Mathematics for Economics — Dynamic Optimization, Behavioral Economics Math, and Networks

Dynamic Optimization

Dynamic programming:

developed by Richard Bellman
breaks multi-period decision problems into recursive steps
the Bellman equation: V(x) = max_u { F(x, u) + β·V(f(x, u)) } V(x): the value function at state x β: the discount factor (0 < β < 1) F(x, u): the current-period payoff f(x, u): the state-transition function
the optimal policy function: u* = arg max { F + β·V }
applications: capital accumulation, consumption-savings decisions, resource depletion problems

Continuous-time optimal control:

Pontryagin’s Maximum Principle
state variable x(t) and control variable u(t)
the Hamiltonian: H(x, u, λ) = F(x, u) + λ·f(x, u) λ: the costate variable (shadow price — the marginal value of the state variable)
optimality conditions: ∂H/∂u = 0 (optimizing the control variable) λ̇ = -∂H/∂x (the costate equation of motion) ẋ = ∂H/∂λ (the state equation)

The Euler equation:

the optimal condition for consumption-savings decisions
basic form: u’(C_t) = β(1+r)·u’(C_{t+1}) u’(·): marginal utility / β: the discount rate for time preference / r: the interest rate
intuition: the marginal utility of consuming today equals the present value of the return from saving until tomorrow
constant relative risk aversion utility (CRRA): u(C) = C^(1-σ) / (1-σ) the Euler equation becomes: ΔC/C = (1/σ)(r - ρ) σ: the coefficient of relative risk aversion / ρ: the rate of time preference

The Ramsey-Cass-Koopmans model:

the standard dynamic macroeconomic model
households: an infinitely lived utility-maximizing agent
the balanced growth path: the long-run equilibrium level of capital and consumption
the saddle path: pins down the initial level of consumption if C₀ is too high, capital gets depleted; if too low, capital is over-accumulated

Stochastic Processes and Financial Mathematics

Stochastic processes:

a sequence of random variables that changes over time
the Markov property: the current state alone determines future states (the past doesn’t matter) P(X_{t+1} | X_t, X_{t-1}, …) = P(X_{t+1} | X_t)

Markov chains:

a Markov process with discrete states and discrete time
transition matrix P: P_ij = P(X_{t+1} = j | X_t = i)
economic applications: business cycles: the transition probability between boom and recession credit-rating transition matrices employment-status transitions (employed, unemployed, not in the labor force)
stationary distribution: π·P = π

Brownian motion:

the Wiener process W_t: W_0 = 0 / independent increments / W_t - W_s ~ N(0, t-s) continuous paths, non-differentiable almost everywhere
the Itô correction term: (dW_t)² = dt (differs from ordinary calculus)
the Itô integral and Itô’s Lemma: dF = (∂F/∂t + μ∂F/∂x + ½σ²∂²F/∂x²)dt + σ∂F/∂x·dW_t

The Black-Scholes-Merton equation:

the stock-price model: dS = μS·dt + σS·dW_t (geometric Brownian motion)
the price of a European call option: C = S·N(d₁) - K·e^(-rT)·N(d₂) d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T) d₂ = d₁ - σ√T N(·): the standard normal CDF
the option Greeks: Delta, Gamma, Theta, Vega, Rho
limitations: assumes constant volatility and doesn’t account for price jumps

Vector autoregression (VAR):

captures interdependence between multiple time series
the impulse response function (IRF): the dynamics of variables following a shock
variance decomposition: each variable’s contribution to forecast error
used to analyze macroeconomic shocks and the effects of monetary policy

The Mathematics of Behavioral Economics

Comparing theories of expected utility:

standard expected utility theory (Von Neumann-Morgenstern): EU = Σ p_i · u(x_i) risk aversion: u”>0 (a concave function)
the Allais Paradox: experimental evidence that violates EU theory the certainty effect: overweighting outcomes that are certain

Prospect theory (Kahneman & Tversky):

the value function v(x): gains: concave (risk averse) losses: convex (risk seeking) loss aversion coefficient: λ ≈ 2.25 (losses hurt roughly 2.25 times more than equivalent gains feel good) evaluated relative to a reference point
the probability weighting function w(p): overweighting small probabilities: w(0.01) > 0.01 underweighting large probabilities: w(0.99) < 0.99 an inverse-S-shaped curve

