Title: Microeconomics / misc.

Q: state contingent profits
A: the profits that a firm or investor expects to receive under different states of the world or future outcomes. These profits are contingent on the realization of specific events or conditions. In a state-contingent approach, various possible scenarios are considered, each associated with a specific state of the world. The concept of state-contingent profits is often used in finance and economics to analyze risk management, investment decisions, and the pricing of financial instruments.
For example, imagine an investor who holds a portfolio of stocks. The investor might analyze their state-contingent profits by considering the potential profit they would earn in different economic conditions, such as a strong economy, a recession, or a period of high inflation. By understanding the potential profits under different scenarios, the investor can make more informed decisions about the riskiness of their investments and adjust their portfolio accordingly.

Q: Rational, continuous, locally nonsatiated preferences
A: Preferences that have a utility function, meaning consumers prefer more to less, don't have gaps in preferences, and always prefer a bundle slightly more than a given one.

Q: Utility maximization problem (UMP)
A: The consumer's goal to choose the best consumption bundle given prices and wealth. Maximize utility subject to the constraint p∙x ≤ w.
 If prices are positive and utility is continuous, the utility maximization problem has a solution.

Q: Walrasian demand correspondence/function (x(p, w))
A: The consumption bundle(s) that solve the consumer's utility maximization problem. homogeneity of degree zero in (p, w), Walras' law, and convexity/uniqueness.

Q: Marginal rate of substitution (MRS)
A: The relative "value" of two goods in terms of their contribution to the consumer's utility. Equal to the ratio of partial derivatives of the utility function.

Q: Interior and boundary (corner) solutions
A: An interior solution occurs when all commodities are consumed, and a boundary solution occurs when one or more commodities are not consumed.

Q: Lagrange multiplier (λ) in the UMP
A: Represents the marginal (shadow) value of relaxing the wealth constraint, i.e., the marginal utility of an increase in wealth for the optimal bundle of x.

Q: strict quasiconcavity
A: The curve represents the utility function U = √X.
a property of functions that captures the idea of strictly diminishing returns or strictly increasing preference for diversification. A real-valued function f(x) defined on a convex domain is strictly quasiconcave if, for any two distinct points x and y in the domain, and any α ∈ (0, 1), the following inequality holds:
f(αx + (1 - α)y) > min{f(x), f(y)}
In other words, a function is strictly quasiconcave if, when taking a weighted average of any two distinct points in the domain (where the weights sum to 1), the function's value at the average point is strictly greater than the minimum of the function's values at the individual points. This property ensures that the function exhibits a unique maximum or minimum value, which is useful in various optimization problems, including consumer utility maximization problems in economics.

Q: Concavity of utility function
A: the property that the utility function is shaped such that it exhibits decreasing marginal rates of substitution, implying that the consumer has a preference for specialization. A function is concave if, for any two points x and y in the domain, and any α ∈ [0, 1], the following inequality holds: f(αx + (1 - α)y) ≥ αf(x) + (1 - α)f(y).

Q: Expenditure Minimization Problem (EMP)
A: Chooses a bundle of commodities to attain a given level of utility at the lowest possible cost.

Q: Expenditure Function
A: e(p, u), the optimal value of the function that is being minimized in the EMP.
1. e(p, v(p, w)) = w and v(p, e(p, u)) = u
2.  concave function with respect to prices.

Q: Properties of Uncompensated Demand (marshallian demand)
A: Homogeneous of degree zero, increasing in w, and nonincreasing in pℓ.

Q: Properties of Hicksian Demand
A: Homogeneous of degree zero, increasing in u, and nonincreasing in pℓ.

Q: Roy's Identity
A: The Marshallian demand for a good can be derived from the indirect utility function
x_i(p, w) = - (∂v(p, w) / ∂p_i) / (∂v(p, w) / ∂w)
Where:
- x_i(p, m) is the Marshallian demand function for good i
- v(p, m) is the indirect utility function
- p is the vector of prices
- w is the income
- p_i is the price of good i
- ∂v(p, m) / ∂p_i is the partial derivative of the indirect utility function with respect to the price of good i
- ∂v(p, m) / ∂m is the partial derivative of the indirect utility function with respect to income

Q: Shephard's Lemma
A: a mathematical result in consumer demand theory that relates the expenditure function to the Hicksian (compensated) demand function. It states that the Hicksian demand function for a good can be obtained by differentiating the expenditure function with respect to the price of that good. In mathematical terms, Shephard's Lemma is expressed as:
h_i(p, u) = ∂e(p, u) / ∂p_i
Where:
- h_i(p, u) is the Hicksian demand function for good i
- e(p, u) is the expenditure function
- p is the vector of prices
- u is the utility level
- p_i is the price of good i
- ∂e(p, u) / ∂p_i is the partial derivative of the expenditure function with respect to the price of good i
Shephard's Lemma provides a way to find the Hicksian demand function directly from the expenditure function without having to go through the process of solving the expenditure minimization problem. It is useful in consumer demand analysis and helps to derive demand functions that take into account the substitution effect of price changes while holding utility constant.

Q: Hicksian (Compensated) Demand Function
A: h(p, u), the solution to the EMP, minimizes expenditure while maintaining a given level of utility.

Q: properties of expenditure funciton
A: homogeneous of degree one in p, strictly increasing in u, nondecreasing in pℓ, and concave in p.

Q: relationship between Hicksian Demand and Expenditure Function
A: Hicksian demand function helps to determine the expenditure function. Expenditure function (e) can be calculated using e(p, u) = p∙h(p, u), where p is the price vector and h(p, u) is the Hicksian demand.

Q: Shepherd's Lemma
A: the Hicksian demand vector is the derivative of the expenditure function with respect to prices. In other words, h(p, u) = ∂e(p, u)/∂p.
There are two effects of changes in price on e(p, u): a direct effect, holding h(p, u) constant, and an indirect effect due to substitution. These effects cancel each other out, resulting in h(p, u) = ∂e(p, u)/∂p.

Duality of consumer theory

Q: Money Metric Welfare
A: helps measure the impact of a change in prices on a consumer's utility. It converts utility changes into dollar amounts, making it easier to analyze welfare changes.

Q: Equivalent Variation (EV)
A: EV measures the welfare change when comparing new and previous utility levels evaluated at old prices. It shows how much wealth would have to increase or decrease under old prices to reach the new utility level.
EV = e(p0, U1) - e(p0, U0)

Q: Compensating Variation (CV)
A: measures the welfare change when comparing new and previous utility levels evaluated at new prices. It shows how much wealth must change under new prices to keep the consumer at the old utility level.
CV = e(p1, U1) - e(p1, U0)

Q: Relationship between EV and CV
A: EV(p0, p1, w) = e(p0, u1) - w; CV(p0, p1, w) = w - e(p1, u0). When only one price changes, both EV and CV can be represented by areas to the left of the Hicksian demand curve.

Q: Normal Good and EV > CV
A: If the good for which the price changes is a normal good, EV > CV. Demand for a normal good increases when wealth rises and prices are held constant.

Q: Inferior Good and EV < CV
A: If the good for which the price changes is an inferior good, EV < CV. Demand for an inferior good decreases when wealth rises and prices are held constant.

Q: No Change in Demand with Wealth
A: If the demand for a good does not change with wealth, EV = CV. In this case, the welfare change is the area to the left of the Walrasian (Marshallian) demand curve.

Q: Welfare Evaluation and Hicksian Demand Curves
A: When only one price changes, EV and CV can be shown in terms of area under (or to the left of) the Hicksian demand curve. This area represents the welfare change in monetary terms.

Q: Differentiable Expenditure and Indirect Utility Functions
A: Differentiability allows for the application of calculus techniques and ensures the smoothness of expenditure and indirect utility functions, making them easier to analyze and interpret.

Q: Welfare Impact of Price Changes
A: To measure welfare impact, compare the indirect utility function values v(p1, w) and v(p0, w). A consumer is worse off if v(p1,w) - v(p0, w) < 0.

Q: Reference Price Level
A: used to convert utility into a dollar amount for welfare analysis. The choice of reference price level affects the conversion of utility to money amounts.

Q: EV and CV for Price Increase
A: For a price increase, both EV and CV will be less than 0. The negative values represent a decrease in welfare due to the price increase.

Q: What is the deadweight loss in commodity taxation?
A: Deadweight loss = w – T - e(p0, u1)

Q: The Hicksian demand curve is used to see if the deadweight loss is > 0.
A: What is the role of the Hicksian demand curve in determining deadweight loss

Q: Why is Hicksian demand at u1 instead of u0 when comparing tax policies?
A: Because we are considering the consumer's preferences after the tax has been applied.

Q: What is the difference between using equivalent valuation (EV) and compensating variation (CV) in determining deadweight loss?
A: EV evaluates the consumer's loss of utility under the old prices, while CV shows how much wealth, under the NEW PRICES, must
be given to (taken from) a “planner” to keep the consu-
mer at the OLD UTILITY leve

Q: What is the welfare analysis with partial information?
A: A way to evaluate the welfare impact of a change in prices when only initial prices, new prices, and consumer demand under the initial prices are known.

Q: What is area variation (AV)?
A: AV is the area to the left of the Walrasian demand curve for a given change in prices.

Q: When is the AV measure of welfare effects reasonable?
A: When wealth effects are small.

Q: What are the conditions for AV, EV, and CV to be equal?
A: If wealth effects did not exist.

Q: What is the relationship between AV, EV, and CV for normal and inferior goods?
A: For normal goods: EV > AV > CV; for inferior goods: EV < AV < CV.

