An allocation (x,y) is feasible if…
… the total amount of each commodity consumed = sum of intital endowment of that commodity + total amount of that commodity produced
A feasible allocation (x,y) is Pareto Optimal if…
… 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
Assumptions about the Private Ownership Economy
- every good is traded at publicly known prices
- price-taking behavior by firms and consumers
- consumers trade to maximize their well-being +firms produce and trade to maximize profit
- wealth is derived from individual endowments ω + ownership claims (shares) of profits of firms (which are owned by consumers) θij
Equilibrium in the Private Ownership Economy
Price-taking equilibrium: Walrasian Equilibrium
Walrasian Equilibrium
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 *
Price Equilibrium with Transfers
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
Nonsatiation of preferences
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
First Fundamental Theorem of Welfare Economics
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
First Fundamental Theorem of Welfare Economics - Proof
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
First Fundamental Theorem of Welfare Economics - Central Idea of the Proof
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.
Discussion of First Fundamental Theorem of Welfare Economics
include implicit assumption
- 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
Second Fundamental Theorem of Welfare Economics
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
Quasi-equilibrium with transfers
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 *
Under which conditions can a price quasi-equilibrium w/ transfers be also a price equilibrium w/ transfers?
Second Fundamental Theorem of Welfare Economics - Proof
- Ever
Assumptions about preferences Second Fundamental Theorem of Welfare Economics
Preferences are
- rational
- convex
- LNS
- continuous
Why do we need LNS?
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
When does the Second Welfare Theorem hold?
- Every production set Yj is convex
- every consumption set Xi is convex
- every preference relation is convex, continuous and LNS
Second Fundamental Theorem of Welfare Economics - Discussion
- 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
Hyperplane Theorem Intuition
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.
When is a price quasi equilibrium with transfers also a price equilibrium with transfers?
Walras Law 3
If one market clears at a (non-zero) price, the other will clear as well.
Walras Law 2
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.
Walras Law 1
Agents spend all their budget.
