Advanced Microeconomics Problem Set 3
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Questions and Answers

What condition must be met for an allocation to be Pareto efficient in the context of an exchange economy?

An allocation is Pareto efficient if no agent can be made better off without making at least one other agent worse off.

How do utility functions help in understanding the behavior of economic agents?

Utility functions quantify the preferences of agents, helping to predict their choices based on the allocation of goods.

What is a blocking allocation, and why is it significant in determining core allocations?

A blocking allocation is one where a coalition of agents can reallocate resources to make each member of the coalition better off, thereby signaling that the initial allocation is not in the core.

Explain the significance of aggregate endowment in an exchange economy.

<p>Aggregate endowment represents the total resources available to all agents, which is crucial for determining feasible allocations in the economy.</p> Signup and view all the answers

In the context of the given exercise, what does the Edgeworth box illustrate regarding allocations?

<p>The Edgeworth box illustrates the possible allocations of goods between two agents and shows the set of Pareto efficient allocations within the box.</p> Signup and view all the answers

What is the role of price optimization in achieving equilibrium in an economy with multiple agents?

<p>Price optimization helps to equalize supply and demand across goods, ensuring that resources are allocated efficiently among agents.</p> Signup and view all the answers

How does the identical utility function assumption simplify the analysis of an exchange economy?

<p>The identical utility function assumption allows us to predict that agents will have similar preferences and thus engage in trades that benefit all equally.</p> Signup and view all the answers

What would indicate that a specific allocation is not in the core of the economy?

<p>An allocation is not in the core if there exists a coalition of agents that can find an alternative allocation that strictly improves their utility.</p> Signup and view all the answers

What is the core in the context of an exchange economy?

<p>The core is the set of allocations where no group of agents can profitably trade among themselves.</p> Signup and view all the answers

How is the contract curve related to the core in an Edgeworth box?

<p>The contract curve comprises all tangent points of the indifference curves, and the core is the segment of this curve inside the indifference curves corresponding to a specific endowment.</p> Signup and view all the answers

Explain the significance of the Walrasian equilibrium prices in this economic context.

<p>Walrasian equilibrium prices ensure that the market clears, equalizing supply and demand across all agents in the economy.</p> Signup and view all the answers

What determines the feasibility of an allocation in this exchange economy?

<p>An allocation is feasible if the total amount of goods allocated equals the total endowments available in the economy.</p> Signup and view all the answers

What indicates that an allocation is no longer in the core when doubling the agents?

<p>An allocation becomes non-core when there exists a coalition of agents that can reallocate their endowments to achieve a higher utility.</p> Signup and view all the answers

How do identical agents affect economic equilibrium in this scenario?

<p>Identical agents share the same maximization problems, resulting in identical equilibrium allocations and prices.</p> Signup and view all the answers

Define utility functions in the context of economic agents.

<p>Utility functions represent agents' preferences, mapping combinations of goods to levels of satisfaction or happiness.</p> Signup and view all the answers

What is the relationship between contingent commodities and core allocations?

<p>Contingent commodities allow agents to ensure specific outcomes in uncertain situations, influencing core allocations through enhanced trading possibilities.</p> Signup and view all the answers

How does price optimization relate to agent behavior in an economy?

<p>Price optimization entails agents choosing goods such that their utility is maximized given their budget constraints and prevailing market prices.</p> Signup and view all the answers

Why is it important to assess the relationship between agents' indifference curves when analyzing trade?

<p>Assessing this relationship helps determine potential gains from trade and whether an allocation is efficient or if reallocation could yield better outcomes.</p> Signup and view all the answers

What is the significance of surplus utility when considering allocations among economic agents?

<p>Surplus utility indicates the additional satisfaction agents gain from consuming beyond their initial allocation, suggesting potential for mutually beneficial trades.</p> Signup and view all the answers

How can the aggregate endowment be determined for multiple agents?

<p>The aggregate endowment is found by summing the individual endowments of all agents to understand the total resources available in the economy.</p> Signup and view all the answers

What role does price optimization play in consumer decision-making?

<p>Price optimization helps consumers maximize their utility by determining the best price point at which the quantity they purchase is maximized relative to their budget constraints.</p> Signup and view all the answers

How does the utility function affect an agent's preference for consumption bundles?

<p>The utility function quantifies the satisfaction an agent derives from different bundles of goods, guiding their preference and choice in consumption.</p> Signup and view all the answers

Explain the concept of a blocking coalition in the context of economic trade.

