Economic Applications

Mathematics in Service of Economics

This lesson synthesises the mathematical tools of Units 4 and 9 into their most important economic applications.

Elasticity — the economist's favourite derivative. The price elasticity of demand is:

$$\varepsilon = \frac{\%\Delta Q}{\%\Delta P} = \frac{dQ}{dP} \cdot \frac{P}{Q}$$

Elasticity is a unitless measure of responsiveness: $|\varepsilon| > 1$ means demand is elastic (revenue falls if price rises); $|\varepsilon| < 1$ means inelastic (revenue rises). At $|\varepsilon| = 1$, revenue is maximised.

Cobb-Douglas production — the workhorse of economic modelling:

$$Q = AK^\alpha L^{1-\alpha}$$

• $\alpha$ is capital's share of output (and income, under perfect competition).
• Returns to scale: the exponents sum to 1, so the function exhibits constant returns to scale — doubling all inputs doubles output.
• Marginal products: $MP_K = \alpha Q/K$, $MP_L = (1-\alpha) Q/L$ — proportional to the average product, a defining feature of Cobb-Douglas.

Input-output analysis (Leontief): the economy's total output $\mathbf{x}$ satisfies:

$$\mathbf{x} = A\mathbf{x} + \mathbf{d}$$

where $A$ is the matrix of inter-industry requirements and $\mathbf{d}$ is final demand. Solving:

$$(I - A)\mathbf{x} = \mathbf{d} \qquad \Rightarrow \qquad \mathbf{x} = (I - A)^{-1}\mathbf{d}$$

The Leontief inverse $(I - A)^{-1}$ gives the total output multiplier — including all indirect requirements rippling through the supply chain.

The envelope theorem: in an optimised system, the effect of a parameter change on the optimised value equals the direct effect only — you can ignore the indirect effect through the optimised variables:

$$\frac{dV^*}{d\theta} = \frac{\partial \mathcal{L}}{\partial \theta}\bigg|_{\text{optimum}}$$

This is why $\lambda$ (the Lagrange multiplier) is the shadow price: it is the direct effect of relaxing the constraint on the optimised objective.

The synthesis toolkit

Tip: The envelope theorem is the great labour-saver: at an optimum, a small parameter change affects the optimised value only directly — the induced re-optimisation contributes nothing to first order. That is exactly why $\lambda$ prices the constraint without further calculation.