Mathematical Methods in Economics

Consumer theory is the mathematical foundation of microeconomics. It explains how a rational agent chooses a bundle of goods, maximizing utility under a budget constraint. This theory provides tools for analyzing changes in demand, evaluating the effects of government policy, and welfare.

The consumer chooses a vector of goods x = (x₁,...,xₗ) ∈ ℝᴸ₊, maximizing utility u(x) under a budget constraint. Formally:

Here p = (p₁,...,pₗ) is the price vector, m is monetary income. The constraint p·x ≤ m is a "budget hyperplane" in ℝᴸ.

Marshallian demand: xˡ(p, m) — solution to the consumer's problem. Properties: (1) homogeneous of degree zero: x(tp, tm) = x(p, m) — no "money illusion"; (2) Walras' identity: p·x(p,m) = m (everything is spent).

Equilibrium in a single market is only part of the picture. A change in the price of one good sends waves through all markets. General equilibrium (GE) is a theory describing the simultaneous equilibrium in all markets of the economy. Two fundamental welfare theorems link competitive equilibrium ...

Exchange economy: $I$ consumers, $L$ goods. Consumer $i$ has an initial endowment $\omega_i \in \mathbb{R}^L_+$ and preferences $\succeq_i$.

Competitive equilibrium (Walrasian): price vector $p^*$ and allocation $(x_1^*,...,x_I^*)$ such that: 1. Optimality: $x_i^* \succeq_i x_i$ for all admissible $x_i$ with $p^* \cdot x_i \leq p^* \cdot \omega_i$ 2. Markets clear: $\sum_i x_i^* = \sum_i \omega_i$ (demand = supply for all goods)

Walras' law: $\sum_l p_l(\sum_i x_{il} - \omega_{il}) = 0$ — the total excess demand in value is always $= 0$. Corollary: if $L-1$ markets are in equilibrium, the $L$th is also in equilibrium. Prices are defined only up to a scalar (normalization: $\sum_l p_l = 1$ or $p_1 = 1$).

Game theory is the mathematical language of strategic interactions. When each agent’s outcome depends on the actions of others, classical optimization is insufficient. Game theory, developed by Nash, Aumann, and Shapley (all Nobel laureates), has become a universal tool in economics, political sc...

Nash Equilibrium (1950): a strategy profile (s₁*,...,sₙ*) such that no player can improve their payoff by unilaterally changing their strategy:

Explanation: s_{-i}* = strategies of all players except i. Nash equilibrium is a "rest point": no one wants to deviate.

Nash’s Theorem (1950): Every finite normal form game has an equilibrium in mixed strategies.

Most real markets are not perfectly competitive. Monopoly (one seller), duopoly/oligopoly (a few major players), monopolistic competition (many differentiated sellers) are the main market structures, each with unique pricing strategies.

Monopolist's problem: max_q π = p(q)·q − c(q), where p(q) is the inverse demand function.

Necessary condition: MR = MC, where MR = ∂(p·q)/∂q = p + q·∂p/∂q = p(1 + 1/εₚ). Here, εₚ = ∂q/∂p · p/q < 0 is the price elasticity of demand.

Breakdown: L = 0 → competitive price (p = MC). L → 1 → maximum monopoly power (with vertical demand). Optimal condition: price is set above MC proportionally to the inverse of elasticity.

Production theory studies the technological capabilities of firms, the optimal choice of production factors, and efficiency analysis. Key tools are production functions, cost functions, and quantitative methods for efficiency assessment (DEA, SFA). These tools are used by regulators, banks, and c...

The production function f: ℝᴸ₊ → ℝ₊ maps vectors of factors (xₗ) to maximum output. Key characteristics: returns to scale and elasticity of substitution.

Returns to scale: f(tx) = tˢ f(x) — increasing (s > 1, IRS), constant (s = 1, CRS), decreasing (s < 1, DRS).

Cobb-Douglas: y = AK^α L^β. With competitive markets (p·MPₖ = wₖ): capital share in revenue = α = wₖK/pY, labor share = β. Elasticity of substitution between K and L: σ = 1 (unitary — at any factor prices, their shares in expenses are constant). α + β > 1: IRS (e.g., a large plant). α + β = 1: CR...

Observed consumer demand must be consistent with rational utility maximization. The theory of integrability provides the answer: when is this satisfied? Consumer surplus is a key tool for welfare analysis of price and policy changes. Accurate welfare measurement is critical for evaluating tax ref...

