Key Concepts in Microeconomic Theories

Why This Matters

Microeconomic theory forms the mathematical backbone of how economists model decision-making at the individual level. You're being tested on your ability to translate economic intuition into formal mathematical frameworks—optimization problems, equilibrium conditions, and comparative statics. These aren't just abstract concepts; they're the tools you'll use to analyze everything from consumer behavior to market failures, and they appear repeatedly in both theoretical proofs and applied problem sets.

The concepts in this guide connect through a unifying theme: constrained optimization. Whether a consumer maximizes utility subject to a budget constraint or a firm maximizes profit subject to technology constraints, the mathematical structure is remarkably similar. Don't just memorize formulas—understand why the first-order conditions look the way they do, and how different market structures change the optimization problem. That's what separates students who struggle from those who excel.

---

Consumer Decision-Making

The foundation of demand theory rests on modeling how rational agents choose among alternatives. These models assume consumers have well-defined preferences and face resource constraints, allowing us to derive testable predictions about behavior.

Consumer Theory

• Preference axioms—completeness, transitivity, and continuity allow us to represent preferences with a utility function $U(x_1, x_2, ..., x_n)$
• Budget constraint $p_1x_1 + p_2x_2 \leq m$ defines the feasible consumption set given prices and income
• Indifference curves show combinations yielding equal utility; their slope (MRS) equals the price ratio at optimum

Utility Maximization

• Lagrangian method solves $\max U(x) \text{ s.t. } p \cdot x = m$ yielding first-order conditions where $\frac{MU_i}{p_i} = \lambda$ for all goods
• Marginal utility $MU_i = \frac{\partial U}{\partial x_i}$ measures additional satisfaction from one more unit; diminishing MU explains downward-sloping demand
• Marshallian demand $x^*(p, m)$ emerges from solving the utility maximization problem and forms the basis for demand curve derivation

---

Producer Behavior and Costs

Firms face a parallel optimization problem: maximize output given inputs, or minimize costs given output targets. The mathematical duality between these approaches is a key insight tested in intermediate and advanced coursework.

Production Functions

• Functional form $Q = f(K, L)$ maps inputs (capital, labor) to output; common forms include Cobb-Douglas $Q = AK^\alpha L^\beta$ and CES
• Marginal product $MP_L = \frac{\partial Q}{\partial L}$ shows output gain from one additional unit of labor; diminishing returns occur when $\frac{\partial^2 Q}{\partial L^2} < 0$
• Returns to scale determined by $f(tK, tL)$ relative to $tf(K,L)$—constant, increasing, or decreasing scale properties affect long-run industry structure

Cost Functions

• Total cost $TC(Q) = FC + VC(Q)$ decomposes into fixed costs (independent of output) and variable costs (change with production level)
• Marginal cost $MC = \frac{dTC}{dQ}$ is the cost of producing one additional unit; intersects ATC and AVC at their minimum points
• Cost minimization via Lagrangian yields $\frac{MP_L}{w} = \frac{MP_K}{r}$, meaning firms equalize marginal product per dollar across all inputs

Profit Maximization

• Objective function $\pi = TR - TC = p \cdot Q - C(Q)$ for price-takers; first-order condition $p = MC$ determines optimal output
• Second-order condition requires $\frac{d^2\pi}{dQ^2} < 0$, which holds when MC is increasing at the optimal quantity
• Shutdown rule applies when $p < AVC$ in short run or $p < ATC$ in long run—negative contribution margin means ceasing production is optimal

---

Market Equilibrium and Price Determination

Markets aggregate individual decisions into collective outcomes. Equilibrium analysis examines when and how markets clear, and what happens when conditions change.

Market Equilibrium

• Equilibrium condition $Q^D(p^*) = Q^S(p^*)$ defines the price where quantity demanded equals quantity supplied; solve simultaneously for $p^*$ and $Q^*$
• Stability analysis examines whether markets return to equilibrium after shocks; typically assumes excess demand drives prices up, excess supply drives prices down
• Comparative statics uses implicit differentiation to determine how $\frac{dp^*}{d\theta}$ responds to parameter changes $\theta$ (income, costs, taxes)

Demand and Supply Analysis

• Demand shifters—income, preferences, prices of related goods, expectations—move the entire demand curve; mathematically, these are parameters in $Q^D(p; \theta)$
• Supply shifters—input prices, technology, number of firms—affect the supply curve position; changes in these parameters require re-solving for equilibrium
• Graphical analysis complements algebraic solutions; always check that your mathematical results match the intuition from shifting curves

Elasticity

• Price elasticity of demand $\varepsilon_d = \frac{\partial Q^D}{\partial p} \cdot \frac{p}{Q}$ measures percentage change in quantity per percentage change in price; $|\varepsilon_d| > 1$ is elastic
• Income elasticity $\varepsilon_m = \frac{\partial Q}{\partial m} \cdot \frac{m}{Q}$ classifies goods as normal ($\varepsilon_m > 0$) or inferior ($\varepsilon_m < 0$)
• Cross-price elasticity $\varepsilon_{xy} = \frac{\partial Q_x}{\partial p_y} \cdot \frac{p_y}{Q_x}$ identifies substitutes (positive) versus complements (negative)

---

Market Structures

Different competitive environments change the firm's optimization problem fundamentally. The key mathematical difference lies in whether the firm takes price as given or chooses it strategically.