Q: Aggregate Wealth
A: The total wealth in an economy, considering the wealth of all individuals

Q: Wealth Effect
A: The change in demand for a good resulting from a change in an individual's wealth

Q: Homothetic Preferences
A: Preferences where the ratio of goods consumed remains constant as wealth increases

Q: Quasi-Linear Preferences
A: Preferences where one good is consumed linearly while the consumption of other goods changes
In the quasi-linear utility function U = X + log(Y), both X and Y have an effect on utility, but the effect is different for each variable.
- The variable X has a linear effect on utility. This means that an increase in X by one unit will always increase utility by the same amount, regardless of the current level of X. This reflects a constant marginal utility of X.
- The variable Y, on the other hand, has a logarithmic effect on utility. This means that an increase in Y by one unit will increase utility by less and less as Y gets larger. This reflects a diminishing marginal utility of Y.
So, if we consider the marginal effect (the effect of a small increase in the variable), X has a greater effect on utility when the level of both X and Y is the same. However, if Y is very small, an increase in Y can have a larger effect on utility than an increase in X, due to the logarithmic form of Y in the utility function.

Q: Gorman Form
A: A necessary and sufficient condition for parallel wealth expansion paths; vi(p, wi) = ai(p) + b(p)wi
The Gorman form is a specific type of utility function that has the property of being homothetic. Homothetic preferences are those where the marginal rate of substitution (the rate at which a consumer is willing to trade one good for another) is the same at all points along a ray from the origin. This means that the consumer's preferences are scale invariant: if they prefer bundle A to bundle B, then they will also prefer 2A to 2B, 3A to 3B, and so on.
A utility function is in Gorman form if it can be written as:
U(x) = a(x) + b(x) * v(p)
where:
- x is a vector of quantities of different goods,
- p is a vector of prices of the different goods,
- a(x) and b(x) are functions that depend only on the quantities of goods,
- v(p) is a function that depends only on the prices of goods.
Let's consider a simple example where a consumer consumes two goods, x1 and x2, and the utility function is in Gorman form:
U(x1, x2) = x1 + x2 + p1 * x1 * x2
where p1 is the price of good 1.

Q: Wealth Distribution Rule
A: A rule for distributing wealth among individuals based on prices and aggregate wealth

Q: Positive Representative Consumer
A: An imaginary individual whose preferences generate the observed aggregate demand function for the economy

Q: Social Welfare Function
A: A function that assigns a utility value to each possible vector of utility levels for all consumers in an economy

Q: Normative Representative Consumer
A: A positive representative consumer whose preferences correspond to the social welfare function

Q: Homothetic Preferences and Social Welfare Function
A: W(u1, u2, ... uI) = αiln(ui), with all αi > 0 and αi = 1

Q: Optimal Wealth Distribution with Homothetic Preferences
A: The wealth distribution rule where wi(p, w) = αiw

Q: Aggregate Demand Function
A: x(p, w1, w2, ... wI) = xi(p, wi)

Q: Redistribution of Wealth Condition
A: ∂x𝓁i(p, wi)/∂wi = ∂x𝓁j(p, wj)/∂wj for all ℓ, all p, and all distributions of wealth (w1, w2, ...wI)

Q: Aggregate Indirect Utility Function
A: v(p, w), the maximized value of the social welfare function

Q: Normative vs. Positive Representative Consumers
A: A normative representative consumer requires the existence of a positive representative consumer, but not vice versa; a social welfare function and social planner are necessary for the existence of a normative representative consumer

Q: Simple Lottery (L)
A: A list (vector) of the form L = (p1, p2, ... pN) with pn ≥ 0 for all n and pn = 1.

Q: Compound Lottery
A: A lottery of lotteries, where each simple lottery Lk has probability αk for k = 1, 2, ... K.

Q: Reduced Lottery
A: A simple lottery that generates the identical ultimate distribution of outcomes as a compound lottery.

Q: Continuous Preference Relation
A: Small changes in probabilities do not change the ordering between two lotteries.

Q: Independence Axiom
A: The preference ordering of two combined lotteries depends only on the preference ordering of the first two lotteries, not the third.

Q: Expected Utility Form
A: A utility function U(L) = u1p1 + u2p2 + ... + uNpN for every lottery L = (p1, p2, ... pN).

Q: Linearity
A: A utility function U(L) has an expected utility form if and only if it is linear.

Q: Cardinal Property
A: The ranking of choices remains the same only under LINEAR transformations of the expected utility function.

Q: Ordinal Utility
A: The differences in utilities between different lotteries affect choices, not just their rankings.

Q: Continuity on L
A: A continuous preference relation on the space of simple lotteries L.

Q: Risk Aversion
A: A preference for a certain outcome over a risky outcome with the same expected value.

Q: Certainty Equivalent
A: The certain amount of money that a decision maker would accept in place of a risky outcome.
- Mathematical Expression:
  - Let's say U is the utility function of a risk-averse investor, and let's denote E[U(W)] as the expected utility of a risky wealth level W. The certainty equivalent, CE, is then defined by U(CE) = E[U(W)].
Suppose an investor is considering a gamble that has a 50% chance of paying $100 and a 50% chance of paying $0. The expected value of this gamble is $50. However, if the investor is risk-averse, they may prefer a certain payoff of $40 right now rather than taking the gamble. In this case, the certainty equivalent of the gamble for this particular investor is $40.

Q: Relative Risk Aversion
A: RRA=−c∗U′′(c)/U′(c)
The degree to which an individual is risk-averse, taking into account the size of thegamble.

Q: Bernoulli Utility Function
A: A utility function that incorporates risk preferences, such as risk aversion or risk-seeking behavior.

Q: Indifference Curves
A: Curves that show the decision maker's preferences for all possible sets of lotteries, where all points on the curve are equally preferred.

Q: Straight and Parallel Indifference Curves
A: Indifference curves are straight and parallel if, for every pair of lotteries L and L′ such that L ~ L′, it is also the case that αL + (1-α)L′ ~ L for all α [0, 1].

Q: Allais Paradox
A: An example that challenges the expected utility theory by demonstrating a preference for lotteries that contradicts the independence axiom.

Q: Regret Theory
A: A theory that suggests people choose certain options to avoid potential feelings of regret from making suboptimal choices.

Q: Prospect Theory
A: A psychological and economic theory that describes how people make decisions under uncertainty, incorporating elements like reference dependence, loss aversion, diminishing sensitivity, and probability weighting.

Q: Reference Dependence
A: The idea that people derive utility from gains or losses measured with respect to a reference point, such as their current situation, rather than specific values of money.

Q: Loss Aversion
A: The phenomenon where people are more sensitive to losses, even small ones, than they are to gains of the same magnitude.

Q: Diminishing Sensitivity
A: The concept that people are more sensitive to changes in their situation than they are to levels or magnitudes, resulting in a convex value function for negative values of x and a concave value function for positive values of x.

Q: Probability Weighting
A: The idea that people do not weigh events by their true probabilities but instead use probability weights that "overweight" rare events and "underweight" likely events.

Q: Complications of Prospect Theory
A: Issues that arise when applying prospect theory, such as determining reference points, the validity of laboratory experiments, the necessity of all four elements, and the domains of economics in which it applies.

Q: Cumulative Prospect Theory
A: A version of prospect theory that evaluates gambles using decision weights and a value function.

Q: Value Function (v)
A: A function in prospect theory that is increasing in x and has v(0) = 0, representing the value people assign to gains or losses.

Q: Decision Weights (π):
A: Weights in prospect theory that represent how people weigh events by their probabilities, which do not equal actual probabilities.

Q: Risk Aversion in Gains
A: The preference for a certain gain over a risky gain with the same expected value.

Q: Risk Seeking in Losses
A: The preference for a risky loss over a certain loss with the same expected value.

Q: Rare Events Overweighting
A: The tendency for people to assign higher weights to rare events in their decision-making process.

Q: Common Events Underweighting
A: The tendency for people to assign lower weights to likely events in their decision-making process.

Q: What does first-order stochastic dominance compare between two distributions?
A: It compares the return of one distribution to another.

Q: Define first-order stochastic dominance.
A: Distribution F first-order stochastically dominates distribution G if, for every non-decreasing utility function u, expected utility of F(x) is greater than or equal to the expected utility of G(x).
In this plot, CDF1 and CDF2 represent the cumulative distribution functions of two different random variables. The area where CDF1 is below CDF2 represents the region where the first random variable stochastically dominates the second one. This is because for any given outcome on the x-axis, the cumulative probability of achieving that outcome or less is higher for the first random variable (CDF1) than for the second one (CDF2). This is the essence of first-order stochastic dominance.

Q: What does second-order stochastic dominance compare between two distributions?
A: It compares the risk or dispersion of one distribution relative to another distribution.

Q: Define second-order stochastic dominance.
A: For two distributions F and G with the same mean, F second-order stochastically dominates G if, for every non-decreasing concave function u, the expected utility of F(x) is greater than or equal to the expected utility of G(x).

Q: What is a mean-preserving spread?
A: It's a distribution G derived from F by adding a variable z to x, such that the final payoff is x + z and E[z|x] = 0 for all x. E[x_F] = E[x_G].

Q: How is second-order stochastic dominance related to mean-preserving spreads?
A: If G is a mean-preserving spread of F, then F second-order stochastically dominates G. The converse is also true.

Q: What is an elementary increase in risk?
A: G is an elementary increase in risk relative to F if G is generated from F by taking all the mass that F assigns to an interval [x′, x′′] and transferring it to the endpoints x′ and x′′ in a way that the mean is preserved.

Q: What are the three equivalent statements for second-order stochastic dominance?
A: 1. F second-order stochastically dominates G
2. G is a mean-preserving spread of F
3. Integral of G(t)dt is greater than or equal to the integral of F(t)dt, for all x.

Q: Does first-order stochastic dominance imply second-order stochastic dominance?
A: No, first-order stochastic dominance does not necessarily imply second-order stochastic dominance.

Q: Does second-order stochastic dominance imply first-order stochastic dominance?
A: No, second-order stochastic dominance does not necessarily imply first-order stochastic dominance.

Q: What are the two possibilities for defining a higher return in first-order stochastic dominance?
A: 1. F yields higher expected utility than G for all u(x) with u′(x) > 0
2. For any possible value of x, the probability that x is equal or greater than that value is higher for F than for G.

