Optimization Algorithm
The first method of determining the unique clearing vector is an optimization problem that can easily be solved with a linear program. The objective function and constraints are as follows:
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Note that the objective is to maximize all banks’ total payments where p_i is what bank i pays. In addition, while all variables in the objective function above have a coefficient of 1, the value of the coefficients can be anything greater than 0 in order to attain the same results. However, a coefficient of 1 is appropriate because, if possible, banks pay exactly what they owe.
This algorithm maximizes the objective function subject to the specified constraints. The first constraint is consistent with the limited liability condition outlined above, where a bank pays less than or equal to the cash it has on hand, and the second constraint states that a bank can never pay negative money and will never pay more than they owe. This algorithm thereby maximizes payments and minimizes total debt in the system.
The optimization algorithm can further be used to conduct a sensitivity analysis. A sensitivity analysis is a study of how sensitive the optimal solution is to changes in the parameters. In our project, a sensitivity analysis would entail calculating the shadow prices, or amounts by which the banks’ payments would change given one unit increases in the specified constraints.
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