Unit 5 The rules of the game: Who gets what and why

Extension 5.9 The Pareto efficiency curve

There are many feasible allocations resulting from the interaction between Angela and Bruno; for example, we have considered the allocation that Bruno would impose if he could use force, and the allocation he chooses when he can make a take-it-or-leave-it offer of a contract in which Angela may work the land if she pays him rent in the form of some of the grain produced. In this extension, we work out mathematically the set of allocations that are Pareto efficient: that is, the Pareto efficiency curve.

Pareto efficient, Pareto efficiency

An allocation is Pareto efficient if there is no feasible alternative allocation in which at least one person would be better off, and nobody worse off.

An allocation is Pareto efficient if there is no allocation that Pareto-dominates it: that is, no person can be made better off without making another worse off. To determine a Pareto-efficient allocation between Bruno and Angela, we start by thinking about their preferences—that is, what would make them better off.

As in the previous extensions, Angela’s preferences are represented by a quasi-linear utility function, \(u(t,\ c) = v(t) + c\), where \(t\) represents her daily hours of free time, \(c\) is her consumption of grain, and the function, \(v\), is increasing and concave. The feasible frontier for producing grain is \(y=g(24-t)\), where \(y\) is the amount produced and \(g\) is an increasing and concave production function.

Bruno’s preferences are very simple. He cares only about the amount of grain he receives, which we call \(b\). Higher values of \(b\) make Bruno better off.

The potential outcome of an interaction between Angela and Bruno is an allocation of \(b\) bushels for grain for Bruno, and \(c\) bushels of grain and \(t\) hours of free time for Angela. Given the production technology, all potential allocations \((b, c, t)\) must satisfy:

\[c+b \leq g(24-t)\]

These are the allocations in which the total amount of grain consumed is less than or equal to what is produced. We assume also that \(c\geq 0\) and \(b\geq 0\).

One way to find the Pareto-efficient allocations is to say: ‘suppose we take an allocation in which Bruno receives an amount of grain, \(b\geq 0\). Then it is Pareto efficient if and only if Angela is as well off as possible, given Bruno’s amount of grain.’

For Angela to be as well off as possible when Bruno receives \(b\), she must consume all of the rest of the grain produced: \(c+b = g(24-t)\). So we can find the Pareto-efficient allocations by solving a constrained choice problem.

We will solve the problem by the substitution method. Substituting \(c=g(24-t)-b\) into the objective \(u(t,\ c)= v(t) +c\), all we need to do is:

\[\text{Choose }t \text{ to maximize }v(t) + g(24-t) -b\]

Then, differentiating with respect to \(t\) and setting the derivative to zero gives us the first-order condition:

\[v'(t) =g'(24-t)\]

The first-order condition is the one we discussed in Extension 5.5, where we noted that the assumption that \(v\) and \(g\) are concave functions implies that it has at most one solution. We assume that one exists; therefore consumption is given by

\[c=g(24-t)-b\]

Remember that \(v'(t)=\text{MRS}\) and \(\text{MRT} = g'(t)\); the first-order condition is the familiar equation \(\text{MRT} = \text{MRS}\).

The problem we have solved is the one that Angela would solve if Bruno demanded an amount of rent \(b\), and she could choose \(c\) and \(t\) for herself. Here we have shown that solving this problem for all possible values of \(b\) gives us the set of Pareto-efficient allocations.

In summary, an allocation \((b, c, t)\) is Pareto efficient if and only if \(MRS = MRT\), and \(c+b=g(24-t)\).

Drawing the Pareto efficiency curve

The analysis above shows that for each possible value of \(b\) (grain for Bruno) there is a corresponding outcome \((c,t)\) for Angela so that the allocation \((b,c,t)\) is Pareto efficient. We can draw the Pareto efficiency curve that shows the whole set of Pareto allocations by plotting these points, \((c,t)\) for every value of \(b\) between 0 and \(g(24-t)\).

Figure E5.6 shows the feasible frontier for production and Angela’s indifference curves for the example in Extensions 5.5 and 5.7, in which \(u(t,\ c) =4\sqrt{t} + c\) and \(g(24-t)=2\sqrt{2(24-t)}\).