<p>A blocking coalition occurs when a subset of agents can improve their welfare by refusing to accept a proposed allocation, indicating inefficiency in the allocation.</p> Signup and view all the answers

In terms of contingent commodities, why is the allocation between consumers significant?

<p>Allocating contingent commodities allows agents to hedge against future uncertainties, enabling them to maximize their expected utility over time.</p> Signup and view all the answers

What results when agents 2 and 4 offer part of their consumption to agent 3?

<p>When agents 2 and 4 give up some consumption for agent 3, it can create a new allocation where all agents achieve higher utility.</p> Signup and view all the answers

How is the concept of individual rationality reflected in the allocation among agents?

<p>Individual rationality ensures that each agent prefers their allocation to what they would receive from an outside option, maintaining their utility above a certain threshold.</p> Signup and view all the answers

Why might economic agents prefer to trade rather than consume their initial endowments?

<p>Economic agents may prefer to trade because they can achieve a more favorable consumption bundle, enhancing their overall utility compared to their initial endowments.</p> Signup and view all the answers

In what ways can utility functions be used to evaluate the welfare of agents?

<p>Utility functions can be analyzed to assess the level of satisfaction or happiness agents derive from their consumption choices, which helps measure their welfare status.</p> Signup and view all the answers

How does agent B's demand remain unchanged despite changes in agent A's demand?

<p>Agent B's demand remains unchanged because the parameter ΠB is constant at 4, which means their demand function is not affected by agent A's changes.</p> Signup and view all the answers

What does the equation xA_A + xB = 1 imply about market clearing conditions?

<p>The equation xA_A + xB = 1 implies that the total demand from both agents must equal the total available supply, ensuring that the market clears.</p> Signup and view all the answers

In the context of price optimization, what is the significance of finding that pA = 1 is a solution?

<p>Finding that pA = 1 is a solution indicates that this price allows for equal distribution of resources among agents, maximizing their utility given the demand equations.</p> Signup and view all the answers

How do differing beliefs between agents A and B lead to incomplete insurance?

<p>Differing beliefs about parameters like ΠA and ΠB result in varied demand levels, which ultimately prevent both agents from fully insuring against risks.</p> Signup and view all the answers

What role do utility functions play in the demand equations for agents A and B?

<p>Utility functions dictate the preferences of agents A and B, influencing their choices and demand levels based on the prices set in the market.</p> Signup and view all the answers

What utility function is used for the consumers in the economy with storage?

<p>The utility function used is $u(x_1, x_2) = x_1 x_2$.</p> Signup and view all the answers

In the case where consumers cannot trade, how much does consumer 2 consume in each period?

<p>Consumer 2 consumes his endowment of 1 in period 1 and 9 in period 2.</p> Signup and view all the answers

What is the total endowment available in the economy?

<p>The total endowment is 20 units of the good, with consumer 1 owning 19 and consumer 2 owning 1.</p> Signup and view all the answers

When consumer 1 can trade, what is the impact on his storage decisions?

<p>Consumer 1 may choose to store less or none, depending on the equilibrium prices and trade benefits.</p> Signup and view all the answers

How does the introduction of storage costs ($\delta$) affect consumer 1's consumption choices when trading is allowed?

<p>The presence of storage costs will reduce consumer 1's optimal storage level and might alter his consumption in period 1.</p> Signup and view all the answers

Define the term 'Walrasian equilibrium prices' in the context of this economy.

<p>Walrasian equilibrium prices are the prices at which supply equals demand in the market, allowing for optimal allocations among consumers.</p> Signup and view all the answers

What does the Lagrangian formulation help determine in the maximization problem for consumer 1?

<p>The Lagrangian helps determine the optimal consumption bundle that maximizes utility subject to the budget constraint.</p> Signup and view all the answers

How does consumer 1's utility compare when storage is costly versus when it is not, assuming they can trade?

<p>Consumer 1's utility is generally lower when storage is costly because the inefficiencies from storage reduce overall utility.</p> Signup and view all the answers

What condition must hold for consumer 1 to prefer to trade rather than consume his entire endowment?

<p>Consumer 1 prefers to trade if the utility gained from trading exceeds what can be achieved by consuming his endowment directly.</p> Signup and view all the answers

What can the marginal products derived from the first-order conditions indicate about resource allocation between the two consumers?

<p>Marginal products indicate the rates at which one good can be substituted for another while keeping utility constant.</p> Signup and view all the answers

Consumer 2 consumes a total of 9 units of the good in both periods when trade is not allowed.

