Rules of Differentiation
- Constant function rule: \(f(x) = k ; f'(x) = 0\)
- Linear function rule: \(f(x) = mx + b ; f'(x) = m\)
- Power function rule: \(f(x) = kx^n ; f'(x) = nkx^{n-1}\)
- Sums and differences rule: \(f(x) = g(x) \pm h(x) ; f'(x) = g'(x) \pm h'(x)\)
- Product rule: \(f(x) = g(x) \cdot h(x) ; f'(x) = g(x) \cdot h'(x) + h(x) \cdot g'(x)\)
- Quotient rule: \(f(x) = \frac{g(x)}{h(x)} ; f'(x) = \frac{h(x) \cdot g'(x) - g(x) \cdot h'(x)}{h(x)^2}\)
- Generalized power rule: \(f(x) = [g(d)]^n ; f'(x) = n[g(x)]^{n-1} \cdot g'(x)\)
- Chain rule: \(y=f\left(g(x)\right);\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\)
- Natural exponential function rule: \(f(x) = e^{g(x)} ; f'(x) = e^{g(x)} \cdot g'(x)\)
- Natural log function rule: \(f(x)=\ln\left(g(x)\right);f'(x)=\frac{g'(x)}{g(x)}\)
Rules of Integration
- Constant rule: \(\int (a) dx = ax + c\)
- Variable rule: \(\int (x)dx = \frac{x^2}{2} + c\)
- Reciprocal rule: \(\int (\frac{1}{x}) dx = \ln|x| + c\)
- Exponential rules:
\(\int (e^x)dx= e^x + c\)
\(\int (a^x)dx = \frac{a^x}{\ln(a)} + c\)
- Natural log rule: \(\int \ln(x)dx= x \ln(x) - x +c\)
- Multiplication by a constant: \(\int cf(x) dx = c \int f(x) dx\)
- Power rule: \(\int (x^n) dx = \frac{x^{n+1}}{(n+1)} + c\)
- Sum/difference rule: \(\int (f \pm g)dx = \int (f)dx \pm \int (g) dx\)
- Partial: \(\int g(f(x))dx = g(x) \int f(x)dx - \int g'(\int f(x)dx)dx\)
Optimization of Functions
- First-order condition: Take first derivative, set it equal to zero, and solve for critical points. All such points are candidates for a possible relative maximum or minimum since the function is neither increasing nor decreasing (but at a plateau) at that point.
- Second-order condition: Take the second derivative, evaluate it at the critical points, and check the signs. If:
\(f''(a) < 0\), concave and at a relative maximum
\(f''(a) > 0 \), convex and at a relative minimum
Utility maximization
With Lagrangian:
First, what is the Lagrange multiplier? It gives a shadow price-- marginal utility of one more unit of income.
\(\mathcal{L} = f(x, y) + \lambda[m - g(x, y)]\)
Where \(f(x, y)\) is the function to be maximized/minimized.
\(m - g(x, y)\) has to be zero, because there is the budget restriction:
\(m = x P_x + y P_y \)
Then, take the partial derivatives set to zero:
\(\frac{\partial \mathcal{L}}{\partial x} = 0\)
\(\frac{\partial \mathcal{L}}{\partial y} = 0\)
\(\frac{\partial \mathcal{L}}{\partial \lambda} = 0\)
Use the results to cancel out \(\lambda\). Finally, substitute to solve the remaining 2 equations for the desired variable.
Without Lagrangian:
There are two conditions that must be satisfied for concave utility maximization,
\(MRS_{xy} = \frac{P_x}{P_y} \Longleftrightarrow \frac{MU_x}{P_x} = \frac{MU_y}{P_y}\) (the indifference curve)
\(x P_x + y P_y = m\) (the budget line)
The first condition guarantees that the indifference curve (or MRS, how your utility changes when you substitute one good for another) and budget line share the same slope. The second equation enables us to find the bundle on the highest indifference curve that is tangent to the budget line.
Marginal Concepts
The marginal concept of any economic function can be expressed as the derivative of its total function. For example:
If \(TC = TC(Q)\), then \(MC = \frac{dTC}{dQ}\)
If \(TR = TR(Q)\), then \(MR = \frac{dTR}{dQ}\)
Elasticities
\(E \approx \frac{P}{Q} \cdot \frac {dQ}{dP}\)
Financial Formulas
Compound interest: \(A = P(1+\frac {r}{n})^{nt}\)
Continual compounding: \(A = Pe^{rt}\)
Mortgage payment formula: \(P = L [\frac{c(1+c)^n}{[(1+c)^n - 1]}\)
Perpetuity (payment each period, forever): \(PV = \frac{c}{r}\) where c is payment amount and r is discount rate
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
*Economists' favorite shortcut: \(\ln(1+x) \approx x\)
Solow Model
\(Y(t)=F(K(t),L(t)A(t)\)