Leibniz 5.8.1

The Pareto efficiency curve

For an introduction to the Leibniz series, please see ‘Introducing the Leibnizes’.

There are many feasible allocations resulting from the interaction between Angela and Bruno; for example, we have looked at the allocation that Bruno would impose if he could use force, and at 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 Leibniz we work out mathematically the set of allocations that are Pareto efficient: that is, the Pareto efficiency curve.

Pareto efficient

An allocation with the property that there is no alternative technically feasible allocation in which at least one person would be better off, and nobody worse off.

A feasible 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. So to look for a Pareto-efficient allocation between Bruno and Angela, we start by thinking about their preferences—that is, what would make them better off.

Angela cares about free time, and the grain that she consumes. She has a quasi-linear utility function, and as in Leibniz 5.4.1 we write it as:

$$U(t,\ c) = v(t) + c$$

where $t$ is her daily hours of free time and $c$ the number of bushels of grain she consumes per day, and the function $v$ is increasing and concave.

Bruno’s preferences are very simple. He cares only about the amount of grain he receives, which we call $R$. The higher $R$, the better off Bruno is.

The economically feasible set

Next we work out which allocations are economically feasible. The total amount of grain available to Angela and Bruno is the amount that Angela produces by using some of her time to work the land. As in Leibniz 5.4.1, suppose that when Angela has $t$ hours of free time per day, she produces $g(t)$ bushels of grain. We assume again that $g$ is a decreasing, concave, function: $g'(t)\lt0$ and $g''(t)\lt0$.

$g(t)$ is the maximum amount of grain that Bruno and Angela can consume between them. So the feasible frontier for their interaction is:

$$c+R = g(t)$$

Suppose that if Angela did no work and received survival rations from the government her level of utility would be $u_0$. She will only be willing to work if she is at least as well off by doing so. Her reservation indifference curve is:

$$v(t) + c=u_0$$

Bruno will not be willing to engage in the interaction unless the amount of grain he receives is at least zero: $R\geq0$. (A negative amount of grain would mean giving some additional grain from his own store to Angela.)

The economically feasible set is the set of allocations of grain for Angela, $c$, grain for Bruno, $R$, and free time for Angela, $t$, such that:

$$c+R \leq g(t),\ R\geq 0 \mbox{ and } v(t) + c \geq u_0$$

These are the allocations in which the total amount of grain consumed is less than or equal to what is produced, and Angela and Bruno both receive at least their reservation utility.

The first-order condition for Pareto-efficient allocations

One way to find the Pareto-efficient allocations in the interaction between Angela and Bruno is to say: ‘suppose we take an allocation in which Bruno receives an amount of grain $R\geq 0$. Then it is Pareto efficient if and only if Angela is as well off as possible, given Bruno’s amount of grain, and her utility is at least $u_0$.’

This means, first of all, that in a Pareto-efficient allocation, all of the grain produced must be consumed. So if the amount of grain produced is $g(t)$, and Bruno consumes $R$, Angela must consume all of the rest: $c+R = g(t)$. No spare grain is left lying around.

Secondly, subject to the constraint that all the grain is consumed, $c$ and $t$ must together maximize Angela’s utility. In other words, given $R$, an allocation $(t,\ c,\ R)$ can be Pareto efficient only if we:

$$\text{choose }t \text{ and }c \text{ to maximize }v(t)+c , \text{ subject to the constraint that }c+R=g(t)$$

provided that this gives Angela her reservation utility. This constrained optimization problem can be solved by first using the constraint to substitute for $c$. Since $c=g(t)-R$, all we need to do is:

$$\text{choose }t \text{ to maximize }v(t) + g(t) -R$$

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

$$v'(t) + g'(t)=0$$

We have seen this equation before in Leibniz 5.4.2, 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 and denote it by $t^*$.

Notice that the constrained optimization problem we have solved is the one that Angela would solve if Bruno demanded an amount of rent $R$, and she could choose $c$ and $t$ for herself. The first-order condition is the familiar equation $\text{MRT} = \text{MRS}$, where $\text{MRT} = -g’(t)$ and $\text{MRS} = v’(t)$. In Leibniz 5.4.2 we solved the problem for the case when she was an independent farmer: $R=0$. We have shown here that her choice of $t$ would be the same whatever the amount of rent demanded.

