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Flashcards by Max Schnidman, updated more than 1 year ago
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Created by Max Schnidman
almost 5 years ago
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| Question | Answer |
| Choice Structure | Collection of choice sets and choice rule. \[\mathcal{B}, C(\cdot)\] |
| Weak Axiom of Revealed Preference (Choice Structure) | A choice structure satisfies WARP if for some \[B \in \mathcal{B}: x,y \in B, x\in C(B)\] then for any \[B^\prime \in \mathcal{B}: x,y \in B^\prime, y\in C(B^\prime) \implies x \in C(B^\prime\] |
| Preference Relations | At least as good: \[\succeq\] Strictly preferred: \[\succ\] Indifferent: \[\sim\] |
| Rationality | 1. Completeness: \[\forall x,y \in X, x\succeq y \ or \ y \succeq x\] 2. Transitivity: \[\forall x, y, z \in X, x\succeq y \ and \ y \succeq z \implies x \succeq z\] |
| Arrow's Theorem | If a choice set satisfies WARP, and the choice set includes all subsets of X with up to 3 elements, then a unique rational preference relation exists. |
| Relation between WARP and Rationality | Rationality implies WARP, but not the converse. |
| Monotone Preferences | \[y > > x \implies y \succ x\] |
| Strictly Monotone Preferences | \[y\ge x, y \ne x \implies y \succ x\] |
| Local nonsatiation | \[\forall x, \epsilon \ge 0, \exists y \ s.t. \ ||y-x||\le \epsilon, y\succ x\] |
| Indifference Curves | \[UC(x) = \{y\in x, y\succeq x\}\] \[LC(x) = \{y\in x, x\succeq y\}\] \[IC(x) = \{y\in x, y\sim x\}\] |
| Convexity of Preferences | \[\forall x \in X, y \succeq z, z\succeq x \implies \alpha y + (1-\alpha) z \succeq x\] |
| Homothetic Preferences | \[x\sim y \implies \alpha x \sim \alpha y\] |
| Quasilinear Preferences | \[1. (t,y) \succ (t^\prime, y) \forall t> t^\prime\] \[2. (t,y) \succ (t^\prime, y) \implies (t + d,y) \succ (t^\prime + d, y)\] |
| Continuous Preferences | \[x_n \succeq y_n \implies x \succeq y\] \[IC = UC \cap LC\] |
| Lexiographic Preferences | \[x \succeq y \ if\] \[1. x_1 > y_1\] \[2. x_1 = y_1, x_2 \ge y_2\] Cannot be represented by utility function |
| Utility Function | 1. Ordinal (except under lotteries) 2. Not unique 3. Requires rational, continuous preferences |
| Marginal Rate of Stubstitution | \[-\frac{\frac{\delta u}{\delta x_1}}{\frac{\delta u}{\delta x_2}}\] Representation-Agnostic |
| Relations between Preferences and Utility Functions | 1. Continuous preferences imply a continuous function. 2. Monotone preferences imply monotone utility functions 3. Convex preferences imply quasi-concave utility functions 4. Homothetic preference imply HD-1 Utility functions |
| Separable Preferences | \[(x, y) \succeq (x^\prime, y) \equiv (x, y^\prime) \succeq (x^\prime, y^\prime)\] |
| Strongly Separable Preferences | \[u(x_0, ..., x_n) = u_1(x_0, x_1) + u_2(x_2, ..., x_n)\] |
| Additive Preferences | \[u(x_0, ..., x_n) = u(x_1) + ... + u(x_n)\] |
| Quasi-linear Utility | \[u(x_1,...,x_n) = x_1 + u(x_2,...,x_n)\] |
| Budget Set | \[B_{p,w} = \{x\in X, px \le w\}\] Typically convex |
| Utility Maximization Problem | \[\underset{x\in R}{max} u(x) \ s.t. \ px \le w\] |
| Marshallian/Walrasian Demand | \[x(p,w) = argmax u(x) \ s.t. \ px \le w\] |
| Indirect Utility Function | \[V(p,w) = max u(x) = u(x(p,w))\] |
| Properties of U() | 1. If Prices are strictly positive and U is continuous, a solution to the UMP exists. 2. If preferences are LNS, Budget is fully exhausted (constraint binds at equality) \[3. \frac{\delta v}{\delta w} = \lambda\] |
| Properties of Indirect Utility | 1. V is HD-0 in p,w 2. V is non-increasing in p and non-decreasing in w 3. V is continuous in p,w 4. V is quasiconvex in p,w |
| Properties of Walrasian Demand | 1. X is HD-0 \[2. px = w \forall x \in x(p,w)\] (Walras's Law) 3. If U is quasi-concave, X is a convex set. If U is strictly quasi-concave, X is singleton |
