Term 2 Lecture 10

Question 1

Summary

Answer 1

• once market reaches equilibrium, no way to make everyone better off through trade
• if individual demands are interdependent, like with public goods/ externalities, then efficiency can be promoted by public intervention

Question 2

Walraian equilibrium in exchange economies have the general property of being

Answer 2

Pareto efficient
- no feasible allocation that all consumers are better off

Question 3

Proof of Pareto efficiency in Walrasian Equilibrium
- known as the first fundamental theorem of welfare economics

Answer 3

Proof by contradiction:
- allocation r1, r2, such that r1 preferred to current q1, etc
- means these bundles could not have been affordable to equilibrium prices p, by WARP
- SUM(pirih) >/ SUM(piqih) = SUM(piwih), with at least one of these inequalities being strict, as at least one consumer strictly better off
- this would mean total cost of the new allocation is greater than the total income available - impossible.

Question 4

The locus of Pareto efficient allocations is known as

Answer 4

The contract curve and is illustrated for a two good economy using the Edgeworth-Bowley box
- those allocations on the contract curve that are Pareto superior to the initial endowment allocation are known as the CORE

Question 5

The core

Answer 5

• a subset of the contract curve which is better than the initial endowments
• if an allocation is in the core, it means both parties would voluntarily trade to reach it instead of staying with original endowments
• core represents fairer or more mutually beneficial trades that improve utility

Question 6

Why may some Pareto-efficient outcomes be more socially desirable than others

Answer 6

Even if all outcomes on the contract curve are efficient, some may be unequal or unfair, depending on how resources were distributed initially - questions on distributional equity.

Question 7

Second Fundamental Theorem of Welfare Economics

Answer 7

Says any Pareto-efficient allocation can be achieved as a competitive equilibrium, but only if we start with the right initial endowments
- means in theory, society could choose which PE outcome it wants and then redistribute initial resources, through taxes, once this is done, market can reach desired outcome efficiently through trade

Question 8

What assumption does the second theorem depend on?

Answer 8

On convex preferences, essentially means consumers prefer balanced, diversified bundles of goods, rather than extreme allocations

Question 9

How to determine PE allocations in an exchange economy?
- setup

Answer 9

• total resources (endowment) in the economy is denoted as k = SUM(wh), so total amount of goods available is the sum off what each household initially owns
• assume we are maxing the utility of one consumer, while ensuring the others received a specified level of utility u^h, for h = 2,3,4…
• quantity of goods left for household 1 is what remains after all others take their share, q1 = K - SUM(qh)

Question 10

How to determine PE allocation in exchange economy
- optimisation problem

Answer 10

Max u1(K - SUM(qh)), s.t. Uh(qh) = u^h
- want to max the utility of household 1
- restricted by the fact that all other households must receive at least their required utility u^h

Question 11

How to determine PE allocation in an exchange economy
- first order conditions

Answer 11

Du1/Dqi1 = (lamdaH)(DuH/DqiH), H = 2,3…
- implies:
((DuH/DqiH)/(DuH/DqjH)) = ((DuG/DqiG)/(DuG/DqjG))
- ensures all households have the same MRS between any too goods
- this is the condition for Pareto efficiency, means at optimal allocation, no one can be made better off without making someone worse off

Question 12

Competitive markets naturally lead to PE allocations as they

Answer 12

force consumers to adjust their consumption so that all individuals’ MRS values match the common price ratio
- so no further trade can improve efficiency

Question 13

Fundamental Welfare Theorems for Equilibria with Production:

Answer 13

The previous 2 theorems still hold in the presence of production as long as certain assumptions are extended
- suppose a competitive equilibrium allocation is not PE, so production plans x1, x2 and alternative allocation r1, r2 that is Pareto superior
- SUM(pirih) >/ SUM(piwih) + SUM(thetahk).SUM(piyik), where theta is ownerships of shares of firms by consumer h, sums to 1
- SUM(pi).SUM(rih) > SUM(pi).(SUM(wih) + SUM(yik))
Feasibility Conditon: SUM(rih) = SUM(wih) + SUM(xik)
- sub into 2 lines above, you get that the equilibrium production plants max profits because it suggests that the alternative production plan produces a strictly greater total value, which means equilibrium firms aren’t profit maximising, which contradicts the definition of comp eq.

