General Equilibrium

Question 1

An allocation (x,y) is feasible if…

Answer 1

… the total amount of each commodity consumed = sum of intital endowment of that commodity + total amount of that commodity produced

Question 2

A feasible allocation (x,y) is Pareto Optimal if…

Answer 2

… there is no other allocation (x’, y’) that Pareto dominates it. That is if there is no feasible allocation (x’, y’) such that x’≽x for all i and x’≻x for some i.

• x’≽x for all i = every consumer is at least as happy with x’ as they are with x
• x’≻x for some i = at least one consumer is strictly happier with allocation x’ than with x
• an allocation is pareto optimal if there is no waste: it is impossible to make any consumer strictly better off without making some other consumer worse off
• no distributional (fairness) issue

Question 3

Assumptions about the Private Ownership Economy

Answer 3

1. every good is traded at publicly known prices
2. price-taking behavior by firms and consumers
3. consumers trade to maximize their well-being +firms produce and trade to maximize profit
4. wealth is derived from individual endowments ω + ownership claims (shares) of profits of firms (which are owned by consumers) θij

Question 4

Equilibrium in the Private Ownership Economy

Answer 4

Price-taking equilibrium: Walrasian Equilibrium

Question 5

Walrasian Equilibrium

Answer 5

Given a private ownership economy,
an allocation (x * ,y * ) and a price-vector p= (p_1, … p_L) constitute a Walrasian Equilibrium IF

1. For every firm j, the production vector y * j maximizes profits in within the set f possible production vectors Yj that the firm can choose from. Price is given but firm chooses level of output that maximises profits.

p yj ≤ p yj * for all yj ∈ Yj

2. For every consumer i, the consumption vector xi * is the most preferred bundle given their budget. Their budget is the value of inital endowment ω + income from shares θij

–> see equation

3. Market clearing conditions: sum of all consumers’ chosen consumption vectors xi * = sum of all intital endowments ω + sum of all firms’ chosen production vectors y_j *

Σ xi * = ω + Σ yj *

Question 6

Price Equilibrium with Transfers

Answer 6

Given an allocation (x * , y * ) and a price vector p=(p1,…,pL) constitute a price equilibrium with transfers IF there is an assignment of wealth levels (w1,…,wI) with Σ wi = p ω + Σ p y * such that

1. For every firm j, the production vector y * j maximizes profits in within the set f possible production vectors Yj that the firm can choose from. Price is given but firm chooses level of output that maximises profits.

p yj ≤ p yj * for all yj ∈ Yj

2. For every consumer i, the consumption vector xi * is the most preferred bundle given their wealth wi and prices.

3. For every good l, the market clears when sum of all consumers’ chosen consumption vectors xi * = sum of all intital endowments ω + sum of all firms’ chosen production vectors y_j *

Σ xi * = ω + Σ yj *

• it shows how prices can balance out supply and demand in an economy even when some people are given additional wealth through transfers
• emphasizes that the wealth levels of consumers can be determined in different ways, not just through the market or their initial wealth
• allows for a more general understanding of how different policies, like taxes or subsidies, can affect the economy and still reach an equilibrium where supply matches demand

Question 7

Nonsatiation of preferences

Answer 7

The preference relation ≽ in the consumption set Xi is locally nonsatiated IF for every xi ∈ Xi and every ε > 0, there is another x’i ∈ Xi such that ||x’-x || ≤ ε and x’ ≻ x

= there is always a more preferred bundle

Question 8

First Fundamental Theorem of Welfare Economics

Answer 8

IF preferences are locally non satiated +
IF (x * , y * , p) is a price equilibrium with transfers,
THEN the allocation (x * , y * ) is Pareto Optimal.

Any Walrasian Equilibrium is Pareto optimal.

• in a market, where everyone is a price-taker and where markets clear, we reach a pareto efficient alloation.
• why LNS? It ensures that there is always a desire for more of at least some goods and that consumers will continue to trade to improve their wellbeing
• why price equilbrium with transfers? Allowing for broader real-world scenarios

Question 9

First Fundamental Theorem of Welfare Economics - Proof

Answer 9

Proof by contradiction:

• by definition of price-equilbrium with transfers (preference maximisation) , IF there is a bundle more preferred than our optimal consumption choice xi ≻ xi * , then this must be unaffordable p xi > wi
• under LNS, there is a bundle within distance ε that is weakly preferred to the optimal bundle xi ≽ i * but this can be at best just affordable p xi ≥ w
• Now suppose there is an allocation (x,y) that Pareto dominates our equilbrium choices (x * , y * ). This means: xi ≽ xi * for all i and xi ≻ xi * for some i.
• Given LNS, this pareto dominating bundle must violate the affordability condition = it is unaffordable p xi > w
• This is contradicting BECAUSE the sum of all values p xi for all i (total expenditure) must be greater than the sum of wealth Σ p xi > Σ wi = p ω + Σ p yj *
• If the new allocation (x,y) were to be feasible, it would violate the market clearing condition as it would require more resources than supplied
• HENCE, no such pareto optimal allocation can be feasible given the competitive equilibrium prices and wealth distribution - so this must be pareto optimal

Question 10

First Fundamental Theorem of Welfare Economics - Central Idea of the Proof

Answer 10

At any feasible allocation (x,y) ,
the cost of the total consumption bundles (x1,..,xI) evaluated at the prices p
= to the social wealth at those prices p ω + Σ p yj

Hence, (x,y) cannot be resource feasible and dominate (x * , y * ). (x * , y * ) must be pareto optimal.

