Uncertainty, Game Theory, Production, Costs, and Market Equilibrium for Microeconomics

Uncertainty

How individuals make decisions when outcomes are uncertain.

Game Theory

How individuals make strategic decisions when their payoffs depend on the actions of others.

Consumer Theory

Utility maximization, demand, income/substitution effects.

Producer Theory

Profit maximization, cost minimization, supply/demand.

Markets

Partial and general equilibrium.

Expected Utility

A model using random variables and probability distributions to evaluate uncertain outcomes.

Fair Gamble

Has an expected value of zero, such as flipping a coin for $1.

St. Petersburg Paradox

Illustrates that people are unwilling to pay much to play a game with an infinite expected monetary value.

Bernoulli's Resolution

Introduced expected utility to explain why individuals avoid fair gambles.

VNM Utility Function

Unique up to positive affine transformations, formalized by von Neumann and Morgenstern.

Risk Aversion

Most people avoid fair gambles due to diminishing marginal utility of wealth.

Concave Utility Function

Reflects risk aversion, where expected utility of a fair gamble is less than the utility of the expected value.

Certainty Equivalent (CE)

The amount of certain wealth that gives the same utility as the gamble.

Absolute Risk Aversion

Measured by r(W) = −U ′′(W) / U ′(W).

Relative Risk Aversion

Measured by R(W) = W · r(W) = −W U ′′(W) / U ′(W).

Managing Risk

Includes strategies such as insurance, diversification, flexibility, and information acquisition.

Game

Consists of a set of players, action sets, strategy sets, payoff functions, information structure, and common knowledge of the game rules.

Simultaneous Moves

Players act at the same time.

Sequential Moves

Players act in a sequence.

Imperfect Information

Players lack full knowledge.

Asymmetric/Incomplete Information

One player knows more than others.

Action/Strategy Sets

Can be discrete or continuous.

Repetition

One-shot or repeated.

Simultaneous-Move Games

Represented in normal form (payoff matrix).

Nash Equilibrium (NE)

A strategy profile (s∗1, . . . , s∗n) where each player's strategy is a best response to the others.

Sequential-Move Games

Represented in extensive form (game tree).

Subgame Perfect Nash Equilibrium (SPNE)

A strategy profile that is a NE in every subgame.

Prisoner's Dilemma (Normal Form)

Two suspects choose to Fink or Silent.

Payoffs in Prisoner's Dilemma

(Fink, Fink): (1, 1), (Fink, Silent): (3, 0), (Silent, Fink): (0, 3), (Silent, Silent): (2, 2).

NE in Prisoner's Dilemma

(Fink, Fink) — both players fink, even though (Silent, Silent) is better for both.

Battle of the Sexes (Extensive Form)

Sequential version: One player moves first, the other responds.

Backward induction

A method used to solve sequential games.

Production Function

Describes the maximum output q achievable from inputs: q = f(k, l).

Marginal Product (MP)

The additional output from one more unit of an input, holding others constant.

Diminishing Marginal Product

As more of an input is used, its MP eventually decreases.

Average Product (AP)

Output per unit of input: APl = q/l = f(k, l)/l.

Isoquant

Shows all combinations of k and l that produce the same output q0: f(k, l) = q0.

Rate of Technical Substitution (RTS)

The slope of the isoquant: RTS(l for k) = -dk/dl = MPk/MPl.

Returns to Scale

How output changes when all inputs are scaled by the same factor t > 1.

Constant Returns to Scale (CRTS)

f(tk, tl) = tf(k, l).

Increasing Returns to Scale (IRTS)

f(tk, tl) > tf(k, l).

Decreasing Returns to Scale (DRTS)

f(tk, tl) < tf(k, l).

Cobb-Douglas Function

q = kαlβ is CRTS if α + β = 1.

Elasticity of Substitution

Measures how easily the firm can substitute between inputs: σ = %∆(k/l)

Perfect substitutes

σ = ∞: Linear isoquants

Fixed proportions

σ = 0: No substitution

Cobb-Douglas case

σ = 1

Technical Progress

Shifts the production function upward: q = A(t)f(k, l), dA/dt > 0

Growth Accounting Equation

Gq = GA + eq,kGk + eq,lGl where Gx = d ln x/dt and eq,x is the elasticity of output with respect to input x.

Accounting costs

Out-of-pocket expenses, historical costs, depreciation.

Economic costs

Opportunity cost — the value of the next best use.

Total cost

C = wl + vk

Cost Minimization

min l,k wl + vk s.t. f(k, l) = q0

Lagrangian

L = wl + vk + λ[q0 −f(k, l)]

First-order conditions

∂L/∂l = w −λfl = 0, ∂L/∂k = v −λfk = 0, ∂L/∂λ = q0 −f(k, l) = 0

Marginal cost of production

λ is the marginal cost of production.

Contingent Input Demands

The input demands derived from cost minimization are contingent on the output level: lc = lc(w, v, q), kc = kc(w, v, q)

Cobb-Douglas Production Function

Let q = kαlβ.

Total Cost Function

C(w, v, q) = wlc + vkc

Average Cost

AC = C/q

Marginal Cost

MC = ∂C/∂q

Short Run

At least one input is fixed (e.g., k = ¯k)

Long Run

All inputs are variable.

Short-Run Costs

SC = v¯k + wl, where l depends on q and ¯k

Shephard's Lemma

∂C/∂w = lc, ∂C/∂v = kc

Shifts in Cost Curves

Input price changes shift cost curves.

Total Revenue

R(q) = p(q) · q

Profit

π(q) = R(q) − C(q)

First-Order Condition

dπ/dq = 0 ⇒ dR/dq = dC/dq ⇒ MR = MC

Marginal Revenue

MR = dR/dq = p(q) + q · dp/dq

Perfect Competition

dp/dq = 0 ⇒ MR = p

Monopoly

dp/dq < 0 ⇒ MR < p

Short-Run Supply for Price-Taking Firms

For price-taking firms: P = MR

Profit Maximization in Short-Run

P = SMC (Short-Run Marginal Cost)

Short-Run Supply Curve

Positively sloped segment of SMC above minimum SAVC

Short-Run Cost

SC = vk1 + wq1/βk−α/β

Short-Run Marginal Cost

SMC = wβ q(1−β)/βk−α/β

Profit Maximization Equation

Set P = SMC, q = (w^(-β/(1−β)) * k^(α/(1−β)) * P^(β/(1−β)))^β

Properties of Supply Function

Increasing in P, shifts left with w ↑, shifts right with k1 ↑, unaffected by v (fixed cost)

Marginal Revenue Product

MRP = P · MP

Input Demand Functions

k(P, v, w), l(P, v, w)

Substitution Effect

Change input mix for given output

Output Effect

Change in optimal output level

Properties of Profit Function

Homogeneous of degree 1 in all prices, nondecreasing in P, nonincreasing in v, w, convex in P

Market Demand

X = Σ xi(px, py, Ii)

Price Elasticity

eD,P = ∂D/∂P · P/Q

Cross-Price Elasticity

eD,P' = ∂D/∂P' · P'/Q

Income Elasticity

eD,I = ∂D/∂I · I/Q

Classification of Elasticities

Elastic: eD,P < −1, Inelastic: 0 > eD,P > −1, Unit elastic: eD,P = −1

Time Periods in Supply Response

Very Short Run (VSR): Quantity fixed, no supply response; Short Run (SR): Existing firms adjust quantity, no entry/exit; Long Run (LR): New firms can enter/exit, flexible supply response

Perfect Competition Assumptions

1. Large number of firms producing homogeneous product; 2. Each firm maximizes profits; 3. Price-taking behavior; 4. Perfect information

Short-Run Market Supply

Horizontal summation of individual firm supply curves

Equilibrium Price Determination

Intersection of market demand and supply curves

Comparative Statics

Analyze how equilibrium changes when underlying conditions change

Effects of Elasticity on Comparative Statics

Elastic demand: Price changes slightly, quantity changes significantly

Long-Run Competitive Equilibrium

Firms produce where P = LRMC

Long-Run Supply Curves

Constant Cost Industry: Horizontal LS curve

Consumer Surplus

Area below demand curve and above price

Producer Surplus

Area above supply curve and below price

Total Surplus

Sum of consumer and producer surplus