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

5.9 Case 3 continued: Negotiating to a Pareto-efficient sharing of the surplus

Under the new law that changes workers’ rights, Angela’s reservation position has improved, and Bruno offers her contract N on her new reservation indifference curve ICN (Figure 5.18 below). But Angela has an opportunity to do better still, because allocation N is not Pareto efficient. This doesn’t mean it would be better to go back to the Pareto efficient contract L—that would make her worse off. It means that there are other allocations that both parties would prefer to N. Both Angela and Bruno could be better off if they could negotiate successfully.

Pareto improvement
A change that benefits at least one person without making anyone else worse off. See also: Pareto dominant, Pareto criterion.

Figure 5.18 shows that in contract N, where Angela has more free time than in Cases 1 and 2, her indifference curve is flatter and the feasible frontier is steeper: MRS at N < MRT at M. Her marginal rate of substitution between grain and free time is lower than the rate at which she can transform free time into grain. And whenever MRS and MRT are unequal, there is the potential for a Pareto improvement. In particular, when MRS < MRT, Angela could transform some of her free time into grain, producing more grain than she would need to compensate her for the loss of free time (in other words, to keep her on ICN). So if her free time were reduced, the extra grain could make both Angela and Bruno better off.

In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). A downward-sloping, concave curve connects points (0, 64) A (16, 46), M (19.5, 35) and (24, 0) and is labelled feasible frontier. There are two parallel, downward-sloping, convex curves. One passes through point L (16, 23) and is labelled IC2. The other passes through point N (19.5, 23) and is labelled IC_N. IC_N lies above IC2 at all points. The slope of the feasible frontier at M is steeper than the slope of IC_N at N.
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Figure 5.18 After the new laws are implemented, MRT > MRS.

Within the lens-shaped area between ICN and the feasible frontier, the surplus is maximized where Angela has 16 hours of free time. Angela’s indifference curves are parallel, so at 16 hours the MRS on every indifference curve is equal to the MRT.

Just as before in Cases 1 and 2, an allocation where MRS = MRT is Pareto efficient.

Negotiation

We will suppose that the new law allows a longer work day if both parties voluntarily agree, provided that the fallback position is a four and a half hour day if no agreement is reached.

Bruno has offered contract N. Angela could respond with a counter-offer: she could suggest a contract with eight hours of work (16 hours of free time) to increase the surplus, and a way of splitting the surplus that makes them better off than at N.

Figure 5.19 shows what might happen.

In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). A downward-sloping, concave curve connects points (0, 64) A (16, 46), M (19.5, 35) and (24, 0) and is labelled feasible frontier. There are two parallel, downward-sloping, convex curves. One passes through point L (16, 23) and is labelled IC2. The other passes through points P (16, 30) and N (19.5, 23) and is labelled IC_N. IC_N lies above IC2 at all points. Point R has coordinates (16, 34). The vertical distance between points A and R is the same as between points M and N.
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Figure 5.19 Bargaining to restore Pareto efficiency.

The maximum joint surplus: In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). A downward-sloping, concave curve connects points (0, 64) A (16, 46), M (19.5, 35) and (24, 0) and is labelled feasible frontier. There are two parallel, downward-sloping, convex curves. One passes through point L (16, 23) and is labelled IC2. The other passes through points P (16, 30) and N (19.5, 23) and is labelled IC_N. IC_N lies above IC2 at all points.
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The maximum joint surplus

When Angela works for Bruno, the surplus is maximized at 16 hours of free time, where MRT = MRS. The surplus is 16 bushels, (AP = 16) compared to 12 at contract N (MN = 12). All allocations on AP are Pareto efficient, and there is a potential gain in rents of four bushels.

Which Pareto-efficient allocations make Angela better off?: In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). A downward-sloping, concave curve connects points (0, 64) A (16, 46), M (19.5, 35) and (24, 0) and is labelled feasible frontier. There are two parallel, downward-sloping, convex curves. One passes through point L (16, 23) and is labelled IC2. The other passes through points P (16, 30) and N (19.5, 23) and is labelled IC_N. IC_N lies above IC2 at all points.
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Which Pareto-efficient allocations make Angela better off?

