Some experimentation on assumptions of competitive equilibrium yields very interesting results. Suppose in an exchange economy, the first individual has a year long utility function:
(1) U = X1*Y1
And the second individual:
(2) Z1= X2*Y2 during summertime
(3) Z2=X2*(Y2)^2 during wintertime
Suppose the first individual was endowed with the basket w1(90,80), and the second individual w2(10,20).
Solving for X1,Y1,X2,Y2 where individuals maximize their utility given their constraints each season gives the following table:
(Py/Px is a price ratio)
While it is not possible to compare allocations during Summer and Winter seasons, we can compare Summers of the each year. The second individual who has cyclical preferences is worse off, not only he was unlucky to be poor at the beginning, but also he is getting poorer in consequent years. Inequality between them gradually expanding. Nevertheless, all points are Pareto efficient.
Is it realistic to assume that the second individual has different preferences during different seasons? One explanation could be that people's needs during summer are different than in winter, and they frequently trade their goods to match their urgent needs in both periods. Especially if they are poor.
It is also curious that once preference of the individual has shifted, it is impossible to get to the original basket anymore. Given that utility function is Cobb-Douglas type and that other individual's preference does not change.
(each dot represents X1, Y1 allocation in particular season, they are approaching to 100 ).
(1) U = X1*Y1
And the second individual:
(2) Z1= X2*Y2 during summertime
(3) Z2=X2*(Y2)^2 during wintertime
Suppose the first individual was endowed with the basket w1(90,80), and the second individual w2(10,20).
Solving for X1,Y1,X2,Y2 where individuals maximize their utility given their constraints each season gives the following table:
| Years | Season | X1 | Y1 | X2 | Y2 | Py/Px | Px/Py |
| Endowement | 90 | 80 | 10 | 20 | |||
| 1 | Summer | 85.00 | 85.00 | 15.00 | 15.00 | 1.0000 | 1.0000 |
| Winter | 80.95 | 89.47 | 19.05 | 10.53 | 0.9048 | 1.1053 | |
| 2 | Summer | 85.21 | 85.21 | 14.79 | 14.79 | 1.0000 | 1.0000 |
| Winter | 81.21 | 89.63 | 18.79 | 10.37 | 0.9061 | 1.1037 | |
| 3 | Summer | 85.42 | 85.42 | 14.58 | 14.58 | 1.0000 | 1.0000 |
| Winter | 81.46 | 89.78 | 18.54 | 10.22 | 0.9073 | 1.1022 | |
| 4 | Summer | 85.62 | 85.62 | 14.38 | 14.38 | 1.0000 | 1.0000 |
| Winter | 78.82 | 93.71 | 21.18 | 6.29 | 0.8412 | 1.1888 | |
| 5 | Summer | 86.26 | 86.26 | 13.74 | 13.74 | 1.0000 | 1.0000 |
| Winter | 79.70 | 94.01 | 20.30 | 5.99 | 0.8477 | 1.1796 | |
| 6 | Summer | 86.85 | 86.85 | 13.15 | 13.15 | 1.0000 | 1.0000 |
| Winter | 80.50 | 94.29 | 19.50 | 5.71 | 0.8538 | 1.1713 | |
(Py/Px is a price ratio)
While it is not possible to compare allocations during Summer and Winter seasons, we can compare Summers of the each year. The second individual who has cyclical preferences is worse off, not only he was unlucky to be poor at the beginning, but also he is getting poorer in consequent years. Inequality between them gradually expanding. Nevertheless, all points are Pareto efficient.
Is it realistic to assume that the second individual has different preferences during different seasons? One explanation could be that people's needs during summer are different than in winter, and they frequently trade their goods to match their urgent needs in both periods. Especially if they are poor.
It is also curious that once preference of the individual has shifted, it is impossible to get to the original basket anymore. Given that utility function is Cobb-Douglas type and that other individual's preference does not change.
