An example of a possible, though not necessarily optimal, allocation occurs when Angela has 12 hours of free time, meaning she works for 12 hours. In this scenario, she produces 54 bushels of grain. A contract could specify that she keeps 36 of these bushels as her wage, which would leave the remaining 18 bushels for Bruno as his profit.

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Figure 5.13 presents a comparative analysis of outcomes through two diagrams. The first diagram, with axes for Angela's free time (0-24 hours) and grain production (0-70 bushels), illustrates the concave feasible frontier running from (0, 64) to (24, 0). It features two convex indifference curves: IC1, representing Angela's reservation utility under coercion, and a higher curve, IC2, for when she has a choice. The feasible set of agreements in the choice scenario is the area enclosed by IC2 and the feasible frontier. The second diagram shows Bruno's potential share of grain (0-40 bushels) against Angela's free time. It contrasts two hump-shaped curves: a lower curve for the coercion scenario, which starts at (8, 0), peaks at (16, 22), and ends at (23.5, 0); and a higher curve for the choice scenario, which starts at (7, 0), peaks at (16, 31), and ends at (24, 0).

Detailed Description of Figure 5.13 - Comparing Outcomes in Case 1 (Force) and Case 2 (Choice)

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Angela's reservation option improves significantly in Case 2 relative to Case 1. In Case 2, her fallback is seeking other work, providing a higher utility than her alternatives in Case 1, such as forced labor. This distinction is visually captured in Figure 5.13, where her reservation indifference curve for Case 2, labeled IC2, is shown to be higher than IC1, the curve representing her reservation option under forced labor.

Angela's Reservation Option in Case 2 vs. Case 1 (IC2 vs. IC1)

In the Angela-Bruno model, the feasible set includes all allocations of grain and free time from which Bruno can choose. On a diagram with Angela's free time (0-24 hours) on the horizontal axis and bushels of grain on the vertical axis, this set is visually represented by the lens-shaped area between two curves. The upper boundary is the concave feasible frontier, which shows the maximum possible output and connects points such as (0, 64) and (24, 0). The lower boundary is Angela's reservation indifference curve, labeled IC1, which is a convex curve that intersects the frontier in two places. Any allocation within this area is feasible because it is both technically possible (on or below the frontier) and acceptable to Angela (on or above her reservation indifference curve).

Feasible Set in the Angela-Bruno Model

Bruno's potential share of the grain is compared under two distinct institutional frameworks. When coercion is permitted and Angela's fallback is revolt, Bruno's share is depicted by a hump-shaped curve that begins at 8 hours of free time, reaches a maximum of 22 bushels at 16 hours, and ends at 23.5 hours. In contrast, when force is prohibited and Angela has an alternative employment option, his share is shown by a higher curve that starts at 7 hours of free time (yielding zero profit), peaks at 31 bushels (also at 16 hours), and terminates at 24 hours of free time (returning to zero profit). This visual comparison shows that Bruno's maximum obtainable profit is higher under the 'no force' institutional arrangement (31 bushels) than under the 'force' arrangement (22 bushels).

Bruno's Share under Coercion vs. No Coercion

Allocation at 12 Hours of Free Time

The allocation resulting from the take-it-or-leave-it contract is point L, where Angela has 16 hours of free time (working 8 hours) and both she and Bruno receive 23 bushels of grain. This outcome is Pareto efficient, meaning there is no alternative change that could make either person better off without making the other worse off. The reason for its efficiency is that it maximizes the joint surplus at the point where Angela's Marginal Rate of Substitution (MRS) on her reservation indifference curve equals the Marginal Rate of Transformation (MRT) on the feasible frontier.

Allocation L as a Pareto-Efficient Outcome

In the model, the optimal number of work hours is identical across three distinct situations: when Angela is an independent farmer, when she is coerced by Bruno (Case 1), and when she accepts a take-it-or-leave-it offer from him (Case 2). In all these cases, the chosen allocation involves 8 hours of work, which corresponds to 16 hours of free time.

Optimal Work Hours Across Different Scenarios

In the scenario where Angela has the freedom to accept or reject offers (Case 2), the feasible set represents all possible contract agreements. This set is graphically defined as the area bounded above by the concave feasible production frontier and below by Angela's improved reservation indifference curve, IC2. Any allocation within this region is both technically achievable and provides Angela with at least the minimum utility she requires to voluntarily agree to the contract.

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