Hyperbolic discounting:

standard exponential discounting: δ = 1/(1+r) (time-consistent)
hyperbolic discounting: D(t) = 1/(1+kt) the near-future discount rate exceeds the far-future discount rate present bias: a reversal between choosing now vs. later
the quasi-hyperbolic (β-δ) model: U = u₀ + β·Σ δ^t·u_t (β < 1: present bias)
applications: self-control problems around pensions, quitting smoking, and dieting

The mathematics behind nudges:

choice architecture: modeling the default effect default x₀, switching cost c probability of switching p = f(v(x₁) - v(x₀) - c)
libertarian paternalism: preserving freedom of choice while steering toward better choices automatic enrollment defaults for retirement savings
informational nudges: providing social-norm comparison information “92% of your neighbors are already saving energy”

Network Economics

Network theory basics:

a graph G = (V, E): a set of nodes V and a set of edges E
degree: the number of edges connected to a node
path length, clustering coefficient, and centrality measures
scale-free networks: degree distribution follows a power law: P(k) ~ k^(-γ) hub nodes exist: the internet, social networks the Barabási-Albert model: preferential attachment
the small-world phenomenon: high clustering plus short average path length the Watts-Strogatz model

Network externalities:

more users → higher utility for other users
direct externalities: phones, messaging apps (benefits from direct connections)
indirect externalities: platforms (benefits from a growing set of complements)
critical mass: the tipping point beyond a minimum number of users, growth becomes self-sustaining the maximization condition: u_i = v(n) - p ≥ 0

Platform economics:

two-sided markets: a platform connecting two distinct user groups pricing structure: one side free, the other side paying, is common cross-network externalities: growth on one side raises value on the other
the Rochet-Tirole model: the optimal pricing structure: P_i = c_i - (n_j · ∂v_j/∂n_i) / (∂v_i/∂n_i)
the tendency toward monopolization: winner-takes-all dynamics the market dominance of large tech platforms

Information economics and search theory:

moral hazard and adverse selection: insurance market failures
the Diamond search model: labor-market search costs and matching efficiency the search equilibrium between job seekers and firms
matching theory: the Gale-Shapley algorithm: stable matching the 2012 Nobel Prize in Economics (Roth and Shapley): applications to real markets school assignment, organ donation, and residency matching

Frequently Asked Questions

Q. How can the dynamic optimization in mathematical economics apply to real-life consumption-savings decisions? A. The Euler equation is the core of the optimal consumption-savings decision. Intuitively, it works like this: “If you cut consumption by one dollar today and save it, it becomes (1+r) dollars next year with interest. If the utility from using that money to boost next year’s consumption exceeds the utility of spending that dollar today, you should save — otherwise, spending today is optimal.” In the Euler equation u’(C_t) = β(1+r)·u’(C_{t+1}), if β(1+r) = 1, consumption stays smooth (consumption smoothing). If β(1+r) > 1, it’s optimal to increase future consumption (save more); if it’s less than 1, it’s optimal to increase consumption today. In real life, the hyperbolic discounting problem (present bias) interferes with this optimal plan. The pattern of making “just this one exception” over and over — in retirement savings, dieting, or study plans — comes directly from present bias, where β < 1. The fix is precommitment devices, like automatic transfers or automatic retirement-plan enrollment, that automate saving on behalf of your future self.

Q. Can the Black-Scholes equation accurately predict real option prices? A. The Black-Scholes equation is arguably the most influential formula in financial history, but it has important limitations in real markets. Its assumptions don’t quite match reality. It assumes constant volatility, but volatility itself actually fluctuates (the volatility smile/skew). It assumes stock prices move continuously, but in reality, sudden jumps and crashes happen. It assumes no transaction costs or taxes, which also doesn’t match reality. That’s why practitioners rely on the concept of implied volatility: plugging an observed market option price back into the Black-Scholes formula and solving for σ yields a different σ for each maturity and strike price (the volatility smile). Modern derivatives trading supplements Black-Scholes with stochastic volatility models (like the Heston model), jump-diffusion models, and local volatility models. Even so, Black-Scholes remains an essential tool for computing hedge ratios, managing risk, and benchmarking market volatility (as in the VIX calculation).