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Pricing Algorithm

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Sales of the object is a Poisson process. The probability of a sale occurring in the time interval t to t+dt is

where p is the price of the object and is the average rate of sales at some fixed price p via the demand curve. A simple linear demand curve will be assumed:

where is a constant equal to the rate of sales at optimum price . The optimum price is the price at which the rate of income is maximum. For prices above the sales rate will be zero.

The pricing algorithm will be to have a linearly decreasing price, decreasing to zero at time or until a sale is made, at which point the price jumps to times the sale price, and again begins a linear decline. That is, if where is the sale price, then the price p as a function of time t after that sale is

for
for

divides into two cases. When is greater than , then remains zero until , at which point it begins to rise linearly. It does so until , at which point it remains at

for
for
for

When p_n is less than , rises linearly until , at which point it remains at .

for
for

Approximate Equilibrium

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Depending on the initial price, the price function will take a certain amount of time to equilibrate. (This does not mean it is constant, of course, only that its average behavior gives no clue as to the amount of time elapsed since time zero.)

An approximate equilibrium condition is that the average time between sales is such that the price after a sale decays to the price before the sale.

These are two equations in two unknowns ( and ). Solving:


Note the problems when

Exact equilibrium

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The probability that the price is p at time t+dt is the probability that the price was at time t and a sale was not made, plus the probability that the price was at time t and that a sale was made. Normalizing to unity and

or

or

The sale price

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The sale price is a random variable, but its not a Poisson process. The sale price probability is dependent on the previous sale price.

Given that the last sale price was at time the probability that the next sale will occur between time t and t+dt is

will be zero when

or, equivalently,

as long as that t>0. For then, we have:

and for

and that sale price will be

The expected value of is

Given a sale at [p,0], and given that there is a sale at time t, what is the probability distribution for that sale price? The sale is not necessarily the first sale after the original.

References

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