Public Economics
Lecture 2: Market Efficiency
Dr. Jonathan St¨abler
Masaryk University
Faculty of Economics and Administration
Fall 2024
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Outline of Lecture 2
1. Pareto-Efficiency and Welfare Economics
2. Welfare Theorem 1
2.1 Competitive Equilibrium
2.2 Exchange Efficiency
2.3 Production Efficiency
2.4 Product Mix Efficiency
3. Welfare Theorem 2
3.1 Definition
3.2 Equity and Efficiency
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Outline
1. Pareto-Efficiency and Welfare Economics
2. Welfare Theorem 1
2.1 Competitive Equilibrium
2.2 Exchange Efficiency
2.3 Production Efficiency
2.4 Product Mix Efficiency
3. Welfare Theorem 2
3.1 Definition
3.2 Equity and Efficiency
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Efficiency of Private Markets
Private markets can lead to efficient resource allocation:
▶ Adam Smith’s ”invisible hand”: self-interest unintentionally promotes public
interest.
▶ Entrepreneurs drive efficiency and innovation by responding to profit opportunities.
▶ If a good passes the ”market test” (value > production cost), it will be produced
without government intervention.
▶ Competition eliminates inefficiencies and ensures better resource allocation.
But: Market failures (e.g., unemployment, pollution) can prevent efficiency.
▶ The public sector can address these failures and enhance overall efficiency.
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Welfare Economics and Pareto Efficiency
▶ Welfare economics deals with normative questions: what to produce, how to
produce, for whom, and who decides.
▶ How can we measure market efficiency and resource allocation?
Pareto efficiency
A resource allocation is Pareto efficient when it’s NOT possible to make someone better
off without making someone else worse off
− If we can find a way to make some people better off without making anybody else
worse off, we have a Pareto improvement.
− If an allocation allows for a Pareto improvement, it is called Pareto inefficient.
− if an allocation is such that no Pareto improvements are possible, it is called
Pareto efficient.
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Pareto Improvements and the Pareto Principle
▶ Pareto improvements occur when some are made better off without making
anyone worse off.
Example: A profitable bridge project is only a Pareto improvement if no one is
negatively impacted.
▶ Not all changes meet this standard due to indirect effects (e.g., noise from
infrastructure).
▶ Economists seek ”packages” of changes that together lead to Pareto
improvements.
Example: Reducing steel tariffs + raising taxes slightly to subsidize the steel
industry.
▶ The Pareto principle suggests pursuing all Pareto improvements.
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Pareto Efficiency and Individualism
▶ Pareto efficiency is individualistic, focusing on each individual’s welfare rather than
addressing inequality.
▶ A change that benefits the rich but leaves the poor unaffected can still be a Pareto
improvement.
▶ Each individual’s perception of his or her own welfare that counts.
−→ It respects consumer sovereignty: individuals are considered the best judges of their
own needs and interests.
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Pareto-Efficiency
Pareto efficiency implies the following:
▶ All resources are used efficiently in the production.
▶ All resources are allocated without waste.
▶ All trades, where both parties are (weakly) better off have already happened.
Important
Pareto efficiency does not imply an equal or fair distribution of resource allocations.
Example: A change that makes the rich better off but leaves the poor unaffected is still
a Pareto improvement.
▶ Pareto efficient allocations can be inequitable and socially undesirable.
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The Fundamental Theorems of Welfare Economics:
1. Theorem: If the economy is competitive, it is Pareto efficient.
▶ Competitive markets tend to automatically reach efficiency. There can be many
efficient Pareto efficient allocations.
2. Theorem: Every Pareto efficient resource allocation can be obtained through
a competitive market process with an initial redistribution of wealth.
▶ What if we want to obtain a particular distribution? If we transfer money from one
individual to another, then let the market forces work, we will obtain another
(different) Pareto efficient allocation.
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Decentralized vs. Centralized Systems
▶ Decentralized system: Decisions about production and consumption are made by
firms and individuals.
▶ Centralized system: A single planner or agency makes all economic decisions
(e.g., former Soviet Union, North Korea, Cuba).
▶ The second theorem shows that, with ideal conditions, decentralized competitive
markets can be as efficient as centralized systems.
▶ The role of government should be limited to the redistribution of resources.
A word of caution: theorems are logical propositions in which the conclusion follows
from the assumptions.
− Purely decentralized (private) systems do not work perfectly and there is need for
public sector involvement.
− Centralized system also don’t work (e.g., absence of price signal −→ difficulty to
determine what is needed, concentration of power −→ dictatorship).
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What are your questions now?
Let’s look more closely at the two welfare theorems.
