Which sounds great, except that my intuition screams that maximizing U should depend on b. So what’s up there?
Your condition ab<1 is incomplete; you appear to be implicitly assuming a constant b.
Consider the equation:
U—abU = U1 + aV1
Therefore, U(1 - ab) = U1 + aV1
Now consider holding U1, V1 and a constant and changing b (but keeping to ab<1). Since U1, V1 and a are constant, the product U(1-ab) is constant. Thus, U and (1-ab) are inversely proportional; a decrease in the value of (1-ab) results in an increase in the value of U. A decrease in (1-ab) is caused by an increase in b.
Thus, an increase in b results in an increase in U. This goes to infinity the closer ab gets to 1; that is, the closer b gets to 1/a.
As a general strategy, picking random values for all-but-one variable and using some graphing software, like gnuplot, to plot the effects of the last variable will generally help to visualise this sort of thing.
Your condition ab<1 is incomplete; you appear to be implicitly assuming a constant b.
Consider the equation:
U—abU = U1 + aV1
Therefore, U(1 - ab) = U1 + aV1
Now consider holding U1, V1 and a constant and changing b (but keeping to ab<1). Since U1, V1 and a are constant, the product U(1-ab) is constant. Thus, U and (1-ab) are inversely proportional; a decrease in the value of (1-ab) results in an increase in the value of U. A decrease in (1-ab) is caused by an increase in b.
Thus, an increase in b results in an increase in U. This goes to infinity the closer ab gets to 1; that is, the closer b gets to 1/a.
As a general strategy, picking random values for all-but-one variable and using some graphing software, like gnuplot, to plot the effects of the last variable will generally help to visualise this sort of thing.