Question 14

Explanation of Efficiency of Equilibria with Production
- problem setup

Answer 14

• consumption allocation and production plans are chosen to maximise the utility of one consumer, e.g. h = 1
• utilities of other consumers are fixed at u^h
• technical feasibility constraints: I^K(Qk,-Lk) = 0, as if the form is operating efficiently on its PPF, function evaluates to 0

Question 15

Explanation of Efficiency of Equilibria with Production
- the Pareto efficiency problem

Answer 15

Ensure: other consumers maintain utility u^h and production remains technically feasible
MAX u ( SUM(yk) - SUM(qh)), h = 2,3,…H
S.t. Uh(qh) = u^h and I^k(yk) = 0, where yk = (Qk, -Lk)

Question 16

Explanation of Efficiency of Equilibria with Production
- FOC conditions for efficiency

Answer 16

Du1/Dqi = (lambda)Duh/Dqih = u^k.(Dtk/Dyk)
- lambda is the Lagrange multiplier for the consumer utility constraints
- u^k is the Lagrange multiplier for the firm’s production feasibility constraint
- FOC conditions show that marginal utility of consumption is aligned across consumers, and the MB of output equals the MC in terms of production constraints.

Question 17

Equating MRS and MRT from FOC

Answer 17

(Duh/Dqih)/(Duh/Dqjh) = (DTk/Dyik)/(DTk/Dykj)
- LHS is the MRS, which measures consumer tradeoff between goods i and j
- RHS is the MRT, describing how firms convert inputs into different outputs
- in a competitive equilibrium, firms and consumers face the same price ratios, so MRS = MRT automatically.

Question 18

Public goods are

Answer 18

Non-rivalrous and Non-excludable

Question 19

Mathematical model for Public Goods

Answer 19

• one private good, denoted as qh, consumed individually by each person h
• one public good, denoted as Q, which is consumed collectively
  Uh = uh( qh ,Q )

Question 20

Tradeoff between private and public goods:

Answer 20

Economy has fixed total resource M
M = SUM(qh) + PQ
- P is the cost in private goods to produce one unit of the public good

Question 21

Efficient supply of public goods
- problem

Answer 21

Max u1( M - SUM(qh - PQ,Q)), s.t. Uh(qh,Q) = u^h

Question 22

Efficient supply of public goods
- FOC

Answer 22

Private good allocation condition: MU of consumption for consumer 1 should equal that of other consumers
- Du1/Dq1 = (lambda^h).Duh/Dqh
Public good allocation condition: MC of providing P should equal the sum of all the individuals’ MB
- P.(Du1/Dq1) = Du1/DQ + SUM(lambda^h).(Duh/DQ)

Question 23

Samuelson condition

Answer 23

Efficient provision of a public good requires that the sum of individual MRS equals the MC of providing the Public Good

P = SUM((Duh/DQ)/(Duh/Dqh))
- LHS is the MC per unit of the public good
- RHS is the sum of MB across all individuals

Question 24

Samuelson conditon restated for private and public goods:

Answer 24

Private: MRS between two goods should be the same for all consumers
Public: instead of equating each individuals MRS to the cost P, the sum of all individuals’ MRS must equal P
- here MRS represents the tradeoff each individual is willing to make between the public and private good
- since public goods are shared, each consumer benefits from its provision, so the marginal valuations must be summed.

Question 25

Diagrammatic explanation for a two person economy

Answer 25

1. Fix the utility of individual A 2. Use economy’s resource constraint to determine remaining resources for B, for different levels of public good Q 3. Plot the max possible utility for individual B given Q 4. The efficient allocation is found where the tangency between this curve and the individuals IC occurs 5. This satisfies SC: P - MRSb = MRSa, meaning the total willingness to pay for the public good must match the cost of providing it

Question 26

Private contributions model

Answer 26

- individuals start with an endowment of the private good wh, and each can contribute some to a public good - Gh is contribution of individual h, Q = Gh + SUM(g^g), where g is not h - each person’s utility depends on their remaining private good, wh - PGh and the total amount of Q Max uh(wh - PGh, Gh + SUM(g^g)), here each person takes others’ contribution as a given and decides how much you contribute leading to some NE

Question 27

Why private provision is inefficient

Answer 27

- some wont contribute at all - free riders - those who do will choose P = (Duh/DQ)/(Duh/Dqh), so each person equates their own MRS to P - but SC requires the sum of all MRS values to equal P, so we have under provision, which then leads to public provision through taxation.