Allocation P gives Angela the same utility as N. Allocations between P and A are better for Angela than N (she would be on a higher indifference curve).

Which counter-offers on AP might Bruno accept?: In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). A downward-sloping, concave curve connects points (0, 64) A (16, 46), M (19.5, 35) and (24, 0) and is labelled feasible frontier. There are two parallel, downward-sloping, convex curves. One passes through point L (16, 23) and is labelled IC2. The other passes through points P (16, 30) and N (19.5, 23) and is labelled IC_N. IC_N lies above IC2 at all points. Point R has coordinates (16, 34). The vertical distance between points A and R is the same as between points M and N.
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Which counter-offers on AP might Bruno accept?

At allocation R, Bruno would get the same rent as in contract N (12 bushels, the distance between M and N). Allocations between R and P are better for Bruno than N.

What counter-offer will Angela make?: In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). A downward-sloping, concave curve connects points (0, 64) A (16, 46), M (19.5, 35) and (24, 0) and is labelled feasible frontier. There are two parallel, downward-sloping, convex curves. One passes through point L (16, 23) and is labelled IC2. The other passes through points P (16, 30) and N (19.5, 23) and is labelled IC_N. IC_N lies above IC2 at all points. Point R has coordinates (16, 34). The vertical distance between points A and R is the same as between points M and N.
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What counter-offer will Angela make?

Angela could respond to Bruno’s offer of N with a counter-offer anywhere on PR. Suppose she decides to offer allocation R, four bushels above P. It gives Bruno the same rent as N, and her own utility will be four bushels higher.

A win-win agreement: In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). A downward-sloping, concave curve connects points (0, 64) A (16, 46), M (19.5, 35) and (24, 0) and is labelled feasible frontier. There are two parallel, downward-sloping, convex curves. One passes through point L (16, 23) and is labelled IC2. The other passes through points P (16, 30) and N (19.5, 23) and is labelled IC_N. IC_N lies above IC2 at all points. Point R has coordinates (16, 34). The vertical distance between points A and R is the same as between points M and N.
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A win-win agreement

Bruno might respond that he would accept a contract halfway between P and R, sharing the gain from moving to 16 hours equally. Angela thinks this is reasonable. She accepts a wage of 32 bushels for eight hours’ work, and Bruno gets the remaining 14 bushels she produces.

Through negotiation, Angela and Bruno agree to an allocation between points P and R, with a wage of 32 bushels for eight hours of work. Compared with N, they are both better off. This is the outcome in Case 3.

Since there is scope to go back and forth in the negotiation, it is not the only possible outcome. But we can say that they are likely to reach an agreement on PR with a wage above 30 and below 34.

The change from the outcome in Case 2 to the final one in Case 3 is summarized in Figure 5.20. It consists of two distinct steps:

  • From L to N, the outcome is imposed by new legislation. This is definitely not win-win: Bruno loses because he gets less grain at N than at L. Angela benefits from greater structural power, raising her reservation position.
  • But once at the legislated outcome N, they both have bargaining power because N is not Pareto efficient. They voluntarily agree to a Pareto-efficient contract with longer working hours. This change is win-win. They share the gains from the negotiation.
Case 2: Contract L Case 3: Contract N Case 3: Outcome
Angela’s free time 16 hours 19.5 hours 16 hours
Angela’s income 23 bushels 23 bushels 32 bushels
Bruno’s income 23 bushels 12 bushels 14 bushels
Angela’s change in utility +7 bushels +2 bushels
Bruno’s change in utility –11 bushels +2 bushels

Figure 5.20 The change in outcomes from Case 2 to Case 3.

Pareto efficiency, and the Pareto efficiency curve

We now know that there are many Pareto-efficient allocations that could result from the interaction between Angela and Bruno, including all the outcomes from Cases 1, 2, and 3.

