User:PAR/Work3

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

:$$

dP=\lambda(p)dt\,

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

:$$

\lambda(p)=2\lambda_0\left(1-\frac{p}{2p_0}\right)\,\,\,\,(0 \le p<2p_0)

:$$

\lambda(p)=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(p \ge 2p_0)

where $\backslash lambda\_0$ is a constant equal to the rate of sales at optimum price $p\_0$. The optimum price $p\_0$ is the price at which the rate of income $p\backslash lambda$ is maximum. For prices above $2\; p\_0$ the sales rate will be zero.

The pricing algorithm will be to have a linearly decreasing price, decreasing to zero at time $\backslash tau$ or until a sale is made, at which point the price jumps to $1+\backslash alpha$ times the sale price, and again begins a linear decline. That is, if $p\_n=p\_s(1+\backslash alpha)$ where $p\_s$ is the sale price, then the price p as a function of time t after that sale is

:$p(t)=p\_n\backslash left(1-\backslash frac\{t\}\{\backslash tau\}\backslash right)\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,$ for $t\backslash le\backslash tau$

:$p(t)=0\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,\backslash ,$ for $t\backslash ge\backslash tau$

$\backslash lambda(t)$ divides into two cases. When $p\_n$ is greater than $2p\_0$, then $\backslash lambda(t)$ remains zero until $p\_n=2p\_0$, at which point it begins to rise linearly. It does so until $t=\backslash tau$, at which point it remains at $2\backslash lambda\_0$

:$\backslash lambda(t)=0\backslash ,$ for $0\backslash le\; t\; \backslash le\; \backslash tau(1-2p\_0/p\_n)$

:$\backslash lambda(t)=2\backslash lambda\_0\backslash left(1-\backslash frac\{p\_n\}\{2p\_0\}\backslash left(1-\backslash frac\{t\}\{\backslash tau\}\backslash right)\backslash right)$ for $\backslash tau(1-2p\_0/p\_n)\backslash le\; t\; \backslash le\; \backslash tau$

:$\backslash lambda(t)=2\backslash lambda\_0\backslash ,$ for $t\; \backslash ge\; \backslash tau$

When p_n is less than $2p\_0$, $\backslash lambda(t)$ rises linearly until $t=\backslash tau$, at which point it remains at $2\backslash lambda\_0$.

:$\backslash lambda(t)=2\backslash lambda\_0\backslash left(1-\backslash frac\{p\_n\}\{2p\_0\}\backslash left(1-\backslash frac\{t\}\{\backslash tau\}\backslash right)\backslash right)$ for $0\backslash le\; t\; \backslash le\; \backslash tau$

:$\backslash lambda(t)=2\backslash lambda\_0\backslash ,$ for $t\; \backslash ge\; \backslash tau$

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 $(T=1/\backslash overline\{\backslash lambda\}(t))$ is such that the price after a sale decays to the price before the sale.

:$$

p_s=p_s(1+\alpha)(1-T/\tau)\,

:$$

\frac{1}{T}=?

These are two equations in two unknowns ($p\_s$ and $t$). Solving:

:$$

T=\tau\frac{\alpha}{1+\alpha}\,

:$$

\frac{p_s}{p_o}=?

Note the problems when

The probability that the price is p at time t+dt is the probability that the price was $pe^\{t/\backslash tau\}$ at time t and a sale was not made, plus the probability that the price was $p/(1+\backslash alpha)$ at time t and that a sale was made. Normalizing to unity $p\_0$ and $\backslash lambda\_0$

:$$

P(p,t+dt)

=(1-\lambda(pe^{dt/\tau})dt)P(pe^{dt/\tau},t) + \lambda(p/(1+\alpha)) P(p/(1+\alpha),t)dt\,