Q: What is the goal of first-order stochastic dominance?
A: To attach a precise meaning to claims such as "distribution F has an unambiguously higher return than distribution G."

Q: How does first-order stochastic dominance relate to utility functions?
A: First-order stochastic dominance is based on comparing the expected utility of two distributions for non-decreasing utility functions.

Q: How is second-order stochastic dominance defined for risk-averse decision makers?
A: Any decision maker who is risk averse prefers the distribution F over distribution G.

Q: State-Dependent Utility
A: A framework that allows utility to depend not only on money but also on the "state of nature" associated with the random outcome that "causes" a specific amount of money to be received.

Q: State-Dependent Preferences and the Extended Expected Utility Representation
A: A utility function that takes the extended expected utility form, with a different utility function for each possible state s.

Q: Subjective Probability Theory
A: A theory that derives probabilities based on observed behavior, assuming that individuals behave rationally based on their subjective probabilities.

Q: State Uniform Preferences
A: Preferences on state lotteries are state uniform if s = s' for any s and s', implying that two state-specific utility functions can differ only by a linear transformation.

Q: Subjective Expected Probability Theorem
A: If the preference relation on L satisfies continuity and extended independence axioms and derived state preferences are state uniform, then there are unique probabilities and a unique utility function (up to linear transformations).

Q: Production Possibility Set (PPS)
A: Represents all feasible combinations of outputs and inputs. In general, PPS ⊂ R

Q: Long-Run and Short-Run production
A: Long-Run production allows for maximum flexibility in reallocating all commodities, while Short-Run production has some commodities that cannot be reallocated.

Q: Input Requirements Set (IRS)
A: All combinations of inputs capable of producing a combination of output: IRS(q)={z∈R :(q,-z)∈PPS}.

Q: Input Distance Function
A: Measures how much we need to scale inputs proportionally up or down to efficiently produce q

Q: Feasibility condition
A: The feasibility condition for qq and zz in the input distance function is typically defined as follows:
1. Non-negativity: All inputs and outputs are non-negative. That is, for all ii, zi≥0zi​≥0 and qi≥0qi​≥0. This reflects the physical reality that you can't have negative quantities of inputs or outputs.
2. Technical feasibility: The input-output combination (q,z)(q,z) is technically feasible. That is, it's possible to produce the output vector qq using the input vector zz, given the current level of technology.
In mathematical terms, the feasibility condition can be represented as:
D(q,z)≤1D(q,z)≤1
An input distance function represents the maximum proportional reduction in all inputs that is possible, given the current level of technology, without changing the level of output.

Q: Homogeneous property of input distance function
A: The input distance function's property of being proportional to the scale of inputs.
homogeneous of degree one in input quantities. This property reflects the idea that if you scale up all inputs by a certain factor, the maximum proportional reduction in inputs that is possible without changing the level of output also scales up by the same factor.

Q: Output Distance Function
A: A function that measures the proportional scaling of output vector q to achieve efficient production with inputs z.

Q: Returns to Scale in Production
A: The study of economies of scale and the implications of scale on the production possibilities set, resulting in a taxonomy of production possibilities sets in terms of non-increasing, non-decreasing, and constant returns to scale.

Q: Non-Increasing Returns to Scale (NIRS)
A: A Production Possibilities Set exhibits NIRS if any feasible production vector can be scaled down: if y ∈ PPS, then αy ∈ PPS for all α ∈ [0, 1].

Q: Non-Decreasing Returns to Scale (NDRS)
A: A Production Possibilities Set exhibits NDRS if any feasible production vector can be scaled up: if y ∈ PPS, then εy ∈ PPS for all ε ≥ 1.

Q: Constant Returns to Scale (CRS)
A: A Production Possibilities Set exhibits CRS if any feasible production vector can be scaled up and down: if y ∈ PPS, then τy ∈ PPS for all τ ≥ 0.

Q: Elasticity of Scale
A: A measure of local returns to scale that indicates the percentage change in outputs given the percentage change in inputs when outputs and inputs change proportionally.

Q: Marginal Rate of Technical Substitution (MRTS)
A: The amount by which one factor of production must increase to maintain the same level of production when another factor decreases, assuming technically efficient production.

Q: Marginal Rate of Transformation (MRT)
A: The amount by which one output must decrease, while another output increases to maintain efficient production with the same level of inputs.

Q: PPS Assumption A1
A: The production possibilities set is Nonempty (PPS ≠ ∅). This means there is something that can be produced.

Q: PPS Assumption A2
A: The production possibilities set is Closed. This is a technical assumption that implies the PPS contains its boundary, so max and min are used in analysis instead of sup and inf.

Q: PPS Assumption A3
A: The production possibilities set satisfies Free Disposal. If y ∈ PPS and y' ≤ y, then y' ∈ PPS. This means we can produce less output with the same inputs, or use more inputs to produce the same output.

Q: Strong vs. Weak Free Disposal
A: Strong free disposal means you can always produce less of one or more outputs with the same amount of inputs or the same amount of output with more of one or more inputs. Weak free disposal means you can always produce proportionally less output with the same amount of inputs or the same amount of output with proportionally more inputs.

Q: What is weak free disposal?
A: a property that allows inputs to be reduced or increased without affecting the production possibilities. It allows to establish additional properties of distance functions.

Q: How do input and output distance functions characterize feasible production with weak free disposal?
A: Input and output distance functions can be used to characterize feasible production by setting the input distance function equal to one when inputs satisfy weak free disposal or by setting the output distance function equal to one when outputs satisfy weak free disposal.

Q: What are the assumptions about the production possibilities set?
A: convexity assumptions that deal with input requirement sets (A4.I), feasible output sets (A4.O), strictly convex input requirement sets (A4.IS), and strictly convex feasible output sets (A4.OS).

Q: No Free Lunch
A: if y ∈ PPS and y ≥ 0, then y = 0 or if z = 0, then q = 0. It means you cannot get something from nothing.

Q: Feasibility of Inactivity
A: states that 0 ∈ PPS, which means that doing nothing is always a feasible option in the production possibilities set.

Q: Irreversibility
A: states that if y ∈ PPS and y ≠ 0, then -y ∉ PPS. It means that if a production process has been executed (e.g., producing shirts from cotton), the process cannot be reversed (e.g., turning shirts back into cotton).

Q: Why is the Feasibility of Inactivity assumption more applicable in the long-run than the short-run?
A: The assumption may not hold true in the short-run because some commodities might be restricted from production or use. In the long-run, there are no fixed factors or constraints, making inactivity a more feasible option.

Q: what does it mean when a function is concave in prices, wealth, or other?
A: concave in prices, it means that the function exhibits diminishing marginal returns with respect to prices. In other words, as the prices of goods increase, the rate at which expenditure increases becomes slower. This property is closely related to the concept of consumer preferences being convex, which indicates that consumers prefer a diversified consumption bundle. if its second derivative is less than or equal to zero.
Mathematically, if e(p, u) is the expenditure function, concavity in prices implies that for any two price vectors p and q, and any α ∈ [0, 1], the following inequality holds:
E(αp+(1−α)q)≥αE(p)+(1−α)E(q)
This property is important in consumer demand theory because it guarantees that the expenditure function has certain desirable properties, such as being continuous and differentiable. Additionally, it ensures that the Hicksian (compensated) demand functions derived from the expenditure function, according to Shephard's Lemma, will be well-behaved and consistent with consumer preferences. Concavity in prices also implies that the substitution effect of a price change is diminishing, meaning that consumers are less sensitive to price changes as prices increase.

Q: the equation of colluding and deviating
A: π_C / (1 - δ) ≥ π_D + δπ_N / (1 - δ)

Q: What does it mean if a preference relation monotonic?
A: A preference relation is monotonic if for all bundles x and y in the set X, if x is at least as good as y (x ≥ y), then x is preferred to y (x <j y). This means that a consumer prefers more of a good to less.

Q: What does it mean if a preference relation is monotone increasing?
A: A preference relation is monotone increasing if for all bundles x and y in the set X, if x is strictly better than y (x > y), then x is strictly preferred to y (x < y). This implies that the consumer prefers more to less, and the relation does not permit horizontal or vertical portions of indifference curves.

Q: What does it mean if a preference relation is strongly monotonic?
A: A preference relation is strongly monotonic if for all bundles x and y in the set X, if x is at least as good as y (x ≥ y) and x is not equal to y (x ≠ y), then x is strictly preferred to y (x y). This condition rules out bands of indifference and vertical and horizontal portions of indifference curves.

Q: What does it mean if a preference relation is weakly convex?
A: Weakly convex preferences admit thick indifference curves.
A preference relation is weakly convex if for all bundles x and y in the set X and for all t in the interval [0, 1], if x is preferred to y (x < y), then the convex combination tx + (1 − t)y is preferred to y. This implies that the consumer prefers averages and combinations to extremes.

Q: What does it mean if a preference relation is convex?
A: A preference relation is convex if for all bundles x and y in the set X and for all t in the open interval (0, 1), if the consumer is indifferent between x and y (x ∼ y), then the convex combination tx + (1 − t)y is preferred to y. This indicates the consumer's preference for diversified bundles. Allows for flat spots

Q: What does it mean if a preference relation is strictly convex?
A: rule out both thick indifference curves and flat sections.
A preference relation is strictly convex if for all bundles x and y in the set X and for all t in the open interval (0, 1), if x is strictly preferred to y (x < y), then the convex combination tx + (1 − t)y is strictly preferred to y. This means that the consumer always prefers a mixture of goods to extremes.

Q: What is a utility function for a preference relation
A: A utility function for a preference relation is a continuous function Uj : X → R such that for all bundles x and y in the set X, the utility of x is greater than or equal to the utility of y (Uj (x) ≥ Uj (y)) if and only if x is at least as good as y (x < y). This function provides a numerical representation of the consumer's preferences.