To be Pareto efficient, an allocation must have two important properties:

  • The MRT on the feasible frontier is equal to the MRS on Angela’s indifference curve.
  • No grain is wasted: all the grain produced is consumed by Angela or Bruno.

We have demonstrated the first property by arguing that if MRS is not equal to MRT, a Pareto improvement is possible if Angela’s hours of work are changed, while if MRS = MRT, no Pareto improvement is available. The diagrams show that when MRS ≠ MRT, the surplus can be increased; if MRS = MRT, it cannot.

The second property, which holds at all the allocations we have considered, means that no Pareto improvement can be achieved simply by changing the amounts of grain they each consume. If it holds, then if one consumed more, the other would get less. If it doesn’t hold, some grain is not being consumed, and consuming it would make at least one of them better off.

You may also hear it called the contract curve, even in situations where there is no contract, which is why we prefer the more descriptive term Pareto efficiency curve.

Pareto efficiency curve
The set of all allocations that are Pareto efficient. The Pareto efficiency curve is sometimes called the ‘contract curve’, even though it is not necessary for any contract to be involved. See also: Pareto efficiency.

The set of all Pareto-efficient allocations is called the Pareto efficiency curve. In our model, it is the set of allocations with 16 hours of free time, shown in Figure 5.21. It is a vertical straight line, because of our assumption that Angela’s indifference curves are parallel; if we had made a different assumption about her preferences, the Pareto efficiency curve would have had a different shape.

In this diagram, the horizontal axis shows Angela’s hours of free time, and ranges between 0 and 24. The vertical axis shows bushels of grain, and ranges between 0 and 70. Coordinates are (hours, bushels). A downward-sloping, concave curve connects points (0, 64), R (16, 46), and (24, 0) and is labelled feasible frontier. A downward-sloping, convex curve passes through point S (16, 38) and is labelled IC_s. The slope of the feasible frontier at R and IC_s5 at S is the same. A vertical line going through points R and S is labelled Pareto efficiency curve.
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Figure 5.21 The Pareto efficiency curve.

At any allocation on this line, such as allocation S, the MRS (the slope of ICS) is equal to the MRS at R. At S, Angela gets 38 bushels and Bruno gets eight bushels; different allocations on the line correspond to different ways of splitting the grain between them.

Question 5.6 Choose the correct answer(s)

Figure 5.19 shows the outcome from the interaction between Angela and Bruno.

Read the following statements and choose the correct option(s).

  • The allocation at A Pareto-dominates the one at L.
  • Angela’s marginal rate of substitution is equal to the marginal rate of transformation at all points on the Pareto efficiency curve.
  • The midpoint of AL is the most Pareto-efficient allocation.
  • Angela and Bruno are indifferent between all the points on the Pareto efficiency curve, because they are all Pareto efficient.
  • All points on the Pareto efficiency curve (allocations at 16 hours of free time) are Pareto efficient, so none of them is Pareto-dominated. (Comparing A and L, Bruno prefers L and Angela prefers A.)
  • The Pareto efficiency curve, by definition, joins all the points where MRS = MRT (all allocations at 16 hours of free time).
  • All the points on AL are Pareto efficient. It does not make any sense to say that one point on AL is more efficient than another.
  • All the points on the Pareto efficiency curve are Pareto efficient, but Bruno and Angela are not indifferent. Some points (like A) are better for Angela, while others (like L) are better for Bruno.

Question 5.7 Choose the correct answer(s)

In Figure 5.19, suppose that Angela and Bruno are at allocation N, where she receives 23 bushels of grain for four and a half hours of work.

From the figure, we can conclude that:

  • All the points on MN are Pareto efficient.
  • Any point in the area between R, P and N would be a Pareto improvement.
  • Any point between P and L would make Angela better off because it is on the Pareto efficiency curve.
  • They would both be indifferent between all points on RP.
  • Along MN, MRS < MRT. So MN is not Pareto efficient—there are other allocations where both would be better off.
  • In area RPN, Angela is on a higher indifference curve than ICN, and Bruno has more grain than MN, so both are better off.
  • Points on PL are Pareto efficient, but below P, Angela is on a lower indifference curve than at N, so she would be worse off.
  • Points on RP are all Pareto efficient, but Bruno and Angela are not indifferent. He prefers points nearer to P, and she prefers points nearer to R.