<p>True (A)</p> Signup and view all the answers

Consumer 1's utility function is defined as $u(x_1, x_2) = x_1 + x_2$.

<p>False (B)</p> Signup and view all the answers

When storage is costly, the optimal storage decision for consumer 1 does not depend on the value of $eta$.

<p>False (B)</p> Signup and view all the answers

The first-order condition derived from the Lagrangian for consumer 1 includes a constraint that $x_2$ must be greater than or equal to 20.

<p>False (B)</p> Signup and view all the answers

The total endowment of the good in this economy is 20 units in total.

<p>True (A)</p> Signup and view all the answers

Walrasian equilibrium prices are achieved only when consumers are not allowed to trade.

<p>False (B)</p> Signup and view all the answers

Consumer 1 is better off when storage is allowed but costly compared to when no trade occurs.

<p>False (B)</p> Signup and view all the answers

Consumer 1 stores 9 units of a good while receiving a utility of 100.

<p>True (A)</p> Signup and view all the answers

If p1 < p2, the firm would make zero profits.

<p>False (B)</p> Signup and view all the answers

The firm optimally chooses y = (0, 0) when p1 > p2.

<p>True (A)</p> Signup and view all the answers

For any given allocation, m1 + m2 equals 40p1 when there is a contradiction in prices.

<p>False (B)</p> Signup and view all the answers

The first-order conditions for consumer optimization can be expressed as x2i = λp1 and x1i = λp2.

<p>False (B)</p> Signup and view all the answers

With equilibrium price conditions, p2 x2i equals p1 x1i.

<p>True (A)</p> Signup and view all the answers

Feasibility of the equilibrium allocation is ensured when y = (0, 0).

<p>False (B)</p> Signup and view all the answers

In equilibrium, p1 must equal 2p2 to satisfy all conditions.

<p>True (A)</p> Signup and view all the answers

X1i = 1 and x2i = 2 indicate a possible production scenario.

<p>False (B)</p> Signup and view all the answers

The equilibrium allocation exists only if $δ \geq 12$.

<p>True (A)</p> Signup and view all the answers

Agent A receives happiness of 0 if Argentina wins.

<p>False (B)</p> Signup and view all the answers

Agent B believes that Argentina is more likely to win if $πA = 3/4$ and $πB = 1/4$.

<p>False (B)</p> Signup and view all the answers

If both agents agree that Brazil is more likely to win with $πA = πB = 1/4$, the utility of Agent A would be higher than that of Agent B when betting.

<p>False (B)</p> Signup and view all the answers

The utility of agent A is derived from the expression $h^2$.

<p>False (B)</p> Signup and view all the answers

The equation $x21 + x22 = ω12 + ω22 + y2$ is part of the feasibility requirement for the economy.

<p>True (A)</p> Signup and view all the answers

The proposed happiness that Agent A wagers for Agent B is a function of the outcome of the World Cup.

<p>True (A)</p> Signup and view all the answers

In the context of the Edgeworth box, an optimistic agent always prefers to bet higher amounts on their expected outcome.

<p>True (A)</p> Signup and view all the answers

The utility of consumer 1 in autarky is represented as $\frac{(1 + 19δ)^2}{4δ}$.

<p>True (A)</p> Signup and view all the answers

The values of $a$ and $b$ in the laundry trade are based on the mutual preferences for happiness between agents A and B.

<p>True (A)</p> Signup and view all the answers

The statement 'aggregate endowment is zero' indicates aggregate uncertainty.

<p>True (A)</p> Signup and view all the answers

If the total endowment across states is constant, then there is no aggregate uncertainty.

<p>True (A)</p> Signup and view all the answers

A utility function without the parameter $ ext{Π}$ implies that the agents only care about commodity A.

<p>False (B)</p> Signup and view all the answers

The maximum utility of an agent occurs when the budget constraint is breached.

<p>False (B)</p> Signup and view all the answers

Setting the price of commodity B equal to 1 simplifies the price optimization problem for agents.

<p>True (A)</p> Signup and view all the answers

The utility functions for agents A and B are different due to differing initial endowments.

<p>False (B)</p> Signup and view all the answers

The aggregate endowment affects individual consumption decisions regardless of prices.

<p>False (B)</p> Signup and view all the answers

If the price of commodity A increases, it will lead to an increase in the quantity of commodity B demanded, all else being equal.

<p>False (B)</p> Signup and view all the answers

In the given economic model, agents can guarantee their utility without trading.

<p>False (B)</p> Signup and view all the answers

If the commodity endowments of two agents are equal, their utility functions must also provide equal utilities.

<p>False (B)</p> Signup and view all the answers

In this economic model, the relationship between the variables suggests that $mA = pA$ and $mB = 1$ implies a direct correlation between prices and endowments.

<p>True (A)</p> Signup and view all the answers

The formula $xA_i = rac{mA}{pA + 1 - Π imes Πi} imes (pA)^2$ accurately calculates the consumption of agent A based on price optimization.

<p>False (B)</p> Signup and view all the answers

The expressions shown suggest that in equilibrium, $ΠA$ and $ΠB$ must both equal 4 for the allocations to be valid.

<p>True (A)</p> Signup and view all the answers

The total endowment of the economy is represented as 20 units, which is consistent across all calculations involving agent consumption.

<p>False (B)</p> Signup and view all the answers

Agent B's consumption behavior is indicated to be independent of changes made by agent A in the presented equations.

<p>True (A)</p> Signup and view all the answers

The steady state resource constraint can be expressed as kt+1 = (1 − δ)kt + f(kt, 1) − ______

<p>ct</p> Signup and view all the answers

In the steady state, the equation c̄ = k̄ ______ − δk̄ describes the relationship between consumption and capital.

<p>α</p> Signup and view all the answers

The modified-golden-rule capital stock is characterized by maximizing long-run ______.

<p>consumption</p> Signup and view all the answers

The first steady state equation simplifies to 1 = β(1 + αk̄ ______ − δ).

<p>α−1</p> Signup and view all the answers

Nullclines in the steady state are represented graphically with k on the ______ axis and c on the vertical axis.

<p>horizontal</p> Signup and view all the answers

The steady state Euler equation can be expressed as u(ct) = β(1 + fk(kt+1, 1) − ______)u′(ct+1)

<p>δ</p> Signup and view all the answers

In per capita terms, the production function is given by f(kt, 1) = k̄______.

<p>α</p> Signup and view all the answers

The evolution of capital in the economy is influenced by the rate of ______ and the output produced.

<p>depreciation</p> Signup and view all the answers

The steady state is represented when kt+1 equals ______.

<p>kt</p> Signup and view all the answers

In the North-east quadrant of the phase diagram, both kt and ct are greater than their steady state values k∗ and c∗, which indicates ______ growth.

<p>positive</p> Signup and view all the answers

The saddle path represents how the economy converges to the ______ state.

<p>steady</p> Signup and view all the answers

In the context of capital accumulation, a shock leading to the destruction of capital stock will lower ______.

<p>income</p> Signup and view all the answers

The nullclines in the phase diagram are defined by the values k∗ and c∗ which tell us where ______ dynamics balance.

<p>capital</p> Signup and view all the answers

Consumption c0 is chosen such that the bundle {k0, c0} lies on the ______ path.

<p>saddle</p> Signup and view all the answers

If kt < k∗ and ct < c∗, it indicates that the economy is in a state of ______.

<p>underperformance</p> Signup and view all the answers

To calculate the adjustment path following a shock, one must consider how kt+1, ct, wt, and Rt will ______.

<p>respond</p> Signup and view all the answers

The transversality condition is key to determining the ______ path in capital accumulation models.

<p>optimal</p> Signup and view all the answers

In steady state, the system can continue without ______ change.

<p>parameter</p> Signup and view all the answers

The steady state wage depends positively on a: ______

<p>firm's productivity</p> Signup and view all the answers

The dynamics of capital and consumption are illustrated through the use of ______ in economic models.

<p>nullclines</p> Signup and view all the answers

In the context of capital accumulation, the ______ indicates the optimal level of capital that maximizes consumption in the long run.

<p>Golden Rule Capital Stock</p> Signup and view all the answers

Steady state consumption is influenced by the rate of ______ and the dynamics of the economy.

<p>capital accumulation</p> Signup and view all the answers

The initial nullclines and saddle paths in a graphical analysis are typically represented in ______.

<p>black</p> Signup and view all the answers

In the context of capital accumulation dynamics, the equation related to the resource constraint is given by kt+1 = kt (1 + rt − ______) + wt − ct.

<p>δ</p> Signup and view all the answers

The condition that defines the steady state capital is when kt+1 = ______.

<p>kt</p> Signup and view all the answers

The golden rule capital stock maximizes steady state ______.

<p>consumption</p> Signup and view all the answers

Steady state consumption is directly influenced by the level of ______.

