Chapter 14: Production Possibility Frontiers


 Brenda Parsons
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1 Chapter 14: Production Possibility Frontiers 14.1: Introduction In chapter 8 we considered the allocation of a given amount of goods in society. We saw that the final allocation depends upon the initial allocation. The obvious next question then is the determination of this initial allocation. This obviously depends upon the choices open to society. It is to this question that we now turn: what determines the production possibilities open to a society? We consider two different scenarios. In the first the society consists of two individuals, each of whom can produce various combinations of two goods, using what we term a linear technology. We examine how the two individuals should divide up the work between them so that society (the two of them together) might produce the maximum output possible. In the second scenario, we have two firms, each producing one good, and with a fixed amount of inputs between them. We consider the optimal use of the two inputs so that, once again, output is maximised. In both scenarios we find the production possibility frontier available to society and discuss the factors which determine its shape. 14.2: Linear Technologies We start with a nice simple story. The society consists of two individuals, A and B, and two goods, 1 and 2, which both A and B can produce. We tell a very simple story about each of their production capabilities. For each of the goods there is a maximum amount that each individual can produce if they devote all their time to the production of that good say m 1 for good 1 and m 2 for good 2. (These maximum amounts can differ from individual to individual, as we shall see.) So if the individual works all day on good 1 he or she can produce m 1 of good 1 and nothing of good 2; if the individual works all day on good 2 he or she can produce m 2 of good 2 and nothing of good 1. Moreover let us assume that the working process is such that if he or she works half the day on good 1 and half on good 2 then he or she can produce m 1 /2 of good 1 and m 2 /2 of good 2. More generally if he or she spends a fraction a of the day working on good 1 and a fraction (1a) of the day working on good 2, then he or she can produce a quantity am 1 of good 1 and a quantity (1a)m 2 of good 2. This is called a linear technology. Let us now consider a specific example. Let us start with individual A and suppose that his or her maximum quantities are 120 units of good 1 and 60 units of good 2. Given what we have assumed above individual A s production possibility frontier is shown in the following figure, which has the quantity of good 1 produced along the horizontal axis and the quantity of good 2 along the vertical axis.
2 The line indicates the possibilities open to individual A assuming he or she works all day 1. So, for example, the midpoint represents what he or she would produce if half the day were spent on the production of each good. We make the same kinds of assumptions about individual B though we assume that he or she is less productive than A in the production of both of the goods. So, for individual B, we assume maximum quantities of good 1 and good 2 equal to 20 and 40 respectively. We thus get the following production possibility frontier for individual B. Working all day on good 1 he or she produces 20 of good 1 and none of good 2; working all day on good 2 he or she produces 40 of good 2 and none of good 1; working half the day on each, he or she produces 10 of good 1 and 20 of good 2. And so on. We note that B is absolutely worse than A in the production of both goods though we can say that B is relatively better than A at producing good 2 and A is relatively better than B at producing good 1. How can we say this? We note that for individual A, for every extra unit of good 1 that he or she produces he or she has to produce 0.5 units less of good 2, whereas for B, for every extra unit of good 1 that he or she produces he or she has to produce 2 units less of good 2. So, in terms of the cost of producing good 1 (as measured by the amount of good 2 that has to be given up to produce the extra amount of good 1) it is relatively low for A. In this sense then A is relatively better than B at producing good 1. We can do all this the other way around. We note that for individual B, for every extra unit of good 2 that he or she produces he or she has to produce 0.5 units less of good 1, whereas for A, for every extra unit of good 2 that he or she produces he or she has to produce 2 units less of good 1. So, in terms of the cost of producing good 2 (as measured by the amount of 1 From now on we shall simply take this as given. If he or she works less than all day then point within the triangle will the result.
