Economic Applications

Elasticity & Production

Price elasticity of demand measures responsiveness: $\varepsilon = \frac{\%\Delta Q}{\%\Delta P} = \frac{dQ}{dP} \cdot \frac{P}{Q}$. It is unitless. If $|\varepsilon| > 1$, demand is elastic and revenue falls if price rises. If $|\varepsilon| < 1$, it is inelastic and revenue rises. Revenue is maximised at $|\varepsilon| = 1$.

Cobb-Douglas production models output as $Q = AK^\alpha L^{1-\alpha}$. Here, $\alpha$ is capital's share of output. Since exponents sum to 1, it exhibits constant returns to scale: doubling inputs doubles output. Marginal products are $MP_K = \alpha Q/K$ and $MP_L = (1-\alpha) Q/L$, always proportional to average products.

Input-Output & Envelope Theorem

Input-output analysis (Leontief) relates total output $\mathbf{x}$, inter-industry matrix $A$, and final demand $\mathbf{d}$ via $\mathbf{x} = A\mathbf{x} + \mathbf{d}$. Solving gives the Leontief inverse: $\mathbf{x} = (I - A)^{-1}\mathbf{d}$, which accounts for all supply chain ripples.

The envelope theorem states that in an optimised system, a parameter's effect on the optimised value $V^*$ equals the direct effect only: $\frac{dV^*}{d\theta} = \frac{\partial \mathcal{L}}{\partial \theta}\Big|_\text{opt}$. Indirect effects through variables vanish at the optimum.