Extension 5.9 The Pareto efficiency curve

This extension builds on the three previous ones, Extensions 5.4, 5.5, and 5.7. We determine the set of Pareto-efficient allocations for Angela’s and Bruno’s interaction. This can again be done by formulating the problem in terms of constrained choice, and using calculus to solve it. Lastly, we find the Pareto efficiency curve in a different case in which Angela’s preferences are not quasi-linear.

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 \(b\geq 0\).

The assumption that \(c\) and \(b\) are non-negative rules out the possibility that grain can be stored or obtained elsewhere. For example, \(b\) would be less than zero if Bruno received none of the grain produced and also gave Angela some grain that he had kept from last year.

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.

Pareto-efficient allocations

An allocation, \((c, b, t)\), is Pareto efficient if and only if it is a solution of the constrained choice problem:

Choose \(t\) and \(c\) to maximize \(u(t,\ c)\) subject to the constraint, \(c+b=g(24-t)\).

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.

In this diagram, the horizontal axis shows Angela’s hours of free time, denoted t, ranging from 0 to 24, and the vertical axis shows bushels of grain, denoted c, ranging from 0 to 20. Coordinates are (hours of free time, bushels of grain). Angela’s feasible frontier is a downward-sloping concave curve that connects the points (0, 13.86) and (24, 0) and has the equation c = 2 times the square root of 2 times 24 minus t. Angela’s highest possibility utility level is denoted by a downward-sloping convex curve that is tangent to the feasible frontier at the point P1 (16, 8). The slope of Angela’s indifference curves is the same at 16 hours of free time. The Pareto efficiency curve is the line connecting P1 to point P0 (16, 0). Points P2 (16, 5) and P3 (16, 2) are on the Pareto efficiency curve.
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Figure E5.6 The Pareto efficiency curve (quasi-linear preferences).

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\(b=1\), saywe 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 \(\alpha = \frac{8}{13}\).

In this diagram, the horizontal axis shows Angela’s hours of free time, denoted t, ranging from 0 to 24, and the vertical axis shows bushels of grain, denoted c, ranging from 0 to 20. Coordinates are (hours of free time, bushels of grain). Angela’s feasible frontier is a downward-sloping concave curve that connects the points (0, 14.4) and (24, 0). Three downward-sloping convex curves that do not intersect are some of Angela’s indifference curves. The uppermost curve is tangent to the feasible frontier at (16, 8). The slope of the uppermost curve at this point is the same as the slope of the middle curve at 12 hours of free time, and the slope of the lowest curve at 8 hours of free time.
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Figure E5.7 The indifference curves for the case when Angela has Cobb–Douglas preferences.

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).

In this diagram, the horizontal axis shows Angela’s hours of free time, denoted t, ranging from 0 to 24, and the vertical axis shows bushels of grain, denoted c, ranging from 0 to 20. Coordinates are (hours of free time, bushels of grain). Angela’s feasible frontier is a downward-sloping concave curve that connects the points (0, 14.4), P1 (16, 8), and (24, 0). The Pareto efficiency curve connects the points P0 (0, 0), P2 (10, 3.13), and P1, and has the equation c = t squared divided by 32. The vertical distance between P2 and the feasible frontier is 8.78, denoted as b.
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Figure E5.8 The Pareto efficiency curve (Cobb–Douglas preferences).

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.

Exercise E5.4 The Pareto efficiency curve

Find and sketch the Pareto efficiency curve under the following scenarios:

  1. The individual’s preferences are Cobb–Douglas, with the utility function \(u(t,c)= t^{0.5} c^{0.5}\), and their production function is \(c(t)=100 \text{ ln } (25-t)\).
  2. The individual’s utility function is \(u(t,c)=\sqrt{t} + \sqrt{c}\) and their production function is \(c(t) = 2 \sqrt{48-2t}\).