3 good 1 that has to be given up to produce the extra amount of good 2) it is relatively low for B. In this sense then B is relatively better than A at producing good 2. The issue that we now discuss is that of the maximum output of the two individuals combined. We want to find the highest production possibility production of society. Well, the answer to this depends upon how they divide up the work between them. Actually the end points of the production possibility frontier for society are easy to find: if society wants the maximum amount of good 1 then both individuals have to work all day on good 1 this gives us a total output of good 1 of ( ) = 140; if society wants the maximum amount of good 2 then both individuals have to work all day on good 2 this gives us a total output of good 1 of ( ) = 100. It is in between these two extremes that difficulties lie. One possibility is that they simply divide up the work equally in the following sense: if A works half the day on good 1 then so does B; more generally, if A works a fraction a of the day on good 1 then so does B. This gives us the following production possibility frontier for society: The top line is society s production possibility frontier if A and B share the work equally. The second line is A s production possibility frontier and the bottom line B s. We have already discussed the extremes (140, 0) and (0, 100). Points in between are found by the sharing work equally rule. For example the point at the middle of society s frontier (70, 50) is reached by each working half the day on good 1 and half on good 2 so A produces 60 of good 1 and 30 of good 2 and B produces 10 of good 1 and 20 of good 2: jointly they produce 70 of good 1 and 50 of good 2. A further example is the point ¼ of the way along: (105, 25). This is reached by A working ¾ of the day on good 1 and ¼ on good 2 and producing 90 of good 1 and 15 of good 2, and by B working ¾ of the day on good 1 and ¼ on good 2 and producing 15 of good 1 and 10 of good 2. You might be asking yourself whether this is a particularly clever way for the two individuals to share the work. After all we have already realised that A is relatively good at producing good 1 and that B is relatively good at producing good 2. Might it not be better for each individual to specialise in the production of the good which they are relatively better at producing? We can easily see that the answer to this question must be yes by considering what happens at the point where each individual specialises in the good at which he or she is relatively good. If A works all day on good 1 he or she produces 120 of good 1 and 0 of good 2; if B works all day on good 1 then he or she produces 40 of good 2 and 0 of good 1. Between the two of them they produce 120 of good 1 and 40 of good 2 that is society reaches the point (120, 40). What do we notice about this point (120, 40)? Yes that it lies outside the production possibility frontier for society that we have drawn above.
4 If we apply this principle that each individual specialises in the production of the good in which he or she has a relative advantage, we get the society production possibility frontier shown below: We have already discussed the extremes and the point (120, 40) which is the point at the angle on society s frontier. This kink point divides up society s frontier into two regions the region above the kink and the region below the kink. We see that in the region above the kink, society is consuming more than 40 units of good 2 more than the amount of good 2 that can be produced by the expert in good 2, namely individual B. So in this region above the kink B spends all day on good 2 while A produces the rest. In the region below the kink, society is consuming more than 120 units of good 1, more than the amount of good 1 that can be produced by the expert in good 1, namely individual A. So in this region below the kink A spends all day on good 1 while B produces the rest. This way of dividing up the work between them is the most efficient possible in the sense that it leads to the highest possible production possibility frontier for society 2. It is instructive to look at this frontier and see where it comes from. There is a part which corresponds to the frontier for A shifted vertically upwards a distance 40 (corresponding to B specialising in good 2) and a part which corresponds to the frontier for B shifted horizontally rightwards a distance 120 (corresponding to A specialising in good 1). You will notice that the frontier is concave. Note that it must always be so if individuals specialise in what they are relatively good at producing. You might now be able to work out what happens if a third individual enters the society. His or her effect on society s production possibility frontier clearly depends upon his or her frontier. Let us suppose that he or she is between A and B in terms of productive ability both absolutely and relatively: his frontier going from (50, 0) to (0. 50) as in the figure Note that at all levels of production A is relatively better than B in the production of good 1 and B is relatively better than A in the production of good 2. So A should produce as much of good 1 as possible and B as much of good 2 as possible (though obviously this depends upon where one wants to be on society s frontier).
