Process of Reaching Equilibrium Through Trade:

You  have  already seen the task assigned  to  an  auctioneer for  deciding  the equilibrium prices. We  apply the same logic to our example of food (F) and shelter (S)  for reading the  contract curve. See that  in  the position of initial endowment, all  the  goods  are  consumed.  In  this sense, the market clears.

However, such an  allocation is not Pareto efficient. Therefore, the auctioneer announces some price after which  both  the parties trade. What they preferred at such  prices might  be  Pareto efficient but not  necessarily market clearing. So, there must be  re-auction at new set of prices. To  understand  the  price  adjustment mechanism  of  the auctioneer, which ultimately produces an equilibrium, you may  recall the option of charging the relative  price. According to the  position  of  excess  demand, price,  in  our  example P_F/P_S, will have to be varied. When the auctioneer get the price ratio correct, the market  clears and  there  is  no excess demand or  supply for any good.

Example: (Due  to Edgar Preugschat Lecture Notes)

We continue with an economy with 2 agents, 2 goods presented in the Edgeworth box. Present consumer i's problem as

It  will  useful  to  recall  the  utility maximisation  problem  seen  in  case of individual  consumer.  It  was  seen that the  slope  of  the  budget  line  was determined by  the negative of the price ratio. If you bring in the contrast now, the increment depends on both, the prices and the endowments  of the goods. So, to tind the budget  line, we need to find the endowment point  in  the two-,
goods diagram. Then, given the slope by  the price ratio, draw the budget  line through the endowment point.

Note that you can consume your endowment independent price. So,

 Now  we bring  together the  budget  line of two consumers in  the  Edgeworth box.  In  Figure 18.5  the two 2-goods diagrams are put-together. The size of the box is determined  by  the endowments. 

The axes  for F  and  S  are  of  length equals to the sum of  the endowments,  i.e.,  the length of  the F side is and the length of  the s side is