<p>capital</p> Signup and view all the answers

In nullclines analysis, the intersection points of the ______ define the steady states of the system.

<p>nullclines</p> Signup and view all the answers

The equation u′(ct) = β(1 + rt+1 − ______)u′(ct+1) is part of the core equation in the economic dynamics.

<p>δ</p> Signup and view all the answers

The resource constraint of the economy includes the term f(kt, ______) to represent production.

<p>1</p> Signup and view all the answers

The transversality condition is important for ensuring that initial capital choices lead to a ______ trajectory.

<p>sustainable</p> Signup and view all the answers

In the Ramsey model, the representative agent aims to maximize lifetime ______.

<p>utility</p> Signup and view all the answers

To find the steady state consumption, one needs to consider the relationship between capital, ______, and output.

<p>investment</p> Signup and view all the answers

The steady state consumption falls short of consumption at the modified-golden-rule capital stock due to the consumer not maximizing lifetime ______.

<p>utility</p> Signup and view all the answers

In the Ramsey model, the dynamics of consumption and capital can be studied through the ______ analysis.

<p>nullcline</p> Signup and view all the answers

The trajectory of capital over time is often represented using a ______ path in phase diagrams.

<p>saddle</p> Signup and view all the answers

The resource constraint represents the curve for which ______ stays constant.

<p>capital</p> Signup and view all the answers

Optimality conditions derived from the Euler equation define levels of capital for which consumption is ______.

<p>constant</p> Signup and view all the answers

Steady state capital is ideally at the modified-golden-rule level where the marginal utility of consumption is ______.

<p>maximized</p> Signup and view all the answers

When analyzing capital dynamics, we often look at the relationship between current consumption and ______ consumption.

<p>future</p> Signup and view all the answers

The steady state occurs when the accumulation of ______ is in equilibrium with the depreciating stock.

<p>capital</p> Signup and view all the answers

Higher future consumption can result from deciding to invest more capital today rather than increasing current ______.

<p>consumption</p> Signup and view all the answers

The ______ condition indicates when resources are allocated optimally in terms of capital and consumption.

<p>optimality</p> Signup and view all the answers

Flashcards

Pareto Efficiency

An allocation where no one can be made better off without making someone else worse off.

Blocking Coalition

A group of agents who can improve their outcomes by forming a new agreement.

Core of an Economy

The set of allocations that cannot be blocked by any coalition.

Pareto Inefficient Allocation

An allocation where there is an alternative allocation that makes at least one agent better off without harming others.

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Blocking Allocation

A new allocation that improves the outcomes of members of a blocking coalition without harming others.

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Exchange Economy

An economic environment where agents trade goods with each other.

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Utility Function

A function that represents the satisfaction or preference of an agent.

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Endowment

The initial amount of goods each agent possesses.

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Consumer Allocation

A specific distribution of goods or resources to individuals.

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Feasible Allocation

A distribution of resources that satisfies all given constraints and limitations.

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ω2 + ω3 + ω4 = (1,2)+(3,1)+(1,2)

A mathematical representation adding consumption values of consumers 2, 3, and 4. The numbers are x-coordinates and y-coordinates of the different consumption allocations.

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x2 + x3 + x4 = (8/6, 13/8) + (14/6, 14/8) + (8/6, 13/8)

Adding consumption of consumers 2,3, and 4. This involves vector addition and calculation. Note the fractions representing amounts received by each consumer.

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u2(x2)

Utility function of consumer 2 at their given allocation (x2.)

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Utility

Measure of satisfaction or happiness derived from consuming certain goods.

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x1

Allocation to consumer 3. It refers to a specific quantity allocated to consumer 3. It is compared to other allocations (represented as x2 and x4, etc.)

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Allocation x2,x3,x4

Shows allocation quantities for consumers 2, 3, and 4 in fractions.

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Allocation Feasibility

Examines whether combining allocations (x2+x3+x4) is within the permitted limits.

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What happens to agent B's demand?

Agent B's demand remains unchanged because their belief about the probability of state B remains the same, as reflected by ΠB = 4/4.

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How does agent A's demand change?

Agent A's demand changes because their belief about the probability of state A shifts, reflected in the updated ΠA = 3/4, leading to a different demand function.

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What is the significance of the equation for pA?

The equation for pA represents the market clearing condition, ensuring that the total demand for the good equals the total supply (which is 1). The equation provides a way to solve for the equilibrium price of the good.

