Contract Curve - Simple Mathematical Analysis

Simple Mathematical Analysis

In the case of two goods and two individuals, the contract curve can be found as follows. Here refers to the final amount of good 2 allocated to person 1, etc., and refer to the final levels of utility experienced by person 1 and person 2 respectively, refers to the level of utility that person 2 would receive from the initial allocation without trading at all, and and refer to the fixed total quantities available of goods 1 and 2 respectively.

subject to:

x_{1}^{1}+x_{1}^{2} \leq \omega_{1}^{tot}
x_{2}^{1}+x_{2}^{2} \leq \omega_{2}^{tot}
u^2(x_{1}^{2},x_{2}^{2}) \geq u_{0}^{2}

This optimization problem states that the goods are to be allocated between the two people in such a way that no more than the available amount of each good is allocated to the two people combined, and the first person's utility is to be as high as possible while making the second person's utility no lower than at the initial allocation (so the second person would not refuse to trade from the initial allocation to the point found); this formulation of the problem finds a Pareto efficient point on the lens, as far as possible from person 1's origin. This is the point that would be achieved if person 1 had all the bargaining power. (In fact, in order to create at least a slight incentive for person 2 to agree to trade to the identified point, the point would have to be slightly inside the lens.)

In order to trace out the entire contract curve, the above optimization problem can be modified as follows. Maximize a weighted average of the utilities of persons 1 and 2, with weights b and 1 – b, subject to the constraints that the allocations of each good not exceed its supply and subject to the constraints that both people's utilities be at least as great as their utilities at the initial endowments:

subject to:

x_{1}^{1}+x_{1}^{2} \leq \omega_{1}^{tot}
x_{2}^{1}+x_{2}^{2} \leq \omega_{2}^{tot}
u^1(x_{1}^{1},x_{2}^{1}) \geq u_{0}^{1}
u^2(x_{1}^{2},x_{2}^{2}) \geq u_{0}^{2}

where is the utility that person 1 would experience in the absence of trading away from the initial endowment. By varying the weighting parameter b, one can trace out the entire contract curve: If b = 1 the problem is the same as the previous problem, and it identifies an efficient point at one edge of the lens formed by the indifference curves of the initial endowment; if b = 0 all the weight is on person 2's utility instead of person 1's, and so the optimization identifies the efficient point on the other edge of the lens. As b varies smoothly between these two extremes, all the in-between points on the contract curve are traced out.

Note that the above optimizations are not ones that the two people would actually engage in, either explicitly or implicitly. Instead, these optimizations are simply a way for the economist to identify points on the contract curve.

Read more about this topic:  Contract Curve

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