Assumptions:  We assume that Persons A and B are both producing goods X and Y in the amounts shown in the first two columns of the table above. For simplicity, we assume that each individual expends the various factors in constant proportions, so that his or her cost can be expressed in terms of a single factor (indicated in the table's last three columns). It is assumed that all variables in the table are greater than zero and that the quantities of goods are large enough that the indivisibility of units will not affect this proof. Finally, we posit that the production costs of the two goods for the two individuals are not exactly proportional—that is: