Panhellenic Economics & Informatics cheat sheet

Consumer optimum: MRS (x1,x2) = p1/p2; for Cobb-Douglas U = x1^a x2^b, expenditure shares are a/(a+b) and b/(a+b).

Slutsky decomposition: total effect = substitution effect + income effect; the Hicksian (compensated) demand slope is always non-positive.

Perfect competition short-run shutdown rule: produce if p ≥ AVC; long-run exit if p < min ATC, so long-run equilibrium has p = min ATC.

Monopoly profit maximization: MR = MC, where MR = p(1 + 1/ε); markup rises as demand becomes less elastic (|ε| → 1).

Lerner index: (p − MC)/p = 1/|ε|; it equals 0 under perfect competition and 1/|ε| under monopoly pricing.

Elasticity: ε = (dQ/dp)(p/Q); revenue rises with price only when |ε| < 1, and is maximized where |ε| = 1.

Cost relations: MC crosses AVC and ATC at their minima; ATC = AFC + AVC; AFC falls continuously with output.

Cournot duopoly with linear demand p = a − b(q1+q2) and equal marginal cost c: qi = (a − c)/(3b), p = (a + 2c)/3.

Pareto efficiency in exchange requires MRS equal across consumers; in production, MRTS equal across firms.

National accounts identity: Y = C + I + G + NX; disposable income Yd = Y − T + TR, and C = c0 + c1·Yd.

Simple Keynesian multiplier: 1/(1 − c1(1 − t)) with proportional tax rate t; with only autonomous consumption, multiplier = 1/(1 − c1).

IS relation: Y = C(Y − T) + I(Y, i) + G; LM relation: M/P = Y·L(i); the central bank sets i in the modern policy-rate version.

Fisher equation: i ≈ r + π^e; ex post real rate r = i − π. Money neutrality holds only in the long run.

Quantity theory: MV = PY; with constant V, inflation ≈ money growth − real output growth.

Solow steady state: sf(k) = (n + g + δ)k; steady-state output per effective worker rises with s and falls with n, g, δ.

Unemployment: u = U/(U + E); Okun-type rule links output gaps to cyclical unemployment; natural rate is frictional plus structural.

Open economy: uncovered interest parity i ≈ i* + expected depreciation; real exchange rate ε = EP*/P; net exports fall as ε rises.

Phillips curve: π = π^e − β(u − un) + v; in the long run, expectations adjust so u returns to un.

OLS slope: β̂ = Cov(X,Y)/Var(X); intercept α̂ = Ȳ − β̂X̄; the line passes through (X̄, Ȳ).

Gauss-Markov: under linearity, random sampling, no perfect collinearity, zero conditional mean, and homoskedasticity, OLS is BLUE.

Standard error of β̂ is larger when residual variance is larger, X variation is smaller, and n is smaller.

t-test: t = (β̂ − β0)/SE(β̂); reject H0 at 5% two-sided if |t| > 1.96 for large samples.

95% confidence interval: β̂ ± 1.96·SE(β̂); interval widens with higher confidence level and smaller n.

R² = 1 − SSR/SST; adding regressors never lowers R², so use adjusted R² = 1 − (1−R²)(n−1)/(n−k−1).

Omitted variable bias: bias direction equals sign of the omitted variable's effect on Y times its correlation with the included regressor.

Heteroskedasticity leaves OLS unbiased but invalidates usual standard errors; use robust (White) standard errors.

Classical measurement error in X attenuates the OLS slope toward zero (attenuation bias).

Two's complement: n-bit range −2^(n−1) to 2^(n−1) − 1; negation is bitwise NOT plus 1; overflow wraps modulo 2^n.

Floating point (IEEE 754): value = (−1)^s × 1.mantissa × 2^(exponent−bias); double has 64 bits, 52-bit mantissa, bias 127.

Boolean algebra: De Morgan's laws ¬(A∧B) = ¬A∨¬B and ¬(A∨B) = ¬A∧¬B; XOR is 1 when inputs differ.

Combinational vs sequential circuits: sequential circuits add state via flip-flops and are clocked; combinational output depends only on current inputs.

Amdahl's law: speedup = 1/((1 − p) + p/s) where p is the parallel fraction and s the parallel speedup.

Cache locality: temporal locality reuses recently accessed data; spatial locality reuses nearby addresses; hit rate = hits/accesses.

Process vs thread: processes have separate address spaces; threads of a process share heap and code but have their own stacks and registers.

TCP guarantees ordered, reliable delivery via sequence numbers and ACKs; UDP is connectionless and does not retransmit.

Asymptotic order: O(1) < O(log n) < O(n) < O(n log n) < O(n²) < O(2^n); drop constants and lower-order terms.

Binary search requires a sorted array and runs in O(log n); each step halves the remaining interval.

Merge sort is O(n log n) worst case and stable; quicksort averages O(n log n) but is O(n²) worst case with poor pivots.

Master theorem for T(n) = aT(n/b) + f(n): compare f(n) with n^(log_b a) to pick the case.

Dynamic programming applies when subproblems overlap and have optimal substructure; memoization is top-down, tabulation bottom-up.

Dijkstra's algorithm finds single-source shortest paths with non-negative weights in O((V+E) log V) using a binary heap.

Greedy algorithms need the greedy-choice property and optimal substructure; interval scheduling by earliest finish time is optimal.

Hash tables give average O(1) lookup but degrade to O(n) under heavy collisions; load factor triggers resizing.

Relational model: a relation is a set of tuples over a schema; a candidate key is a minimal superkey, and one candidate key is chosen as primary key.

SQL logical clause order: FROM → WHERE → GROUP BY → HAVING → SELECT → ORDER BY; WHERE filters rows before aggregation, HAVING filters groups after.

ACID: atomicity, consistency, isolation, durability; isolation levels trade consistency against concurrency (read uncommitted to serializable).

Normalization: 1NF atomic attributes; 2NF no partial dependency on a composite key; 3NF no transitive dependency; BCNF every determinant is a superkey.

Relational algebra: selection σ, projection π, join ⋈; natural join equates all same-named attributes and removes duplicates.

Indexes speed up lookups and range scans but slow writes and consume space; a B-tree index supports equality and range predicates.

Entity-relationship model: entities, attributes, relationships with cardinality; a many-to-many relationship maps to a junction table.

Transactions and locking: two-phase locking guarantees serializability; deadlock occurs when transactions wait on each other's locks.

Compound interest: FV = PV(1 + r)^n; continuous compounding FV = PV·e^(rt); present value PV = FV/(1 + r)^n.

Net present value: NPV = Σ CFt/(1 + r)^t − C0; accept a project when NPV > 0; IRR sets NPV = 0.

Annuity present value: PV = C·[1 − (1 + r)^(−n)]/r; perpetuity PV = C/r, growing perpetuity PV = C/(r − g).

Matrix product AB requires columns of A equal rows of B; (AB)^T = B^T A^T; A is invertible iff det(A) ≠ 0.

Linear system Ax = b has a unique solution iff det(A) ≠ 0; otherwise check consistency via rank.

Lagrange method: maximize f subject to g = c by solving ∇f = λ∇g together with the constraint.

Expected value E(X) = Σ x·P(x); variance Var(X) = E(X²) − [E(X)]²; for independent variables variances add.

Growth rate over n periods: g = (final/initial)^(1/n) − 1; the rule of 72 approximates doubling time as 72/r percent.