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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