Q: What is an allocation in the context of consumption?
A: An allocation x = (x1, . . . , xm) ∈ Rmn is an assignment of a consumption bundle to each consumer. The allocation is feasible if there is enough of each good to go around, or if the sum of the quantities consumed for each good i (Σ xi) is less than or equal to the sum of the initial endowment of each good i (Σ ωi) for all goods i.

Q: What is the demand correspondence for a consumer j with preferences and endowment ωj?
A: The demand correspondence for a consumer j, φj (p, ωj), is the set of all consumption bundles x in the budget set Bj(p, ωj) that are most preferred by the consumer. In other words, x is in the demand correspondence if it is preferred to all other bundles y in the budget set, or x < y for all y in Bj (p, ωj).

Q: What does it mean for demands to be homogeneous of degree 0 in prices?
A: Demands are homogeneous of degree 0 in prices if scaling all prices by a positive factor t does not change the demand. That is, if t > 0, for all consumers j and for all consumption bundles x in the set X, the demand correspondence at scaled prices tp is the same as the demand correspondence at the original prices p, or φj (tp, ωj) = φj (p, ωj).

Q: What is the normalization of price vectors in the context of general equilibrium theory?
A: Normalization of price vectors in general equilibrium theory is a process to transform any vector of prices into a standard form. One common normalization is to the unit simplex, which is done by dividing each element of the price vector by the sum of all elements. This results in a vector of prices that sum to 1. The relevant simplex for n goods, denoted Sn-1, is the set of all price vectors in Rn+ such that the sum of each price multiplied by itself equals 1.

Q: What is a consumer's offer curve?
A: A consumer's offer curve describes the bundles of goods demanded for any possible price vector, given the consumer's endowment. This curve is traced out in the quantity space as the price vector changes, rotating through the endowment vector. The optimal consumption bundle, determined by the tangency of an indifference curve to the budget line, moves along this curve as prices change. This concept requires preferences to be strictly convex and differentiable.

Q: What does it mean for an allocation x' to Pareto dominate another allocation x?
A: The allocation x' Pareto dominates x if for every consumer j, x'j is at least as preferred as xj (x'j <j xj), and for at least one consumer j, x'j is strictly preferred to xj (x'j <j xj).

Q: What does it mean for a feasible allocation x to be weakly Pareto optimal (WPO)?
A: there is no other feasible allocation that would make all individuals in the economy strictly better off. That is, there isn't an alternative allocation that improves everyone's situation.
A feasible allocation x is weakly Pareto optimal (WPO) if there does not exist another feasible allocation x' such that for each consumer j, x'j is strictly preferred to xj (x'j >j xj).

Q: What does it mean for a feasible allocation x to be strongly Pareto optimal (SPO)?
A: if there is no other feasible allocation that would make all individuals in the economy at least as well off and at least one individual strictly better off. In other words, you cannot find an alternative allocation that would improve someone's situation without making someone else worse off.
This is a strong condition because it implies that the allocation is not only Pareto efficient (no other allocation makes everyone at least as well off) but also that no individual could be made strictly better off without someone else being made worse off. Therefore, in a strongly Pareto optimal allocation, any change would inevitably lead to a loss for someone.
It's important to note that Pareto optimality does not make any statements about fairness or income distribution. It only considers the efficient use of resources: a situation is Pareto optimal if there is no way to rearrange things to make someone better off without making someone else worse off.
A feasible allocation x is strongly Pareto optimal (SPO) if there does not exist another allocation x' that is feasible and Pareto dominates x. That is, there is no feasible x' such that for every consumer j, x'j is at least as preferred as xj (x'j <j xj), and for at least one consumer j, x'j is strictly preferred to xj (x'j >j xj).

Q: What is the contract curve in the context of an exchange economy?
A: In an exchange economy, the contract curve is the set of allocations in the Edgeworth box that are Pareto optimal. In a two-person, two-good exchange economy, if utility functions are differentiable and if demanded quantities are always strictly positive, an interior allocation is on the contract curve.

Q: What is a Walrasian equilibrium
A: Given an exchange economy consisting of a set of agents and their endowments and preferences, a Walrasian equilibrium is a pair (p∗, x∗) such that (i) the allocation is feasible (the sum of demanded goods equals the sum of endowments for each good), and (ii) the allocation x∗j is optimal for each agent j (if another bundle x'j is strictly preferred, it must be unaffordable given endowment and prices).

Q: What is the definition of excess demand?
A: For each consumer and for each good, excess demand for good i is defined as the difference between the consumer’s demanded quantity for the good and her endowment of the good: zi_j(p, ωj) = φi_j(p, ωj) − ωi_j.

Q: What is Walras's Law (strong form)?
A: Walras's Law in its strong form states that if preferences satisfy local nonsatiation, then for any price vector p, the dot product of p and the excess demand vector z(p) equals 0. This means the total value of excess demand in the economy must be zero.

Q: What is Walras's Law (weak form)?
A: Walras's Law in its weak form states that if preferences are monotonic, then for any price vector p, the dot product of p and the excess demand vector z(p) is less than or equal to 0. This means the total value of excess demand in the economy is non-positive.

Q: What is a Walrasian equilibrium (Definition 19)?
A: The price vector p* is an equilibrium price vector if the vector of excess demands at p* is non-positive, i.e., z(p*) ≤ 0.

Q: What happens to the price of a good at a Walrasian equilibrium if the good is in excess supply?
A: At a Walrasian equilibrium, if a good is in excess supply (zi(p*) < 0), then the price of the good must be zero. This is a consequence of Walras's law.

Q: What is the First Theorem of Welfare Economics and what does it mean?
A: The First Welfare Theorem states that any Walrasian equilibrium, given as (p∗, x∗), must be Pareto optimal, provided that preferences are locally nonsatiated. It means that, in such a state of Walrasian equilibrium, there does not exist another allocation that is feasible and Pareto dominates x∗. Mathematically, this can be stated as:
For an allocation x to contradict the theorem, it would have to be feasible, i.e., ∑j ωj ≥ ∑j xj, and satisfy the conditions p∗xj ≥ p∗x∗j for all consumers and p∗xk > p∗x∗k for at least one consumer. But these lead to a contradiction:
p∗ ∑j xj > p∗ ∑j x∗j (from the optimality of x∗)
= p∗ ∑j ωj (from the equilibrium condition)
≥ p∗ ∑j xj (from the feasibility of x)
This contradiction implies that our initial assumption of x∗ not being Pareto optimal is false. Therefore, the First Welfare Theorem asserts that market equilibrium corresponds to an efficient allocation of resources, where no individual can be made better off without making another worse off.

Q: What is the Second Welfare Theorem and its significance?
A: The Second Welfare Theorem states that for any Pareto optimal allocation x∗, there exists a price vector p∗ and a possible redistribution of endowments, such that (p∗, x∗) becomes a Walrasian equilibrium for the economy. This theorem implies that any efficient allocation can be achieved by competitive markets, provided the initial endowments of individuals are suitably adjusted.
The theorem assumes an exchange economy E with Ω = ∑j ωj ≠ 0 and where preferences are continuous, convex, and strongly monotonic. It is proven by assuming a redistribution of the initial endowment such that ̃ωj = x∗j for each j. The proof follows three steps:
1. Define the aggregate strict upper set A relative to x∗, which is nonempty, open, and convex due to convexity of preferences.
2. By the separating hyperplane theorem, there exists a price vector p∗ ≠ 0 with p∗ · a ≥ p∗ · Ω for all a ∈ cl(A), where p∗ > 0.
3. Show that (p∗, x∗) is a Walrasian equilibrium for the redistributed economy ̃E. This is done by showing that x∗j is optimal for every j.
This theorem emphasizes the importance of initial endowment distribution for market outcomes, and provides the theoretical justification for policies aimed at redistributing wealth to achieve a socially desirable outcome.

Q: What are Lindahl prices and Lindahl equilibrium?
A: Lindahl prices are a concept in public economics that represent the prices paid by individuals for their consumption of public goods. In a Lindahl equilibrium, each individual pays a different price for consuming the public good, which is determined by their marginal rate of substitution between the public good and a private good.
The Lindahl equilibrium is achieved when the sum of individual contributions, as determined by their Lindahl prices, equals the cost of supplying the public good. This equilibrium is efficient and mimics a market-like solution for the provision of public goods, which are traditionally challenging due to issues like non-excludability and non-rivalry.
Lindahl prices and equilibrium provide a theoretical framework for understanding how public goods might be efficiently provided and financed, although they can be difficult to implement in practice due to information requirements and negotiation processes.

Q: What is the Vickrey-Clarke-Groves (VCG) mechanism?
A: The VCG mechanism is a scheme used to determine the optimal level of a public good based on people's truthful inputs. The VCG mechanism can lead to truth-telling and the optimal provision of the public good, but it may not balance the budget, meaning it's not Pareto optimal.

Q: How does a Vickrey auction work?
A: A Vickrey auction involves multiple bidders submitting sealed bids for an object. The highest bidder wins the object but pays the second-highest bid amount. The incentive structure of a Vickrey auction means that truth-telling is a dominant strategy. No matter what everyone else does, the payoff from bidding one's true value is always at least as great as the payoff from any other bid.

Q: What is the underlying assumption of the VCG mechanism?
A: The VCG mechanism requires that all preferences are quasi-linear. This ensures that there's no direct connection between a participant's bid and their transfer amounts, making truth-telling a dominant strategy.

Q: What does the VCG mechanism consist of?
A: The VCG mechanism consists of two parts: a binary decision rule and a transfer scheme. The binary decision rule is such that if the sum of all bids is non-negative, the project is built; otherwise, it isn't. The transfer scheme affects only pivotal individuals - those whose bids could change the outcome from "build" to "not build" or vice versa.

Q: What are the potential downsides of the VCG mechanism?
A: While the VCG mechanism encourages truth-telling and can determine the optimal level of public good provision, it's not Pareto optimal. There's no guarantee that the sum of all transfers equals zero, which is a requirement for Pareto optimality. Additionally, the VCG mechanism can raise revenue through VCG taxes, which can introduce inefficiencies because these taxes aren't necessary to fund the public good.