Question 11

Discussion of First Fundamental Theorem of Welfare Economics

include implicit assumption

Answer 11

• equilibria in market economies are efficient: cannot make someone better off wihtout hurting someone else
• However, the result does not imply that the equilibria are “fair” (optimal from a social welfare perspective)
• this is the motivation for government intervention
• Implicit assumption: perfect competition, full information, no externalities in production or consumption (no public goods), rational decision making
• If assumptions are violated, market may not be pareto efficient

Question 12

Second Fundamental Theorem of Welfare Economics

Answer 12

Assuming convexity (!) (both firm’s technology Y and consumer’s preference are convex),
a planner can achieve any desired Pareto Optimal allocation by appropriately distributing wealth with a lump-sum tax through the competitive market process.

This might fail with the price equilbirum with transfers:
* Consumers preferences are not convex
* Firm’s technology is not convex
* strictly better consumption plans cost the same

–> Hence, introduction of quasi equilibrium with transfers!

Theorem:
Consider an economy (specified below in the picture),
suppose every Yj is convex and
every preference relation is convex and locally nonsatiated.

Then for ever pareto optimal allocation (x * , y * ), there is a price vector p=(p1,…,pL) ≠0
such that (x * , y * , p) is a price quasiequilibrium with transfers

Question 13

Quasi-equilibrium with transfers

Answer 13

Purpose:
relaxes the assumption that consumers must strictly prefer their equilibrium bundle over all other affordable bundles. Instead, it allows for between the chosen bundle and other affordable bundles, provided the chosen bundle is still the most preferred within the budget constrained.

1. Firm maximises the same as in PEWT:
For every firm j, the production vector y * j maximizes profits in within the set f possible production vectors Yj that the firm can choose from. Price is given but firm chooses level of output that maximises profits.

p yj ≤ p yj * for all yj ∈ Yj

2. Relaxed consumer behavior assumption
Even when there is a bundle strictly preferred over the competitive equilibrium choice xi * it can still be affordable (at best just but still). xi * does not need to be strictly preferred to other bundle xi - consumers are allowed to be indifferent between different bundles in their consumption set.
xi ≻ xi * then p xi ≥ w

3. Like in PEWT: For every good l, the market clears when sum of all consumers’ chosen consumption vectors xi * = sum of all intital endowments ω + sum of all firms’ chosen production vectors y_j *

Σ xi * = ω + Σ yj *

Question 14

Under which conditions can a price quasi-equilibrium w/ transfers be also a price equilibrium w/ transfers?

Question 15

Second Fundamental Theorem of Welfare Economics - Proof

Answer 15

1. Ever

Question 16

Assumptions about preferences Second Fundamental Theorem of Welfare Economics

Answer 16

Preferences are

• rational
• convex
• LNS
• continuous

Question 17

Why do we need LNS?

Answer 17

Otherwise indifference curves would be thick = consumer doesnt have strong preferences of one good for another, therefore may not participate in trade that would lead to a market competitive outcome

Question 18

When does the Second Welfare Theorem hold?

Answer 18

• Every production set Yj is convex
• every consumption set Xi is convex
• every preference relation is convex, continuous and LNS

Question 19

Second Fundamental Theorem of Welfare Economics - Discussion

Answer 19

• Price taking behavior Planning authority which wishes to implement a particular PE allocation, must be sure that supportng prices are taken by consumers and producers - if this doesnt hold (e.g. because agents are of negligible size) authority must enforce prices
• Full Information Planning authority must have good information to identify PE allocations (know consumer behavior very well)
• Impossibility of lump sum welath transfers Planning authority must have power to enfore wealth transfers –> extensive lump sum wealth transfer is rather impossible

Question 20

Hyperplane Theorem Intuition

Answer 20

A hyperplane seperates two disjoint convex sets:

• In the context of the Second Welfare Theorem, the hyperplane represents prices.
• If you can separate the production set from the consumption set with a hyperplane, it means that there exists a set of prices that can balance what’s produced with what’s consumed.
• This balance allows the market to reach an efficient outcome where resources are allocated in a way that no one can be made better off without making someone else worse off (Pareto efficiency).

or in simpler words:
The hyperplane theorem helps prove that if you can “draw a line” (set prices) that balances what people want with what can be made, then it’s possible to adjust everyone’s wealth (through redistribution) to achieve any fair and efficient distribution of goods and services in the economy. This “line” ensures that everyone gets what they want as per their preferences and available production technologies, and nobody has a reason to trade further outside of this equilibrium.

Question 21

When is a price quasi equilibrium with transfers also a price equilibrium with transfers?

Question 22

Walras Law 3

Answer 22

If one market clears at a (non-zero) price, the other will clear as well.

Question 23

Walras Law 2

Answer 23

The total (monetary) value of the excess demands for all goods is always zero at any prices (depending on prices).

At market clearing, excess demand must be 0.

Question 24

Walras Law 1

Answer 24

Agents spend all their budget.