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What does the solution pA = 1 signify?

The solution pA = 1 indicates that in equilibrium, the price of the good will be 1. This equilibrium price arises due to the different beliefs of agents A and B about the probabilities of the states.

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Why is there incomplete insurance?

The different beliefs of agents A and B about the probabilities of the states prevent them from fully insuring against the risk. This results in a situation where both agents only partially hedge against the risk.

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Intertemporal Utility

A function that represents a consumer's satisfaction from consuming goods over multiple time periods.

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Storage for one agent

The ability of one agent to transform one unit of a good in period one into one unit of the same good in period two.

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Maximization problem

Finding the combination of goods that gives the consumer the highest possible utility, subject to a budget constraint.

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Lagrangian

A mathematical function used to solve constrained optimization problems.

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First-Order Conditions (FOC)

Equations that are derived by taking the derivative of the Lagrangian with respect to each variable and setting it equal to zero.

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Production Economy

An economy where firms produce goods and services using inputs like labor and capital.

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Walrasian Equilibrium Prices

Prices in a market where the supply and demand for each good are equal, ensuring that all markets clear.

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Costly Storage

When storing goods incurs a cost, meaning that only a fraction of the stored goods can be retrieved in the next period.

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Optimal Choice

The combination of goods that maximizes a consumer's utility, taking into account storage costs and trade opportunities.

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Contract Curve

The set of all Pareto efficient allocations in an Edgeworth box, representing points where agents' indifference curves are tangent.

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What does it mean for an allocation to be in the core?

An allocation is in the core if no coalition of agents can find another allocation that makes everyone in the coalition better off, without hurting anyone else.

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What happens to the core when there are more identical agents in the economy?

The core shrinks with more identical agents. This is because it becomes easier for coalitions to form and find a better deal.

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How do we know if an allocation is in the core?

We need to check if any coalition can create a new allocation that makes everyone better off (without hurting those who are not part of the coalition).

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Edgeworth Box

A graphical tool used to represent the feasible allocations of goods between two agents with fixed endowments.

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Indifference Curve

A curve representing all combinations of two goods that provide a consumer with the same level of utility.

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What happens to the indifference curves in the core?

In the core, the indifference curves of all agents are tangent to each other at the allocation point. This shows that no one can be made better off without making someone else worse off.

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Why are allocations in the core important?

Allocations in the core represent stable and ‘fair’ outcomes, where no group has an incentive to deviate and create a better outcome for themselves.

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What happens to the core as the number of agents in the economy increases?

The core shrinks as the number of agents increases. This is because it becomes easier to form coalitions and find new allocations that can benefit a group without harming others.

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Zero Profit Condition

In equilibrium, the firm must make zero profits. This is because in a perfectly competitive market, firms cannot earn positive profits in the long run.

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Consumer Utility Maximization

Consumers aim to maximize their utility (satisfaction) subject to a budget constraint. They allocate their income to purchase goods that provide the highest level of utility.

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Budget Constraint

The constraint that limits a consumer's spending to their available income. They can only buy quantities of goods that fit within their budget.

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Equilibrium Allocation

An allocation of goods where all agents are maximizing their utility and all markets clear. No one wants to trade or reallocate their goods.

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Feasibility

Refers to whether an allocation of goods is possible given the available resources and constraints. It essentially checks if there's enough 'stuff' to go around.

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Contradiction

When two statements or conditions cannot both be true simultaneously. In economics, this often indicates a flaw in a model or assumption.

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Market Clearing Condition

The condition required for an equilibrium in a market. It means that the quantity supplied equals the quantity demanded for a good.

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State B Probability

The probability of state B occurring, represented by ΠB. In this case, ΠB = 4/4, indicating that state B is equally likely as state A.

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Agent A's Demand

The amount of good that agent A wants to buy, which is influenced by their belief about the probability of state A (ΠA). When ΠA changes, their demand for the good also changes.

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pA Equation

An equation representing the market clearing condition, ensuring that the total demand for the good equals the total supply (which is 1). This equation is used to determine the equilibrium price (pA).

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pA = 1

The equilibrium price of the good in the market is 1. This price arises because agents A and B have different beliefs about the probabilities of the states.

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Incomplete Insurance

A situation where agents cannot fully insure against risk because they have different beliefs about the probabilities of states. In this scenario, both agents can only partially hedge against the risk.

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Aggregate Uncertainty

A situation where the total endowment of goods across different states of the world is not constant.

