Random Helpful Tidbits:
- Economists' favorite shortcut: \(\ln(1+x) \approx x\)
- \(\ln(x^n) = n \cdot \ln (x)\)
- Solow problem: \(k(t) = [(Y(0) - \frac{s}{\delta}) e^{-\alpha \delta t} + \frac{s}{\delta}]^2\)
- Test-like question: max \(\int_0 ^T \frac{C_t ^{1 - \theta}}{1 - \theta} dt\) where \(\dot K = rK - c ; K(0) = \bar K\). We would have first period consumption divided by real interest rate minus growth rate, times the growth rate: \(K_0 = \frac{C_0}{r - g} (1 - e^{-(r - g)T})\) and consumption growing continuously at rate of g: \(C_t = C_0 e^{gt}\). To solve, \(H = \frac{C^{1 - \theta}}{1 - \theta} + \lambda (rK - c)\). \(\frac{\partial H}{\partial c} = 0 = c^{- \theta} - \lambda \Longrightarrow c^{- \theta} \). \(\dot \lambda - \theta c^{- \theta - 1} \cdot \dot c\). \(\frac{\partial H}{\partial K} = - \dot \lambda = \lambda r \Longrightarrow \frac{\dot \lambda}{\lambda} = r\). Hence, \(r = \frac{\dot \lambda}{\lambda} = \frac{- \theta c^{- \theta - 1} \cdot \dot c}{c^{- \theta}}\). \(\therefore \frac{\dot c}{c} = \frac{r}{\theta}\).
- \(\frac{MB}{MC} = r\) ?
- No worries.. Mathematical Economics is for Man, not Man for Mathematical Economics