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Outline
1. Pareto-Efficiency and Welfare Economics
2. Welfare Theorem 1
2.1 Competitive Equilibrium
2.2 Exchange Efficiency
2.3 Production Efficiency
2.4 Product Mix Efficiency
3. Welfare Theorem 2
3.1 Definition
3.2 Equity and Efficiency
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Competitive Equilibrium
▶ In deciding how much to demand and supply, individuals and firms equate the
marginal benefit (MB) with the marginal cost (MC).
▶ The market will autonomously tend to reach the equilibrium point E, where demand
equals supply.
Stiglitz and Rosengard (2015)
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Demand Curve: Consumer’s Problem
▶ Consumer receives utility U(x1) from consuming good x1
▶ Each unit of good x1 costs p1 and hence creates costs C = x1p1
How much does she consume?
argmax
x1
U(x1) − x1p1
FOC :
∂(U(x1) − x1p1)
∂x1
=
∂U(x1)
∂x1
− p1 = 0
⇐⇒
∂U(x1)
∂x1
marginal benefit of consumption
= p1
marginal costs of consumption
SOC :
∂2(U(x1) − x1p1)
∂x2
1
< 0
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Demand Curve: Consumer’s Problem
However, consumers not only consume one good, but face trade-offs between spending
their income on the product/ service of interest x1 and all other products and services,
represented by x2. Additionally, they have an upper limit of how much they can spend
m.
▶ Consumer receives utility U(x1, x2) from consuming goods x1 and x2
▶ She faces the budget constraint p1x1 + p2x2 ≤ m
▶ where pi is the price of good i and m the income
argmax
x1,x2
U(x1, x2)
s.t.p1x1 + p2x2 ≤ m
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Demand Curve: Consumer’s Problem
Source: Varian () Figure 2.3
▶ Budget constraint: the amount of
income a consumer can spend on
various goods.
▶ Change in one of the prices (here p1),
changes the slope of the budget line:
−p1
p2
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Demand Curve: Consumer’s Problem
Source: Varian (2010) Figure 3.11
▶ Indifference curve: Gives the
combination of x1 and x2, which give
the same level of utility.
▶ Marginal rate of substitution (∆x2
∆x1
):
The amount of x1 that an individual is
willing to give up in exchange for a unit
of x2.
▶ Optimal Choice: Where the slope of
the budget line equals the MRS :
−∆x2
∆x1
= −p1
p2
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Consumer’s Problem – The Lagrangian Method
argmax
x1,x2
U(x1, x2); s.t.p1x1 + p2x2 ≤ m
Set up the Lagrangian function and maximize it:
L = U(x1, x2) − λ(m − p1x1 − p2x2)
∂L
∂x1
=
∂U
∂x1
+ λp1 = 0 −→
∂U
∂x1
1
p1
= −λ (1)
∂L
∂x2
=
∂U
∂x2
+ λp2 = 0 −→
∂U
∂x2
1
p2
= −λ (2)
∂L
∂λ
= m − p1x1 − p2x2 = 0 −→ m = p1x1 + p2x2 (3)
Combining (1) and (2) leads to:
−
∂U/∂x1
∂U/∂x2
Marginal Rate of Substitution (MRS)
= −
p1
p2
Slope of Budget Line
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Demand Curve: Individual Demand
Source: Varian (2010) Figure 6.9
▶ Indifference curve: Gives the combination of
x1 and x2, which give the same level of utility.
▶ If p1 decreases, demand for x1 increases.
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Demand Curve: Individual Demand
Source: Mankiw (2018)
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Demand Curve: Market Demand
Source: Mankiw (2018)
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Supply Curve: Firm’s Problem
Suppose a firm faces perfect competition:
▶ Large number of consumers and producers
▶ Free market entry and exit
▶ No transaction costs
▶ Complete information
=⇒ Each firm’s output (qi ) is marginal and does not affect the market price P(Q).
Q =
n
i=1
qi and lim
n→∞
qi
Q
= 0
=⇒ The firm takes the market price as given (exogenous market price).
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Supply Curve: Firm’s Problem
The firm takes the market price P as given (exogenous price) and maximizes its profit π
by choosing the quantity q to produce, given the cost function C(q).
π
profit
= Pq
revenue
− C(q)
variable costs
− F
fixed costs
FOC :
∂π
∂q
= P −
∂C(q)
∂q
= 0
⇐⇒ P
marginal revenue
=
∂C(q)
∂q
marginal costs (MC)
SOC :
∂2π
∂q2
< 0
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Supply Curve: Firm’s Problem
Source: Varian (2010) Figure 22.2
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Supply Curve: Individual Supply
Source: Mankiw (2018)
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Supply Curve: Market Supply
Source: Mankiw (2018)
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Competitive Equilibrium
▶ In deciding how much to demand and supply, individuals and firms equate the
marginal benefit (MB) with the marginal cost (MC).