Q: What are the possible outcomes and incentives for a participant in a VCG mechanism?
A: In a VCG mechanism, a participant's payoff varies based on whether their valuation plus the sum of others' valuations is positive or negative. The four potential cases yield different payoffs and incentives, but the consistent finding is that truth-telling is the best strategy. Lying may not affect the outcome or may lead to a less favorable outcome, so truth-telling becomes the best approach.

Q: What is a binary decision rule in the context of the Vickrey-Clarke-Groves (VCG) mechanism?
A: In the VCG mechanism, a binary decision rule is a part of the decision-making process. If the sum of all bids is non-negative, the project is built; if the sum is negative, the project is not built. This rule turns individual bids into a collective decision.

Q: What is a transfer scheme in the context of the Vickrey-Clarke-Groves (VCG) mechanism?
A: The transfer scheme in the VCG mechanism is the second component, which only affects pivotal individuals - those whose bids could change the outcome from "build" to "not build" or vice versa. This scheme determines the amount each participant contributes or receives based on the collective decision.

Q: What are the key components in the general VCG mechanism?
A: It involves a planner choosing a Pareto optimal level of public good, and consumers revealing their utility functions. Taxes are assigned according to the cost of the public good, the reported utilities of others, and a function that may depend on the reported utilities by others but not the consumer's own. It leads to ex-post efficiency, but it can't guarantee a balanced budget.

Q: What does it mean for the VCG mechanism to be "ex post efficient" but not "ex ante efficient"?
A: It means that truth telling is a dominant strategy, the Pareto-optimal level of the public good is selected, and there is enough money to provide the public good (ex post efficiency). However, it doesn't guarantee a balanced budget as there can be excess contributions, making it not ex ante efficient.

Q: What is the significance of the Rj(M-j) function in the VCG mechanism?
A: This function, carefully chosen, can ensure feasibility, meaning the total taxes collected are at least as large as the cost of producing the public good. However, it can't ensure exact equality, making the budget unbalanced.

Q: What is the main idea in the Bergstrom-Blume-Varian model?
A: It provides a model of "crowding out," the idea that government provision of a public good can supplant its private provision. It particularly considers the effects of wealth redistributions on equilibrium contribution levels.

Q: What does the Bergstrom, Blume, and Varian theorem state?
A: It assumes consumers have convex preferences and contributions are originally in a Nash equilibrium. It states that a redistribution of income among contributing consumers that does not make anyone lose more than his original contribution leads to a new Nash equilibrium. In this new equilibrium, every consumer changes his gift by exactly the change in his income, and each consumer consumes the same amount of public and private goods as before the redistribution

Q: What does Theorem 4 from Bergstrom, Blume, and Varian suggest about wealth distribution?
A: Any change in wealth distribution that does not decrease the aggregate wealth of current contributors will either increase or maintain the equilibrium supply of the public good. If the wealth of contributors increases, the supply of the public good necessarily increases. If redistribution among current contributors increases the supply of the public good, then the set of contributors must decrease. Any simple transfer of income from another consumer to a currently contributing consumer will either increase or maintain the supply of the public good.

Q: What are the key limitations of the VCG mechanism?
A: VCG mechanism cannot guarantee a balanced budget (ex ante efficiency). It also doesn't work unless we are willing to assume that utility functions are quasi-linear. And any method of returning the surplus to consumers will cause the incentive properties to unravel.

Q: What is a finite set of alternatives in social choice theory?
A: A finite set of alternatives is a limited list of options, denoted by X, that can be chosen by individuals in a society.

Q: What is a profile in social choice theory?
A: A profile is a vector of preference orderings, one for each individual in the set J. It's denoted {Pj}nj=1.

Q: What is Arrow's social welfare function (SWF)?
A: Arrow's social welfare function, denoted f ({Pj}nj=1), is a function that maps a profile into an ordering P that embodies the preferences of the society.

Q: What is the unrestricted domain condition (U) in Arrow's social welfare function?
A: The unrestricted domain condition is satisfied if the domain of f includes all logically possible profiles.

Q: What does it mean for a social welfare function to be dictatorial (D)?
A: A social welfare function is dictatorial if there is an individual j such that for all profiles, for every pair of alternatives x and y in X, if x is preferred over y by individual j, then x is preferred over y in the social ordering.

Q: What is the Pareto principle (P) in the context of a social welfare function?
A: A social welfare function satisfies the Pareto principle if for every pair of alternatives x, y ∈ X, if every individual prefers x over y, then x is preferred over y in the social ordering.

Q: What does it mean for a social welfare function to be independent of irrelevant alternatives (I)?
A: A social welfare function is independent of irrelevant alternatives if for every pair of alternatives x and y, the social ordering of x and y depends only on the individuals' pairwise ordering of x and y. It states that the ranking between two alternatives should not be affected by the introduction or removal of other alternatives.

Q: What does it mean for a set of individuals to be semi-decisive for alternatives x and y?
A: A set of individuals D ⊂ J is semi-decisive for alternatives x and y if when all individuals in D prefer x over y, and all individuals not in D prefer y over x, then x is preferred over y in the social ordering.

Q: What does it mean for a set of individuals to be decisive for alternatives x and y?
A: A set of individuals D ⊂ J is decisive for alternatives x and y if when all individuals in D prefer x over y, then x is preferred over y in the social ordering, regardless of the preferences of individuals not in D.

Q: What's the difference between a semi-decisive and decisive group in social welfare function context?
A: A decisive group gets its way in the social ranking of x and y regardless of what non-members think. A semi-decisive group gets its way only if everyone else disagrees with them in the ranking.

Q: What does Arrow's Impossibility Theorem state?
A: Arrow's Impossibility Theorem states that if preferences Pj are transitive and the set of alternatives X has at least 3 members, then any social welfare function f({Pj}nj=1) satisfying the conditions of Independence of Irrelevant Alternatives (I), Unrestricted Domain (U), and the Pareto principle (P), is dictatorial.

Q: What is the first step in the proof of Arrow's Impossibility Theorem?
A: The first step of the proof is to show that there is some pair of alternatives a, b ∈ X for which a certain individual j ∈ J is semidecisive. This involves establishing that a person j exists who is semidecisive over some pair a, b.

Q: What is the second step in the proof of Arrow's Impossibility Theorem
A: The second step of the proof is to show that the individual j, who was shown to be semidecisive over a pair of alternatives a and b in step 1, is decisive over all x, y ∈ X. This implies that person j is a dictator.

Q: In the proof of Arrow's theorem, what is shown in case i.) where xPj aPj b?
A: In case i.) where xPj aPj b (x is better than both a and b for person j), it is shown that because only j’s preferences over x and b are known, but still j gets his or her way in ordering this pair, j is decisive over x and b.

Q: In the proof of Arrow's theorem, what is shown in case ii.) where aPj bPj x?
A: In case ii.) where aPj bPj x (x is worse than either a or b for person j), it is shown that because only j’s preferences over a and x are known, but still j gets his or her way in ordering this pair, j is decisive over x and a.

Q: In the proof of Arrow's theorem, what is shown in case iii.) where aPj xPj b?
A: In case iii.) where aPj xPj b (x is between a and b), it is shown that there is another pair in the trio {a, x, b} over which j is SD, and using this fact we can still show that j is decisive over either x and a or x and b.

Q: What is the key point of Arrow's theorem proof?
A: The key point of Arrow's theorem's proof is to show that a social welfare function that satisfies (U), (P), and (I) must always work, no matter the profile. If it satisfies these conditions, it implies decisiveness, and hence, the function must be dictatorial.

Q: Arrow’s Theorem - Second Proof
A: - Consider an alternative, c, that every person places at the very bottom of their ranking.
- According to the Pareto Principle (P), the social ranking must also place c at the very bottom.
- Move c to the top of person 1’s ranking, then person 2’s ranking, and so on, until c is at the top of everyone's ranking. This is permissible according to Unrestricted Domain (U).
- There must be a person, j*, such that moving c to the top of j*'s ranking caused c to move from last to something higher than last in the social ranking.
- c must jump from the bottom to the top of the social ranking when it moves to the top of j*'s ranking.

Q: The Liberal Paradox (Definition)
A: - A liberal society might desire individual decision power over some outcomes, which could lead to non-Pareto optimal decisions.
- Amartya Sen's Minimal Liberalism (L*) axiom states that there should be at least two individuals each with at least one pair of alternatives over which they are decisive.

Q: The Liberal Paradox (Theorem)
A: - Sen's paradox is that we cannot guarantee a decision function that satisfies Unrestricted Domain (U), Pareto Principle (P), and Minimal Liberalism (L*).
- This is because there are profiles where two people can't both be decisive over the same pair, or where preferences lead to contradictions due to minimal liberalism and Pareto.
- Even if a situation arises where minimal liberalism and Pareto are incompatible, any candidate decision function would fail as it would have to work for this case. Hence, it's impossible to satisfy all three conditions simultaneously.
1. Universal Domain: This condition requires that the social choice mechanism should be able to handle any set of individual preferences.
2. Pareto Efficiency: This condition requires that if every individual prefers one outcome to another, then the social choice mechanism should also prefer the first outcome.
3. Minimal Liberty: This condition requires that each individual should have some degree of personal autonomy or liberty. Specifically, each individual should be able to decide between at least two alternatives in a way that the society must respect.

Q: What is the Median-Voter Theorem?
A: The Median-Voter Theorem states that in a system with a set of alternatives ordered along a single dimension (like political ideology or cost of a project) and an odd number of voters, simple majority voting works fine given the voters have single-peaked preferences. There exists a unique Condorcet (majority-rule) winner, which is the median of the voters' peaks.

