Optimal sale time (where price of good grows over time): given \(P_t = P_0 e^{zt}\) where z is declining rate of growth over time and t is time, best time to sell is when \(\frac{\dot p}{p} = r \rightarrow t= \frac{\alpha}{r}\). Economically, this will mean that the larger the \(\alpha\), the longer you wait to sell.
Continuous-time annuity (sum paid out for a period or forever): \(FV = c \cdot [ \frac{(1 + i) ^n - 1}{i}]\) where c is the payment, i is the interest rate, and n is number of payments. For continuous time, the bracketed expression becomes: \(\frac{e^{rt} - 1}{e^r - 1}\) . And \(PV = \frac{365}{r}(1 - e^{-rt})\) .
First-Order Linear Differential Equations with Constant Coefficient and Constant Term
\(y=w(t)\frac{dy}{dt}+u(t)\)
The Homogeneous Case: when u and w are constant functions and w = 0
\(\frac{dy}{dt} y +ay =0\)
General solution (arbitrary constant A): \(y(t)= Ae^{-at} y(t)\)
Definite solution (define A): \(y(t)= y(0) e^{-at} y(t)\)
The Nonhomogeneous Case:
\(\frac{dy}{dt} +ay =b \)
Complementary function: \(y_c = Ae^{-at}\)
Particular integral: \(y_p = \frac{b}{a}\)
\(y(t) = y_c + y_p = Ae^{-at} + \frac{b}{a}\)
Then, if we have the initial condition:
\(y(t) = [y(0) - \frac{b}{a}] e^{-at} + \frac{b}{a}\)
*If we have something like \(\dot y + 0y = 5\), then the rate of change of y doesn't depend on the initial condition, just multiply it by the constant and add: \(y(t) = 5t + 100\)
Basic Firm Profit Maximization & Derived Demand