▶ The market will autonomously tend to reach the equilibrium point E, where demand
equals supply. [video]
Stiglitz and Rosengard (2015)
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Are there any questions?
From focusing on a single market, let’s zoom out and look at the welfare of the
population overall and the production possibilities overall.
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From Competitive Equilibrium to Economic Efficiency
▶ So far, we explored the concept of competitive equilibrium in a single market.
▶ Key Point: In a competitive equilibrium, prices adjust so that supply equals
demand, ensuring no excess supply or demand in that market.
▶ Link to efficiency of an entire economy:
▶ Competitive equilibrium is a necessary condition for economic efficiency.
▶ Now, we extend this idea to evaluate efficiency across the whole economy.
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Analyzing Economic Efficiency
Three Aspects of Pareto Efficiency:
1. Exchange Efficiency:
▶ Goods must be allocated to those who value them most.
▶ Example: chocolate ice cream to a person who prefers chocolate, vanilla to someone
who prefers vanilla.
2. Production Efficiency:
▶ Output maximized with the available resources.
▶ Production efficiency occurs when an economy cannot produce more of one good
without sacrificing another (given fixed resources).
3. Product Mix Efficiency:
▶ Goods produced should match individual preferences.
▶ Example: if people prefer ice cream over apples and it’s cheaper to produce, more ice
cream should be produced.
Together, these conditions lead to Pareto Efficiency of the entire economy.
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Outline
1. Pareto-Efficiency and Welfare Economics
2. Welfare Theorem 1
2.1 Competitive Equilibrium
2.2 Exchange Efficiency
2.3 Production Efficiency
2.4 Product Mix Efficiency
3. Welfare Theorem 2
3.1 Definition
3.2 Equity and Efficiency
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Exchange Efficiency: Utility Possibility Curve
Stiglitz and Rosengard (2015)
▶ The utility possibilities curve represents the maximum utility that can be achieved
by two consumers.
▶ A Pareto efficient economy operates along this curve, where improving one person’s
utility would reduce another’s.
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Exchange Efficiency: Utility Possibility Curve
Stiglitz and Rosengard (2015)
▶ 1st theorem: When markets are competitive, the economy lies on the utility
possibilities curve and is Pareto efficient.
▶ 2nd theorem: Any point on the curve can be reached by redistributing initial
wealth.
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Budget Constraints: Crusoe’s Example
Stiglitz and Rosengard (2015)
▶ Crusoe has $100 to spend on apples and oranges. Apples cost $1 each, and oranges
cost $2 each.
▶ Crusoe’s budget constraint shows the trade-off: for every orange he buys, he must
give up two apples.
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Indifference Curves: Crusoe’s Preferences
Stiglitz and Rosengard (2015)
Figure 3.4
▶ Indifference curves represent combinations of goods yielding the
same utility; e.g., curve I0 indicates combinations equally attractive
to 80 apples and 18 oranges (point A). Points A, B, C,D, E are all
equally attractive.
▶ The slope of the indifference curve reflects the marginal rate of
substitution (MRS), showing how many apples Crusoe is willing to
give up for an additional orange.
▶ As Crusoe consumes more oranges, the MRS diminishes; he
requires less apples to compensate for giving up oranges.
▶ The optimal choice occurs where the highest indifference curve is
tangent to the budget constraint
−→ −MRS = −p1
p2
(point E).
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Exchange Efficiency
▶ Exchange efficiency occurs when goods are distributed so that no one can be made better off
without making someone else worse off.
▶ It ensures that there is no room for further trades that would benefit both parties.
▶ Example:
− Crusoe is willing to trade 1 orange for 1 apple.
− Friday, on the other hand, is willing to trade 3 apples for 1 orange.
− They agree to a trade: Crusoe gives Friday 1 orange, and Friday gives Crusoe 2 apples.
− After the trade, both are better off: Crusoe gets 2 apples (but only wanted 1), and Friday only gives
up 2 apples (but was willing to give 3).
▶ After this trade, Crusoe and Friday move to new points on their indifference curves, each with a
new marginal rate of substitution (MRS).
− Crusoe likely desires more apples to compensate for giving up oranges (as he has fewer oranges and
more apples now).
− Friday would be willing to give up fewer apples for a new orange (as he has more oranges and less
apples now).
− Trade is efficient (and will occur) until both have the same marginal rate of substitution.