Slutsky equation: Marshallian demand $x_l(p,m)$ is related to Hicksian demand $h_l(p,u)$:

$ \frac{\partial x_l}{\partial p_k} = \frac{\partial h_l}{\partial p_k} - x_k \cdot \frac{\partial x_l}{\partial m} $

Decryption of the three components: the left side is the observed change in Marshallian demand as $p_k$ rises. The first term is the "substitution effect" (the Hicksian change at fixed utility, only the price effect). The second is the "income effect": an increase in $p_k$ reduces real income, wh...

Formulas

Dynamic programming (DP) is a method for solving intertemporal optimization problems based on Bellman's principle of optimality: the optimal strategy on any subpath of an optimal trajectory is itself optimal. In economics, DP is used for the analysis of growth, savings, firm management, and asset...

$ \max_{c(t)} \int_0^\infty e^{-\rho t} u(c(t))\,dt \quad \text{subject to} \quad \dot{k} = f(k) - c - \delta k, \quad k(0)=k_0, \quad c \geq 0 $

Pontryagin's Maximum Principle: Hamiltonian $H = u(c) + \mu[f(k) - c - \delta k]$. Optimality conditions: $\partial H/\partial c = 0 \to u'(c) = \mu$ (marginal utility of consumption = price of capital). $\dot{\mu} = \rho\mu - \partial H/\partial k \to \dot{\mu} = (\rho + \delta - f'(k))\mu$.

Here $\sigma(c) = -c u''(c)/u'(c)$ is the intertemporal elasticity of substitution (IES). When $u = \ln c$: $\sigma = 1$, so $\dot{c}/c = f'(k) - \delta - \rho$. Intuition: consumption grows if the return on capital $f'(k)$ exceeds $\delta + \rho$ (returns exceed depreciation plus impatience).

Models of overlapping generations (OLG) make it possible to analyze long-term issues: capital accumulation, pension systems, government debt. In contrast to the representative agent model with an infinite horizon, OLG allows for dynamically inefficient equilibria and Pareto improvements through r...

Structure: Each period, a new generation is born with measure 1. The population growth rate is $n$: generation $t$ has measure $(1+n)^t$. An agent lives for 2 periods: “young” ($t$) receives wage $w_t$, makes savings $s_t$; “old” ($t+1$) consumes $(1+r_{t+1})s_t$.

$ \max_{c_{1t},\ c_{2,t+1}} u(c_{1t},\ c_{2,t+1}) \quad\text{subject to}\quad c_{1t} + s_t = w_t,\quad c_{2,t+1} = (1+r_{t+1})s_t $

Euler condition: $u'_1(c_1) = (1+r_{t+1})\beta\cdot u'_2(c_2)$. The savings function: $s_t = s(w_t, r_{t+1})$ — depends on wage and return.

Formulas

Asset pricing theory unites the microeconomics of optimal intertemporal choice with the mathematics of martingales. The fundamental theorem connects the absence of arbitrage to the existence of a measure under which discounted prices are martingales. This is the foundation of all modern financial...

Formulation: n risky assets, expected returns μ ∈ ℝⁿ, covariance matrix Σ. The problem is to minimize risk for a given expected return:

Here, w is the portfolio weights vector. The KKT conditions yield: w* = λΣ⁻¹μ + γΣ⁻¹1, where λ, γ are Lagrange multipliers. The efficient frontier is a parabola in the (σₚ, μₚ) space, σₚ = √(wᵀΣw).

Two Fund Theorem: All efficient portfolios are linear combinations of any two efficient portfolios. In practice: construct the frontier knowing only two points.

Asymmetric information refers to situations where one participant knows more than another. Akerlof, Spence, and Stiglitz (Nobel Prize 2001) systematized these situations, showing that they generate fundamental market failures. These models explain insurance market crises, banking panics, labor ma...

Used car model: N cars, share q of “lemons” (v_L = 1) and (1−q) of “peaches” (v_H = 2). Sellers know the quality, buyers do not. Sellers’ reservation prices: 1.5 (peach), 1 (lemon).

Adverse selection mechanism: Buyer offers E[v] = 2 − q·1 = 2 − q (average value). If q > 0.5: E[v] < 1.5 → peach sellers leave (better to keep). Now the market = only lemons: E[v] = 1. Buyer offers 1. The market functions, but only for lemons.

Complete collapse: with q > 0.5 the peach market disappears — “bad” goods drive out good ones (Gresham). DWL: all surplus from trading peaches (at least 0 to 0.5 per unit) is lost.

When employers cannot observe productivity, equilibria arise in which agents strategically convey information through "costly signals." The Spence model is a classic example of equilibrium signaling, demonstrating that education can be valuable not as human capital, but as a pure signal of ability.

Structure: Two types of workers: H (productivity θ_H) and L (θ_L < θ_H). The employer does not know the type. The worker chooses the level of education e. Payment = E[θ|e] (expected productivity given observable e). Education costs: c_H(e) = e/θ_H < c_L(e) = e/θ_L. The key condition: education is...

