Lecture 14

PRODUCTION POSSIBILITY FRONTIERS

John Hey

Linear Technologies

We examine in this lecture the optimal production possibilities of an economy.  We start very simple with an economy producing two goods and with two people (two firms) each of whom are capable of producing the two goods. We assume to begin with an extremely simple form of technology for the two people (firms) - that which is linear  in a sense that we will definee shortly.

There are two individuals in the economy - Individual A and Individual B - and two goods - Good 1 and Good 2.  Both Individual A and Individual B can produce Goods 1 and 2 - but they differ in their absolute and relative skills of so doing. Consider first Individual A and suppose for simplicity that the time spent working is fixed.  Let us suppose that If he or she devotes all available time to Good 1 he or she can produce 120 units; and that if he or she devotes all available time to Good 2 s/he can produce 60 units.  Moreover for each extra unit of Good 2 produced Individual A produces 2 less units of Good 1. Then the implied Production Possibility Set for this individual is thus as given in Figure 14.1. The blue line is the production possibility frontier  for Individual A.

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plot01;

Now consider Individual B and again suppose for simplicity that the time spent working is fixed. Let us suppose that If he or she devotes all available time to Good 1 he or she can produce 20 units; and if he or she devotes all available time to Good 2 s/he can produce 40 units.  Moreover for each extra unit of Good 1 produced Individual B produces 2 less units of Good 2. The implied Production Possibility Set for this individual is thus as given in Figure 14.2, and the implied production possibility frontier is the red line.

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plot02;

It is clear that relative to Individual A Individual B is absolutely  worse in producing output, though is relatively  better than A at producing Good 2. Individual A is relatively better than B at producing Good 2.

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Now let us consider the combined Production Possibility Set of the two Individuals - that is for the society as a whole. Clearly this depends upon how they split the production between them.  Let us examine the case where they share the work equally - if A works half the time on Good 1 and half the time on Good 2, then so does B.  If A spends one-quarter ot his time on Good 1 and three-quarters on Good 2, then so does B.  This leads to the following (green) society   production possibility frontier .

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plot04;

But is that a sensible thing to do? Might it not be better if they SPECIALISED?  For example, suppose that A specialises in producing Good 2 and B in Good 1 - by which is meant that A produces as much of Good 2 as possible - with B producing the residual - and B produces as much of Good 1 as possible - with A producing the residual. We get Figure 14.5.

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plot05;

But that is clearly crazy - for example,  it is worse than shared production.  Moreover it is clear that A should specialise in Good 1 and B in Good 2 - for A is RELATIVELY better than B at producing Good 1 while B is RELATIVELY better than A at producing Good 2. Consider the implications.

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In fact this black line is the best society can do - in terms of the production of the two goods. It should be noted that each Individual specialises in the production of the good at which he or she is relatively  better at producing

Now let us add a third individual - Individual C with the (yellow) production possibility frontier. It can be seen that in absolute terms C is between A and B in terms of producing output and, as it happens in this example, is also between A and B in terms of his or her relative abilities. So we have that A is relatively best at producing Good 1, then C and then B. And, of course, B is relatively the best at producing Good 2, then C and then A.  

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If we let society adopt the best specialisation we have the following (best) production possibily frontier for society.

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To make what is happening clearer, let us colour the various segments of the society production possibility frontier

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We can see that there is a segement of the society ppf (production possibility frontier) corresponding to each individual and these segments go in the appropriate order of specialisation. This means that the ppf is as far out as is possible. Moreover it implies that the ppf must be concave .  Furthermore, as we add more individuals to the society, the society ppf will consist of more and more segments, each corresponding to an individual and arranged in descending order of the slope. Now note that the slope measures the (technical) rate of substitution - and this determines the relatively strengths of the individuals in the production of the two goods.

Clearly as society gets bigger - the ppf gets further out - and, in general, becomes more smoothly concave.

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Non-Linear Technologies

There is another way to get nice smoothly concave society production functions - when the underlying technologies are not linear.

Consider a different scenario, in which there are are still two goods produced but now we assume that there are two firms - Firm A and Firm B -  in the society - each producing one of the two goods. Let us suppose that both firms need two inputs to produce their respective goods and that there is a fixed amount of each of these two inputs in the society. We can represent this economy using an Edgeworth Box.  The width of the box shows the total amount of the first input in the economy and the height of the box shows the total amount of the second input in the economy.  We can now represent the possibilities open to the two firms using the isoquant analysis of lecture 10.  Let us measure the amounts of the two inputs that Firm A has from the usual origin at the bottom left hand corner of the box and let us measure the amounts of the two inputs that Firm B has from the top right had corner of the box.

In this example the fixed amounts of the two inputs are 100 and 100.  The technology of firm A is Cobb-Douglas with parameters  and . The technology of firm B is also Cobb-Douglas with parameters  and . In other words, the production function of the good produced by Firm A - call it output 1 - is given by  and the production function of the good produced by Firm B - call it output 2 - is given by . Note that both firms have decreasing returns to scale

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plot10;

Obviously the best thing for society to do is to produce somewhere along the yellow contract curve  - this is the locus of technically efficient points. It will be seen that as we move along the contract curve from the bottom leftt hand corner to the top right hand corner the quantity of output  1 increases while the quantity of output 2 decreases. We can substitute the values of  and  into the two production functions to find out exactly how    falls as  rises. Let us get Maple to do this for us and show the resulting relationship between    and   along the contract curve. This, of course, defines the production possibility frontier in this society.

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We can now plot the relationship between the quantity of output 1 and the quantity of output 2.  This is graph 14.12 below. It should be noted that this production possibility frontier is concave - this is a consequence of both firms having decreasing returns to scale.

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plot12;

The Effect of Returns to Scale on the Production Possibility Frontier

Suppose now the returns to scale worsen.  But the ratio of the exponents  and  remain constant for both firms. The Edgeworth Box looks the same but note the values of the outputs attached to the points on the contract curve.

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plot13;

If we now plot the implied production possibility frontier we see that it is more concave than it was before.

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plot14;

Let us now look at what happens if both firms have constant returns to scale .  But again the ratio of the exponents remain the same. Again the Edgeworth Box looks unchanged but the values of the outputs attached to the various points on the contract curve change.

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As a consequence of the constant returns to scale for both firms the production possibility frontier is now linear.

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Let us now consider the case of increasing returns to scale  for both firms - but again keeping the ratios of the exponents constant.  The Edgeworth Box looks the same but the output values associated with the various points on the contract curve change.

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Now the production possibility frontier is convex  - because of the increasing returns to scale.

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Finally consider the case where one firm has increasing returns to scale  and the other decreasing returns.   What do you think happens?

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Indeed - the production possibility frontier has a concave section and a convex section.

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plot20;

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In reality, the usual case is that both firms have decreasing returns to scale - so that the production possibily frontier of society is concave.

Summary

In a linear economy society's production possibility frontier is piecewise linear and concave.

In an economy with decreasing returns to scale everywhere society's production possibility frontier is concave.

In an economy with constant returns to scale everywhere society's production possibility frontier is linear .

In an economy with   increasing returns to scale everywhere society's production possibility frontier is convex.

(The last three results assume a convex technology - that is the isoquants are convex.)