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Contingent Commodities

Goods whose value depends on the state of the world, meaning their price and desirability vary depending on the outcome.

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Initial Endowment

The amount of each good that agents start with before any trading occurs.

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Price of a Commodity

The amount of money needed to buy one unit of a specific good in a given state of the world.

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Agent's Budget Constraint

A limitation on an agent's spending based on their income and the prices of goods.

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Equilibrium Price

The price of a good in a market where supply and demand are equal.

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Different Beliefs

When different agents in an economy have varying assessments of the probabilities of future events.

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Good 2 Feasibility

The condition that the total consumption of good 2 by both consumers (x21 + x22) must equal the total supply of good 2, which includes the initial endowments and any additional production.

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Equilibrium Condition

The situation where the combined consumption of good 2 by both consumers (x21 + x22) equals the sum of the initial endowment and any additional production (ω12 + ω22 + y2), satisfying the feasibility constraint.

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Consumer 1's Utility in Autarky

When δ ≥ 1/2 , consumer 1's utility remains the same as in autarky (no trade), suggesting consumer 1 only benefits when the firm shuts down.

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What is the meaning of πA and πB?

πA represents the belief of agent A that Argentina will win, and πB represents the belief of agent B that Argentina will win. These beliefs influence their decisions about trading and their willingness to accept risks.

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What is the World Cup Final scenario illustrating?

This scenario demonstrates how individuals with different beliefs about the probability of an event (Argentina winning the World Cup) can engage in trade and bet on the outcome, seeking to maximize their utility based on their beliefs.

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Storage in One-Agent Economy

The ability of one consumer to transfer a good from period one to period two, effectively storing it for later consumption.

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Consumer Maximization with Storage

A consumer's goal is to find the most satisfying combination of goods for both periods, considering that they can store some of their current endowment for future consumption.

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Lagrangian for Storage Problem

A mathematical function that helps solve the consumer's maximization problem, taking into account the storage constraint and the utility function.

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Walrasian Equilibrium with Storage

The prices of goods in a market where supply and demand are balanced, taking into account the possibility of storage by one consumer.

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Optimal Choice with Costly Storage

The best combination of consumption in two periods for a consumer, accounting for the cost of storing goods.

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Trade and Storage in Equilibrium

The prices of goods in a market where both consumers can trade and one consumer can store, taking into account the potential cost of storage.

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Steady State

A state in which the economy's capital stock and consumption remain constant over time.

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Saddle Path

The unique path an economy takes to reach the steady state, determined by the transversality condition and the equations of the model.

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Transversality Condition

A condition that ensures the economy's capital stock does not grow indefinitely, but approaches a finite limit.

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Jump Variable

A variable that can change abruptly at the initial point in time, such as consumption, to jump to a point on the saddle path.

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Capital Stock Shock

A sudden change in the amount of capital stock, such as a decrease due to an earthquake or a technological advancement.

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Wealth Effect

The impact on consumption due to a change in wealth, such as a decrease in capital stock leading to lower consumption.

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Income Effect

The impact on consumption due to a change in income, such as lower capital stock reducing income and leading to lower consumption.

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Adjustment Path

The process by which the economy moves back to the steady state after a shock.

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Household Perspective

The way households react to shocks, adjusting consumption and saving in response to changes in wealth and income.

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Model-Inconsistent Shock

A shock that violates the assumptions of the economic model used for analysis.

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Resource Constraint

An equation that shows the limitation on total resources available for production and consumption in an economy.

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Euler Equation

A mathematical equation that describes the optimal consumption decision of a representative household across time periods, ensuring they maximize their utility.

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Capital Stock

The total amount of physical assets, such as machinery and buildings, used in production.

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Production Function

A mathematical relationship showing how much output can be produced using different combinations of inputs, like labor and capital.

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Interest Rate

The cost of borrowing money or the return on lending it.

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Discount Rate

A rate used to calculate the present value of future cash flows, reflecting the time preference of consumers.

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General Equilibrium

A state in an economy where all markets are in balance, all agents are maximizing their utility, and there are no incentives for further changes.

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Steady State Resource Constraint

An equation that shows how the steady state level of capital stock (k̄) is determined by the balance between investment (f(k̄, 1) - δk̄) and consumption (c̄).

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Steady State Euler Equation

An equation that represents the condition for optimal consumption over time. It equates the marginal utility of consumption in the current period to the discounted marginal utility of consumption in the next period. It captures the balance between current and future consumption.

