 3p 6. You will see that the aggregate supply becomes zero also a price of 2. This is a second example where the aggregate function has the same form as the individual functions. But this is not always the case. Consider the more general case of linear individual supply functions. We present this case graphically as the algebra is messy, while it is clear from the graphs what is happening. We start with A. His or her supply function is linear. The same is true for B. When we aggregate we have to be careful. Remember that we are adding up horizontally. Note that A does not start selling until the price is above 4 whereas B does not start selling until the price is above 2. Therefore when looking at the aggregate, when the price is between 2 and 4, only B is in the market and therefore the aggregate function in this price range is simply B s supply. For prices above 4 however both A and B are in the market. We thus get the aggregate supply of figure Note that above a price of 4, A s supply is linear in the price and so is B s so the aggregate demand curve must also be linear in the price.
9 Note the kink at the price of 4. This is the price at which A enters the market. You will note that, whereas the individual supply curves are linear the aggregate supply curve is piecewise linear it has a different form from the functions from which it was aggregated. This is almost certainly going to be the case. Note moreover that while the aggregate supply is a function of the price of the good it will, in general, also be a function of all the individual incomes. Only in very exceptional circumstances will aggregate supply be a function of the price of the good and aggregate income. To show this let us do a little algebra. Suppose there are two goods, good 1 and good 2, with prices p 1 and p 2, and two individuals A and B with incomes m A and m B, and we denote the supply for individual A for good 1 as q 1 A and so on, then, in general we have that the supplies are given by q 1 A = f 1 A (p 1, p 2, m A ) and q 1 B = f 1 B (p 1, p 2, m B ) for good 1 and similarly for good 2, where f 1 (.) and f 2 (.) are two functions whose form depends upon the preferences of the two individuals. The aggregate supply good 1 is therefore given by: Q 1 = q 1 A + q 1 B = f 1 (p 1, p 2, m A ) + f 2 (p 1, p 2, m B ) Only in very special circumstances can we write this as Q 1 = f 1 (p 1, p 2, m) where m = m A + m B is the aggregate income of the two individuals. One circumstance in which we can do this is when the individual functions are linear in the variables and the coefficient on income is the same for both individuals. We then have and so the aggregate supply is q 1 A = a 0 + a 1 p 1 +a 2 p 2 + cm A and q 1 B = b 0 + b 1 p 1 +b 2 p 2 + cm B Q 1 = (a 0 + b 0 ) + (a 1 + b 1 ) p 1 + (a 2 + b 2 ) p 2 + cm So the aggregate supply is dependent only on the prices and the aggregate income. In other words the distribution of income between the two individuals does not affect aggregate supply. The reason is simple: if we take money away from A and give it to B then A increases his or her supply of the good while B decreases it by exactly the same amount. Note that if the coefficients on income in the individual equations are different then the aggregate supply depends not only on the aggregate
10 income but also on its distribution. We get the result that the distribution of income in society may affect the aggregate supply. 17.5: Aggregating Seller Surplus This section does for supply what section 17.3 did for demand. The methods are the same and the conclusions are the same. Again these conclusions are important. You should be able to deduce by now that our result about the seller s surplus survives aggregation. If you recall well you will realise that we have already answered this question as far as a discrete good is concerned. Let us recall the story using the first supply function above. We take a particular example. Suppose that A and B both can sell any amount that they wish at a price of 8. What are their individual supplies and surpluses? Well A supplies 2 units and has a surplus on the first unit of 5 and a surplus on the second of 4 (recall this his or her reservation prices are 3, 4 and 12) giving a total surplus of 9 at a price of 8. B supplies 1 unit and his or her surplus is 2 ( recall his or her reservation prices are 6 and 9). The total surplus of the two individuals is 11. Let us now look at the surplus as measured with respect to the aggregate supply curve. Again take a price of 8. The aggregate supply curve is that of figure and the surplus measured in the usual way the area between the price paid and the aggregate supply curve is illustrated in figure If we add up the areas it is equal to = 11. This is exactly equal to the aggregate surplus as measured above. Indeed when we think about it clearly must be so: we have a step at every reservation price and the difference between the reservation price and the price received is the surplus or profit on that unit. If you know some mathematics you will realise that this must always be the case irrespective of the form of the supply curve. We are adding horizontally to get the aggregate supply curve and it must be the case that the areas also aggregate. Consider the linear supply curve example we presented graphically above. Take a price received of 8. What is A s surplus? The area between the price received of 8 and A s supply curve as given in figure This area is a triangle with base 12 and height 4 the surplus therefore is 24. As for B his or her surplus is the area between the price received of 8 and B s supply curve as given in figure This area is a triangle with base 12 and height 6 the surplus is therefore 36. The total surplus is ( ) = 60.
11 Now consider the aggregate supply curve figure and work out the area between 8 and this supply curve as in figure What is this area? Well, we can chop it up into two triangles formed by extending the first linear segment of the supply until it reaches a price of 8. The left hand triangle has a base of 12 and a height of 6 (is this familiar?) and the right hand triangle a base of 12 and height 4 (is this familiar?). The total area is thus ½ x 12 x 6 + ½ x 12 x 4 = 60 exactly as before. Indeed for the reasons that we have already stated this must always be the case. So we have the important result that: The surplus as measured from the aggregate supply curve in the usual way (by measuring the area between the price received and the aggregate supply curve) is always equal to the aggregate surplus found by aggregating the individual surpluses. Or more simply, in the following shorthand: The aggregate of the surpluses is always equal to the surplus of the aggregate. This result is very important. What it means is that if all we are interested in is the aggregate surplus (and not in its distribution) then all we need to know is the aggregate supply curve we do not need to know the individual supply curves. 17.6: Summary This has been an important chapter. In particular it allows us to extend our key results on surpluses to aggregate functions. In particular: The aggregate demand curve is the horizontal sum of the individual demand curves. The form of the aggregate demand curve may well be different from the form of the individual demand curves. The aggregate consumer surplus, defined as the area between the aggregate demand curve and the price paid, is always exactly equal to the sum of the individual surpluses. The aggregate supply curve is the horizontal sum of the individual supply curves. The form of the aggregate supply curve may well be different from the form of the individual supply curves.
12 The aggregate seller surplus, defined as the area between the aggregate supply curve and the price received, is always exactly equal to the sum of the individual surpluses.
13
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