Monday, December 13, 2004

Gary Becker Rules

I really think that Dr. Becker has some of the most interesting ideas I've ever read, about economics or otherwise. A prof at my school has recently written a comment on one of Becker's papers, 1991's A Note on Restaurant Pricing and Other Examples of Social Influences on Price. In the paper, Becker argues that demand curves could conceviably be upward sloping, given the following:

D = Sum[di(p, D)] = F(p, D), given that dF/dp <>0 (I can't make partial derivative symbols on here...)

He then takes the differential of D-F(p,D) = 0 and gets:
dD - dF/dp dp - dF/dD dD = 0, which is rearranged to yield:
dD/dp = [(dF/dp) / (1-dF/dD)] > 0 iff dF/dD > 1

He then looks at what would happen if dF/dD > 1, which yields upward sloping demand curves. Here is my problem with the argument:
If demand is equal to F(p,D), and dF/dp < 0, then how can market demand be upward sloping? In other words, if each individual demand curve is downward sloping, how can the sum of all of these curves be upward sloping? It would seem to me that we have really just found a bound for the df/dD term at 1.

My professor has a problem with looking at an increase in income's effect on demand. In this case. It leads to an odd situation: income goes up, but in order for this to reflect a higher value for a normal good we have to shift demand to the left. But this would mean that people will buy a smaller quantity at each price. How can this be?

I have begun writing about the following way of incorporating bandwagon effects:
Let y be the proportion of some population that purchases a good or service. Then dy/dp is the change in this proportion with respect to a change in the price. If we wish to allow for bandwagon effects, we can model using the Ricatti Equation:
dy/dp = (1-y)[g(p) + ky]

Here we can alter g(p) to allow for only downward sloping demand curves and still include bandwagon effects, or we can let the demand curves slope upwards over intervals... all kinds of stuff. Anyway, I'm beginning with the simpler: dy/dp = (1-y)[-rp+ky], where r and k are constants.


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