| Roy's Identity | \[x_i(p,w) = -\frac{\frac{\delta v(p,w)}{\delta p_i}}{\frac{\delta v(p,w)}{\delta w}}\] |
| Expenditure Minimization Problem | \[\underset{x \in R^+}{min} px \ s.t. \ u(x) \ge \Bar{u}\] |
| Hicksian (Compensated) Demand | \[h(p, \Bar{u}) = argmin px s.t. \ u(x) \ge \Bar{u}\] |
| Expenditure Function | \[e(p, \Bar{u}) = min px = ph\] |
| Properties of Expenditure Function | 1. Non-decreasing in p 2. Strictly increasing in utility 3. HD-1 in p 4. Concave in p 5. Continuous in (p,u) |
| Shephard's Lemma for EMP | \[\frac{\delta e}{\delta p_i} = h(p_i, \Bar{u})\] |
| Dualities of Consumer Theory | \[1. e(p, v(p,w)) = w\] \[2. v(p, e(p, \Bar{u})) = \Bar{u}\] \[3. x_i(p, e) = h_i(p, \Bar{u})\] \[4. h_i(p, v(p,w)) = x_i(p,w)\] |
| Money Metric Utility Function | \[w(p,x) = e(p,u(x))\] |
| Money-Metric Indirect Utility Function | \[\mu(p,q,w) = e(p, v(q,w))\] |
| Normal Good | \[\frac{\delta x_i}{\delta w} > 0\] |
| Inferior Good | \[\frac{\delta x_i}{\delta w} < 0\] |
| Luxury Good | \[\epsilon_{i,w} = \frac{\delta ln(x_i)}{\delta ln(w)} > 1\] |
| Necessity Good | \[\epsilon_{i,w} = \frac{\delta ln(x_i)}{\delta ln(w)} < 1\] |
| Homothetic Preferences in Utility | \[\epsilon_{i,w} = \frac{\delta ln(x_i)}{\delta ln(w)} = 1\] |
| Law of Demand | \[\frac{\delta x_i}{\delta p_1} \le 0\] |
| Giffen Good | \[\frac{\delta x_i}{\delta p_1} > 0\] |
| Slutsky Equation | \[\frac{\delta x_i(p,w)}{\delta p_k} = \frac{\delta h_i(p,v(p,w))}{\delta p_k} - \frac{\delta x_i(p,w)}{\delta w}x_k(p,w)\] |
| Substitute Goods | \[\frac{\delta h_i(p,v(p,w))}{\delta p_k} \ge 0\] Gross if unequal |
| Complement Goods | \[\frac{\delta h_i(p,v(p,w))}{\delta p_k} \le 0\] Gross if unequal |
| Slutsky Substitution Matrix | \[Dh(p)\] Symmetric NSD \[\frac{\delta h_i}{\delta p_i} \le 0\] |
| Equivalent Variation | \[e(p^0, u^1) - e(p^0, u^0) \equiv e(p^0, u^1) - m \equiv \int_{p^0}^{p^1} h_1(p, u^1) dp\] |
| Compensating Variation | \[e(p^1, u^1) - e(p^1, u^0) \equiv m - e(p^1, u^0) \equiv \int_{p^0}^{p^1} h_0(p, u^0) dp\] |
| Change in Consumer Surplus | \[\int_{p^1}^{p^0} x_i(p, w) dp\] |
| Relationship between measures of Welfare Evaluation for normal goods | \[CV \le \Delta CS \le EV\] Flip for Inferior Goods Equal for Quasilinear Preferences |
| Laspeyres Index | \[\frac{p^1x^0}{p^0x^0} = L(p^0, p^1, x^0)\] Overstates compensation relative to expenditure function |
| Passche Index | \[\frac{p^1x^1}{p^0x^1} = P(p^0, p^1, x^0)\] Understates compensation relative to expenditure function |
| Aggregate Demand (Consumer Theory) | \[\bar{x}_i(p, w_1, ..., w_n) = \sum x_k^i(p, w_1)\] |
| Gorman Form | \[V^i(p, w_i) = a^i(p) + b(p)w_i\] |
| Social Welfare Function | Assigns a number to each utility profile |
| Direct Revealed Preference | \[x^tR^Dz: p^tx^t \ge p^tz\] |
| Strictly Direct Revealed Preference | \[x^tp^Dz: p^tx^t > p^tz\] |
| Weak Axiom of Revealed Preference (Preference Relations) | \[x^tR^Dx^s \implies \sim x^sR^Dx^t\] \[p^tx^s \le p^tx^t \implies p^sx^t > p^sx^s\] |
| Indirect Revealed Preference | \[x^1R^Dx^2, ...x^{n-1}R^Dx^n \implies x^1Rx^n\] |
| Generalized Axiom of Revealed Preference | \[x^tRx^s \not\implies x^sP^Dx^t\] \[x^tRx^s \implies p^sx^t \ge p^sx^s\] |
| Afirat's Theorem | A finite set of demand data satisfies GARP iff a continuous, monotonic, and concave utility function rationalizes the data. |
| Lottery | Probability vector: \[L = (p_1, ..., p_n), p_1 \ge 0, \sum p_i = 1\] |
| Compound Lotteries | Lottery over Lotteries |
| Properties of Lotteries | 1. Lottery Space is convex 2. Requires consequentialist worldview 3. Assumes probabilities are known |