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Modified Golden Rule Capital Stock

The level of capital stock that maximizes consumption in the steady state. This is not necessarily the same as the level of capital that maximizes total output, as it also takes into account the cost of capital depreciation.

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Cobb-Douglas Production Function

A production function that describes a relationship where output is proportional to the product of capital and labor, each raised to a power.

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CRRA Utility Function

A utility function that exhibits Constant Relative Risk Aversion. It means that an individual's aversion to risk remains constant regardless of their wealth.

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What is the impact of a higher depreciation rate on the steady state level of capital?

A higher depreciation rate leads to a lower steady state level of capital. This is because more capital is lost each period, requiring more investment to maintain a constant capital stock.

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What is the role of the discount factor (β) in the steady state Euler equation?

The discount factor (β) reflects how patient a consumer is. A higher β implies a higher value for future consumption relative to present consumption. Thus, a higher β leads to a higher level of steady state capital.

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How does the steady state Euler equation relate to the idea of intertemporal optimization?

The steady state Euler equation reflects the principle of intertemporal optimization, which means that individuals make decisions about consumption and saving across multiple time periods to maximize their overall satisfaction. It ensures that the marginal utility of consumption is balanced across periods.

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Steady State Wage

The wage level that prevails in a long-run equilibrium where capital and labor inputs are constant.

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Dominating Wealth Effect

A situation where a change in technology leads to a larger increase in consumption than in investment, due to the belief that future wealth will be higher.

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Dominating Substitution Effect

A situation where a change in technology leads to a larger increase in investment than in consumption, due to the belief that future wages will be higher.

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Permanent Technology Shock

A sudden and permanent change in the productivity of an economy that alters the production function.

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Modified Golden Rule

The level of capital stock where steady state consumption is maximized, not utility.

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Discount Rate (β)

The rate at which future consumption is valued less than current consumption. Smaller β means greater preference for present consumption.

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Steady State Consumption

The level of consumption that remains constant over time in a long-run equilibrium.

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Phase Diagram

A graphical representation of the dynamics of an economic model, showing how variables (like capital and consumption) evolve over time.

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Nullcline

A line in the phase diagram where one variable is constant.

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Technology Shock

A sudden change in the production technology of an economy.

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Dynamics

The study of how economic variables change over time.

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Study Notes

Problem Set 3 Solutions

  • Exercise 4.1: Economy with Storage for one Agent
    • Examines an economy with two consumers, a single good, and two time periods.
    • Consumers possess intertemporal utility functions (u(x1, x2) = x1x2) and endowments (w₁ = (19,1) & w2 = (1,9)).
    • Consumer 1 can store the good, transforming period 1 units into period 2 units.
    • Part 1: Consumers cannot trade.
      • Calculates individual consumption levels.
      • Assesses consumer well-being.
      • Determines the amount of storage.
    • Part 2: Consumers can trade.
      • Calculates Walrasian equilibrium prices.
      • Determines the amount of storage.
      • Assesses consumer well-being.
    • Part 3: Storage is costly.
      • Each stored unit recovers at a reduced rate (d) in the second period.
      • Determines optimal choices if consumers cannot trade.
      • Evaluates the well-being of consumer 1.
    • Part 4: Storage is costly, but consumers can trade.
      • Calculates equilibrium prices.
      • Identifies when storage occurs.
      • Determines when consumer 1 benefits from trade.

Additional Information

  • Mathematical notation: Includes mathematical equations showing calculations of consumer utility, equilibrium prices, and storage.
  • Agent 1's optimization: Shows Agent 1's maximization problem with storage constraints & cost considerations.
  • Feasibility: The model considers situations where storage doesn't occur or is less than it would be in a non-costly model.
  • Consumer 2: Consumer 2's choice/optimization is simpler as they have no storage option.
  • Equilibrium prices: The role of equilibrium prices in determining storage behavior.
  • Part 2 (continued): Deals with the firm's production side and the impact of prices on storage.
  • Part 3 (continued): The impact of storage costs on consumer choices.

Other Exercises

  • The document also contains information on further exercises (e.g., exercise 4.2 on Argentina and Brazil) related to similar macroeconomic situations concerning consumption over time and with uncertainty.
  • Detailed equilibrium conditions and explanations of the core are presented.

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Description

This quiz focuses on Exercise 4.1, which analyzes an economy with storage for one agent over two time periods. It covers consumer utility functions, individual consumption levels, and the implications of trade and storage costs on consumer well-being. Each part of the problem set addresses trade scenarios and the impact of storage choices on optimal decision-making.

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