:$$

=(1-\lambda(p)dt)P(p(1+dt/\tau),t) + \lambda(p/(1+\alpha)) P(p/(1+\alpha),t)dt\,

:$$

=(1-\lambda(p)dt)P(p+pdt/\tau,t) + \lambda(p/(1+\alpha)) P(p/(1+\alpha),t)dt\,

or

:$$

P(p,t)+\frac{\partial P}{\partial t}dt=\left(1-\lambda(p)dt\right)\left(P(p,t)+\frac{\partial P}{\partial p}\frac{p}{\tau}dt\right) + \lambda(p/(1+\alpha)) P(p/(1+\alpha),t)dt\,

or

:$$

\frac{\partial P}{\partial t}=\lambda(p)P(p,t)+\frac{p}{\tau}\frac{\partial P}{\partial p}+ \lambda\left(\frac{p}{1+\alpha}\right) P\left(\frac{p}{1+\alpha},t\right)\,

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 $p\_i$ at time $t\_i$ the probability that the next sale will occur between time t and t+dt is

:$$

P(t)dt = e^{-\overline{\lambda}t}\lambda(t)dt

:$$

\lambda(t)=2\lambda_0\left(1-\frac{p_i(1+\alpha)e^{-t/\tau}}{2p_0}\right) \,\,\,\,\,\,\,\mathrm{for}\,\,p_i(1+\alpha)e^{-t/\tau}\le 2p_0\mathrm{\,\,zero\,\,otherwise}

$\backslash lambda(t)$ will be zero when

:$$

p_i(1+\alpha)e^{-t/\tau} \ge 2p_0

or, equivalently,

:$$

t \le \tau\ln\left(\frac{p_i(1+\alpha)}{2p_0}\right)

as long as that t>0. For $p\_i(1+\backslash alpha)e^\{-t/\backslash tau\}\backslash le\; 2p\_0$ then, we have:

:$$

\overline{\lambda}=\frac{1}{t}\int_0^t \lambda(t')dt' =

2\lambda_0\left(1+\frac{\tau}{t}\,\frac{ p_i(1+\alpha)\left(e^{-t/\tau}-1\right)}{2p_0}\right)

and for $p\_i(1+\backslash alpha)e^\{-t/\backslash tau\}\; >\; 2p\_0$

:$$

\overline{\lambda}=\frac{1}{t}\int_{t_x}^t \lambda(t')dt' =

2\lambda_0\left(1-\frac{t_x}{t}+\frac{\tau}{t}\,\frac{ p_i(1+\alpha)\left(e^{-t/\tau}-e^{-t_x/\tau}\right)}{2p_0}\right)

and that sale price will be

:$p\_\{i+1\}(t)=p\_i(1+\backslash alpha)e^\{-t/\backslash tau\}\backslash ,$

The expected value of $p\_\{i+1\}$ is

:$$

\langle p_{i+1}\rangle = \int_0^\infty P(t)p_{i+1}(t)dt = \int_0^\infty p_i(1+\alpha)e^{-t/\tau}e^{-\overline{\lambda}t}\lambda(t)dt

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.

• [http://www.springerlink.com/content/q870032g57u05715/]
• [http://portal.acm.org/citation.cfm?id=1018409.1018759&coll=GUIDE&dl=GUIDE&type=series&idx=SERIES134&part=series&WantType=Proceedings&title=AGENTS&CFID=70591674&CFTOKEN=97332196]
• [http://ecommerce.hostip.info/pages/865/Pricing-DYNAMIC-PRICING.html]
• [http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?tp=&arnumber=1243848&isnumber=27871]
• [http://www.vjolt.net/vol6/issue2/v6i2-a11-Weiss.html]
• [http://www.managingchange.com/dynamic/overview.htm]
• [http://nymag.com/nymetro/news/bizfinance/columns/bottomline/1778/]
• [http://www.meiss.com/download/RM-Maglaras-Meissner.pdf]
• [http://www.jstor.org/pss/2661464]
• [http://cat.inist.fr/?aModele=afficheN&cpsidt=17941386 Dynamic pricing with real-time demand learning K.Y. Lin]