RULE Dual Degree Economics Entrance cheat sheet

Consumer optimum: MRS = Px/Py; for Cobb-Douglas U = x^a y^b, expenditure shares are a/(a+b) and b/(a+b).

Slutsky decomposition: total effect = substitution effect (always opposite to price change) + income effect (sign depends on normal/inferior status).

Perfect competition short-run shutdown rule: shut down if P < minimum AVC; long-run exit if P < minimum ATC.

Monopoly profit maximization: MR = MC, with MR = P(1 + 1/ε) for price elasticity ε < 0; markup is higher when demand is less elastic.

Price elasticity of demand: ε = (ΔQ/Q)/(ΔP/P); revenue rises with price when |ε| < 1 and falls when |ε| > 1.

Nash equilibrium: a strategy profile where no player can gain by unilateral deviation; mixed-strategy equilibrium equates expected payoffs across pure strategies.

Pareto efficiency: no reallocation can make someone better off without making another worse off; competitive equilibrium is Pareto efficient under complete markets and no externalities.

Cost curves: MC crosses ATC and AVC at their minimum points; ATC = AFC + AVC; AFC declines continuously with output.

GDP identity (expenditure): Y = C + I + G + NX; in an open economy, S − I = NX (national saving minus investment equals net exports).

Quantity theory: MV = PY; in long-run growth form, %ΔM + %ΔV = %ΔP + %ΔY, with %ΔV ≈ 0 in the long run.

Solow model steady state: sf(k) = (n + g + δ)k, where s is saving rate, n population growth, g technological progress, δ depreciation; long-run growth in output per worker equals g.

IS-LM: IS: Y = C(Y − T) + I(r) + G; LM: M/P = L(Y, r); fiscal expansion shifts IS right, monetary expansion shifts LM right.

Phillips curve (expectations-augmented): π = π^e − β(u − u_n) + v; in the long run π = π^e and u = u_n.

Taylor rule: i = r* + π + 0.5(π − π*) + 0.5(y − y*), linking policy rate to inflation gap and output gap.

Money multiplier: m = (1 + c)/(r + e + c), where c is currency-deposit ratio, r reserve requirement, e excess reserves; m falls when c or e rises.

Purchasing power parity (relative): %ΔE ≈ π_domestic − π_foreign, where E is the nominal exchange rate in domestic currency per foreign currency.

Derivative rules: d/dx[x^n] = n x^(n−1); product rule (fg)' = f'g + fg'; quotient rule (f/g)' = (f'g − fg')/g^2; chain rule d/dx[f(g(x))] = f'(g(x))g'(x).

Constrained optimization: set up Lagrangian L = f(x,y) + λ(m − p_x x − p_y y); first-order conditions ∂L/∂x = ∂L/∂y = ∂L/∂λ = 0 yield tangency condition f_x/f_y = p_x/p_y.

Second-order condition for a maximum of f(x,y): f_xx < 0 and f_xx f_yy − (f_xy)^2 > 0 (Hessian negative definite).

Elasticity from logarithms: if ln Q = a + b ln P, then b is the constant elasticity of Q with respect to P.

Definite integral as area: ∫a^b f(x) dx gives the area under f between a and b; consumer surplus = ∫{P*}^{P_max} Q(P) dP.

Matrix inverse for 2×2: A = [[a,b],[c,d]], A^(−1) = (1/(ad−bc))[[d,−b],[−c,a]], provided determinant ad − bc ≠ 0.

Growth rates: if Y = A K^α L^(1−α), then g_Y = g_A + α g_K + (1−α) g_L (log-differentiation).

Present value of a perpetuity: PV = C/r; of a growing perpetuity: PV = C/(r − g) for r > g; annuity factor = (1 − (1+r)^(−n))/r.

OLS slope: β̂ = Cov(X,Y)/Var(X) = Σ(x_i − x̄)(y_i − ȳ) / Σ(x_i − x̄)^2; intercept α̂ = ȳ − β̂ x̄.

Gauss-Markov assumptions: linear in parameters, random sampling, no perfect collinearity, zero conditional mean E[u|X] = 0, homoskedasticity; under these OLS is BLUE.

Standard error of the mean: SE = s/√n; 95% confidence interval: x̄ ± 1.96 SE (large n).

t-statistic for a coefficient: t = β̂/SE(β̂); reject H0: β = 0 at 5% two-sided if |t| > 1.96 (large samples).

R² = 1 − SSR/SST = ESS/SST; adjusted R² penalizes additional regressors and can fall when a regressor is added.

Omitted variable bias: bias in β̂ equals β_omitted × (coefficient from regressing omitted variable on included regressor); sign determined by product of the two effects.

Heteroskedasticity: OLS remains unbiased and consistent but inefficient; use robust (White) standard errors for valid inference.

p-value: probability, under H0, of observing a test statistic at least as extreme as the computed one; reject H0 when p-value < significance level α.

Necessary vs sufficient: 'A only if B' means A → B (B necessary); 'A if B' means B → A (B sufficient); 'A if and only if B' means A ↔ B.

Contrapositive equivalence: A → B is logically equivalent to ¬B → ¬A; converse (B → A) and inverse (¬A → ¬B) are not equivalent.

Correlation does not imply causation; three conditions for causal inference are association, temporal precedence, and ruling out confounders.

Common fallacies to flag: ad hominem (attacking person), straw man (misrepresenting an argument), false dilemma (only two options), post hoc (sequence treated as cause), circular reasoning.

Argument structure: premises support a conclusion; an argument is valid if the conclusion must follow from the premises, sound if valid and premises are true.

Base-rate neglect: P(A|B) = P(B|A)P(A)/P(B) (Bayes); a positive test in a low-prevalence population can still imply low posterior probability.

Sample selection bias arises when the observed sample is not random with respect to the outcome; survivorship bias is a common form.

Quantifier logic: ¬(∀x P(x)) ≡ ∃x ¬P(x); ¬(∃x P(x)) ≡ ∀x ¬P(x); watch for scope shifts in negation.

Hedging verbs for empirical claims: 'suggests', 'indicates', 'is consistent with'; avoid 'proves' unless the design supports causal identification.

Reporting results: state estimate, uncertainty, and sample - e.g., 'the coefficient is 0.42 (SE = 0.11, n = 250, p < 0.01)'.

Signposting in essays: 'First', 'Moreover', 'In contrast', 'Therefore', 'However' signal logical relations; each paragraph should advance one claim.

Citation conventions: in-text author-date (Author, year) or numeric [1]; direct quotations require page numbers; paraphrases still require citation.

Avoid nominalization overload; prefer active voice where the actor is known ('We estimate…' rather than 'An estimation was performed…').

Precision in economic writing: distinguish 'demand' from 'quantity demanded', 'inflation' from 'price level', and 'growth rate' from 'level'.

Cohesion devices: pronoun reference, lexical repetition of key terms, and parallel structure; sentence fragments and comma splices are common errors to eliminate.

Abstract structure: context (1 sentence), gap (1), method (1), key result (1–2), implication (1); keep to 150–250 words for a standard economics paper.