Q: What are single-peaked preferences?
A: In the context of voting, preferences are considered single-peaked if there is a peak, such that (i.) for a < b ≤ peak, b is preferred over a, and (ii.) for peak ≤ a < b, a is preferred over b. This assumes the options are ordered on a single dimension.

Q: What is the Gibbard-Satterthwaite Theorem?
A: The Gibbard-Satterthwaite theorem is a result in voting theory which asserts that every deterministic, non-dictatorial, and strategy-proof voting system must be unresponsive in some way. In other words, for any system with three or more choices, no voting method is immune to strategic voting unless it's a dictatorship.

Q: What does strategyproof mean in the context of the Gibbard-Satterthwaite Theorem?
A: A social choice function is considered strategyproof (or nonmanipulable) if every member of society prefers the outcome obtained by truthfully revealing their preferences to any outcome that could be achieved by unilaterally lying and reporting false preferences.

Q: What does it mean for a social choice function to be dictatorial according to the Gibbard-Satterthwaite Theorem?
A: In the Gibbard-Satterthwaite theorem, a social choice function is considered dictatorial if there's a single member whose preferred choice is selected as the social choice regardless of the preferences of others. This member is termed the "dictator."

Q: What is the definition of strong monotonicity in voting?
A: A voting rule is strongly monotonic if, for all profiles and an outcome, when an outcome is pushed up in ranking, the selected outcome either remains the same or becomes the pushed up outcome. It means that improving the position of an outcome in the rankings will not lead to a decrease in its chance of being selected.

Q: How does the Gibbard-Satterthwaite theorem relate to the concept of strong monotonicity?
A: The Gibbard-Satterthwaite theorem makes use of the concept of strong monotonicity. It asserts that a social choice function is strategyproof if and only if it is dictatorial. The proof involves showing that if a function is strategyproof, it implies strong monotonicity, which in turn, due to the Muller-Satterthwaite theorem, implies dictatorship.

Q: What is a fair allocation in exchange economy context?
A: In an exchange economy, an allocation is considered fair if it is both Pareto optimal and envy-free. Pareto optimal means no individual can be made better off without making someone else worse off. Envy-free means no consumer wishes she had received another consumer's bundle.

Q: What does envy-free mean in the context of an exchange economy?
A: In an exchange economy, an allocation is envy-free if no consumer prefers another consumer's bundle to their own. Formally, an allocation x is envy-free if for each consumer i, the utility of consumer i's allocation (Ui(xi)) is greater than or equal to the utility of consumer j's allocation (Ui(xj)) for all other consumers j.

Q: How is an envy-free allocation represented geometrically?
A: Geometrically, an allocation x in an exchange economy is envy-free if its reflection about the midpoint of the Edgeworth box lies outside the lens formed by drawing an indifference curve through xi for each person. This graphical representation is a useful way to visualize the envy-free condition.

Q: What is Pareto optimal in an exchange economy context?
A: In an exchange economy, an allocation is Pareto optimal if there's no other feasible allocation that could make at least one individual better off without making any other individual worse off. It represents a state of allocation where resources cannot be reallocated to make someone better off without making someone else worse off.

Q: Constant Returns to Scale:
A: - Definition: An economic production process has constant returns to scale if increasing all inputs by some proportion results in an output increase by the same proportion.
- Mathematical Expression: If a function F is homogenous of degree k, then it exhibits constant returns to scale if k = 1. In other words, if we scale all inputs by a factor λ, F(λx) = λF(x), for all x.

Q: Increasing Returns to Scale:
A: - Definition: A production process has increasing returns to scale if an increase in all inputs by a certain factor leads to a greater than proportionate increase in output.
- Mathematical Expression: If a function F is homogenous of degree k, then it exhibits increasing returns to scale if k > 1. This implies that scaling all inputs by a factor λ leads to an output increase by a factor λ^k, where λ^k > λ.

Q: Decreasing Returns to Scale:
A: - Definition: A production process has decreasing returns to scale if increasing all inputs by a certain factor leads to a less than proportionate increase in output.
- Mathematical Expression: If a function F is homogenous of degree k, then it exhibits decreasing returns to scale if k < 1. This means that if we scale all inputs by a factor λ, the output will increase by a factor λ^k, where λ^k < λ.

Q: Subjective Beliefs about Probability
A: - Definition: A producer's subjective beliefs about the probability of different states reflect their personal estimation of the likelihood of various outcomes or states of the world occurring. These beliefs guide their decision-making process when they face uncertainty in production.
State a:
φa = [W a(π)] / [Wa(π)+ Wb(π) ]
For state b:
φb = [W b(π)] / [Wa(π)+ Wb(π) ]

Q: Absolute Risk Premium
A: ARA=−U′′(c)/U′(c)
- Definition: The absolute risk premium is the difference between the expected value of a risky prospect and the certain amount that an individual would accept instead of the risky prospect. It measures how much money the individual needs to be compensated for bearing risk.
- Mathematical Expression: Given a risky prospect with expected value E(W) and a certainty equivalent CE, the absolute risk premium is calculated as ARP = E(W) - CE.
- Example: If a producer believes that a risky investment has an expected return of $100 (E(W) = $100), but would accept a certain payoff of $85 (CE = $85) instead of the risky investment, the absolute risk premium is $100 - $85 = $15. This is the extra amount they need to be compensated for taking on the risk.

Q: Unconditional Demand vs. Conditional Demand
A: - Unconditional Demand: This refers to the demand for a good or service without taking into account any other variables or constraints. It is simply the quantity of a good or service that consumers are willing and able to purchase at a given price.
- Conditional Demand: This refers to the demand for a good or service that is contingent on or influenced by certain conditions or variables, such as the prices of other goods, the consumer's income, tastes, or preferences. Conditional demand often arises in the context of consumer theory, where it is used to analyze the consumer's demand for a good given a budget constraint and a set of prices for all goods.
- In mathematical terms, the unconditional demand function can be written as Qd = f(P), where Qd is the quantity demanded and P is the price. The conditional demand function can be written as Qd = f(P, Y, Pother), where Y represents income and Pother represents the prices of other goods.

Q: Definition of a static game.
A: In game theory, a static game is a game where all players play simultaneously one time. It's also known as a single-shot game.

Q: What is complete information in game theory?
A: In game theory, complete information refers to the condition where all players have the same information about the game. This includes the rules, payoffs, and strategies available to each player.

Q: What is the normal form (matrix form) representation of a game?
A: The normal form (matrix form) is a representation of a game that includes all possible strategies and their associated payoffs for each player. It's used for games with no order of moves and no issues with asymmetric information.

Q: Notation for players and strategies in game theory.
A: Players are represented as i = 1, 2, ..., n; i ∈ N. A pure strategy for player i is represented as si, and the set of possible pure strategies for player i is represented as si ∈ Si. The complete set of strategy profiles is represented as s = (s1, s2, ..., sn), s ∈ S where S = X Si.

Q: Definition of utility in game theory.
A: In game theory, utility refers to the satisfaction or payoff that a player receives from a particular outcome. It's usually represented as Ui(s) for player i's ranking of various possible outcomes. The set of all utility functions is represented as U = {U1, U2, ..., Un}.

Q: What is Nash Equilibrium?
A: A Nash Equilibrium is a situation in a game where no player can improve their payoff by unilaterally changing their strategy, given the strategies of the other players. It is the core concept of game theory and is often used in economics.

Q: What are the different solution concepts in static games of complete information?
A: There are several solution concepts for static games of complete information. These include the Nash Equilibrium, Dominant Strategy Equilibrium, Iterated Strict Dominance, Rationalizable Strategies (not actually a solution concept but a way of eliminating strategies), and the Maxi-min solution.

Q: What are some possible refinements to the Nash Equilibrium?
A: Some possible refinements to the Nash Equilibrium include focal points as suggested by Schelling, and Pareto dominance, where one outcome may be preferable because it improves the payoffs of all players. These refinements can help predict the outcomes of games with multiple equilibria.

Q: Strategies
A: Notation: si is a pure strategy (action) for player i. si ∈ Si is the set of possible pure strategies for player i. s = (s1, s2, ..., sn), s ∈ S where S = X Si.

Q: Common Knowledge in a Game
A: We assume that the rules of the game are common knowledge - every player knows the rules of the game, and knows that other players know the rules on the game.

Q: Dominant Strategy Equilibrium
A: A type of equilibrium where a player's dominant strategy is the one that delivers the best payoff no matter what strategies the other players choose.

Q: Iterated Strict Dominance
A: A concept in which we sequentially remove strategies that are dominated by other strategies.

Q: Rationalizable Strategies
A: These are strategies that remain for a player after taking into consideration the best responses to all possible beliefs about the other players' strategies.

Q: Existence Theorem for Finite Games
A: According to Nash (1950), every finite game has a Nash equilibrium in the game with mixed strategies.

Q: Mixed Strategy Nash Equilibrium
A: A mixed strategy profile is a Nash equilibrium if, for all players i ∈ N, Ui(σi*,σi*) ≥ Ui(si,σi*) for all si ∈ Si.
a concept in game theory where no player can improve their payoff by unilaterally changing their strategy, assuming the other players keep their strategies unchanged

Q: How do you find the best response function?
A: To find the best response function, take the first order condition of the utility function (Ui) with respect to xi and set it equal to zero.

Q: What is a dominant strategy equilibrium?
A: a dominant strategy equilibrium occurs when each player's strategy is their best response to the strategies of all other players, regardless of what the other players do. In other words, a player has a dominant strategy if that strategy is the best choice, no matter what strategies the other players choose.
For example, consider a game between two players, where each player can choose either strategy A or strategy B. If Player 1's best response to both Player 2's strategy A and strategy B is strategy A, then strategy A is a dominant strategy for Player 1. If Player 2 also has strategy A as a dominant strategy, then (A, A) is a dominant strategy equilibrium.

Q: What is market power?
A: Market power refers to the ability of a firm to influence the price of a product or service in the market, often as a result of the firm having a significant share of the market.

