Let us characterise the efficient allocations diagrammatically. We can plot both the factory's and the residents' indifference curves on the same Edgeworth Box diagram, with money on the horizontal axis and production and absence of Q on the vertical axis. The total amount of money in this economy is:
This means that the length of the horizontal axis of the Edgeworth Box is M. On the other hand, the maximum amount of Q that can be produced in this economy is Q, so the length of the vertical axis is Q. We can 'flip'
Figure 3.3.1 The Edgeworth Box representation with F's production creates a negative external cost for R
Figure 3.3.2 A move from point "0" to point "1" is a Pareto improvement
the residents' indifference curves and draw them on this Edgeworth Box diagram. This is done in Figure 3.3.1.
We are now in a position to characterise the Pareto-optimal levels of production. An allocation is Pareto optimal if it is not possible to find some other allocation that makes one individual better off without making another individual worse off. Consider a move from point 0 to point 1 in Figure 3.3.2. This move makes both the factory and the residents better off, since each would be on higher indifference curves than at point 0. This move from 0 to 1 constitutes a Pareto improvement, and so point 0 could not have been efficient.
Figure 3.3.3 Point 0' is Pareto optimal, but point 0" is not
Note that Pareto-improving moves like this are always possible in this example, unless the indifference curves are tangent to each other, as they are at point 0' in Figure 3.3.3. Any move away from point 0' (say, to point 0") potentially makes both parties worse off, and can certainly not make them better off. Hence, point 0', where the slopes of the two indifference curves are equal, is efficient.6
The slope of the factory's indifference curve at any particular point is the marginal rate of substitution of money for Q at that point - the amount of additional production of Q that the factory must be allowed as compensation for a one dollar reduction in the amount of money that they consume:
Similarly, the slope of the residents' indifference curve at any particular point is the marginal rate of substitution of money for Q - Q at that point. It is the amount of 'absence of Q' (i.e. the reduction in Q) that the resident must be given to compensate them for a one dollar reduction in the amount of money that they consume:
Figure 3.3.4 The contract curve
Figure 3.3.4 suggests that the set of points where these slopes are equal are efficient. In other words, if
then the allocation is Pareto optimal. Note that in general, this condition is sufficient, but not necessary, for efficiency. In other words, there may be other points where this condition does not hold, but which are also efficient.
Notice also that, in general, there are many levels of production of Q at which the marginal rates of substitution are equal. This contract curve is plotted in Figure 3.3.4. It is the set of possible allocations that the parties could reach if they exhausted all potential gains from trade.