5 As before, society s frontier has a section corresponding to each individual and arranged in such a way that production is maximised. You should be able to verify the following: You will see that: (1) if society wants a lot of good 1 then both A and C specialise in its production; (2) if society wants a lot of good 2 then both B and C specialise in its production; (3) in between A specialises in good 1 and B specialises in good 2. We note that the frontier remains concave. Adding more individuals adds more segments (always one for individual) but keeps the frontier concave. Note that the slope of any segment measures the marginal rate of substitution between the two goods and in each segment this mrs is the mrs of some individual (which is constant for any one individual). 14.3: NonLinear Technologies The interesting point about the material above is that although individual technologies are linear in the sense that I have described, the frontier for society is concave. It is concave because of the specialisation of the individuals. In this section we also derive (under certain assumptions) a concave frontier for society, but starting from a different perspective. Here we use the material of chapters 10 through 13. We assume that the society consists of two firms, firm 1 and firm 2, each producing a particular good: firm 1 produces output 1 and firm 2 produces output 2. They produce the goods using two inputs, input 1 and input 2, the total quantities of which are fixed and given in society. The problem that society faces then is the allocation of these two inputs in such a way that the output is as large as possible in some sense. To solve this problem it is useful to use an Edgeworth box. In the past we have used this to discuss the allocation of two goods between two individuals; here we use it to discuss the allocation of two inputs between two firms. We measure one firm from one origin and the other from the other. The
6 horizontal axis indicates the quantity of input 1 and the vertical axis the quantity of input 2. The width of the box is the total quantity of input 1 in the economy and the height of the box the total quantity of input 2 in the economy. A point in the box indicates an allocation of the two inputs between the two firms. This determines the outputs of the two firms. We have inserted into the box isoquants for the two firms those convex from the bottom left origin those for firm 1; those convex from the top right origin those for firm 2. Now we know that the efficient points are along the contract curve which is inserted into the above figure whatever society does it should choose a point on the contact curve. Once we are on the contract curve there is a tradeoff: moving from the bottom left origin to the top right origin increases the output of firm 1 and decreases the output of firm 2. At each point along the contract curve we can calculate the output of firm 1 and the output of firm 2. An example is given in the figure. Here I am assuming a production function for firm 1 given by a CobbDouglas with parameters a = 0.56 and b = 0.24 and a production function for firm 2 that is also CobbDouglas but now with parameters a = 0.48 and b = Note that both firms have decreasing returns to scale. If we now plot the output of firm 2 against the output of firm 1 along the contract curve we get the production possibility frontier of the society 3. 3 This may all be very familiar. In fact, we did almost the same in chapter 9 when deriving the utility possibility frontier for society.
7 We see that the frontier is concave. The reason for this is that the technology of each firm has decreasing returns to scale: as we move up the contract curve the output of firm 1 rises, but at an increasingly slower rate, while the output of firm 2 falls, at an increasingly faster rate. Clearly the degree of the decreasing returns to scale will influence the concavity of the frontier: if we take, instead of a = 0.56 and b = 0.24 for firm 1 and a = 0.48 and b = 0.32 for firm 2, we take a = 0.49 and b = 0.21 for firm 1 and a = 0.42 and b = 0.24 for firm 2 (notice that the ratio of a to b for each firm has not changed so the isoquants look the same but their sum is smaller), we have for the frontier: This is considerably more concave. It might be obvious now that if both technologies display constant returns to scale we have a linear frontier: as we move along the contact curve from the bottom origin the output of firm 1 rises linearly from zero while the output of firm 2 falls linearly to zero. If we take the above example but instead of a = 0.56 and b = 0.24 for firm 1 and a = 0.48 and b = 0.32 for firm 2, we take a = 0.7 and b = 0.3 for firm 1 and a = 0.6 and b = 0.4 for firm 2 (notice that the ratio of a to b for each firm
8 has not changed so the isoquants look the same but their sum is now unity), we get the following frontier for society: (Note that in this example the total quantities of the two inputs are 100 and 100, so the total output with these technologies if all of both inputs is used is 100.) You can probably anticipate what will happen if both firms have increasing returns to scale. As we move up the contract curve from the bottom left origin then the output of firm 1 rises at an increasingly faster rate while the output of firm 2 falls at an increasingly slower rate. If we take the original example above but instead of a = 0.56 and b = 0.24 for firm 1 and a = 0.48 and b = 0.32 for firm 2, we have a = 0.7/0.9 and b = 0.3/0.9 for firm 1 and a = 0.6/0.9 and b = 0.4/0.9 for firm 2 (notice that the ratio of a to b for each firm has not changed so the isoquants look the same but their sum is now greater than unity), we get the following frontier for society: Finally let us consider a sort of hybrid case one in which firm 1 has increasing returns to scale while firm 2 has decreasing returns to scale. What do you think will happen?