Q: How do monopolies and oligopolies behave strategically?
A: They behave strategically vis-à-vis consumers and other firms by controlling prices and quantities, engaging in price discrimination and non-linear pricing, differentiating products, and influencing entry and deterrence.

Q: What is the profit maximization condition for a monopolist?
A: The monopolist chooses quantity to maximize profits, fulfilling the first order necessary condition: P(Q) + P'(Q)Q - C'(Q) = 0, or marginal revenue equals marginal cost. This can be rewritten in terms of elasticity.

Q: What is the second order condition for profit maximization in monopoly?
A: The second order condition, which ensures a concave profit function, is 2P'(Q) + P''(Q)Q - C''(Q) ≤ 0.

Q: How is the welfare of a monopoly measured?
A: Welfare in a monopoly is measured as the total surplus, which includes both consumer surplus (U(Q) - P(Q)Q) and producer surplus (P(Q)Q - C(Q)). The monopoly leads to a deadweight loss, signifying unrealized gains from trade.

Q: What is perfect price discrimination in the context of monopoly?
A: Perfect price discrimination occurs when a monopolist sets the price for each consumer equal to their valuation of the good, effectively capturing all the consumer surplus. This requires complete information about each consumer's demand.

Q: What is the relationship between monopoly inefficiency and information problem?
A: The inefficiency of a monopoly can be viewed as an information problem rather than just market power, as perfect price discrimination (which maximizes total surplus) requires complete information about consumer demand.

Q: What is a two-part tariff in the context of perfect discrimination?
A: A two-part tariff is a pricing strategy where a monopolist charges a fixed fee (known as a "gate fee") and a price per unit. The total tariff is denoted as T(Q) = A + pQ, where 'A' is the fixed fee and 'p' is the price per unit.

Q: How can a monopolist extract all of the rent using a two-part tariff?
A: The monopolist sets an efficient price, where the price equals the marginal cost, and then sets 'A' (the fixed fee) to make the constraint binding. This constraint is known as the individual rationality constraint, which ensures that the consumer gains non-negative utility from purchasing the good.

Q: What is multi-market price discrimination?
A: Multi-market price discrimination occurs when a monopolist sells a single product in different markets, each at a different price. This price is dependent on the elasticity of demand in each market. The monopolist aims to maximize profits by charging the highest prices in markets with the most inelastic demand.

Q: What are some issues to consider in an oligopoly market structure?
A: In an oligopoly, firms must consider strategic behavior. Questions to consider include: Do firms set prices (Bertrand model) or quantities (Cournot model)? Do they make decisions simultaneously or sequentially? Are the products identical or imperfect substitutes?

Q: What is a Cournot oligopoly?
A: In a Cournot oligopoly, there are 'N' firms that simultaneously choose their quantities. The outputs of all firms are perfect substitutes. Each firm's profit function includes the inverse market demand function and the firm's cost function. The equilibrium in this setting, known as the Cournot (Nash) equilibrium, is achieved when each firm plays a best response given other firms’ strategies.

Q: How is the Nash equilibrium achieved in an oligopoly with linear demand and constant marginal cost?
A: Each firm's best response function is determined by maximizing payoffs for a given strategy of rival players. The Nash equilibrium is achieved when all firms satisfy their best response functions. In a duopoly (N=2), the equilibrium quantities for both firms are equal to (a – c)/3, where 'a' is the maximum price the consumer is willing to pay and 'c' is the constant marginal cost.

Q: Define a Bertrand Oligopoly.
A: A Bertrand Oligopoly involves 'N' firms indexed by 'i' where each firm simultaneously chooses price, pi. All firms have constant marginal costs of production, c. The market demand curve is Q = D(p), and the products are perfect substitutes, meaning that consumers choose based on price alone.

Q: What are some potential solutions to the Bertrand Paradox?
A: The Bertrand Paradox suggests that even minimal competition should lead to perfect competition, which is often not observed in reality. Some potential solutions to this paradox include the introduction of increasing marginal costs, product differentiation, or the idea of repeated play in dynamic games.

Q: How does increasing marginal cost affect the Nash equilibrium in a Bertrand Oligopoly?
A: When marginal costs rise with production, price equal to marginal cost is no longer the Nash equilibrium solution. In a duopoly where each firm has identical costs and where marginal costs rise with production, firms might start by setting pi = pj = P = C'(D(P)/2). Here, C' denotes the marginal cost.

Q: How is the Bertrand Equilibrium reached?
A: The Bertrand Equilibrium is reached through a process of continuous undercutting. If a firm sets a price above the marginal cost, the best response for another firm is to set its price just below that. This leads to a situation where no firm has an incentive to set a price below the marginal cost, leading to the Bertrand Equilibrium.

Q: What is Cournot Equilibrium?
A: In a simultaneous move game, firms choose their quantity simultaneously. The Nash equilibrium (where the reaction functions of the firms cross) is the Cournot equilibrium.

Q: What is Stackelberg Equilibrium?
A: Stackelberg game is a sequential move game. The firm that moves first is the leader and takes into account the reaction function of the follower. The point where the leader's and the follower's reaction functions meet is the Stackelberg equilibrium.

Q: Comparing Cournot and Stackelberg equilibrium
A: The firm moving first (Stackelberg leader) gets to a higher value iso-profit contour compared to Cournot equilibrium, producing more and earning more. The follower produces and earns less.

Q: Definition of Strategies in Extensive Form Games
A: Pure strategies specify action at all nodes. Behavior (mixed strategies) specifies probabilities of each action at all nodes.

Q: Issues with Nash Equilibrium in dynamic games
A: In a dynamic game context, Nash Equilibrium may include strategies that are not credible or sensible.

Q: What is Subgame Perfect Nash Equilibrium?
A: It is a refinement of Nash Equilibrium that rules out Nash equilibria that rely on non-credible threats. It is a Nash Equilibrium in all subgames.

Q: What is a Two-Stage Game?
A: A Two-Stage Game is an economic situation involving an investment stage followed by a strategic competition stage. Examples include capital investment followed by competition among rival firms, business contracts followed by competition, and government policies followed by firms' decisions about prices or quantities.

Q: What is the solution method for the “Generic” Two Stage Model?
A: Stage 2 is solved first to find optimal choices given the investment made in stage 1. This establishes a Nash equilibrium. The investment decision is then made in stage 1 to maximize the payoff considering the expected Nash equilibrium in stage 2.

Q: What does the strategic effect depend on in a Two-Stage Model?
A: The strategic effect, or how investment changes other players' strategy, depends upon the sign of the effect of a change in Firm 2's output on Firm 1's profit, how investment shifts the best response function of Firm 1, and the slope of the reaction function for Firm 2.

Q: What are strategic substitutes and strategic complements?
A: Strategic substitutes occur when the best response functions are downward sloping, such as in Cournot competition, and strategic complements occur when best response functions are upward sloping, such as in Bertrand competition.

Q: What is a repeated game?
A: A repeated game is a strategic interaction that occurs multiple times, rather than being a one-time event. Players in a repeated game can observe previous actions and make their current decisions based on previous outcomes.

Q: What are history-dependent strategies in a repeated game?
A: History-dependent strategies are those where a player's action at a certain stage is determined by the previous actions of all players. This allows for complex interactions like retaliation and reciprocity.

Q: What is the "Chain Store Paradox"?
A: The "Chain Store Paradox" is an example of a repeated game where an incumbent firm faces potential rival entrants in different markets. Despite expectations, the subgame perfect equilibrium is for the incumbent to collude in every round.

Q: What is the Folk Theorem in infinitely repeated games?
A: The Folk Theorem intuitively suggests that any individually rational payoffs can be supported as a subgame perfect equilibrium in an infinitely repeated game. The theorem originated from interests in the 1980s about whether players can cooperate in a repeated prisoner's dilemma game.

Q: What is the concept of the "trigger strategy" in an infinitely repeated game?
A: The trigger strategy is a mechanism in which a player cooperates as long as the other player cooperates, but defects indefinitely if the other player defects in any prior round.

Q: What is a worst perfect equilibrium?
A: A worst perfect equilibrium gives a player the lowest possible payoff as part of a subgame perfect equilibrium. It is achieved through a two-part punishment scheme that front loads punishment in the first stage.

Q: What are "renegotiation-proof" strategies in repeated games?
A: Renegotiation-proof strategies are those that restrict punishment to payoffs on the Pareto frontier. These strategies may provide a weaker deterrence than other available strategies and may support fewer subgame perfect equilibria.

Q: What is a bargaining game?
A: Bargaining games occur anytime a surplus needs to be split among multiple players, such as during trade negotiations, business deals, employment contracts, and household chores distribution.

Q: What is the "Rubenstein bargaining game" or the "alternating offer bargaining game"?
A: It's a type of structured bargaining where players take turns making offers on how to split a prize. If an offer is rejected, the turn goes to the other player and continues until an agreement is struck.

Q: How is the subgame perfect equilibrium solved in a game with T stages?
A: The process starts by analyzing the last stage of the game, where the player who makes the last offer keeps everything and offers the other player nothing. This pattern is repeated backwards through the stages, adjusting the share to reflect the discount factor at each stage.

Q: What happens to the offer when the number of rounds remaining goes to infinity (i.e., an infinite horizon game)?
A: In an infinite horizon game, the share going to player i is δi/(δi+δj) as the horizon goes to infinity. Player j's share is δj/(δi+δj). The player making the offer sets their offer at these values, and the other player accepts if the offer is less than or equal to these values.

Q: What is a subgame perfect equilibrium strategy in bargaining games?
A: It is a strategy in which each player's actions are optimal given their rival’s strategy. In bargaining games, offering a share equal to δi/(δi+δj) and accepting any offer less than or equal to this value constitutes a subgame perfect equilibrium strategy.

Q: What are the main findings from bargaining game experiments?
A: Bargaining experiments like those conducted by Binmore, Shaked and Sutton (1985) and Ochs and Roth (1989) have found that players often make offers close to a 50-50 split initially but can learn to play more strategically over time. However, players often don't consider future rounds, which can limit the predictive power of subgame perfect equilibrium.