The utility function is quasi-linear, so we know that all indifference curves have the same slope for a given value of \(t\). And this means (as shown in the previous extensions) that the solution to the first-order condition is \(t=16\)—whatever the values of \(c\) and \(b\) are. So all Pareto-efficient points lie on the vertical line at \(t=16\), and the total amount produced is eight bushels of grain.

To find them all, consider first the Pareto-efficient allocation when \(b=0\): it is on the vertical line at the point where Angela consumes all the grain produced; that is, point P₁, where \(c=8\). If Bruno gets a small amount—\(b=1\), say—we move down the vertical line to the point where \(c=7\). As his share increases we move further down the line and Angela gets less. At P₂ , Bruno’s share is three and Angela’s is five. The Pareto efficiency curve ends at P₀ , where \(b=8\) and \(c=0\) (Angela’s consumption cannot be negative). The Pareto efficiency curve is all the points between P₁ and P₀ .

When preferences are not quasi-linear

Our assumption of quasi-linearity simplifies all the cases that we have analysed in this unit. But we can use the same methods for cases where Angela’s utility is not quasi-linear. Then, we find that the Pareto efficiency curve really is a curve, and not a vertical line.

To illustrate, we determine the Pareto efficiency curve for the case in which her utility function has the Cobb–Douglas form:

\[u(t,\ c) = t^{\alpha} c^{1- \alpha}\]

and the production function is the increasing and concave function, \(f(h)=(48h-h^2)/40\), where \(h\) is hours of work, so the feasible frontier for production is \(y=g(24-t)\) which is:

\[y=\frac{(576-t^2)}{40}\]

Figure E5.7 shows some of the indifference curves, and the feasible frontier, for this example. To keep the numbers simple in the calculations below, we have chosen \(\alpha = \frac{8}{13}\).

We have marked the points on these three indifference curves where the slope is the same as the slope of the feasible frontier. These occur at a different value of \(t\) on each curve, whereas in the quasi-linear case in Figure E5.6, they all occur at the same value of \(t\).

To calculate the MRS, we use the formula from Extension 3.3:

\[\text{MRS} = \frac{\partial u}{\partial t} \left/ \frac{\partial u}{\partial c} \right. = \frac{\alpha c}{(1-\alpha)t}\]

The MRT between Angela’s free time and her production of grain is, as usual, the absolute value of the slope of the (downward-sloping) feasible frontier:

\[\text{MRT} = -\frac{dy}{dt} = \frac{t}{20}\]

The set of Pareto-efficient allocations \((b, c, t)\) is found as above by solving the problem:

\[\text{choose } t \text{ and } c \text{ to maximize } u(t,\ c) \text{ subject to the constraint } c+b=g(24-t)\]

and the solution satisfies the same two conditions (the first-order condition and the constraint):

\[MRT=MRS \text{ and } c+b=g(24-t)\]

With the particular functions in this example the two equations are:

\[\frac{t}{20} = \frac{\alpha c}{(1-\alpha)t} \text{ and } c+b=\frac{576-t^2}{40}\]

Assuming that \(\alpha = \frac{8}{13}\), as above, the first-order condition can then be rearranged to obtain:

\[c=\frac{t^2}{32}\]

This is the equation of the Pareto efficiency curve—the set of Pareto-efficient points. It tells us that \(c\) is a quadratic function of \(t\), which passes through the origin, and increases for \(t > 0\).

To understand what is going on here it is helpful to plot this curve (the upward-sloping line in Figure E5.8) together with the feasible frontier (the downward-sloping line). To keep the figure simple we haven’t drawn Angela’s indifference curves. But if we did so, then at every value of \(t\) along the upward-sloping line, the slope of the indifference curve (MRS) is the same as the slope of the feasible frontier (MRT).

The point P₁, where the two lies cross, is the solution to the two equations for the case \(b=0\). You can check that in this case \(t=16\) and \(c=8\). Angela has eight hours of free time, and produces and consumes eight bushels; Bruno gets nothing. Each point on the Pareto efficiency curve below the feasible frontier is a solution to the first-order condition where the amount of grain produced (on the feasible frontier) is shared between them. For example, at P₂ Angela has ten hours of free time and produces 11.9 bushels of grain; she gets 3.13 and Bruno gets 8.78 (to two decimal places).

At P₀ (the origin), Angela has no free time; she produces a large amount of grain and consumes nothing. Of course, this extreme outcome could not happen, since she would be unable to live. But if it did, it would be Pareto efficient.