9 Exactly the production possibility frontier for society has a convex section and a concave section. 14.4: Summary In this chapter we have constructed the production possibility frontier for society under different scenarios. The first scenario was when the production technology at the individual level is what we called linear. In this economy society's production possibility frontier is piecewise linear and concave. It is concave because everyone specialises in the production of the good in which they have a relative advantage. We then moved on to consider what might be termed a more normal society in which there are firms producing the output and they are competing for a fixed supply if inputs. On the assumption that the technology has convex isoquants we showed the following important result. In an economy with increasing, constant or decreasing returns to scale everywhere society's production possibility frontier is convex, linear or concave everywhere Should firms choose the same technology? If firms can choose their technology, should they choose the same as others or should they be different? The answer is that, if they are competing for scarce inputs, they should be different. We are going to show this here in the context of a simple example; this can be generalised. The example also illustrates some of the key concepts that emerge from this chapter. Consider a simple society in which there are two inputs into the production process, Input 1 and Input 2. Suppose the quantities of these two inputs are fixed. Suppose that there are two firms, Firm A and Firm B, who respectively produce Good A and Good B. Suppose the technology of Firm A is fixed it has a perfect 1with2 complements technology which displays constant returns to scale. Hence q units of Input 1 combined with 2q units of Input 2 produce q units of Good A. Firm B has a choice of technology. Both display constant returns to scale and both are perfect complements: one is perfect 1with2 complements (like Firm A); the other is perfect 2with1 complements. Which should it use? We shall answer this by looking at the production possibilities
10 open to the society under the two technologies. Intuitively it may be thought that, with limited input resources, that Firm B should use a different technology from that of Firm A, so that the competition for resources is less. We shall show this formally in a particular example in such a way that you should be able to generalise the result. Suppose Firm B uses the same technology as Firm A. We can illustrate the output possibilities in an Edgeworth Box which has Input 1 on the horizontal axis and Input 2 on the vertical axis. The width of the box is the total quantity of Input 1 that is available in the economy; the width of the box is the total quantity of Input 2 that is available in the economy. We assume that both of these are 100. We measure the quantities used by Firm A from the usual origin and the quantities used by Firm B from the upper righthand corner. Any point in the box represents an allocation of the inputs to the two firms, and we can calculate the output of the two firms at any point in the box. Consider the following figure. The blue lines represent the isoquants of Firm A for outputs of 1, 2, 3, 4, 5, and 6. They have angles at the points (1,2), (2,4), (3,6), (4,8) (5,10) and (6,12). Clearly the output of Firm A around the bottom and the lefthand side of the box is zero. The line marked AA joins up the corners of these isoquants. The red lines represent the isoquants of Firm B for outputs of 1, 2, 3, 4, 5, and 6. (Do recall that we measure the inputs going to Firm B from the top righthand corner.) They have angles at the points (11,10), (10,8), (9,6), (8,4) (7,2) and (6,0). Clearly the output of Firm B around the top and the righthand side of the box is zero. The line marked BB joins up the corners of these isoquants.