Drawing the Pareto efficiency curve

We have shown that, given Bruno’s consumption of grain $R$, there is exactly one Pareto-efficient allocation: Angela’s free time is $t^*,$ the solution to the first-order condition:

$$v'(t^*) + g'(t^*)=0$$

and her consumption of grain is $g(t^*)-R$. To find this allocation, we started by arbitrarily fixing $R$. In fact, there are infinitely many feasible values for $R$, and for each one, there is a corresponding Pareto-efficient allocation.

The set of all Pareto-efficient allocations—that is, the Pareto efficiency curve—is the set of all points $(t,\ c,\ R)$ that satisfy the first-order condition, and all of the constraints that determine the economically feasible set. So it is given by the conditions:

$$t=t^*, \quad c+R=g(t^*), \quad c+v(t^*)\geq u_0, \quad R\geq 0$$

In the example we saw in Figure 5.8 of the text, which we reproduce here as Figure 1, $t^*=16$. The Pareto-efficient allocations lie on the vertical line $t=16$, and the amount of grain produced is $g(16)=9$ bushels. The points on this vertical line are the points where Angela’s MRS between free time and grain is equal to the MRT. But not all points on the line satisfy the other constraints. The set of all Pareto-efficient allocations lie between the points C and D. In the allocation represented by point G, for example, Bruno obtains an amount of grain $R$ given by GC, and Angela obtains the rest.

Allocations with higher $R$ and lower $c$ are closer to D. They are more beneficial for Bruno and less so for Angela. Allocations closer to C are better for Angela and worse for Bruno.

The result that Angela’s free time is the same at all Pareto efficient allocations is a consequence of her having a quasi-linear utility function and would not hold for alternative forms of $U$. In other cases, the Pareto efficiency curve would lie within the lens-shaped region in Figure 1, but would not be a vertical straight line.

The Pareto efficiency curve: Two examples

Our first example illustrates the principles of the previous two sections using specific utility and production functions. We then give a second example in which Angela’s utility function is not quasi-linear and the Pareto efficiency curve is really a curve, and not a vertical line.

1. Specific utility and production functions

As in Leibniz 5.4.2, we assume that the feasible frontier and Angela’s utility function are given by:

$$g(t)=2\sqrt{48 -2t}, \quad U(t,\ c) = 4\sqrt{t} + c$$

Then the terms denoted by $g'(t)$ and $v'(t)$ above are respectively:

$$\frac{d}{dt} \left(2(48 -2t)^{1/2}\right) = -2(48 -2t)^{-1/2}, \quad \frac{d}{dt} \left(4t^{1/2}\right) = 2t^{-1/2}.$$

Thus $t^*$, Angela’s daily hours of free time at every Pareto-efficient allocation, is the solution to the equation:

$$\frac{2}{\sqrt{t}} - \frac{2}{\sqrt{48-2t}} = 0$$

This equation implies that:

$$t=48-2t$$

Therefore $t^*=16$, and the amount of grain produced is $g(t^*)= 2\sqrt{48 -32}=8$. Thus in this example each Pareto-efficient allocation $(t,\ c,\ R)$ must be such that:

$$t=16, \quad c+R=8$$

In addition, both Angela and Bruno must receive at least their reservation utility. Angela’s reservation utility is attained when she does no work $(t = 24)$ and consumes $2$ bushels of grain a day, provided by the government. Thus her reservation utility is $4\sqrt{24} + 2$. So a Pareto-efficient allocation, with $t^*=16$, must satisfy:

$$4\sqrt{16} +c\geq 4\sqrt{24}+2$$

Solving this inequality gives us:

$$c\geq 5.596$$

to three decimal places. Bruno’s amount of grain $R$ must be at least zero. The maximum possible value of $R$ is the amount that gives Angela as little as possible. Since $c+R=8$, this is $R=8-5.596=2.404$ to three decimal places. In summary, the possible range for Bruno is:

$$0 \leq R \leq 2.404$$

2. Cobb-Douglas utility

In this example, Angela’s utility function has the Cobb-Douglas form:

$$U(t,\ c) = t^{\alpha} c^{1- \alpha}$$

where $0\lt \alpha \lt1$. To make the numbers work out nicely, we assume $\alpha= \dfrac{8}{13}$. Angela’s marginal rate of substitution (MRS) between free time and consumption of grain is given by the usual formula (recall Leibniz 3.4.1):

$$\text{MRS} = \frac{\partial U}{\partial t} \left/ \frac{\partial U}{\partial c} \right. = \frac{\alpha c}{(1-\alpha)t} = \frac{8c}{5t}$$