| Preferences over Lotteries | 1. Rationality 2. Continuity: \[\forall L \in \mathcal{L}, \alpha L+ (1-\alpha)L^\prime \sim L^{\prime\prime}\] 3. Independence: \[L \succeq L^\prime \equiv \alpha L + (1-\alpha) L^{\prime\prime} \succeq \alpha L^\prime + (1-\alpha) L^{\prime\prime} \succeq\] |
| Expected Utility | \[U(L) = \sum p_i u_i\] |
| Von-Neumann Morgensterm Theorem | A rational preference relation over lotteries is continuous and independent iff it admits an expected utility representation |
| Properties of expected utility | 1. Linear 2. Only preserved under increasing linear transformations 3. Cardinality matters |
| Risk Aversion/Jensen's Inequality | \[\int u(x) dF(x) \le U(E_F) = u(\int x dF(x))\] U is concave |
| Risk Loving | \[\int u(x) dF(x) \ge U(E_F) = u(\int x dF(x))\] U is convex |
| Risk-Neutral | \[\int u(x) dF(x) = U(E_F) = u(\int x dF(x))\] U is linear |
| Risk Premium | \[\eta(F, u) = E_F - c(F,u)\] where c is the certainty equivalent |
| Coefficient of Absolute Risk Aversion | \[r(x) = -\frac{u^{\prime\prime}(x)}{u^\prime(x)}\] |
| More Risk Averse | 1. When agent b prefers lottery F to certain out come X, agent a does as well \[2. c(F, u_b) \le c(F,u_a) \equiv \eta(F, u_b) \ge \eta(F,u_a) \] 3. a is "more concave" than b \[4. r_b(x) \ge r_a(x) \forall x\] |
| Declining Absolute Risk Aversion | CARA is decreasing in wealth |
| Coefficient of Relative Risk Aversion | \[\rho(x) = xr(x)\] |
| Insurance Problem | \[\underset{\alpha \ge 0}{max} \pi u(w - \alpha q - D + \alpha) + (1-\pi) u(w - \alpha q)\] |
| First-Order Stochastic Dominance | \[G(x) \le F(x)\] i.e. G pays unambiguously more than F \[\int u(x) dG(x) \ge \int u(x) dF(x)\] |
| Second-Order Stochastic Dominance | \[\int G(t)dt \le \int F(t) dt\] G is "less risky" than F F is a mean-preserving spread of G For a concave utility, \[\int u(x)dF(x) \le \int u(x) dG(x)\] |
| Portfolio Problem for safe and risky asset | \[\underset{\alpha}{max} \int u(\alpha(z-r) + wr) dF(z)\] |
| Assumptions of Producer Theory | 1. Firms are price-takers 2. Technology is exogenous 3. Firms maximize profits |
| Properties of Production Sets | \[1. Y \ne \emptyset \] 2. Y is closed \[3. y \ge 0, y \in Y \implies y = 0\] \[4. 0 \in Y\] \[5. y \in Y, y^\prime \le y \implies y^\prime \in Y\] \[6. y \in Y, y \ne 0 \implies y \in Y\] |
| Nonincreasing returns to scale | \[y \in Y \implies \alpha y \in Y \ \forall \alpha \in [0,1]\] \[f(tx) \le tf(x) \ \forall t \ge 1\] |
| Nondecreasing returns to scale | \[y \in Y \implies \alpha y \in Y \ \forall \alpha \ge 1\] \[f(tx) \ge tf(x) \ \forall t \ge 1\] |
| Constant returns to scale | \[y \in Y \implies \alpha y \in Y \ \forall \alpha \ge 0\] \[f(tx) = tf(x) \ \forall t \ge 0\] |
| Convexity of production sets | \[y, y^\prime \in Y \implies \alpha y + (1-\alpha) y^\prime \in Y\] |
| Transformation Function | \[T: R^n \to R \ s.t.\] \[T(y) < 0 \equiv inefficient\] \[T(y) = 0 \equiv efficient\] \[T(y) > 0 \equiv infeasible\] |
| Marginal Rate of Transformation (MRT) | \[\frac{\delta T/\delta y_j}{\delta T/\delta y_k}\] |
| Input Requirement Set | \[V(y) = \{ x \in R^n: y \le f(x)\}\] |
| Isoquant | \[V(y) = \{ x \in R^n: y = f(x)\}\] |
| Marginal Rate of Technical Substitution (MRTS) | \[\frac{\delta f/\delta x_j}{\delta f/\delta x_k}\] |
| Homogeneous function | \[f(tx) = t^k f(x)\] |
| Homothetic function | Positive, monotonic transformation of HD-1 function |
| Elasticity of substitution | \[ \sigma = \frac{MRTS}{(x_k/x_j)} \frac{\delta(x_k/x_j)}{\delta MRTS}\] |
| Elasticity of complementarity | \[\frac{1}{\sigma}\] |
| Profit maximization problem | \[max py \ s.t. y \in Y\] |
| Profit function | \[\pi(y)\] \[\pi(p,w) = pf(x^*) - wx^*\] |
| Supply correspondence | \[y(p) = \{y \in Y: py = \pi(p)\}\] |
| Weak Axiom of Profit Maximization | \[p^t y^t \ge p^t y^s\] |
| Rationalization | \[y^t \in y^*(p^t)\] |
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