Perfect Competition

• Price-taking behavior means firms face horizontal demand at market price $p$; profit maximization yields $p = MC$ as the supply condition
• Long-run equilibrium requires $p = MC = ATC_{min}$, implying zero economic profit and production at minimum efficient scale
• Allocative efficiency achieved because $p = MC$ ensures resources flow to their highest-valued uses; no deadweight loss exists

Monopoly

• Market power allows the monopolist to choose price; faces downward-sloping demand $p(Q)$ and maximizes $\pi = p(Q) \cdot Q - C(Q)$
• Marginal revenue $MR = p + Q\frac{dp}{dQ} = p\left(1 + \frac{1}{\varepsilon_d}\right)$ lies below demand curve; optimum occurs where $MR = MC$
• Deadweight loss arises because $p > MC$ at monopoly output; the welfare triangle $\frac{1}{2}(p_m - MC)(Q_c - Q_m)$ measures efficiency loss

---

Strategic Interaction and General Analysis

When agents' decisions affect each other, simple optimization gives way to strategic reasoning. Game theory provides the mathematical framework for analyzing these interdependencies.

Game Theory

• Normal form representation specifies players, strategies, and payoffs; Nash equilibrium occurs when no player can unilaterally improve their payoff
• Nash equilibrium formally: strategy profile $s^*$ where $u_i(s_i^*, s_{-i}^*) \geq u_i(s_i, s_{-i}^*)$ for all players $i$ and strategies $s_i$
• Applications include oligopoly models (Cournot, Bertrand), bargaining, and mechanism design; dominant strategy equilibria are stronger but rarer

General Equilibrium Theory

• Walrasian equilibrium requires prices $p^*$ such that all markets clear simultaneously: $\sum_i x_i^*(p^*, \omega_i) = \sum_i \omega_i$ for all goods
• Welfare theorems establish efficiency properties; First Theorem states competitive equilibria are Pareto efficient under standard assumptions
• Edgeworth box visualizes two-person, two-good exchange; contract curve shows all Pareto efficient allocations

---

Market Failures and Extensions

When standard assumptions fail, markets may not achieve efficient outcomes. Understanding these failures mathematically helps identify appropriate policy interventions.

Externalities and Public Goods

• Externalities occur when $MSC \neq MC$ (negative) or $MSB \neq MB$ (positive); Pigouvian taxes/subsidies set $t = MEC$ to internalize external effects
• Public goods satisfy non-excludability and non-rivalry; efficient provision requires $\sum_i MRS_i = MRT$, but private markets underprovide
• Coase theorem suggests bargaining can achieve efficiency regardless of initial property rights allocation, if transaction costs are zero

Risk and Uncertainty

• Expected utility $EU = \sum_s \pi_s U(x_s)$ weights utility across states by probabilities; risk aversion corresponds to concave utility ($U'' < 0$)
• Risk premium measures willingness to pay to eliminate risk; equals $E[x] - CE$ where $CE$ is the certainty equivalent satisfying $U(CE) = EU$
• Insurance markets emerge when risk-averse agents trade with less risk-averse parties; moral hazard and adverse selection create market imperfections

---

Quick Reference Table

---

Self-Check Questions

1. Both utility maximization and cost minimization use Lagrangian methods. What is the economic interpretation of the Lagrange multiplier $\lambda$ in each case, and why does this interpretation matter for policy analysis?

2. Compare the profit-maximizing conditions for a perfectly competitive firm versus a monopolist. Why does the monopolist's condition involve marginal revenue rather than price, and what does this imply for market efficiency?

3. If you observe that a 10% price increase leads to a 15% decrease in quantity demanded, what is the price elasticity of demand? Would a firm facing this elasticity want to raise or lower its price to increase revenue?

4. Explain how the First Welfare Theorem connects perfect competition to Pareto efficiency. What assumptions must hold, and which market failures violate these assumptions?

5. In a Cournot duopoly, each firm chooses quantity taking the other's quantity as given. How does the Nash equilibrium output compare to the perfectly competitive output and the monopoly output? What economic intuition explains this ordering?