Competitive Equilibrium

#economics #micro

Oh, Hyunzi. (email: wisdom302@naver.com)
Korea University, Graduate School of Economics.
2024 Spring, instructed by prof. Koh, Youngwoo.

Basic Model

: goods.
: consumers.
: firms.

Consumers

: set of consumers.
: consumer 's consumption set.
: consumer 's consumption of good .
: consumer 's consumption bundle.
: consumer 's preference defined on .
: consumer 's utility function representing .

Producers

: set of producers.
: firm 's output for good .
- if , then good is produced as output.
- if , then good is used as input.
: firm 's production plan.
: firm 's transformation function.
: firm 's production possibility set.
- set of feasible production plans.
- if , then the production plan is called feasible.
: Aggregate production possibility set.

Allocation

: total endowment, initial goods in the economy.
: allocation in the economy.
- it is assumed that every resource belongs to someone.
: total aggregated demand for good .
: total net available good , net supply of good

Definition (feasible allocation).

Given , an allocation is feasible if

Ownership

: consumer 's endowment for good .
- : consumer 's endowment
- : endowment of good .
- : vector of all consumer's endowment
- note that .
: consumer 's owned portion of firm .
- : each firm is fully owned by consumers.
- : endowment income of consumer
- : shared income to consumer

Pareto Efficient Allocations

Definition (Pareto efficient).

An feasible allocation is Pareto Efficient (or Pareto optimal) if there does not exist any other feasible allocation such that

is Pareto Efficient if it is impossible to change from the point to make the consumer strictly better off without making someone worse off.
Pareto Efficient does not concern with distributional issues.

Proposition (solution for Pareto efficient).

An allocation is Pareto Efficient if and only if it solve the following maximization problem for some utility levels :

Tip

Note that the equalities hold at optimum since

is continuous on , thus .
demand for good equals to its net supply, for all .
is continuous on , thus .

where the second equality guarantees the unique for given .

Lemma (PE and Equality).

Let the allocation be Pareto Efficient. Then MRS and MRTS for any goods are equal across all consumer and producer .

Proof.Let denote Lagrangian multiplier. then we have Define , and assume the interior solution ().
F.O.C. i.e. using the gradient vectors, where , and .

Therefore,

MRS for any goods must be equalized across all consumers :
MRTS for any goods must be equalized across all producers :
MRS and MRTS for any goods must be equalized across all consumer and producer :

this completes the proof. □

Competitive Equilibrium

Definition (Competitive (or Walrasian) Equilibrium).

A feasible allocation and price constitute a competitive (or Walrasian) equilibrium if

Profit maximization: where and .
Utility maximization: where is total income of consumer . and is the solution for UMP.
Market clearing: which means no excess demand nor supply.

Proposition (HOD1 in

of CE).

If and constitutes a competitive equilibrium, then so do and for any .

Theorem (Walras law in CE).

If and satisfy

Market clearing:
Consumer's budget constraints:

then the market of good is also cleared, i.e.

Proof.From the market clearing condition, for all , we have by adding up over consumers, which leads to then by the market clearing condition, we have , and so the . therefore we have market clearing on good . □

Two-goods Economy

Example (Two-good economy).

Suppose there are two goods ().

initial endowment: .
price of good is normalized: .
consumer 's quasi-linear utility function: where , , and .
firm 's production set: where the cost function is , .

Proof.let the price is given as .

CE conditions:

Profit maximization: for each ,
- Lagrangian thus and by CS condition, , and .
- F.O.C.
  - if and : then .
  - if and : then .
Utility maximization: for each ,
- Lagrangian: thus
- F.O.C.
Market clearing: by the ^2c4b14 Theorem 7 (Walras law in CE), ISTS that the market clears for the good , i.e.

Let is the CE allocation for given if and only if the F.O.C. conditions hold with equality. which means

where , for all .
where , for all .
.

note that these conditions are determined independent to and .

AD function:
since and is bounded, we can derive the aggregate demand function for good from 2) .

the Walrasian demand function for consumer is where if .

the Aggregate demand function is which has following properties:

is continuous
is non-increasing in all
is strictly decreasing in any .

Pasted image 20240412154006.png

AS function:
since and , we can derive the aggregate supply function for good from 1) .

the supply function for firm is where if .

the Aggregate supply function is which has following properties:

is continuous
is non-decreasing in all
is strictly increasing in any .

Pasted image 20240412154124.png

Equilibrium: at such that , if .
Pasted image 20240412155209.png

Note that

Marginal Cost function: .
Marginal social benefit: .

this completes the answer. □

Robinson Crusoe Economy

Example (Robinson Crusoe Economy).

Robinson, who lives in a desert island, works in the daytime (Robinson producer; RP) and consumes "consumption" and "leisure" in the remaining time (Robinson consumer; RC).

RC's initial endowment: .
RC's utility: where is amount of leisure and is amount of consumption.
RC is the only provider of labor: he owns the firm.
RP's production function: where is the supply of labor.

Proof.let and denote the price of consumption and wage, respectively.

CE conditions:

Profit maximization: RP solves,
- F.O.C: assuming an interior solution,
- profit function:
Utility maximization: RC solves,
- F.O.C:
Market clearing: by letting for normalization, therefore,

Pasted image 20240412161400.png

this completes the answer. □

Welfare Theorems

Welfare Theorem

FWT(CE PE): any competitive equilibrium is Pareto efficient if markets are complete, i.e. if the following conditions are satisfied:
1. complete set of market: every relevant good is traded in a market at publicly known prices.
2. price takers: every households and firms act perfectly competitive.
SWT(PE CE): any Pareto efficient outcome can be achieved as competitive equilibrium if the following conditions hold:
1. household preference and firm production sets are convex.
2. markets are complete
3. every agents are price takers
4. appropriate lump-sum transfers of wealth are arranged

Theorem (First Welfare Theorem with Production).

If and constitute a competitive equilibrium, then is Pareto efficient.

Proof.
RTA: suppose that is CE but not PE.
since it is not PE, there exists a feasible such that this implies that note that as is CE, it satisfies market clearing for all , which means then summing up over , we have this implies contradicting the profit maximization condition of CE.
therefore, must be PE. □

CE implies PE

since CE is PMP & UMP, there exists no other feasible allocation that can strictly increases the utility.

Theorem (Second Welfare Theorem with Production).

Suppose that for some and is Pareto efficient. Then, there exist with and such that

such that , for all .
, for all .

Proof.Consider a hypothetical economy . and

define .
let be Pareto efficient.
let be Competitive equilibrium for .

WTS#1: is feasible.

from , we have . thus, for any , we have
since is CE, by the definition definition, it is also feasible. therefore, where the last equality holds by the market clearing of WTS#2: is CE at given .
as is PE in original economy, we have which implies .
thus, which implies .
as holds for all consumer , we have , thus for all , holds.
then, since is CE, which implies profit maximization, we have therefore,
by letting , and , we have the desired result of and

WTS#3: . this hold since this completes the proof. □

PE implies CE

At PE, there is no other feasible allocation that can strictly increase utility. Thus there exists some PE that is also CE, where we can achieve by rearranging the wealth by lump-sum treasfering.