Stock Speculation: In a classic paper on the theory of conflict, L.F. Richardson claimed that the proportion p of a population advocating war or other aggressive action at a time t satisfies
p(t) = (Ce^kt) / (1 + Ce^kt)
where k and C are positive constants. Speculative day-trading in the stock market can be regarded as “aggressive action.” Suppose that initially, (1/200) of total daily market volume is attributed to day-trading and that 4 weeks later, the proportion is (1/100). When will the proportion be increasing most rapidly? What will the proportion be at that time?
El valor del impuesto de un producto equivale a los 10/11 del costo de importación. Si el producto en cuestión vale 22 entonce los impuestos valen:
(de su respuesta en forma decimal)
surely as a calculus student, you can determine C and k:
at t=0,
C/(1+C) = 1/200
C = 1/199
Since you say daily volume, I assume t is in days, so
at t=28,
(1/199)e^28k/(1+(1/199)e^28k) = 1/100
k ≈ 0.025 = 1/40
p(t) = 1/199 e^(t/40)/(1+1/199 e^(t/40))
increasing most rapidly when dp/dt has a max. That is, when p" = 0
p' = 199/40 e^(t/40) / (1+1/199 e^(t/40))^2
p" = c(e^(t/40)-199)/(1+1/199 e^(t/40))^3
where c is a bunch of nonzero junk
p' has a max at 40 log199 ≈ 212 days
now just find p(212)
for the derivatives, wolframalpha is your friend:
http://www.wolframalpha.com/input/?i=p%28t%29+%3D+1%2F199+e^%28t%2F40%29%2F%281%2B1%2F199+e^%28t%2F40%29%29+for+0%3C%3Dt%3C%3D400
http://www.wolframalpha.com/input/?i=199%2F40+e^%28t%2F40%29+%2F+%281%2B1%2F199+e^%28t%2F40%29%29^2
To determine when the proportion will be increasing most rapidly and what the proportion will be at that time, we need to find the maximum value of the function p(t) = (Ce^kt) / (1 + Ce^kt).
To do this, we can take the derivative of p(t) with respect to t and set it equal to zero, since the maximum or minimum of a function occurs when its derivative is zero.
First, let's simplify the function by multiplying the numerator and the denominator by e^(-kt):
p(t) = Ce^kt / (1 + Ce^kt) * e^(-kt) / e^(-kt)
This simplifies to:
p(t) = C / (e^(-kt) + C)
Now, let's take the derivative of p(t) with respect to t:
dp/dt = d/dt (C / (e^(-kt) + C))
Using the quotient rule, we have:
dp/dt = -Ck * e^(-kt) / (e^(-kt) + C)^2
To find the maximum or minimum, we set dp/dt equal to zero:
-Ck * e^(-kt) / (e^(-kt) + C)^2 = 0
Since k and C are positive constants, we can ignore the negative sign and focus on the numerator:
e^(-kt) = 0
However, there is no exponential function that equals zero for any value of t. Therefore, there is no solution to dp/dt = 0, which means that p(t) does not have a maximum or minimum value.
This implies that the proportion p(t) is continuously increasing or decreasing over time. Therefore, there is no specific time at which the proportion will be increasing most rapidly.
As for the proportion at that time, since there is no maximum value, we cannot determine a specific proportion at any given time. The proportion will continue to change over time according to the formula p(t) = (Ce^kt) / (1 + Ce^kt).