11 Where is the contract curve in this society? The contract curve, remember, is the locus of efficient points in the sense that, on the contract curve, it is not possible to increase the output of one firm without decreasing the output of the other firm, while off the contract curve it is possible to increase the output of one firm without decreasing the output of the other. In this particularly economy, the contract curve is, in fact, an area the area coloured yellow in the above figure. This is a consequence of the technologies we have assumed. The simplest way to show this is to first state what the production possibility frontier is in this particular case and then relate it back to the figure above. We will show that the production possibility frontier has the form of the lower straight line in the following figure. The end points are easy to explain. The point on the production possibility frontier in which the output of Firm A is 6 and that of Firm B is 0 comes from the point (6,12) in the Edgeworth Box. The point on the production possibility frontier in which the output of Firm A is 0 and that of Firm B is 6 comes from the point (0,6) in the Edgeworth Box. Actually this is slightly misleading any point on the straight line between (6,12) and (12,12) in the Edgeworth Box gives outputs of 6 for Firm A and 0 for Firm B and any point on the straight line between (0,0) and (6,0) gives outputs of 0 for Firm A and 6 for Firm B. The crucial point is that, as we move along the straight line from (6,12) to (12,12), or along the straight line from (0,0) to (6,0), we are moving along both an isoquant of Firm A and along an isoquant of Firm B. That is why the outputs of both firms remain constant. This gives us a clue as to why the contract curve is the area shaded in the Edgeworth Box. Once inside the shaded area, moving horizontally does not change the output of either firm we are moving along their isoquants. However moving diagonally or vertically increases the output of one firm and decreases the output of the other firm. We cannot increase the output of both firms if we are inside the shaded area. Points inside the shaded area are efficient. If we are outside the shaded area, things are different. If we are to the left of it, moving rightwards increases the output of Firm A but does not change that of Firm B, while moving downwards increases the output of Firm B but does not change that of Firm A. If we are to the right of it, moving leftwards increases the output of Firm B but does not change that of Firm A, while moving upwards increases the output of Firm A but does not change that of Firm B. So points outside the shaded area are inefficient. The production possibility frontier comes from the contract curve. You should be able to verify that: at all points where the output of Firm A is 1 the output of Firm B is 5; at all points where the output of Firm A is 2 the output of Firm B is 4; at all points where the output of Firm A is 3 the output of Firm B is 3; and so on. This verifies the production possibility frontier that is illustrated above.
12 Now let us repeat the analysis with the alternative technique available to Firm 2: perfect 2with1 complements. The Edgeworth Box below shows the appropriate analysis. The line marked AA is again the line joining the corners of the isoqants of Firm A and the line BB that joining the corners of the isoquants of Firm B. Notice that this latter is different from the earlier case considered because of the different technology. Once again the blue lines are the isoquants of Firm A (0 output at the origin and then 1, 2, 3, 4, 5 and 6) and the red lines those of Firm B (0 output at B s origin and then 1, 2, 3,,4, 5 and 6). The contract curve in this case is once again an area though a different area, that shaded in the figure above. You should go through the same reasoning as we went through above and convince yourself that the shaded area is indeed the contract curve. Let us now construct the production possibility frontier. The extremes are once again easy: at the bottom lefthand corner of the box (and indeed along the straight line from (0,0) to (0,6)) the output of Firm A is zero and that of Firm B is 6; at the top righthand corner of the box (and indeed along the straight line from (6,12) to (12,12)) the output of Firm A is 6 and that of Firm B is 0. The middle point is also easy: at the point (4,8) where the lines AA and BB intersect, the output of both firms is 4. Other points you can verify: in the bottom triangle of the contract curve, whenever the output of Firm A is 2, the output of Firm B is 5, and so on; in the top triangle of the contract curve, whenever the output of Firm A is 5, the output of Firm B is 2, and so on. The production possibility frontier is the kinked frontier in the second figure above. What do we notice: that the production possibility frontier when Firm B uses a different technology from that of Firm A is everywhere further out than when both use the same technology. It is therefore unambiguously better for the society for the two firms to have different technologies. The
13 reason is simple but it is important that you understand it: the inputs in society are limited, and the less the competition for them, the better. Alternatively, the more efficiently they are used, the better. Clearly this is a special case, but the important messages generalises. You should try to do this yourself. However, other features of this special case do not generalise though they are of interest. Let us go back to the last graph above and ask ourselves: where would competitive trading of the two inputs take us? We know in general that it will take us to the point on the contract curve where the two priceoffer curves intersect. In the context of this example, the priceoffer curve of Firm A is the line AA and the priceoffer curve of Firm B is the line BB. The competitive equilibrium is at the point (4,8). This is a rather special point in the contract curve. Why? Because at that point, and only at that point, in the contract curve, there is no waste of inputs. Everywhere else some of one of other inputs are unused because their employment does not increase output. So, at the competitive equilibrium, no input resources are being wasted. Do you think that this is always the case? Note there is no competitive equilibrium when the two firms have the same technology (see the first graph and ask yourself where the priceoffer curves intersect). Why is that?
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