▶ Exchange efficiency requires that after trade, Crusoe and Friday have the same marginal rate of
substitution (MRS).
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Exchange Efficiency
Stiglitz and Rosengard (2015)
▶ The Edgeworth–Bowley box shows all allocations of goods between Crusoe (bottom left) and
Friday (top right).
▶ Pareto efficiency is achieved at point E, where Friday’s highest indifference curve is tangent to
Crusoe’s, equalizing their marginal rates of substitution. [[click for video]]
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Outline
1. Pareto-Efficiency and Welfare Economics
2. Welfare Theorem 1
2.1 Competitive Equilibrium
2.2 Exchange Efficiency
2.3 Production Efficiency
2.4 Product Mix Efficiency
3. Welfare Theorem 2
3.1 Definition
3.2 Equity and Efficiency
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Production Efficiency: Production Possibilities Frontier
Stiglitz and Rosengard (2015)
▶ The Production possibility frontier shows all possibly combinations of production levels of apples
and oranges.
▶ The production possibilities frontier illustrates the trade-off between producing apples and oranges.
▶ Production efficiency occurs when an economy cannot produce more of one good without
sacrificing another (given fixed resources).
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Production Efficiency
Stiglitz and Rosengard (2015)
Figure 3.7
▶ Assume that all we need for producing anything is land and labor.
▶ The isocost line represents combinations of inputs (e.g., land and
labor) that yield the same total cost.
▶ Isoquants show combinations of inputs that produce the same
output level, similar to how indifference curves illustrate consumer
preferences.
▶ The marginal rate of technical substitution (MRTS) reflects the
amount of one input that must be increased to compensate for a
decrease in another input.
▶ The optimal input choice occurs where the highest isoquant is
tangent to the isocost line:
−→ − MRTS = −p1
p2
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Production Efficiency
▶ Production efficiency occurs when firms allocate resources such that no firm can increase output
without reducing output for another firm.
▶ It ensures that there is no room for further reallocations of inputs that would benefit both firms.
▶ Example:
− Firm A is willing to trade 1 unit of labor for 2 units of land.
− Firm B, on the other hand, is willing to trade 3 units of land for 1 unit of labor.
− They agree to a trade: Firm A gives Firm B 1 unit of labor, and Firm B gives Firm A 2 units of land.
− After the trade, both firms are better off: Firm A acquires 2 units of land (but only wanted 1), while
Firm B gives up only 2 units of land (but was willing to give up 3).
▶ After this trade, both firms move to new points on their isoquants, each with a new marginal rate
of technical substitution (MRTS).
− Firm A likely desires more land to compensate for having given up some labor.
− Firm B would be willing to give up fewer units of land for additional labor, as it now has more labor
and fewer land units.
− An exchange in production inputs is efficient (and will continue) until both firms have the same
marginal rate of technical substitution.
▶ Production efficiency requires that after the trade, Firm A and Firm B have the same marginal rate
of technical substitution (MRTS).
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Production Efficiency
Stiglitz and Rosengard (2015)
▶ The Edgeworth–Bowley box illustrates all allocations of inputs (land and labor) between Firm O
(bottom left) and Firm O’ (top right).
▶ Pareto efficiency is achieved at point E, where Firm O’s highest isoquant is tangent to Firm O”s,
equalizing their marginal rates of technical substitution (MRTS).
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Outline
1. Pareto-Efficiency and Welfare Economics
2. Welfare Theorem 1
2.1 Competitive Equilibrium
2.2 Exchange Efficiency
2.3 Production Efficiency
2.4 Product Mix Efficiency
3. Welfare Theorem 2
3.1 Definition
3.2 Equity and Efficiency
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Product Mix Efficiency
▶ Product mix efficiency considers both technical feasibility and individuals’ preferences when it
comes to choosing the best mix of apples and oranges.
▶ The slope of the production possibilities curve is called the marginal rate of transformation
(MRT).
▶ Under perfect competition, MRT = MRS.
Stiglitz and Rosengard (2015)
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Basic Conditions for Pareto Efficiency
1. Exchange Efficiency: The marginal rate of substitution between any two goods
must be the same for all individuals.
2. Production Efficiency: The marginal rate of technical substitution between any
two inputs must be the same for all firms.
3. Product Mix Efficiency: The marginal rate of transformation must equal the
marginal rate of substitution.
1. Theorem: If the economy is competitive, it is Pareto efficient...
...because it satisfies all three conditions.