Separating equilibrium: H chooses e*, L chooses 0. Self-selection conditions: H is rational — θ_H − e*/θ_H ≥ θ_L (profit from signal ≥ zero-choice). L does not imitate — θ_L ≥ θ_H − e*/θ_L. Range: e* ∈ [θ_L(θ_H − θ_L), θ_H(θ_H − θ_L)].

Intuitive Criterion (Cho & Kreps, 1987): Eliminates "excess" equilibria. If e' > e*, only H can rationally deviate (L never benefits). The employer, upon observing e', should believe this is H → best response: pay θ_H → L does not benefit from imitation at e' → equilibrium with e* = θ_L(θ_H − θ_L...

Formulas

Theory of principal-agent studies how to structure contracts under asymmetric information about effort. Applications cover corporate governance (CEO–shareholders), insurance, regulation, government procurement, healthcare. Holmstrom and Milgrom (Nobel Prize 2016) laid the foundation of this theory.

Structure: The principal (risk-neutral, P-firm) hires an agent (risk-aversion r, Arrow-Pratt coefficient). Effort e ∈ [0,∞) is unobservable. Output: y = e + ε, ε ~ N(0, σ²). Linear contract: w(y) = α + β·y (fixed part α + variable β·y).

Agent's optimality condition (IC): differentiating with respect to e: e*(β) = β (when c(e) = e²/2).

Principal's problem: max_{α,β} E[y − w] = e*(β) − α − β·e*(β) = (1−β)·β − α. Substituting IR: max_β β − β²/2 − rβ²σ²/2 = β − (1/2 + rσ²/2)β².

Financial mathematics in continuous time is based on the stochastic calculus of Ito. The Black-Scholes-Merton formula (1973)—the first analytical result for derivative pricing—earned the 1997 Nobel Prize and transformed the financial industry.

Standard Brownian motion (Wiener process) W_t: Four properties: W₀ = 0, trajectories are continuous, increments are independent (W_t − W_s ⊥ W_s − W₀ for 0 ≤ s < t), increments are normally distributed: W_t − W_s ~ N(0, t−s).

Key relation: (dW_t)² = dt (in the sense of convergence in probability). This distinguishes stochastic calculus from deterministic: the "square of an infinitesimal" does not vanish.

Ito process: dX_t = μ(X_t, t)dt + σ(X_t, t)dW_t. μ — drift (deterministic component), σ — diffusion (random component). Examples: Geometric Brownian Motion (GBM): dS = μS dt + σS dW — stock price model. CIR process: dr = κ(θ−r)dt + σ√r dW — short rate model (always positive).

Modern portfolio theory (Markowitz, 1952) and CAPM (Sharpe, Lintner, Mossin, 1964–66) are two cornerstones of quantitative finance that created the passive investment industry. The Nobel Prizes awarded to Markowitz and Sharpe (1990) recognized their fundamental role.

Problem: min_w wᵀΣw subject to wᵀμ = μ₀, wᵀ1 = 1. n assets, expected returns μ, covariance matrix Σ. KKT conditions: w* = λΣ⁻¹μ + γΣ⁻¹1 — a parametric family. Efficient frontier: a parametric curve (σₚ(w*), μₚ(w*)) — a parabola in the (σ,μ)-space.

Two-Fund Theorem: All efficient portfolios are linear combinations of any two points on the efficient frontier. Practically: there is no need to enumerate all n assets — it suffices to use two base portfolios.

Tangency portfolio with risk-free r: T = Σ⁻¹(μ − r·1)/cᵀΣ⁻¹(μ − r·1) — maximizes the Sharpe Ratio SR = (μ_p − r)/σ_p. Capital Market Line (CML): μ_p = r + SR·σ_p.

Standard financial theory assumes a rational investor. Behavioral finance documents systematic deviations of real decisions from rationality and explains why irrationality persists—because arbitrage is limited. Kahneman and Tversky (Nobel Prize in Economics, 2002) laid the psychological foundatio...

EMH (Fama, 1970): Weak form: prices reflect all historical information. Semi-strong: all public data. Strong: all information, including insider.

Weak form test: Momentum (12-month) violates the weak EMH. Post-earnings announcement drift (PEAD): prices continue to move in the direction of the surprise 30–90 days after the quarterly report—violates the semi-strong EMH.

Excess volatility (Shiller, 1981): Var(P) ≫ Var(PV(D)) — stock prices are much more volatile than the present value of dividends. With rational expectations: σ_P ≤ σ_{PV(D)}. Data: violated by a factor of 5–10. “The most important econometric evidence in the history of finance” (Cochrane).