I see. Thank you Nick. I was confused by the idea that utility could be proportional. e.i.
100%U[$1′000′000] = { 50%U[$1′000′000] } 2
because when you put 2 50%U[$1′000′000] the utility was less than 100%*U[$1′000′000]. But that was because U[$1′000′000] = U[$1′000′000] is not always true depending on if it’s the 1st or 2nd million.
U[$1′000′000] going from $0 to $1′000′000 is not the same as
U[$1′000′000] going from $1′000′000 to $2′000′000.
Back to Alais:
1A. 100%U[$24,000] + 0%U[$0]
1B. 33⁄34U[$27,000] + 1⁄34U[$0]
2A. 34%U[$24,000] + 66%U[$0]
2B. 33%U[$27,000] + 67%U[$0]
In 1A, U[$24,000] is going from $0 to $24,000.
In 1A, U[$0] is going from $0 to $0.
In 1B, U[$27,000] is going from $0 to $27,000.
In 1B, U[$0] is going from $0 to $0.
In 2A, U[$24,000] is going from $0 to $24,000.
In 2A, U[$0] is going from $0 to $0.
In 2B, U[$27,000] is going from $0 to $27,000.
In 2B, U[$0] is going from $0 to $0.
Looks like all the variables, U[money] = U[money] hold.
So if 1A > 1B then
100%U[$24,000] + 0%U[$0] > 33⁄34U[$27,000] + 1⁄34U[$0]
add 66%U[$0] (which makes the total percentages add up to 100%)
34%( 100%U[$24,000] + 0%U[$0] ) + 66%U[$0] > 34%( 33⁄34U[$27,000] + 1⁄34U[$0] ) 66%*U[$0]
I see. Thank you Nick. I was confused by the idea that utility could be proportional. e.i. 100%U[$1′000′000] = { 50%U[$1′000′000] } 2 because when you put 2 50%U[$1′000′000] the utility was less than 100%*U[$1′000′000]. But that was because U[$1′000′000] = U[$1′000′000] is not always true depending on if it’s the 1st or 2nd million. U[$1′000′000] going from $0 to $1′000′000 is not the same as U[$1′000′000] going from $1′000′000 to $2′000′000.
Back to Alais:
1A. 100%U[$24,000] + 0%U[$0]
1B. 33⁄34U[$27,000] + 1⁄34U[$0]
2A. 34%U[$24,000] + 66%U[$0]
2B. 33%U[$27,000] + 67%U[$0]
In 1A, U[$24,000] is going from $0 to $24,000. In 1A, U[$0] is going from $0 to $0. In 1B, U[$27,000] is going from $0 to $27,000. In 1B, U[$0] is going from $0 to $0. In 2A, U[$24,000] is going from $0 to $24,000. In 2A, U[$0] is going from $0 to $0. In 2B, U[$27,000] is going from $0 to $27,000. In 2B, U[$0] is going from $0 to $0. Looks like all the variables, U[money] = U[money] hold.
So if 1A > 1B then 100%U[$24,000] + 0%U[$0] > 33⁄34U[$27,000] + 1⁄34U[$0]
multiply by 34% 34%( 100%U[$24,000] + 0%U[$0] ) > 34%( 33⁄34U[$27,000] + 1⁄34U[$0] )
add 66%U[$0] (which makes the total percentages add up to 100%) 34%( 100%U[$24,000] + 0%U[$0] ) + 66%U[$0] > 34%( 33⁄34U[$27,000] + 1⁄34U[$0] ) 66%*U[$0]
algebra 34%U[$24,000] + 0%U[$0] + 66%U[$0] > 33%U[$27,000] + 1%U[$0] + 66%U[$0]
more algebra 34%U[$24,000] + 66%U[$0] > 33%U[$27,000] + 67%U[$0]
meaning 2A > 2B if 1A > 1B
/sigh, all this time to rediscover the obvious.