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Outline
1. Pareto-Efficiency and Welfare Economics
2. Welfare Theorem 1
2.1 Competitive Equilibrium
2.2 Exchange Efficiency
2.3 Production Efficiency
2.4 Product Mix Efficiency
3. Welfare Theorem 2
3.1 Definition
3.2 Equity and Efficiency
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Welfare Theorem 2
1. Theorem: If the economy is competitive, it is Pareto efficient.
2. Theorem: Every Pareto efficient resource allocation can be obtained through
a competitive market process with an initial redistribution of wealth.
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Welfare Theorem 2
▶ Even with no market failures, free market outcomes might generate substantial
inequality.
▶ Inequality is often viewed as the biggest issue with market economies.
2nd Welfare Theorem:
▶ Any Pareto Efficient allocation can be achieved by:
1. Suitable redistribution of initial endowments
(e.g., individualized lump-sum taxes based on individual characteristics, not behavior)
2. Then, letting markets work freely.
=⇒ No conflict between efficiency and equity.
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Example: 2nd Welfare Theorem
▶ Suppose the economy is populated by 50% disabled people who are unable to work
(earning $0) and 50% able people who can work and earn $100.
▶ Free market outcome: Disabled individuals have $0, able individuals have $100.
2nd Welfare Theorem:
▶ The government can identify the disabled and the able (even if the able do not
work).
▶ Redistribution: The government can tax the able by $50 (regardless of whether
they work) and give $50 to each disabled person.
▶ The able will continue working because if they don’t, they would still have to pay
$50 and earn zero income.
▶ Both able and disabled individuals have $50 in the end.
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Example: Welfare Theorem 2
▶ Suppose that households supply labour (L) and consume a single good (X)
▶ Two types of households 1 and 2:
▶ Different productivity levels (wage): w1 and w2, with w1 < w2
▶ Same utility from consumption and free time (1-L):
logU1
= logU2
= αlogX + (1 − α)log(1 − L) where α ∈ (0, 1)
▶ Household 1 consumes X1 units of the product, and household 2 X2. Together,
they cannot consume more than they produce and thus face the production
constraint: X1 + X2 = w1L1 + w2L2
argmax
L1,L2,X1,X2
logU1
+ logU2
s.t.X1 + X2 = w1L1 + w2L2
Source: Atkinson and Stiglitz (2015)
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Example: Solving
Solving this problem (see Atkinson and Stiglitz (2015), p.285ff), the utility levels for
household 1 and 2 are:
▶ U1 = Awα
1
▶ U2 = Awα
2
The more productive household (w2 > w1) receives more utility, because they work more
efficiently and hence can either buy more products and/ or enjoy more free-time.
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Example: Welfare Theorem 2
▶ The more productive household 2
(w2 > w1) receives more utility than
household 1 (point N).
▶ How can we reach other Pareto
efficient allocations on this possibility
frontier, (e.g. point E)?
=⇒ 2nd Welfare Theorem: ’simply’ transfer
some initial wealth from household 2 to
household 1.
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Example: Welfare Theorem 2
2nd Welfare Theorem: every Pareto efficient resource allocation can be obtained
through a competitive market process with an initial redistribution of wealth.
▶ Let there be a lump-sum transfer T, where household 2 transfers some wealth T to
household 1.
▶ Then the individual budget constraints are: X1 = w1L1 + T and X2 = w2L2 − T
−→ Household 2 has to work more to reach the same consumption level
−→ Household 1 has to work less to reach the same consumption level
Solving this new problem (see Atkinson and Stiglitz (2015), p.285ff) yields the new
individual utility levels:
▶ U1 = A(w1 + T)w
−(1−α)
1
▶ U2 = A(w2 − T)w
−(1−α)
2
Plugging in any T will move us on the utility possibility frontier.
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Why would we want to redistribute wealth?
▶ Equality means each individual or group of people is given the same resources or
opportunities.
▶ Equity recognizes that each person has different circumstances and allocates the
exact resources and opportunities needed to reach an equal outcome.
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Readings for Next Week
3 Market Failure
Main Reading: Stiglitz and Rosengard (2015) Chapter 4: Market Failure
Recommended Reading: Varian (2010) Chapter 37: Asymmetric Information (37.1,
37.2, 37.3, 37.4, and 37.5)
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Thank you and see you next week!
Jonathan.Stabler@econ.muni.cz
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References I
Atkinson, A. B. and Stiglitz, J. E. (2015). Lectures on public economics: Updated
edition, Princeton University Press.
Mankiw, N. G. (2018). Intermediate macroeconomics (8th edition).
Stiglitz, J. E. and Rosengard, J. K. (2015). Economics of the public sector (4th edition).
Varian, H. (2010). Intermediate microeconomics. a modern approach (8th edition)., NY:
University of California at Berkeley .
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