Q: What are the limitations of subgame perfect equilibrium as a predictive tool?
A: Although subgame perfect equilibrium provides an optimal strategy, it may not accurately predict how people actually play in real-world scenarios. Players often only look ahead one or two rounds and don't instinctively employ backward induction, limiting the application of subgame perfect equilibrium as a predictive tool.

Q: arrow's theorem four axiom's
A: Arrow's Impossibility Theorem, named after economist Kenneth Arrow, is a fundamental result in social choice theory. It states that it's impossible to design a social welfare function that satisfies all of the following four axioms simultaneously when there are three or more distinct alternatives:
1. Unrestricted Domain (or Completeness): The social welfare function should be able to rank any set of individual preferences into a social preference order. In other words, for any possible set of individual preference orders, the function should be able to produce a corresponding social preference order.
2. Pareto Efficiency (or Unanimity): If every individual prefers one alternative to another, then the social preference order should also prefer that alternative. This axiom respects the principle that if everyone agrees that A is better than B, then the social preference should reflect that A is indeed better than B.
3. Independence of Irrelevant Alternatives (IIA): The social preference between two alternatives should depend only on the individual preferences between those two alternatives, and not on individual preferences about other alternatives. This axiom ensures that adding or removing other alternatives doesn't change the social preference order between the existing alternatives.
4. Non-Dictatorship: There should not be a single individual who can determine the social preference order regardless of the preferences of others. This axiom ensures that the social preference order isn't solely determined by one person's preferences.
Arrow's theorem shows that no social welfare function can satisfy all these axioms at the same time when there are three or more alternatives, which has profound implications for the theory of fair decision making, such as voting systems.

Q: What is the key difference between weak and strong Pareto optimality?
A: the difference lies in the strictness of the better-off condition. In a weakly Pareto optimal allocation, we cannot find an alternative allocation that strictly improves everyone's situation. In contrast, a strongly Pareto optimal allocation does not allow for an alternative that would strictly improve at least one individual's situation without making anyone else worse off.
In other words, the weakly Pareto optimal condition is less restrictive than the strongly Pareto optimal condition. In a weakly Pareto optimal allocation, it's possible that there may exist an alternative allocation that makes some people better off and no one worse off. However, in a strongly Pareto optimal allocation, this isn't possible.

Proof of strong walras law: Suppose that, for each j, j satisfies local nonsatiation. For any price vector pp, it must be true that p⋅z(p)=0p⋅z(p)=0. What is the proof for this?

Q: Walras' Law Weak Form
A: Suppose that, for each jj, jj is monotonic. For any price vector pp, it must be true that p⋅z(p)≤0p⋅z(p)≤0.
If each jj is monotonic, it means that each consumer jj prefers more of any good to less. Therefore, the excess demand z(p)z(p) cannot be positive for any good that has a positive price, because that would mean the consumer wants more of the good than they can afford, which contradicts the assumption of monotonicity.
Therefore, the dot product p⋅z(p)p⋅z(p) must be less than or equal to zero, because it represents the sum of the price of each good times the excess demand for that good, and each term in the sum must be non-positive.
This flashcard helps you remember the reasoning that, under monotonic preferences, the dot product of the price vector and the excess demand vector is less than or equal to zero.

Q: How would you prove monotonicity?
A: Monotonicity in the context of economics, specifically consumer theory, is a property of preference relations rather than something you typically "solve for". It's an assumption about how a consumer's preferences behave.
If you have a utility function that represents a consumer's preferences, you can check for monotonicity by examining the partial derivatives of the utility function with respect to each good. If all the partial derivatives are positive, then the utility function represents monotonic preferences. This is because a positive partial derivative means that the utility increases as the quantity of the good increases, holding the quantities of all other goods constant.
For example, consider a simple utility function of two goods, x and y:
Both of these are positive for all x,y>0, so this utility function represents monotonic preferences.
Remember, this is a simplification and real-world preferences may not always be monotonic. There can be satiation points or cases where more of a good is not preferred.

Q: Strictly Monotonic Preferences
A: A consumer's preferences are strictly monotonic if the consumer strictly prefers more of any good to less. That is, given two bundles of goods A and B, if A has more of at least one good and no less of any other goods compared to B, then the consumer strictly prefers A to B. This is a stronger condition than weak monotonicity.

Q: Weakly Monotonic Preferences:
A: A consumer's preferences are weakly monotonic if the consumer prefers more of any good to less, or is at least indifferent. That is, given two bundles of goods A and B, if A has more of at least one good and no less of any other goods compared to B, then the consumer either prefers A to B or is indifferent between the two.

Q: monotonically increasing
A: f, for all xx and yy in the domain of the function, if x<yx<y then f(x)≤f(y)f(x)≤f(y). In other words, as you move from left to right along the x-axis, the function never decreases; it either stays the same or increases.
If the function strictly increases as you move from left to right (meaning if x<y then f(x)<f(y), then the function is said to be "strictly monotonically increasing".
In the context of economics, if a utility function is monotonically increasing, it means that a consumer's utility increases or stays the same as they consume more of a good or goods. This is often referred to as the "more is better" assumption.

Explain and prove the First Theorem of Welfare Economics.

First Anomaly of Welfare theorem 1:
xplain what happens to the First Welfare Theorem if one of the consumers has a satiation point or "bliss point," a bundle at which utility reaches a global maximum.

Explain and prove the Second Theorem of Welfare Economics.

Explain and prove the existence of equilibrium with strict convexity.

Q: Explain the Existence II: Convexity theorem in the context of an exchange economy.
A: The Existence II: Convexity theorem is a proof of the existence of an equilibrium in an exchange economy, even when preferences are not strictly convex. This theorem removes the need for the assumption of strict convexity and Walras's law, which were required in the Existence I: Strict Convexity theorem.
The theorem uses the concept of a correspondence, which is a mapping whose value for a given input is a set of possible outputs. This is necessary because when preferences are merely convex (not strictly), demand may not be a function and can be multi-valued.
The proof uses Kakutani's fixed-point theorem, which is a generalization of Brouwer's fixed-point theorem for correspondences. It states that if a correspondence maps a compact, convex set into itself and is upper hemi-continuous (u.h.c.), then it has a fixed point. This theorem is used to show that the aggregate demand correspondence and the auctioneer's payoff correspondence both have fixed points, which leads to the existence of a Walrasian equilibrium.
The theorem also uses Berge's theorem, which gives conditions under which the solution to a maximization problem is upper hemi-continuous and the value function is continuous.
In summary, the Existence II: Convexity theorem shows that an equilibrium exists in an exchange economy even when preferences are only convex, not strictly convex. This is achieved by using the concepts of correspondences, upper hemi-continuity, and fixed-point theorems.
Key Concepts:
- Convexity
- Correspondence
- Upper Hemi-Continuity
- Kakutani's Fixed-Point Theorem
- Berge's Theorem
- Walrasian Equilibrium

Explain the steps and rationale in proving Arrow's Impossibility Theorem

Q: What is the Liberal Paradox and prove it
A: The Liberal Paradox, proposed by Amartya Sen, is a result in social choice theory which states that there is no social decision function that can simultaneously satisfy conditions of unrestricted domain (U), Pareto principle (P), and minimal liberalism (L*).
Definitions:
- Pareto Principle (P): If every individual in a society prefers one alternative to another, then the society as a whole should prefer that alternative.
- *Minimal Liberalism (L)**: There are at least two individuals in a society such that for each of them there is at least one pair of alternatives over which they are decisive. If they prefer one alternative to another, then society should prefer that alternative as well.

Q: Median-Voter Theorem
A: The Median-Voter Theorem is a result in social choice theory which states that if preferences are single-peaked and the number of voters is odd, there is a unique Condorcet (majority-rule) winner for any single-peaked profile. This winner is the median of the peak preferences.

Strategyproof Social Choice Function

Q: Relative Production-Risk Premium
A: The relative production-risk premium is a measure used in economics to characterize technological risk. It is denoted as apP(θ,p,r,I) and is defined as the ratio of expected revenue to the certainty equivalent CE(p,r,I):
In this context, a technology is considered inherently risky if apP(θ,p,r,I)>1. This means that the absolute risk premium exceeds the certainty equivalent, indicating a higher level of risk associated with the technology.

Q: Absolute Production-Risk Premium
A: The production-risk premium is a measure used in economics to characterize technological risk. It is denoted as ARP(θ,p,r,I) and is defined as the difference between the expected revenue R(θ,I) and the certainty equivalent CE(p,r,I) :
This measure is explicitly written as a function of the probabilities used to calculate the mean revenue. The production-risk premium is similar to the risk premium discussed in the context of choice under uncertainty, but it specifically measures technological risk, while the risk premium measures risk in general.

Q: Brouwer's Fixed-Point Theorem and Its Proof
A: Theorem:
Brouwer’s fixed-point theorem: If f : Sn−1 → Sn−1 is a continuous function, then there is some x ∈ Sn−1 such that x = f (x).
Proof:
For n ≥ 3 Brouwer’s theorem is a deep topological result. However, for n = 2, it can be proved using only the intermediate-value theorem.
1. The domain and range of f can be associated with the unit interval [0, 1].
2. For any x ∈ [0, 1] we must have f (x) ∈ [0, 1].
3. Consider the function g(x) = f (x) - x. A fixed point of f is any x at which g(x) = 0.
4. We know that g(0) ≥ 0 because f (0) ∈ [0, 1].
5. We also know that g(1) ≤ 0 because f (1) ∈ [0, 1].
6. Because f is continuous, the intermediate-value theorem guarantees that there is some x ∈ [0, 1] such that g(x) = f (x) - x = 0.
This proof is significant in the context of economic theory because it can be used when preferences are strictly convex, simplifying demands into functions rather than correspondences. This ensures that if demands are functions, they are also continuous.