Let the equation of the feasible frontier be $y=g(t)$. To avoid messy algebra, we assume a different form for $g$ from the first example:

$$g(t)= \frac{576 - t^2}{40}$$

Though the graph of this function is not the same as the frontier of the previous example, it has the same general shape and again passes through the points $(24, 0)$ and $(16, 8)$. The marginal rate of transformation 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:

$$\text{MRT} = - g'(t) = \frac{t}{20}$$

Given Bruno’s rent $R$, Pareto efficiency of the allocation $(t,\ c,\ R)$ requires that we

$$\text{choose }t \text{ and }c \text{ to maximize }U(t,\ c), \text{subject to the constraint that }g(t) - c =R$$

The first-order condition for maximization, $\text{MRT} = \text{MRS}$, implies that $t^2/4 =8c$. Rearranging,

$$c=\frac{t^2}{32}$$

The points satisfying this condition in the positive quadrant of the $t-c$-plane form the upward-sloping branch of a parabola. The Pareto efficiency curve is the section of this curve that is economically feasible.

At the Pareto efficient allocation that is minimally acceptable to Bruno, he receives no rent and Angela consumes all the grain that is produced. Thus, at that allocation,

$$c=\frac{t^2}{32} \, \text{ and } \, c= \frac{576 - t^2}{40}$$

Substituting the first equation into the second and multiplying by $40$, we see that $5t^2/4 = 576 - t^2$. Hence:

$$t^2 = \frac{-76}{9/4} = 64 \times 4$$

It follows that $t = 16$ and $c = t^2/32 = 8$: the Pareto efficient allocation in which Bruno receives no grain (that is, Angela pays no rent) is the same as in the previous example. (We chose the numbers to make this happen.)

We now find the Pareto-efficient allocation that is only just acceptable to Angela. As in the previous example, she can obtain her reservation utility by doing no work and consuming $2$ bushels of grain per day. Thus, at the Pareto-efficient allocation where Angela also obtains her reservation utility,

$$c=\frac{t^2}{32} \, \text{ and } \, t^{\alpha} c^{1- \alpha} = 24^{\alpha} \times 2^{1- \alpha}$$

Substituting the first equation into the second, $t^{2- \alpha} \! \left/ 32^{1- \alpha} \right. = 24^{\alpha} \times 2^{1- \alpha}$. Hence:

$$t^{2- \alpha} = 24^{\alpha} \times 64^{1- \alpha} = 3^{\alpha} \times 8^{2- \alpha}$$

Since $\alpha= \dfrac{8}{13}$, $\, \dfrac{\alpha}{2- \alpha}= \dfrac{4}{9}$. Therefore:

$$t = 3^{4/9} \times 8 = 13.036, \quad c = \frac{t^2}{32}= 5.311, \quad y = \frac{576 - t^2}{40}= 10.152$$

all to three decimal places. Bruno’s rent $y-c$ is $4.841$ to three decimal places. Since Angela obtains her reservation utility, this is the highest rent at which Angela will be willing to work on Bruno’s land.

To summarize: in this case Angela’s utility function is not quasi-linear. Her hours of free time, and hence of work, are not constant across Pareto efficient allocations. The Pareto efficiency curve in the $tc$-plane is not a vertical line but rather the upward-sloping branch of the parabola $c=t^2/32$ corresponding to allocations that are economically feasible. This goes from the point $(13.036,\ 5.311)$, where Angela obtains her reservation utility and Bruno his maximal rent of $4.841$ bushels per day, to the point $(16,\ 8)$, where Angela consumes all the grain she produces, and Bruno receives no rent.

To make sure you understand this example fully, try drawing the curve, and comparing it with the one in the quasi-linear case.